Modern Geometry: A Celebration of the Work of Simon Donaldson

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Proceedings of Symposia in

PURE MATHEMATICS Volume 99

Modern Geometry: A Celebration of the Work of Simon Donaldson ˜ Vicente Munoz Ivan Smith Richard P. Thomas Editors

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Volume 99

Modern Geometry: A Celebration of the Work of Simon Donaldson ˜ Vicente Munoz Ivan Smith Richard P. Thomas Editors

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Nathalie Wahl, 2017 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Proceedings of Symposia in

PURE MATHEMATICS Volume 99

Modern Geometry: A Celebration of the Work of Simon Donaldson ˜ Vicente Munoz Ivan Smith Richard P. Thomas Editors

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

2010 Mathematics Subject Classification. Primary 32J25, 32L05, 53C07, 53C44, 53D35, 53D40, 53D50, 57R55, 57R57, 57R58.

Library of Congress Cataloging-in-Publication Data Names: Mu˜ noz, V. (Vicente), 1971– editor. | Smith, Ivan, 1973– editor. | Thomas, Richard P., 1972– editor. Title: Modern geometry : a celebration of the work of Simon Donaldson / Vicente Mu˜ noz, Ivan Smith, Richard P. Thomas, editors. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Proceedings of symposia in pure mathematics ; volume 99 | Includes bibliographical references. Identifiers: LCCN 2017052437 | ISBN 9781470440947 (alk. paper) Subjects: LCSH: Donaldson, S. K. | Manifolds (Mathematics) | Four-manifolds (Topology) | Geometry. | Topology. | AMS: Several complex variables and analytic spaces – Compact analytic spaces – Transcendental methods of algebraic geometry. msc | Several complex variables and analytic spaces – Holomorphic fiber spaces – Holomorphic bundles and generalizations. msc | Differential geometry – Global differential geometry – Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills). msc | Differential geometry – Global differential geometry – Geometric evolution equations (mean curvature flow, Ricci flow, etc.). msc | Differential geometry – Symplectic geometry, contact geometry – Global theory of symplectic and contact manifolds. msc | Differential geometry – Symplectic geometry, contact geometry – Floer homology and cohomology, symplectic aspects. msc | Differential geometry – Symplectic geometry, contact geometry – Geometric quantization. msc | Manifolds and cell complexes – Differential topology – Differentiable structures. msc | Manifolds and cell complexes – Differential topology – Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants. msc | Manifolds and cell complexes – Differential topology – Floer homology. msc Classification: LCC QA613 .M6345 2018 | DDC 516/.07–dc23 LC record available at https://lccn.loc.gov/2017052437 DOI: http://dx.doi.org/10.1090/pspum/099

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Contents

Preface

vii

Graded linearisations Gergely B` erczi, Brent Doran, and Frances Kirwan

1

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations Aliakbar Daemi and Kenji Fukaya 23 Weinstein manifolds revisited Yakov Eliashberg

59

Remarks on Nahm’s equations Nigel Hitchin

83

Conjectures on counting associative 3-folds in G2 -manifolds Dominic Joyce

97

Toward an algebraic Donaldson-Floer theory Jun Li

161

Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories Hiraku Nakajima 193 An overview of knot Floer homology ´ th and Zolta ´ n Szabo ´ Peter Ozsva

213

Descendants for stable pairs on 3-folds Rahul Pandharipande

251

The Dirichlet problem for the complex homogeneous Monge-Amp`ere equation ¨m Julius Ross and David Witt Nystro 289 K¨ahler-Einstein metrics ´ bor Sz´ Ga ekelyhidi

331

Donaldson theory in non-K¨ahlerian geometry Andrei Teleman

363

Two lectures on gauge theory and Khovanov homology Edward Witten

393

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Preface Simon Donaldson has been one of the central figures in modern geometry for thirty-five years, and remains as active today as ever. His work has revolutionised numerous fields; the breadth of the essays in this volume are testament to his profound influence across different areas of differential and algebraic geometry, and its connections to topology, to analysis and to theoretical physics. Simon Kirwan Donaldson was born on August 20th, 1957, in Cambridge, U.K. He attended secondary school at Sevenoaks in Kent, and was a mathematics undergraduate at Pembroke College, Cambridge, before going on to doctoral work under the joint supervision of Michael Atiyah and Nigel Hitchin at Oxford. After his DPhil degree, Donaldson became a Research Fellow at All Souls College, Oxford, and then (with a year at the Institute for Advanced Study in Princeton as intermission) the Wallis Professor at Oxford. He remained in Oxford until 1997, then spent one year at Stanford, California, before returning to the U.K. with a Chair at Imperial College, London. In 2014 he joined the Simons Center for Geometry and Physics at Stony Brook, and now divides his time between there and Imperial. Donaldson was an invited speaker at the 1982 ICM in Warsaw, and was awarded the Fields Medal at the 1986 ICM in Berkeley. Amongst his many other awards are the King Faisal International Prize (2006), the Nemmers Prize (2008), the Shaw Prize (2009, joint with Cliff Taubes), and the Breakthrough Prize (2015). He was knighted in the 2012 New Year Honours list for services to mathematics. Whilst still a graduate student, in 1982, Donaldson overturned the world of low-dimensional topology, bringing to bear methods from classical gauge theory and the Yang-Mills equations – ideas later recast by Witten in terms of quantum field theory – to prove new constraints on the topology of smooth four-dimensional manifolds, the nature of which have no analogue in either lower or higher dimensions. Celebrated results in this period include: the diagonalisability theorem1 for the intersection forms of definite four-manifolds; the disproof of the fourdimensional s-cobordism conjecture and introduction of his polynomial invariants of four-manifolds; the Donaldson-Uhlenbeck-Yau (DUY) theorem describing the solutions of the Hermitian-Yang-Mills equations on K¨ahler manifolds; and his work on Nahm’s equations and monopoles. Whilst his work in low-dimensional topology dominated four-manifold theory from 1982–1994, Donaldson later made profound contributions to three quite different areas. In 1996 he introduced Lefschetz pencils into symplectic topology, proving the first general existence theorem for symplectic hypersurfaces. At the core of this 1 The frontispiece to this volume, painted by Nathalie Wahl, merges Simon’s childhood passion for sailing with an abstracted version of the renowned image of the cobordism underlying the diagonalisability theorem. Readers might look for hints of other theorems hidden in the painting!

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viii

PREFACE

work is an estimated transversality or quantitative Sard theorem, established via a novel h-principle based on analytical methods of approximately holomorphic geometry. In an attempt to fit Floer’s symplectic-geometric invariants into the formalism of topological quantum field theory, in analogy with the expected and known structures for gauge-theoretic Floer homology, he introduced the triangle product in Lagrangian Floer cohomology, and the quantum category of a symplectic manifold – the cohomological version of which became the Fukaya category, central in mirror symmetry. At around the same time, Donaldson laid out a program in higherdimensional gauge theory suggesting generalisations of both instanton theory and Lagrangian Floer theory to G2 and Spin(7)-manifolds, a program in rapid current development. In the mid 1990s, Donaldson began studying the existence question for constant scalar curvature K¨ahler metrics – the higher-dimensional analogue of the constant curvature metrics on Riemann surfaces provided by the uniformisation theorem. Over the following two decades, he introduced a huge array of new ideas into this part of complex differential geometry, partly based on intuitions derived from infinite-dimensional moment maps and ideas around geometric quantisation. He eventually successfully resolved (in 2013, with Xiuxiong Chen and Song Sun) the existence question for K¨ ahler-Einstein metrics on Fano manifolds, as conjectured by Yau and Tian – a landmark achievement, once again binding together ideas from algebraic geometry and from infinite-dimensional analysis. Whilst the DUY theorem relied essentially on the link between stability of bundles and the existence of special-curvature connections, the results in complex geometry establish a “more non-linear” analogue, reformulating the existence of K¨ ahler-Einstein metrics in terms of the stability of the varieties themselves. Donaldson will give the opening lecture at the ICM in Rio in 2018, the 4th ICM which he will address. Donaldson’s influence on mathematics reaches very much further than his body of published results. He has had a huge number of graduate students (44 students and 132 descendents so far, according to the Mathematics Genealogy database). Our own extraordinarily priviliged experiences of being his students were that one was not just given a thesis problem, one was given a whole raft of problems, early entry to an intellectual landscape which other people had scarcely begun to think of populating. Donaldson suggested key examples which paved routes through these uncharted territories and made them familiar, generously leaving the impression one had surveyed and discovered the contours of the theory for oneself. Many people have worked on his suggestions without formally being his students or postdocs: he has always been incredibly generous with his ideas, and equally generous in stepping back from credit. His gentleness and kindness are renowned, and he has been a unique role model to generations of those who have learned from him, listened to his lectures and seminars2 , or had the privilege of being party to one of his many informal asides, questions or car-ride reflections. The editors wish to thank Zak Turcinovic for help with the typesetting. Vicente Mu˜ noz, Ivan Smith, Richard Thomas 2 In the early 1990s, at his “Geometry and Analysis” seminar at Oxford, instead of inviting a speaker, Donaldson would sometimes talk about a result which excited him, outlining the proof he imagined the author had given. Often this had no resemblance to the actual work, and opened up an entirely new perspective.

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01735

Graded linearisations Gergely B`erczi, Brent Doran, and Frances Kirwan Dedicated to Simon Donaldson on the occasion of his 60th birthday, with warm thanks for the inspiration he has provided to generations of mathematicians. Abstract. When the action of a reductive group on a projective variety has a suitable linearisation, Mumford’s geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford’s GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford’s GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder–Narasimhan type).

In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford’s geometric invariant theory (GIT) to construct and study such quotient varieties [32]. The aim of this article is to describe how Mumford’s GIT can be extended effectively to actions of a large family of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way. When a linear algebraic group over an algebraically closed field k of characteristic 0 is a semi-direct product H = U R of its unipotent radical U and a reductive subgroup R ∼ = H/U which contains a central one-parameter subgroup λ : Gm → R whose adjoint action on the Lie algebra of U has only strictly positive weights, we will see that any linearisation for an action of H on a projective variety X becomes graded if it is twisted by an appropriate (rational) character, and then many of 2010 Mathematics Subject Classification. Primary 14L24, 13A50. Early work on this project was supported by the Engineering and Physical Sciences Research Council [grant numbers GR/T016170/1,EP/G000174/1]. Brent Doran was partially supported by Swiss National Science Foundation Award 200021-138071. c 2018 American Mathematical Society

1

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` GERGELY BERCZI, BRENT DORAN, AND FRANCES KIRWAN

the good properties of Mumford’s GIT hold. Many non-reductive linear algebraic group actions arising in algebraic geometry are actions of groups of this form: for example, any parabolic subgroup of a reductive group has this form, as does the automorphism group of any complete simplicial toric variety [11], and the group of k-jets of germs of biholomorphisms of (Cp , 0) for any positive integers k and p [6]. Example 0.1. The automorphism group of the weighted projective plane P(1, 1, 2) with weights 1,1 and 2 is Aut(P(1, 1, 2)) ∼ =RU where R ∼ = (GL(2) × Gm )/Gm ∼ = GL(2) is reductive and U ∼ = (k+ )3 is unipotent 2 2 with elements given by (x, y, z) → (x, y, z + λx + μxy + νy ) for (λ, μ, ν) ∈ k3 . Example 0.2. Under composition modulo tk+1 we have a group G(k) whose elements are k-jets of germs of biholomorphisms of (C, 0): {t → φ(t) = a1 t + a2 t2 + . . . + ak tk | aj ∈ C, a1 = 0}. G(k) is isomorphic to a group of matrices of the form ⎫ ⎧⎛ ⎞ a1 a2 ... ak ⎪ ⎪ ⎪ ⎪ ⎬ ⎨⎜ 2 ⎟ ⎜ 0 (a1 ) . . . p2k (a) ⎟ : a1 ∈ C∗ , a2 , . . . ak ∈ C , ⎝ ⎠ ... ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ k 0 0 . . . (a1 ) where the (i, j)th entry pij (a) is a polynomial in a1 , . . . , ak . This reparametrisation group G(k) has a one-parameter multiplicative subgroup Gm = C∗ (represented by φ(t) = a1 t) and unipotent radical U(k) (represented by φ(t) = t + a2 t2 + . . . + ak tk ) with G(k) ∼ = U(k)  C∗ . In Mumford’s classical geometric invariant theory the GIT quotient X//G = ˆL (X) = ∞ H 0 (X, L⊗k )) for an action of a reductive ˆL (X)G ) (where O Proj(O k=0 group G on a projective variety X with respect to an ample linearisation L is a projective completion of the geometric quotient X s /G of the stable set X s . When X is nonsingular then the singularities of X s /G are very mild, since the stabilisers of stable points are finite subgroups of G. If X ss = X s the singularities of X//G are ˜ typically more severe, but X//G has a ‘partial desingularisation’ X//G [27] which s is also a projective completion of X /G and is itself a geometric quotient ˜ ˜ ss /G X//G =X ˜ ss = X ˜ s of a G-equivariant blow-up X ˜ of X. When X is by G of an open subset X ss ss ˜ ˜ ˜ ss is obtained nonsingular then so is X , and G acts on X with finite stabilisers. X ss from X by successively blowing up along the subvarieties of semistable points stabilised by reductive subgroups of G of maximal dimension and then removing the unstable points in the resulting blow-up. So in the best case in classical GIT we have X ss = X s = ∅, and then s ˆL (X)G ) is simultaneously a projective variety and a geoX /G = X//G = Proj(O s metric quotient of X by the action of G. More generally when X s = ∅ then ˜ the geometric quotient X s /G has a projective completion X//G which is itself a ˜ ss /G of an open subset of a G-equivariant blow-up of X. geometric quotient X Moreover using the Hilbert–Mumford criteria for (semi)stability, which allow us to determine which points of X are stable and which are semistable for the G-action without having to know the G-invariant sections of powers of L, together with the

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GRADED LINEARISATIONS

3

≈

≈

˜ ss = X ˜s explicit blow-up construction, we can give effective descriptions of X s , X s ˜ and thus their geometric quotients X /G and X//G. This is the picture which can be generalised to the action of a non-reductive linear algebraic group, given a graded linearisation of the action. The immediate problem which arises when trying to extend classical GIT to non-reductive linear algebraic groups H is that in general we cannot define a projecˆL (X)H is not necessarily finitely genˆL (X)H ) because O tive variety X//H = Proj(O ˆL (X)H ) exists as a scheme. Nonetheless erated as a graded algebra, although Proj(O an analogue of classical GIT for non-reductive linear algebraic group actions is described in [1, 15]. Here it is shown that if H is a linear algebraic group over k acting linearly on a projective variety X with respect to an ample line bundle L, then X has open subvarieties X s (the locus of ‘stable points’) and X ss (‘semistable points’) with a geometric quotient X s → X s /H and an ‘enveloping quotient’ X ss → X H. Furthermore there is a diagram ˆL (X)H ) X  −− → Proj(O open −→ X H semistable Xss open −→ X s /H stable X s

≈

≈

≈

≈

≈

≈

ˆL (X)H is finitely where the vertical inclusions are of open subvarieties, and if O H ˆ generated then X H = Proj(OL (X) ) as in the reductive case. However this picture is less helpful than in the case of classical GIT in three significant respects: ˆL (X)H firstly X H is not necessarily a projective variety; secondly (even when O H ˆL (X) ) is a projective variety) the is finitely generated and so X H = Proj(O H-invariant morphism X ss → X H is not necessarily a categorical quotient, and its image is not in general a subvariety of X H, only a constructible subset; and thirdly there are in general no obvious analogues of the Hilbert–Mumford criteria for (semi)stability. We can see the second of these issues arising in simple examples, when the additive group Ga = k+ acts on a projective space Pn via a linear representation (see Example 1.3 below). It follows from Jordan canonical form that the representation of Ga extends to a representation of SL(2). This enables us to identify Pn Ga with the reductive GIT quotient (P2 × Pn )// SL(2),

≈

≈

≈

≈

≈

and thus to see that in general the quotient morphism qGa : (Pn )ss,Ga −→ Pn Ga fails to be surjective. Twisting the representation of the Borel subgroup B ∼ = Ga  Gm of SL(2) by a character χ : B → Gm = k∗ (whose kernel must contain Ga ) changes the linearisation but not the action of B on Pn to give an enveloping quotient Pn χ B = (Pn Ga )//χ Gm . It turns out that for appropriate choice of (rational) character χ the complement of the image of (Pn )ss,Ga in Pn Ga becomes unstable for the Gm -action and the morphism qB : (Pn )ss,B,χ −→ Pn χ B

≈

to the projective variety Pn generally.

χ

B is surjective. This phenomenon occurs more

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4

` GERGELY BERCZI, BRENT DORAN, AND FRANCES KIRWAN

Definition 0.3. Let us call a unipotent linear algebraic group U graded unipotent if there is a homomorphism λ : Gm → Aut(U ) with the weights of the Gm action on Lie(U ) all strictly positive. For such a homomorphism λ let ˆ = U  Gm = {(u, t) : u ∈ U, t ∈ Gm } U be the associated semi-direct product of U and Gm with multiplication (u, t) · (u , t ) = (u(λ(t)(u )), tt ). When L is very ample, and so induces an embedding of X in a projective space Pn , we can choose coordinates on Pn such that the action of Gm on X is diagonal, given by ⎞ ⎛ r t 0 0 ... 0 ⎜ 0 tr 1 . . . 0 ⎟ ⎟ t → ⎜ ⎠ ⎝ ... rn 0 0 ... t where r0 ≤ r1 ≤ · · · ≤ rn . The lowest bounded chamber for this linear Gm -action is the closed interval [r0 , rj ] where r0 = · · · = rj−1 < rj ≤ · · · ≤ rn , with interior the open interval (r0 , rj ), unless the action of Gm on X is trivial; when the action is trivial so that r0 = r1 = · · · = rn we will say that [r0 , r0 ] is the lowest bounded chamber and it is its own interior. Note that in the situation above, if Gm acts trivially then so does U . Let L be a very ample linearisation with respect to a line bundle L → X of the ˆ on an irreducible projective variety X. Let χ : U ˆ → Gm be a character action of U ˆ with kernel containing U ; we will identify such characters χ with integers so of U that the integer 1 corresponds to the character which defines the exact sequence ˆ -action by multiplying the ˆ → Gm . We can twist the linearisation of the U U →U ˆ -action to L by such a character; this will leave the U -linearisation on lift of the U ˆ on X unchanged. Note that a linearisation L of U ˆ with L and the action of U respect to L induces a linearisation L⊗m with respect to the line bundle L⊗m , for any integer m ≥ 1, such that twisting L by χ corresponds to twisting L⊗m by mχ; GIT will be essentially unaffected. We call a character χ rational, if cχ lifts to a ˆ as above for a sufficiently divisible positive integer c. character of U ˆ acts linearly (with respect to an ample line bundle L) on a Suppose that U projective. By choosing an appropriate rational character we can obtain a GIT picture with many of the good properties of the reductive case, as the following result demonstrates. Theorem 0.1 ([2, 3]). Let U be graded unipotent acting linearly on an irreducible projective variety X with respect to an ample line bundle L, and suppose ˆ = U  Gm . Suppose also that semistability that the linear action extends to U coincides with stability in the sense that x ∈ Zmin ⇒ StabU (x) = {e} where Zmin is the union of those connected components of the fixed point set X Gm where Gm acts on the fibres of L∗ with minimum weight. Then the linearisation ˆ on X can be twisted by a rational character of U ˆ so that 0 lies for the action of U in the interior of the lowest bounded chamber for the linear Gm action on X, and for this twisted action

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GRADED LINEARISATIONS

5

≈

≈

ˆ 0 ⊗cm U ˆL⊗c (X)Uˆ = ⊕∞ ˆ -invariants is finitely gen(i) the algebra O ) of U m=0 H (X, L erated for any sufficiently divisible integer c > 0, so that the enveloping quotient ˆ = Proj(O ˆL⊗c (X)Uˆ ) is a projective variety; X U  ˆ ˆ (ii) X ss,U = X s,U has a Hilbert–Mumford description as u∈U uX s,Gm , and X ˆ = X s,Uˆ /U ˆ is a geometric quotient of X s,Uˆ by U ˆ. U Moreover, even when the condition that semistability should coincide with staˆ of an bility fails, there is a projective completion of a geometric quotient by U ˆ ˆ s,U open subvariety of X (conjecturally X /U ), which is itself a geometric quotient ˜ ss,Uˆ /U ˆ by U ˆ of an open subset X ˜ ss,Uˆ = X ˜ s,Uˆ of a U ˆ -equivariant blow-up X ˜ of X X.

If we are interested in constructing quotients of open subsets of X by the action ˆ , then we can apply these results to the diagonal action of U ˆ on X ×P1 , of U , not of U 1 ˆ where U acts on P via   t 0 (0.1) (u, t) → 0 1 with kernel U , and the linearisation is L tensored with OP1 (m) for m >> 1. This gives us a U -invariant open subset X sˆ,U of X with a geometric quotient X sˆ,U /U ˆ of the open subvariety by U which is isomorphic to the geometric quotient by U sˆ,U 1 × {[1 : 1]} of X × P ; moreover it has a projective completion which Gm (X ˆ of an open subvariety of a U ˆ -equivariant blow-up of is a geometric quotient by U 1 X × P . Furthermore in this set-up there are Hilbert–Mumford-like criteria for (semi)stability. This motivates the following definitions. Definition 0.4. An extended linearisation L of an action of a linear algebraic group H on a projective variety X is given by the data: (a) a line bundle L on X; ˆ = H  Gm of H by Gm ; (b) a semi-direct product H ˆ and a lift of the H-action ˆ (c) an extension of the H-action on X to H to L. Given an extended linearisation L and a rational number q ∈ Q, define the ‘q-hat-stable’ locus X sˆ,q = X sˆ,q,L to be the H-invariant open subvariety of X determined by ˆ

X sˆ,q × {[1 : 1]} = (X × P1 )s,H ∩ (X × {[1 : 1]}) ˆ acts on P1 as at (0.1) above with its linearisation on OP1 (1) twisted by q, where H ˆ ˆ and (X ×P1 )s,H is defined with respect to the induced linearisation for the H-action 1 on X × P on L tensored with OP1 (m) for m >> 1. We then have a geometric quotient X sˆ,q /H by H which is isomorphic to an ˆ ˆ open subvariety of (X × P1 )s,H /H. Remark 0.2. Given a linearisation in the classical sense of an action of a linear algebraic group H on a projective variety X with respect to a line bundle L, we ˆ = H × Gm and Gm acts trivially have a ‘trivial extended linearisation’ for which H on X and on L. Then if q ∈ (0, 1) the q-hat-stable locus X sˆ,q coincides with the stable locus defined as in [1] for the action of H on X with the given linearisation, while if q ∈ / [0, 1] the q-hat-stable locus X sˆ,q is empty.

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6

` GERGELY BERCZI, BRENT DORAN, AND FRANCES KIRWAN

Definition 0.5. A linear algebraic group with graded unipotent radical is a linear algebraic group H with unipotent radical U , equipped with a semi-direct ˆ = H  Gm such that the adjoint action of Gm on the Lie algebra of product H U has only strictly positive weights and the induced conjugation action of Gm on H/U is trivial. A graded linearisation L of an action of H on a projective variety X is then ˆ such an extended linearisation in the sense of Definition 0.4 for this choice of H, ˆ ˆ that the H-linearisation is twisted by a rational character of H so that 0 lies in the interior of the lowest bounded chamber for the Gm action; we will assume that the line bundle L on X is ample unless stated otherwise. Given a graded linearisation L, the ‘hat-stable’ locus X sˆ = X sˆ,L is the 0-hat-stable locus X sˆ,0 as defined in Definition 0.4 when q = 0. Remark 0.3. When H is a linear algebraic group with graded unipotent radical U and L is a graded linearisation for an action of H on a projective variety X (with respect to an ample line bundle L on X), then we can apply Theorem 0.1 to the ˆ on X × P1 as above, and then apply classical GIT and the partial action of U desingularisation construction of [27] to the induced action of the reductive group ˆ U ˆ∼ H/ = H/U . Thus the geometric quotient X sˆ/H by H has a projective completion ˆ of an open subset of a H-equivariant ˆ which is a geometric quotient by H blow-up 1 sˆ of X × P . Furthermore the geometric quotient X /H by H and its projective completion can be described using Hilbert–Mumford-like criteria combined with an explicit blow-up construction. Remark 0.4. Definitions 0.4 and 0.5 can be extended to define T -extended linearisations and T -graded linearisations for the actions of linear algebraic groups with T -graded unipotent radical, for any torus T . The layout of this article is as follows. In §1 we will review GIT with classical linearisations [1, 15, 32]. In §2 we will describe extended, graded and torus-graded linearisations and the associated geometric invariant theory for these. Finally §3 describes some potential applications, including the construction of moduli spaces of ‘unstable’ objects, such as unstable projective curves or unstable sheaves over a fixed nonsingular projective variety (with suitable fixed discrete invariants in each case, involving their singularities or Harder–Narasimhan type).

1. GIT with classical linearisations 1.1. GIT for reductive groups. In Mumford’s classical Geometric Invariant Theory a linearisation (more precisely, an ample linearisation) of an action of a reductive group G on an irreducible projective variety X over an algebraically closed field k of characteristic 0 is given by an ample line bundle L on X and a lift of the action to L; when X is embedded in a projective space Pn and L = O(1), the action ˆL (X) = ∞ H 0 (X, L⊗k ) is is given by a representation ρ : G → GL(n + 1) and O k=0 k[x0 , . . . , xn ]/IX where IX is the ideal generated by the homogeneous polynomials

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GRADED LINEARISATIONS

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which vanish on X. We consider the picture: ˆL (X) = ∞ H 0 (X, L⊗k ) (X, L)  O k=0 |  | | ↓ ˆL (X)G algebra of invariants. X//G  O ˆL (X)G is finitely generated as Since G is reductive, the algebra of G-invariants O ˆ L (X)G ). The a graded algebra with associated projective variety X//G = Proj(O G ˆ ˆ inclusion of OL (X) in OL (X) determines a rational map X − − → X//G which fits into a diagram X 

−− →

semistable

Xss

onto −→

X//G 

stable

Xs

−→

X s /G

X//G ||

projective variety

open

where X s and X ss are open subvarieties of X, the GIT quotient X//G is a categorical quotient for the action of G on X ss via the G-invariant surjective morphism φG : X ss → X//G, and φG (x) = φG (y) ⇔ Gx ∩ Gy ∩ X ss = ∅. Remark 1.1. When k = C then G is reductive if and only if it is the complexification G = KC of a maximal compact subgroup K, and then X//G = μ−1 (0)/K for a suitable ‘moment map’ μ for the action of K. The subsets X ss and X s of X for a linear action of a reductive group G with respect to an ample linearisation are characterised by the following properties (see [32, Chapter 2], [34]). Proposition 1.2. (Hilbert–Mumford criteria for reductive group actions) (i) A point x ∈ X is semistable (respectively stable) for the action of G on X if and only if for every g ∈ G the point gx is semistable (respectively stable) for the action of a fixed maximal torus T of G. (ii) A point x ∈ X with homogeneous coordinates [x0 : . . . : xn ] in some coordinate system on Pn is semistable (respectively stable) for the action of a maximal torus T of G acting diagonally on Pn with weights α0 , . . . , αn if and only if the convex hull Conv{αi : xi = 0} contains 0 (respectively contains 0 in its interior). The GIT quotient X//G is a projective completion of the geometric quotient X s /G of the stable set X s . When X is nonsingular then the singularities of X s /G are very mild, since the stabilisers of stable points are finite subgroups of G. If X ss = X s = ∅ the singularities of X//G are typically more severe, but X//G has ˜ a ‘partial desingularisation’ X//G which is also a projective completion of X s /G and is itself a geometric quotient ˜ ˜ ss /G X//G =X ˜ ss = X ˜ s of a G-equivariant blow-up X ˜ of X [27]. by G of an open subset X

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` GERGELY BERCZI, BRENT DORAN, AND FRANCES KIRWAN

˜ ss is obtained from X ss by successively blowing up along the subvarieties X of semistable points stabilised by reductive subgroups of G of maximal dimension and then removing the unstable points in the resulting blow-up, as follows. We suppose that X has some stable points. There exist semistable points of X which are not stable if and only if there exists a non-trivial connected reductive subgroup of G fixing a semistable point. Let r > 0 be the maximal dimension of a reductive subgroup of G fixing a point of X ss and let R(r) be a set of representatives of conjugacy classes of all connected reductive subgroups R of dimension r in G such that ss = {x ∈ X ss : R fixes x} ZR is non-empty. Then



ss GZR

R∈R(r) ss is a disjoint union of nonsingular closed subvarieties of X of G on  . The action ss ss ss which can X lifts to an action on the blow-up X(1) of X along R∈R(r) GZR ss be linearised so that the complement of X(1) in X(1) is the proper transform of the −1 ss ss ss subset φ (φ(GZR )) of X where φ : X → X//G is the quotient map (see [27] 7.17). Here we use the linearisation with respect to (a tensor power of) the pullback of the ample line bundle L on X perturbed by a sufficiently small multiple of the exceptional divisor E(1) . This will give us an ample line bundle on the blow-up ψ : X(1) → X , and if the perturbation is sufficiently small it will have the property that s ss ψ −1 (X s ) ⊆ X(1) ⊆ X(1) ⊆ ψ −1 (X ss ) = X(1) , s ss and the stable and semistable subsets X(1) and X(1) will be independent of the ss choice of perturbation. Moreover no point of X(1) is fixed by a reductive subgroup ss of G of dimension at least r, and a point in X(1) is fixed by a reductive subgroup R of dimension less than r in G if and only if it belongs to the proper transform of ss the subvariety ZR of X ss .  ss Remark 1.3. In fact in [27] X itself is blown up along the closure R∈R(r) GZR  ss of R∈R(r) GZR in X (or in a projective completion of X ss with a G-equivariant morphism to X which is an isomorphism over X ss ). This gives us a projective ¯ (1) → X restricting to ψ : X(1) → X where ¯ (1) and blow-down map ψ¯ : X variety X −1 ss ¯ ψ (X ) = X(1) . We can then choose a sufficiently small perturbation of the pull¯ (1) of the linearisation on X which provides an ample linearisation of the back to X ¯ (1) such that ψ¯−1 (X s ) ⊆ X ¯s ⊆ X ¯ ss ⊆ ψ¯−1 (X ss ) = X(1) , and projective variety X (1) (1) moreover the restriction of the linearisation to X(1) is obtained from the pullback of L by perturbing by a sufficiently small multiple of the exceptional divisor E(1) . ss ss to obtain X(2) such that If r > 1 the same procedure can be applied to X(1) ss no reductive subgroup of G of dimension at least r − 1 fixes a point of X(2) . After ˜ ss repeating this enough times, we obtain X ss = X ss , X ss , X ss , . . . , X ss = X (0)

(1)

(2)

(r)

˜ ss . such that no reductive subgroup of G of positive dimension fixes a point of X ss ˜ ˜ /G can be obtained from X//G by blowing up along the Similarly X//G = X ss of X ss proper transforms of the images ZR //N in X//G of the subvarieties GZR in decreasing order of dim R.

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GRADED LINEARISATIONS

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Thus when a reductive group G acts linearly on an irreducible projective variety X with respect to an ample linearisation, we can summarise the GIT output when X s = ∅ as follows: i) The best case is when X ss = X s = ∅, and then the GIT quotient X//G = X s /G is a projective variety which is a geometric quotient of the open subvariety X s of X. Furthermore the stabiliser in G of every x ∈ X s is finite, so if X is nonsingular then X//G has at worst orbifold singularities. ii) When X ss = X s = ∅ then the GIT quotient X//G is a projective completion of the geometric quotient X s /G. Typically the singularities of X//G are significantly more serious than those of X s /G, but X s /G has another projective completion ˜ ˜ s /G which is a ‘partial desingularisation’ of X//G in the sense described X//G =X above. 1.2. GIT for non-reductive groups. Now let X be an irreducible projective variety over an algebraically closed field k of characteristic 0 and let H be a linear algebraic group, with unipotent radical U , acting on X with an ample linearisation of the action (that is, an ample line bundle L on X and a lift of the action to L). First we will define stability and semistability for the linear action of the unipotent group U .  Definition 1.1. (cf. [15] §4 and [15] 5.3.7). Let I = m>0 H 0 (X, L⊗m )U and for f ∈ I let Xf be the U -invariant affine open subset of X where f does not vanish, with O(Xf ) its coordinate ring. A point x ∈ X is called semistable for the linear action of the unipotent group U if there exists some f ∈ I which does not is finitely generated as a graded algebra. The vanish at x, and such that O(Xf )U  U -semistable locus of X is X ss,U = f ∈I f g Xf where I f g = {f ∈ I | O(Xf )U is finitely generated }.  The stable locus of X for the linear action of U is X s,U = f ∈I lts Xf where I lts = {f ∈ I f g | the quotient map qU : Xf −→ Spec(O(Xf )U ) is a locally trivial geometric quotient}.

≈

The enveloped quotient of X ss,U by the linear U -action is qU : X ss,U → qU (X ss,U ), ˆL (X)U ) is the natural morphism of schemes and where qU : X ss,U → Proj(O ss,U qU (X ) is a dense constructible subset of the enveloping quotient  Spec(O(Xf )U ) X U= f ∈I f g

of X ss,U .

≈

≈

≈

≈

ˆL (X)U is finitely generated then X U is the projective Remark 1.4. If O U ˆL (X) ). Note that even in this case qU (X ss,U ) is not necessarily a variety Proj(O subvariety of X U (see for example [15] §6). The enveloping quotient X U has quasi-projective open subvarieties (‘inner enveloping quotients’ X//◦ U ) which contain the enveloped quotient qU (X ss ) and have ample line bundles pulling back to positive tensor powers of L under the natural map qU : X ss → X U (see [1] for details).

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The H-semistable set X ss = X ss,H , enveloped and enveloping quotients and inner enveloping quotients H

≈

qH : X ss → qH (X ss ) ⊆ X//◦ H ⊆ X

for the linear action of H are defined exactly as for the unipotent case in Definition 1.1 and Remark 1.4 (cf. [1]). However the definition given in [1] of the stable set X s = X s,H for the linear action of H combines the unipotent and reductive cases as follows. Definition 1.2. Let H be a linear algebraic group acting on an irreducible variety X and L → X a linearisation for the action. The stable locus is the open subvariety  Xf Xs = f ∈I s



of X ss , where I s ⊆ r>0 H 0 (X, L⊗r )H is the subset of H-invariant sections f of tensor powers of L satisfying the following conditions: (1) the open set Xf is affine (this is automatically true when X is projective); (2) the action of H on Xf is closed with all stabilisers finite groups; and (3) the restriction of the U -enveloping quotient map qU : Xf → Spec(O(Xf )U ) is a principal U -bundle for the action of U on Xf . Remark 1.5. When H is reductive or unipotent these definitions of X ss,H and coincide with those already given. X s,H

Example 1.3. Let Ga = k+ act linearly on Pn via a representation on kn+1 . We can choose coordinates in which the generator of Lie(Ga ) has Jordan normal form with blocks of size k1 + 1, . . . , kq + 1. The linear Ga action therefore extends to the reductive group G = SL(2) with    1 a :a∈k G Ga = 0 1 ∼ q Symki (k2 ). In fact in this case the Ga -invariants via the identification kn+1 = i=1 are finitely generated by the Weitzenb¨ock theorem [13], so we have Ga = Proj((k[x0 , . . . , xn ])Ga ).

≈

Pn

The Weitzenb¨ock theorem can be proved by considering the identification of Gspaces G ×G Pn ∼ = (G/Ga ) × Pn ∼ = (k2 \ {0}) × Pn a

via (g, x) → (gGa , gx), composed with the inclusions (k2 \ {0}) × Pn ⊆ k2 × Pn ⊆ P2 × Pn . We choose a linearisation for the diagonal G-action on P2 × Pn given by L = OPn (1) tensored with OP1 (m) for m >> 1. Then restricting G-invariant sections of tensor powers of this linearisation to {1} × Pn defines an isomorphism onto the algebra of Ga -invariant sections of tensor powers of L, and we have Ga = Proj((k[x0 , . . . , xn ])Ga ∼ = (P2 × Pn )//SL(2).

≈

Pn

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GRADED LINEARISATIONS

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≈

We can see how to interpret (Pn )s = (Pn )s,Ga and (Pn )ss = (Pn )ss,Ga as well as the morphism (Pn )ss,Ga −→ Pn Ga from the diagram n P2 × P

−− → P2 × Pn //G || −− → Pn Ga || −→ Pn Ga

≈

0 : 1]} × Pn Pn ∼ = {[1 : 

≈

n ss (P )

−→

(Pn )s

Ga is not onto when

≈

In particular the morphism (Pn )ss −→ Pn

(Pn )s /Ga .

P = P(Sym (k )) = { n unordered points on P1 } n

n

2

for n ≥ 3. When n = 3 then (P3 )ss = (P3 )s = is{ 3 unordered points on P1 , at most one at ∞}

≈

≈

while its image in P3 Ga = (P3 )s /Ga  P3 //SL(2) is the open subset (P3 )s /Ga which does not include the ‘boundary’ points coming from 0 ∈ k2 ⊆ P2 . When n = 4 then (P4 )ss = (P4 )s and the image of (P4 )ss in P4 Ga is a constructible subset but not a subvariety. Let    a b B= : a ∈ Gm , b ∈ k ∼ = Ga  Gm 0 a−1 be the standard Borel subgroup acting on Pn via a linear representation q of SL(2), n+1 n+1 ∼ ki 2 . Then k on k = i=1 Sym (k ) ⊗ k(ri ) where B acts on k(r) = k as multiplication by a character χr . Twisting the representation of B on kn+1 by a character χ changes the linearisation but not the action of B on Pn to give = (Pn

Ga )//χ Gm .

≈

χB

≈

Pn

Ga become

≈

For appropriate χ, in the example above the ‘boundary points’ in P3 unstable for the Gm action and we have a surjective morphism χ B.

≈

(P3 )ss,B,χ −→ P3

It turns out, as will be discussed next, that this is a special case of a more general phenomenon. 1.3. GIT for linear algebraic groups with graded unipotent radicals. Recall from Definition 0.5 that a linear algebraic group with graded unipotent radical is a linear algebraic group H with unipotent radical U , equipped with a ˆ = H  Gm such that the adjoint action of Gm on the Lie semi-direct product H algebra of U has only strictly positive weights and the induced conjugation action of Gm on H/U is trivial. Remark 1.6. Suppose that H = U  R where the reductive group R = H/U itself contains a central one-parameter subgroup whose conjugation action on the Lie algebra of U has all weights strictly positive. Then corresponding semi-direct ˆ and H ˆ can be constructed such that U ˆ is isomorphic to a subgroup of products U ˆ and any linear action of H on a projective variety X can be extended to a linear H, ˆ We will call this situation an ‘internal grading’ for the unipotent action of H. radical of H. We will call this situation an ‘internal grading’ for the unipotent radical of H.

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≈

≈

ˆ on a projective variety X which is linear with respect Given any action of H to an ample line bundle L on X, it is shown in [2, 3] that provided: (i) we are willing to replace L with a suitable tensor power L⊗m , with m ≥ 1 ˆ by a suitable sufficiently divisible, and to twist the linearisation of the action of H ˆ (rational) character of H with kernel containing H, and moreover (ii) ‘semistability coincides with stability’ for the action of the unipotent radical U, ˆ then the H-invariants form a finitely generated algebra. Moreover in this situaˆ tion the natural quotient morphism qH from the semistable locus X ss,H to the ˆ ˆ enveloping quotient X H is surjective, and expresses the projective variety X H ˆ ˆ ˆ as a categorical quotient of X ss,H . Furthermore this locus X ss,H = X s,H can be described using Hilbert–Mumford criteria. In [3] it is also shown that when the condition that semistability coincides with stability for the unipotent radical is not satisfied, but is replaced with the weaker condition that the stabiliser in U of a generic point in X is trivial, or equivalently min dim(StabU (x)) = 0,

(1.1)

x∈X

≈

≈

ˆ then there is a sequence of blow-ups of X along H-invariant subvarieties (similar to ˆ with an induced that of [27] when H is reductive) resulting in a projective variety X ˆ linear action of H satisfying the condition that semistability coincides with stability ˆ for the unipotent radical U . In this way we obtain a projective variety X × P1 H ˆ of a H-invariant ˆ which is a categorical quotient by H open subset of a blow-up of X × k and contains as an open subset a geometric quotient of an H-invariant open subset X sˆ,H of X by H, where the geometric quotient X sˆ,H /H and the projective ˆ have descriptions in terms of Hilbert–Mumford-like criteria and variety X × P1 H the explicit blow-up construction. Remark 1.7. In fact this can be generalised to the case when min dim(StabU (x)) > 0

x∈X

[3–5].

The description of the condition we need the action of the unipotent radical U of H to satisfy as ‘semistability coincides with stability’ is a rather loose one. To describe it more precisely, let L → X be a very ample linearisation of the action of ˆ on an irreducible projective variety X. Let χ : H ˆ → Gm be a character of H ˆ with H kernel containing H; such characters χ can be identified with integers so that the ˆ → integer 1 corresponds to the character which fits into the exact sequence H → H 0 ∗ Gm . Let ωmin be the minimal weight for the Gm -action on V := H (X, L) and let Vmin be the weight space of weight ωmin in V . Suppose that ωmin < ωmin +1 < ˆ ≤H ˆ · · · < ωmax are the weights with which the one-parameter subgroup Gm ≤ U acts on the fibres of the tautological line bundle OP((H 0 (X,L)∗ ) (−1) over points of the connected components of the fixed point set P((H 0 (X, L)∗ )Gm for the action of Gm on P((H 0 (X, L)∗ ); since L is very ample X embeds in P((H 0 (X, L)∗ ) and the line bundle L extends to the dual OP((H 0 (X,L)∗ ) (1) of the tautological line bundle OP((H 0 (X,L)∗ ) (−1). Without loss of generality we may assume that there exist at least two distinct such weights, since otherwise the action of the unipotent radical U of H on X is trivial, and so the action of H is via an action of the reductive group R = H/U and reductive GIT can be applied.

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ˆ as Let χ be a rational character of Gm (lifting to a rational character of H above) such that ωmin < χ < ωmin +1 ; we will call rational characters χ with this property adapted to the linear action of ˆ and we will call the linearisation adapted if ωmin < 0 < ωmin +1 ; we will call H, ˆ if χ = ωmin , and the linearisation χ borderline adapted to the linear action of H ˆ on X with borderline adapted if ωmin = 0. The linearisation of the action of H ⊗c respect to an ample line bundle L for a sufficiently divisible positive integer c such that cχ is a character can be twisted by this character; effectively the weights ωj are replaced with ωj − χ and this twisted linearisation is adapted in the sense s,Gm above. Let Xmin + denote the stable set in X for the linear action of Gm with ˆ containing respect to this adapted linearisation and for a maximal torus T of H s,T Gm , let Xmin + denote the stable set in X for the linear action of T with respect to the adapted linearisation; by the theory of variation of (classical) GIT [14, 40], s,Gm s,T Xmin + and Xmin + are independent of the choice of adapted rational character χ. Let  ˆ s,U s,Gm s,Gm ˆ Xmin uXmin + + = X \ U (X \ Xmin + ) = u∈U

ˆ -sweep (or equivalently the U -sweep) of the complement be the complement of the U s,Gm of Xmin + , and let  ˆ s,H s,T Xmin uXmin + = +, h∈H

while

 Zmin = X ∩ P(Vmin ) =

x∈X

Gm

x is a Gm -fixed point and acts on L∗ |x with weight ωmin



and 0 Xmin = {x ∈ X |

lim

t→0, t∈Gm

t · x ∈ Zmin }.

0 ˆ -invariant and X s,Uˆ = X 0 \ U Zmin . Note that Xmin is U min min + The condition that ‘semistability coincides with stability’ for the linear action of ˆ required in [2] is slightly stronger than that required in [3]; in [3] the hypothesis U ˆ -linearisation L → X is that needed for the U

(C∗ )

StabU (z) = {e} for every z ∈ Zmin .

≈

Theorem 1.8. [3] Let H be a linear algebraic group over k with unipotent ˆ = H  Gm be a semidirect product of H by Gm with subgroup radical U . Let H ˆ U = U  Gm , where the conjugation action of Gm on U is such that all the weights of the induced Gm -action on the Lie algebra of U are strictly positive, while the ˆ acts induced conjugation action of Gm on R = H/U is trivial. Suppose that H linearly on an irreducible projective variety X with respect to an ample line bundle L, and that the linearisation is adapted in the sense above. Suppose also that the ˆ on X satisfies the condition (C∗ ). Then linear action of U ˆ ˆ s,U s,U ˆ ˆ (i) the open subvariety Xmin + of X has a geometric quotient X U = Xmin + /U ˆ which is a projective variety, while by U

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ˆ

≈

s,H (ii) the open subvariety Xmin + of X has a categorical quotient X which is also a projective variety.

ˆ by H ˆ H

≈

≈

Remark 1.9. In order to prove this theorem it is helpful to strengthen slightly the requirement that the linearisation is adapted. This strengthening does not alter ˆ ˆ s,U s,H ˆ ˆ Xmin + or Xmin + or their quotients X U and X H. The proof in [2] (which is then strengthened in [3]) that, if a suitable version of the condition that semistability ˆ 0 ⊗cm U coincides with stability is satisfied, the algebras of invariants ⊕∞ ) m=0 H (X, L and ˆ ˆ 0 ⊗cm H 0 ⊗cm U ⊕∞ ) = (⊕∞ ) )R m=0 H (X, L m=0 H (X, L

≈

≈

≈

≈

≈

≈

≈

≈

ˆ = X s,Uˆ /U ˆ are finitely generated (and thus that the enveloping quotients X U min + ˆ are the associated projective varieties) proceeds by induction on the and X H dimension of U and requires that the linearisation is twisted by a ‘well adapted’ rational character χ. More precisely, it is shown in [2] that, given a linear action ˆ on X with respect to an ample line bundle L, there exists > 0 such that of H ˆ with kernel if χ is a rational character of Gm (lifting to a rational character of H containing H) with ωmin < χ < ωmin + , and if a suitable ‘semistability coincides with stability’ condition is satisfied, then ˆ ˆ 0 ⊗cm U 0 ⊗cm H the algebras of invariants ⊕∞ ) and ⊕∞ ) are finitely m=0 H (X, L m=0 H (X, L ˆ and X H ˆ are the associated progenerated, and the enveloping quotients X U ˆ s,U ˆ ˆ a categorical jective varieties with X U a geometric quotient of Xmin + and X H ˆ s,H ˆ is the reductive GIT quotient of X U ˆ by the quotient of Xmin H + . Here X ˆ U ˆ ∼ induced action of the reductive group H/ = R with respect to the linearisation induced by a sufficiently divisible tensor power of L.

≈

≈

≈

Applying Theorem 1.8 with X replaced by X × P1 , with respect to the tensor power of the linearisation L (over X) with OP1 (M ) (over P1 ) for M >> 1, gives us a ˆ which is a categorical quotient by H ˆ of an H-invariant ˆ projective variety (X×P1 ) H ˆ s,U open subvariety of X ×k. This open subvariety is the inverse image in (X ×P1 )min + ˆ )ss,R of (X × P1 ) U ˆ = (X × P1 )s,Uˆ /U ˆ, of the R-semistable subset ((X × P1 ) U min + and contains as an open subvariety a geometric quotient by H of an H-invariant open subvariety X sˆ,H of X.

min +

≈

Remark 1.10. Here X sˆ,H can be identified in the obvious way with X sˆ,H ×{[1 : ˆ U 1]} which is the intersection with X×{[1 : 1]} of the inverse image in (X×P1 )s, min + = ˆ ˆ )s,R of (X × P1 )ss,fg,U of the R-stable subset ((X × P1 ) U

≈

ˆ ˆ s,U s,U 0 0 ˆ = ((Xmin ˆ∼ ˆ (X × P1 ) U × k∗ )  (Xmin + × {0}))/U = (Xmin /U )  (Xmin + /U ).

≈

≈

≈

ˆ can This geometric quotient X sˆ,H /H and its projective completion (X × P1 ) H be described using Hilbert–Mumford-like criteria, by combining the description of ˆ as the geometric quotient (X × P1 )s,Uˆ /U ˆ with reductive GIT for (X × P1 ) U min + ˆ. the induced linear action of the reductive group R = H/U on (X × P1 ) U Theorem 1.8 describes the good case when semistability coincides with stability ˆ . Theorem 1.12 below, which is proved in [3], applies to for the linear action of U

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GRADED LINEARISATIONS

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ˆ provided that the much weaker condition that the any adapted linear action of H, stabiliser in the unipotent radical U of a generic x ∈ X is trivial. Remark 1.11. In fact this weaker hypothesis can itself be removed. It is shown in [3] that Theorem 1.8 is still true when condition (C∗ ), that semistability coincides ˆ , is replaced with the weaker condition with stability for U ˜ ∗) (C

dim(StabU (z)) = min dim(StabU (x)) for every z ∈ Zmin . x∈X

Theorem 1.12 is then still valid without the hypothesis that the stabiliser in the unipotent radical U of a generic x ∈ X is trivial, provided that the condition (C∗ ) ˜ ∗ ) in its statement. is replaced with (C Theorem 1.12 is a non-reductive analogue of the partial desingularisation construction for reductive GIT described at the end of §1.1.

≈

≈

≈

Theorem 1.12. Let H be a linear algebraic group over k with graded unipotent ˆ = H  Gm be the extension of H by Gm which defines the radical U and let H ˆ acts linearly on an irreducible projective variety X with grading. Suppose that H respect to an adapted ample linearisation. Suppose also that StabU (x) = {e} for generic x ∈ X. ˆ Then there is a sequence of blow-ups of X along H-invariant projective subvarieties (the first of which is the closure in X of the locus where the stabiliser 0 ˆ with an in U has maximal dimension in Xmin ) resulting in a projective variety X ˆ adapted linear action of H (with respect to a power of an ample line bundle given by tensoring the pullback of L with small multiples of the exceptional divisors for the blow-ups) which satisfies the condition (C∗ ), so that Theorem 1.8 applies. ˆ Moreover there is a sequence of further blow-ups along H-invariant projective subvarieties appearing as the closures of H-sweeps of connected components of fixed ˜ satisfying point sets of reductive subgroups of H, resulting in a projective variety X ˆ is the ˆ ˜ the same conditions as X and in addition that the enveloping quotient X H ˆ s, H ˆ and ˆ of the H-invariant ˆ ˜ ˆ H geometric quotient by H open subset X min + . Both X ˜ H ˆ are projective completions of the geometric quotient by H ˆ of the H-invariant ˆ X ˆ

s,H open subset Xmin + of X which can be identified via the blow-down map with the ˆ s,H ˜ complement in Xmin + of the exceptional divisors.

≈

≈

ˆ on X × P1 ) as above, By considering the action of H × P1 (and similarly on X ˆ which is a categorical quotient by H ˆ of we obtain a projective variety X × P1 H ˆ a H-invariant open subset of a blow-up of X × k and contains as an open subset a geometric quotient of an H-invariant open subset X sˆ,H of X by H, where the ˆ have descripgeometric quotient X sˆ,H /H and its projective completion X × P1 H tions in terms of Hilbert–Mumford-like criteria, the explicit blow-up construction used to obtain X × P1 from X × P1 and an analogue of S-equivalence. 2. Extended, graded and torus-graded linearisations Recall from Definition 0.4 that an extended linearisation L of an action of a linear algebraic group H on a projective variety X is given by the data: (a) a line bundle L on X (usually assumed to be ample);

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` GERGELY BERCZI, BRENT DORAN, AND FRANCES KIRWAN

ˆ = H  Gm of H by Gm ; (b) a semi-direct product H ˆ and a lift of the H-action ˆ (c) an extension of the H-action on X to H to L. Recall also that given an extended linearisation L and a rational number q ∈ Q, we define the q-hat-stable locus X sˆ,q = X sˆ,q,L to be the H-invariant open subvariety of X determined by ˆ

X sˆ,q × {[1 : 1]} = (X × P1 )s,H ∩ (X × {[1 : 1]}) ˆ acts on P1 as at (0.1) above with its linearisation on OP1 (1) twisted by where H ˆ ˆ q, and (X × P1 )s,H is defined with respect to the induced linearisation for the H1 action on X × P on L tensored with OP1 (m) for m >> 1. We then have a geometric ˆ quotient X sˆ,q /H by H which is isomorphic to the open subset ((X × P1 )s,H ∩ (X × ˆ ˆ of (X × P1 )s,H /H ˆ for this choice of linearisation. (k \ {0})))/H Remark 2.1. Given a linearisation in the classical sense of an action of a linear algebraic group H on a projective variety X with respect to a line bundle L, we ˆ = H × Gm and Gm acts trivially have a ‘trivial extended linearisation’ for which H on X and on L. If q ∈ (0, 1) then the stable locus for the action of Gm on X × P1 ˆ with respect to the induced linearisation for the H-action on X × P1 on L tensored with OP1 (m) is X × (k \ {0}). Thus taking m >> 1 the q-hat-stable locus X sˆ,q coincides with the stable locus defined as in [1] for the action of H on X with the given linearisation. Similarly if q ∈ / [0, 1] the q-hat-stable locus X sˆ,q is empty for this linearisation. Recall from Definition 0.5 that a linear algebraic group with graded unipotent radical is a linear algebraic group H with unipotent radical U , equipped with a ˆ = H  Gm such that the adjoint action of Gm on the Lie semi-direct product H algebra of U has only strictly positive weights and the induced conjugation action of Gm on H/U is trivial. Recall also that a graded linearisation L of an action of H on a projective variety X is then an extended linearisation in the sense of ˆ such that the H-linearisation ˆ Definition 0.4 for this choice of H, is twisted by a ˆ so that 0 lies in the interior of the lowest bounded chamber rational character of H for the Gm action. Given a graded linearisation L, the ‘hat-stable’ locus X sˆ = X sˆ,L is the 0-hat-stable locus X sˆ,0 as defined in Definition 0.4 when q = 0. Remark 2.2. When H is a linear algebraic group with graded unipotent radical U and L is a graded linearisation for an action of H on a projective variety X (with respect to an ample line bundle L on X), then we can apply Theorems 1.8 and ˆ on X × P1 . Thus the geometric quotient X sˆ/H by H has 1.12 to the action of H ˆ of an open subset of a projective completion which is a geometric quotient by H 1 ˆ a H-equivariant blow-up of X × P . Furthermore the geometric quotient X sˆ/H by H and its projective completion can be described using Hilbert–Mumford-like criteria combined with the explicit blow-up construction. Thus the data of the graded linearisation gives us a GIT-like quotient with most of the good properties which hold in the reductive case. Now let T be a torus defined over k. Definitions 0.4 and 0.5 can be generalised to define T -extended linearisations, and T -graded linearisations for the actions of linear algebraic groups with T -graded unipotent radical. Definition 2.1. A T -extended linearisation L of an action of a linear algebraic group H on a projective variety X is given by the data:

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GRADED LINEARISATIONS

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(a) a line bundle L on X (usually assumed to be ample); ˆ = H  T of H by T ; (b) a semi-direct product H ˆ and a lift of the H-action ˆ (c) an extension of the H-action on X to H to L. Given an extended linearisation L for the action of H on X, and a projective toric variety Y = T y0 with an ample linearisation LT for the action of T on Y , we can define the ‘(Y, LT )-hat-stable’ locus X sˆ,Y,LT ,L to be the H-invariant open subvariety of X determined by ˆ

X sˆ,Y,LT ,L × {y0 } = (X × Y )s,H ∩ (X × {y0 }) ˆ

where (X × Y )s,H is defined as in [1] with respect to the induced linearisation for ˆ the H-action on X × Y with respect to the linearisation L tensored with L⊗m for T m >> 1. Remark 2.3. We then have a geometric quotient X sˆ,Y,LT ,L /H by H which is ˆ ˆ isomorphic to an open subset of (X × Y )s,H /H for a choice of linearisation as in Definition 2.1. Definition 2.2. A linear algebraic group with T -graded unipotent radical is a linear algebraic group H with unipotent radical U , equipped with ˆ = H  T such that the induced conjugation action of T i) a semi-direct product H on H/U is trivial, and ii) a non-empty open rational cone C in the Lie algebra of T such that the adjoint action on the Lie algebra of U of any one-parameter subgroup of T whose derivative at the identity lies in C has only strictly positive weights. A T -graded linearisation L of an action of H on a projective variety X is then ˆ with a T -extended linearisation in the sense of Definition 2.1 for this choice of H, ˆ ˆ whose kernel contains H, the H-linearisation twisted by a rational character of H in such a way that 0 lies in the interior of the lowest bounded chamber for some one-parameter subgroup of T whose derivative at the identity lies in the cone C. When T is the one-parameter multiplicative group Gm and Y = P1 , and LT is the linearisation of the Gm -action on OP1 (1) given by the representation (0.1), then we recover the definitions of extended and graded linearisations given above. Remark 2.4. When H is a linear algebraic group with T -graded unipotent radical U and L is a T -graded linearisation for an action of H on a projective variety X with respect to an ample line bundle L on X, then an analogous picture to that of Remark 2.2 holds [5]. Thus the geometric quotient X sˆ,Y,LT ,L /H has a ˆ of an open subset of a projective completion which is a geometric quotient by H ˆ H-equivariant blow-up of the product of X with the toric variety Y . Furthermore the geometric quotient X sˆ,Y,LT ,L /H and its projective completion can be described using Hilbert–Mumford-like criteria combined with the geometry of the toric variety T , the rational cone C and the blow-up construction. 3. Applications In this section we will describe some linear actions of non-reductive groups where GIT for suitable graded linearisations, obtained as in Remark 2.2, behaves better than GIT for classical linearisations.

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` GERGELY BERCZI, BRENT DORAN, AND FRANCES KIRWAN

≈

Example 3.1. The first of these are the famous Nagata counterexamples to Hilbert’s 14th problem [33], which provide examples of linear actions of unipotent groups U on projective space such that the corresponding algebras of U -invariants are not finitely generated. In these examples the linear action extends to a linear ˆ = U Gm by Gm such that the action of Gm by conjugation action of an extension U on the Lie algebra of U has all its weights strictly positive, and StabU (x) = {e} ˆ gives us a for generic x, so Theorem 1.12 applies, and the quotient (X × P1 ) U projective completion of a geometric quotient by U of a U -invariant open subset of the projective space X which can be determined by Hilbert–Mumford-like criteria. We can regard this as the GIT quotient of the projective space by the graded unipotent group U with respect to the induced graded linearisation. Note that the ˆ -invariants on X × P1 restrict to U -invariants on X, so the grading is picking out U for us a finitely generated subalgebra of the algebra of U -invariants, and thus a tractable GIT quotient. Example 3.2. Recall that the automorphism group of the weighted projective plane P(1, 1, 2) = (k3 \ {0})/Gm , for Gm acting linearly on k3 with weights 1, 1, 2, is given by Aut(P(1, 1, 2)) ∼ =RU 3 ∼ ∼ where R = GL(2) is reductive and U = (Ga ) is unipotent, with elements (λ, μ, ν) ∈ (k)3 acting on P(1, 1, 2) via [x, y, z] → [x, y, z + λx2 + μxy + νy 2 ]. The central one-parameter subgroup Gm of R ∼ = GL(2) acts on the Lie algebra of U with all positive weights, and the associated semi-direct product ˆ = U  Gm U can be identified with a subgroup of Aut(P(1, 1, 2)). Thus any ample linearisation for an action of Aut(P(1, 1, 2)) on a projective variety X becomes a graded linearisation in a natural way. It therefore follows from Theorem 1.12 that whenever H = Aut(P(1, 1, 2)) acts linearly on a projective variety X and StabU (x) = {e} for generic x ∈ X, then there is a geometric quotient by H of an open subset of X described by Hilbert–Mumford-like criteria, with a projective completion which is ˜ of X. a categorical quotient of an open subset of an H-equivariant blow-up X Indeed the same is true for the automorphism group of any complete simplicial toric variety. For it was observed in [2] using the description in [11] that the automorphism group H of any complete simplicial toric variety is a linear algebraic group with a graded unipotent radical U ; there is a grading defined by a one parameter subgroup Gm of H acting by conjugation on the Lie algebra of U with all weights strictly positive, and inducing a central one-parameter subgroup of R = H/U . Thus Theorems 1.8 and 1.12 (and if necessary Remark 1.11) can be applied. Example 3.3. Suppose now that k = C and consider k-jets at 0 of holomorphic maps from Cp to a complex manifold Y for any k, p ≥ 1. It was observed in [6] that the group G(k,p) of k-jets of holomorphic reparametrisations of (Cp , 0) has a graded unipotent radical U(k,p) such that the grading is defined by a one-parameter subgroup of G(k,p) acting by conjugation on the Lie algebra of U(k,p) with all weights strictly positive, and inducing a central one-parameter subgroup of the reductive group G(k,p) /U(k,p) . So Theorems 1.8 and 1.12, with Remark 1.11, can be applied to any linear action of the reparametrisation group G(k,p) .

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GRADED LINEARISATIONS

19

Example 3.4. Finally let G be a reductive group over an algebraically closed field k of characteristic zero, acting linearly on a projective variety X with respect to an ample line bundle L. Associated to this linear G-action and an invariant inner product on the Lie algebra of G, there is a stratification  X= Sβ β∈B

of X by locally closed subvarieties Sβ , indexed by a partially ordered finite subset B of a positive Weyl chamber for the reductive group G, such that (i) S0 = X ss , and for each β ∈ B  (ii) the closure of Sβ is contained in γβ Sγ , and (iii) Sβ ∼ = G ×Pβ Yβss where Pβ is a parabolic subgroup of G acting on a projective subvariety Y β of X with an open subset Yβss which is determined by the action of the Levi subgroup Lβ of Pβ with respect to a suitably twisted linearisation [22, 26]. Here the original linearisation for the action of G on L → X is restricted to the action of the parabolic subgroup Pβ over Y β , and then twisted by the rational character β of Pβ which is almost adapted for a central one-parameter subgroup of the Levi subgroup Lβ acting with all weights strictly positive on the Lie algebra of the unipotent radical of Pβ . So Pβ is a linear algebraic group with graded unipotent radical; indeed, its unipotent radical is graded by the torus which is the centre of a Levi subgroup of Pβ . Thus to construct a quotient by G of (an open subset of) an unstable stratum Sβ , we can study the linear action on Y β of the parabolic subgroup Pβ , and apply Theorems 1.8 and 1.12, with Remark 1.11. In this situation Yβ is equal to (Y β )0min (in the notation introduced in §1.3, immediately before Theorem 1.8), and we have a retraction pβ : Yβ → Zβ where pβ (y) =

lim

t→0, t∈Gm

t·y

for y ∈ Yβ and Zβ plays the role of Zmin in §1.3. Since the rational character β of Pβ is borderline adapted, not adapted, we have Yβss = p−1 β (Zβ

ss,Lβ

)

and the reductive GIT quotient Zβ //Lβ is the categorical quotient of Yβss by Pβ . However this is certainly not a geometric quotient (because the closure of every Pβ ss,L orbit in Yβss meets Zβss ). Zβ //Lβ is also the categorical quotient of Pβ Zβ β ⊆ Yβss by Pβ , and ss,Lβ

Pβ Zβ

ss,Lβ

= Uβ Zβ

where Uβ is the unipotent radical of Pβ . On the other hand if we modify the linear action of Pβ on Y β by an adapted rational character, given by (1 + δ)β for 0 < δ 0 such that the following identity holds for all β ∈ π2 (X, L): c · μL (β) = ω(β).

(3.1)

The minimal Maslov number of L is defined to be: inf{μL (β) | β ∈ π2 (X, L), ω(β) > 0}. Following Floer’s original definition [Flo88b], Oh constructed Lagrangian Floer homology for a pair L0 and L1 of monotone Lagrangians, if one of the following conditions holds [Oh93]: (m.a) The minimal Maslov numbers of L0 and of L1 are both strictly greater than 2. (m.b) The Lagrangian submanifold L1 is Hamiltonian isotopic to L0 . Lagrangian Floer homology can be enriched when there is a group action on the underlying symplectic manifold. Such constructions have been carried out in various ways in the literature. (See Remark 3.3). Let a compact Lie group G act on X, preserving the symplectic structure ω. We fix a G-equivariant almost complex structure J which is compatible with ω. Note that the space of all such almost complex structures is contractible because the set of all G-invariant Rie∗ (M ) for a G-space M denotes the mannian metrics is convex. In the following, HG G-equivariant cohomology of M with coefficients in R. In the case that M is just ∗ ∗ . The group HG (M ) has the structure of a a point, this group is denoted by HG ∗ module over HG [Bor60]. Theorem 3.2. Let L0 , L1 be G-equivariant spin Lagrangian submanifolds of X. Suppose they are both monotone and satisfy either (m.a) or (m.b). Then there ∗ -module HFG (L0 , L1 ), called G-equivariant Lagrangian Floer homology of is a HG L0 and L1 . In the case that the intersection L0 ∩ L1 is clean, there exists a spectral ∗ sequence whose E2 page is HG (L0 ∩ L1 ) and which converges to HFG (L0 , L1 ). Recall that two submanifolds L0 and L1 of a smooth manifold M have clean intersection, if N = L0 ∩ L1 is a smooth submanifold of M and for any x ∈ N , we have Tx N = Tx L0 ∩ Tx L1 . Sketch of the proof. We assume that the intersection L0 ∩ L1 is a disjoint union of finitely many G-orbits G · p for p ∈ A. A pseudo-holomorphic strip u : R × [0, 1] → X is a map that satisfies the following Cauchy-Riemann equation: (3.2)

∂t u + J∂τ u = 0

We are interested in the moduli space of pseudo-holomorphic maps u which satisfy the following boundary condition: (3.3)

u(R × {0}) ⊂ L0 , lim u(t, τ ) ∈ G · p,

t→+∞

u(R × {1}) ⊂ L1 lim u(t, τ ) ∈ G · q.

t→−∞

We will denote the homology classes of all such maps by H(p, q). For a fixed ◦

β ∈ H(p, q), let M(p, q; β; L0 , L1 ) be the moduli space of pseudo-holomorphic maps satisfying (3.3) and representing β, where we identify two maps u and u ◦

if u (t, τ ) = u(t + t0 , τ ) for some t0 ∈ R. Note that M(p, q; β; L0 , L1 ) is invariant with respect to the action of the group G. We also assume that this space is cut

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ATIYAH-FLOER CONJECTURE

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out transversely by Equation (3.2). This moduli space can be compactified to a cornered manifold M(p, q; β; L0 , L1 ) using stable map compactification [FO99, Definition 10.3]. Codimension one boundary components of this space can be identified with the union of the fiber products: (3.4)

M(p, r; β1 ; L0 , L1 ) ×G·r M(r, q; β2 ; L0 , L1 )

where r ∈ A and β1 #β2 = β. Here # : H(p, r) × H(r, q) → H(p, q) is the concatenation of homology classes. Monotonicity and (m.a) or (m.b) are the main ingredients to proving these claims about stable map compactification. The classifying space BG and the universal bundle EG over BG can be approximated by finite dimensional manifolds BG(N ), EG(N ). To be more precise, suppose EG(N ) is a principal G-bundle over a manifold BG(N ) such that the homotopy groups of EG(N ) vanish up to degree N . We consider the approximate Borel construction M(p, q; β; L0 , L1 ) ×G EG(N ). Taking asymptotic values as t → ±∞, we obtain two evaluation maps as below: ev−∞

ev+∞

G · p ×G EG(N ) ←−−−− M(p, q; β; L0 , L1 ) ×G EG(N ) −−−−→ G · q ×G EG(N ). If ev+∞ is a submersion, then we can define an operator: (3.5)

dp,q;β : Ω∗ (G · p ×G EG(N )) → Ω∗ (G · q ×G EG(N ))

between the space of differential forms by: (3.6)

dp,q;β (h) = (ev+∞ )! (ev∗−∞ h),

where (ev+∞ )! is integration along the fiber. Characterization of codimension one boundary components in (3.4) implies that:   ±dp,r;β1 ◦ dr,q;β2 . (3.7) d ◦ dp,q;β ± dp,q;β ◦ d = r

β1 +β2 =β

 Here d is the usual de Rham differential. Therefore, the map δN = d + dp,q;β defines a differential, i.e., it satisfies δN ◦ δN = 0. Taking the limit N → ∞, we obtain equivariant Floer homology as the limit. In general, it might be the case that M(p, q; β; L0 , L1 ) is not a smooth manifold or ev+∞ is not a submersion. Then we can use the theory of Kuranishi structures and continuous families of perturbations on Kuranishi spaces to prove the same conclusion. In fact, following [Fuk17b], we obtain a G-equivariant Kuranishi structure on M(p, q; β; L0 , L1 ) and hence a Kuranishi structure on M(p, q; β; L0 , L1 ) ×G EG(N ). Then we can define a system of perturbations on these Kuranishi structures which give rise to a map as in (3.5) between the spaces of differential forms that satisfy (3.7).  The elements of the moduli space M(p, q; β; L0 , L1 ) can be regarded as solutions of a Fredholm equation which is defined on an infinite dimensional space and takes values in another infinite dimensional space. Roughly speaking, a Kuranishi structure on this moduli space replaces these infinite dimensional spaces with spaces of finite dimensions. To be a bit more detailed, a Kuranishi structure is a covering of the moduli space with Kuranishi charts. For a point p in the moduli space, a Kuranishi chart in a neighborhood of p is a quadruple (V, E, s, ψ) such that V is a manifold, E is a vector bundle, s is a section of E and ψ is a homeomorphism from s−1 (0) to an open neighborhood of p in the moduli space. In general, we might need to work in the case that V and E are orbifold and orbi-bundle. Another part of the

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ALIAKBAR DAEMI AND KENJI FUKAYA

data of a Kuranishi structure is the set of coordinate change maps which explain how to glue different Kuranishi charts together. In order to get smooth spaces, we need to perturb the zero sets of the sections of Kuranishi charts in a consistent way, and continuous families of perturbations give a systematic way to achieve this goal. For a more detailed definition of Kuranishi structures and continuous families of perturbations, we refer the reader to [FOOO09a, Definition A1.5] and [FOOO15]. Remark 3.3. In this section, we discussed an approach to equivariant Lagrangian Floer homology which is given by applying the Borel construction after taking finite dimensional reduction. This approach was proposed independently by the second author in [Fuk96b, Section 7] and Viterbo. There are alternative approaches to equivariant Lagrangian Floer homology which also use the Borel construction but avoid virtual techniques. These approaches give rise to similar results as Theorem 3.2 under more restrictive assumptions. In the case that G = Z/2Z, Floer homology coupled with Morse homology on EG is used in [SS10] by Seidel and Smith to define equivariant Lagrangian Floer homology. More recently, Hendricks, Lipshitz and Sarkar employed homotopy theoretic methods to define Lagrangian Floer homology in the presence of the action of a Lie group [HLS16b, HLS16a]. There are also various other equivariant theories for other Floer homologies (see, for example, [Don02, KM07, AB96]). 4. Lagrangian Floer Theory in a Smooth Divisor Complement Let (X, ω) be a compact symplectic manifold and D be a codimension 2 submanifold. We assume that (X, D) is a K¨ahler manifold in a neighborhood of D, and D is a smooth divisor in this neighborhood. Definition 4.1. Let L1 and L2 be compact subsets of X \ D. We say L1 is Hamiltonian isotopic to L2 relative to D if there exists a compactly supported time dependent Hamiltonian H : (X\D) × [0, 1] → R so that the Hamiltonian diffeomorphism ϕ : X \ D → X \ D generated by H sends L1 to L2 , that is, ϕ(L1 ) = L2 . Definition 4.2. We say L ⊂ X \ D is monotone if (3.1) holds for β ∈ H2 (X \ D, L). The minimal Maslov number of L relative to D is defined as: inf{μL (β) | β ∈ π2 (X \ D, L), ω(β) > 0}. R general,  Λ0 ,

In the universal Novikov ring with ground ring R, consists of λi formal sums c T where ci ∈ R, λi ∈ R≥0 λi = +∞, and T is a i i , limλi→∞ i c T where ci ∈ R, λi ∈ R, formal parameter. Similarly, ΛR consists of i i R limi→∞ λi = +∞. If R is a field then Λ is also a field. Theorem 4.3. ([DF]) Let L0 , L1 be compact, monotone and spin Lagrangian submanifolds of X \D. We assume that (m.a) or (m.b) holds for these Lagrangians. Then there is a vector space HF (L0 , L1 ; X \ D) over ΛQ which is called the Lagrangian Floer homology of L0 and L1 relative to D, and satisfies the following properties: (i) If L0 is transversal to L1 then we have rankΛQ HF (L0 , L1 ; X \ D) ≤ #(L0 ∩ L1 ). (ii) If

Li

is Hamiltonian isotopic to Li in X \ D for i = 0, 1 then ∼ HF (L , L ; X \ D) HF (L0 , L1 ; X \ D) = 0

1

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ATIYAH-FLOER CONJECTURE

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(iii) If either L1 = L0 or π1 (L0 ) = π1 (L1 ) = 0, then we can take Q as a coefficient ring instead of the Novikov filed ΛQ . (iv) If L0 = L1 = L holds, then there exists a spectral sequence whose E 2 page is H(L; Q) and which converges to HF (L, L; X \ D). Remark 4.4. The main point in Theorem 4.3 is that we do not assume Li to be a monotone Lagrangian submanifold in X, for i = 1, 2. The general theory of [FOOO09a, FOOO09b] says that there is an obstruction to defining Floer homology HF (L0 , L1 ). The Floer homology HF (L0 , L1 ; X \ D) uses only holomorphic disks which ‘do not intersect’ D. Therefore, the situation is similar to monotone Lagrangian Floer homology due to Oh [Oh93]. If X \ D is convex at infinity, then [FOOO09a, FOOO09b] imply that we can define Floer homology HF (L0 , L1 ; X \ D) satisfying the properties mentioned in Theorem 4.3. Note that in Theorem 4.3, we do not impose any kind of convexity assumption for X \ D. The specialization of the construction of Theorem 4.3 to the case where Li is exact and the homology class of each component of D is proportional to the Poincar´e dual of [ω] is given in [She15]. Sketch of the proof. We assume that L0 is transversal to L1 . Let p, q ∈ ◦

L0 ∩ L1 . We consider the moduli space M(p, q; β; L0 , L1 ) of pseudo-holomorphic maps to X\D which satisfy (3.2) and (3.3) for G = {1}. Following Floer [Flo88b] and Oh [Oh93] (see also [FOOO09a, Chapter2]), we can define HF (L0 , L1 ; X \ D) if we obtain a compactification MRGW (p, q; β; L0 , L1 ) of our moduli space ◦

M(p, q; β; L0 , L1 ) with the following properties: (I) The compactification MRGW (p, q; β; L0 , L1 ) carries a Kuranishi structure with boundary and corner. (II) The codimension one boundary of this moduli space is identified with the union of (4.1)

MRGW (p, r; β1 ; L0 , L1 ) × MRGW (r, q; β2 ; L0 , L1 )

for various r ∈ L1 ∩ L2 and β1 , β2 with β1 + β2 = β. The (virtual) dimension d(β) of MRGW (p, q; β; L0 , L1 ) is determined by the homology class β and satisfies d(β) = d(β1 ) + d(β2 ) + 1 for the boundary component in (4.1). We fix a multisection3 (or equivalently a multivalued perturbation) which is transversal to 0 and which is compatible with the description of the boundary as in (4.1). Note that transversality implies that its zero set is the empty set when the virtual dimension is negative. Therefore, the zero set is a finite set in the case that the virtual dimension is 0. Assuming d(β) = 0, let #MRGW (p, q; β; L0 , L1 ) be the number (counted with sign and multiplicity) of the points in the zero set of the perturbed moduli space. Then we define:  #MRGW (p, q; β; L0 , L1 )[q]. ∂[p] = q,β

Here the sum is taken over all q ∈ L0 ∩L1 and homology classes β such that d(β) = 0. In the case L0 = L1 or π1 (L0 ) = π1 (L1 ) = 0 the right hand side is a finite sum. Otherwise we use an appropriate Novikov ring and put the weight T ω(β) on each of the terms of the right-hand side, so that the right-hand side converges in the T -adic 3 See

[FOOO09b, Definition A1.21])

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34

ALIAKBAR DAEMI AND KENJI FUKAYA

topology. As it is customary with other Floer theories, we can show ∂ ◦ ∂ = 0 using the moduli spaces associated to the homology classes β with d(β) = 1. (See, for example, [Flo88b,Oh93].) The proofs of parts (ii) and (iii)) are also similar to the proof of the corresponding statements in the case of usual monotone Lagrangian Floer homology. If X\D is convex at infinity, then we can let MRGW (p, q; β; L0 , L1 ) be the ◦

closure of M(p, q; β; L0 , L1 ) in the moduli space of stable holomorphic maps to X. In this case, monotonicity can be used to show that (4) gives all the configurations appearing in the boundary of MRGW (p, q; β; L0 , L1 ). In case we do not assume monotonicity, disk bubbles can occur as the other type of boundary component. (See [FOOO09a, Subsection 2.4.5] for example.) The stable map compactification in [FOOO09b, Subsection 7.1.4] does not give a compactification MRGW (p, q; β; L0 , L1 ) with the required properties. The issue is that in the stable map compactification a stable map with a sphere bubble which is contained completely in the divisor D is included. At the points of such stable maps, the limits of the following two kinds of sequences of stable maps are mixed up. (A) A limit of a sequence of pseudo-holomorphic disks ui : (D2 , ∂D2 ) → (X, L) such that ui (D2 ) ∩ D = ∅. (B) A limit of a sequence of stable maps ui : (Σi , ∂Σi ) → (X, L), where Σi is a disk plus sphere bubbles, and such that ui (Σi ) ∩ D = ∅. We need to include (A) in our moduli space but (B) is not supposed to be an element of the moduli space. As it is shown in Figure 1, elements given as the limit points of type (A) and type (B) can be mixed up in the stable map compactification. Here all the sphere bubbles in the figure are contained in D. The numbers written in the sphere bubbles S 2 are the intersection numbers [S 2 ] ∩ D. The numbers written at the roots of the sphere bubble are the intersection multiplicities of the disk with the divisor D. The configuration shown as (a) is a limit of disks as in (A) above since 2 + (−2) = 1 + (−1) = 0. The configuration shown as (b) is not a limit of disks as in (A),since 2 + (−1) = 0 = 1 + (−2). However, these two configurations can intersect in the limit, which is the stable map shown as (c) in the figure. Note that a limit of the configuration (b) in the figure can split into two pieces as shown in the figure.Then the union of the disk component together with sphere bubble rooted on it is not monotone. Thus if we include (b), then there will be trouble in showing (4.1). The idea to resolve this issue is to use a compactification which is different from the stable map compactification (in X). We use the compactification used in relative Gromov-Witten theory, where the limits of type (A) and type (B) are clearly separated. (See [LR01, IP03, Li01, Li02, GS13, Par12, Teh17].) Namely in this compactification configuration (c) in the figure comes with additional information so that the limits of type (A) and of type (B) become different elements in this compactification. Using this fact we can then show the above properties (I)(II).  5. The Atiyah-Floer Conjecture Floer’s original instanton Floer homology is an invariant of 3-manifolds which have the same integral homology as the 3-dimensional sphere [Flo88a]. Given

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ATIYAH-FLOER CONJECTURE

35

Figure 1. (a) and (b) mixed up. an integral homology sphere M , he introduced a chain complex C∗ (M ) with the differential ∂, whose homology is defined to be instanton Floer homology and is denoted by I ∗ (M ). The chain complex C∗ (M ) is a free group generated by nontrivial flat SU(2)-connections.4 Since any SU(2)-bundle over a 3-manifold can be trivialized, these connections all have the same topological type. The differential ∂ is defined by considering the moduli space of instantons on the trivial SU(2)-bundle P over R × M . To be a bit more detailed, fix a product metric on R × M corresponding to a fixed metric on M . The Hodge ∗-operator decomposes the space of 2-forms to anti-self-dual and self-dual forms. Then an instanton on R × M is a connection A on P such that: (5.1)

F + (A) = 0

||F (A)||2 < ∞

where F + (A) and ||F (A)||2 are respectively the self-dual part and the L2 norm of the curvature of A. It is shown in [Flo88a] that for any instanton A, there are flat connections a+ and a− such that5 : (5.2)

lim A|{t}×M = a± .

t→±∞

Translation in the R-direction and SU(2)-bundle automorphisms act on the space of instantons. The quotient space of instantons satisfying (5.2) with respect to these two actions is denoted by M(a− , a+ ; M ). Moreover, if we require ||F (A)||22 to be equal to a fixed real number E, then the resulting space is denoted by M(a− , a+ ; E; M ). The differential ∂(a) for a non-trivial flat connection a is defined as:  ∂(a) = #M(a, b; E; M ) · b where the sum is over all E and b that M(a, b; E; M ) is 0-dimensional. Here #M(a, b; M ) denotes the signed count of the points in the 0-dimensional space M(a, b; E; M ). In general, we might need to perturb the equation in (5.1) as the 4 These flat connections are critical points of a Chern-Simons functional. Here we are assuming that the Chern-Simons functional is Morse in an appropriate sense. In general we need to perturb the Chern-Simons functional to ensure that the critical points are non-degenerate. 5 We still assume that the Chern-Simons functional is Morse. But this fact is true even in the more general case that this assumption does not hold. (See [Don02, Chapter 4] or [MMR94, Chapter 4].)

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36

ALIAKBAR DAEMI AND KENJI FUKAYA

space of flat connections on M and the space of instantons on R × M might not be cut out by transversal equations. There are several other versions of instanton Floer homology in the literature. The trivial connection on an integral homology sphere M does not play any role in the definition of I ∗ (M ). An alternative version of this invariant, constructed in [Don02], uses the moduli spaces M(a, b, E; M ) where a or b could be the trivial ∗ ∗ -module6 . connection. We will write I (M ) for this invariant, which is an HPU(2) Next, we aim to construct a version of symplectic instanton Floer homology which is conjecturally isomorphic to I ∗ (M ) for an integral homology sphere M . We shall apply a combination of the constructions of the previous two sections to the following pair of a symplectic manifold and a smooth divisor, introduced in (2.4) and (2.5):  (5.3) (X, D) := (R(Σ, p, θ), μ−1 (θ)/S 1 ) with 0 < θ < 12 . This version of symplectic instanton Floer homology can be regarded as an equivariant version of a variation of the construction in [MW12]. (See Remark 5.10.) Fix a Heegaard splitting of the 3-manifold M : M = Hg0 ∪Σg Hg1 .

(5.4)

˜ gi , p) According to Proposition 2.6, we can form the Lagrangian submanifolds R(H of X associated to this Heegaard splitting. The following Lemma about the intersection of these Lagrangians can be proved using holonomy perturbations [Tau90, Flo88a, Don02, Her94]. We omit the details here: ˜ gi , p) in Lemma 5.1. There are Hamiltonian isotopies of the Lagrangians R(H X\D to submanifolds with clean intersection. Moreover, we can assume that each connected component of the intersection of the perturbed Lagrangians is either a point which consists of the trivial connection or a single PU(2)-orbit. ˜ i , p) provided by Suppose Li denotes the perturbation of the Lagrangian R(H g ˜ gi , p) is monotone in X\D [MW12], the Lagrangian Li is also Lemma 5.1. Since R(H monotone in X\D. The manifold Li is diffeomorphic to the Cartesian product of g copies of SU(2) [MW12]. In particular, it can be equipped with a spin structure. The intersection of L0 and L1 can be decomposed as:  Ra (5.5) L0 ∩ L1 = {θ} ∪ a∈A

where Ra ∼ = PU(2). Here θ denotes the trivial connection. Let A+ = A ∪ {θ} and Rθ = {θ}. ◦

For a, b ∈ A, define M(a, b; β; L0 , L1 ) to be the moduli space of maps u : R × [0, 1] → X \ D which satisfy the analogues of (3.2), (3.3) and represent the homology class β ∈ H(a, b). As before, we also identify two maps u and u if u (τ, t) = u(τ + τ0 , t) for some τ0 ∈ R. There is an obvious PU(2) action on this moduli space. We can also form the restriction maps: (5.6)

◦

ev−∞ : M(a, b; β; L0 , L1 ) → Ra ,

◦

ev+∞ : M(a, b; β; L0 , L1 ) → Rb .

A combination of the proofs of Theorems 3.2 and 4.3 can be used to prove the following Proposition: 6 The

original notation for this invariant in [Don02] is HF (M ).

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ATIYAH-FLOER CONJECTURE

37 ◦

Proposition 5.2. There exists a compactification of M(a, b; β; L0 , L1 ), denoted by MRGW (a, b; β; L0 , L1 ), which satisfies the following properties: (i) This space has a Kuranishi structure with corners. The PU(2) action ◦

of M(a, b; β; L0 , L1 ) extends to MRGW (a, b; β; L0 , L1 ) and the Kuranishi structure is PU(2)-equivariant. The evaluation maps in (5.6) also extend to MRGW (a, b; β; L0 , L1 ) and are underlying maps of PU(2)-equivariant weakly submersive maps.7 (ii) Let d(β) be the virtual dimension of MRGW (a, b; β; L0 , L1 ). For any d, there are only finitely many choices of β such that MRGW (a, b; β; L0 , L1 ) is nonempty and d(β) = d. There also exists deg : A → Z/8Z such that deg(θ) = 0 and for any a ∈ A, b ∈ A+ , β ∈ H(a, b), we have: d(β) ≡ deg(b) − deg(a) + 2 mod 8.

(5.7)

Moreover, if b ∈ A+ and β ∈ H(θ, b) then: d(β) ≡ deg(b) − 1 mod 8.

(5.8)

(iii) The codimension one boundary components of MRGW (a, b; β; L0 , L1 ) consist of fiber products (5.9)

MRGW (a, c; β1 ; L0 , L1 ) ×Rc MRGW (c, b; β2 ; L0 , L1 ), where the union is taken over c ∈ A+ and β1 ∈ H(a, c), β2 ∈ H(c, b) with β1 #β2 = β.

Remark 5.3. The characterization of codimension one boundary components in (5.9) implies that if c = θ, then: d(β1 #β2 ) = d(β1 ) + d(β2 ) − 3 and if c = θ, then: d(β1 #β2 ) = d(β1 ) + d(β2 ). This is consistent with the identities in (5.7) and (5.8). Analogous to the construction of Section 3, we can use the compactification provided by this proposition to define a Lagrangian Floer homology group : (5.10)

HFPU(2) (L0 , L1 ; X \ D)

for an integral homology sphere. This Lagrangian Floer homology group is a mod∗ ∗ = HSU(2) . The following conjecture states that this module is a ule over HPU(2) 3-manifold invariant. This invariant can be regarded as a version of symplectic ∗ instanton Floer homology and is denoted by I symp (M ). ∗ Conjecture 5.4. The HPU(2) -modules in (5.10) for different choices of Heegaard splitting are isomorphic to each other.

Remark 5.5. We hope to address Conjecture 5.4 in the same way as in the proof of the corresponding result in [MW12]. (The result of [MW12] can be regarded as a non-equivariant version of Conjecture 5.4.) Following the arguments in [MW12] requires us to consider quilted Floer homology of Lagrangian correspondences such that each Lagrangian correspondence is from a pair (X1 , D1 ) of 7 See

[FOOO11, Definition 32.1] for its definition.

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38

ALIAKBAR DAEMI AND KENJI FUKAYA

a symplectic manifold and a smooth divisor to another such pair (X2 , D2 ). Consequently, we need to study the moduli space of holomorphic curves for the pair (X1 × X2 , (D1 × X2 ) ∪ (X1 × D2 )). The space (D1 × X2 ) ∪ (X1 × D2 ) is a normal crossing divisor in X1 × X2 . The extension of the theory of Section 4 to normal crossing divisors is the content of a work in progress and its details have not been completely worked out yet. (See Conjecture 8.5 and Remark 8.6.) However, as it is explained in [Fuk17c, Section 12], we can use a different compactification of holomorphic discs whose target is the product X1 × X2 . This compactification is denoted by M and is discussed in [Fuk17c, Section 12]. In this compactification, the sphere bubbles on two factors are studied separately. It is plausible that adapting this construction to our set up allows us to avoid the case of holomorphic discs in the complement of normal crossing divisors and to work only with smooth divisors. There is an alternative version of symplectic instanton Floer homology constructed by the moduli spaces MRGW (a, b; β; L0 , L1 ). The PU(2) action on the space MRGW (a, b; β; L0 , L1 ) is free unless a = b = θ. If a = b = θ, then the action is still free unless β = 0, which is the homology class of the constant map. The moduli space MRGW (θ, θ; 0; L0 , L1 ) consists of a single element. Therefore, the quotient space M

RGW

(a, b; β; L0 , L1 ) := MRGW (a, b; β; L0 , L1 )/PU(2)

has an induced Kuranishi structure. Proposition 5.2 can be used to verify the following lemma: RGW

Lemma 5.6. For a, b ∈ A, the boundary of M of two types of spaces:

(a, b; β; L0 , L1 ) is the union

(1) The direct product: RGW

M

RGW

(a, c; β1 ; L0 , L1 ) × M

(c, b; β2 ; L0 , L1 )

for c ∈ A, β1 ∈ H(a, c) and β2 ∈ H(c, b) such that β1 #β2 = β. (2) The quotient of the union of direct products MRGW (a, θ; β1 ; L0 , L1 ) × MRGW (θ, b; β2 ; L0 , L1 ) by the diagonal PU(2) action. Here the union is taken over β1 ∈ H(a, θ) and β2 ∈ H(θ, b) with β1 #β2 = β. We pick a system of PU(2) invariant multi-sections over each moduli space MRGW (a, b; β; L0 , L1 ) that is compatible with the description of the boundaries in (5.9). This is equivalent to choosing a system of multi-sections over various RGW M (a, b; β; L0 , L1 ) that is compatible with the description of the boundaries in Lemma 5.6. In the case that d(β) = 0 and a, b = θ, Lemma 5.6 and the compatibility of the multi-sections show that the zero set of the multi-section in the moduli space RGW M (a, b; β; L0 , L1 ) is a compact 0-dimensional space. Therefore, we can count the number of points in this space (with signs) to define: (5.11)

#M

RGW

(a, b; β; L0 , L1 ).

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ATIYAH-FLOER CONJECTURE

39

Now we are ready to define another version of symplectic instanton Floer homology for integral homology spheres. Define:  ∗ (M ) := Q[a] (5.12) Csymp a∈A

and (5.13)

∂[a] :=



 RGW #M (a, b; β; L0 , L1 ) [b].

b∈A,β∈H(a,b)

where the sum is over all b ∈ A and β ∈ H(a, b) such that d(β) = 0. Another application of Lemma 5.6 and the compatibility of the multi-sections show that ∂ 2 = 0. To be a bit more detailed, the terms in ∂ 2 (a), for a non-trivial flat connection a, are in correspondence with the boundary points of the 1-dimensional RGW (a, b; β; L0 , L1 ) which are of type (1) in Lemma 5.11. For moduli spaces M a 1-dimensional moduli space, the space of boundary points of type (2) is empty, because each component of the space of type (2) boundary points has dimension at least 3. ∗ (M ). The homology of the chain complex in (5.12) and (5.13) is denoted by Isymp ∗ Note that our definition of Isymp (M ) resembles Floer’s instanton homology I ∗ (M ) in the sense that the trivial connection θ does not enter into the definition of the corresponding chain complex. The following is the analogue of Conjecture 5.4. The same comment as in Remark 5.5 applies to this conjecture. ∗ (M ) is an invariant of the integral homology Conjecture 5.7. The group Isymp ∗ sphere M . That is to say, the homology of the chain complex (Csymp (M ), ∂) is independent of the choice of Heegaard splitting. ∗ (M ), we only need the moduli Remark 5.8. In the course of defining Isymp spaces of virtual dimension 1 or 0. Therefore, we do not need to prove the smoothness of the coordinate change maps of our Kuranishi structure. We also do not need to study triangulations of the zero set of our multi-sections. For example, we can discuss in the same way as in [FOOO15, Section 14].

The first part of the following conjecture can be regarded as a rigorous formulation of the original version of the Atiyah-Floer conjecture for integral homology spheres. In Section 6, we sketch a plan for the proof of the first part of Conjecture 5.9. Conjecture 5.9. For any integral homology sphere M , the vector spaces ∗ ∗ ∗ (M ) are isomorphic to each other. The HPU(2) -modules I (M ) I ∗ (M ) and Isymp ∗

and I symp (M ) are also isomorphic to each other. Remark 5.10. One can forget the PU(2)-action on MRGW (a, b; β; L0 , L1 ) and apply the construction of the previous section to define (non-equivariant) Lagrangian Floer homology for the Lagrangians L0 and L1 in the complement of D. The resulting Floer homology is essentially the same 3-manifold invariant as the version of symplectic instanton Floer homology that is constructed in [MW12]. There is also an analogue of the Atiyah-Floer conjecture for this invariant. It is conjectured in [MW12] that this invariant is isomorphic to an alternative version of instanton ! (M ). Floer homology, defined in [Don02] and denoted by HF

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40

ALIAKBAR DAEMI AND KENJI FUKAYA

6. Atiyah-Floer Conjecture and Moduli Space of Solutions to the Mixed Equation In this section we propose a program to prove Conjecture 5.9. The main geometrical input in this program is a moduli space which is a mixture of the moduli space of Anti-Self-Dual connections and pseudo-holomorphic curves. Here we describe the version introduced in [Lip14]. Similar moduli spaces appeared in [Fuk98]. Analogous mixed moduli spaces are also being used by Max Lipyanskiy and the authors to prove an SO(3)-analogue of the Atiyah-Floer conjecture [DFL]. Suppose M is an integral homology sphere and a Heegaard splitting as in (5.4)  is fixed for M . Therefore, we can form the symplectic manifold X = R(Σ, p, θ) i ˜ g , p). For the sake of exposition, we and the Lagrangian submanifolds Li = R(H assume that L0 and L1 have clean intersection. Recall that Lemma 5.1 states that in general we can perturb these Lagrangians by Hamiltonian isotopies to ensure that this assumption holds. Let the domain W in the complex plane C be given as in Figure 2. We also

∂ 0 W+

∂ 1 W+ W+ C W−

τ ∂ 0 W−

∂ 1 W− t

Figure 2. The domain W decompose this domain into two parts W− and W+ as in the figure and let C = W− ∩ W+ . Using the coordinate t, τ in the figure, the line C is the part τ = 0. The domain W has four boundary components, denoted by ∂0 W− , ∂1 W− , ∂0 W+ , ∂1 W+ , and four ends as below: (6.1)

{(t, τ ) | τ ∈ [−1, 1], t < −K0 },

{(t, τ ) | τ ∈ [−1, 1], t > K0 },

{(t, τ ) | t ∈ [−1, 1], τ < −K0 },

{(t, τ ) | t ∈ [−1, 1], τ > K0 }.

We fix a Riemannian metric gW on W which coincides with the standard Riemannian metric on the complex plane where |t| or |τ | is large and outside a small neighborhood of ∂0 W+ ∪ ∂1 W+ . We also require that the metric is isometric to (−ε, 0] × R on a small neighborhood of ∂0 W+ , ∂1 W+ . Fix a product metric on the product 4-manifold W+ × Σg . We glue Hg0 × R and Hg1 × R to the boundary components Σg × ∂0 W+ and Σg × ∂1 W+ of W+ × Σg , respectively. We will denote the resulting 4-manifold with Y+ (cf. Figure 3). The manifold Y+ has three ends and one boundary component which is Σg × C. The three ends correspond to the part t → ±∞ and τ → +∞, and they can be identified

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ATIYAH-FLOER CONJECTURE

41

M

Hg1 × R

Hg0 × R

Σg × W+ C τ

Σg × ∂ 0 W +

Σg × ∂ 1 W+ t

Figure 3. The 4-dimensional manifold Y+ with: (6.2)

Hg0 × (−∞, −K0 ),

Hg1 × (K0 , +∞),

(Hg0 ∪Σg Hg1 ) × (K0 , +∞).

We extend the product Riemannian metric on Σg × W+ to Y+ so that the ends in (6.2) have the product Riemannian metric. Note that Hg0 ∪Σg Hg1 in (6.2) is the integral homology sphere M . Consider the decomposition:  L0 ∩ L1 = {θ} ∪ Ra a∈A

as in (5.5). The set A is identified with the set of irreducible flat connections on the trivial SU (2)-bundle over M . Definition 6.1. Let a, b ∈ A. We say the pair (u, A) satisfies the mixed equation, if they satisfy the following properties. The first two conditions are constraints on the map u: (1.1) u : W− → X \ D is a holomorphic map with finite energy. Here X and D are given in (5.3), and the energy of u is defined to be: " u∗ ω W0

with ω being the symplectic form of X. (1.2) The map u satisfies the boundary conditions u(∂0 W− ) ⊂ L0 and u(∂1 W− ) ⊂ L1 . Moreover, we require that for t ∈ [−1, 1], we have: lim u(τ, t) = p ∈ Ra .

τ →−∞

Here p is an element of Ra which is independent of t. The next two conditions are on the connection A:

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42

ALIAKBAR DAEMI AND KENJI FUKAYA

(2.1) A is a connection on the trivial SU(2)-bundle over Y+ which satisfies the anti-self-duality equation F + (A) = 0 and its energy, given by:

" |FA |2 dvol, Y+

is finite. (2.2) For τ > K0 , let Aτ denote the restriction of A to (Hg0 ∪Σg Hg+ )×{τ } ∼ = M. The connection Aτ on M converges to the flat connection b as τ goes to +∞. The last three conditions are matching conditions for u and A on the borderline C:  : X → su(2) is the moment (3.1) If (t, 0) ∈ C, then u(t, 0) ∈ μ −1 (0), where μ map of Theorem 2.4. (3.2) The restriction of A to Σg × {(t, 0)} ⊂ ∂X+ , denoted by A(t,0) , is flat for any (t, 0) ∈ C. (3.3) The gauge equivalence class of the flat connection A(t,0) coincides with the equivalence class [u(t, 0)] of u(t, 0) in μ −1 (0)/PU(2) = R(Σ). (See Theorem 2.4.) Definition 6.2. Suppose (u, A), (u , A ) are two pairs that satisfy the mixed equation. These two elements are equivalent, if there exists a gauge transformation g on Y+ and h ∈ PU(2) such that: A = g∗ A

u = hu.

◦

We will write M(W− , Y+ , L0 , L1 ; a, b; E) for the space of equivalence classes of pairs (u, A) satisfying the mixed equation and the following energy constraint: " " E= u∗ ω + FA 2 . W0

Y+ ◦

We wish to show that the moduli space M(W− , Y+ , L0 , L1 ; a, b; E) behaves nicely and it can be compactified in a way that we can use it to construct an isomor∗ (M ). This requires us to generalize the analytical phism between I ∗ (M ) and Isymp results of [Lip14, DFL]. The matching condition in Definition 6.2 can be regarded as a Lagrangian boundary condition associated to a Lagrangian correspondence from the infinite dimensional space of SU(2) connections over Σ to X. A similar infinite dimensional Lagrangian correspondence appears in [Lip14, DFL]. However, the Lagrangian correspondence in the present context is singular. Therefore, prov◦

ing the required analytical results for the moduli space M(W− , Y+ , L0 , L1 ; a, b; E) (such as Fredholm theory, regularity and compactness) seems to be more challenging. Nevertheless, we conjecture that this moduli space satisfies these properties and it can be compactified to a space M(W− , Y+ , L0 , L1 ; a, b; E). This compactification M(W− , Y+ , L0 , L1 ; a, b; E) is expected to have a virtual fundamental chain whose boundary is the union of the following two types of spaces. The first type is: (6.3)

RGW

M

(a, c; β; L0 , L1 ) × M(W− , Y+ , L0 , L1 ; c, b; E2 )

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ATIYAH-FLOER CONJECTURE

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with c ∈ A and ω[β] + E2 = E, and the second type is: (6.4)

M(W− , Y+ , L0 , L1 ; a, c; E1 ) × M(c, b; E2 ; M ) RGW

with c ∈ A and E1 + E2 = E. Here M (a, c; β; L0 , L1 ) and M(c, b; E2 ; M ) are the moduli spaces that appeared in Section 5. Assuming the existence of the compactification M(W− , Y+ , L0 , L1 ; a, b; E) with ∗ (M ) → C ∗ (M ) as follows: the above properties, we define a map Φ : Csymp  #M(W− , Y+ , L0 , L1 ; a, b; E)[b]. Φ(a) = b,E

where the sum is over all choices of E and b such that M(W− , Y+ , L0 , L1 ; a, b; E) is 0-dimensional. The signed number of points in this 0-dimensional moduli space is denoted by #M(W− , Y+ , L0 , L1 ; a, b; E). By a standard argument applying to the 1-dimensional moduli spaces M(W− , Y+ , L0 , L1 ; a, b; E) and using the description of the boundary of this moduli space in (6.3), (6.4), we can conclude that this implies that Φ is a chain map. The energy 0 part of the moduli space M(W− , Y+ , L0 , L1 ; a, b; 0) is empty if a = b and has one point if a = b. It implies that Φ induces an isomorphism between corresponding Floer homologies. Remark 6.3. Note that in Definition 6.1, we do not assume any particular asymptotic boundary conditions on the ends where t → ±∞. In fact, the finiteness of the energy should imply that the pair (u, A) converges to a constant map and to a flat connection on Hgi as t → ±∞. Therefore, the choices of Definition 6.1 ˜ g0 , p) on these ends. imply asymptotic convergence to the fundamental class of R(H This particular choice of the asymptotic boundary condition at t → ±∞ is very important for showing that Φ induces an isomorphism in homology. In fact, we use it to show that the contribution of the lowest energy part to Φ is the identity map. Remark 6.4. The map Φ is defined in a similar way to some chain maps which ˜ g0 , p) appear in [Fukb]. In the definition of these chain maps the Lagrangian R(H is replaced with arbitrary Lagrangian submanifold of the underlying symplectic manifold. However, the idea that such maps can be used to construct isomorphisms is inspired by Lekili and Lipyanskiy’s work in [LL13], where the methods of [Fukb] are revived in a similar context. Remark 6.5. The special case of the SO(3)-Atiyah-Floer conjecture for mapping tori of surface diffeomorphisms was proved in the seminal work of Dostoglou and Salamon [DS94, DS07]. Their proof uses an adiabatic limit argument and is based on the following crucial observation. Consider the 4-manifold Σg × W , where W is a surface, and let the metric on Σg degenerate. Then the ASD equation turns into the holomorphic curve equation from W to the space R(Σg ) of flat connections on Σg . Later, Salamon proposed a program for the original version of the AtiyahFloer conjecture using a similar adiabatic limit argument [Sal95] and this approach was pursued further by Salamon and Wehrheim [SW08, Weh05a, Weh05b]. The extension of the adiabatic limit argument to the general case of the SO(3)-analogue of the Atiyah-Floer conjecture is also being investigated by David Duncan [Dun12]. The adiabatic limit argument has the potential advantage of finding a relationship between the moduli spaces involved in gauge theory and symplectic geometry, and not only a relationship at the level of Floer homologies. The drawback is one has to face complicated analytical arguments. We believe the approach discussed in

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this section (and the corresponding one in [DFL] for the SO(3)-analogue of the Atiyah-Floer conjecture) has less analytical difficulties because it uses the functorial properties of Floer homologies. A similar phenomenon appears in the proof of the connected sum theorem for instanton Floer homology of integral homology spheres where the “functorial” approach [Fuk96a, Don02] seems to be easier than the adiabatic limit argument [Li94]. 7. Yang-Mills Gauge Theory and 3-Manifolds with Boundary In Section 5, we sketched the construction of I symp (M ), as a module over ∗ , for an integral homology sphere M . This invariant is defined by considering HPU(2) Yang-Mills gauge theory on principal SU(2)-bundles. It is natural to ask to what extent this construction can be generalized to arbitrary 3-manifolds and arbitrary choice of principal bundles. In the following conjecture, suppose G is given as in Section 2. Conjecture 7.1. Suppose E is a G-bundle over a 3-manifold M . Then there ∗ ∗ is a HG -module I symp (M, E) which is an invariant of the pair (M, E). In the case that M is an integral homology sphere and G = PU(2), this invariant matches with the construction of Section 5. To be more precise, we expect that the above invariant is defined using Lagrangian Floer homology on appropriate moduli spaces of flat G-connections over Riemann surfaces. We shall propose a plan for the construction of this invariant in Section 9. The tools from symplectic topology required for this construction are discussed in the next section. On another level of generalization, one can hope to define symplectic instanton Floer homology for 3-manifolds with boundary. In order to state the expected structure of symplectic instanton Floer homology for 3-manifolds with boundary, we need to recall the definition of A∞ -categories: Definition 7.2. Fix a commutative ring R. An A∞ -category C consists of a set of objects OB(C ), a graded R-module of morphisms C (c, c ) for each pair of objects # c, c ∈ OB(C ), and the structural operations mk : ki=1 C (ci−1 , ci ) → C (c0 , ck ) of degree k − 2 for each k ≥ 1. The multiplication maps mk are required to satisfy the following relations: (7.1)



k 1 −1

(−1)∗ mk1 (x1 , . . . , xi , mk2 (xi+1 , . . . , xk2 ), . . . , xk ) = 0

k1 +k2 =k+1 i=0

where ∗ = i +

i

j=1 deg xj .

Let M be a 3-dimensional manifold whose boundary is decomposed as below: ∂M = −Σ1  Σ2 . where −Σ1 denotes the 3-manifold Σ1 with the reverse orientation. Suppose also E is a G-bundle on M whose restriction to Σi is denoted by Fi . We shall say (M, E) is a cobordism from (Σ1 , F1 ) to (Σ2 , F2 ) and we shall write: (7.2)

(M, E) : (Σ1 , F1 ) → (Σ2 , F2 ).

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ATIYAH-FLOER CONJECTURE

45

Conjecture 7.3. (A-1) For any G-bundle F over a Riemann surface, there H∗ 8 is a unital filtered A∞ -category I(Σ, F) over the ring Λ0 G . The A∞ category associated to (−Σ, F) is I(Σ, F)op , the opposite A∞ category of I(Σ, F)9 . Moreover, if Σ is the disjoint union of two surfaces Σ1 , Σ2 , and the restriction of E to Σi is Ei , then we have the identification: I(Σ, F) ∼ = I(Σ1 , F1 ) ⊗ I(Σ2 , F2 ). Here the right hand side is the tensor product of filtered A∞ -categories. (See [Amo16, Fuk17c].) (A-2) For any pair as in (7.2), there is a filtered A∞ functor10 : I(M,E) : I(Σ1 , F1 ) → I(Σ2 , F2 ). The A∞ -functor associated to (−M, E) is the adjoint functor of I(M, E)11 . (A-3) For i = 1, 2, let (Mi , Ei ) be a 3-dimensional cobordism from (Σi , Fi ) to (Σi+1 , Fi+1 ). Let (M, E) be the result of composing these cobordisms along (Σ2 , F2 ). Then: (7.3)

I(M,E) ∼ = I(M2 ,E2 ) ◦ I(M1 ,E1 ) . Here ◦ is the composition of filtered A∞ -functors and ∼ = is the homotopy equivalence of filtered A∞ functors12 from I(Σ1 , F1 ) to I(Σ2 , F2 ).

The following conjecture extends Conjecture 7.3 to the case that at least one of the ends of (M, E) is empty: Conjecture 7.4. Let (M, E) be as in (7.2): (B-1) If Σ1 = ∅, then I(M,E) is an object of I(Σ2 , F2 ). (B-2) If Σ2 = ∅, then I(M,E) is a filtered A∞ functor from I(Σ1 , F1 ) to CH, where CH is the DG category of chain complexes. H∗ (B-3) If Σ1 = Σ2 = ∅, then I(M,E) is a chain complex over Λ0 G . The next conjecture is an extension of part (A-3) of Conjecture 7.3 to the case that one of the boundary components is empty: Conjecture 7.5. Let (M1 , E1 ) and (M2 , E2 ) be as in part (A-3) of Conjecture 7.3: (C-1) If Σ1 = ∅, then: (7.4)

I(M,E) ∼ = I(M2 ,E2 ) (I(M1 ,E1 ) )

This is a homotopy equivalence of objects in the category I(Σ3 , F3 ). (C-2) If Σ2 = ∅, then (7.3) as the homotopy equivalence of A∞ functors from the category I(Σ1 , F1 ) to the category CH holds. (C-3) If Σ1 = Σ2 = ∅, then (7.4) as a chain homotopy equivalence between chain complexes holds. 8A

filtered A∞ category is unital if it has a strict unit. opposite A∞ category is defined by reversing the direction of arrows. See [Fuk02, Definition 7.8]. 10 In the terminology of [Fuk17c], I (M,E) is a strict filtered A∞ functor. 11 See [Fuk17c] for the definition of adjoint functor of a filtered A functor. ∞ 12 Two filtered A ∞ functors are homotopy equivalent if they are homotopy equivalent in the functor category. (See [Fuk02, Theorem 7.55].) 9 The

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46

ALIAKBAR DAEMI AND KENJI FUKAYA

8. Lagrangians and A∞ -categories For a given symplectic manifold (X, ω), we can define an A∞ -category over the universal Novikov ring ΛR 0 , which is usually denoted by Fuk(X, ω, L). The objects of this category Fuk(X, ω, L) are defined using immersed Lagrangian submanifolds, and L denotes a collection of such Lagrangians.  is an immersed Lagrangian submanifold of X given by ι : L  → Suppose L   X where the self-intersections of X are transversal. Define CF (L, L) to be the  ΛR ) where L  is the fiber product of the map  ×X L,  ×X L cohomology group H ∗ (L 13    and a free abelian ι with itself . Therefore, CF (L, L) is the direct sum of H ∗ (L)  Z), let  group generated by the self-intersection points of L. For β ∈ H2 (X, ι(L);  β) be the compactified moduli space of pseudo-holomorphic disks with k+ Mk+1 (L;  β) are required to represent 1 boundary marked points. The elements of Mk+1 (L; the homology class β and need to satisfy the Lagrangian boundary condition. The  β) has to be mapped to ι(L),  and away from boundary of an element of Mk+1 (L;  the marked points it can be lifted to L. (See Figure 4 for a schematic picture and [FOOO09a, Definition 2.1.27] and [AJ10, Section 4] for the precise definitions  β) can be used to form the of these moduli spaces.) The moduli space Mk+1 (L; following diagram:  ko  ×X L) (L

(ev1 ,...,evk )

 β) Mk+1 (L;

ev0

  ×X L) / (L

where evi for 0 ≤ i ≤ k, is the evaluation map at the ith marked point. A standard ‘pull-up-push-down construction’ applied to these diagrams for various choices of β  L)  ⊗k → CF (L,  L)  for any k ≥ 0. determines a map mk : CF (L,

X X X

X

X

X

X X

X

X

X

Figure 4. The operation mk Next, let L be a finite family of immersed spin Lagrangian submanifolds of X.  1 , ι1 ) and (L  2 , ι2 ) of this We say that this family is clean if for any two elements (L   family the fiber product L1 ×X L2 is a smooth manifold and the tangent space at  2 is given by the fiber product of the tangent spaces of L  1 ×X L  1 and each point of L  ˜ ˜ L1 . Here we include the case L1 = L2 . For any two such elements of L, we define 1, L  2 ). Then the construction of  2 ) to be the cohomology group H ∗ (L  1 ×X L CF (L 13 To be more precise, one needs to start with a chain model for this cohomology group. As it is shown in [FOOO09a], this chain model can be replaced with the cohomology groups by an algebraic argument.

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ATIYAH-FLOER CONJECTURE

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the previous paragraph can be modified to define the following maps for any k ≥ 0  i , ιi )}0≤i≤k of elements of L [FOOO09a, AJ10]: and for any sequence {(L mk :

k $

 i−1 , L  i ) → CF (L 0, L k ) CF (L

i=1

The maps mk satisfy the analogues of the A∞ -relations in (7.1). However, the map m0 does not have to vanish in general14 . Therefore, these maps cannot immediately be used to define an A∞ -category. This issue can be fixed with the  ι), an element b ∈ aid of bounding cochains. For an immersed Lagrangian (L, odd  H (L, Λ0 ) is called a bounding cochain if it is divisible by T ε for a positive ε, and it satisfies the following Maurer-Cartan equation: ∞ 

(8.1)

mk (b, . . . , b) = 0,

k=0

 b) where L  is an eleAn object of the category Fuk(X, ω, L) is a pair (L, ment of L and b is a bounding cochain. The module of morphisms for two ob 0 , b0 ) and (L  1 , b1 ) is defined to be CF (L 0 , L  1 ). The structural map m b : jects (L k #k  i−1 , L  i ) → CF (L 0, L  k ) for a sequence of objects {(L  i , ιi , bi )}0≤i≤k is CF ( L i=1 also defined as follows: 

⊗lk−1 ⊗lk ⊗l1 0 mk+l0 +···+lk (b⊗l (8.2) mbk (p1 , . . . , pk ) := 0 , p1 , b1 , . . . , bk−1 , pk , bk ) l0 ≥,...,lk ≥0

Using the results of [FOOO09a, FOOO09b], it is shown in [FOOO10, AFO+, Fuk17c] that Fuk(X, ω, L) is an A∞ -category in the case that L consists of only embedded Lagrangians. The more general case of immersed Lagrangians is treated in [AJ10]. Suppose L0 and L1 are two monotone and embedded Lagrangians in X that satisfy the condition (m.a) of Section 3. Then the map m0 : Λ0 → CF (Li , Li ) vanishes and we can associate the trivial bounding cochain to each of these Lagrangians. The map m1 : CF (L0 , L1 ) → CF (L0 , L1 ) defines a differential. The homology of this chain complex is the same as Oh’s Lagrangian Floer homology for monotone Lagrangians [Oh93]. We can also consider equivariant version of the category Fuk(X, ω, L). The following theorem provides the main ingredient for the equivariant construction: Theorem 8.1. Let G be a Lie group acting on (X, ω). Let L be a clean collection of immersed Lagrangians which are equivariant with respect to the action  i , ιi )}0≤i≤k of elements of L, there exists a H ∗ -linear of G. For any sequence {(L G homomorphism: mG k :

k $

∗  ∗   i , Λ0 ) → HG  k , Λ0 ) HG (Li−1 ×X L (L0 ×X L

i=1

14 An

A∞ -category with a non-vanishing m0 is called a curved A∞ -category.

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48

ALIAKBAR DAEMI AND KENJI FUKAYA

which satisfies the A∞ -relations in (7.1). Moreover, the following diagram commutes: #k mG k ∗  ∗    i=1 HG (Li−1 ×X Li , Λ0 ) −−−−→ HG (L0 ×X Lk , Λ0 ) ⏐ ⏐ ⏐ ⏐ (8.3) & & #k mk ∗  ∗    i=1 H (Li−1 ×X Li , Λ0 ) −−−−→ H (L0 ×X Lk , Λ0 ). Here the vertical arrows are canonical maps from equivariant cohomology to deRham cohomology. Sketch of the proof. For the simplicity of exposition, assume that the immersed Lagrangians are the same. We use the equivariant Kuranishi structure on the space Mk+1 (L; β) [Fuk17b] and an approximation of the universal principal G-bundle EG → BG to obtain: Mk+1 (L; β) ×G EG(N ) SSS i SSSS iiii i i i SSS i i i i ev0 SSSS tiiii (ev1 ,...,evk ) S)  k ×G EG(N )  ×X L) L ×G EG(N ) (L   ×G EG(N ))k  ×X L) ((L where evi , for 0 ≤ i ≤ k, is the evaluation map at the i-th marked point. By a formula similar to (3.6), we can define operations: ⊗k → H(L ×G EG(N )). mG,N k,β : H(L ×G EG(N ))

Taking the limit N → ∞, we obtain the operation mG k,β between the equivariant  ω(β) G = T m is the required A∞ operation.  cohomology groups. Then mG k k,β β odd (L; Λ0 ) is a G-equivariant bounding Definition 8.2. An element b ∈ HG ε cochain, if b is divisible by T for a positive ε and b satisfies (8.1), where mk is replaced with mG k.

The following theorem claims the existence of the G-equivariant analogue of Fuk(X, ω, L). The geometric content of this theorem is given in Theorem 8.1: Theorem 8.3. Let L be a clean collection of G-equivariant immersed Lagrangian submanifolds of (X, ω). There exists a (filtered) A∞ -category FukG (X, ω, L) whose objects are pairs of the form (L, b) where L ∈ L and b is a G-equivariant

are also defined by applying the anabounding cochain. The structural maps mb,G k logue of the formula of (8.2) to the maps mG . k The above theorem can be regarded as a generalization of the results of Section 3 on G-equivariant Lagrangian Floer homology. Similarly, the techniques of [DF] can be used to extend the results of Section 4. More generally, we can also consider the A∞ -category associated to G-invariant Lagrangians in the complement of a smooth divisor: Theorem 8.4. Let (X, ω) be a symplectic manifold with a Lie group G acting on X by symplectomorphisms. Let D be a G-invariant smooth divisor in X such

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ATIYAH-FLOER CONJECTURE

49

that X admits a K¨ ahler structure in a neighborhood of D compatible with the symplectic form ω. Let L be a clean collection of G-equivariant immersed Lagrangian submanifolds of (X, ω). (i) There are operations: (8.4)

mG k :

k $

 i−1 , L  i ) → CF (L 0, L k ) CF (L

i=1

 i , ιi )}0≤i≤k of elements of L. These operations satisfy for any sequence {(L A∞ -relations. (ii) There exists a (filtered) A∞ -category FukG (X\D, ω, L) whose objects are pairs of the form (L, b) where L ∈ L and b is a bounding cochain in  L)  with respect to the operators defined in Item (i). The structural CF (L, operations of FukG (X\D, ω, L) are given by modifications of the operators in (8.4) as in (8.2). Conjecture 8.5. Theorems 4.3 and 8.4 still hold in the case that D is a normal crossing divisor with respect to a K¨ ahler structure in a neighborhood of D which is compatible with ω. Remark 8.6. As in Section 4, we need to use a non-standard compactification of the moduli spaces of pseudo-holomorphic disks in X \ D to prove Theorem 8.4. It is plausible that the compactification appearing in relative Gromov-Witten theory for the complements of normal crossing divisors [GS13] can be employed to prove Conjecture 8.5. The analysis of Gromov-Witten theory for the complement of normal crossing divisors is much more subtle than the case of the complements of smooth divisors, and we would expect that similar phenomena happen in the construction of Lagrangian Floer homology and the category Fuk(X\D, ω). 9. Cut-down Extended Moduli Spaces for Other Lie Groups In this section, we discuss a plan to address the conjectures in Section 7. Fix a Lie group G as in Section 2 and let T be a maximal torus of G whose Lie algebra is denoted by t. The Lie group G acts on g and the quotient space can be identified with the quotient t/W of t by the Weyl group W . Let t+ ⊂ t be a Weyl chamber of G. Then t+ is a fundamental domain for the action of the Weyl group on t, i.e., we can identify t/W with t+ . The quotient map from g to t+ is denoted by Q. We will also write tZ for the integer lattice in t. Thus T is equal to the quotient t/tZ . The dual lattice of tZ is denoted by t∗Z . The action of the Weyl group on t induces actions of this group on the lattices tZ and t∗Z . For a finite subset N = {α1 , . . . , αn } of t∗Z , we define: (9.1)

◦

ΔN (ε) = {ξ ∈ t | ∀α ∈ N , α(ξ) < ε} ◦

Let ΔN (ε) be the closure of ΔN (ε). The intersection of these open and closed ◦

+ polytopes with the Weyl chamber t+ is denoted by Δ+ N (ε) and ΔN (ε).

Condition 9.1. The set N is required to satisfy the following conditions: (1) N is invariant with respect to the action of the Weyl group. (2) ΔN (ε) is compact.

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ALIAKBAR DAEMI AND KENJI FUKAYA

(3) If α ∈ N , ξ ∈ ΔN (ε), σ ∈ W satisfy α(ξ) = ε and σ(ξ) = ξ, then σ(α) = α. (4) For any vertex v of ΔN (ε) the set {α ∈ N | α(v) = ε} is a Z basis of t∗Z . Example 9.2. For G = PU(3), we can assume that  t is the set of diagonal matrices with diagonal entries (2πiθ0 , 2πiθ1 , 2πiθ2 ) with θi = 0. Suppose αi ∈ tZ is the map that assigns θi to a diagonal matrix of this form. We may take N to be the set that consists of θi and −θi for i = 1, 2, 3. The Weyl chamber and the set ΔN (ε) is illustrated in Figure 5.

α2 = ΔN ( )

t+

α1 =

Figure 5. Wely chamber and symplectic cut Remark 9.3. A set of similar conditions for polytopes in t+ are introduced by Woodward in [Woo96]. For example, Condition 9.1 (2) implies the requirements of [Woo96, Definition 1.1]. Condition 9.1 (4) also asserts that Δ+ N (ε) is Delzant. (See [Woo96, page 5].) However, our requirements are more restrictive. For example, we only consider polytopes in t+ which contain the origin. In fact, our definition is essentially the same as the outward-positive condition in [MT12].   For the following proposition, let R(Σ, F, p) and μ : R(Σ, F, p) → g be given as in (2.2). This proposition is a consequence of well-established results on non-abelian symplectic cutting [Woo96, Mei98, MT12]: Proposition 9.4. Let ε be a positive real number. If ε is small enough, then  there exists a compact symplectic manifold R(Σ, N , F, p; ε) with a Hamiltonian G  action and a moment map μ  : R(Σ, N , F, p; ε) → g, which satisfies the following properties: (i) The image of the map Q ◦ μ  is equal to Δ+ N (ε). ◦

◦

(ii) The open subspaces (Q ◦ μ)−1 (Δ+ )−1 (Δ+ N (ε)) and (Q ◦ μ N (ε)) are symplectomorphic.  Sketch of the proof. Now we let R(Σ, N , F, p; ε)0 denote the subspace ◦

 (Q ◦ μ)−1 (Δ+ N (ε)) of R(Σ, N , F, p). According to Proposition 2.2, this space has a symplectic structure if ε is small enough. We compactify this space into a closed

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ATIYAH-FLOER CONJECTURE

51

symplectic manifold in the following way. Given x ∈ μ−1 (ΔN (ε)), we may assume without loss of generality that there is k such that: αi (μ(x)) = ε if and only if 1 ≤ i ≤ k. The map Φi : μ−1 (t) → R defined as: Φi (y) = αi (μ(y))  F, p) of x. We can be extended as a smooth function to a neighborhood Ux ⊂ R(Σ, k define a real-valued function fi on Ux × C as follows: fi (y, ξ1 , . . . , ξk ) = ε2 − φi (y) − |ξi |2 . Then the function f := (f1 , . . . , fk ) : Ux × Ck → Rk is the moment map for a Hamiltonian action of T k on Ux × Ck . The symplectic quotient f −1 (0)/T k con tains a dense subset which is symplectomorphic to R(Σ, N , F, p; ε)0 ∩ Ux . To  be a bit more detailed, if we map y'∈ R(Σ, N , F, p; ε)0 ∩ Ux to the equivalence class of (y, ξ1 . . . , ξk ) where ξi = |ε2 − Φi (y)|, then we obtain an open em bedding of R(Σ, N , F, p; ε)0 ∩ Ux into f −1 (0)/T k . Condition 9.1 (4) shows that −1 U x := f (0)/T k is a smooth manifold. (See [Woo96, Proposition 6.2].) We can glue U x for various choices of x ∈ μ−1 (Δ(ε)) to obtain the desired symplectic  manifold R(Σ, N , F, p; ε).    N , F, p; ε). We Let D denote the complement of R(Σ, N , F, p; ε)0 in X := R(Σ,  expect that R(Σ, N , F, p; ε) admits a K¨ ahler structure compatible with the symplectic structure of X denoted by ω such that D forms a normal crossing divisor in this neighborhood. Therefore, in the light of Conjecture 8.5, we make the following conjecture: Conjecture 9.5. There is an A∞ -category Fuk G (X\D, ω) associated to (X, ω) and D as above, where the objects of this category form a family of G-equivariant immersed Lagrangian submanifolds of X\D, and the morphisms of this category are constructed by holomorphic maps to X. The homotopy equivalence type of this category is independent of N and ε. Conjecture 9.6. Suppose Σ is a connected Riemann surface. Then the A∞ category has the properties of the category I(Σ, F) in Conjecture 7.3. For a disconnected Σ, we can take the tensor product of categories associated to the connected components. To elaborate on this proposal, let (M, E) be a cobordism from the empty pair to (Σ, F). As in the case of handlebodies discussed in Section 2, we can associate  to (M, E) a subspace of R(Σ, F, p) which lives in μ−1 (0). Therefore, it can be also  regarded as a subspace of X = R(Σ, N , F, p; ε). A holonomy perturbation of this subspace can be used to turn this space into an immersed Lagrangian submanifold  (M,E) . L Conjecture 9.7. There exists a bounding cochain bM in  (M,E) ).  (M,E) ×X L HG ( L  (M,E) determines an obTogether with bM , the immersed Lagrangian submanifold L G ject of Fuk (X\D, ω).

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If the construction of the various moduli spaces can be carried out as expected, then the proof of Conjecture 9.7 is expected to be similar to the similar result in the case of SO(3)-bundles. (See [Fuk15, Theorem 1.1 (1)] and [Fuk17a].) Let (Mi , Ei ) be a pair such that: ∂(M1 , E1 ) = (Σ, F) = −∂(M2 , E2 ). Therefore, we can glue these two pairs to form a closed manifold M and a G (M ,E ) , b1 ) and bundle E over M . Conjecture 9.7 asserts that there are pairs (L 1 1 G  (M ,E ) , b2 ) of objects of Fuk (X\D, ω). (L 2 2 Conjecture 9.8. The Lagrangian Floer homology  (M ,E ) , b1 ), (L  (M ,E ) , b2 ); X \ D) HFG ((L 1 1 2 2 is an invariant of (M, EM ). 10. Admissible Bundles and Instanton Floer Homology Let E be a hermitian vector bundle of rank N over a 3-manifold M . Then the vector bundle E is determined up to isomorphism by its first Chern class. The pair (M, E) is called an admissible pair if there is an oriented embedded surface S in M such that the pairing of c1 (E) and the fundamental class of S is coprime to N . We will also write E for the PU(N )-bundle associated to E. Note that M in this case is not an integral homology sphere. Floer’s instanton homology can be extended to the case where (M, E) arises from an admissible pair [Flo95, KM11]. We will write I ∗ (M, E) for this version of instanton Floer homology. (See [DX17, Section 3.1] for a review of the general properties of I ∗ (M, E).) The proposal of the previous section to define symplectic instanton Floer homology can be also specialized to admissi∗ (M, E) to denote this conjectural ble pairs. We shall keep using the notation Isymp invariant. There is yet another approach to define symplectic instanton Floer ho∗ mology of (M, E) in this context, temporarily denoted by Isymp (M, E), which avoids the technical difficulties of equivariant Floer homology in divisor complements. The ∗ (M, E) and current section concerns the relationship between the invariants Isymp ∗  Isymp (M, E). ∗ ∗ The definition of Isymp (M, E) follows a similar route as Isymp (M, E). Suppose F is a hermitian vector bundle of rank N over an oriented Riemann surface Σ such that the evaluation of c1 (F ) is coprime to N . Then the pair (Σ, F ) is called an admissible pair. Let F be the PU(2)-bundle associated to F . Then the moduli space of flat connections R(Σ, F), defined in Section 2, is a smooth K¨ ahler manifold for this choice of F. Let M be a 3-manifold with boundary Σ and E be a hermitian  vector bundle on M extending F . Then we define R(M, E) to be the space of all elements of R(Σ, F) represented by flat connections on F that can be extended to E. This space can be perturbed to an immersed Lagrangian submanifold of  R(Σ, F) which we still denote by R(M, E) [Her94]. The moduli space of solutions to the mixed equation can be also used to define a bounding cochain b(M,E) for  this Lagrangian [Fuk15, Theorem 1.1 (1)]. Therefore, (R(M, E), b(M,E) ) defines an object of Fuk(R(Σ, F)). Next, let (M, E) be an admissible pair. There is an embedded Riemann surface Σ in M such that removing Σ from M gives a disconnected manifold, and the pair given by Σ and F := E|Σ is admissible. Let M1 and M2 be the closure of

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the connected components of M \Σ and Ei := E|Mi . We can assume that the La 1 , E1 ) and R(M  2 , E2 ) have clean intersection by applying holonomy grangians R(M ∗ (M, E) is defined to be the Lagrangian perturbations to one of them. Then Isymp   2 , E2 ), b(M ,E ) ). Floer homology of the two elements (R(M1 , E1 ), b(M1 ,E1 ) ) and (R(M 2 2 This Lagrangian Floer homology is independent of the choice of Σ. Conjecture 10.1. For an admissible pair (M, E), the vector spaces I ∗ (M, E), ∗ ∗ (M, E) and Isymp (M, E) are isomorphic to each other. Isymp The part of the above conjecture about the relationship between I ∗ (M, E) ∗ (M, E) is what we previously referred to as the SO(3)-analogue of the and Isymp Atiyah-Floer conjecture. In the case that the Lagrangians involved in the definition ∗ of Isymp (M, E) are embedded, the SO(3)-Atiyah-Floer conjecture is addressed in [DFL]. The more general case will be treated in another forthcoming paper following the strategy proposed in [Fuk15]. In the following, we discuss some general results in symplectic Floer homology which are related to the part of Conjecture ∗ ∗ 10.1 about the existence of isomorphism between Isymp (M, E) and Isymp (M, E). ∗ Once the definition of the invariant Isymp (M, E) is fully developed, we hope that these general results give a proof for this part of the above conjecture. We first need an extension of the category Fuk(X, ω, L) for a clean collection of immersed Lagrangian submanifolds L in a symplectic manifold (X, ω). Suppose b ∈ H even (X; Λ0 ) with b ≡ 0 mod T ε .15 Then the A∞ operations mk associated to L can be deformed by b to mbk as in [FOOO09a, Definition 3.8.38]. Such deformations of the A∞ structure of Fuk(X, ω, L) are called Lagrangian Floer theory with bulk deformation. Roughly speaking, we deform mk to mbk using the holomorphic disks which hit a cycle that is Poincar´e dual to b. Bounding cochains of this deformed structure are also defined in the same way as in (8.1). Consequently, there is a (filtered) A∞ -category Fuk(X, ω, L, b) whose objects are pairs of a Lagrangian L ∈ L and a bounding cochain b with respect to the b-deformed (filtered) A∞ structure [FOOO09a, Definition 3.8.38]. Situation 1. Suppose a Hamiltonian action of a Lie group G on a symplectic manifold (X, ω) is given. Let μ : X → g∗ denote the moment map of this action. Let the action of G on μ−1 (0) be free. Then the quotient Y = μ−1 (0)/G is a symplectic manifold with a symplectic form ω [MW74]. Let L be a clean collection of G ι  ) ∈ L, we assume equivariant immersed Lagrangian submanifolds. For each (L, L −1  ⊂ μ (0). Then L := (L/G,   is free and ι  (L) [iL ]) is that the G action on L L an immersed Lagrangian submanifold of Y . The collection of all such immersed Lagrangians of Y is denoted by L. Finally we assume that the following Lagrangian: (10.1)

{(x, y) ∈ X × Y | x ∈ μ−1 (0), y = [x]}

is spin. Theorem 10.2. There exists b ∈ H even (Y ; Λ0 ) such that the two filtered A∞ categories FukG (X, ω, L) and Fuk(Y, ω, L, b) are homotopy equivalent16 . If (10.1) 15 The condition b ≡ 0 mod T ε is not necessary. However, we need a slightly delicate argument to prove the convergence of operators. See for example, [FOOO11, Definition 17.8]. For our application in this paper it suffices to consider the case when this extra condition is satisfied. 16 See [Fuk02, Definition 8.5].

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is a monotone Lagrangian with minimal Maslov number > 2, then b can be chosen to be zero. Remark 10.3. The element b in Theorem 10.2 is related to the quantum Kirwan map introduced by Woodward in [Woo15a, Woo15b, Woo15c]. Theorem 10.2 is also closely related to the results of Tian and Xu, written or announced in a series of papers [TX16]. Both the works of Woodward and Tian-Xu (as well as various other related works such as [GW13]) are based on the study of gauged sigma models [Rie99, CGS00]. On the other hand, the second author’s proof, which will appear in [Fuka], uses equivariant Kuranishi structures and relies on the idea of employing Lagrangian correspondence and cobordism arguments in a similar way as in [Fukb, LL13]. We were informed by Max Lipyanskiy that he had similar ideas to use Lagrangian correspondences and cobordism arguments instead of gauged sigma models. Conjecture 10.4. Suppose (X, ω) and L are given as in Situation 1. Moreover, assume that there exists a G-invariant normal crossing divisor D ⊂ X \μ−1 (0) such that X \ D is monotone. Let (10.1) be a monotone Lagrangian submanifold of (X \ D) × Y . Then the filtered A∞ category FukG (X \ D, ω, L) is homotopy equivalent to Fuk(Y, ω, L). Remark 10.5. The above generalization of Theorem 10.2 is related to Conjec ture 10.1. By picking X = R(Σ, N , F, p, ε), this conjecture implies the predicted ∗ ∗ relationship between Isymp (M, E) and Isymp (M, E) in Conjecture 10.1. The main difficulty with this conjecture is to define FukG (X \ D, ω, L) for the case that D is a normal crossing divisor. Existence of this A∞ -category in the special case that D is a smooth divisor is the content of Theorem 8.4. A combination of the techniques used in verifying Theorems 8.4 and 10.2 proves the above conjecture in the special case that D is a smooth divisor. References [AB96]

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Aliakbar Daemi, Kenji Fukaya, and Max Lipyanskiy, Lagrangians, SO(3)-instantons and mixed equations and Lagrangians, SO(3)-instantons and Atiyah-Floer conjecture, In preparation. [Don02] S. K. Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, vol. 147, Cambridge University Press, Cambridge, 2002. With the assistance of M. Furuta and D. Kotschick. MR1883043 [DS94] S. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2) 139 (1994), no. 3, 581–640, DOI 10.2307/2118573. MR1283871 [DS07] S. Dostoglou and D. A. Salamon, Corrigendum: “Self-dual instantons and holomorphic curves” [Ann. of Math. (2) 139 (1994), no. 3, 581–640; MR1283871], Ann. of Math. (2) 165 (2007), no. 2, 665–673, DOI 10.4007/annals.2007.165.665. MR2299744 [Dun12] D. L. Duncan, Compactness results for neck-stretching limits of instantons, arXiv:1212.1547 (2012). [DX17] A. Daemi and Y. Xie, Sutured Manifolds and Polynomial Invariants from Higher Rank Bundles, arXiv:1701.00571 (2017). [Flo88a] A. Floer, An instanton-invariant for 3-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240. MR956166 [Flo88b] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR965228 [Flo95] A. Floer, Instanton homology and Dehn surgery, The Floer memorial volume, Progr. Math., vol. 133, Birkh¨ auser, Basel, 1995, pp. 77–97. MR1362823 [FO99] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933–1048, DOI 10.1016/S0040-9383(98)00042-1. MR1688434 [FOOO09a] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR2553465 [FOOO09b] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR2548482 [FOOO10] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Anchored Lagrangian submanifolds and their Floer theory, Mirror symmetry and tropical geometry, Contemp. Math., vol. 527, Amer. Math. Soc., Providence, RI, 2010, pp. 15–54. MR2681791 [FOOO11] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory, arXi:1105.5123, to appear in AMS Memoir (2011). [FOOO15] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamental chain: Part 1, arXiv:1503.07631 (2015). [Fuka] Kenji Fukaya, Equivariant Floer homology and equivariant Gromov-Witten invariants, In preparation. , Floer homology for 3-manifolds with boundary I, Unpublished, but a draft [Fukb] is available at http://www.math.kyoto-u.ac.jp/%7Efukaya/fukaya.html. [Fuk96a] K. Fukaya, Floer homology of connected sum of homology 3-spheres, Topology 35 (1996), no. 1, 89–136, DOI 10.1016/0040-9383(95)00009-7. MR1367277 , Geometry of gauge fields, Lectures on geometric variational problems [Fuk96b] (Sendai, 1993), Springer, Tokyo, 1996, pp. 43–114. MR1463751 [Fuk98] K. Fukaya, Anti-self-dual equation on 4-manifolds with degenerate metric, Geom. Funct. Anal. 8 (1998), no. 3, 466–528, DOI 10.1007/s000390050064. MR1631255 [Fuk02] K. Fukaya, Floer homology and mirror symmetry. II, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999), Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 31–127. MR1925734 , SO(3)-Floer homology of 3-manifolds with boundary 1, arXiv:1506.01435 [Fuk15] (2015). [Fuk17a] K. Fukaya, Categorification of invariants in gauge theory and symplectic geometry, Jpn. J. Math. 13 (2018), no. 1, 1–65, DOI 10.1007/s11537-017-1622-9. MR3776467 , Lie groupoid, deformation of unstable curve, and construction of equivariant [Fuk17b] Kuranishi charts, arXiv:1701.02840 (2017). [DFL]

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[Fuk17c] [Gol84] [GS13] [GW13] [Her94]

[HLS16a] [HLS16b]

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, Unobstructed immersed Lagrangian correspondence and filtered A∞ functor, arXiv:1105.5123 (2017). W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225, DOI 10.1016/0001-8708(84)90040-9. MR762512 M. Gross and B. Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510, DOI 10.1090/S0894-0347-2012-00757-7. MR3011419 E. Gonzalez and C. Woodward, Gauged Gromov-Witten theory for small spheres, Math. Z. 273 (2013), no. 1-2, 485–514, DOI 10.1007/s00209-012-1016-x. MR3010172 C. M. Herald, Legendrian cobordism and Chern-Simons theory on 3manifolds with boundary, Comm. Anal. Geom. 2 (1994), no. 3, 337–413, DOI 10.4310/CAG.1994.v2.n3.a1. MR1305710 K. Hendricks, R. Lipshitz, and S. Sarkar, A simplicial construction of G-equivariant Floer homology, arXiv:1609.09132 (2016). K. Hendricks, R. Lipshitz, and S. Sarkar, A flexible construction of equivariant Floer homology and applications, J. Topol. 9 (2016), no. 4, 1153–1236, DOI 10.1112/jtopol/jtw022. MR3620455 J. Huebschmann, Symplectic and Poisson structures of certain moduli spaces. I, Duke Math. J. 80 (1995), no. 3, 737–756, DOI 10.1215/S0012-7094-95-08024-7. MR1370113 E.-N. Ionel and T. H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96, DOI 10.4007/annals.2003.157.45. MR1954264 L. C. Jeffrey, Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann. 298 (1994), no. 4, 667–692, DOI 10.1007/BF01459756. MR1268599 P. Kronheimer and T. Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR2388043 P. B. Kronheimer and T. S. Mrowka, Knot homology groups from instantons, J. Topol. 4 (2011), no. 4, 835–918, DOI 10.1112/jtopol/jtr024. MR2860345 W. Li, Floer homology for connected sums of homology 3-spheres, J. Differential Geom. 40 (1994), no. 1, 129–154. MR1285531 J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR1882667 J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR1938113 Max Lipyanskiy, Gromov-Uhlenbeck Compactness, arXiv:1409.1129 (2014). Y. Lekili and M. Lipyanskiy, Geometric composition in quilted Floer theory, Adv. Math. 236 (2013), 1–23, DOI 10.1016/j.aim.2012.12.012. MR3019714 A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of CalabiYau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218, DOI 10.1007/s002220100146. MR1839289 E. Meinrenken, Symplectic surgery and the Spinc -Dirac operator, Adv. Math. 134 (1998), no. 2, 240–277, DOI 10.1006/aima.1997.1701. MR1617809 J. W. Morgan, T. Mrowka, and D. Ruberman, The L2 -moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, II, International Press, Cambridge, MA, 1994. MR1287851 J. Martens and M. Thaddeus, On non-Abelian symplectic cutting, Transform. Groups 17 (2012), no. 4, 1059–1084, DOI 10.1007/s00031-012-9202-9. MR3000481 J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR0402819 C. Manolescu and C. Woodward, Floer homology on the extended moduli space, Perspectives in analysis, geometry, and topology, Progr. Math., vol. 296, Birkh¨ auser/Springer, New York, 2012, pp. 283–329, DOI 10.1007/978-0-8176-82774 13. MR2884041 Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993, DOI 10.1002/cpa.3160460702. MR1223659 B. Parker, Exploded manifolds, Adv. Math. 229 (2012), no. 6, 3256–3319, DOI 10.1016/j.aim.2012.02.005. MR2900440 I. M. i Riera, Yang-Mills-Higgs theory for symplectic fibrations, arXiv:9912150 (1999).

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Simons Center for Geometry and Physics, State University of New York, Stony Brook, New York 11794 Email address: [email protected] Simons Center for Geometry and Physics, State University of New York, Stony Brook, New York 11794–and–Center for Geometry and Physics, Institute for Basic Sciences (IBS), Pohang, Korea Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01737

Weinstein manifolds revisited Yakov Eliashberg To Simon Donaldson with admiration Abstract. This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.

1. Weinstein manifolds, domains, cobordisms We begin with a notion of a Liouville domain. Let (X, ω) be a 2n-dimensional compact symplectic manifold with boundary equipped with an exact symplectic form ω. A Liouville structure on (X, ω) is a choice of a primitive λ, dλ = ω, called Liouville form such that λ|∂X is a contact form and the orientation of ∂X by the form λ ∧ dλn−1 |∂X coincides with its orientation as the boundary of symplectic manifold (X, ω). The vector field Z, that is ω-dual to λ, i.e. ι(Z)ω = λ, is also called Liouville. It satisfies the condition LZ ω = ω which means that its flow is conformally symplectically expanding. The contact boundary condition is equivalent to the outward transversality of Z to ∂X. A Liouville domain X can always  by attaching a cylindrical end: be completed to a Liouville manifold X  := X ∪ (∂X × [0, ∞)) X  as equal to es (λ|∂X ) on the attached end. We will be conand extending λ to X stantly going back and forth between these two tightly related notions of Liouville domains and Liouville manifolds. Given a Liouville structure L = (X, ω, Z) we say that a Liouville structure L = (X  , ω, Z) is obtained by a radial deformation from L if there exists a function  is the image of X under the time 1 map ψ : X →X   → R such that X  ⊂ X h:X  of the flow of the vector field hZ on the completion X. The completions of the radially equivalent Liouville domains L and L are canonically isomorphic. The space of Liouville structures for (X, ω) is convex, and hence any two Liouville structures are canonically homotopic. Given a homotopy of completed Liou ωt , λt ) there exists an isotopy φt : X  →X  such that φ∗t ωt = ω0 . ville structures (X, ∗ Moreover, one can always arrange that φt λt = λ0 + dHt , see [11], Sections 11.1 and 11.2. In particular on completed Liouville manifolds it is always sufficient to Partially supported by NSF grant DMS-1505910. c 2018 American Mathematical Society

59

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consider homotopies fixing the symplectic form, and, moreover, changing the Liouville form by adding an exact form. Homotopic non-completed Liouville domains are symplectomorphic up to radial deformation. Given a Liouville domain L = (X, ω, λ) consider a compact set  Z−t (X), Core(L) = t>0

the attractor of the negative flow of the Liouville vector field Z. We will call Core(L) the core, or the skeleton of the Liouville structure L. While Core(L) has obviously its 2n-dimensional Lebesgue measure equal to 0, it still can be pretty large if no extra conditions are imposed on the Liouville structure. For instance, McDuff constructed in [31] a Liouville structure on T ∗ Sg \ Sg for a closed surface Sg of genus g > 1, whose core has codimension 1. However, the situation changes if one requires existence of a Lyapunov function for the Liouville vector field Z. A Weinstein structure on a domain X is a Liouville structure L together with a function φ : X → R which is Lyapunov for the Liouville vector field Z, i.e. (L1) dφ(Z) > c||Z||2 for a positive constant c and some Riemann metric on X.

Figure 1.1. Skeleton of a Weinstein domain Note that condition (L1) implies that Core(X, λ) is the union of Z-stable manifolds of critical points of φ (i.e. points converging to the critical locus in forward time). In [11] it was required in addition that φ is either Morse or generalized Morse (i.e. may have death-birth critical points). Under these assumptions it was shown in [11], see also [15, 20], that (L2) the core is stratified by isotropic for λ, and hence for ω submanifolds. F. Laudenbach proved, see [30], that if the flow of Z is Morse-Smale (i.e. stable and unstable manifolds of critical points intersect transversely) and near critical points the vector field Z is gradient with respect to an Euclidean metric, then

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the skeleton can be further Whitney substratified. It is likely that the Whitney condition also holds if near its zeroes the vector field Z is gradient with respect to any Riemannian metric. However, as far as know, this was never verified in the literature. The Whitney condition need not hold if eigenvalues of the linearization of Z at critical points have non-vanishing imaginary parts, as a spiraling phenomenon of trajectories may occur.1 Condition (L2) holds for a much more general class of taming functions (e.g. when φ is Morse-Bott), and hence for the the purposes of this paper we will take the following working definition of a Weinstein structure, extending the class considered in [11]: W = (X, λ, Z, φ) is Weinstein if it satisfies conditions (L1) and (L2) with the Whitney condition and also condition (L3) there exists a smooth family of Weinstein structures Wt = (X, λt , φt ), t ≥ 0 such that (λ, φ) = (λ0 , φ0 ) and φt is Morse for t > 0. Problem 1.1. Which conditions (or maybe none?) on φ and Z are needed to deduce (L2) and (L3) from (L1)? E. Giroux and independently A. Oancea suggested to me that a good sufficiently general condition on a Weinstein structure could be to require that near critical points it is generated by a J-convex function with respect to some (not necessarily integrable) almost complex structure J, see [11], Chapter 1, for the details. Remark 1.2. Note that not every closed subset C of a symplectic manifold which is stratified by isotropic strata may serve as the skeleton for an appropriately chosen Weinstein structure on a neighborhood of C (compatible with the given ambient symplectic form). Examples of this kind exist already in R2 . For instance, let C := {x = 0, y ≥ 0} ∪ {x = y 2 , y ≥ 0} ∪ {y = 0, x ≥ 0} ∪ {y = x2 , x ≥ 0} be the union of 4 arcs emanating from the origin. Then there is no Liouville structure on a neighborhood U  0 which has C ∩ U as a part of its skeleton. Indeed it is straightforward to check that for any 1-form λ vanishing on C ∩ U one has (dλ)0 = 0. Problem 1.3. Find a necessary and sufficient condition on a compact subset C of a symplectic manifold to serve as the skeleton of some a) Liouville, or b) Weinstein structure on its neighborhood. In particular, is it true that a Whitney stratified subset C which is the skeleton of a Liouville structure on its neighborhood also serves as the skeleton of a Weinstein structure? It is also useful to consider a notion of a Weinstein cobordism. This is a cobordism (W, ∂− W = Y− , ∂+ W = Y+ ) endowed with a Liouville form λ, whose Liouville vector field Z is outward transverse to ∂+ W and inward transverse to ∂− W , and a Lyapunov (i.e. satisfying condition (L1)) function φ : W → R for the field Z. We also postulate (L3) and an analog of condition (L2) for the core of the Weinstein cobordism, which we define in that case as the stable manifold of the critical locus of φ. We will also be considering Weinstein cobordisms between manifolds with 1I

thank Francois Laudenbach for the discussion of the involved issues.

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Figure 1.2. Sutured Weinstein cobordism W with corners. boundary ∂± W . We will view these cobordisms as sutured manifolds with a corner along the suture, see Fig. 1.2. More precisely, we assume that the boundary ∂W is presented as the union of two manifolds ∂− W and ∂+ W with common boundary ∂ 2 W = ∂+ W ∩ ∂− W along which it has a corner. The vector field Z transversely enters W through ∂− W and exits through ∂+ W , but of course, in this case the function φ cannot be chosen constant on ∂− W and ∂+ W . While any two Weinstein structures on the same symplectic manifold are (canonically) homotopic as Liouville structures, the problem of existence of a Weinstein homotopy is widely open.  λ1 , φ1 ) be two completed Weinstein  λ0 , φ0 ) and (X, Problem 1.4. Let (X,  ω). Are they homotopic as Weinstructures on the same symplectic manifold (X, stein structures? In particular,  ω, λ, φ) be a completed Weinstein structure, and Problem 1.5. Let W = (X,   f : X → X a symplectomorphism. Is the pull-back Weinstein structure f ∗ W is Weinstein homotopic to W? The Weinstein structure notion was introduced in [20] as a symplectic counterpart of the notion of Stein complex structure, and inspired by the work of A. Weinstein [45], see also [11, 15, 16]. I discussed the notions and problems considered in this paper with many people. I am especially grateful to Daniel Alvarez-Gavela, Oleg Lazarev, David Nadler, Sheel Ganatra, Vivek Shende, Laura Starkston and Kyler Siegel for contributing many ideas and suggestions for improvement of the current text. I am very grateful to the anonymous referee for critical remarks and many useful suggestions. Special thanks to Nikolai Mishachev for making the pictures. 2. Weinstein hypersurfaces and Weinstein pairs Weinstein hypersurfaces are special cases of Liouville hypersurfaces introduced by Avdek in [3]. This and other related notions discussed in this paper are also similar to “stops” of Sylvan, [42] and Liouville sectors of Ganatra-Pardon-Shende, [26]. Related constructions are also considered in Ekholm-Lekili’s paper [14].

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Weinstein hypersurfaces in a contact manifold. Let (Y, ξ) be a contact manifold. A codimension 1 submanifold Σ ⊂ Y with boundary is called Weinstein hypersurface if there exists a contact form λ for ξ such that (Σ, λ|Σ ) is compatible with a Weinstein structure on Σ, i.e. dλ|Σ is symplectic and the Liouville vector field ZΣ on Σ dual to the Liouville form λ|Σ is outward transverse to ∂Σ and admits a Lyapunov function φ : Σ → R. The Reeb vector field for λ is transverse to Σ and the boundary ∂Σ of a Weinstein hypersurface Σ is a codimension two contact submanifold of (Y, ξ). Though the induced Weinstein structure on Σ depends on the choice of a contact form, its skeleton is independent of this choice. Indeed, the Liouville fields for the Liouville structures λ and f λ for a positive f > 0 are proportional. In fact, as it is computed in Lemma 12.1 in [11] the form f λ is Liouville if and only if k := inf(f + df (Z)) > 0, where Z is the Liouville form for λ, and in that case the Liouville vector field for f λ is equal to k1 Z. Moreover, the space of functions f for which f λ is Liouville (and hence in the considered case Weinstein) is contractible. It follows that the skeleton Core(Σ, λ|Σ ) is a stratified subset of Y which consists of strata which are isotropic, and in the top dimension n − 1 are Legendrian for the contact structure ξ. Example 2.1. (i) Weinstein thickening of a Legendrian submanifold. Let Λ ⊂ (Y, ξ) be a Legendrian submanifold. Then it admits a Darboux neighborhood U (Λ) isomorphic to (J 1 (Λ), dz−pdq), q ∈ Λ, ||p||2 +z 2 ≤ ε2 . Then Σ(Λ) := U (Λ) ∩ {z = 0} is a Weinstein hypersurface symplectomorphic to the cotangent ball bundle of Λ. Up to Weinstein isotopy the Weinstein thickening Σ(Λ) is independent of all the choices. 2 (ii) Pages of open books. According to Giroux’s theorem [25], any contact manifold admits an open book decomposition whose pages are Weinstein hypersurfaces. (iii) Halves of convex hypersurfaces. Recall that a hypersurface Σ in a contact manifold is called convex if it admits a transverse contact vector field, see [20, 24]. The set D of points where the contact vector field is tangent to the contact plane field, called a dividing set, is generically a smooth hypersurface which divides Σ into two Liouville manifolds. In many interesting examples these Liouville manifolds are, in fact, Weinstein, and hence serve a rich source of Weinstein hypersurfaces. Given two Legendrian isotopic submanifolds Λ0 , Λ1 ⊂ (Y, ξ) their Weinstein thickenings Σ(Λ0 ) and Σ(Λ1 ) are isotopic as Weinstein hypersurfaces. Problem 2.2. Is the converse true? Here by isotopy we mean an isotopy of unparameterized submanifolds. Note that an isotopy of Weinstein hypersurfaces carries Λ0 to an exact La 1 ⊂ Σ(Λ1 ). Moreover, there is a symplectomorphism grangian submanifold Λ  1 ) = Λ1 . Hence, the positive answer to Problem ψ : Σ(Λ1 ) → Σ(Λ1 ) such that ψ(Λ 2.2 would follow from the positive resolution of the following special case of the nearby Lagrangian conjecture: Lagrangians which are images of the 0-section under a global symplectomorphism are Hamiltonian isotopic to the 0-section. 2 Warning: unlike the case of a Legendrian isotopy, an isotopy of Weinstein hypersurfaces does not extend in general to an ambient contact diffeotopy.

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If the contact manifold (Y, ξ) is symplectically fillable then one can prove that the Legendrian algebras LHA(Λ0 ) and LHA(Λ1 ) are isomorphic3 . It is likely that this claim could be generalized to the case of a general contact manifold (Y, ξ). Problem 2.3. Is there an analog of the Legendrian algebra LHA(Λ) for a general Weinstein hypersurface? Let us return to the case of the Legendrian homology algebra of a Legendrian submanifold Λ and pick a contact form λ such that its Reeb vector field is tangent to the contact submanifold Δ := ∂Σ(Λ). We also choose an almost complex structure J on ξ such that ξ ∩ T (Δ) are J-invariant. This allows us to define a deformation (A[t], D) of the Legendrian differential algebra (A, ∂) := LHA(Λ) as  follows. For a generating chord c ∈ A define D(c) = (∂k c)tk , where ∂0 = ∂ and k≥0

∂k c counts holomorphic curves with the intersection index k with the symplectization of Δ. This symplectization is a complex hypersurface in the symplectization of Y , and hence k ≥ 0. The sum defining differential D is finite due to the Gromov compactness. Problem 2.4. Explore whether the above construction yields a genuinely new invariant of a Legendrian submanifold. Given a Weinstein hypersurface Σ ⊂ Y we slightly extend it to a larger Wein ⊃ Σ such that on Σ  \ Σ the Liouville form λ can be written stein hypersurface Σ  has a neighborhood U  difas tλ|∂Σ , t ∈ [1, 1 + ε]. The extended hypersurface Σ ∗  feomorphic to Σ × (−ε, ε) such that λ|U = π (λ|Σ  ) + du where u is the coordinate  → Σ.  Note that the corresponding to the second factor and π the projection U  level sets {u = const} are translates of Σ under the Reeb flow of the contact form  → R which is equal to 0 on Σ and to t − 1 λ. Pick a non-negative function h : Σ   . The neighborhood U (Σ) near ∂ Σ and set U (Σ) = Uε (Σ) := {h2 + u2 ≤ ε2 } ⊂ U will be called the contact surrounding of a Weinstein hypersurface Σ. Proposition 2.5. Contact manifolds Y \ U (Σ), Y \ Σ and Y \ Core(Σ, λ|Σ ) are contactomorphic. Let us first recall a few basic facts about convex hypersurfaces in contact manifolds. If a germ ξ of a contact structure along a closed hypersurface V in a (2n − 1)-dimensional manifold admits a transverse contact vector field v then we canonically can construct a contact structure ξ on V × R which is invariant with respect to translations along the second factor and whose germ along any slice V ×t, t ∈ R, is isomorphic to ξ. We will call ξ the invariant extension of the convex germ ξ. Lemma 2.6. Let V be a closed (2n − 2)-dimensional manifold and ξ a contact structure on Y = V × [0, ∞) which admits a contact vector field v inward transverse to V ×0 and such that its trajectories intersecting V ×0 fill the whole manifold Y (we  do not require v to be complete). Then (Y, ξ) is contactomorphic to (V × [0, ∞), ξ), where ξ is the invariant extension of the germ of ξ along V × 0. Moreover, for any compact set C ⊂ Y , Int C ⊃ V × 0, there exists a contactomorphism h : (Y, ξ) →  which is equal to the identity on V × 0 and which sends the contact (V × [0, ∞), ξ) ∂ . vector field v|C to the vector field ∂t 3I

thank Sheel Ganatra and Tobias Ekholm for the discussion of this problem.

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Proof. It is sufficient to construct a complete contact vector field v on V ×[0, 1) which coincides with v on C and whose trajectories intersecting V × 0 fill the whole manifold V ×[0, 1). We will construct it using the following inductive process. Take ∞  a sequence of compact sets C0 = C, C1 , . . . , which exhausts Y , i.e. Cj = Y and 0

Cj ⊂ Int Cj , j = 0, 1, . . . . Let v1 be a contact vector field obtained by cutting off v outside C0 but inside C1 . Let h1 be the time T1 > 1 flow map of v1 , where T1 0 the is chosen sufficiently large to ensure that h1 (V × 0) ⊂ C1 \ C0 . Denote by C domain bounded by V × 0 and h1 (V × 0) and by v1 the contact vector field equal to 0 and to the push-forward vector field (h1 )∗ v on Y \ C 0 . Let v2 be a contact v1 on C vector field obtained by cutting off v1 outside C1 but inside C2 and denote by h2 the time T2 > T1 + 1 flow of v2 , where T2 is chosen such that h2 (V × 0) ⊂ C2 \ C1 . 1 the domain bounded by V × 0 and h2 (V × 0) and by v2 the contact Denote by C 1 and to the push-forward vector field (h1 )∗ v1 on Y \ C 1 . vector field equal to v2 on C Continuing this process we construct a sequence of contact vector fields v1 , v2 , . . . , which stabilize on compact sets C1 , C2 , . . . and converge to the contact vector field v on Y with the required properties.  ∂ Proof of Proposition 2.5. The contact vector field v = −ZΣ −u ∂u is transverse to ∂U (σ) and retracts U (Σ) to Core(Σ, λΣ ), and hence the contact structure on U (Σ) \ Core(Σ, λΣ ) is canonically isomorphic to ∂U (Σ) × [0, ∞) endowed with the invariant extension ξ of the germ of contact structure  ξ along ∂U (Σ). On the ∂Uδ (Σ) = U (Σ) \ Σ. other hand, v is transverse to ∂Uδ (σ) for each δ ≤ ε and δ∈(0,ε]

Hence, applying Lemma 2.6 we conclude that (U (Σ) \ Σ, ξ) is contactomorphic to  and the claim follows. (∂U (Σ) × [0, ∞), ξ),  Remark 2.7. One of the corollaries of Lemma 2.6 is that any open domain in  the standard contact (R2n+1 , dz + xi dyi − yi dxi ) which is star-shaped with respect ∂ ∂ ∂ + xi ∂x + yi ∂y is contactomorphic to R2n+1 . On to the contact vector field 2z ∂z i i 3 the other hand, in the standard contact R any open domain diffeomorphic to R3 is contactomorphic to R3 , see [17]. Problem 2.8. Is there a domain in the standard contact R2n+1 , n > 2, which is diffeomorphic to the closed ball, has convex in contact sense boundary, but whose interior is not contactomorphic to the standard R2n+1 ? Or even are there any open domains in the standard contact R2n+1 , n > 2, which are diffeomorphic but not contactomorphic to R2n+1 ? Weinstein pairs. A Weinstein pair (W, Σ) consists of a Weinstein domain W = (X, λ, φ) together with a Weinstein hypersurface (Σ, λ|Σ ) in its boundary ∂X. Equivalently, a Weinstein pair can be viewed as a Weinstein manifold with cylindrical end, together with a Weinstein hypersurface in its ideal contact boundary. Let Λ = Core(Σ) be the skeleton of Σ and   := Z −t (Λ) Λ t≥0

be its saturation by the trajectories of the Liouville vector field Z. The union  Core(X, Σ) := Core(X) ∪ Λ is called the core, or the skeleton of the Weinstein pair.

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It turns out that it is possible to modify the Liouville form λ on X in a neighborhood of Σ in X to make the attractor of the modified Liouville vector field equal to the skeleton Core(X, Σ). Given a Weinstein pair (W, Σ), W = (X, ω, λ, Z, φ), let U = U (Σ) ⊂ ∂X be its contact surrounding. Denote by ZΣ the Liouville field dual to λ|Σ and by φΣ its Lyapunov function. A Liouville form λ0 , the corresponding Liouville vector field Z0 for ω on X and a smooth function φ0 : X → R are called adjusted to the structure of the pair if (see Fig. 2.1) • Z0 is tangent to ∂X on U (Σ) and transverse to ∂X elsewhere; ∂ • Z0 |U(Σ) = ZΣ + u ∂u ;  −t • the attractor Z0 (X) of the Liouville vector field −Z0 coincides with t≥0

the core Core(X, Σ) of the Weinstein pair; • the function φ0 : X → R is Lyapunov for Z0 and such that φ0 |U(Σ) = φΣ + u2 and φ0 has no critical values ≥ ε2 = φ0 |∂U(Σ) . Proposition 2.9. Given a Weinstein pair (W, Σ), W = (X, λ, φ), there exist a Liouville form λ0 for ω and a function φ0 : X → R such that • λ0 , φ0 are adjusted to (W, Σ); • λ0 coincides with λ outside a neighborhood of Σ;  φ of (λ0 , φ0 ) to a slightly bigger domain X ⊃ Moreover, there exists an extension λ, (   (  X such that the W := (X, λ, φ) is a Weinstein domain and Core(W) = Core(W, Σ). To construct the adjusted Liouville field Z0 let us write the form λ near ∂X as s(du + λΣ ) near U (Σ). Note that the Hamiltonian vector field Y for a function su ∂ ∂ + u ∂u + ZΣ , and hence by appropriately cutting near U (Σ) coincides with −s ∂s off the function su outside a neighborhood of U (Σ) and subtracting the differential dg of the resulting function g to the Liouville form λ we get the Liouville form λ0 with the required properties. Note that the form λ0 |U is no more contact. Instead, λ0 |U = π ∗ (λ|Σ ). Suppose that λ0 , φ0 are adjusted to the Weinstein pair (W, Σ). Recall that φ0 |∂U(Σ) = ε2 . Denote X0 = {φ0 ≤ ε2 }. We note that φ0 has no critical points in X \ Int X0 , and hence X0 is a manifold with boundary with a corner along ∂U (Σ) which is homeomorphic to X. We will sometime refer to (X0 , λ0 , φ0 ) as the cornered version of the Weinstein pair (W, Σ).4 For instance, the cornered version of the standard Weinstein ball B 2n is the cotangent ball bundle of Dn . Thus, it is always possible to go back and forth between the original and adjusted (cornered) versions of a Weinstein pair, and we will be using the term “Weinstein pair” for both versions. Remark 2.10. There are several other useful adjustments of a Weinstein pair structure. Ekholm and Lekili in [14], Section B.3, are doing a similar to the cornered version construction by deforming the boundary ∂X near U (Σ) without changing Z, as on Fig. 2.2. Without defining here Sylvan’s stop structure we just say that for a given Weinstein pair there is a contractible space of choices of stop structures on the completion. 4 The completion of the cornered version of a Weinstein pair is a special case of a Liouville sector in the sense of [26].

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Figure 2.1. Modifications of a Weinstein pair structure.

Figure 2.2. Ekholm-Lekili deformation of ∂X. One can also transform a Weinstein pair into a Weinstein cobordism whose negative boundary is U (Σ), see Fig. 2.1: Let (X0 , λ0 , Z0 , φ0 ) be the cornered adjusted version of a Liouville pair structure (W, Σ), as in Proposition 2.9. There exists a Liouville form λ− on X0 such that (X0 , λ− , φ0 ) is a sutured Weinstein cobordism structure with ∂− X0 = U (Σ), and Core(X0 , λ− , φ0 ) = Core(W, Σ). To obtain such a form λ− one subtracts from λ the differential of the appropriately cut off function 2su instead of the function su used to modify λ into λ0 . 3. Operations on Weinstein pairs 3.1. Splitting and gluing of Weinstein pairs. Let W = (X, λ, Z, φ) be a Weinstein domain. A hypersurface (P, ∂P ) ⊂ (X, ∂X) is called splitting for W if it satisfies the following conditions: - ∂P splits the boundary ∂X into two parts, ∂X = Y− ∪ Y+ with ∂Y− = ∂Y+ = Y+ ∩ Y− = ∂P (and respectively, P divides X into two parts X+ and X− with ∂X− = P ∪ Y− , ∂X+ = P ∪ Y+ and X+ ∩ X− = P ; - the Liouville vector field Z is tangent to P ; - there exists a hypersurface (S, ∂S) ⊂ (P, ∂P ) which is Weinstein for the restricted Liouville form λ|S , tangent to the vector field Z and intersects all leaves of the characteristic foliation F of the hypersurface P ; we will

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refer to S as the Weinstein soul of the splitting hypersurface P and denote it by Soul(P ). Note that the latter condition together with Lemma 2.6 imply that P is contactomorphic to the contact surrounding of its Weinstein soul. It follows that (S, λ|S , φ|S ; ∂S) is a codimension two Weinstein subdomain of X and Core(S, λ|S , φ|S ) = Core(W) ∩ P. Moreover, (W± ; S), where W± := (X± , λ|X± , φ|X± ) are cornered Weinstein pairs and Core(W± ; S) = Core(W) ∩ X± . The gluing construction reverses the splitting. This operation was considered by Avdek in [3] in the context of Liouville hypersurfaces. Let (W, Σ) and (W , Σ ) be two Weinstein pairs and (X0 , λ0 , φ0 ), (X0 , λ0 , φ0 ) their cornered forms. Let F : (Σ, λ|Σ , φ|Σ ) → (Σ , λ |Σ , φ |Σ ) be a Weinstein isomorphism. We extend F to a contactomorphism U (Σ) → U (Σ ), still denoted by F , and use it to define a domain X  X  := X0  X0 /{(x ∈ U (Σ)) ∼ (F (x) ∈ U (Σ )). F

Then the Liouville forms λ0 and λ0 , as well as Lyapunov functions φ0 : X0 → R and φ0 : X0 → R, can be glued together to define a Weinstein structure (W, Σ) ∪(W , Σ ) := (XF , λF , φF ), see Fig. 3.1. F

Figure 3.1. Gluing of Weinstein pairs. Note that Core(XF , λF , φF ) = Core(X, Σ)

∪

F|Core(Σ)

Core(X , Σ ).

Note that the constructed Weinstein domain XF contains U (Σ) as its splitting hypersurface. Applying the above described splitting construction we get back the Weinstein pairs (W, Σ) and (W , Σ ). The gluing of Weinstein pairs is a generalization of the Legendrian surgery construction (or rather Weinstein handle attachment). When Σ = Σ(Λ) for a Legendrian Λ ⊂ ∂X, X  = B 2n and Σ = Σ(Λ0 ), where Λ0 is the Legendrian unknot in S 2n−1 = ∂B, then (XF , λF , φF ) is the Weinstein n-handle attachment to X along Λ. Conversely, the general gluing operation (W, Σ) ∪(W , Σ ) can be F

decomposed into a sequence of subcritical and critical handle attachments. To do that, one fixes first a Weinstein handle decomposition of Σ, and then for each handle of index k of this decomposition one needs to attach a handle of index k + 1 to the glued domains. For instance, for a handle of index 0 centered at a point p ∈ Σ one attaches a handle of index 1 along an arc connecting the point p ∈ Σ with its image p = F (p) ∈ Σ under the gluing map.

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Both, splitting and gluing constructions can be naturally generalized to the relative setting. Let (W, Σ) and (W , Σ ) be two Weinstein pairs. Suppose that Σ and Σ are split by splitting hypersurfaces T ⊂ Σ and T  ⊂ Σ as Σ = Σ− ∪ Σ+ and Σ = Σ− ∪ Σ+ and we are given a Weinstein isomorphism F : Σ+ → Σ− . Then the result of the partial gluing is the pair (W, Σ) ∪  (W , Σ ) which F,T,T

consists of the Weinstein domain (W, Σ− ) ∪(W , Σ+ ) together with the Weinstein hypersurface Σ−

F

∪

F |Soul(T )

Σ+ ⊂ X ∪ X  which is the result of gluing the Weinstein F

pairs (Σ+ , Soul(T )) and (Σ− , Soul(T  )) using the Weinstein isomorphism F |Soul(T ) . The reverse operation to the partial gluing of Weinstein pairs is a splitting of a Weinstein pair (W, Σ), W = (X, λ, φ), along a splitting hypersurface (P, Q := ∂P ) ⊂ (X, ∂X) for the Weinstein domain X where in addition P satisfies the following condition: - Q intersects Σ transversely, Q ∩ Σ = Soul(Q) and Q ∩ Σ is a splitting hypersurface for Σ, which splits it into Σ+ and Σ− ;  − ) and (X+ , Σ  + ), The result of this splitting are two Weinstein pairs (X− , Σ  where the Weinstein hypersurface Σ± ⊂ ∂X± = Y± ∪ P is the result of gluing of Weinstein pairs (Σ± , Soul(Q ∩ Σ)) and (Soul(P ), Soul(Q ∩ Σ)). As in the absolute case, the gluing operation of Weinstein pairs glues their skeleta along the skeleta of glued hypersurfaces. Conversely, a splitting of the skeleton of a Weinstein domain lifts to a splitting of a Weinstein domain into two Weinstein pairs. 3.2. Product and Stabilization of Weinstein pairs. Given two Weinstein pairs (W, Σ) and (W , Σ ), where W = (X, λ, φ), W = (X  , λ , φ ) we define their product as the Weinstein pair (W, Σ) × (W , Σ ) := (X × X  , λ ⊕ λ , (Σ × X  ; Σ × Σ )  (X × Σ , Σ × Σ )). Id









Here (Σ × X ; Σ × Σ )  (X × Σ , Σ × Σ )) is the result of gluing of two Weinstein Id

pairs by the identity map between the Weinstein hypersurfaces Σ × Σ ⊂ ∂(X × Σ ) and Σ × Σ ⊂ ∂(Σ × X  ). We note that Core ((W, Σ) × (W , Σ )) = Core(W, Σ) × Core(W , Σ ). In the case when (X  , Σ ) is the Weinstein pair (T ∗ Dk , T ∗ S k−1 ) the product operation is called the stabilization (or k-stabilization). It was first proposed in a slightly different form by M. Kontsevich, [28]. The core of the k-stabilized pair (W, Σ) is equal to Core(W, Σ) × Dk . It is important to stress the point that the result of the stabilization is always a Weinstein pair with a non-empty hypersurface in the boundary, even if we begin with the absolute case of a Weinstein domain. 3.3. Weinstein homotopy as a Weinstein pair. Consider a Weinstein structure W0 := (X, ω, λ0 , φ0 ) and its 1-stabilization Wst := W × T ∗ I, viewed as a Weinstein pair (X × T ∗ I, λ0 + udt, X × 0 ∪ X × 1). Consider a Weinstein homotopy Wt := (X, λt = λ0 + dht , φt ), t ∈ [0, 1]. We assume, in addition, that t h˙ 1 = h˙ 0 = 0, where we denoted h˙ t := dh dt (t). This condition can always be arranged by a re-parameterization of the homotopy. Consider the product X × T ∗ I with the symplectic form Ω := ω ⊕ du ∧ dt , where (u, t) are canonical coordinates on T ∗ I

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 := λt + (u + h˙ t )dt is a (so that u = 0 defines the 0-section). Note that the 1-form λ ˙ ˙  Liouville form for Ω. Indeed, dλ := dλt + dt ∧ dht + dht ∧ dt + du ∧ dt = ω + du ∧ dt.  X = λ1 .  X = λ0 and λ| We have λ| 0 1 Proposition 3.1. There exists a function φ = X × T ∗ I → R such that   := λt + (u + h˙ t )dt, φ;  X0 ∪ X1 (X × T ∗ I, λ where X0 := X × {t = u = 0}, ; X1 := X × {t = 1, u = 0}, is a Weinstein pair. We call this pair the concordance generated by the homotopy Wt .  Liouville vector field is given by the Proof. Note that the corresponding to λ  = Zt + (u + h˙ t ) ∂ , where Zt is the Liouville vector field corresponding formula Z ∂u to λt . Define the function φ by the formula φ = φt + k2 (u + h˙ t )2 , where a positive constant k will be chosen later. Then we have  Z)  = dφt (Zt ) + k(u + h˙ t )2 + k(u + h˙ t )dh˙ t (Zt ). dφ( Not that |dφt (Zt ) ≥ a||Zt ||2 and |dh˙ t (Zt )| ≤ b||Zt || for some constants a, b > 0. Denoting X := ||Zt ||, Y := u + h˙ t we can write  Z)|  ≥ a||Zt ||2 + k(u + h˙ t )2 − bk|u + h˙ t |||Zt || |dφ( aX 2 + kY 2 − bkXY. The quadratic form aX 2 + kY 2 − bkXY is positive definite if b2 k2 − 4ak < 0 or k < 4a b2 . Under this condition, which can be arranged by choosing the constant k  Z)|  2 for positive constants c,   ≥ c(X 2 + Y 2 ) ≥  c||Z|| c. sufficiently small, we get |dφ( This concludes the proof.   is equal to Remark 3.2. The critical point locus of φ (= the zero locus of Z)  = {(x, t, u); x is a critical point of φt , u = h˙ t (x), t ∈ [0, 1]}. C The stable manifold of a critical point (x0 , t0 , u0 ) projects to the stable manifold of the critical point x0 of φt0 . Its u-coordinate can be found by solving the inhomogeneous linear ODE du(γ(s)) = u(γ(s)) + h˙ t0 (γ(s)) ds with the asymptotic boundary condition lim u(γ(s)) = u0 , where γ(s) is a trajecs→∞ tory of Xt0 converging to the critical point x0 . 4. Looseness and Flexibility Let us recall that in contact manifolds of dimension 2n − 1 ≥ 5 there is a local modification construction for Legendrian submanifolds, called stabilization5 , see [11, 16, 35]. This operation can be performed in an arbitrarily small neighborhood of any point of a Legendrian. Moreover, it can also be performed without changing the formal Legendrian isotopy class of the Legendrian submanifold. In her 2012 paper [35] Emmy Murphy called a Legendrian submanifold loose if it is isotopic to a stabilization of another Legendrian submanifold, and showed that loose Legendrians 5 The

term “stabilization” is used here in a completely different sense than in Section 3.2.

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satisfy an h-principle: any two loose formally isotopic Legendrians can be connected by a Legendrian isotopy. The notion of flexibility, see [11], for Weinstein cobordisms is tightly related to the looseness property of Legendrian knots. One first defines flexibility for elementary Weinstein cobordisms, i.e. Weinstein cobordisms (W, ω, Z, φ) without any Z-trajectories connecting critical points of the Lyapunov function φ. An elementary 2n-dimensional, n > 2, Weinstein cobordism (W, ω, X, φ) is called flexible if the attaching spheres of all index n handles form in ∂− W a loose Legendrian link (i.e. each sphere is loose in the complement of the others). A Weinstein structure is called flexible if it is homotopic to one which can be decomposed into elementary flexible cobordisms. As it was shown by E. Murphy and K. Siegel in [37] existence of a decomposition into flexible elementary cobordisms really depends on the choice of a particular Weinstein structure in the given homotopy class. Moreover, there exist non-flexible Weinstein domains which become flexible after attaching an n-handle. Flexible Weinstein structures are indeed flexible: they abide a number of hprinciples. Theorem 4.1. (i) ([11]) Any two flexible Weinstein structures on a given smooth cobordism are homotopic as Weinstein structures provided that the corresponding symplectic forms are in the same homotopy class of non-degenerate (but not necessarily closed) 2-forms. (ii) ([11]) Let (X, ω, Z, φ) be any flexible Weinstein structure and φt , t ∈ [0, 1], be a family of generalized Morse functions such that φ0 = φ. Then there exists a homotopy (X, ωt , Zt , φt ) of Weinstein structures. (iii) ([22]) Let (X± , ω± , Z± , φ± ) be two Weinstein structures. Suppose that the structure (X− , ω− , Z− , φ− ) is flexible and that there exists an embedding f : X− → X+ such that the forms ω− and f ∗ ω+ are homotopic as non-degenerate (but not necessarily closed) 2-forms. Then there exists a t t , Z− , φt− ), t ∈ [0, 1], beginning homotopy of Weinstein structures (X− , ω− 0 0 0 with (X− , ω− = ω− , Z− = Z− , φ− = φ− ) and an isotopy f t : X− → X+ 1 beginning with f 0 = f such that (f 1 )∗ ω+ = ω− . At first glance Theorem 4.1 implies that symplectic topology of flexible Weinstein manifolds is quite boring. This is also confirmed by the fact that symplectic homology in all its flavors of a flexible Weinstein manifold is trivial. However, as we will see below in Section 7 the contact boundaries of flexible Weinstein domains have a rich contact topology. The looseness property of a Legendrian submanifold can be naturally extended to Weinstein hypersurfaces of contact manifolds. A Weinstein hypersurface Σ of a contact manifold Y of dimension 2n + 1 ≥ 5 is called loose if for each n-dimensional strata S of the skeleton Core(Σ) there is a ball BS ⊂ Y \ (Core(Σ) \ S) such that BS ∩ S is loose in BS relative ∂(BS ∩ S). A canonical Weinstein thickening of a loose Legendrian knot is loose. However, it is unclear whether looseness is preserved under Weinstein isotopy. Problem 4.2. Is looseness property preserved under a Weinstein isotopy of Σ. In particular, suppose that a Weinstein thickening Σ(Λ) of a Legendrian knot Λ is

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isotopic in the class of Weinstein hypersurfaces to a loose Weinstein hypersurface. Does this imply that Λ itself is loose? Proposition 4.3. Let (W, Σ) and (W , Σ ) where W = (X, λ, φ), W = (X  , λ , φ ), be two Weinstein pairs and  := (Σ × X  ; Σ × Σ )  (X × Σ , Σ × Σ )) (W, Σ) × (W , Σ ) := (X × X  , λ ⊕ λ , Σ Id

 is loose in ∂(X × X  ). be their product. Suppose that Σ is loose in ∂X. Then Σ Indeed, this is straightforward from the following fact: given any contact manifold (Y, {α = 0}), a Liouville manifold (U, μ), a loose Legendrian Λ ⊂ Y and a Lagrangian L ⊂ U with μ|L = 0, then the Legendrian Λ × L ⊂ (Y × U, {α ⊕ μ = 0}) is loose as well. Let us stress the point that while flexibility of a Weinstein manifold is its intrinsic property, the looseness of a Weinstein hypersurface depends on its embedding in the contact manifold. However, the above fact about the looseness of a product shows that flexibility always implies looseness (I thank the referee for this argiment). Proposition 4.4. Let (Y, ξ) be a contact manifold of dimension ≥ 7, and Σ ⊂ Y a flexible Weinstein hypersurface. Then Σ is loose. Indeed, let α be a contact form for ξ which restricts to a Liouville form μ on Σ. Consider a Weinstein subdomain Σ0 ⊂ Σ and let a Lagrangian disc Δ ⊂ Σ \ Σ0 be attached to Σ0 along a loose Legendrian sphere Λ := ∂Δ ⊂ ∂Σ0 . In a neighborhood U ⊃ ∂Σ0 in Σ the Liouville form μ can be written as sβ, s ∈ (1 − ε, 1 + ε) for a  of U in Y the contact form α can contact form on ∂Σ0 , and on a neighborhood U  can be viewed as the be written as dt + sβ = s(udt + β), |t| < ε, u = 1s . Hence, U product of the contact manifold (∂Σ0 , β) and a Liouville subdomain   1 1 Q := {(u, t) ∈ − , × (−ε, ε)} ⊂ (R2 , udt), 1+ε 1−ε  = Λ × {t = 0, 1 < u ≤ 1]} ⊂ Σ0 × Q. Hence looseness of attaching while Δ ∩ U 1−ε spheres of top index Weinstein handles of Σ implies looseness of their Lagrangian cores viewed as Legendrian submanifolds of Y . The notion of flexibility naturally extends to Weinstein pairs. A Weinstein pair (W = (X, λ, φ), Σ) is called flexible if it is flexible viewed as a cobordism between ∂X− = U (Σ) and ∂+ X = X \ Int U (Σ), see Remark 2.10. It is straightforward to see that flexibility is preserved under the stabilization construction. However, the converse is not clear. Problem 4.5. Suppose that the stabilization of a Weinstein pair is flexible. Does this imply that the Weinstein pair itself is flexible? More generally, does existence of a homotopy between stabilizations of two Weinstein (pair) structures implies existence of a homotopy between the structures themselves? Attaching a critical handle along a loose Legendrian knot to a flexible Weinstein domain by definition preserves its flexibility. This generalizes to the following

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Proposition 4.6. Let (W = (X, λ, φ), Σ) and (W = (X  , λ , φ ), Σ ) be two Weinstein pairs. Let Σ and Σ be decomposed as Σ = Σ− ∪ Σ+ , Σ = Σ− ∪ Σ+ by splitting hypersurfaces T ⊂ Σ and T  ⊂ Σ , see Section 3.1. Suppose that - there exists a Weinstein isomorphism F : Σ+ → Σ− , - Σ− is loose in ∂X and - pairs (W, Σ− ) and (W , Σ+ ) are flexible. Then the glued pair (W, Σ) ∪  (W , Σ ) is flexible. In particular, the result of gluF,T,T

ing of two flexible Weinstein domains along Weinstein hypersurfaces one of which is loose is flexible. This follows from the fact that the gluing operations of two Weinstein pairs can be decomposed into a sequence of handle attachments, and the looseness assumption for the Weinstein hypersurface in one of the glued parts implies that all the critical handles are attached along loose knots. As a corollary Proposition 4.6 implies the following generalization of the following result of E. Murphy and K. Siegel, [37]: Proposition 4.7. The product of two Weinstein pairs, one of which is flexible, is flexible. Indeed, the product of two Weinstein pairs can always be built by a sequence of gluing of various stabilizations of the first pair. 5. Lagrangian submanifolds of Weinstein domains In this section we discuss exact Lagrangian submanifolds in a Weinstein domain (X, λ, φ). The Lagrangians will always be assumed either closed or with Legendrian boundary in ∂L ⊂ ∂X. Let Σ(∂L) be the Weinstein thickening of the (possibly empty) Legendrian boundary ∂L. A Lagrangian L is called regular, see [18], if the Weinstein pair (X, Σ(∂L)) admits a skeleton which contains L. Problem 5.1. Are there non-regular exact Lagrangians? The problem is widely open. While no examples of non-regular Lagrangians are known, in the opposite direction in the case of a closed exact Lagrangian L in a general Weinstein domain X it is even unknown whether L realizes a non-zero homology class in Hn (X) (which is a necessary condition for its regularity). If L ⊂ X is regular then by removing its tubular neighborhood N (L) one gets a Weinstein cobordism XL := (W \ N (L), ∂− XL := ∂N (L) \ ∂X, ∂+ XL := ∂X \ N (L)) (between manifolds with boundary if ∂L = ∅) whose negative boundary is the unit cotangent bundle of L. The Lagrangian L is called flexible, see [18]), if the cobordism XL is flexible. It was shown in [18] that any flexible (X, λ) admits a surprising abundance of flexible Lagrangians with non-empty Legendrian boundary. In particular, Theorem 5.2. Let L be an n-manifold with non-empty boundary, equipped with a fixed trivialization η of its complexified tangent bundle T L ⊗ C. Then there exists a flexible Lagrangian embedding with Legendrian boundary (L, ∂L) → (B 2n , ∂B 2n ) where B 2n is the standard symplectic 2n-ball, realizing the trivialization η. In particular, any 3-manifold with boundary can be realized as a flexible Lagrangian submanifold of B 6 with Legendrian boundary in ∂B 6 .

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6. Symplectic topology of Weinstein manifolds While flexible Weinstein structures enjoy a full parametric h-principle, there is plenty of symplectic rigidity and fine symplectic invariants of non-flexible ones. I will not discuss in this survey any such invariants and just mention that until recently most examples of formally homotopic but not symplectomorphic Weinstein manifolds were distinguished by their (possibly appropriately deformed) symplectic cohomology. For instance, there are infinitely many non-symplectomorphic Weinstein structures on R2n for any n > 2 ([33, 41]) and by taking connected sums of these examples with flexible Weinstein manifolds one gets infinitely many nonsymplectomorphic Weinstein structures on any given “almost Weinstein” (i.e. an almost complex manifold of homotopy type of a half-dimensional CW-complex) manifolds, see [2]. Note that Theorem 5.2 can also be used for constructing exotic Weinstein structures. In particular, Theorem 6.1 ([18]). Let L be a closed 3-manifold. Then there exists a unique up so symplectomorphism Weinstein structure W(L) = (ωL , ZL , φL ) on T ∗ S 3 which contains L as its flexible Lagrangian submanifold in the homology class of the 0section (with Z/2-coefficients in the non-orientable case). Moreover, infinitely many of these W(L) are pairwise non-symplectomorphic. Note that there exists only 1 homotopy class of almost complex structures on T ∗S3. While the symplectic structure of W(L) carries a lot of information about the topology of L, the following problem is open: Problem 6.2. Suppose W(L) is symplectomorphic to W(L )? Does it imply that L is diffeomorphic to L ? The famous ”nearby Lagrangian problem” asks whether there is a unique up to Hamiltonian isotopy exact closed Lagrangian submanifold in the standard T ∗ M for a closed M . Though in this form the answer is unknown except for M = S 2 and T 2 , see [12,27], the answer is positive up to simple homotopy equivalence, [1], and hence according to Smale, Freedman and Perelman for M = S n up to homeomorphism, and for some dimensions, e.g. n = 3, 5, 6, 12, even up to diffeomorphism, [34]. As it was pointed out to me by O. Lazarev, one can show using methods of [9] that certain exotic T ∗ S n may contain several not homotopy equivalent regular closed exact Lagrangian submanifolds. Problem 6.3. Can the uniquenes results from [1] be extended to a more general class of Weinstein structures on T ∗ S n ? The proof of Theorem 5.2 yields also the following slightly stronger result. Theorem 6.4. Let (X, ω, λ, φ) be a 6-dimensional Weinstein domain such that φ has exactly 1 critical point of index 3 (and any number of critical points of smaller indices). Suppose also that the symplectic vector bundle (T X, dλ) is trivial. Then there exists a Weinstein structure (ωX , λX , φX ) on T ∗ S 3 which admits an embedding (X, ω, λ, φ) → (T ∗ S 3 , ωX , λX , φX ) onto a Weinstein subdomain with a flexible complement.

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7. Topology of Weinstein fillings Contact manifolds appeared as boundaries of Weinstein domains are called Weinstein fillable. The fact that a Weinstein filling has a homotopy type of a half-dimensional CW-complex imposes constraints on the topology of its contact boundary and the stable almost complex class which can be realized by Weinstein fillable contact structures on a given smooth manifold. This question was studied in detail by Bowden-Crowley-Stipsicz in [5, 6]. In particular, they showed that there are classes of homotopy spheres which do not admit any Weinstein fillable contact structure. Given a contact manifold (Y, ξ) one can try to describe (symplectic) topology of its Weinstein fillings. In this section we discuss this problem for contact manifolds of dimension 2n − 1 > 3, see [40] for a survey of results for 3-dimensional manifolds. First of all notice that the fact that X retracts to its n-dimensional skeleton implies that the inclusion Y = ∂X → X is (n − 1)-connected, and in particular, if Y is a homotopy sphere then X is (n − 1)-connected. It turns out that some contact structures know much more about the topology of their fillings. Theorem 7.1 ([31]). Any Weinstein filling of the standard contact sphere (S 2n−1 , ξstd ) is diffeomorphic to the ball B 2n . Generalizing Theorem 7.1 K. Barth, H. Geiges and K. Zehmisch proved in [7]: Theorem 7.2. All Weinstein fillings of a simply connected contact manifold admitting a subcritical filling are diffeomorphic. In fact, both Theorems 7.1 and 7.2 hold in a stronger form for a more general class of symplectic, and not necessarily Weinstein fillings. We also note that while it follows from Theorem 4.1 that all completed subcritical Weinstein fillings of a given contact manifold are symplectomorphic (we note that the (n − 1)-connectedness of the inclusion map ∂X → X implies that the homotopy class of an almost complex structure on a subcritical manifold is determined by the homotopy class of its restriction to the boundary), it is unknown for n > 2 whether all completed fillings of a contact manifold admitting a subcritical filling (e.g. the standard contact sphere) are symplectomorphic. The following theorem of Oleg Lazarev constrains topology of flexible Weinstein manifolds. Theorem 7.3 ([29]). All flexible fillings of of a contact manifold (Y, ξ) with c1 (Y, ξ) = 0 have canonically isomorphic integral homology. In particular, as Lazarev observed, Theorem 7.3 together Smale’s classification of 2-connected 6-manifolds from [43] and the fact that π3 (O/U ) = 0 yield a complete classification of flexibly fillable contact structures on S 5 . Corollary 7.4 ([29]). There exists a sequence ξn , n = 0, 1, . . . , of pairwise non-contactomorphic contact structures on S 5 such that • any flexibly fillable contact structure on S 5 is contactomorphic to one of the structures from this sequence; • the contact structure ξ0 is standard; • for n ≥ 1 the contact sphere (S 5 , ξn ) admits a unique up to symplecton

morphism flexible Weinstein filling diffeomorphic to (# S 3 × S 3 ) \ B 6 . 1

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There are further constraints on the topology of flexible Weinstein fillings. In particular, Theorem 7.5 ([19]). Let (S 4n−1 , ξ) be a flexibly fillable contact structure. Then the signature of its flexible filling is uniquely determined by the contact structure ξ. Problem 7.6. Does a contact structure (Y, ξ) remember a) the diffeomorphism type of its flexible Weinstein filling (X, ω, Z, φ)? b) the almost symplectic homotopy class [ω] of the symplectic structure ω? We note that the diffeomorphism type of X together with the homotopy class [ω] determine a flexible Weinstein structure up to Weinstein homotopy, and hence the positive answer to a) and b) would imply that the contact structure (Y, ξ) remember the symplectomorphism type of the completion of its flexible filling. 8. Nadler’s program of arborealization A priori, a skeleton of a Weinstein domain can have very complicated singularities. However, David Nadler conjectured that up to Weinstein homotopy the singularities of the skeleton can be reduced to a finite list in any dimension, see [38]. For 2n-dimensional symplectic Weinstein manifolds the list of Nadler’s singularities, which he calls arboreal, are enumerated by decorated rooted trees with ≤ n + 1 vertices. It is remarkable that the singularity of each given type has a unique symplectic realization. Nadler also proposed in [39] a procedure for arborealization of the skeleton of a Weinstein structure. His procedure replaces a given Weinstein structure by another one whih an arboreal skeleton. Nadler proved in [39] that the constructed Weinstein manifold has microlocal sheaf-theoretic invariants equivalent to those of the Weinstein manifold. Conjecturally this implies that the wrapped Fukaya categories are also the same for the original and modified Weinstein manifold. However, it is unclear whether Nadler’s modification yields a Weinstein structure which is homotopic, or even symplectomorphic to the original one. In an ongoing joint project [23] with David Nadler and Laura Starkston we are exploring a somewhat different strategy for arborealization of the Weinstein skeleton via a Weinstein homotopy using simplification of singularities type technique in the ´ spirit of a recent paper of D. Alvarez-Gavela, [4]. In some special cases this program was already carried out by Starkston in [44]. In this section we discuss the arboreal singularities with more detail and give precise statements of some of the results from [23]. 8.1. Definition of an arboreal singularity. While we define below arboreal models as closed properly embedded subsets of the standard symplectic vector space, we are interested only in germs of these models at the origin. Consider a tree T with ≤ n + 1 vertices and a fixed vertex R, the root. Suppose in addition that all edges, except the terminal ones are decorated with ±1. We will denote by ε the decoration, and by |T | the total number of vertices. With each decorated rooted tree (T, ε) we associate a unique up to symplectomorphism model A(T, ε, m) ⊂ R2m = T ∗ Rm in each dimension m ≥ n of the skeleton. The models will be stratified by strata which are isotropic for the Liouville form pdq. In dimension m > n we have A(T, ε, m) = A(T, ε, n) × Rm−n ⊂ T ∗ Rn × T ∗ Rm−n = T ∗ Rm .

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Figure 8.1. Arboreal singularities labeled by rooted decorated trees. The picture represents Lagrangian skeleta themselves, and not their front projections. Free boundaries of vertical strata form Legendrian trees, while their traces at the horizontal plane are fronts of these trees.

The model A(T, ε, n) will be defined inductively in n. For a tree T which consists of one vertex we define A(T, 0) to be a point (in the 0-dimensional symplectic space T ∗ R0 ), and respectively A(T, n) = Rn ⊂ T ∗ Rn . As it was already stated above, the Liouville form pdq vanishes on each stratum of the model A(T, ε, n) ⊂ T ∗ Rn . Hence, if we view T ∗ R2n as a (Weinstein) hypersurface {z = 0} in the contact space (R2n+1 = T ∗ Rn × R, pdq + dz), then all strata of A(T, ε, n) ⊂ T ∗ Rn are also isotropic for the contact form pdq + dz. However, unless A(T, ε, n) is a Lagrangian plane, the front projection (p, q, z) → (q, z) is very degenerate, because it collapses the image to the hyperplane {z = 0}. We want to deform the model A(T, ε, n) in R2n+1 to make the front projection more generic. To do that, consider a contactomorphism S : R2n+1 → R2n+1 given by the formula S(p1 , . . . , pn , q1 , q2 , . . . , qn , z) = (p1 , . . . , pn , q1 + p1 , q2 , . . . , qn , z −

p21 ). 2

Then S −1 (p1 , . . . , pn , q1 , q2 , . . . , qn , z) = (p1 , . . . , pn , q1 − p1 , q2 , . . . , qn , z + Denote + (T, ε, n) := S(A(T, ε, n)), A − (T, ε, n) := S −1 (A(T, ε, n)). A

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± (T, ε, n) are stratified by isotropic for the contact form dz + pdq strata. The sets A + (T, n) = A(T, n). If |T | = 1 we have A Suppose that we already defined models for all decorated rooted trees (T, ε) with |T | ≤ n. Consider a rooted tree (T, ε) with |T | = n + 1. By removing the root R and all edges adjacent to R we get k decorated trees (T1 , ε), . . . , (Tk , εk ) with |T1 | = n1 , . . . , |Tk | = nk , n1 + · · · + nk = n. For each of them we choose as its root the vertex which was connected in T to R. Let σj = ±1 be the decoration of the edge which was connecting the root R with the root of the tree Tj , j = 1, . . . , k. Consider already defined models A(T1 , ε1 , n − 1), . . . , A(Tk , εk , n − 1) ⊂ T ∗ Rn−1 × R. Denote N0 := 0, Nj :=

j 

ni , j = 1, . . . , k − 1. For each j = 0, . . . , k − 1

i=1

consider the hyperplane Πj = {pNj +1 = 1} in R2n = T ∗ Rn with the Liouville form n n   λ = pj dqj . Note that Πj is transverse to the Liouville vector field Z = pj ∂p∂ j , 1 1  or equivalently λ|Πj = dqNj +1 + pi dqi is a contact form. Cyclically i∈{1,...,n},i=Nj +1

ordering coordinates qNj +2 , . . . , qn , q1 , . . . , qNj and taking the coordinate qNj +1 as z we identify Πj with T ∗ Rn−1 × R. Consider Asign(σj ) (Tj , εj , n − 1) ⊂ Πj . Denote B(T, ε, n) := {(tp, q) ∈ T ∗ Rn ; t ∈ [0, ∞), (p, q) ∈

k 

sign(σj ) (Tj , εj , n − 1))}. A

j=1

Note that B(T, ε, n) ∩ {p = 0} is the union of front projections of Legendrian sign(σj ) (Tj , εj , n − 1)), and B(T, ε, n) is the positive conormal of this complexes A stratified set co-oriented by the vector field ∂qN∂ +1 . Finally, we define j

A(T, ε, n) := {p = 0} ∪ B(T, ε, n). Singularities of the form A(T, ε, n) where (T, ε) is a decorated rooted tree are called primary arboreal. Note that up to linear symplectomorphism the result of the above construction is independent of the ordering of the trees T1 , . . . , Tk . Indeed, the corresponding symplectomorphism is the symplectization of the linear automorphism of Rn appropriately permuting the coordinates q1 , . . . , qn . As an example, let us explicitly construct the models shown on Fig, 8.1. For a tree with 2 vertices we take the standard symplectic R2 with coordinates (p, q). Then Π = {p = 1}. For the 1-vertex tree T1 the model A(T1 , 0) coincides is the  1 , 0) = A(T1 , 0). Hence B(T, 1) = {(t, 0), t ≥ 0} point {p = 1, q = 0} ∈ Π and A(T is the positive p-semi-axis, and A(T, 1) = {p = 0} ∪ B(T, 2), is the union of the coordinate line q with this semi-axis, as it is shown on the left side of Fig. 8.1. For the rooted tree with three vertices and the central root, as on the lower picture in Fig. 8.1, each of the trees T1 , T2 has 1 vertex. Hence, Π1 = {p1 = 1}, Π2 = {p2 = 1}, and identifying this hyperplanes with the standard contact R3 we get A(T1 , 1) = {p2 = q1 = 0} ⊂ Π1 and A(T2 , 1) = {p1 = q2 = 0} ⊂ Π2 . Therefore, A(T, 2) = {p = 0} ∪ {p2 = q1 = 0, p1 ≥ 0} ∪ {p1 = q2 = 0, p2 ≥ 0}.

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Finally, consider the right models on Fig. 8.1. The models are contained in the standard symplectic R4 with canonical coordinates (p1 , q1 , p2 , q2 ), and we have Π = Π1 = {p1 = 1} The tree T1 in this case consists of two vertices, and identifying Π with the standard symplectic R2 , we find that ± (T1 , 1) = {q1 = p2 = 0} ∪ {q1 = ∓p22 , p2 = ±q2 , p2 ≥ 0}. A Note that the second stratum in the union can also be written as {q1 = ∓q22 , p2 = ±q2 , p2 ≥ 0} Thus we have B(T, +1, 2) = {p2 = 0, q1 = 0, p1 ≥ 0} ∪ {q1 = − B(T, −1, 2) = {p2 = 0, q1 = 0, p1 ≥ 0} ∪ {q1 =

q22 , p2 = p1 q2 , p1 , p2 ≥ 0}, 2

q22 , p2 = −p1 q2 , p1 , p2 ≥ 0} 2 q2

Note that B(T, ±1, 2) ∩ {p = 0} = {q2 = 0} ∪ {q1 = ∓ 22 } is the front of the ± (T1 , 1), while B(T, ±1, 2) is the positive conormal of this front Legendrian tree A co-oriented by the vector field ∂q∂ 1 . A general arboreal singularity is associated to a double decorated rooted tree with an additional decoration β which assigns 0 or 1 to all terminal vertices of the tree T . We extend β to all vertices by setting β(v) = 0 for all non-terminal vertices. Primary arboreal singularities correspond to the case when the decoration β is identically 0.  We denote |β| := β(v), where the sum is taken over all terminal vertices v of the tree T . With each double decorated tree (T, ε, β) we associate a unique up to symplectomorphism model A(T, ε, β, m) ⊂ T ∗ Rm for each m ≥ |T | + |β| − 1. In dimension m ≥ n := |T | + |β| − 1 we have A(T, ε, β, m) = A(T, ε, β, n) × Rm−n ⊂ T ∗ Rn × T ∗ Rm−n = T ∗ Rm . The model A(T, ε, β, m) ⊂ T ∗ Rm with m = |T | + |β| − 1 is defined by a similar inductive procedure as for primary arboreal singularities, beginning with A(T, ε, β, 1) = {p = 0, q ≥ 0} ⊂ T ∗ R for |T | = 1 and |β| = 1. Every model A(T, ε, β, m) ⊂ T ∗ Rm can be presented as a union of Lagrangian sheets Lv enumerated by vertices of the graph T . Denote by d(v) the distance between v and the root. Then Lv is diffeomorphic to the quadrant {(x1 , . . . , xn ) ∈ Rn ; x1 , . . . xk ≥ 0},

k = d(v) + β(v).

Note that the model A(T, ε, β, m) inherits a smooth structure (i.e. the algebra of smooth functions) from the ambient space R2n . By an n-dimensional arboreal complex we mean a set covered by charts diffeomorphic to one of the models A(T, ε, β, n). Hence, every arboreal complex can be canonically stratified by strata ST,ε,β of dimension n − |T | − |β| + 1. A diffeomorphism f : C → C  between two arboreal complexes induces a diffeomorphism between the corresponding strata, but not every continuous map f : C → C  which is a diffeomorphism on the corresponding strata is a diffeomorphism of arboreal complexes C and C  .

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8.2. Main results. Proposition 8.1 ([23]). For each arboreal complex C there exists a unique up to symplectomorphism Weinstein domain W(C) = (X, ω, Z, φ), “the cotangent bundle” of C such that C = Core(X, ω, Z). Any two such Weinstein structures (X, ω, Z, φ) and (X, ω, Z  , φ ) are homotopic through a family of Weinstein structures with a fixed core. Theorem 8.2 ([23]). (i) Any Weinstein structure is homotopic to a Weinstein structure with an arboreal skeleton. (ii) Let Wt , t ∈ [0, 1] be a Weinstein homotopy such W0 and W1 have arboreal skeleta. Then there exists a Weinstein pair structure (W; W0 ∪ W1 ) on X × T ∗ I with an arboreal skeleton which is homotopic to the Weinstein pair associated to the homotopy Wt (see Section 3.3). Under some topological constraints on the manifold X one can further restrict the list of necessary singularities. Theorem 8.3 ([23]). Let W = (X, ω, Z, φ) be a Weinstein structure. Suppose that a) the manifold X is (n − 2)-connected; b) there exists a field of Lagrangian planes τ ⊂ T X; in other words, T X with its homotopically canonical almost complex structure is isomorphic to the complexification of a real n-dimensional vector bundle. ( = Then the Weinstein structure W is homotopic to a Weinstein structure W  whose skeleton is an arboreal complex with singularities of type (T, ε, β)  φ) (X, ω, Z, where the distance from the root of the tree T to any other vertex is no more than 2 and the decoration ε takes only positive values. References [1] M. Abouzaid and T. Kragh, Simple homotopy equivalence of nearby Lagrangians, arXiv:1603.05431. [2] M. Abouzaid and P. Seidel, Altering symplectic manifolds by homologous recombination, arXiv:1007.3281. [3] R. Avdek, Liouville hypersurfaces and connect sum cobordisms, arXiv:1204.3145. ´ [4] D. Alvarez-Gavela, The simplification of singularities of Lagrangian and Legendrian fronts, arXiv:1605.07259, to appear in Invent. Math. [5] J. Bowden, D. Crowley, and A. I. Stipsicz, The topology of Stein fillable manifolds in high dimensions I, Proc. Lond. Math. Soc. (3) 109 (2014), no. 6, 1363–1401, DOI 10.1112/plms/pdu028. MR3293153 [6] J. Bowden, D. Crowley, and A. I. Stipsicz, The topology of Stein fillable manifolds in high dimensions, II, Geom. Topol. 19 (2015), no. 5, 2995–3030, DOI 10.2140/gt.2015.19.2995. With an appendix by Bernd C. Kellner. MR3416120 [7] K. Barth, H. Geiges and K. Zehmisch, The diffeomorphism type of symplectic fillings, arXiv:1607.03310. [8] F. Bourgeois, T. Ekholm, and Y. Eliashberg, Effect of Legendrian surgery, Geom. Topol. 16 (2012), no. 1, 301–389, DOI 10.2140/gt.2012.16.301. With an appendix by Sheel Ganatra and Maksim Maydanskiy. MR2916289 [9] C. Cao, N. Gallup, K. Hayden, and J. M. Sabloff, Topologically distinct Lagrangian and symplectic fillings, Math. Res. Lett. 21 (2014), no. 1, 85–99, DOI 10.4310/MRL.2014.v21.n1.a7. MR3247041 [10] K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 2, 115–142, DOI 10.1007/s100970100036. MR1911873

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[11] K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and back, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. MR3012475 [12] G. Dimitroglou Rizell, E. Goodman, and A. Ivrii, Lagrangian isotopy of tori in S 2 × S 2 and CP 2 , Geom. Funct. Anal. 26 (2016), no. 5, 1297–1358, DOI 10.1007/s00039-016-0388-1. MR3568033 [13] T. Ekholm, Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology, Perspectives in analysis, geometry, and topology, Progr. Math., vol. 296, Birkh¨ auser/Springer, New York, 2012, pp. 109–145, DOI 10.1007/978-0-8176-8277-4 6. MR2884034 [14] T. Ekholm and Y. Lekili, Duality between Lagrangian and Legendrian invariants, arXiv:1701.01284. [15] Y. Eliashberg, Symplectic geometry of plurisubharmonic functions, Gauge theory and symplectic geometry (Montreal, PQ, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 488, Kluwer Acad. Publ., Dordrecht, 1997, pp. 49–67. With notes by Miguel Abreu. MR1461569 [16] Y. Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Internat. J. Math. 1 (1990), no. 1, 29–46, DOI 10.1142/S0129167X90000034. MR1044658 [17] Y. Eliashberg, Classification of contact structures on R3 , Internat. Math. Res. Notices 3 (1993), 87–91, DOI 10.1155/S107379289300008X. MR1208828 [18] Y. Eliashberg, S. Ganatra and O. Lazarev, Flexible Lagrangians, arXiv:1510.01287. [19] Y. Eliashberg, S. Ganatra and O. Lazarev, Topology of flexible fillings, in preparation. [20] Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 135–162, DOI 10.1090/pspum/052.2/1128541. MR1128541 [21] Y. Eliashberg and M. Gromov, Lagrangian intersection theory: finite-dimensional approach, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 186, Amer. Math. Soc., Providence, RI, 1998, pp. 27–118, DOI 10.1090/trans2/186/02. MR1732407 [22] Y. Eliashberg and E. Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013), no. 5, 1483– 1514, DOI 10.1007/s00039-013-0239-2. MR3102911 [23] Y. Eliashberg, D. Nadler and L. Starkston, in preparation. [24] E. Giroux, Convexit´ e en topologie de contact (French), Comment. Math. Helv. 66 (1991), no. 4, 637–677, DOI 10.1007/BF02566670. MR1129802 [25] E. Giroux, G´ eom´ etrie de contact: de la dimension trois vers les dimensions sup´ erieures (French, with French summary), Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 405–414. MR1957051 [26] S. Ganatra, J. Pardon and V. Shende, Covariantly functorial Floer theory on Liouville sectors, arXiv:1706.03152. [27] R. Hind, Lagrangian spheres in S 2 × S 2 , Geom. Funct. Anal. 14 (2004), no. 2, 303–318, DOI 10.1007/s00039-004-0459-6. MR2060197 [28] M. Kontsevich, Symplectic geometry of homological algebra, http://www.ihes.fr/ $\sim$maxim/TEXTS/Symplectic$_-$AT2009.pdf. [29] O. Lazarev, Contact manifolds with flexible fillings, arXiv:1610.04837. [30] F. Laudenbach, On the Thom–Smale complex, an Appendix to Bismut-Zhang, An extension of a Theorem by Cheeger and M¨ uller, Ast´ erisque 205(1992). [31] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991), no. 3, 651–671, DOI 10.1007/BF01239530. MR1091622 [32] M. Maydanskiy and P. Seidel, Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol. 3 (2010), no. 1, 157–180, DOI 10.1112/jtopol/jtq003. MR2608480 [33] M. McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), no. 4, 1877–1944, DOI 10.2140/gt.2009.13.1877. MR2497314 [34] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537, DOI 10.2307/1970128. MR0148075 [35] E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245.

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[36] E. Murphy, Closed exact Lagrangians in the symplectization of contact manifolds, arXiv:1304.6620. [37] E. Murphy and K. Siegel, Subflexible symplectic manifolds, arXiv:1510.01867. [38] D. Nadler, Arboreal singularities, Geom. Topol. 21 (2017), no. 2, 1231–1274, DOI 10.2140/gt.2017.21.1231. MR3626601 [39] D. Nadler, Non-characteristic expansion of Legendrian singularities, arXiv:1507.01513. [40] B. Ozbagci, On the topology of fillings of contact 3-manifolds, http://home.ku.edu.tr/ $\sim$bozbagci/SurveyFillings.pdf. [41] P. Seidel and I. Smith, The symplectic topology of Ramanujam’s surface, Comment. Math. Helv. 80 (2005), no. 4, 859–881, DOI 10.4171/CMH/37. MR2182703 [42] Z. A. Sylvan, On partially wrapped Fukaya categories, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–University of California, Berkeley. MR3427304 [43] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399, DOI 10.2307/2372978. MR0153022 [44] L. Starkston, Arboreal Singularities in Weinstein Skeleta, arXiv:1707.03446. [45] A. Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251, DOI 10.14492/hokmj/1381413841. MR1114405 Department of Mathematics, Stanford University, Stanford California 94305 Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01738

Remarks on Nahm’s equations Nigel Hitchin Dedicated to Simon Donaldson on the occasion of his 60th birthday Abstract. Nahm’s equations are viewed in a more general context where they appear as a vector field on a moduli space of O(2)-twisted Higgs bundles on the projective line. Zeros of this vector field correspond to torsion-free sheaves on a singular spectral curve which we translate in terms of a smooth curve in three-dimensional projective space. We also show how generalizations of Nahm’s equations are required when the spectral curve is non-reduced and deduce the existence of non-classical conserved quantities in this situation.

1. Introduction Nahm’s equations are the reduction of the self-dual Yang-Mills equations from four to one dimension and have played an important role in many parts of geometry and physics – from the original study of magnetic monopoles to a vast collection of constructions of hyperk¨ ahler metrics. In fact, since Euler’s equations for a spinning top form the simplest example one could say that they entered the scene centuries ago. They are equations for a triple of n × n matrix-valued functions of t: dT1 = [T2 , T3 ], dt

dT2 = [T3 , T1 ], dt

dT3 = [T1 , T2 ]. dt

In [11] they also appeared naturally in the context of generalized complex structures. The moduli space of generalized holomorphic bundles on the projective line has an action of a one-parameter group of the fundamental B-field symmetry which is omnipresent in generalized geometry. In the simplest case this turns out to be equivalent to evolution via Nahm’s equations. It was a throwaway remark in [11], but here we consider it in more detail and in particular look for fixed points in the moduli space. The equations are integrable in the sense that they correspond (up to conjugation) to a linear flow on the Jacobian of an algebraic curve, the spectral curve. Put like that it seems as if there are no fixed points, but what happens is that they occur for singular or reducible spectral curves and therefore have a different flavour from the more standard treatment of Nahm’s equations. We show, using the twistor theory that lies behind the geometry of monopoles, that a fixed point corresponds to a curve together with a line bundle in projective 3-space. The situation where the spectral curve is non-reduced involves further features, both for Nahm’s equations and the fixed points. We consider the case where it is c 2018 American Mathematical Society

83

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a ribbon and in doing so introduce new conserved quantities for Nahm’s equations beyond the coefficients of the equation of the spectral curve. 2. Co-Higgs bundles and Nahm’s equations 2.1. Generalized geometry. One of the basic features of generalized geometry is the extra symmetry beyond diffeomorphisms provided by the action of a closed 2-form, the B-field. The concept of a generalized complex structure (as in [10],[6]) includes an ordinary complex structure and then closed 2-forms of type (1, 1) preserve the generalized complex structure and hence transform naturally associated geometrical objects. The notion of a generalized holomorphic bundle on a generalized complex manifold becomes quite concrete for an ordinary complex structure ([6],[11]): Definition 2.1. Let M be a complex manifold. A generalized holomorphic bundle is a holomorphic vector bundle V together with a holomorphic section φ of End V ⊗ T such that φ ∧ φ = 0 as a section of End V ⊗ Λ2 T . Replacing T by T ∗ gives us Higgs bundles, so these are also called co-Higgs bundles. We shall retain this terminology since “generalized” will be used in a different way later on. We adopt the differential-geometric approach to holomorphic bundles by considering a fixed C ∞ vector bundle V and a holomorphic structure A 2 = 0. defined by an operator ∂¯A : Ω0 (M, V ) → Ω01 (M, V ) with ∂¯A If B is a closed (1, 1)-form then the interior product of the matrix-valued vector field φ with B gives iφ B ∈ Ω01 (M, End V ) and the B-field transform is the new ¯ holomorphic structure defined by the ∂-operator ∂¯B = ∂¯ + iφ B ¯ = 0, ∂φ ¯ = 0, φ ∧ φ = 0 show on the same C ∞ bundle V . The three conditions ∂B 2 = 0 which is the integrability condition for the holomorphic structure. The that ∂¯B last two show that φ, which is unchanged, is holomorphic with respect to this new ¯ then the pairs (V, ∂, ¯ φ) and (V, ∂¯B , φ) are structure. As shown in [11] if B = ∂θ holomorphically equivalent. 2.2. Nahm’s equations. We shall consider co-Higgs bundles in the onedimensional case of P1 , studied in some detail in [13]. In this case there is only a one-dimensional choice of Dolbeault cohomology class in H 1 (P1 , K) for B. Choose a generator [ω]. Generically, if c1 (V ) = 0 the bundle V will be a trivial rank n bundle and then we can write d φ = (φ0 + φ1 z + φ2 z 2 ) dz where the φi are constant n × n matrices. Theorem 2.2. Let (V, ψ) be a rank n co-Higgs bundle over P1 with V holomorphically trivial and B a (1, 1)-form whose integral is non-zero. Then if t lies in a neighbourhood of 0 ∈ C over which the holomorphic structure ∂¯tB is trivial, there is a t-dependent choice of trivialization in which ψ is represented by φ(t) and the components of φ(t) satisfy the equations 1 1 dφ1 dφ2 dφ0 = − [φ1 , φ0 ], = [φ0 , φ2 ], = [φ1 , φ2 ] dt 2 dt dt 2 and φ(0) = ψ.

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85

Remark 2.3. Writing φ0 = −(T1 + iT2 ), φ1 = −2iT3 , φ2 = −(T1 − iT2 ) gives Nahm’s equations dT1 dT2 dT3 = [T2 , T3 ], = [T3 , T1 ], = [T1 , T2 ]. dt dt dt Proof. Triviality of the bundle V means the existence of a gauge transformation g(t) such that ¯ tiψ B = g −1 ∂g

(2.1)

Any two are related by g˜ = hg where for each t, h(t) is a constant matrix. The Higgs field ψ then defines a t-dependent φ(t) by ψ = g −1 φg

(2.2)

Differentiating (2.1) with respect to t gives ¯ + g −1 ∂¯g˙ iψ B = −g −1 gg ˙ −1 ∂g or, conjugating by g, ¯ −1 + ∂¯gg ¯ gg ˙ −1 ∂gg ˙ −1 = ∂( ˙ −1 ). iφ B = −gg For B take the standard volume form ω=

dzd¯ z (1 + z z¯)2

then

1 (φ0 + φ1 z + φ2 z 2 )d¯ z. (1 + z z¯)2 Integrating to give a regular integral gives ¯ gg ∂( ˙ −1 ) = iφ B =

gg ˙ −1 =

−1 φ0 (φ0 + φ1 z + φ2 z 2 ) + + c(t) z(1 + z z¯) z

for a choice of constant matrix c. Take c = φ1 /2 and then (2.3)

gg ˙ −1 =

φ1 −1 φ0 (φ0 + φ1 z + φ2 z 2 ) + + z(1 + z z¯) z 2

Differentiating (2.2) with respect to t gives ˙ + g −1 φg˙ ˙ −1 φ + g −1 φg 0 = −g −1 gg or φ˙ = [gg ˙ −1 , φ] and substituting from (2.3) we obtain ) * φ0 φ1 φ˙ = + ,φ . z 2 

Equating coefficients of z gives the result.

Remark 2.4. The choice of c gives the symmetrical form of Nahm’s equations arising from their origin where the Ti lie in a compact Lie algebra, and P1 is endowed with the real structure z → −1/¯ z . Taking c = 0 instead gives the equations dφ0 = [φ0 , φ1 ], dt

dφ1 = [φ0 , φ2 ], dt

dφ2 =0 dt

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If V has degree k where 0 < k < n then the generic splitting type of a holomorphic structure is V = Ok (1) ⊕ On−k and the rank k subbundle is uniquely determined. The structure group then reduces to a parabolic subgroup, the subgroup of GL(n, C) preserving a k-dimensional subspace. Now the Higgs field has the form   A B φ= C D where, in the affine coordinate z, A, B, C, D are matrix-valued polynomials of degree 2, 3, 1, 2 respectively. We can then write d dz where φ2 lies in the parabolic subalgebra and φ3 in its nilradical. Applying the B-field action as above gives an integral φ = (φ0 + φ1 z + φ2 z 2 + φ3 z 3 )

gg ˙ −1 =

−1 φ0 (φ0 + φ1 z + φ2 z 2 + φ3 z 3 ) + z(1 + z z¯) z

which is regular at the origin. But gg ˙ −1 =



α β γ δ



where α, δ are functions, β is a C ∞ section of O(1) and γ of O(−1). A term of the form z k /z(1 + z z¯) extends smoothly to a section of O(m) if k ≤ m + 2, so given the degrees of A, B, C, D this is well-defined on P1 . With the constant c = 0 the equations are: (2.4)

dφ0 = [φ0 , φ1 ], dt

dφ1 = [φ0 , φ2 ], dt

dφ2 = [φ0 , φ3 ], dt

dφ3 = 0. dt

3. Moduli spaces and the Nahm flow 3.1. Moduli spaces. Just as in the case of Higgs bundles, one can introduce the notion of stability into our situation and construct moduli spaces [13]. A coHiggs bundle (V, φ) on P1 is stable if for any φ-invariant holomorphic subbundle U ⊂ V , deg U/ rk U < deg V / rk V . In the case of equality the pair is semi-stable. Since φ-invariance implies that U is also preserved by ∂¯B = ∂¯ + iφ B stability is clearly invariant under B-field transforms. The space of S-equivalence classes of co-Higgs bundles (where S-equivalence means replacing the Harder-Narasimhan filtration of a semistable bundle by its graded version) is a well-defined non-compact algebraic variety and, as with vector bundles themselves, when the degree and rank are coprime it is smooth. Moreover, as with Higgs bundles, the coefficients ak of the characteristic polynomial det(x − φ) = xn + a1 xn−1 + · · · + an define a proper map to a vector space W = H 0 (P1 , O(2)) ⊕ H 0 (P1 , O(4)) ⊕ · · · ⊕ H 0 (P1 , O(2n)). The B-field action therefore defines a canonical holomorphic vector field on this moduli space and we shall call this more general action from now on the Nahm flow.

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87

Example 3.1. Take V to be of rank 2 and degree (−1). Then in [13] it is shown that the moduli space of stable co-Higgs bundles with tr φ = 0 is the universal elliptic curve S = {(z, w, c0 , c1 , . . . , c4 ) : w2 = c0 + c1 z + · · · + c4 z 4 }. More invariantly, S ⊂ O(2) × H 0 (P1 , O(4)) is the divisor of w2 − π ∗ q(z) where w is the tautological section of π ∗ O(2) on the total space of π : O(2) → P1 and q = c0 + c1 z + · · · + c4 z 4 is a section of O(4). From [13] stability implies that V ∼ = O ⊕ O(−1) and so, as above, the Higgs field is of the form   a b φ= c d where in particular c ∈ H 0 (P1 , O(1)). If c = 0 then O ⊂ V is invariant which contradicts stability, so c has a unique zero z0 . Since a is a section of O(2) there is a map from the moduli space M to S by setting w = a(z0 ), c(z) = det φ and this is in fact an isomorphism. Considering the Nahm flow, the last equation in (2.4) gives φ3 = const. and this, as the coefficient of z 3 , is strictly upper triangular so we may take it to be   0 1 φ3 = 0 0 From the other equations we obtain, with c(z) = c0 + zc1 , a(z) = a0 + a1 z + a2 z 2 , c˙0 = 2(c1 a0 − a1 c0 ),

c˙1 = −2a2 c0

and since z0 = −c0 /c1 this gives z˙0 = −2(a0 + a1 z0 + a2 z02 ) = −2a(z0 ). Thus, at the points where (z, c0 , . . . , c4 ) are local coordinates on S the vector field is ∂ w . ∂z The parameter z fails to be part of a coordinate system if w = 0 in which case w is a coordinate and since w2 = q(z) the vector field has the local form q  (z) ∂ . 2 ∂w 3.2. Fixed points of the Nahm flow. In the example above a zero of the vector field occurs where w = 0 and from (3.1) we then have q  (z) = 0 which is when the elliptic curve w2 = q(z) is singular. Note that it also vanishes if q ≡ 0: this is where the Higgs field is nilpotent. To see this in more generality, we note that in the original generalized geometry formulation, we have the pair (∂¯tB , ψ) where the holomorphic structure is varying and ψ is fixed and so clearly det(w − ψ) is constant, so that the vector field is always tangent to the fibres of the map M → W . This means that the curve in O(2) defined by the equation det(w −φ) = 0, the spectral curve S, is fixed along the flow. In particular, the coefficients of the characteristic polynomial are constants of integration of Nahm’s equations. A naive treatment of the integrability of Nahm’s equations as in [9] assumes that the spectral curve S ⊂ O(2) is smooth. In this case the co-Higgs bundle (V, φ) is obtained from a line bundle L on S as the direct image V = π∗ L, φ = π∗ w, (3.1)

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where w is again the tautological section of π ∗ O(2), and then L is the cokernel of φ − w : π ∗ V (−2) → π ∗ V . If V is of rank n then the genus of S is g = (n − 1)2 , its canonical bundle KS ∼ = π ∗ O(2n − 4). By Grothendieck-Riemann-Roch if L has degree d then deg V = d+n−n2 , so the original Nahm equations require d = n2 −n. The bundle V is then trivial if and only if V (−1) has no sections which is when L(−1) of degree g − 1 does not lie on the theta-divisor of S. The Nahm flow then consists of tensoring L by the one-parameter group of line bundles Ut = exp(tw[ω]) ∈ H 1 (O(2), O∗ ) restricted to S. Here ω is the standard (1, 1)-form used in Theorem 1 and [ω] ∈ H 1 (P1 , O(−2)) its cohomology class. The product with the tautological section w of O(2) on its total space gives w[ω] ∈ H 1 (O(2), O). Then L → LUt is a one-parameter group of translations in the Jacobian of S. Moreover, as in [8], the class w[ω] is always non-zero if rk V > 1 and hence the flow has no fixed points. However, even in the original appearance of Nahm’s equations for this author [8], singular and reducible spectral curves are allowed, for example in the construction of axi-symmetric monopoles. Subsequent treatments of similar moduli spaces [3], [15],[14], identify the fibre as a compactified Jacobian parametrizing stable (in an appropriate sense) rank one torsion-free sheaves on the spectral curve. There is a large literature on compactified Jacobians but if we assume that the curve is reduced, then following [1], a torsion-free sheaf is given by the direct image of a line bundle on some partial normalization S  of S. The generalized Jacobian H 1 (S  , O∗ ) of a singular curve is still a group so a fixed point of the Nahm flow must be represented by the direct image of a line bundle on a normalization f : S  → S for which the class f ∗ w[ω] = 0 ∈ H 1 (S  , O). Example 3.2. In the example above the singular elliptic curves w2 − q(z) = 0 are normalized by P1 and H 1 (P1 , O) = 0 so any degree zero line bundle is trivial. Determining all such partial normalizations is seemingly a difficult task, but there is a more geometrical approach which we adopt now, and takes us back to the twistor theory of R4 and R3 . 3.3. Twistor spaces and liftings. Penrose’s twistor theory encodes the Euclidean geometry of R4 in the holomorphic geometry of the complex 3-manifold O(1) ⊕ O(1) → P1 . The points of R4 correspond to holomorphic sections which are real with respect to an antiholomorphic involution with no fixed points. We are not concerned with reality here however. Any orientation-preserving Euclidean motion of R4 induces a holomorphic action on the twistor space, and in particular the one-parameter group of translations (x0 , x1 , x2 , x3 ) → (x0 + t, x1 , x2 , x3 ). The twistor space is the complement of a line in P3 : in homogeneous coordinates (z0 , z1 , z2 , z3 ) we remove the line z0 = z1 = 0 and then [z0 , z1 ] ∈ P1 defines the projection. The free holomorphic action is then (3.2)

(z0 , z1 , z2 , z3 ) → (z0 , z1 , z2 + tz0 , z3 − tz1 )

and the invariant section w = z1 z2 + z3 z0 of O(2) identifies the quotient by the action with the total space of O(2). As a principal C-bundle over O(2) it defines a class α ∈ H 1 (O(2), O). The quotient of R4 by the translation is R3 and each section of O(1) ⊕ O(1) → 1 P projects to a section w = a0 z02 + a1 z0 z1 + a2 z12 = a(z0 , z1 ) of O(2) → P1 . The three-dimensional space of such real sections is the twistor interpretation of the

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Euclidean geometry of R3 as in [7]. A fixed section w = a(z0 , z1 ) of O(2) → P1 has a one-parameter family of inverse images in O(1) ⊕ O(1) and these sweep out a surface z1 z2 + z3 z0 − a(z0 , z1 ) = 0. Adding in the line z0 = z1 = 0 gives a smooth projective quadric in P3 and the inverse images form one of the two families of lines. Remark 3.3. According to [7], a holomorphic vector bundle on O(2) trivial on each real section corresponds to a solution to the Bogomolny equations FA = ∗∇φ on R3 . The class exp α ∈ H 1 (O(2), O∗ ) gives A = 0, φ = 1. Proposition 3.4. The class α is (up to a multiple) the same as the class w[ω] which gives the Nahm flow. Proof. Note that over the open set U0 where z0 = 0 we have a section of the action defined by (z0 , z1 , 0, z3 ) and similarly over U1 where z1 = 0 we have ˇ cocycle in H 1 (O(2), O) defining it is provided by the (z0 , z1 , z2 , 0). Thus a Cech value of t on U0 ∩ U1 which relates these two sections. This is t = w/z0 z1 . Now z0 z1 is the section of O(2) on P1 vanishing at 0 and ∞ and using the affine parameter z = z1 /z0 , and identifying O(2) with the tangent bundle this is the vector field zd/dz. But its inverse, the form dz/z, is a cocycle on U0 ∩ U1 which is a generator of H 1 (P1 , K), so the class w[ω] is represented by w/z0 z1 .  It follows that if C is a partial normalization of S on which the pull-back of the class w[ω] is zero, then a choice of trivialization lifts it to a map into the principal C-bundle over S. This is then a curve in P3 which misses the line z0 = z1 = 0. Conversely any such curve projects to a curve S in O(2) and points in C which lie in the same orbit of the C-action map to singular points of S. This way C is a partial normalization of S and by construction the class w[ω] is trivial on C. The direct image of any line bundle on C is a torsion-free sheaf on S, and taking the direct image on P1 we have a rank n co-Higgs bundle where n = deg C, whose equivalence class in the moduli space is fixed by the Nahm flow. 3.4. Commuting pairs. In the generic case where the bundle V on P1 is trivial, the Nahm flow yields Nahm’s equations dT2 dT3 dT1 = [T2 , T3 ], = [T3 , T1 ], = [T1 , T2 ]. dt dt dt and a zero of the induced vector field in the moduli space consists of matrices (T1 , T2 , T3 ) where a fourth matrix T0 satisfies [T0 , T1 ] = [T2 , T3 ],

[T0 , T2 ] = [T3 , T1 ],

[T0 , T3 ] = [T1 , T2 ].

Remark 3.5. If T0 , T1 , T2 , T3 lie in the Lie algebra of a compact Lie group G with a bi-invariant metric then these equations are equivalent to the vanishing of the hyperk¨ ahler moment map μ : g ⊗ H → g ⊗ R3 for the adjoint action of G on the flat hyperk¨ ahler manifold g ⊗ H. However we are dealing here with the complex case – there are no non-trivial solutions for a compact group. We can see this by interpreting the equations as giving a translation-invariant solution to the self-dual Yang-Mills equations on R4 , or equivalently a translation-invariant solution to the Bogomolny equations ∗dA φ = FA on R3 . Quotienting by a lattice in R3 we have a solution on the 3-torus, but the Bianchi identity gives 0 = dA FA = dA ∗ dA φ. Integrating (dA ∗ dA φ, φ) and using Stokes’ theorem we get dA φ = FA = 0.

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To link this up with the above spectral curve approach we collect T1 , T2 , T3 , as in Theorem 1, into a co-Higgs field φ = φ0 + φ1 z + φ2 z 2 and obtain (with ψ = T0 ) * ) φ1 φ0 + , φ = 0. −ψ + z 2 Hence the term φ− = −zψ + φ0 + zφ1 /2, which is linear in z, defines a matrix with entries in H 0 (P1 , O(1)) which commutes with φ. Consider also φ1 + φ2 z + ψ 2 which is a similar section. This also commutes with φ since φ− + zφ+ = φ. Thus (φ+ , φ− ) defines a matrix-valued section ϕ of O(1) ⊕ O(1), and since [φ+ , φ− ] = 0 we have ϕ ∧ ϕ = 0 – rather like a higher-dimensional Higgs field. Following this approach (due to Simpson [15] and in this context as in [11]) it defines a sheaf with compact support on the total space of O(1) ⊕ O(1), or P3 \ P1 . If we denote by x, y the tautological sections of O(1) on the two factors then x acts by φ+ , y by φ− and the sheaf is supported on the variety defined by φ+ =

det(u(x − φ+ (z)) + v(y − φ− (z))) = 0 for all u, v. Roughly speaking it is the common cokernel of the family of commuting matrices u(x − φ+ (z)) + v(y − φ− (z)) and is a rank 1 sheaf supported on the curve C ⊂ P3 \ P1 above. Moreover, since φ− + zφ+ = φ we have, putting u = z, v = 1 det(xz + y − φ) = 0 which with w = xz + y is the equation of the spectral curve S of φ. This provides the projection to S ⊂ O(2). 3.5. Rank 2. Consider the basic example where φ takes values in sl(2, C). We take the equations for a fixed-point of the Nahm flow in the form 1 1 [ψ, φ0 ] = [φ0 , φ1 ], [ψ, φ1 ] = [φ0 , φ2 ], [ψ, φ2 ] = [φ1 , φ2 ]. 2 2 Since φ becomes nilpotent at some point, without loss of generality we can take   0 1 φ0 = . 0 0 The first equation gives ψ + φ1 /2 = aφ0 . Substituting in the second we get φ2 − aφ1 = bφ0 and in the third (a2 + b)[φ0 , φ1 ] = 0 so either [φ0 , φ1 ] = 0 or a2 + b = 0. In the first case, φ1 and φ2 are multiples of φ0 which means φ is nilpotent which we consider later. So with a2 + b = 0 we have ψ = aφ0 − φ1 /2 and φ2 = aφ1 − a2 φ0 . This means φ− = (1 − az)φ0 + zφ1 ,

φ+ = a(1 − az)φ0 + azφ1

so φ+ = aφ− . = 2z(1 − az) tr φ0 φ1 + z 2 tr φ21 . So the curve C ⊂ P \ P has the equation in affine coordinates Moreover tr φ2− 3 1 y = ax,

x2 = 2z(1 − az) tr φ0 φ1 + z 2 tr φ21 .

If tr φ0 φ1 = 0, this is a nonsingular conic in the plane y = ax. Suppose (x, y) and (x + t, y − zt) lie on C. Then since y = ax, z = −a and x2 = −2a(1 + a2 ) tr φ0 φ1 + a2 tr φ21

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so in general there are two such points and the image S has a double point where w = xz + y = 0. If the right hand side is zero, then the vector field is tangential to C and the image has a cusp. If tr φ0 φ1 = 0 the curve C is a pair of lines meeting in one point x = y = z = 0. The image is a pair of sections of O(2) meeting at (w, z) = (0, 0) and (w, z) = (0, −a). In all cases these are partial normalizations with H 1 (C, O) = 0. Now turn to the other zero of the vector field in the example above: where φ is nilpotent and the characteristic polynomial is w2 . The spectral curve in this case is the zero section of O(2) with multiplicity 2: its first order neighbourhood. Let X be the curve w2 = 0 and P1 = Xred the reduced curve, then there is an exact sequence of sheaves 0 → O(−2) → OX → O → 0 1 1 1 ∼ H (P , O(−2)) = ∼ C. In fact our class w[ω] is a generator. and H (X, OX ) = Remark 3.6. In the twistor theory of monopoles the Higgs field φ for a solution of the Bogomolny equations is precisely the obstruction to extending the trivialization of the corresponding holomorphic vector bundle on O(2) to the first order neighbourhood. In our case φ = 1 which is everywhere non-vanishing and hence is a non-zero element of H 1 (X, OX ). We learn nothing more about the co-Higgs bundle from the spectral curve, but there is extra information in the Higgs field φ = a(z)φ0 . In a neighbourhood of a point where z = 0 the cokernel of φ defines an invertible sheaf on X, generated by the cokernel of the constant matrix φ0 . This is no longer true where a(z) vanishes and all we get is a rank one torsion-free sheaf. Although the curve is smooth, we are in a similar situation to the general case and we can define φ+ = (a1 /2 + za2 )φ0 ,

φ− = (a0 + za1 /2)φ0

where φ− + zφ+ = φ. Then φ− , φ+ map S into a curve C ⊂ O(1) ⊕ O(1) with equation x2 = 0 = y 2 . Moreover if a21 − 4a0 a2 = 0, φ+ , φ− have no common zero and the cokernels define a line bundle on C whose direct image on S is the required torsion-free sheaf. As above, the class w[ω] is trivial on C and so we have a fixed point of the Nahm flow. When a(z) has a double zero we take the direct image of a torsion-free sheaf, which is still invariant under tensoring by the line bundles. 4. Ribbons 4.1. Ribbons and line bundles. The previous example is part of a more general picture where the spectral curve is non-reduced. We restrict attention to multiplicity 2 and a smooth reduced curve: this is called a ribbon [2]. Definition 4.1. A ribbon X on S is a curve such that Xred ∼ = S and the ideal sheaf I of S in X is invertible and satisfies I 2 = 0. This is an abstract ribbon. We are concerned with a curve X defined by det(w − φ) = p(w, z)2 = 0 in O(2) so that I is the conormal bundle of S defined by p(w, z) = 0. Simpson’s results on the moduli spaces of sheaves imply [5] that if det(w −φ) = 0 defines a ribbon X in the surface O(2), then the co-Higgs bundle is defined by the direct image of one of two types of sheaves: • a rank 2 vector bundle E on the reduced curve Xred = S

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• a generalized line bundle on X, a torsion-free sheaf which is free of rank one outside a divisor D ⊂ S. The first case is rank one since OS (E) and OX have the same dimension as OS modules, or equivalently the rank term in the Hilbert polynomial is 1. In the second case it was shown in [2] that there is a canonical blow up f : X  → X of X at the points of D giving a ribbon X  and the generalized line bundle is then f∗ L for a line bundle L on X  . Remark 4.2. The first case occurs naturally in the Higgs bundle description of the moduli space of representations of a surface group into certain real Lie groups associated to the quaternions [12]. Example 4.3. In Section 3.5 the cokernel of a(z)φ0 defines a generalized line bundle on the ribbon w2 = 0: the first order neighbourhood of the zero section S. There we lifted the curve to a quadric surface in P3 where, if the two zeros of a(z) were distinct, we had a line bundle. The blow-up in this case is achieved in the ambient surface O(2), which compactifies to the Hirzebruch surface P(O ⊕ O(2)). The zero section S ⊂ P(O ⊕ O(2)) has self-intersection 2 and blowing up the two zeros of a(z) this becomes zero. But the two P1 fibres now have self-intersection −1 and can thus be blown down giving the quadric surface P1 × P1 . We see from this that the Nahm flow is obtained by either tensoring the rank 2 bundle E on S by Ut or the line bundle L by f ∗ Ut . When S is smooth and has genus > 0 there are clearly no fixed points, and the example in Section 3.5 shows what happens in the case of genus 0. 4.2. Ribbons and conserved quantities. We shall discuss here the implications for Nahm’s equations themselves when the spectral curve is a ribbon. The simplest case is where V is the direct image of a rank 2 bundle E on S. This is when the Higgs field φ has 2-dimensional eigenspaces, or equivalently p(φ) = 0. The Nahm flow is then described by E → E ⊗ Ut and so the projective bundle P(E) on S is an invariant of the flow – a geometric conserved quantity. The case of a generalized line bundle occurs when the generic eigenspaces are one-dimensional. To see what this means for the Nahm flow, we follow the approach of Lucas Branco, who considers in his Oxford DPhil thesis [4] the Higgs bundle case. In our language we suppose then that we have a co-Higgs bundle (V, φ) on P1 where rk V = 2m, det(w − φ) = p2 (w) and p = 0 defines a smooth curve S. The two cases of rank one sheaves on a ribbon correspond to whether the generic minimal polynomial is p (the first case) or p2 . In the latter case, the kernel of p(φ) ∈ H 0 (P1 , End V (2m)) defines a φ-invariant subbundle W1 ⊂ V . Since S is irreducible, there are no further invariant subbundles and since the generic minimal polynomial is of degree m we have rk W1 = m. Thus V is an extension of co-Higgs bundles 0 → W1 → V → W2 → 0 where W1 , W2 have the same spectral curve S. With respect to a C ∞ -splitting we can therefore write     ∂¯1 β ϕ1 ψ ¯ φ= ∂V = 0 ∂¯2 0 ϕ2

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where ∂¯V φ = 0 implies ¯ + ϕ12 β = 0. ∂ψ

(4.1)

In this equation for β ∈ Ω01 (P , Hom(W2 , W1 )) we define ϕ12 β = βϕ2 − ϕ1 β. Since both ∂¯V and φ preserve W1 the B-field action ∂¯V → ∂¯V + iφ B also preserves the structure of an extension. 1

Remark 4.4. Since W remains an extension, we have two Nahm flows corresponding to the bundles W1 , W2 . If W is stable then W1 has negative degree −d and so deg W2 = d. This means (unless m divides d) that the Nahm flows on W1 , W2 generically correspond to the equations (2.4) rather than the original Nahm equations, even if W itself is holomorphically trivial. Consider the holomorphic map ϕ12 : Hom(W2 , W1 ) → Hom(W2 , W1 )(2). Its kernel and cokernel are in fact holomorphic vector bundles. To see this note that ϕ1 and ϕ2 are Higgs fields with the same spectral curve S, so there are line bundles L1 , L2 on S whose direct images are W1 , W2 and the Higgs fields are the direct images of w : L1 → L1 (2), w : L2 → L2 (2). For a small open set U ⊂ P1 , L1 and L2 are isomorphic on π −1 (U ) and hence W1 ∼ = W2 = W and ϕ1 = ϕ2 = ϕ. Trivializing O(2) over U , ker ϕ12 can be identified with the sheaf of centralizers of a holomorphic matrix ϕ. Since S is assumed smooth, ϕ is regular and the space of centralizers is spanned by 1, ϕ, . . . , ϕm−1 . So globally ker ϕ12 is a rank m holomorphic vector bundle and the same holds for the cokernel. Equation (4.1) now says that the projection ψ˜ of ψ to coker ϕ12 is holomorphic. Now the B-field action changes the holomorphic structure to ∂¯ + iφ ω and the ¯ + ϕ12 ωψ. Since ϕ12 ωψ is induced operator on ψ ∈ Ω0 (P1 , Hom(W2 , W1 )(2)) is ∂ψ trivial on the cokernel the B-field action induces the same holomorphic structure on coker ϕ12 . In our formalism φ is unchanged and so the holomorphic section ψ˜ unchanged. This can therefore be considered as a conserved quantity under the Nahm flow. (Strictly speaking ψ is defined by the extension rather than the bundle V itself and so the invariant is the section up to a constant multiple). Remark 4.5. A more sophisticated interpretation of the above is via the second spectral sequence of the hypercohomology for the complex of sheaves ϕ12 : O(Hom(W2 , W1 )) → O(Hom(W2 , W1 )(2)) [4]. If ψ projects to zero in coker ϕ12 then ψ = ϕ12 θ for some θ ∈ Ω0 (P , Hom(W2 , W1 )). But θ can be used to change the C ∞ splitting making ψ = 0. In this case   ϕ1 0 φ= 0 ϕ2 ˜ invariant by the flow, must be and p(φ) ≡ 0 and we are back to the first case, so ψ, part of the data of a generalized line bundle on X. We shall see next what it is in a more concrete fashion next. A local holomorphic section of L∗1 L2 on S defines a map from L1 to L2 commuting with the scalar multiplication by w ∈ H 0 (S, π ∗ (O(2))). The direct image therefore intertwines φ1 and φ2 and it follows that ker ϕ12 ∼ = π∗ (L∗1 L2 ). Then ∗ ∼ ∗ (coker ϕ12 ) = π∗ (L1 L2 )(−2). Relative duality gives ∼ π∗ (L1 L∗ KS )(4) = ∼ π∗ (L1 L∗ )(2m) ∼ (π∗ (L∗ L2 ))∗ (2) = (4.2) coker ϕ12 = 1

1

2

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Thus the projection ψ˜ defines a non-zero holomorphic section s of L1 L∗2 (2m) on S, and this vanishes on a divisor D, and since ψ was really only defined up to a multiple it is the divisor which is the conserved quantity. Note that if 1 , 2 are the degrees of the line bundles then −d = deg W1 = 1 + m − m2 , d = deg W2 = 2 + m − m2 and so deg D = 1 − 2 + 2m2 = −2d + 2m2 , so 0 < d < m2 . Proposition 4.6. Let (w = λ, z = a) be a point of the divisor D on the curve S. Then the λ-eigenspace of φ(a) has multiplicity 2. Proof. The application of relative duality in Equation 4.2 identifies coker ϕ12 with ker ϕ21 for a homomorphism ϕ21 from W2 to W1 (2m). In fact, as we have seen, locally ϕ12 can be considered as a holomorphic m × m matrix acting as x → [ϕ, x]. Using the invariant inner product tr(xy), ker ϕ is the orthogonal complement of the image of ad ϕ. This maps to coker ϕ isomorphically unless the kernel of (ad ϕ)2 has dimension greater than m. But these points correspond to the discriminant locus of p, giving the ramification points of π : S → P1 , which provide the twist in the relative duality formula. Pulling back π ∗ (L1 L∗2 (2m)) to S there is the natural evaluation map π ∗ (L1 L∗2 (2m))(λ,a) → L1 L∗2 (2m)(λ,a) and a point (λ, a) of D is where the global section s of L1 L∗2 (2m) vanishes which ˜ maps the cokernel L2 of φ2 − λ means that at this point the direct image of s, ψ, to zero in the cokernel L1 of ϕ1 − λ. Equivalently, Im ψ ⊆ Im(ϕ1 − λ). Let v2 be a λ-eigenvector of ϕ2 at z = a then ψv2 = (ϕ1 − λ)v0 for some v0 and then        ϕ1 0 v0 λv1 + ψv2 − ψv2 v0 = =λ . −v2 −λv2 −v2 0 ϕ2 Together with (v1 , 0) where ϕ1 v1 = λv1 these span a two-dimensional eigenspace.  The proposition shows that the divisor D corresponds to the points of S at which the generalized line bundle on X fails to be locally free. This data is conserved by the Nahm flow. One may say that for a reduced curve, the singularities are part of the characteristic equation of φ and clearly conserved under the flow. For the ribbon it is the singularities of the sheaf which are conserved. References [1] V. Alexeev, Compactified Jacobians and Torelli map, Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, 1241–1265. MR2105707 [2] D. Bayer and D. Eisenbud, Ribbons and their canonical embeddings, Trans. Amer. Math. Soc. 347 (1995), no. 3, 719–756, DOI 10.2307/2154871. MR1273472 [3] A. Beauville, M. S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169–179, DOI 10.1515/crll.1989.398.169. MR998478 [4] L. Branco, H iggs bundles, Lagrangians and mirror symmetry, Oxford DPhil thesis (2018), arXiv: 1803.0611. [5] D. Chen and J. L. Kass, Moduli of generalized line bundles on a ribbon, J. Pure Appl. Algebra 220 (2016), no. 2, 822–844, DOI 10.1016/j.jpaa.2015.07.019. MR3399392 [6] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75–123, DOI 10.4007/annals.2011.174.1.3. MR2811595 [7] N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), no. 4, 579–602. MR649818

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[8] N. J. Hitchin, On the construction of monopoles, Comm. Math. Phys. 89 (1983), no. 2, 145–190. MR709461 [9] N. Hitchin, Riemann surfaces and integrable systems, Integrable systems (Oxford, 1997), Oxf. Grad. Texts Math., vol. 4, Oxford Univ. Press, New York, 1999, pp. 11–52. Notes by Justin Sawon. MR1723385 [10] N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308, DOI 10.1093/qjmath/54.3.281. MR2013140 [11] N. Hitchin, Generalized holomorphic bundles and the B-field action, J. Geom. Phys. 61 (2011), no. 1, 352–362, DOI 10.1016/j.geomphys.2010.10.014. MR2747007 [12] N. Hitchin and L. P. Schaposnik, Nonabelianization of Higgs bundles, J. Differential Geom. 97 (2014), no. 1, 79–89. MR3229050 [13] S. Rayan, Co-Higgs bundles on P1 , New York J. Math. 19 (2013), 925–945. MR3158239 [14] D. Schaub, Courbes spectrales et compactifications de jacobiennes (French), Math. Z. 227 (1998), no. 2, 295–312, DOI 10.1007/PL00004377. MR1609069 [15] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective ´ variety. I, Inst. Hautes Etudes Sci. Publ. Math. 79 (1994), 47–129. MR1307297 Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01739

Conjectures on counting associative 3-folds in G2 -manifolds Dominic Joyce In honour of Simon Donaldson on his 60th birthday Abstract. There is a strong analogy between compact, torsion-free G2 -manifolds (X, ϕ, ∗ϕ) and Calabi–Yau 3-folds (Y, J, g, ω). We can also generalize (X, ϕ, ∗ϕ) to ‘tamed almost G2 -manifolds’ (X, ϕ, ψ), where we compare ϕ with ω and ψ with J. Associative 3-folds in X, a special kind of minimal submanifold, are analogous to J-holomorphic curves in Y . Several areas of Symplectic Geometry – Gromov–Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories – are built using ‘counts’ of moduli spaces of J-holomorphic curves in Y , but give an answer depending only on the symplectic manifold (Y, ω), not on the (almost) complex structure J. We investigate whether it may be possible to define interesting invariants of tamed almost G2 -manifolds (X, ϕ, ψ) by ‘counting’ compact associative 3folds N ⊂ X, such that the invariants depend only on ϕ, and are independent of the 4-form ψ used to define associative 3-folds. We conjecture that one can define a superpotential Φψ : U → Λ>0 ‘counting’ associative Q-homology 3-spheres N ⊂ X which is deformation-invariant in ψ for ϕ fixed, up to certain reparametrizations Υ : U → U of the base U = Hom(H3 (X; Z), 1 + Λ>0 ), where Λ>0 is a Novikov ring. Using this we define a notion of ‘G2 quantum cohomology’. We also discuss Donaldson and Segal’s proposal from their 2011 work to define invariants ‘counting’ G2 -instantons on tamed almost G2 -manifolds, with ‘compensation terms’ counting weighted pairs of a G2 -instanton and an associative 3-fold, and suggest some modifications to it.

Contents 1. Introduction 2. Geometry of G2 -manifolds 2.1. G2 -manifolds 2.2. Calabi–Yau 3-folds and G2 -manifolds 2.3. Calibrated submanifolds 2.4. G2 -instantons 2.5. Tamed almost-G2 -manifolds 2.6. Moduli spaces of associative 3-folds 2.7. Associative 3-folds with boundary in coassociatives 3. How to orient moduli spaces of associatives 2010 Mathematics Subject Classification. Primary 53C38; Secondary 53C07, 53D45. Partly funded by a Simons Collaboration Grant on ‘Special Holonomy in Geometry, Analysis and Physics’. c 2018 American Mathematical Society

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3.1. Flags and flag structures 3.2. Canonical flags of associatives, and orientations 4. An index 1 singularity of associative 3-folds 4.1. A family of SL 3-folds in C3 4.2. Desingularizing immersed associative 3-folds 5. Another index 1 associative singularity 5.1. Three families of SL 3-folds in C3 5.2. Associative 3-folds with singularities modelled on L0 5.3. Algebraic topology of desingularizations using Las 6. U(1)-invariant associative 3-folds in R7 6.1. Associative 3-folds and J-holomorphic curves 6.2. Associative 3-folds with boundary in coassociatives 7. A superpotential counting associative 3-folds 7.1. Set up of situation and notation 7.2. Six kinds of wall-crossing behaviour 7.3. Definition of the superpotential 7.4. How Φψ depends on choices, and on ψ 7.5. Our main conjecture 7.6. G2 quantum cohomology 7.7. Generalizations 8. Remarks on counting G2 -instantons 8.1. The Donaldson–Segal programme 8.2. Canonical orientations for moduli of G2 -instantons 8.3. P  -flags, and canonical P  -flags 8.4. Problems with counting G2 -instantons 8.5. A suggestion for how to modify Donaldson–Segal References

1. Introduction Let (Y, ω) be a compact symplectic manifold. Several areas of Symplectic Geometry — Gromov–Witten invariants [19, 25, 66], Quantum Cohomology [66], Lagrangian Floer cohomology [17, 18], Fukaya categories [73], and so on — involve choosing an almost complex structure J on Y compatible with ω, ‘counting’ moduli spaces M of J-holomorphic curves in Y satisfying some conditions, and using the ‘numbers’ [M]virt and homological algebra to define the theory. A remarkable feature of these theories is that although the family J of possible choices of J is infinite-dimensional, and two J1 , J2 in J may be very far apart, nonetheless the theory is independent of choice of J (up to a suitable notion of equivalence), and so depends only on (Y, ω). These areas are related to String Theory, and are driven by conjectures made by physicists. Oversimplifying rather, String Theorists tell us that if (Y, J, g, ω) is a Calabi–Yau 3-fold, then the String Theory of Y (a huge structure) has a ‘topological twisting’, the ‘A model’, a smaller and simpler theory. The A model depends only on the symplectic manifold (Y, ω), not on the other geometric structures J, g, Ω, and encodes data including the Gromov–Witten invariants, Quantum Cohomology, and Fukaya category of (Y, ω).

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We wish to explore the possibility that an analogue of these ideas may work for compact G2 -manifolds. As in §2, if (X, g) is a Riemannian 7-manifold with holonomy group G2 then X has a natural closed 3-form ϕ and Hodge dual closed 4-form ∗ϕ, in a local normal form that we call ‘positive’ 3- and 4-forms. There are two classes of special submanifolds in X, ‘associative 3-folds’ N ⊂ X calibrated by ϕ, and ‘coassociative 4-folds’ C ⊂ X calibrated by ∗ϕ. There is a well known analogy: Calabi–Yau 3-folds (Y, J, h) (1.1) J-holomorphic curves in Y

↔ Torsion-free G2 -manifolds (X, ϕ, ∗ϕ) ↔ associative 3-folds in X

(Special) Lagrangian 3-folds in Y ↔ coassociative 4-folds in X. Torsion-free G2 -manifolds (X, ϕ, ∗ϕ) are a rather restrictive class. Following Donaldson and Segal [15, §3–§4], we will work with the much larger class of tamed almost-G2 -manifolds, or TA-G2 -manifolds, (X, ϕ, ψ), which have a closed G2 3-form ϕ and a compatible closed G2 4-form ψ on X, but need not have ψ = ∗ϕ. We call ϕ, ψ good if they extend to a TA-G2 -manifold (X, ϕ, ψ). Then we can extend the analogy (1.1), adding the lines: (1.2)

Symplectic form ω on Y

↔ Good 3-form ϕ on X

(Almost) complex structure J on Y

↔ Good 4-form ψ on X

Symplectic manifold (Y, ω) with compatible almost complex structure J

↔ TA-G2 -manifold (X, ϕ, ψ).

Here we compare ϕ with ω and ψ with J because the notion of associative 3-fold N in (X, ϕ, ψ) depends only on X, ψ, not on ϕ, but N has volume [ϕ] · [N ] 3 (X; R) and [N ] ∈ H3 (X; Z). Following analogy (1.1)–(1.2), and being for [ϕ] ∈ HdR very optimistic, one might hope to construct: (a) Gromov–Witten type invariants GWψ,α ∈ Q counting associative 3-folds N in a TA-G2 -manifold (X, ϕ, ψ) in homology class [N ] = α ∈ H3 (X; Z). (b) A ‘quantum cohomology algebra’ QH ∗ (X; Λ0 ) for a TA-G2 -manifold (X, ϕ, ψ), defined by modifying usual cohomology H ∗ (X; Λ0 ) by terms involving counting associative 3-folds in X passing through given cycles. (c) ‘Floer cohomology groups’ or ‘Fukaya categories’ for coassociative 4-folds C in (X, ϕ, ψ), defined by counting associative 3-folds N in X with boundary ∂N ⊂ C, as discussed by Leung, Wang and Zhu [59, 60]. We particularly want anything we define to be unchanged by continuous de3 (X; R), as this formations of (ϕ, ψ) which fix the cohomology class [ϕ] = γ in HdR is our analogue of symplectic theories being independent of choice of almost complex structure J, and is our criterion for having found an interesting, ‘topological’ theory, in the style of invariant theories in Symplectic Geometry. Our conjectural answers to these are: (a) We outline how to define numbers GWψ,α ∈ Q ‘counting’ associative Q-homology 3-spheres N in (X, ϕ, ψ) with [N ] = α ∈ H3 (X; Z) and ψ generic. These GWψ,α depend on arbitrary choices, and are not invariant 3 (X; R). under deformations of (ϕ, ψ) fixing [ϕ] ∈ HdR However, we expect the family of GWψ,α for all α ∈ H3 (X; Z) to have some interesting deformation-invariant features, as in Conjecture 1.1. In particular, the GWψ,α should be combined in a superpotential Φψ : U →

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Λ>0 as in (1.3) which is independent of choices up to reparametrization by a class of automorphisms of the base U. (b) If this superpotential Φψ has a critical point θ ∈ U, we explain how to define ‘G2 quantum cohomology’ QHθ∗ (X; Λ0 ), a supercommutative algebra over the Novikov ring Λ0 , which is a deformation of H ∗ (X; Λ0 ), expected to be deformation-invariant up to isomorphism. If a critical point θ exists, we say that (X, ϕ, ψ) is unobstructed. This is a condition similar to Lagrangian Floer cohomology of a Lagrangian being unobstructed in Fukaya, Oh, Ohta and Ono [17, 18]. (c) We expect that it is not possible to construct a deformation-invariant version of Lagrangian Floer theory for coassociatives C in X, based on counting associatives N in X with ∂N ⊂ C, for reasons explained in §6.2. The next conjecture explains (a) in more detail. We need the following notation. Let F be the field Q, R or C. Write Λ for the Novikov ring over F: , + ∞ αi : ci ∈ F, αi ∈ R, αi → ∞ as i → ∞ , Λ= i=1 ci q with q a formal variable. Then Λ is a commutative F-algebra. Define v : Λ → R # {∞} by v(λ) is the least α ∈ R with the coefficient of q α in λ nonzero for λ ∈ Λ \ {0}, and v(0) = ∞. Write Λ0 ⊂ Λ for the subalgebra of λ ∈ Λ with v(λ)  0, and Λ>0 ⊂ Λ0 for the ideal of λ ∈ Λ with v(λ) > 0. Then Λ is a complete non-Archimedean field in the sense of Bosch, G¨ untzer and Remmert [10, §A], with valuation λ = 2−v(λ) , so we can consider rigid analytic spaces over Λ as in [10, §C]. These are like schemes over Λ, except that polynomial functions on schemes are replaced by convergent power series. Conjecture 1.1 (see Conjecture 7.4). Let X be a compact, oriented 7-manifold. Consider 1 + Λ>0 ⊂ Λ as a group under multiplication in Λ. Write . U = Hom H3 (X; Z), 1 + Λ>0 for the set of group morphisms θ : H3 (X; Z) → 1 + Λ>0 . By choosing a basis e1 , . . . , en for H3 (X; Z)/torsion, where n = b3 (X), we can identify U ∼ = Λn>0 by ∼ θ = (λ1 , · · · , λn ) if θ(ei ) = 1 + λi for i = 1, . . . , n, where Λ>0 is the open unit ball in Λ in the norm  . . We regard U as a smooth rigid analytic space over Λ, as in Bosch, G¨ untzer and Remmert [10]. 3 (X; R), and write F γ for the set of closed 4-forms ψ on X such Let γ ∈ HdR 3 (X; R), for which that there exists a closed 3-form ϕ on X with [ϕ] = γ ∈ HdR (X, ϕ, ψ) is a TA-G2 -manifold, with the given orientation on X. Let ψ ∈ F γ be generic. Then we can define a superpotential Φψ : U → Λ>0 , of the form  GWψ,α q γ·α θ(α), (1.3) Φψ (θ) = α∈H3 (X;Z):γ·α>0

where GWψ,α ∈ Q is a weighted count of associative Q-homology 3-spheres in (X, ϕ, ψ) with homology class α. The GWψ,α are not independent of choices, and are not invariant under deformations of ψ in F γ . So they are not enumerative invariants in the usual sense. Nonetheless, the whole superpotential Φψ does have the following invariance property. If ψ0 , ψ1 are generic elements of the same connected component of F γ (we allow ψ0 = ψ1 ), and Φψ0 , Φψ1 are any choices for the superpotentials for ψ0 , ψ1 , then there is a quasi-identity morphism Υ : U → U , a special kind of isomorphism of rigid analytic spaces defined in §7.1, with Φψ1 = Φψ0 ◦ Υ.

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Here we work over the Novikov ring Λ>0 , as in [17, 18], as our theory involves infinite sums such as (1.3), but we do not know these sums converge in the usual sense, so we have to use formal power series. If we knew all our formal sums converged, we could work over R or C instead, with q ∈ R, C small. Conjecture 1.1 implies that any information which depends on Φψ only up to reparametrizations by quasi-identity morphisms Υ : U → U is deformationinvariant. For example, the least A > 0 such that GWψ,α = 0 for α ∈ H3 (X; Z) with γ · α = A should be deformation-invariant, and the values of GWψ,α for all α ∈ H3 (X; Z) with γ · α = A should also be deformation-invariant. Section 7.6 outlines how to define a ‘G2 quantum cohomology algebra’ QHθ∗ (X; Λ0 ) depending on a critical point θ of Φψ in U , which should be deformation-invariant. The message of this paper is both positive and negative. On the positive side, there is (the author believes) some nontrivial deformation-invariant information from counting associatives. On the negative side, not that much information survives – much less than for J-holomorphic curves in Symplectic Geometry – and conjectures more optimistic than Conjecture 1.1 are likely to be false. The reasoning behind Conjecture 1.1 is as follows. Let (X, ϕt , ψt ), t ∈ [0, 1] be a smooth 1-parameter family of TA-G2 -manifolds. Then as in §2.7 we can form moduli spaces M(N , α, ψt ) of compact associative 3-folds N in (X, ϕt , ψt ) of diffeomorphism type N and homology class [N ] = α ∈ H3 (X; Z). To define enumerative invariants for associative 3-folds which are the same for (X, ϕ0 , ψ0 ) and (X, ϕ1 , ψ1 ), we need to understand how the moduli spaces M(N , α, ψt ) can change as t increases through [0, 1]. The typical reason why moduli spaces change is that for some t0 ∈ (0, 1) there exists a family Nt for t ∈ [0, t0 ], where Nt for t ∈ [0, t0 ) is a compact associative 3-fold in (X, ϕt , ψt ) in homology class α depending smoothly on t, and Nt0 = limt→t0− Nt is a singular associative 3-fold, and no Nt for t ∈ (t0 , 1] exist, so that a point in M(N , α, ψt ) disappears as t crosses t0 in [0, 1]. Let us suppose that (X, ϕt , ψt ), t ∈ [0, 1] is a generic 1-parameter family. Then the singularities of Nt0 are not arbitrary. To each singularity type S of associative 3folds we can assign an index ind S, which is the codimension in which singularities of type S occur in families of associative 3-folds over generic families of G2 -structures. In our problem Nt0 can only have index 1, so it is enough for us to understand index 1 singularities of associative 3-folds. Sections 4–5 and 7.2 describe several kinds of index 1 singularity of associative 3-folds. These are the only kinds the author knows, and may perhaps be the only kinds there are. They all definitely change the number of associative 3-folds, and so mean that na¨ıve counts of associative 3-folds cannot be deformation-invariant. In §7 we assume that moduli spaces of compact associatives in (X, ϕ, ψ) have good compactness, smoothness, and orientation properties, and that their only boundary behaviour comes from the six kinds of index 1 singularity described in §7.2. Under these very strong assumptions, we explain how by counting associative 3-folds in cunning ways, we can still extract deformation-invariant information from the numbers of associative 3-folds as in Conjecture 1.1, as we arrange that the changes under index 1 singularities cancel out. As in [15], G2 -instantons on a TA-G2 -manifold (X, ϕ, ψ) are connections A on principal G-bundles P → X whose curvature FA satisfies FA ∧ ψ = 0. In our

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analogy (1.1)–(1.2), we can add the line: Hermitian–Yang–Mills vector bundles on Y ↔ G2 -instantons on (X, ϕ, ∗ϕ). Donaldson and Segal [15, §6.2] proposed a programme to define invariants counting G2 -instantons, which would hopefully be unchanged under deformations of (ϕ, ψ), and would be analogues of Donaldson–Thomas invariants of Calabi–Yau 3-folds [47, 54]. It is currently under investigation by Menet, Nordstr¨ om, S´a Earp, Walpuski, and others [68, 71, 72, 77–80]. As in [15, §6.2], to define invariants of (X, ϕ, ψ) unchanged under deformations of ψ will require the inclusion of ‘compensation terms’ counting solutions of some equation on associative 3-folds N in X, to compensate for bubbling of G2 -instantons on associative 3-folds. Section 8 discusses several aspects of this programme. We make a proposal for how to define canonical orientations for G2 -instanton moduli spaces, based on the ideas in §3 on orienting associative moduli spaces. And we argue that counting G2 -instantons on (X, ϕ, ψ) in a deformation-invariant way should only be possible if counting associative 3-folds in (X, ϕ, ψ) is ‘unobstructed’ – the superpotential Φψ has a critical point θ, as in (b) above – and we choose some such θ, similar to choosing a ‘bounding cochain’ for a Lagrangian in the Lagrangian Floer theory of Fukaya, Oh, Ohta and Ono [17, 18]. On the relation with String Theory and M-theory, we can ask: Question 1.2. Is there some good notion of ‘topological twisting’ for M-theory or String Theory on TA-G2 -manifolds (X, ϕ, ψ), which includes the superpotential Φψ , and G2 quantum cohomology QHθ∗ (X; Λ0 ), and modified Donaldson–Segal invariants, proposed above? See de Boer et al. [5–7] for a discussion of topological G2 -strings. Superpotentials Φ counting associative 3-folds similar to those in Conjecture 1.1 were discussed in M-theory by Acharya [1, 2] and Harvey and Moore [23]. Throughout §2–§7 we state conjectures on how the author expects the mathematics to work. These are not of uniform difficulty. For some of them, the author or one of his friends could easily write down a proof, if we were not too busy writing grant proposals. However, our main conjecture includes some aspects which are seriously difficult, and the author has no idea how to prove: • Implicit in Conjecture 1.1 is the idea that the only index one singularities of associative 3-folds (i.e. the only singularities that can occur in associatives in generic 1-parameter families of TA-G2 -manifolds (X, ϕt , ψt ), t ∈ [0, 1]) are those described in §7.2. This is difficult because it requires some measure of control over all possible singularities of associative 3-folds, as described using Geometric Measure Theory, for instance. • A proper understanding of the multiple cover phenomena for associatives in §7.2(F) also looks rather difficult, but is essential for Conjecture 1.1. We emphasize that this paper is very speculative, and little in it is actually proved. There are a few bits which are both new and more-or-less rigorous, in particular, some ideas on TA-G2 -manifolds in §2.5, and on canonical flags, flag structures, and orientations for associative moduli spaces M(N , α, ψ) in §3. This paper is similar to the author’s paper [30], which made conjectures on invariants counting special Lagrangian 3-folds in Calabi–Yau 3-folds. Acknowledgements. This research was partly funded by a Simons Collaboration Grant on ‘Special Holonomy in Geometry, Analysis and Physics’. I would like

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to thank Bobby Acharya, Robert Bryant, Alexsander Doan, Simon Donaldson, Mark Haskins, Andriy Haydys, Johannes Nordstr¨ om, Matthias Ohst, and Thomas Walpuski for helpful conversations, and a referee for useful comments. 2. Geometry of G2 -manifolds We begin by introducing G2 -manifolds and associative and coassociative submanifolds. Some references for §2.1–§2.3 are the author’s books [29, 46]. Parts of §2.5–§2.6 on TA-G2 -manifolds and on associative moduli spaces are new. 2.1. G2 -manifolds. Let (X, g) be a connected Riemannian n-manifold, and fix a basepoint x ∈ X. The holonomy group Hol(g) of g is the group of isometries of Tx X generated by parallel transport around smooth loops γ : [0, 1] → X with γ(0) = γ(1) = x. We consider Hol(g) to be a (Lie) subgroup of O(n), defined up to conjugation by elements of O(n). Then Hol(g) is independent of the choice of base point x. The possible holonomy groups were classified by Berger [4] in 1955. If X is simply-connected and g is irreducible and nonsymmetric, then Hol(g) is one of SO(n),

U(m), SU(m) (n = 2m, m  2),

Sp(m), Sp(m) Sp(1) (n = 4m, m  2),

G2 (n = 7),

or

Spin(7) (n = 8).

We are concerned with the exceptional holonomy group G2 in 7 dimensions. In 1987, Bryant [11] first used the theory of exterior differential systems to show that locally there exist many metrics with holonomy G2 . In 1989, Bryant and Salamon [12] found explicit examples of complete metrics with holonomy G2 on noncompact manifolds. Constructions of compact 7-manifolds with holonomy G2 were given by the author [27–29] in 1993 and 2000, by Kovalev [55] in 2000, and by Corti, Haskins, Nordstr¨om and Pacini [13] in 2012. Let (x1 , . . . , x7 ) be coordinates on R7 . Write dxij...l for the exterior form dxi ∧ dxj ∧ · · · ∧ dxl on R7 . Define a 3-form ϕ0 on R7 by (2.1)

ϕ0 = dx123 + dx145 + dx167 + dx246 − dx257 − dx347 − dx356 .

The subgroup of GL(7, R) preserving ϕ0 is the exceptional Lie group G2 . It is compact, connected, simply-connected, semisimple and 14-dimensional, and it also preserves the Hodge dual 4-form (2.2)

∗ϕ0 = dx4567 + dx2367 + dx2345 + dx1357 − dx1346 − dx1256 − dx1247 ,

the Euclidean metric g0 = dx21 +· · ·+dx27 , and the orientation on R7 . The subgroup of GL(7, R) preserving ∗ϕ0 is {±1} × G2 , but the subgroup preserving ∗ϕ0 and the orientation on R7 is G2 . Let X be a 7-manifold, and ϕ ∈ Γ∞ (Λ3 T ∗ X) a smooth 3-form on X. We call ϕ positive if for each x ∈ X there exists an isomorphism Tx X ∼ = R7 identifying ϕ|x with ϕ0 in (2.1). This is an open condition on ϕ. If ϕ is positive then the set of isomorphisms Tx X ∼ = R7 identifying ϕ|x ∼ = ϕ0 for all x ∈ X is a principal subbundle Pϕ of the frame bundle F → X of X with structure group G2 . That is, Pϕ is a G2 -structure on X. This gives a 1-1 correspondence between positive 3-forms and G2 -structures on a 7-manifold X. Similarly, we call a 4-form ψ ∈ Γ∞ (Λ4 T ∗ X) positive if for each x ∈ X there exists an isomorphism Tx X ∼ = R7 identifying ψ|x with ∗ϕ0 in (2.2). If we fix an orientation on X, the set of oriented isomorphisms Tx X ∼ = R7 identifying ψ|x ∼ = ∗ϕ0

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for all x ∈ X is a G2 -structure Pψ on X. This gives a 1-1 correspondence between positive 4-forms and G2 -structures on an oriented 7-manifold X. A G2 -manifold is a 7-manifold X with a G2 -structure P . As above P corresponds to positive 3- and 4-forms ϕ, ∗ϕ, and by an abuse of notation we call (X, ϕ, ∗ϕ) a G2 -manifold. A G2 -manifold (X, ϕ, ∗ϕ) has an associated Riemannian metric g and orientation. Proposition 2.1. Let (X, ϕ, ∗ϕ) be a G2 -manifold, with associated metric g. Then the following are equivalent: (i) Hol(g) ⊆ G2 , and ϕ is the induced 3-form, (ii) ∇ϕ = 0 on X, where ∇ is the Levi-Civita connection of g, and (iii) dϕ = d(∗ϕ) = 0 on X. We call ∇ϕ the torsion of the G2 -structure Pϕ . If ∇ϕ = 0 then (X, ϕ, ∗ϕ) is called torsion-free. If g has holonomy Hol(g) ⊆ G2 , then g is Ricci-flat. Theorem 2.2. Let (X, g) be a compact Riemannian 7-manifold with Hol(g) ⊆ G2 . Then Hol(g) = G2 if and only if π1 (X) is finite. In this case the moduli space of metrics with holonomy G2 on X, up to diffeomorphisms isotopic to the identity, is a smooth manifold of dimension b3 (X). 2.2. Calabi–Yau 3-folds and G2 -manifolds. Let (Y, J) be a compact complex 3-manifold admitting K¨ ahler metrics, with trivial canonical bundle KY ∼ = OY . Yau’s proof of the Calabi Conjecture implies that each K¨ahler class on Y contains a unique Ricci-flat K¨ahler metric h. Then h has holonomy group Hol(h) ⊆ SU(3). We call (Y, J, h) a Calabi–Yau 3-fold. The Levi-Civita connection ∇ of h preserves J, h, the K¨ahler form ω of h, and a holomorphic volume form Ω in H 0 (KY ), which we can scale to have length |Ω| = 23/2 . Then at each point y ∈ Y , there is an isomorphism of complex vector spaces Ty Y ∼ = C3 identifying h|y , ω|y , Ω|y with h0 , ω0 , Ω0 , where h0 = |dz1 |2 + |dz2 |2 + |dz3 |2 , (2.3)

and

ω0 = 2i (dz1 ∧ d¯ z1 + dz2 ∧ d¯ z2 + dz3 ∧ d¯ z3 ), Ω0 = dz1 ∧ dz2 ∧ dz3 ,

with (z1 , z2 , z3 ) the complex coordinates on C3 . Calabi–Yau 3-folds and G2 -manifolds are connected in the following way. Identify R7 ∼ = R×C3 by (x1 , . . . , x7 ) ∼ = (x1 , x2 +ix3 , x4 +ix5 , x6 +ix7 ). Then g0 , ϕ0 , ∗ϕ0 in §2.1 are related to h0 , ω0 , Ω0 in (2.3) by (2.4)

g0 = dx21 + h0 , ϕ0 = dx1 ∧ ω0 + Re Ω0 , ∗ϕ0 = 12 ω0 ∧ ω0 − dx1 ∧ Im Ω0 .

Therefore, if (Y, J, h) is a Calabi–Yau 3-fold with K¨ ahler form ω and holomorphic volume form Ω, if we define X = R × Y or X = S 1 × Y , with x the coordinate on R or S 1 = R/Z, and set g = dx2 + h, ϕ = dx ∧ ω + Re Ω, ∗ϕ = 12 ω ∧ ω − dx ∧ Im Ω, then (X, ϕ, ∗ϕ) is a torsion-free G2 -manifold with metric g. There is a strong analogy between torsion-free G2 -manifolds and Calabi–Yau 3-folds. 2.3. Calibrated submanifolds. The next definition is due to Harvey and Lawson [22].

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Definition 2.3. Let (X, g) be a Riemannian manifold, and ϕ a closed k-form on X. We call ϕ a/ calibration if for every x ∈ X and k-dimensional subspace / V ⊆ Tx X we have /ϕ|V /  1. If ϕ is a calibration, we say that an oriented kdimensional submanifold N in X is calibrated with respect to ϕ if ϕ|Tx N = volTx N for all x ∈ N . Calibrated submanifolds are naturally oriented, and a compact calibrated submanifold N is volume-minimizing in its homology class, with volume [ϕ] · [N ], so calibrated submanifolds are minimal submanifolds. Calibrated geometry is a natural companion to the theory of holonomy groups. If (X, g) is a Riemannian manifold with special holonomy H ⊂ O(n), it will have natural constant k-forms ϕ corresponding to H-invariant k-forms in Λk (Rn )∗ , and if ϕ is rescaled appropriately it is a calibration. Thus, manifolds with special holonomy have interesting special classes of minimal submanifolds. Let (X, ϕ, ∗ϕ) be a torsion-free G2 -manifold, with metric g and 4-form ∗ϕ. Then as in Harvey and Lawson [22, §IV]: (a) ϕ is a calibration on (X, g), and its calibrated submanifolds are called associative 3-folds. (b) ∗ϕ is a calibration on (X, g), and its calibrated submanifolds are called coassociative 4-folds. If C is a 4-dimensional submanifold of X then C is coassociative (with some unique orientation) if and only if ϕ|C = 0. Examples of compact associative 3-folds and coassociative 4-folds in compact 7manifolds with holonomy G2 can be found in the author [29, §12.6]. Similarly, there are three kinds of calibrated submanifolds in a Calabi–Yau 3-fold (Y, J, h) with K¨ ahler form ω and holomorphic volume form Ω: (A) J-holomorphic curves, that is, 2-submanifolds Σ ⊂ Y calibrated w.r.t. ω. (B) Special Lagrangian 3-folds, or SL 3-folds, with phase eiθ , for θ ∈ R, that is, 3-submanifolds L ⊂ Y calibrated w.r.t. cos θ Re Ω + sin θ Im Ω. In particular, SL 3-folds with phase 1 are calibrated w.r.t. Re Ω, and SL 3-folds with phase i are calibrated w.r.t. Im Ω. When we do not specify a phase, we mean phase 1. (C) Complex surfaces, that is, 4-submanifolds S ⊂ Y calibrated w.r.t. 12 ω ∧ ω. Remark 2.4. From (2.4), we deduce the following relation between calibrated submanifolds in a Calabi–Yau 3-fold Y (or in Y = C3 ), and calibrated submanifolds in the G2 -manifold R × Y (or in R7 = R × C3 ): (i) If Σ is a J-holomorphic curve in Y then R×Σ is associative 3-fold in R×Y . (ii) If L is an SL 3-fold in Y with phase 1 then {x} × L is an associative 3-fold in R × Y for each x ∈ R. (iii) If L is an SL 3-fold in Y with phase i then R × L is a coassociative 4-fold in R × Y . (iv) If S is a complex surface in Y then {x} × S is a coassociative 4-fold in R × Y for each x ∈ R. This will be important to us because a great deal is known about examples and properties of singularities of SL 3-folds, as in [29–46], and from Remark 2.4(ii) we can deduce many examples of singularities of associative 3-folds. Examples of singular associative 3-folds in R7 which do not come from special Lagrangians in C3 can be found in Lotay [63–65].

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2.4. G2 -instantons. Let (X, ϕ, ∗ϕ) be a compact, torsion-free G2 -manifold. As in [29, §10.3], the 2-forms Λ2 T ∗ X on X have a natural splitting Λ2 T ∗ X = Λ27 ⊕Λ214 , where Λ27 , Λ214 are vector subbundles of Λ2 T ∗ X with ranks 7,14, and Λ214 is the kernel of the vector bundle morphism Λ2 T ∗ X → Λ6 T ∗ X mapping α → α∧(∗ϕ). Let G be a compact Lie group (we usually take G = SU(2)), π : P → X a principal G-bundle, and A a connection on P , with curvature FA . Following Donaldson and Segal [15], we call (P, A) a G2 -instanton, with structure group G, if the component of FA in ad(P ) ⊗ Λ27 is zero, or equivalently, if FA ∧ (∗ϕ) = 0. Write M(P, ∗ϕ) for the moduli space of gauge equivalence classes [A] of G2 instanton connections A on P . The deformation theory of A, and hence the local description of M(P, ∗ϕ) near [A], is controlled by the elliptic complex 0

/ Γ∞ (ad P )

0o

Γ (ad P ⊗ Λ T X) o

dA

(2.5)

/ Γ∞ (ad P ⊗ T ∗ X) (−∧∗ϕ)◦dA

∞

7

∗

dA

 Γ (ad P ⊗ Λ6 T ∗ X). ∞

Here infinitesimal gauge transformations live in Γ∞ (ad P ), infinitesimal changes δA to A live in Γ∞ (ad P ⊗ T ∗ X), and FA+δA ∧ ∗ϕ lives in Γ∞ (ad P ⊗ Λ6 T ∗ X). For any connection A on P , as dA FA = 0 and d(∗ϕ) = 0 we have dA (FA ∧ ∗ϕ) = 0, and the linearization of this equation at A, δA lies in Γ∞ (ad P ⊗ Λ7 T ∗ X). Because the deformation theory of G2 -instantons comes from an elliptic complex (2.5), which has index 0, the moduli spaces M(P, ∗ϕ) are well behaved, in the same way that moduli spaces of associative 3-folds in §2.6 are well behaved: except at points [A] with nontrivial stabilizer groups, M(P, ∗ϕ) should be a derived manifold of virtual dimension 0 in the sense of [8, 9, 48–53, 74], and if ∗ϕ is suitably generic then M(P, ∗ϕ) should be a manifold of dimension 0. There is a topological formula for the L2 -norm FA L2 of the curvature of a G2 -instanton. When G = SU(2) this is (2.6)

FA 2L2 = −4π 2 ([ϕ] ∪ c2 (P )) · [X],

where c2 (P ) is the second Chern class of P . We will discuss G2 -instantons and the Donaldson–Segal programme [15] further in §8. 2.5. Tamed almost-G2 -manifolds. So far we have focused on torsion-free G2 -manifolds (X, ϕ, ∗ϕ), with dϕ = d(∗ϕ) = 0. But for our purposes, these are too restrictive, for two reasons: • We want to discuss structures invariant under deformations of ϕ, ∗ϕ. On a compact 7-manifold X, torsion-free G2 -structures (ϕ, ∗ϕ) come in finite-dimensional families as in Theorem 2.2, so deformation-invariance amongst torsion-free G2 -structures is not a powerful statement. 3 (X; R), Even worse, we will want to fix the cohomology class [ϕ] ∈ HdR and then there are no torsion-free deformations at all. • We hope that choosing (ϕ, ∗ϕ) generic will simplify the problem (e.g. ensure all associative 3-folds N ⊂ X are unobstructed). But this is only plausible if we choose (ϕ, ∗ϕ) from an infinite-dimensional family. The obvious answer is to relax the condition dϕ = 0 or d(∗ϕ) = 0 on (X, ϕ, ∗ϕ), but there would be a cost to this, as the next remark explains.

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Remark 2.5. Here are the important consequences of allowing dϕ = 0 or d(∗ϕ) = 0 for the theories of associative 3-folds and coassociative 4-folds discussed in §2.3, and for G2 -instantons in §2.4: (a) If dϕ = 0 then a compact associative 3-fold N ⊂ X has volume given by 3 (X; R) and [N ] ∈ H3 (X; Z): the topological formula, for [ϕ] ∈ HdR (2.7)

vol(N ) = [ϕ] · [N ]. If dϕ = 0 then [ϕ] no longer makes sense. This matters to us for two reasons. Firstly, if dϕ = 0 then in a moduli space M(N , α, ∗ϕ) of associative 3-folds N in class α ∈ H3 (X; Z), we might have a sequence [Ni ]∞ i=1 in M(N , α, ∗ϕ) with vol(Ni ) → ∞ as i → ∞, and then there could be no limit point limi→∞ [Ni ] in M(N , α, ∗ϕ). Thus, the lack of a volume bound may cause moduli spaces M(N , α, ∗ϕ) to be noncompact (though they could also be noncompact for other reasons). Secondly, as in (1.3) we hope to combine invariants GWψ,α counting associatives N in a formal power series weighted by q vol(N ) = q γ·α , and this is only sensible with a topological formula for vol(N ). (b) McLean’s moduli theory for compact associative 3-folds N in §2.6 works fine if dϕ = 0 = d(∗ϕ). However, the linear elliptic operator D : Γ∞ (ν) → Γ∞ (ν) need only be self-adjoint if d(∗ϕ) = 0. As in Remark 3.15 below, we need D to be self-adjoint for the ‘canonical flag’ of N defined in §3 to be well behaved, and this is important for our proposal in Conjecture 1.1. (c) As in (a), if d(∗ϕ) = 0 then as in (2.7) a compact coassociative 4-fold C ⊂ X has volume given by the topological formula

(2.8)

vol(C) = [∗ϕ] · [C]. If d(∗ϕ) = 0 then [∗ϕ] no longer makes sense, and the lack of a volume bound could cause moduli spaces of coassociatives to become noncompact. (d) McLean’s moduli theory for compact coassociative 4-folds C in §2.6 relies on the alternative definition that C is coassociative if ϕ|C = 0. If dϕ = 0 then the deformation theory of C is no longer part of an elliptic complex, so coassociatives will not form well behaved moduli spaces. (e) If dϕ = 0 then as in (a),(c) a G2 -instanton (P, A) has a topological formula (2.6) for the L2 -norm of its curvature. This may be important in proving compactness of moduli spaces M(P, ∗ϕ). (f) Moduli theory for G2 -instantons A uses FA ∧ (∗ϕ) = 0. If d(∗ϕ) = 0 then the deformation theory of A is no longer part of an elliptic complex (2.5), so as in (d), G2 -instantons will not form well behaved moduli spaces.

Therefore we do not want to sacrifice either condition dϕ = 0 or d(∗ϕ) = 0. Instead we will do something more complicated: we will work with a version of the ‘tamed almost-G2 -manifolds’ introduced by Donaldson and Segal [15, §3–§4], for the same reasons as us. Our treatment using (i)–(iii) is new. Definition 2.6. A tamed almost-G2 -manifold or TA-G2 -manifold (X, ϕ, ψ) is a 7-manifold X equipped with a closed positive 3-form ϕ and a closed positive 4-form ψ satisfying a compatibility condition. As in §2.2, ϕ corresponds to a G2 structure Pϕ on X, and this induces an orientation on X. Using this orientation, ψ corresponds to a G2 -structure Pψ on X. Write gϕ , gψ for the metrics induced by Pϕ , Pψ . We require that the following equivalent conditions should hold:

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(i) For all x ∈ X and all oriented 3-planes V ⊂ Tx X which are associative with respect to the G2 -structure Pψ , we have ϕ|V > 0. (ii) For all x ∈ X and all oriented 4-planes W ⊂ Tx X which are coassociative with respect to the G2 -structure Pϕ , we have ψ|W > 0. (iii) There do not exist x ∈ X, an oriented 3-plane V ⊂ Tx X which is associative with respect to Pψ , and an oriented 4-plane W ⊂ Tx X which is coassociative with respect to Pϕ , such that V ⊂ W ⊂ Tx X. To show that (i)–(iii) are equivalent, suppose (iii) does not hold, so there exist V ⊂ W ⊂ Tx X as in (iii). Then ϕ|W = 0 as W is ϕ-coassociative, so ϕ|V = 0, and (i) does not hold. Also V ⊂ W with V ψ-associative and W a 4-plane imply that ψ|W = 0, so (ii) does not hold. Hence (i),(ii) both imply (iii). Suppose (i) does not hold. Then by connectedness, either (a) ϕ|V < 0 for all x ∈ X and ψ-associative V ⊂ Tx X, or (b) there exist x ∈ X and ψ-associative V ⊂ Tx X with ϕ|V = 0. But for (a), by reversing the orientation used to define Pψ we would get ϕ|V > 0 for all x, V , so that (i) holds after all. In fact (a) is impossible, as we chose Pϕ , Pψ to have the same orientation. Thus there exists a ψ-associative V ⊂ Tx X with ϕ|V = 0. By [22, Th. IV.4.6] there is a unique ϕ-coassociative W ⊂ Tx X with V ⊂ W , so (iii) does not hold. Thus (iii) implies (i). A similar argument shows that (iii) implies (ii), so (i)–(iii) are equivalent. Observe that if X is compact, then (i),(ii) are equivalent to: (i) There exists a constant K > 0 such that for all x ∈ X and all oriented 3-planes V ⊂ Tx X which are associative with respect to Pψ , we have g g volVψ  Kϕx |V , where volVψ ∈ Λ3 V ∗ is the volume form defined using the metric gψ |x on Tx X and the orientation on V . (ii) There exists a constant K  > 0 such that for all x ∈ X and all oriented 4-planes W ⊂ Tx X which are coassociative with respect to Pϕ , we have g g volWϕ  K  ψx |W , where volWϕ ∈ Λ4 W ∗ is the volume form defined using the metric gϕ |x on Tx X and the orientation on W . Note that we can have Pϕ = Pψ , and Pϕ = Pψ if and only if (X, ϕ, ψ) is a torsion-free G2 -manifold (X, ϕ, ∗ϕ). For (X, ϕ, ψ) to be a TA-G2 -manifold is an open condition on pairs (ϕ, ψ) of a closed 3-form ϕ and a closed 4-form ψ on X. Thus the family of TA-G2 -structures on X is infinite-dimensional, if it is nonempty. Following [15], we extend the definitions of associative 3-folds, coassociative 4-folds and G2 -instantons to TA-G2 -manifolds: Definition 2.7. Let (X, ϕ, ψ) be a TA-G2 -manifold. Then: (i) An associative 3-fold N ⊂ X is a 3-submanifold N in X which is associative with respect to the G2 -structure Pψ . (ii) A coassociative 3-fold C ⊂ X is a 4-submanifold C in X which is associative with respect to the G2 -structure Pϕ . (iii) A G2 -instanton (P, A) on X, with structure group G for G a compact Lie group, is a principal G-bundle π : P → X and a connection A on P whose curvature FA satisfies FA ∧ ψ = 0. All the issues in Remark 2.5(a)–(f) work out nicely with these definitions. For (a), if (X, ϕ, ψ) is a compact TA-G2 -manifold, so that Definition 2.6(i) holds for some K > 0, and N ⊂ X is a compact associative 3-fold, then for each x ∈ N

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g

we have volTψx N  Aϕx |V , so integrating over N yields a topological volume bound generalizing (2.7): volgψ (N )  K[ϕ] · [N ].

(2.9)

For (b), as dψ = 0 the elliptic operator D in §2.6 is self-adjoint. For (c), if X is compact and C ⊂ X is compact coassociative then as for (2.9) we get a topological volume bound generalizing (2.8), for K  > 0 as in Definition 2.6(ii) : volgϕ (C)  K  [ψ] · [C]. For (d), as dϕ = 0, moduli spaces of coassociatives are well behaved. For (e), if (P, A) is a G2 -instanton with group G = SU(2), as in [15] we can show that FA 2L2  −K  ([ϕ] ∪ c2 (P )) · [X], generalizing (2.6), for K  > 0 depending on (X, ϕ, ψ) similar to K in Definition 2.6(i) . For (f), as dψ = 0, moduli spaces of G2 -instantons are well behaved. Proposition 2.8. (a) Let X be a compact oriented 7-manifold and ψ a closed + ∞ := ϕ ∈ Γ (Λ3 T ∗ X) : (X, ϕ, positive 4-form on X. Then C X,ψ , ψ) is a TA-G2 , + ∞ 3 ∗ (Λ T X) : dϕ = 0 . manifold is an open convex cone in ϕ ∈ Γ , + 3 (X; R). Hence KX,ψ := [ϕ] : ϕ ∈ C X,ψ is an open convex cone in HdR (b) Let X + be a compact 7-manifold and ϕ a closed positive ,3-form on X. Then C X,ϕ := ψ ∈ Γ∞ (Λ4 T ∗ X) : (X, ϕ, ψ) is a TA-G2 -manifold is an open convex + , cone in ψ ∈ Γ∞ (Λ4+T ∗ X) : dψ = 0, . 4 Hence KX,ϕ := [ψ] : ψ ∈ C X,ϕ is an open convex cone in HdR (X; R). Proof. Suppose ϕ1 , ϕ2 ∈ C X,ψ , and let t1 , t2  0 with (t1 , t2 ) = (0, 0). Consider the 3-form ϕ = t1 ϕ1 + t2 ϕ2 on X. It is closed as ϕ1 , ϕ2 are, and it satisfies Definition 2.6(i) as ϕ1 , ϕ2 do, and from this we can deduce that ϕ is positive. Thereso ϕ ∈ C X,ψ , and C X,ψ is a convex cone fore+ (X, ϕ, ψ) is also a TA-G2 -manifold, , in ϕ ∈ Γ∞ (Λ3 T ∗ X) : dϕ = 0 . Openness holds as Definition 2.6(i) is an open condition on ϕ, proving (a). Part (b) is similar.  Definition 2.9. Let X be a 7-manifold. A closed positive 3-form ϕ on X will be called good if there exists a 4-form ψ on X with (X, ϕ, ψ) a TA-G2 -manifold. Similarly, a closed positive 4-form ψ on X will be called good if there exists a 3-form ϕ on X with (X, ϕ, ψ) a TA-G2 -manifold. For compact X, to be good is an open condition on closed 3- and 4-forms ϕ, ψ. Remark 2.10. We can now extend our analogy between Calabi–Yau 3-folds (Y, J, h) and G2 -manifolds (X, ϕ, ∗ϕ), adding the lines: Symplectic form ω on Y

↔

Good 3-form ϕ on X

(Almost) complex structure J on Y

↔

Good 4-form ψ on X

Symplectic manifold (Y, ω) with compatible almost complex structure J

↔

TA-G2 -manifold (X, ϕ, ψ).

Then Proposition 2.8(a) is an analogue of the fact that K¨ahler forms ω on a fixed complex manifold (Y, J) form an open convex cone in the closed real (1,1)-forms on Y , and KX,ψ is an analogue of the K¨ ahler cone of (Y, J). Also Proposition 2.8(b) is analogous to the fact that the family of almost complex structures J compatible with a fixed symplectic form ω on Y form an infinite-dimensional contractible space.

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Suppose we can show some structure we define for TA-G2 -manifolds (X, ϕ, ψ), e.g. G2 quantum cohomology in §7.6, is unchanged under deformations of (X, ϕ, ψ) fixing ϕ. If so, this structure depends only on X and the good 3-form ϕ, as Proposition 2.8(b) shows that the family of ψ compatible with ϕ is connected. This is the analogue of the Gromov–Witten invariants, Lagrangian Floer cohomology, etc. of a symplectic manifold (Y, ω) being independent of almost complex structure J. In fact our theories will manifestly depend only on ψ and the cohomology class [ϕ] ∈ H 3 (X; R), so if they are independent of ψ up to deformation, then they depend only on (X, ϕ) up to deformations fixing [ϕ]. 2.6. Moduli spaces of associative 3-folds. Much of this paper concerns moduli spaces of associative 3-folds M(N , α, ψ) in a TA-G2 -manifold (X, ϕ, ψ). We will use the following notation. Definition 2.11. Consider compact, oriented 3-manifolds N . Write [N ]D or N for the equivalence class of N under the equivalence relation N ∼ N  if there exists an orientation-preserving diffeomorphism δ : N → N  . We call [N ]D an oriented diffeomorphism class. Write D for the set of all oriented diffeomorphism classes, and DQHS ⊂ D for the subset of [N ]D with N a Q-homology sphere, that is, b1 (N ) = b2 (N ) = 0, which is equivalent to H1 (N ; Z) being finite. Let (X, ϕ, ψ) be a TA-G2 -manifold. For each N ∈ D and α ∈ H3 (X; Z), we write M(N , α, ψ) for the moduli space of immersed associative 3-folds i : N → X in (X, ϕ, ψ) which have oriented diffeomorphism type N and homology class α. In more detail, consider pairs (N, i), where: • N is a compact, oriented 3-manifold in oriented diffeomorphism class N ; • i : N → X is an immersed associative 3-fold in (X, ϕ, ψ); • i∗ (ϕ) is a positive 3-form on N with its given orientation; and • i∗ ([N ]) = α ∈ H3 (X; Z). Two such pairs (N, i), (N  , i ) are equivalent, written (N, i) ≈ (N  , i ), if there exists an orientation-preserving diffeomorphism δ : N → N  with i = i ◦ δ. We write [N, i] for the ≈-equivalence class of (N, i). Then just as a set, M(N , α, ψ) is the set of all such [N, i]. We make M(N , α, ψ) into a topological space by choosing N ∈ N , and writing + M(N , α, ψ) ∼ = i ∈ MapC ∞ (N, X) : i is an associative immersion, ,0 i∗ (ϕ) is positive, i∗ ([N ]) = α ∈ H3 (X; Z) Diff + (N ), with Diff + (N ) the group of orientation-preserving diffeomorphisms δ : N → N acting by i → i ◦ δ. Then we give M(N , α, ψ) the quotient-subspace topology coming from the C ∞ -topology on MapC ∞ (N, X). We write M(N , α, ψ)emb ⊆ M(N , α, ψ) for the open subset of [N, i] with i : N → X an embedding. For each [N, i] ∈ M(N , α, ψ) we define the isotropy group Iso([N, i]) to be the subgroup δ ∈ Diff + (N ) with i ◦ δ = i. Then Iso([N, i]) is finite, as N is compact and i an immersion, and Iso([N, i]) = {1} if [N, i] ∈ M(N , α, ψ)emb . We use the notation M(N , α, ψ), omitting ϕ, since as in Definition 2.7 the notion of associative 3-fold in (X, ϕ, ψ) depends only on X, ψ, not on ϕ. Now suppose (X, ϕt , ψt ) : t ∈ F is a smooth family of TA-G2 -manifolds over a base F which is a finite-dimensional manifold, or manifold with boundary. Then we write M(N , α, ψt : t ∈ F ) for the moduli space of pairs + , M(N , α, ψt : t ∈ F ) = (t, [N, i]) : t ∈ F , [N, i] ∈ M(N , α, ψt ) , with topology induced from that on F × MapC ∞ (N, X) as above. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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We want the moduli spaces M(N , α, ψ), M(N , α, ψt : t ∈ F ) to be not just topological spaces, but (in good cases) manifolds or orbifolds, preferably compact and oriented, and (in general) derived manifold or derived orbifolds. The deformation theory of compact associative 3-folds was studied by McLean [67, §5]. He considered compact, embedded associative 3-folds in torsion-free G2 -manifolds, and showed that their moduli space is locally the solutions of a nonlinear elliptic p.d.e. with linearization the twisted Dirac operator D below. Our theorem follows from and extends McLean’s work using standard techniques. Theorem 2.12 (McLean [67, §5], extended). Suppose (X, ϕ, ψ) is a TA-G2 manifold, and i : N → X be a compact, immersed associative 3-fold, with i∗ ([N ]) = α ∈ H3 (X; Z) and [N ] = N ∈ D, so that [N, i] ∈ M(N , α, ψ). Write g for the Riemannian metric on X from the G2 -structure associated to ψ, and ν → N for the normal bundle of N in X, a rank 4 vector bundle, and ∇ν for the connection on ν induced by the Levi-Civita connection of g. Then there is a natural first-order linear elliptic operator D : Γ∞ (ν) → Γ∞ (ν) of index 0, a twisted Dirac operator, which is characterized by the equation " 1 2 ψa1 a2 [b1 b2 (∇νb3 ] v a1 )wa2 (2.10) Dv, w L2 = N

∞

for all v, w ∈ Γ (ν). Here the L -inner product on Γ∞ (ν) is defined using g, and we use the index notation for tensors, contracting together ψ, v, ∇ν w to get a 3-form, which we integrate over the oriented 3-manifold N . Write T N = Ker D and ON = Coker D, as finite-dimensional real vector spaces with dim T N = dim ON . Then the finite group Γ := Iso([N, i]) from Definition 2.11 acts on T N , ON . There exist a Γ-invariant open neighbourhood V of 0 in T N , a Γ-equivariant smooth map Θ : V → ON with Θ(0) = dΘ(0) = 0, an open neighbourhood W of [N, i] in M(N , α, ψ), and a homeomorphism Ψ : Θ−1 (0)/Γ → W with Ψ(0) = [N, i]. We call T N the Zariski tangent space and ON the obstruction space to M(N , α, ψ) at [N, i]. We call N unobstructed if ON = 0. 2

The proof of Theorem 2.12 does not need ψ closed, and does not use ϕ at all. However, if v, w ∈ Γ∞ (ν) then by Stokes’ Theorem and (2.10) we have " 3 0= d[ψa1 a2 b1 b2 v a1 wa2 "N 4 3 dψa1 a2 b1 b2 b3 v a1 wa2 + ψa1 a2 [b1 b2 ∇νb3 ] v a1 wa2 + ψa1 a2 [b1 b2 v a1 ∇νb3 ] wa2 = "N 4 3 1 2 1 2 = dψa1 a2 b1 b2 b3 v a1 wa2 + Dv, w L2 − v, Dw L2 . N

Hence if dψ = 0 we have $Dv, w%L2 = $v, Dw%L2 , giving: Lemma 2.13. In Theorem 2.12, if dψ = 0 (which is included in the definition of TA-G2 -manifold (X, ϕ, ψ)) then D is a self-adjoint linear operator. In §3 we want D to be self-adjoint to define ‘flags’ of unobstructed associative 3-folds, and this is one reason we take ψ closed in TA-G2 -manifolds (X, ϕ, ψ). Derived Differential Geometry is the study of ‘derived manifolds’ and ‘derived orbifolds’. Different versions of derived manifolds are defined by Spivak [74],

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Borisov–Noel [8, 9] and the author [48–53]. The author gives two equivalent notions of derived manifolds and orbifolds: d-manifolds and d-orbifolds [48–50], and m-Kuranishi spaces and Kuranishi spaces [51–53], which are an improved version of Fukaya–Oh–Ohta–Ono’s Kuranishi spaces [18, 19]. Many moduli spaces in differential geometry are known to be derived manifolds or derived orbifolds [50]. Theorem 2.12 implies that M(N , α, ψ) locally has the structure of a derived orbifold/Kuranishi space, since (V, ON , Γ, Θ, Ψ) is a Kuranishi neighbourhood on M(N , α, ψ). The author expects to prove the following conjecture in the next few years, as part of a larger project. Conjecture 2.14. In Definition 2.11 we can give M(N , α, ψ) the structure of a d-orbifold in the sense of [48–50], or a Kuranishi space in the sense of [51–53], of virtual dimension 0, canonical up to equivalence in the 2-categories dOrb, Kur. The open subset M(N , α, ψ)emb ⊆ M(N , α, ψ) of embedded associatives becomes a d-manifold or m-Kuranishi space. Similarly, we can make M(N , α, ψt : t ∈ F) into a d-orbifold or Kuranishi space, with virtual dimension dim F , and with a 1-morphism π : M(N , α, ψt : t ∈ F) → F . Here is a class of immersed submanifolds that will be important to us: Definition 2.15. Let i : N → X be a compact, immersed submanifold. We call N finite-embedded if either i : N → X is an embedding, or else i = ˜ı ◦ π for ˜ → X an embedded submanifold and π : N → N ˜ a finite cover. ˜ı : N In several important moduli problems, by taking the geometric data generic, one can ensure that the moduli spaces are smooth. For example, Donaldson and Kronheimer [14, §4.3] show that if (M, g) is a compact oriented Riemannian 4manifold with b2+ (M ) > 0 then all moduli spaces of SU(2)-instantons on X are smooth, and McDuff and Salamon [66, §3.4] prove that if (S, ω) is a symplectic manifold and J is a generic almost structure on S compatible with ω then all moduli spaces of embedded J-holomorphic curves in S are smooth. Conjecture 2.16. Suppose (X, ϕ, ψ) is a compact TA-G2 -manifold, with ψ generic amongst closed 4-forms on X. Then for all N ∈ D and α ∈ H3 (X; Z), the moduli space M(N , α, ψ) in Definition 2.11 is a finite set. For each [N, i] ∈ M(N , α, ψ), the associative 3-fold N is unobstructed, and N is finite-embedded, as in Definition 2.15. Furthermore, for any A > 0 there are only finitely many pairs (N , α) with M(N , α, ψ) = ∅ and [ϕ] · α  A. Note here that M(N , α, ψ) has virtual dimension 0, and ‘compact smooth 0manifold’ is equivalent to ‘finite set’. McLean [67, §3–§4] also studied moduli spaces of compact special Lagrangian submanifolds, and coassociative 4-folds. These are simpler than the associative case, as they are always smooth manifolds. Theorem 2.17 (McLean [67]). (a) Suppose (Y, J, h) is a Calabi–Yau m-fold, and L ⊂ Y is a compact SL m-fold. Then the moduli space ML of special Lagrangian deformations of L is a smooth manifold of dimension b1 (L). (b) Suppose (X, ϕ, ψ) is a TA-G2 -manifold, and C is a compact coassociative 4-fold in X. Then the moduli space MC of coassociative deformations of C is a smooth manifold of dimension b2+ (C). The proof of part (b) requires dϕ = 0.

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2.7. Associative 3-folds with boundary in coassociatives. If (X, ϕ, ψ) is a TA-G2 -manifold and C ⊂ X is coassociative, we can consider associative 3-folds N ⊂ X with boundary ∂N ⊂ C. Note that associatives N are defined using ψ, and coassociatives C defined using ϕ, but Definition 2.6(iii) ensures that ∂N ⊂ C is a well behaved boundary condition for N . If (X, ϕ, ψ) is a compact TA-G2 manifold, so that Definition 2.6(i) holds for some K > 0, then as in (2.9) we have the topological volume bound volgψ (N )  K[ϕ] · [N ]. 3 (X, C; R), [N ] ∈ H3 (X, C; Z). where now we use relative (co)homology [ϕ] ∈ HdR Gayet and Witt [20] generalized Theorem 2.12 to the boundary case. The dimension of the moduli space is no longer automatically zero.

Theorem 2.18 (Gayet and Witt [20], extended). Let (X, ϕ, ψ) be a TA-G2 manifold, and C ⊂ X a coassociative 4-fold. Suppose N is a compact, immersed associative 3-fold in X with connected boundary ∂N ⊂ C of genus g. Then the deformation theory 5 of N for fixed (X, ϕ, ψ), C is a nonlinear elliptic equation, of index d(N ) := ∂N c1 (ν∂N ) + 1 − g, where νN is the normal bundle of ∂N in C with its natural complex structure. Thus as in Conjecture 2.14 we expect the moduli space MN of deformations of N to be a derived orbifold as in [48–53], of virtual dimension d(N ). Given two nearby coassociatives C1 , C2 in (X, ϕ, ∗ϕ) with C1 ∩ C2 = ∅, Leung, Wang and Zhu [59, 60] prove results on associative 3-folds N in (X, ϕ, ∗ϕ) with boundary ∂N ⊂ C1 # C2 and vol(N ) small. This is intended as a first step towards constructing some kind of Floer theory for coassociative 4-folds C by counting associative 3-folds N with boundary ∂N ⊂ C. We discuss this in §6.2.

3. How to orient moduli spaces of associatives The material of this section is new. Our aim is to construct orientations on the moduli spaces M(N , α, ψ) of associatives in (X, ϕ, ψ) in §2.6, considered as derived orbifolds in the case of Conjecture 2.14, or as orbifolds in the case of Conjecture 2.16. For unobstructed associatives, our construction is rigorous. We will show that any compact associative 3-fold N ⊂ X has a natural flag fN , a partial framing of the normal bundle ν → N , defined in a subtle way using the operator D : Γ∞ (ν) → Γ∞ (ν) from Theorem 2.12. The set Flag(N ) of flags on N is a Z-torsor. Roughly speaking we have Flag(N ) ∼ = Z, and when N is unobstructed we define N to be positively (negatively) oriented if fN ∈ Flag(N ) corresponds to an even (odd) number in Z. In fact things are more complicated, as the isomorphism Flag(N ) ∼ = Z is not canonical. We will define a new algebro-topological structure on X called a flag . structure F . The set of flag structures is a torsor over HomGrp H3 (X; Z), {±1} . Given a flag structure on X, the isomorphism Flag(N ) ∼ = Z is canonical mod 2Z, which is enough to define orientations. Orienting moduli spaces M(N , α, ψ) is important for our programme, since it is essential to count associative 3-folds with signs to have any chance of getting a deformation-invariant answer, as we explain in §7.

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For comparison, Donaldson and Kronheimer [14, §5.4 & §7.1.6] construct orientations on moduli spaces of instantons on a 4-manifold M , and Fukaya–Oh–Ohta– Ono [18, §8] define orientations on moduli spaces of J-holomorphic discs in a symplectic manifold S with boundary in a Lagrangian L. In both cases some extra al2 (M ; R) gebraic topological data is needed, namely an orientation on H 1 (M ; R)⊕H+ in [14], and a relative spin structure for (S, L) in [18]. 3.1. Flags and flag structures. Though we explain the material of this section for 3-submanifolds N in a 7-manifold X, in fact it works in exactly the same way for (2k + 1)-dimensional submanifolds N of a (4k + 3)-manifold X for k = 0, 1, . . . . Definition 3.1. Let X be an oriented 7-manifold, and i : N → X a compact, oriented, immersed 3-manifold in X. Write ν → N for the normal bundle of N in X. Then the orientation on X induces an orientation on the total space of ν. Consider nonvanishing sections s ∈ Γ∞ (ν), so that s(x) = 0 for all x ∈ N . Let s, s be nonvanishing sections. Write 0 : N → ν for the zero section, and γ : [0, 1] × N → ν for the map γ : (t, x) → (1 − t)s(x) + ts (x). Then 0(N ) is a 3-cycle in the homology of ν over Z, and γ([0, 1] × N ) is a 4-chain in the homology of ν, where ∂[γ([0, 1] × N )] is disjoint from 0(N ), and ν is an oriented 7-manifold. Define d(s, s ) ∈ Z to be the intersection number 0(N ) • γ([0, 1] × N ). We have d(s , s) = −d(s, s ) and d(s, s ) = d(s, s ) + d(s , s ) for all nonvanishing sections s, s , s ∈ Γ∞ (ν). Define a flag on N to be an equivalence class [s] of nonvanishing s ∈ Γ∞ (ν), where s, s are equivalent if d(s, s ) = 0. We call (N, [s]) a flagged submanifold. Write Flag(N ) for the set of all flags [s] on N . For [s], [s ] ∈ Flag(N ) we define d([s], [s ]) = d(s, s ) ∈ Z for any representatives  s, s for [s], [s ]. It is not difficult to show that for any [s] ∈ Flag(N ) and any k ∈ Z, there is a unique [s ] ∈ Flag(N ) with d([s], [s ]) = k. We write [s ] = [s] + k. This gives a natural action of Z on Flag(N ) by addition, which makes Flag(N ) into a Z-torsor (that is, the Z-action is free and transitive). For the next parts we restrict to (N, [s]) with N finite-embedded, as in Definition 2.15. We compare flags for homologous 3-submanifolds N1 , N2 . Definition 3.2. Let X be an oriented 7-manifold, and suppose N1 , N2 are compact, oriented, finite-embedded 3-submanifolds in X with [N1 ] = [N2 ] in H3 (X; Z) and N1 ∩ N2 = ∅, and [s1 ], [s2 ] are flags on N1 , N2 . Choose a 4-chain C12 in the homology of X over Z with ∂C12 = N2 − N1 . Let s1 , s2 be representatives for N1 , N2 , and let N1 , N2 be small perturbations of N1 , N2 in the normal directions s1 , s2 . Then N1 ∩ N1 = N2 ∩ N2 = ∅ as s1 , s2 are nonvanishing and N1 , N2 are finite-embedded, and also N1 ∩ N2 = N2 ∩ N1 = ∅ as N1 , N2 are disjoint and N1 , N2 are close to N1 , N2 . Define D((N1 , [s1 ]), (N2 , [s2 ])) to be the intersection number (N2 − N1 ) • C12 in homology over Z. This is well defined as ∂C12 = N2 − N1 , so the 3-cycles N2 − N1 and ∂C12 are disjoint. It is also independent of the choices of C12 and N1 , N2 . We can show that for k1 , k2 ∈ Z we have (3.1)

D((N1 , [s1 ] + k1 ), (N2 , [s2 ] + k2 )) = D((N1 , [s1 ]), (N2 , [s2 ])) − k1 + k2 .

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Proposition 3.3. Let X be an oriented 7-manifold, and (N1 , [s1 ]), (N2 , [s2 ]), (N3 , [s3 ]) be disjoint finite-embedded flagged submanifolds in X. Then (3.2)

D((N1 , [s1 ]), (N3 , [s3 ])) = D((N1 , [s1 ]), (N2 , [s2 ])) + D((N2 , [s2 ]), (N3 , [s3 ])) mod 2.

Proof. Let s1 , s2 , s3 be representatives for [s1 ], [s2 ], [s3 ], and N1 , N2 , N3 be small perturbations of N1 , N2 , N3 in directions s1 , s2 , s3 . Choose 4-chains C12 , C23 over Z in X with ∂C12 = N2 − N1 and ∂C23 = N3 − N2 . Then C13 = C12 + C23 is   with ∂C12 = N2 − N1 . a 4-chain with ∂C13 = N3 − N1 . Also choose a 4-chain C12 Then we have D((N1 , [s1 ]), (N3 , [s3 ])) − D((N1 , [s1 ]), (N2 , [s2 ])) − D((N2 , [s2 ]), (N3 , [s3 ])) = (N3 − N1 ) • (C12 + C23 ) − (N2 − N1 ) • C12 − (N3 − N2 ) • C23  = (N3 − N2 ) • C12 + (N2 − N1 ) • C23 = (N3 − N2 ) • C12 + (N2 − N1 ) • C23      = ∂C23 • C12 + ∂C12 • C23 = ∂(C23 • C12 ) + 2∂C12 • C23 = 0 + 2∂C12 • C23 ,

using the definition of D((Ni , [si ]), (Nj , [sj ])) in the first step, the easy identity  in the third, and that a boundary is zero in (N3 − N2 ) • C12 = (N3 − N2 ) • C12 homology in the sixth. Equation (3.2) follows.  Proposition 3.4. Let X be an oriented 7-manifold, and (N, [s]) be an immersed flagged submanifold in X, and (N  , [s ]), (N  , [s ]) be any two small perturbations of (N, [s]) with N  , N  embedded in X. Then (3.3)

D((N  , [s ]), (N  , [s ])) = 0 mod 2.

Proof. For (N  , [s ]), (N  , [s ]) as in the proposition, choose a generic smooth ˆt : [ˆ 1-parameter family (N st ]) of small perturbations of (N, [s]) for t ∈ [0, 1] with   ˆ1 , [ˆ ˆ s0 ]) = (N , [s ]) and (N s1 ]) = (N  , [s ]). Then by genericness we can (N0 , [ˆ ˆt is embedded suppose that there exist 0 < t1 < t2 < · · · < tk < 1 such that N ˆ for t ∈ [0, 1] \ {t1 , . . . , tk }, and Nti is immersed with a single self-intersection point ˆt , t ∈ [0, 1] crosses itself transversely xi ∈ X for i = 1, . . . , k, such that the family N at xi as t increases through ti . ˇ , [ˇ Choose another compact embedded flagged submanifold (N s]) in X with ˇ ] = [N ] ∈ H3 (X; Z) which is disjoint from N , and hence also disjoint from [N ˆt as these are small perturbations of N . Consider the function N  , N  , N ˇ , [ˇ ˆt , [ˆ st ])) for t ∈ [0, 1] \ {t1 , . . . , tk }. (3.4) t −→ D((N s]), (N ˇ is disjoint from N ˆt , and (N ˆt , [ˆ Since N st ]) deforms continuously in t, this function ˇ , [ˇ s]), is constant in each connected component of [0, 1] \ {t1 , . . . , tk }. Define D((N ˆt , [ˆ ˆt − N ˇ , where Ct depends continuously (N st ])) using a 4-chain Ct with ∂Ct = N ˆt+ , N ˆt− of N ˆt , and on t in [0, 1]. For t close to ti , near xi in X there are two sheets N + − + ˆ hence two sheets ∂Ct , ∂Ct of ∂Ct . As t crosses ti , we see that Nt crosses ∂Ct− ˆt− crosses ∂Ct+ transversely with the same orientation, so that transversely, and N ˇ ˆ D((N , [ˇ s]), (Nt , [ˆ st ])) changes by ±2. Therefore the total change in (3.4) between t = 0 and t = 1 is even, giving ˇ , [ˇ ˇ , [ˇ D((N s]), (N  , [s ])) = D((N s]), (N  , [s ])) mod 2. Equation (3.3) now follows from Proposition 3.3.

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Flag structures are the algebro-topological data we will need in §3.2 to orient moduli spaces of associative 3-folds in (X, ϕ, ψ). Definition 3.5. Let X be an oriented 7-manifold. A flag structure is a map + , (3.5) F : immersed flagged submanifolds (N, [s]) in X −→ {±1}, satisfying: (i) If (N, [s]) is an immersed flagged submanifold and (N  , [s ]) is any small perturbation of (N, [s]) then F (N, [s]) = F (N  , [s ]). (ii) F (N, [s] + k) = (−1)k · F (N, [s]) for all (N, [s]) and k ∈ Z. (iii) If (N1 , [s1 ]), (N2 , [s2 ]) are disjoint finite-embedded flagged submanifolds in X with [N1 ] = [N2 ] in H3 (X; Z) then (3.6)

F (N2 , [s2 ]) = F (N1 , [s1 ]) · (−1)D((N1 ,[s1 ]),(N2 ,[s2 ])) .

(iv) If (N1 , [s1 ]), (N2 , [s2 ]) are disjoint immersed flagged submanifolds then (3.7)

F (N1 # N2 , [s1 # s2 ]) = F (N1 , [s1 ]) · F (N2 , [s2 ]).

Proposition 3.6. Let X be an oriented 7-manifold. Then: (a) There exists a flag structure F on X. (b) If F, F  are flag structures on X then there exists a unique group morphism : H3 (X; Z) → {±1} such that (3.8)

F  (N, [s]) = F (N, [s]) · ([N ])

for all (N, [s]).

(c) Let F be a flag structure on X and : H3 (X; Z) → {±1} a group morphism, and define F  in (3.5) by (3.8). Then F  is a flag structure on X. Parts (a)–(c) imply that.the set FlagSt(X) of flag structures on X is a torsor over HomGrp H3 (X; Z), {±1} . Proof. For (a), let V be the image of the projection H3 (X; Z) → H3 (X; Z2 ). It is a Z2 -vector space, as Z2 is a field. Choose a basis (ei : i ∈ I) for V . The indexing set I is countable, and finite if X is compact. For each i ∈ I, choose an embedded flagged submanifold (Ni , [si ]) in X with [Ni ] = ei in H3 (X; Z2 ). As there are at most countably many Ni , we can choose them to be disjoint. For each i ∈ I, choose δi = ±1. We will construct a flag structure F with F (Ni , [si ]) = δi . Let (N, [s]) be an immersed flagged submanifold in X. Then [N ] ∈ V ⊆ H3 (X; Z2 ), soas the ei are a basis for V there is a unique finite subset J ⊆ I with [N ] = j∈J ej   in H3 (X; Z2 ). Choose a small perturbation (N , [s ]) of (N, [s]) such that N  is embedded in X and disjoint from Nj for all j ∈ J. Observe that Definition 3.2 and Propositions 3.3–3.4 make sense in homology over Z2 as well as over Z, so we can define DZ2 ((N1 , [s1 ]), (N2 , [s2 ])) ∈ Z2 if (N1 , [s1 ]), (N2 , [s2 ]) are embedded submanifolds with [N1 ] = [N2 ] ∈ H3 (X; Z2 ). Thus we may set   6   F (N, [s]) = (−1)DZ2 ((N ,[s ]),( j∈J Nj , j∈J [sj ])) · j∈J δj , 7 since [N  ] = [N ] = [ j∈J Nj ] in H3 (X; Z2 ). Propositions 3.3 and 3.4 imply that this is independent of the choice of perturbation (N  , [s ]), so F (N, [s]) is well defined. From (3.1)–(3.3) and by construction it is not difficult to show that F is a flag structure, proving (a).

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For (b), suppose (N1 , [s1 ]), (N2 , [s2 ]) are immersed flagged submanifolds with [N1 ] = [N2 ] = α ∈ H3 (X; Z). Choose another immersed flagged submanifold (N3 , [s3 ]) with [N ] = α and N disjoint from both N1 , N2 . Then F  (N1 , [s1 ])F (N1 , [s1 ])−1 34 3 4 = F  (N3 , [s3 ])·(−1)D((N3 ,[s3 ]),(N1 ,[s1 ])) · F (N3 , [s3 ])·(−1)D((N3 ,[s3 ]),(N1 ,[s1 ])) −1 = F  (N3 , [s3 ])F (N3 , [s3 ])−1 34 3 4 = F  (N2 , [s2 ])·(−1)D((N2 ,[s2 ]),(N3 ,[s3 ])) · F (N2 , [s2 ])·(−1)D((N2 ,[s2 ]),(N3 ,[s3 ])) −1 = F  (N2 , [s2 ])F (N2 , [s2 ])−1 , by Definition 3.5(iii) for F, F  . Thus F  (N, [s])F (N, [s])−1 depends only on the homology class [N ] ∈ H3 (X; Z). Hence there exists a unique map : H3 (X; Z) → {±1} with F  (N, [s])F (N, [s])−1 = ([N ]), so that (3.8) holds. Dividing (3.7) for F  by (3.7) for F yields ([N1 # N2 ]) = ([N1 ]) · ([N2 ]), so

(α + β) = (α) (β) for α, β ∈ H3 (X; Z), and : H3 (X; Z) → {±1} is a group morphism. This proves (b). Part (c) is easy to check from Definition 3.5.  3.2. Canonical flags of associatives, and orientations. Given any compact, immersed associative i : N → X in a TA-G2 -manifold (X, ϕ, ψ), we will define a flag [s] for N . To do this we will need the notion of spectral flow introduced by Atiyah, Patodi and Singer [3, §7]. Definition 3.7. Let N be a compact manifold, and suppose that for all t ∈ [0, 1] we are given a vector bundle Et → N and a linear first-order elliptic operator At : Γ∞ (Et ) → Γ∞ (Et ), which is self-adjoint with respect to some metrics gt on N and ht on the fibres of Et , where Et , At , gt , ht depend continuously on t ∈ [0, 1]. Then Atiyah et al. [3, §7] define the spectral flow SF(At : t ∈ [0, 1]) ∈ Z. Heuristically, SF(At : t ∈ [0, 1]) ∈ Z is the number of eigenvalues λ of At which cross from λ ∈ (−∞, 0) to λ ∈ [0, ∞) as we deform t from 0 to 1, counted with signs. We need the At to be self-adjoint so that their eigenvalues are real. If E0 = E1 , A0 = A1 then (for simplicity assuming Et , At are smooth in t ∈ S 1 = R/Z) we may define a vector bundle E → N × S 1 by E|N ×{t} = Et ∂ and an elliptic operator A : Γ∞ (E) → Γ∞ (E) by A|N ×{t} = At + ∂t , and then [3, Th. 7.4] shows that SF(At : t ∈ [0, 1]) = ind(A), which may be computed using the Atiyah–Singer Index Theorem. Definition 3.8. Let (X, ϕ, ψ) be a TA-G2 -manifold, and i : N → X be a compact, immersed associative 3-fold in X. Write g for the Riemannian metric on X from the G2 -structure associated to ψ, and ν → N for the normal bundle of N in X. Then Theorem 2.12 defines a first-order linear elliptic operator D : Γ∞ (ν) → Γ∞ (ν), which by Lemma 2.13 is self-adjoint with respect to the metrics induced by g, as we assume dψ = 0 for TA-G2 -manifolds (X, ϕ, ψ). Choose a flag [s] for N , and choose a representative s for [s] which is of constant length 1 for the metric on ν induced by g. Now D is a twisted Dirac operator on N . Another example of a twisted Dirac operator on N is   0 ∗d (3.9) d ∗+∗ d = : Γ∞ (Λ0 T ∗ N ⊕Λ2 T ∗ N ) → Γ∞ (Λ0 T ∗ N ⊕Λ2 T ∗ N ). ∗d d∗

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∼ Λ0 T ∗ N ⊕ Λ2 T ∗ N which It is easy to see that there is a unique isomorphism ν = ∞ 0 ∗ 2 ∗ identifies s with 1 ⊕ 0 in Γ (Λ T N ⊕ Λ T N ), and identifies the symbols of D and d ∗ + ∗ d. Under this identification, D and d ∗ + ∗ d differ by an operator of order zero, since their symbols (first-order parts) agree. Thus we have (3.10) D ∼ = d ∗ + ∗ d + B : Γ∞ (Λ0 T ∗ N ⊕ Λ2 T ∗ N ) −→ Γ∞ (Λ0 T ∗ N ⊕ Λ2 T ∗ N ), for some unique vector bundle morphism - 00 B02 . 0 ∗ 2 ∗ 0 ∗ 2 ∗ (3.11) B= B B20 B22 : Λ T N ⊕ Λ T N −→ Λ T N ⊕ Λ T N. Define a family of self-adjoint first order linear elliptic operators (3.12)

At : Γ∞ (Λ0 T ∗ N ⊕ Λ2 T ∗ N ) −→ Γ∞ (Λ0 T ∗ N ⊕ Λ2 T ∗ N )

∼D for t ∈ [0, 1] by At = d ∗ + ∗ d + tB. Then A0 = d ∗ + ∗ d in (3.9), and A1 = under our isomorphism Λ0 T ∗ N ⊕ Λ2 T ∗ N ∼ = ν. Thus as in Definition 3.7 we have the spectral flow SF(At : t ∈ [0, 1]) ∈ Z. Suppose s, s are non-vanishing sections of ν → N yielding flags [s], [s ], and At : t ∈ [0, 1], At : t ∈ [0, 1] the corresponding families of elliptic operators. Definition 3.1 defines d(s, s ) ∈ Z. By using [3, Th. 7.4] and computing the index of a Dirac-type operator on N × S 1 by the Atiyah–Singer Index Theorem, we can show that (up to the sign of d(s, s )) (3.13)

SF(At : t ∈ [0, 1]) = SF(At : t ∈ [0, 1]) + d(s , s).

This implies that SF(At : t ∈ [0, 1]) depends only on the flag [s], not on the representative s. Also, since Flag(N ) is a Z-torsor as in §3.1, there is a unique flag fN on N , called the canonical flag of N , such that SF(At : t ∈ [0, 1]) = 0 for At : t ∈ [0, 1] constructed using s ∈ fN . It has the property that for any flag [s] for N and family At : t ∈ [0, 1] constructed from s ∈ [s] as above, we have (3.14)

fN = [s] + SF(At : t ∈ [0, 1]).

Remark 3.9. Suppose (X, ϕ, ∗ϕ) is a torsion-free compact G2 -manifold, and N ⊂ X is a compact, unobstructed associative 3-fold in X, and (W, Ω) is an Asymptotically Cylindrical Spin(7)-manifold (not necessarily torsion-free) with Spin(7) 4form Ω, with one end asymptotic to (X × (0, ∞), dt ∧ ϕ + ∗ϕ), and M ⊂ W is a closed, Asymptotically Cylindrical Cayley 4-fold in W , with one end asymptotic to N × (0, ∞) in X × (0, ∞). Ohst [70] studies the deformation theory of M in X. We can interpret [70, Prop. 19] in our language as saying that the moduli space MM of Asymptotically Cylindrical Cayley deformations of M in (W, Ω) has virtual dimension . vdim MM = 12 χ(M ) + σ(M ) − b0 (N ) − b1 (N ) − e(νM , fN ), where χ(M ), σ(M ) are the Euler characteristic and signature of M (the sign of σ(M ) depends on the model for Spin(7) 4-forms Ω, we follow [29, 46]), and νM is the normal bundle of M in W , and e(νM , fN ) is the Euler class of νM relative to the canonical flag fN at infinity in M . That is, e(νM , fN ) is the number of zeroes, counted with signs, of a generic section s of νM → M asymptotic to a nonvanishing section s of the normal bundle νN of N in X with [s ] = fN . Suppose that for u ∈ (− , ) we are given a TA-G2 -manifold (X, ϕu , ψu ) and compact immersed associative Nu in (X, ϕu , ψu ), both varying smoothly with u. Consider how the canonical flag fNu of Nu varies with u ∈ (− , ). Choose su ∈

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Γ∞ (νu ) depending smoothly on u ∈ (− , ) and of constant length 1 in the metric gu associated to ψu , and let At,u : t ∈ [0, 1] be the family of operators associated to (X, ϕu , ψu ), Nu , su in Definition 3.8. Then by (3.14) we have fNu = [su ] + SF(At,u : t ∈ [0, 1]). Here the flag [su ] varies smoothly with u ∈ (− , ), so fNu varies smoothly with u if and only if SF(At,u : t ∈ [0, 1]) is constant in u. Since At,u depends smoothly on t, u the only way SF(At,u : t ∈ [0, 1]) could fail to be constant in u is if either (a) an eigenvalue of A0,u = d ∗u + ∗u d crosses 0 as u varies; or (b) an eigenvalue of A1,u ∼ = Du crosses 0 as u varies. Now by Hodge theory, Ker A0,u ∼ = H 0 (N ; R) ⊕ H 2 (N ; R), which is of constant dimension. Thus (a) is impossible. Hence fNu must vary smoothly with u unless Ker Du = T Nu jumps as u varies. In particular, if Nu is unobstructed for all u ∈ (− , ) then Ker Du = 0, so (b) does not happen. This proves: Proposition 3.10. Suppose that for u ∈ (− , ) we are given a TA-G2 -manifold (X, ϕu , ψu ) and a compact, immersed, unobstructed associative 3-fold Nu in (X, ϕu , ψu ), both varying smoothly with u. Then the canonical flag fNu of Nu varies continuously with u in (− , ). Now we explain how to orient moduli spaces of associatives. Definition 3.11. Let (X, ϕ, ψ) be a TA-G2 -manifold. Choose a flag structure F on X, which is possible by Proposition 3.6(a). The orientations on moduli spaces we define will depend on this choice. Let N be a compact, immersed, unobstructed associative 3-fold in (X, ϕ, ψ). Then Definition 3.8 defines a canonical flag fN for N . Define Or(N ) = F (N, fN ), so that Or(N ) = ±1. If we take ψ to be generic, and assume Conjecture 2.16, then all compact associatives are unobstructed, so this defines maps Or : M(N , α, ψ) → {±1} for all N , α. We think of Or as an orientation on the 0-manifold M(N , α, ψ), since in dimension 0 an orientation is a choice of sign for each point. Note that Or(N ) is not an orientation on N , which already has a natural orientation. Combining Proposition 3.10 and Definition 3.5(i) yields: Corollary 3.12. Suppose that for u ∈ (− , ) we are given a TA-G2 -manifold (X, ϕu , ψu ) and a compact, immersed, unobstructed associative 3-fold Nu in (X, ϕu , ψu ), both varying smoothly with u. Fix a flag structure F on X. Then the orientation Or(Nu ) is constant in u ∈ (− , ). The next conjecture should be proved using similar methods to Fukaya–Oh– Ohta–Ono’s treatment [18, §8] of orientations on Kuranishi space moduli spaces of J-holomorphic discs. Conjecture 3.13. Assume Conjecture 2.14. Then for any TA-G2 -manifold (X, ϕ, ψ) we have Kuranishi spaces M(N , α, ψ), the moduli spaces of associative 3-folds in (X, ϕ, ψ), and for any smooth family of TA-G2 -manifolds (X, ϕt , ψt ) : t ∈ F , we have 1-morphisms of Kuranishi spaces π : M(N , α, ψt : t ∈ F ) → F , interpreted as families of moduli spaces M(N , α, ψt ) over the base F . Choose a flag structure F for X. Using the ideas on canonical flags above, we can construct orientations for the Kuranishi spaces M(N , α, ψ) and coorientations for the 1-morphisms π : M(N , α, ψt : t ∈ F ) → F , for all N , α. These

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(co)orientations are compatible with pullbacks of families (X, ϕt , ψt ) : t ∈ F , and agree with those in Definition 3.11 for unobstructed [N, i] ∈ M(N , α, ψ). The next example describes the typical way in which the author expects orientations of associatives to change discontinuously in a family. Example 3.14. Let (X, ϕs , ψs ) for s ∈ (− 2 , 2 ) be a smooth family of TAG2 -manifolds, and it : N → X for t ∈ (− , ) a family of compact, immersed 3-submanifolds, with Nt := it (N ) associative in (X, ϕt2 , ψt2 ) for s = t2 . Write Dt for the operator D in Theorem 2.12 for Nt . Suppose Nt is unobstructed for t = 0, so that Ker Dt = 0 for t = 0. d it |t=0 is an infinitesimal As t → s = t2 is stationary at t = 0, we see that dt associative 3-fold in (X, ϕ , ψ ), and lies in Ker D0 . We deformation of N0 as an 0 0 2 1d R. Thus, Ker D suppose that Ker D0 = dt it |t=0 ∼ is 0 for t = 0 and R for t = 0. = t This happens because an eigenvalue λ of Dt crosses 0 as t increases through zero, crossing either from λ < 0 to λ > 0, or from λ > 0 to λ < 0. Thus the canonical flag fNt of Nt changes discontinuously by ±1 as t passes through zero. If we fix a flag structure F on X, so that Definition 3.11 defines orientations of compact, unobstructed associative 3-folds, then Or(Nt ) changes sign as t passes through zero. Thus we can suppose that 8 −1, t < 0, Or(Nt ) = 1, t > 0. This does not contradict Corollary 3.12, as N0 is obstructed. When s < 0 we have no associative 3-folds of interest in (X, ϕs , ψs ), but when √ s > 0 we have two compact, unobstructed associative 3-folds Nt , N−t for t = s, with opposite orientations. Thus, if we count associative 3-folds N weighted by orientations Or(N ), the number will not change under this transition, making it plausible that we might get a deformation-invariant answer. Note that the use of spectral flow in defining orientations, so that Or(Nt ) changes sign when eigenvalues of Dt cross zero, is crucial here. If we counted associatives without orientations, the number would not be deformation-invariant. Remark 3.15. We have been discussing associative 3-folds N in a TA-G2 manifold (X, ϕ, ψ), which by definition has dψ = 0. We now consider how the theory changes if we allow dψ = 0. In §2.6, the moduli spaces M(N , α, ψ), McLean’s Theorem 2.12, and Conjectures 2.14 and 2.16 remain unchanged when dψ = 0. However, as in Lemma 2.13 the twisted Dirac operator D in Theorem 2.12 is no longer self-adjoint if dψ = 0, though it does have self-adjoint symbol. This affects the spectral flow term SF(At : t ∈ [0, 1]) in Definition 3.8. For non-self-adjoint operators At of this type, eigenvalues λ are either real, or ¯ in C\R. To define SF(At : t ∈ [0, 1]), we must occur in complex-conjugate pairs λ, λ count eigenvalues that cross the imaginary axis iR in C as t increases from 0 to 1. ¯ in C\R can cross iR at So when dψ = 0 we have a new phenomenon, that a pair λ, λ t ∈ (0, 1), changing SF(At : t ∈ [0, 1]) by ±2. For D to have imaginary eigenvalues does not make N unobstructed, and does not correspond to any qualitative change in the families of associative 3-folds in (X, ϕ, ψ). As a consequence, the analogue of Proposition 3.10 with dψu = 0 should be false: given families (X, ϕu , ψu ) and compact, unobstructed associative 3-folds Nu

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in (X, ϕu , ψu ) varying smoothly with u ∈ (− , ), but allowing dψu = 0, the canonical flag fNu of Nu need not vary continuously with u in (− , ), but can jump by ±2 when conjugate pairs of eigenvalues of Du cross iR. However, because these jumps in canonical flags are even, the analogue of Corollary 3.12 with dψu = 0, and also Conjecture 3.13, should remain true. In conclusion: for associative 3-folds in (X, ϕ, ψ) with dψ = 0, the author expects the theory of orientations on moduli spaces M(N , α, ψ) outlined above to continue to work nicely. But the canonical flags fN lose the continuity property in Proposition 3.10, which is important for our proposal in Conjecture 1.1. ¯ cross iR for Nu , say at λ = is for An aside: when a pair of eigenvalues λ, λ s > 0, the author expects a new S 1 family of Cayley 4-folds Nu × S 1s to appear in the Spin(7)-manifold X × S 1s , where S 1s = R/2πsZ. So one might be able to compensate for this phenomenon by counting Cayley 4-folds in X × S 1s . 4. An index 1 singularity of associative 3-folds We now describe the first of two kinds of singularity of associative 3-folds that will be crucial to our discussion. 4.1. A family of SL 3-folds in C3 . We describe a family of explicit SL 3-folds Kφ,s in C3 . This family was first found by Lawlor [56], was made more explicit by Harvey [21, p. 139–140], and was discussed from a different point of view by the author in [32, §5.4(b)]. Our treatment is based on that of Harvey. Let a1 , a2 , a3 > 0, and define polynomials p(x), P (x) by p(x) = (1 + a1 x2 )(1 + a2 x2 )(1 + a3 x2 ) − 1 and P (x) = Define real numbers φ1 , φ2 , φ3 and s by " ∞ dx ' φk = a k 2 −∞ (1 + ak x ) P (x)

and s =

p(x) . x2

1 (a1 a2 a3 )−1/2 . 3

Clearly φk > 0 and s > 0. But writing φ1 + φ2 + φ3 as one integral and rearranging gives " ∞ " ∞ p (x)dx dw ' = π, φ1 + φ2 + φ3 = =2 2+1 w (p(x) + 1) p(x) 0 0 ' making the substitution w = p(x). So φk ∈ (0, π) and φ1 + φ2 + φ3 = π. This yields a 1-1 correspondence between triples (a1 , a2 , a3 ) with ak > 0, and quadruples (φ1 , φ2 , φ3 , s) with φk ∈ (0, π), φ1 + φ2 + φ3 = π and s > 0. 9 For k = 1, 2, 3 and y ∈ R, define zk (y) by zk (y) = eiψk (y) " y dx ' ψk (y) = ak . 2 −∞ (1 + ak x ) P (x)

2 a−1 k + y , where

Now write φ = (φ1 , φ2 , φ3 ), and define a submanifold Kφ,s in C3 by + , (4.1) Kφ,s = (z1 (y)x1 , z2 (y)x2 , z3 (y)x3 ) : y ∈ R, xk ∈ R, x21 + x22 + x23 = 1 . Our next result comes from Harvey [21, Th. 7.78].

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Proposition 4.1. The set Kφ,s defined in (4.1) is an embedded SL 3-fold in C diffeomorphic to S 2 × R. It is asymptotically conical at rate O(r −2 ) to the union Π0 ∪ Πφ of two special Lagrangian 3-planes Π0 , Πφ given by + , + , Π0 = (x1 , x2 , x3 ) : xj ∈ R , Πφ = (eiφ1 x1 , eiφ2 x2 , eiφ3 x3 ) : xj ∈ R . 3

An easy calculation shows that near Π0 for small s > 0 we have + Kφ,s ≈ (1 + isr −3 )(x1 , x2 , x3 ) + O(s5/3 r −4 ) : (x1 , x2 , x3 ) ∈ R3 , (4.2) , r = (x21 + x22 + x23 )1/2 ' 0 . The next proposition can be proved from Proposition 4.1 and Remark 2.4(ii). Proposition 4.2. Suppose V, V  are 3-dimensional vector subspaces of R7 which are associative, with V ∩ V  = {0}. Then there exists an isomorphism R7 ∼ = R × C3 such that (2.4) holds, which identifies V ⊂ R7 with {0} × Π0 ⊂ R × C3 and V  ⊂ R7 with {0} × Πφ ⊂ R × C3 , for some unique φ = (φ1 , φ2 , φ3 ) in (0, π)3 with φ1 + φ2 + φ3 = π.  Hence there is a family of associative 3-folds KsV,V ⊂ R7 for s > 0 identified  with {0} × Kφ,s ⊂ R × C3 , such that KsV,V is diffeomorphic to S 2 × R, and is  Asymptotically Conical, with cone V ∪ V . This family is independent of the choice of isomorphism R7 ∼ = R × C3 . We could think of V ∪V  as a singular associative 3-fold in R7 with a singularity  at 0, and KsV,V for s > 0 as a family of associative smoothings of V ∪ V  . However, it is more helpful to regard V ∪ V  as a nonsingular, immersed associative 3-fold with a self-intersection point at 0.  Let us describe KsV,V near V \ {0} for small s > 0. From (4.2) we see that we may choose Euclidean coordinates (x1 , x2 , x3 ) on V and (x4 , x5 , x6 , x7 ) on the orthogonal complement V ⊥ in R7 , which we identify with the normal bundle νV of V in R7 , such that (4.3)



KsV,V ≈ Γsζ + O(s5/3 r −4 )

near V \ {0} for small s > 0,

with Γsζ the graph of sζ in ν, where ζ ∈ Γ∞ (νV |V \{0} ) is given by ζ(x1 , x2 , x3 ) = (r −3 x1 , r −3 x2 , r −3 x3 , 0),

r = (x21 + x22 + x23 )1/2 .

Let DV : Γ∞ (νV ) → Γ∞ (νV ) be the operator of Theorem 2.12 for the associative V in R7 . Then D(ζ) = 0 on V \ {0}, since ζ is an associative deformation of V . In fact we can regard ζ as a section of ν on V in currents (a kind of generalized section). Then calculation shows that in currents we have DV (ζ) = 4π δ0 · (0, 0, 0, 1), with δ0 the delta function on V at 0, in the sense of currents. 4.2. Desingularizing immersed associative 3-folds. The next definition sets up notation for a conjecture on an index one singularity of associative 3-folds. Definition 4.3. Suppose that for t ∈ (− , ) we are given a TA-G2 -manifold (X, ϕt , ψt ) and a compact, immersed, unobstructed associative 3-fold it : N → X in (X, ϕt , ψt ), both varying smoothly with t. We write Nt = it (N ). Suppose there are distinct points x± in N with i0 (x+ ) = i0 (x− ) = x in X, and these are the only immersed points in i0 : N → X. We will be interested in two separate cases:

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(a) N is a disjoint union N = N + # N − , where N ± are connected with x+ ∈ N + and x− ∈ N − , and it |N ± are embeddings. (b) N is connected. Write Π+ = dx+ i0 (Tx+ N ) and Π− = dx− i0 (Tx− N ), as associative 3-planes in Tx X, and suppose Π+ ∩ Π− = {0}, so that we have a splitting (4.4)

Tx X = Π+ ⊕ Π− ⊕ $v%R ,

where v ∈ Tx X is chosen uniquely to be of unit length, orthogonal to Π+ ⊕ Π− , with (4.4) compatible with the orientations of Π+ , Π− , $v%R ∼ = R and Tx X. Proposition 4.2 gives a unique family of associative 3-folds Ks , s > 0 in Tx X asymptotic at rate O(r −2 ) to Π+ ∪ Π− . Conjecture 4.4 explains when we expect ˜t in (X, ϕt , ψt ) which is close to i0 (N ) there to exist a compact associative 3-fold N away from x in X, and close to Ks near x, identifying X ∼ = Tx X near x. To state the conjecture we first need to define two real numbers γ, δ. d d it (x+ )|t=0 and dt it (x− )|t=0 lie in Tx X. Define γ ∈ R by Now dt . -d d it (x+ )|t=0 − dt it (x− )|t=0 . γ = v · dt Then γ measures the speed at which the two sheets of Nt near x in X cross each other as t increases through 0 in (− , ). The discussion at the end of §4.1 gives O(r −2 ) sections ζ + of νΠ+ |Π+ \{0} and − ζ of νΠ− |Π− \{0} such that Ks ≈ Γsζ + + O(s5/3 r −4 )

near Π+ \ {0} for small s > 0,

Ks ≈ Γsζ − + O(s5/3 r −4 ) near Π− \ {0} for small s > 0. These ζ ± make sense as currents on all of Π± , and satisfy (4.5)

DΠ+ (ζ + ) = 4π δ0 · v,

DΠ− (ζ − ) = −4π δ0 · v,

where v in (4.4) is a normal vector to both Π+ and Π− . Now let DN0 : Γ∞ (νN0 ) → Γ∞ (νN0 ) be the operator from Theorem 2.12 for N0 in (X, ϕ0 , ψ0 ). It is an isomorphism, as N0 is unobstructed. So its extension to currents is also an isomorphism. Thus there exists a unique current section χ of νN0 such that DN0 (χ) = 4π δx+ · v − 4π δx− · v. Then χ is smooth on N0 \ {x+ , x− }, and from (4.5) we see that χ − ζ + is smooth near x+ , and χ − ζ − is smooth near x− . Near x+ in N , under the splitting (4.4), the section χ ≈ ζ + of νN0 has a pole in the Π− factor in (4.4), but remains continuous in the $v%-factor, so that limx→x+ v · χ(x) exists in R, and similarly limx→x− v · χ(x) exists. Define δ = limx→x+ v · χ(x) − limx→x− v · χ(x) in R. ˜s in (X, ϕ0 , ψ0 ) The point of this is if we try to define an associative 3-fold N ˜ should look like the graph of sχ near by gluing Ks for small s into N0 at x, then N i0 (N ) \ {x} to leading order in s. But the two ends of this graph only fit together to leading order in s if δ = 0, so δ is the first-order obstruction to deforming N0 ˜s in the fixed TA-G2 -manifold (X, ϕ0 , ψ0 ), rather than in to an associative 3-fold N (X, ϕt , ψt ) for some t. To make Conjecture 4.4 simpler, we suppose γ = 0 = δ. This should hold if (X, ϕt , ψt ) : t ∈ (− , ) is a generic 1-parameter family of TA-G2 -manifolds.

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Conjecture 4.4. Work in the situation of Definition 4.3. Then for all sufficiently small t ∈ (− , ) with γδ −1 t < 0 there exists a unique compact, embedded, ˜t in (X, ϕt , ψt ), such that N ˜t is close to N0 away unobstructed associative 3-fold N ˜ from x in X and Nt is close to Ks near x in X, identifying X near x with Tx X ∼ = R7 −1 near 0, where 0 < s ≈ −γδ t to leading order in t. ˜t is the connected sum of N with itself at x+ , x− , so that N ˜t ∼ Topologically, N = 1 2 + − ∼ ˜ N #N in case (a), and Nt = N #(S × S ) in case (b). ˜t exists in (X, ϕt , ψt ) if γδ −1 t  0. No such associative 3-fold N ˜t in §3.2 from that of Nt as We may determine the canonical flag fN˜t of N   follows. Let (N , [s ]) be a flagged submanifold in X with [N  ] = [N0 ] in H3 (X; Z), ˜t for small t. Then in such that N  is disjoint from N0 , and hence from Nt and N the notation of §3.1 we have 8   ˜t , f ˜ )) = D((N, [s ]), (Nt , fN )) + 0, δ < 0, (4.6) D((N, [s ]), (N t Nt 1, δ > 0. If we fix a flag structure on X, so that §3.2 defines orientations Or(N ) of compact, unobstructed associative 3-folds N, then (4.6) implies that 8 δ < 0, ˜t ) = Or(N0 ), Or(N − Or(N0 ), δ > 0. Remark 4.5. (a) Here is why we require s ≈ −γδ −1 t in this conjecture. To ˜t in (X, ϕt , ψt ) by gluing Ks for small s > 0 into Nt for define an associative 3-fold N d ˜ small t near x, then Nt should look like the graph of sχ + t dt it |t=0 near i0 (N ) \ {x} to leading order in s, t. The distance between the two ends of this graph in the R-component in (4.4) is sδ + tγ, by definition of γ, δ in Definition 4.3. As the two ends of the graph must match up, we require that sδ + tγ = 0, to leading order in ˜t exists if γδ −1 t  0. s, t. Since Ks is only defined if s > 0, we expect that no such N (b) Equation (4.6) is a guess, but here is some justification for it. The author expects that the eigenvalues (in any bounded region) and eigenvectors of DN˜t for small t will be close to those of DN0 , except that DN˜t should have one additional eigenvector ξt , with small eigenvalue λt , where we expect ξt ∼ = χ away from x, and d Ks near Ks , with s ≈ −γδ −1 t. ξt ∼ = ds We can estimate this eigenvalue λt by −1/6 λt = ξt −2 |δ|1/6 t−1/6 )−2 · $χ, DN0 χ%L2 ˜t ξt %L2 ≈ (C|γ| L2 · $ξt , DN

= C −2 |γ|1/3 |δ|−1/3 t1/3 · $χ, 4π δx+ · v − 4π δx− · v%L2 . = 4πC −2 |γ|1/3 |δ|−1/3 t1/3 · limx→x+ v · χ(x) − limx→x− v · χ(x) = 4πC −2 |γ|1/3 |δ|−1/3 t1/3 δ. d Ks L2 = Cs−1/6 for Here in the first step we expect ξt L2 to be dominated by  ds −1 C > 0 and s ≈ −γδ t, and $ξt , DN˜t ξt %L2 to be dominated by $χ, DN0 χ%L2 . Hence we expect DN˜t to have one small eigenvalue λ = O(t1/3 ), which is positive if δ > 0 and negative if δ < 0. So by properties of spectral flow, the canonical flag fN˜t of ˜t should increase by 1 as δ increases through 0, and this is the reason for the last N term in (4.6).

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(c) Motivated by a talk on earlier version of these conjectures given by the author in a conference in London in 2012, Nordstr¨ om [69] proved part of Conjecture 4.4. ˜t in (X, ϕt , ψt ) He shows that for for small s > 0 there exists a associative 3-fold N by gluing Ks into N0 for some unique small t ∈ (− , ), but he does not prove that s ≈ −γδ −1 t. A related conjecture for SL 3-folds was stated in [30, §6] and proved in [44, §9], and also independently by Yng-Ing Lee [58], and by Dan Lee [57]. 5. Another index 1 associative singularity Next we describe a second kind of singularity of associative 3-folds. 5.1. Three families of SL 3-folds in C3 . Let G be the group U(1)2 , acting on C3 by (5.1)

(eiθ1 , eiθ2 ) : (z1 , z2 , z3 ) → (eiθ1 z1 , eiθ2 z2 , e−iθ1 −iθ2 z3 )

for θ1 , θ2 ∈ R.

All the G-invariant special Lagrangian 3-folds in C were written down explicitly by Harvey and Lawson [22, §III.3.A], and studied in more detail in [31, Ex. 5.1] and [34, §4]. Here are some examples of G-invariant SL 3-folds which will be important in what follows. 3

Definition 5.1. Define a subset L0 in C3 by + L0 = (z1 , z2 , z3 ) ∈ C3 : |z1 |2 = |z2 |2 = |z3 |2 , (5.2) , Im(z1 z2 z3 ) = 0, Re(z1 z2 z3 )  0 . Then L0 is a special Lagrangian cone on T 2 , invariant under the Lie subgroup G of SU(3) given in (5.1). Let s > 0, and define + L1s = (z1 , z2 , z3 ) ∈ C3 : |z1 |2 − s = |z2 |2 = |z3 |2 , (5.3) , Im(z1 z2 z3 ) = 0, Re(z1 z2 z3 )  0 , + L2s = (z1 , z2 , z3 ) ∈ C3 : |z1 |2 = |z2 |2 − s = |z3 |2 , (5.4) , Im(z1 z2 z3 ) = 0, Re(z1 z2 z3 )  0 , + L3s = (z1 , z2 , z3 ) ∈ C3 : |z1 |2 = |z2 |2 = |z3 |2 − s, (5.5) , Im(z1 z2 z3 ) = 0, Re(z1 z2 z3 )  0 . Then each Las is a G-invariant, nonsingular, embedded SL 3-fold in C3 diffeomorphic to S 1 × R2 , which is Asymptotically Conical (AC ), with cone L0 . Thus the Las for a = 1, 2, 3 are three different families of AC SL 3-folds in C asymptotic to the same SL cone L0 , each family depending on s ∈ (0, ∞). Hence {0} × Las is a nonsingular AC associative 3-fold in R7 = R × C3 as in §2.2, diffeomorphic to S 1 × R2 for a = 1, 2, 3 and s > 0, asymptotic to the singular associative T 2 -cone {0}×L0 . For brevity we write L0 , Las in place of {0}×L0 , {0}× Las . Write νL0 for the normal bundle of L0 in R7 , and DL0 : Γ∞ (νL0 ) → Γ∞ (νL0 ) for the operator in Theorem 2.12. Define sections ζ1 , ζ2 of νL0 by 3

(5.6)

ζ1 : (0, z1 , z2 , z3 ) −→ (0, 13 z¯1−1 , − 16 z¯2−1 , − 16 z¯3−1 ), ζ2 : (0, z1 , z2 , z3 ) −→ (0, − 16 z¯1−1 , 13 z¯2−1 , − 16 z¯3−1 ).

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Then ζ1 , ζ2 are homogeneous O(r −1 ) with DL0 (ζ1 ) = DL0 (ζ2 ) = 0. A similar analysis to (4.3) shows that (5.7)

L1s ≈ Γsζ1 + O(s2 r −2 ), L3s

2 −2

≈ Γ−sζ1 −sζ2 + O(s r

L2s ≈ Γsζ2 + O(s2 r −2 ), )

and

near L0 \ {0} in R for small s > 0. 7

5.2. Associative 3-folds with singularities modelled on L0 . The next definition sets up notation for our conjecture. Definition 5.2. Let (X, ϕt , ψt ) for t ∈ (− , ) be a smooth family of TAG2 -manifolds, and that N0 a compact associative 3-fold in (X, ϕ0 , ψ0 ) with one singular point x, locally modelled on L0 (or {0} × L0 ) in R7 = R × C3 , under an identification Tx X ∼ = R7 . Write ν for the normal bundle of N0 \ {x} in X, and ∞ ∞ D : Γ (ν) → Γ (ν) for the operator in Theorem 2.12. The author [40–44] studied SL m-folds with isolated conical singularities in (almost) Calabi–Yau m-folds, and very similar techniques should work to study singular associative 3-folds of this type. To do the analysis, we should work in weighted Sobolev spaces L2k,λ (ν) in the sense of Lockhart and McOwen [61, 62], where λ ∈ R is a growth rate, so that roughly L2k,λ (ν) contains sections of ν on N0 \ {x} which grow at rate O(r λ ) near x in N0 , where r is the distance to x. Then D extends to an operator on weighted Sobolev spaces (5.8)

Dk,λ : L2k+1,λ (ν) −→ L2k,λ−1 (ν).

Write νL0 for the normal bundle of L0 \ {0} in R7 , and DL0 : Γ∞ (νL0 ) → Γ∞ (νL0 ) for the corresponding twisted Dirac operator. For each λ ∈ R, define + Vλ = s ∈ Γ∞ (νL0 ) : DL0 (s) = 0 and s is homogeneous of order , O(r λ ) under dilations of L0 . Then Vλ is finite-dimensional, isomorphic to the kernel of an elliptic operator on the link T 2 of L0 . Write D L0 ⊂ R for the set of λ with Vλ = 0. Then D L0 is discrete. The Lockhart–McOwen theory implies that Dk,λ in (5.8) is Fredholm if and only if λ ∈ R \ D L0 , where the index, kernel and cokernel of Dk,λ are independent of k ∈ N, and if λ1 , λ2 ∈ R \ D L0 with λ1 < λ2 then  (5.9) ind(Dk,λ1 ) = ind(Dk,λ2 ) + dim Vμ . λ∈D L0 :λ1 0 and t ∈ (− , ). Then near N0 we can write ˜t1 ≈ Γθ for θ ∈ Γ∞ (ν). As N ˜t1 is associative we must have D(θ) = tξ + O(t2 ). N 1 1 ˜ Since Nt approximates Ls near x, from (5.7) we see that θ ≈ sζ1 + O(1). We now show that tγ = $χ, tξ%L2 − 0 = $χ, D(θ)%L2 − $D(χ), θ%L2 + O(t2 )

(5.11)

= (δ1 ζ1 + δ2 ζ2 ) ∧ (sζ1 ) + O(t2 ) = −δ2 s + O(t2 ).

Here one might expect that $χ, D(θ)%L2 = $D(χ), θ%L2 , as D is self-adjoint. However, as χ, θ = O(r −1 ) and ∇χ, ∇θ = O(r −2 ), so that the L2 -inner products between χ, θ and ∇χ, ∇θ are not defined, it turns out that $χ, D(θ)%L2 − $D(χ), θ%L2 = boundary term, where the boundary term is obtained by completing N0 \{x} to a compact manifold ¯0 with boundary ∂ N ¯0 = T 2 , and using Stokes’ Theorem. N The boundary term depends only on the leading terms χ = δ1 ζ1 + δ2 ζ2 + · · · , θ = sζ1 +· · · in V−1 , and may be written in terms of an antisymmetric bilinear form ∧ : V−1 ×V−1 → R, as in the third step of (5.11). Guessing (out of laziness) that this is normalized with ζ1 ∧ζ2 = 1 gives the final step of (5.11). Thus tγ = −δ2 s+O(t2 ), ˜t1 in (i) exists only when γδ −1 t < 0, as s > 0. giving s ≈ −γδ2−1 t, and showing that N 2 Parts (ii),(iii) are similar, using (5.7) for L2s , L3s . (b) A related conjecture for SL 3-folds with singularities modelled on L0 ⊂ C3 was stated in [30, §3.2], and now follows from work of the author [40–44] and Imagi [26]. Proving Conjecture 5.3 should not be that difficult, by adapting known technology for special Lagrangians to the associative case. 5.3. Algebraic topology of desingularizations using Las . In [30, §4] the author discussed starting with a compact SL 3-fold N0 with one singular point locally modelled on L0 ⊂ C3 in (5.2) in an (almost) Calabi–Yau 3-fold (Y, J, h), and desingularizing N0 by gluing in Las ⊂ C3 for a = 1, 2, 3 and small s > 0 ˜sa in Y . In [30, §4.3] from (5.3)–(5.5) to get compact nonsingular SL 3-folds N a ˜s ; Z) from H1 (N0 ; Z). This is a we computed the integral homology groups H1 (N purely topological calculation, and so applies just as well to smoothing associative 3-folds with singularities modelled on L0 ⊂ R7 by gluing in Las ⊂ R7 , as in §5.2. Thus, from [30, §4.2] we deduce: Proposition 5.5. Work in the situation of Conjecture 5.3. Write P = N0 \ B (x), for B (x) a ball of radius about x in X for > 0 small. Then P is a compact, nonsingular 3-manifold with boundary, where ∂P may be identified with G = T 2 in (5.1), since ∂(L0 \ B (x)) is a free G-orbit. Define ρ : Z2 → H1 (P ; Z) to be the composition of natural morphisms Z2

H1 (G; Z)

∼ =

/ H1 (∂P ; Z)

inc∗

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/ H1 (P ; Z).

CONJECTURES ON COUNTING ASSOCIATIVE 3-FOLDS IN G2 -MANIFOLDS

129

Then Ker(ρ) ∼ = Z, so Ker ρ = $(b1 , b2 )%Z for (b1 , b2 ) ∈ Z2 \ {0} unique up to sign. ˜ta ; Z) are determined by the exact sequences Also H1 (N0 ; Z) and H1 (N Z2 Z Z Z

ρ

/ H1 (P ; Z)

/ H1 (N0 ; Z)

/ 0,

n→ρ(n,0)

/ H1 (P ; Z)

/ H1 ( N ˜t1 ; Z)

/ 0,

n→ρ(0,n)

/ H1 (P ; Z)

˜t2 ; Z) / H1 ( N

/ 0,

n→ρ(−n,−n)

/ H1 (P ; Z)

˜t3 ; Z) / H1 ( N

/ 0.

˜t1 ; Z) for a = 1, 2, 3. If H1 (N0 ; Z) is infinite then so are H1 (N Suppose now that H1 (N0 ; Z) is finite. Then we have 8 / / / / / / 1 ˜t ; Z)/ = |b1 | · H1 (N0 ; Z) , b1 = 0, /H1 (N ∞, b1 = 0, 8 / / / / / / / H1 ( N ˜t2 ; Z)/ = |b2 | · H1 (N0 ; Z) , b2 = 0, ∞, b2 = 0, 8 / / / / / /  0, / H1 ( N ˜t3 ; Z)/ = | − b1 − b2 | · H1 (N0 ; Z) , −b1 − b2 = ∞, −b1 − b2 = 0. Hence if we define an invariant I of compact 3-manifolds N by 8/ / /H1 (N ; Z)/, H1 (N ; Z) is finite, (5.12) I(N ) = 0, otherwise, then in all cases in Conjecture 5.3 we have ˜t1 ) + sign(b2 ) · I(N ˜t2 ) + sign(−b1 − b2 ) · I(N ˜t3 ) = 0. (5.13) sign(b1 ) · I(N Note too that for all compact 3-manifolds N1 , N2 we have (5.14)

I(N1 #N2 ) = I(N1 ) · I(N2 ).

Conjecture 5.6. In the situation of Conjecture 5.3, there is some formula re˜t1 , N ˜t2 , N ˜t3 , depending on γ, δ1 , δ2 , b1 , b2 . If we choose lating the canonical flags of N a flag structure on X then the corresponding orientations satisfy   ˜ta ) · I(N ˜ta ) = ˜ta ) · I(N ˜ta ). (5.15) Or(N Or(N ˜ a exists when t < 0 a = 1, 2, 3 : N t

˜ a exists when t > 0 a = 1, 2, 3 : N t

Observe that Conjecture 5.6 is plausible by (5.13), as there are always at least ˜t1 ), Or(N ˜t2 ), Or(N ˜t3 ) for which (5.15) holds. The point two choices of signs Or(N of (5.15) is that as we cross the ‘wall’ t = 0 in the family of TA-G2 -manifolds (X, ϕt , ψt ), the signed weighted count of associative 3-folds does not change. In [30] the author made a similar proposal to define invariants of (almost) Calabi– Yau 3-folds by counting SL 3-folds N weighted by I(N ) in (5.12). Remark 5.7. (a) Let N be a compact oriented 3-manifold. If b1 (N ) = 0 then U(1) the moduli space MN of flat U(1)-connections on N is finite, and is |H1 (N ; Z)| 1 U(1) is a finite number of copies of T b (N ) , so points. If b1 (N ) > 0 then MN U(1) U(1) χ(MN ) = 0. In both cases, χ(MN ) = I(N ) in (5.12).

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In §7 we propose counting associative 3-folds N in (X, ϕ, ψ), with signs, and weighted by I(N ). Thus, we can interpret this as counting associative 3-folds with flat U(1)-connections. This may have an interpretation in String Theory or Mtheory, as counting some kind of brane, such as D3-branes in Type IIB String Theory on the G2 -manifold, or M2-brane instantons in M-theory. (b) The programme of §7 would work using any invariant I of compact oriented 3-manifolds satisfying (5.13)–(5.14), and such that I(N ) = 0 if b1 (N ) > 0. The author expects that I in (5.12) is the unique such invariant. 6. U(1)-invariant associative 3-folds in R7 Next we discuss a class of U(1)-invariant associative 3-folds in R7 which should be amenable to study using analytic techniques, and will provide a large class of examples of singularities of associative 3-folds. Understanding the behaviour of these singularities may help guide any programme for defining invariants by counting associative 3-folds. This class is closely related to the author’s papers [36–39] on U(1)-invariant SL 3-folds in C3 . 6.1. Associative 3-folds and J-holomorphic curves. We will study associative 3-folds N in R7 invariant under the U(1)-action (6.1)

eiθ : (x1 , . . . , x7 ) −→ (x1 , x2 , x3 , cos θ x4 − sin θ x5 , sin θ x4 + cos θ x5 , cos θ x6 + sin θ x7 , − sin θ x6 + cos θ x7 ).

This preserves+ g0 , ϕ0 , ∗ϕ0 on R7 from §2.1., The U(1)-action fixes the associative 3-plane R3 = (x1 , x2 , x3 , 0, 0, 0, 0) : xj ∈ R in R7 . Define U(1)-invariant quadratic polynomials y1 , y2 , y3 on R7 by y1 (x1 , . . . , x7 ) = x24 + x25 − x26 − x27 , y2 (x1 , . . . , x7 ) = 2(x4 x7 + x5 x6 ), y3 (x1 , . . . , x7 ) = 2(x4 x6 − x5 x7 ). Then y12 + y22 + y32 = (x24 + x25 + x26 + x27 )2 . Consider the map Π = (x1 , x2 , x3 , y1 , y2 , y3 ) : R7 −→ R6 . This is U(1)-invariant, and its fibres are exactly the U(1)-orbits in R7 . Hence it descends to a homeomorphism +Π : R7 /U(1) → R6 . The U(1)-fixed locus R3 ⊂ R7 , 3 6 maps to the 3-plane L = R = (x1 , x2 , x3 , 0, 0, 0) : xj ∈ R in R . Note that we should not think of R7 /U(1) as a smooth manifold near the fixed locus R3 ⊂ R7 . The identification R7 /U(1) ∼ = R6 is only topological, not smooth, near R3 , and we should expect singular behaviour near R3 ⊂ R6 . The next proposition relates U(1)-invariant associative 3-folds N in R7 \ R3 to J-holomorphic curves Σ in R6 \ R3 , for a certain almost complex structure J on R6 \ R3 . It is similar to [36, Prop. 4.1]. Proposition 6.1. Let R6 have coordinates (x1 , x2 , x3 , y1 , y2 , y3 ), and write + , 3 L = R = (x1 , x2 , x3 , 0, 0, 0) : xj ∈ R ⊂ R6 . Define u : R6 → [0, ∞) by u(x1 , x2 , x3 , y1 , y2 , y3 ) = (y12 + y22 + y32 )1/2 . Define an almost complex structure J

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CONJECTURES ON COUNTING ASSOCIATIVE 3-FOLDS IN G2 -MANIFOLDS

on R6 \ R3 to have matrix ⎛ 0 0 ⎜ 0 0 ⎜ ⎜ 0 0 (6.2) J =⎜ ⎜2u1/2 0 ⎜ ⎝ 0 2u1/2 0 0

0 0 0 0 0 2u1/2

− 12 u−1/2 0 0 0 0 0

0 1 −1/2 −2u 0 0 0 0

131

⎞ 0 ⎟ 0 ⎟ 1 −1/2 ⎟ −2u ⎟ ⎟ 0 ⎟ ⎠ 0 0

∂ with respect to the basis of sections ∂x , ∂ , ∂ , ∂ , ∂ , ∂ of T (R6 \ R3 ). 1 ∂x2 ∂x3 ∂y1 ∂y2 ∂y3 Suppose N is a U(1)-invariant 3-submanifold in R7 \ R3 , so that Σ = N/U(1) is a 2-submanifold in R6 \ R3 ∼ = (R7 \ R3 )/U(1). Then N is an associative 3-fold 7 3 in R \ R if and only if Σ is a J-holomorphic curve in R6 \ R3 .

Note that J in (6.2) becomes singular when u = 0, that is, on L = R3 ⊂ R6 . + , Example 6.2. Let N be the associative 3-plane (x1 , 0, 0, x4 , x5 , 0, 0) : xj ∈ R in R7 . Then N is U(1)-invariant, and Σ = N/U(1) is the half-plane , + Σ = (x1 , 0, 0, y1 , 0, 0) : x1 ∈ R, y1 ∈ [0, ∞) ∼ = R × [0, ∞), which has boundary ∂Σ ⊂ L ⊂ R6 . This example illustrates the general principle that J-holomorphic curves Σ in R6 with boundary ∂Σ in L ⊂ R6 lift to associative 3-folds N = Π−1 (Σ) without boundary in R7 . Note that J is singular along L. One moral is that we should expect any theory ‘counting’ associative 3-folds N in a TA-G2 -manifold (X, ϕ, ψ) to look more like Lagrangian Floer cohomology [17, 18] (built on counting J-holomorphic curves Σ with boundary in L) than like Gromov–Witten theory [19, 25, 66] (built on counting J-holomorphic curves Σ without boundary). Identify R6 with C3 with complex coordinates (x1 +iy1 , x2 +iy2 , x3 +iy3 ). This corresponds to the complex structure J0 , with matrix ⎛ ⎞ 0 0 0 −1 0 0 ⎜0 0 0 0 −1 0 ⎟ ⎜ ⎟ ⎜0 0 0 0 0 −1⎟ ⎜ ⎟ J0 = ⎜ 0 0⎟ ⎜1 0 0 0 ⎟ ⎝0 1 0 0 0 0⎠ 0 0 1 0 0 0 ∂ , ∂ , ∂ , ∂ , ∂ , ∂ , so that J in (6.2) becomes with respect to the basis ∂x 1 ∂x2 ∂x3 ∂y1 ∂y2 ∂y3 1/2 by 1. This J0 is compatible with the standard symplectic J0 if we replace 2u structure ω0 = dx1 ∧ dy1 + dx2 ∧ dy2 + dx3 ∧ dy3 on R6 , for which L is a Lagrangian submanifold. The next conjecture is not very precise:

Conjecture 6.3. J-holomorphic curves in R6 (with boundary in L) have essentially the same qualitative behaviour as ordinary J0 -holomorphic curves in R6 = C3 (with boundary in L), which are already very well understood. In [36–39] the author studied U(1)-invariant SL 3-folds in C3 , in terms of solutions of a singular nonlinear Cauchy–Riemann equation. These correspond to studying J-holomorphic curves in the R6 above lying in the R4 ⊂ R6 defined by x1 = 0, y1 = a. One moral of [36–39] is that the singular nonlinear Cauchy– Riemann equation behaves exactly like the usual Cauchy–Riemann equation, for

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questions such as existence and uniqueness of solutions with prescribed boundary data. The author expects a similar picture for this more general class. If we accept Conjecture 6.3 then we can give heuristic descriptions of a large class of singularities of associative 3-folds: every kind of singularity of J0 -holomorphic curves in C3 , possibly with boundary in a Lagrangian L, should correspond to a kind of singularity of associative 3-folds. Both the associative singularities in §4–§5 can be made invariant under (5.1), and so interpreted in this framework, as the next two examples show. 6 − ˜ Example 6.4. Consider the J0 -holomorphic curves Σ+ t , Σt and Σs in R with boundary in L, for s  0 and t ∈ R: + , Σ+ t = (x1 , 0, t, y1 , 0, 0) : x1 ∈ R, y1 ∈ [0, ∞) , + , Σ− t = (0, x2 , −t, 0, y2 , 0) : x2 ∈ R, y2 ∈ [0, ∞) , + , ˜ s = (x1 , x2 , 0, y1 , y2 , 0) : (x1 + iy1 )(x2 + iy2 ) = −s, y1 , y2  0 . Σ − Here Σ+ t , Σt do not intersect for t = 0, and when t = 0 they intersect in one ˜ 0 = Σ+ ∪ Σ− , but Σ ˜ s for s > 0 point (0, . . . , 0) in their common boundary. Also Σ 0 0 + − ˜ − ˜ is diffeomorphic to [0, 1] × R. Write Nt , Nt , Ns for the preimages of Σ+ t , Σt , Σs 7 6 3 7 + − under Π : R → R . Then Nt , Nt are affine associative 3-planes R ⊂ R , and ˜s for s > 0 is diffeomorphic to S 2 × R ∼ N = R3 #R3 , and is a distorted version of the V,V  associative 3-fold Ks in §4.1. This is an approximate local model for the index one singularity of associative 3-folds described in §4: we have associative 3-folds Nt+ , Nt− in (X, ϕt , ψt ), which are disjoint for t = 0, and intersect in one point {x} when t = 0. As t passes ˜s diffeomorphic to Nt+ #Nt− . through 0 we create a new associative 3-fold N

˜ s in R6 , where Σ ˜s Example 6.5. Consider the J0 -holomorphic curves Σt and Σ has boundary in L, for s  0 and t ∈ R: + , Σt = (x1 , x2 , 0, x2 , −x1 , t) : x1 ∈ R, y1 ∈ [0, ∞) , + , ˜ s = (x1 , x2 , 0, y1 , y2 , 0) : (x1 + iy1 )2 + (x2 + iy2 )2 = s, x2 y1 − x1 y2  0 . Σ ∼ R2 , which does not intersect L when t = 0, and intersects L in one Then Σt = ˜ 0 = Σ0 , and Σ ˜ s for s > 0 is diffeomorphic to point (0, . . . , 0) when t = 0. Also Σ + , S 1 × [0, ∞), with boundary the circle (x1 , x2 , 0, 0, 0, 0) : x21 + x22 = s in L. ˜s for the preimages of Σt , Σ ˜ s under Π : R7 → R6 . Then N0 = N ˜0 is Write Nt , N 7 1 2 2 ˜ a T -cone in R , and Nt , Ns for s, t = 0 are diffeomorphic to S × R . In fact Nt for t < 0 and Nt for t > 0 differ by a Dehn twist around S 1 ⊂ Nt . So we should regard ˜s , s > 0 as three different families of 3-manifolds Nt , t < 0 and Nt , t > 0 and N 2 1 2 ˜0 . These are distorted versions of the S × R desingularizing the T -cone N0 = N 2 1 2 1 2 associative T -cone L0 and S × R ’s Ls , Ls , L3s in §5.1. 6.2. Associative 3-folds with boundary in coassociatives. Next we use the ideas of §6.1 to discuss associative 3-folds with boundary in a coassociative 4-fold, as in §2.7. Let C be the coassociative 4-plane , + C = (0, x2 , x3 , x4 , x5 , 0, 0) : xj ∈ R ⊂ R7 , which is invariant under the U(1)-action (6.1). Then + , M = C/U(1) = (0, x2 , x3 , y1 , 0, 0) : x2 , x3 ∈ R, y1 ∈ [0, ∞) ∼ = [0, ∞) × R2 .

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CONJECTURES ON COUNTING ASSOCIATIVE 3-FOLDS IN G2 -MANIFOLDS

133

We think of M as a Lagrangian half-plane in R6 ∼ = C3 with boundary in L = R3 ⊂ 3 C . In the language of §2.3, L is special Lagrangian with phase 1, and M is special Lagrangian with phase i. Suppose now that N is a U(1)-invariant associative 3-fold in R7 with ∂N ⊂ C. Then Σ = N/U(1) is a (possibly singular) J-holomorphic curve in R6 , which can have boundary ∂Σ of two kinds. As in Example 6.2, the fixed locus of U(1) in N (which may lie in the interior N ◦ ) gives a boundary component ∂L Σ of Σ in L. And ∂N/U(1) gives a boundary component ∂M Σ of Σ in M . Thus we expect that ∂Σ = ∂L Σ ∪ ∂M Σ ⊂ L ∪ M , where Σ may have codimension 2 corners ∂L Σ ∩ ∂M Σ mapping to L ∩ M . Thus we conclude: Counting associative 3-folds N with boundary ∂N ⊂ C in a coassociative 4-fold C in a TA-G2 -manifold (X, ϕ, ψ), is analogous to counting J-holomorphic curves Σ in a symplectic manifold (Y, ω) with boundary ∂Σ ⊂ L ∪ M, where L is a Lagrangian in Y, and M is another Lagrangian in Y with boundary ∂M ⊂ L. The author does not know of any symplectic theory involving counting Jholomorphic curves with boundary in L ∪ M in this way. If we assume Conjecture 6.3 we can give heuristic models for singularities of U(1)-invariant associative 3-folds N with boundary in C. Here is one with index one: Example 6.6. Let s  0, and consider the J0 -holomorphic map + , fs : Σ = a + ib ∈ C : a, b  0 −→ R6 = C3 , fs : a + ib → (x1 + iy1 , x2 + iy2 , x3 + iy3 ) = (s(a + ib) − (a + ib)3 , (a + ib)2 , 0). + , Then fs maps the boundary component (a, 0) : a ∈ [0, ∞) of Σ to L ⊂ R6 , + , and the boundary component (0, b) : b ∈ [0, ∞) of Σ to M ⊂ R6 , so fs (Σ) is a J0 -holomorphic curve in R6 with boundary in L ∪ M . If s < 0 then fs does not map (0, b) to M for small b > 0, which is why we restrict to s  0. Let Ns be the preimage of fs (Σ) under Π : R7 → R6 . Then Ns for s > 0 is a nonsingular 3-submanifold of R7 diffeomorphic to√[0, ∞) × R2 , with boundary ∂Ns ⊂ C. One interior point of Ns , from a + ib = s, maps to C. Also N0 is homeomorphic to [0, ∞) × R2 , but is not smooth at (0, . . . , 0). These Ns are not associative, since fs is holomorphic with respect to J0 rather than J. But as in Conjecture 6.3, we expect there to exist J-holomorphic maps f˜s with essentially ˜s very like the Ns . the same behaviour as fs , yielding associative 3-folds N ˜ Such Ns , s  0 should provide an example of an index one singularity of associative 3-folds N with boundary in coassociative 4-folds C. That is, singularities of this type occur in codimension one in generic families of TA-G2 -manifolds, and so could cause numbers of associatives N with ∂N ⊂ C to change under deformation. Because of all this, the author expects that it is not possible to define an interesting Floer-type theory for coassociative 4-folds C in (X, ϕ, ψ), suitably deformation-invariant in ϕ, ψ, involving counting associatives N with ∂N ⊂ C, following the analogy of Lagrangian Floer cohomology or Fukaya categories in symplectic geometry, say. But the author is not completely certain.

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7. A superpotential counting associative 3-folds 7.1. Set up of situation and notation. In §7 we will consider the following situation, and use the following notation. Let X be a compact, oriented 7-manifold, 3 and γ ∈ HdR (X; R). Write F γ for the set of closed 4-forms ψ on X such that there 3 (X; R), for which (X, ϕ, ψ) is a exists a closed 3-form ϕ on X with [ϕ] = γ ∈ HdR TA-G2 -manifold, with the given orientation on X. Suppose F γ = ∅. Then F γ is open in the vector space of closed 4-forms on X, and so is infinite-dimensional. We will be discussing moduli spaces M(N , α, ψ) of compact associative 3-folds N in such TA-G2 -manifolds (X, ϕ, ψ), but note as in §2.5 that M(N , α, ψ) depends only on ψ and the orientation on X, not on the choice of ϕ. Given any ψ or ψt , t ∈ [0, 1] in F γ , we generally implicitly suppose we have chosen ϕ or ϕt , t ∈ [0, 1] to make TA-G2 -manifolds (X, ϕ, ψ) or (X, ϕt , ψt ), but this is just for notational convenience, the choices of ϕ, ϕt do not affect anything. We often restrict to ψ which is generic in F γ , as we expect this will simplify the singular behaviour of associatives considerably, as in Conjecture 2.16. Given generic ψ0 , ψ1 in the same connected component of F γ , we can choose a smooth 1-parameter family ψt , t ∈ [0, 1] in F γ connecting ψ0 , ψ1 . We often restrict to a generic 1-parameter family ψt , t ∈ [0, 1], that is, to a family which is generic amongst all smooth 1-parameter families with fixed end-points ψ0 , ψ1 . We expect that this will simplify the singular behaviour of associative 3-folds in (X, ϕt , ψt ) for t ∈ (0, 1) considerably. Fix a flag structure F on X, as in §3.1. Then as in §3.2 we have orientations Or(N ) = ±1 on M(N , α, ψ) at [N ] for all compact, unobstructed associative 3-folds N in (X, ϕ, ψ). Let F be the field Q, R or C. As in §1, write Λ for the Novikov ring over F: , + ∞ αi : ci ∈ F, αi ∈ R, αi → ∞ as i → ∞ , (7.1) Λ= i=1 ci q with q a formal variable. Then Λ is a commutative F-algebra. Define v : Λ → R # {∞} by v(λ) is the least α ∈ R with the coefficient of q α in λ nonzero for λ ∈ Λ \ {0}, and v(0) = ∞. Write Λ0 ⊂ Λ for the subalgebra of λ ∈ Λ with v(λ)  0, and Λ>0 ⊂ Λ0 for the ideal of λ ∈ Λ with v(λ) > 0. Then Λ is a complete non-Archimedean field in the sense of Bosch, G¨ untzer and Remmert [10, §A], with valuation λ = 2−v(λ) , so we can consider rigid analytic spaces over Λ as in [10, §C]. These are like schemes over Λ, except that polynomial functions on schemes are replaced by convergent power series. Consider 1 + Λ>0 ⊂ Λ as a group under multiplication in Λ. Write . U = Hom H3 (X; Z), 1 + Λ>0 for the set of group morphisms θ : H3 (X; Z) → 1 + Λ>0 . By choosing a basis e1 , . . . , en for H3 (X; Z)/torsion, where n = b3 (X), we can identify U ∼ = Λn>0 by θ∼ = (λ1 , · · · , λn ) if θ(ei ) = 1 + λi for i = 1, . . . , n, where Λ>0 is the open unit ball in Λ in the norm  . . We regard U as a smooth rigid analytic space over Λ. A map Υ : U → U will be called a quasi-identity morphism if: (i) Writing Υ(θ) = (Υ1 (λ1 , . . . , λn ), . . . , Υn (λ1 , . . . , λn )) under U ∼ = Λn>0 , each Υi is given by a power series in λ1 , . . . , λn convergent in Λ>0 . (ii) There exists > 0 such that if (λ1 , . . . , λn ), (λ1 , . . . , λn ) ∈ Λn>0 and δ > 0 with λi − λi ∈ q δ · Λ0 for i = 1, . . . , n then Υj (λ1 , . . . , λn ) − λj − Υj (λ1 , . . . , λn ) + λj ∈ q δ+ · Λ0 for j = 1, . . . , n.

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Here (i) implies that Υ is a morphism of rigid analytic varieties. Using (ii) we can show that Υ : U → U is a bijection, and Υ−1 is also a quasi-identity morphism, so that Υ is an isomorphism of rigid analytic varieties. Quasi-identity morphisms are closed under composition, and form a group. 7.2. Six kinds of wall-crossing behaviour. Suppose now that ψ0 , ψ1 ∈ F γ are generic, and ψt , t ∈ [0, 1] is a generic 1-parameter family joining ψ0 , ψ1 . As in §2.6, fix N ∈ D and α ∈ H3 (N ; Z). We want to know how the moduli spaces M(N , α, ψt ) can change over t ∈ [0, 1]. We briefly sketch six conjectural ways in which this can happen, labelled (A)–(F), where (A) comes from §3, (B)–(D) from §4, and (E) from §5. All of (A)–(F) can also happen in reverse, that is, we can replace ψt by ψ1−t . When we say ‘associative 3-folds of interest’, we just mean the family of associative 3-folds in (X, ϕt , ψt ) whose behaviour we are describing. There may of course be many other associative 3-folds in (X, ϕt , ψt ) as well. 7.2(A) Cancelling non-singular associatives with opposite signs. As explained in Example 3.14, we expect the following can happen in generic families ψt , t ∈ [0, 1], for some t0 ∈ (0, 1): • For t ∈ [0, t0 ) there are no associative 3-folds of interest in (X, ϕt , ψt ). • There is a single compact, nonsingular associative 3-fold Nt0 of interest in (X, ϕt0 , ψt0 ). It is obstructed, with ONt0 ∼ = R. • For t ∈ (t0 , 1] there are two compact, nonsingular, unobstructed associative 3-folds Nt+ , Nt− of interest in (X, ϕt , ψt ), with limt→t0 − Nt+ = limt→t0 − Nt+ = Nt0 . They are diffeomorphic to Nt0 and in the same homology class in α ∈ H3 (N ; Z), and have Or(Nt+ ) = 1 and Or(Nt− ) = −1. The canonical flags of Nt+ , Nt− differ by 1, in a suitable sense. Provided we count unobstructed associatives [N ] ∈ M(N , α, ψ) weighted by Or(N ) (possibly multiplied by some 3-manifold invariant I(N )), the count does not change over t ∈ [0, 1] under this transition. 7.2(B) Intersecting associatives Nt±0 give a connect sum Nt+0 #Nt−0 . As explained in Definition 4.3(a) and Conjecture 4.4, we expect the following can happen in generic families ψt , t ∈ [0, 1], for some t0 ∈ (0, 1): • For all t ∈ [0, 1] there are compact, connected, unobstructed associatives Nt+ , Nt− in (X, ϕt , ψt ), depending smoothly on t. For t = t0 we have Nt+ ∩ Nt− = ∅, but Nt+0 ∩ Nt−0 = {x}, and Nt+ , Nt− cross transversely at x with nonzero speed as t increases through t0 . ˜t in • For t ∈ (t0 , 1] there is a compact, unobstructed associative 3-fold N + (X, ϕt , ψt ), depending smoothly on t. It is diffeomorphic to Nt #Nt− , ˜t ] = [Nt+ ] + [Nt− ] in H3 (X; Z), with limt→t − N ˜t = Nt+ ∪ Nt− . with [N 0 0 0 ˜t ) = No such associative of interest exists for t ∈ [0, t0 ]. We have Or(N Or(Nt+ ) · Or(Nt− ) · , where = ±1 according to whether Nt+ crosses Nt− with positive or negative intersection number in X. 7.2(C) Self-intersecting Nt0 gives a connect sum Nt0 #(S 1 × S 2 ). As explained in Definition 4.3(b) and Conjecture 4.4, we expect the following can happen in generic families ψt , t ∈ [0, 1], for some t0 ∈ (0, 1): • For all t ∈ [0, 1] there is a compact, connected, unobstructed associative Nt in (X, ϕt , ψt ), depending smoothly on t. Here Nt0 is immersed, with a self-intersection point x ∈ X, the image of distinct points x+ , x− in Nt0 .

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The two sheets of Nt near x+ , x− cross transversely at x with nonzero speed as t increases through t0 . ˜t in • For t ∈ (t0 , 1] there is a compact, unobstructed associative 3-fold N (X, ϕt , ψt ), depending smoothly on t. It is the self-connect-sum of Nt0 at ˜t ] = [Nt ] in H3 (X; Z), x+ , x− , diffeomorphic to Nt #(S 1 × S 2 ). It has [N ˜ and limt→t0 − Nt = Nt0 . No such associative of interest exists for t ∈ [0, t0 ]. ˜t ∼ ˜t )  1, so N ˜t is not a Q-homology Note that N = Nt #(S 1 × S 2 ) has b1 (N 3-sphere. Thus, if we count only associative Q-homology 3-spheres, the count does not change over t ∈ [0, 1] under this transition. 7.2(D) Self-intersecting Nt0 gives a connect sum Nt0 #Nt0 . Here is a combination of (B),(C) above: • For all t ∈ [0, 1] there is a compact, connected, unobstructed associative Nt in (X, ϕt , ψt ), depending smoothly on t. Here Nt0 is immersed, with a self-intersection point x ∈ X, the image of distinct points x+ , x− in Nt0 . The two sheets of Nt near x+ , x− cross transversely at x with nonzero speed as t increases through t0 . ˜t in • For t ∈ (t0 , 1] there is a compact, unobstructed associative 3-fold N (X, ϕt , ψt ), depending smoothly on t. It is the connect sum of two copies of ˜t ] = 2[Nt ] in H3 (X; Z), Nt0 at x+ , x− , diffeomorphic to Nt #Nt . It has [N ˜ and limt→t0 − Nt = 2Nt0 . No such associative of interest exists for t ∈ ˜t ) = , where = ±1 according to whether Nt near [0, t0 ]. We have Or(N + − x crosses Nt near x with positive or negative intersection number in X. 7.2(E) Three families Nt1 , Nt2 , Nt3 from Nt0 with T 2 -cone singularity. As explained in Definition 5.2 and Conjectures 5.3 and 5.6, we expect the following can happen in generic families ψt , t ∈ [0, 1], for some t0 ∈ (0, 1): • For all t ∈ [0, t0 ) there is a compact, unobstructed associative Nt1 in (X, ϕt , ψt ), depending smoothly on t. • For all t ∈ (t0 , 1] there are compact, unobstructed associatives Nt2 , Nt3 in (X, ϕt , ψt ), depending smoothly on t. • There is a compact associative Nt0 in (X, ϕt0 , ψt0 ) with one singular point at x ∈ X locally modelled on the associative T 2 -cone L0 ⊂ R7 from §5.1. We have limt→t0 − Nt1 = limt→t0 + Nt2 = limt→t0 + Nt3 = Nt0 , where Nta is locally modelled near x on Las ⊂ R7 in §5.1, for |t − t0 | and s > 0 small. • Writing I for the 3-manifold invariant in (5.12), from (5.15) we have (7.2)

Or(Nt1 ) · I(Nt1 ) = Or(Nt2 ) · I(Nt2 ) + Or(Nt3 ) · I(Nt3 ).

If we count unobstructed associatives [N ] ∈ M(N , α, ψ) weighted by Or(N ) · I(N ), equation (7.2) implies that the count does not change over t ∈ [0, 1] under this transition. Note that I(N ) = 0 unless N is a Q-homology sphere, so this is consistent with counting only associative Q-homology 3-spheres, as in (C). 7.2(F) Multiple cover phenomena. This is one of the less satisfactory parts of this paper. The author expects that in generic families ψt , t ∈ [0, 1], for some t0 ∈ (0, 1), ˆt in (X, ϕt , ψt ) for t ∈ (t0 , 1] it can happen that a family of associative 3-folds N can converge as t → t0 to a branched multiple cover of some associative Nt0 in (X, ϕt0 , ψt0 ), where Nt0 may be obstructed, or immersed, or singular. There may be several ways in which this can happen.

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We illustrate this using (B) above. We expect the following can happen in generic families ψt , t ∈ [0, 1], for some t0 ∈ (0, 1): ˜t be as in (B). Then for t ∈ (t0 , 1] there is a compact, • Let Nt± , t0 , x, N ˆt in (X, ϕt , ψt ), depending smoothly on unobstructed associative 3-fold N + ˆt ∼ t. Topologically, we have N Nt+ #k− Nt− #l (S 1 ×S 2 ), where k±  1 k = ˆt is the connect sum of k+ with (k+ , k− ) = (1, 1) and l  0. That is, N + − − + copies of Nt and k copies of Nt at k + k− + l − 1 pairs of points. ˆt converges to a branched multiple cover of Nt+ ∪ Nt− , with • As t → t0 , N 0 0 multiplicity k+ over Nt+0 and multiplicity k− over Nt+0 . There is a 1dimensional singular set S ⊂ Nt+0 ∪ Nt−0 with x ∈ S, probably a union of points x and curves γ with end-points. Over Nt+0 \ S (or Nt−0 \ S), k+ ˆt converge smoothly to Nt± \ S. On the interiors sheets (or k− sheets) of N 0 ˆt should look like a double cover of Nt± branched γ ◦ of curves γ in S, N 0 along γ ◦ , as for branched covers of Riemann surfaces but one dimension higher. At points x or end-points of curves γ in S, the local models for ˆt converges to Nt+ ∪ Nt− are more complicated. how N 0 0 Using the ideas of §6 we can write down heuristic U(1)-invariant local models for ˆt can converge to Nt+ ∪ Nt− , based on branched-cover behaviour for families how N 0 0 of J0 -holomorphic curves in C3 with boundary in L ⊂ C3 . However, the author does not have a conjectural global description of how such multiple cover transitions happen, that is detailed enough to predict how many ˆt of each type (N , α) are created or destroyed in each such transition. associatives N Such a global description would necessarily be complicated. ˆt1 of type (k+ , k− , l1 ) and N ˆt2 In the example above, suppose we have families N 1 1 + − 1 ˆt crosses of type (k2 , k2 , l2 ) for t ∈ (t0 , 1]. If we deform the geometry so that N ˆt2 , then as in (B) above we create a new associative N ˆt1 #N ˆt2 , which is another N ˆt N of type (k+ , k− , l) = (k1+ + k2+ , k1− + k2− , l1 + l2 ). Because of this, the number of ˆt ’s of type (k+ , k− , l) that appear or disappear as t crosses t0 will depend on all N ˆt of type (k+ , k− , l ) for (k+ , k− , l ) < (k+ , k− , l), and the canonical the other N ˆt , and their pairwise ‘linking numbers’. flags of these N We can see (D) as the simplest example of such a multiple cover transition. Similar (but simpler) multiple cover phenomena occur for J-holomorphic curves in symplectic geometry, and do not spoil the deformation-invariance. 7.3. Definition of the superpotential. Work in the situation of §7.1, and assume Conjecture 2.16. Let ψ ∈ F γ be generic. We will define a superpotential Φψ : U → Λ>0 , which is a generating function for Gromov–Witten type invariants GWψ,α counting associative Q-homology spheres N in (X, ϕ, ψ) with [N ] = α ∈ H3 (X; Z), depending on some arbitrary choices. Definition 7.1. For i = 0, . . . , 7, choose elements ei1 , . . . , eibi (X) in Hi (X; Z) such that ei1 , . . . , eibi (X) is a basis for Hi (X; Q), with e71 = [X]. Choose compact, embedded, oriented, generic i-dimensional submanifolds C1i , . . . , Cbii (X) in X with [Cji ] = eij in Hi (X; Z) for j = 1, . . . , bi (X), with C17 = X. By the K¨ unneth theorem, eij  e7−i for j = 1, . . . , bi (X), k = 1, . . . , b7−i (X) is k is represented by a basis for the homology group H7 (X × X; Q), where eij  e7−i k

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the compact, oriented submanifold Cji × Ck7−i in X × X. The diagonal map ΔX : X → X × X, ΔX : x → (x, x), gives a homology class [ΔX (X)] in H7 (X × X; Q). Hence for some coefficients Aijk ∈ Q we have [ΔX (X)] =

7 b i (X) b7−i (X)   i=0 j=1

Aijk eij  e7−i k

in H7 (X × X; Q),

k=1

b (X)

i with (Aijk )j,k=1 the matrix of the intersection form Hi (X; Q) × H7−i (X; Q) → Q. Therefore we can choose an 8-chain D in homology of X × X over Q with  bi (X) b7−i (X) i Ajk · Cji × Ck7−i . (7.3) ∂D = ΔX (X) − 7i=0 j=1 k=1

As ψ ∈ F γ is generic and we assume Conjecture 2.16, for each α ∈ H3 (X; Z) and N ∈ D, the moduli space M(N , α, ψ) is finite and N is finite-embedded and unobstructed for each [N, i] ∈ M(N , α, ψ). By genericness of Cji we can suppose that for all such N we have N ∩ Cji = ∅ for all i = 0, 1, 2, 3 and j = 1, . . . , bi (X). Recall that a tree is a finite, undirected graph Γ which is connected and simplyconnected. A tree Γ has a set V of vertices v, and a set E of edges e joining two vertices v, w. In the next equation, a labelled tree (Γ, [Nv , iv ]v∈V ) is a tree Γ together with an isomorphism class [Nv , iv ] of compact, immersed associative Q-homology spheres iv : Nv → X in (X, ϕ, ψ) for all v ∈ V , so that [Nv , iv ] ∈ M(N , α, ψ) for some N ∈ DQHS and α ∈ H3 (X; Z). Define a superpotential Φψ : U → Λ>0 by  Or(Nv )I(Nv )  1 · q γ·[Nv ] θ([Nv ]) Φψ (θ) = | Aut(Γ, [Nv , iv ]v∈V )| | Iso([Nv , iv ])| v∈V

labelled trees (Γ,[Nv ,iv ]v∈V )

(7.4)



· v

 1 2 (Nv

× Nw + Nw × Nv ) • D

w

 edges • − • in Γ: Nv , Nw are small perturbations of Nv , Nw in directions fNv , fNw

+ similar, but unknown, contributions from multiple covers. Here in the first line, Aut(Γ, [Nv , iv ]v∈V ) is the finite group of automorphisms of Γ preserving the assignment v → [Nv , iv ]. For each v ∈ V , Iso([Nv , iv ]) is as in Definition 2.11, and Or(Nv ) as in §3.2, and I(Nv ) as in (5.12). In the second line, the associatives Nv , Nw have canonical flags fNv , fNw , as in §3.2. We choose representatives sNv ∈ Γ∞ (νNv ), sNw ∈ Γ∞ (νNw ) for fNv , fNw , and take Nv , Nw to be small perturbations of Nv , Nw in normal directions sNv , sNw . Then (Nv × Nw + Nw × Nv ) • D in (7.4) is the intersection number in homology over Q of the 6-cycle Nv × Nw + Nw × Nv and the 8-chain D. This is well defined provided Nv × Nw + Nw × Nv does not intersect ∂D, which is given in (7.3). As above Nv , Nw do not intersect Cji for i = 0, 1, 2, 3, so Nv , Nw also do not intersect Cji as they are close to Nv , Nw . Hence Nv × Nw + Nw × Nv does not intersect  7−i i i in (7.3). i,j,k Ajk · Cj × Ck  To see that Nv × Nw + Nw × Nv does not intersect ΔX (X), as ψ is generic we may divide into cases (i) Nv ∩ Nw = ∅, and (ii) Nv and Nw are finite covers of the same embedded N ⊂ X. In case (i) Nv ∩ Nw = ∅ = Nw ∩ Nv as Nv , Nw are close to Nv , Nw . In case (ii) Nv ∩ Nw = ∅ = Nw ∩ Nv since Nv , Nw have the same image N ⊂ X. So in both cases (Nv × Nw + Nw × Nv ) ∩ ΔX (X) = ∅, and (Nv × Nw + Nw × Nv ) • D is well defined.

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w

Each edge • − • in Γ appears only once in the product in (7.4), that is, we do v w w v not distinguish • − • and • − •. This makes sense as (Nv × Nw + Nw × Nv ) • D is symmetric in v, w. The sum (7.4) is generally infinite, but by the last part of Conjecture 2.16 there are only finitely many labelled trees (Γ, [Nv , iv ]v∈V ) with  6 γ·[Nv ] in (7.4) implies that v∈V γ · [Nv ]  A for any A > 0, so the term v∈V q the sum converges in Λ>0 , and thus the first two lines of (7.4) are well defined. For the third line of (7.4), the idea is to include correction terms which will ensure deformation-invariance of Φψ under the multiple cover phenomena discussed in §7.2(F). As the author does not have a good conjectural description of these phenomena, we cannot yet write down the correction terms explicitly. We will mostly ignore this issue, and just hope things work out nicely. We can also write (7.4) as  (7.5) GWψ,α q γ·α θ(α), Φψ (θ) = α∈H3 (X;Z):γ·α>0 γ·α where GWψ,α ∈ Q is defined by taking  GWψ,α q θ(α) to be the sum of all terms in (7.4) from (Γ, [Nv , iv ]v∈V ) with v∈V [Nv ] = α in H3 (X; Z). Then GWψ,α is a Gromov–Witten type invariant counting associative Q-homology spheres in class α in (X, ϕ, ψ). Note however that the GWψ,α are not independent of the choices of Cji , D, and are not invariant under deformations of ψ in F γ . So they are not enumerative invariants in the usual sense.

Remark 7.2. We can interpret (7.4) as the sum of a ‘main term’ Φmain (θ) ψ coming from trees Γ with one vertex and no edges, and a series of increasingly complex ‘correction terms’ coming from trees Γ with n  2 vertices and n − 1 edges, as n → ∞. The ‘main term’ may be rewritten as    Or(N )I(N ) γ·α Φmain · q θ(α). (θ) = ψ | Iso([N, i])| α∈H3 (X;Z): N ∈DQHS [N,i]∈M(N ,α,ψ) γ·α>0

This is a straightforward weighted count of associative Q-homology 3-spheres. Now Φmain (θ) is not deformation-invariant, because of the wall-crossing behaviour in ψ §7.2(B),(D). The ‘correction terms’ are designed to remedy this. 7.4. How Φψ depends on choices, and on ψ. We now consider how Φψ in §7.3 depends on the arbitrary choices Cji , D in its definition, and how it varies under smooth deformations of ψ in F γ . The next “theorem” depends on the conjectures in §2–§5, and we only sketch the proof. The hypotheses are rather limited and artificial. As in §7.2, we do not have a detailed conjecture for how multiple cover phenomena in §7.2(F) behave. So we exclude them, by just assuming that only wall-crossings of type §7.2(A)–(E) occur. However, the author actually expects that some §7.2(F) phenomena will occur simultaneously with §7.2(A)–(E), and §7.2(F) is needed to cancel interaction terms in (7.4) between pairs of associatives in §7.2(A)–(E). Part (a)(iii) ensures, just by assumption, that these interaction terms are zero. Theorem 7.3. (a) Let ψ0 , ψ1 ∈ F γ be generic, and ψt , t ∈ [0, 1] be a generic smooth 1-parameter family in F γ connecting ψ0 , ψ1 . Suppose that: (i) The only changes to moduli spaces M(N , α, ψt ) as t increases through [0, 1] are those of type §7.2(A)–(E) (and not those of type §7.2(F)).

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(ii) For any A > 0, only finitely many changes happen over t ∈ [0, 1] to all M(N , α, ψt ) with γ · α  A. (iii) If Nt1 , Nt2 are two distinct associatives in (X, ϕt , ψt ) considered in one of §7.2(A)–(E), that do not both exist for all t ∈ [0, 1] (this excludes Nt1 , Nt2 being Nt+ , Nt− in §7.2(B)), and Nt1 , Nt2 are small perturbations of Nt1 , Nt2 in directions fNt1 , fNt2 , then (Nt1 × Nt2 + Nt2 × Nt1 ) • D = 0. (iv) Cji , D in Definition 7.1 are independent of t, and Cji ∩ it (Nt ) = ∅ for all i = 0, . . . , 3, j = 1, . . . , bi (X), t ∈ [0, 1] and [Nt , it ] ∈ M(N , α, ψt ). Define Φψt as in (7.4), but taking the ‘unknown multiple cover contributions’ in the third line to be zero. Then Φψ1 = Φψ0 . (b) Generalize (a) by dropping (iv). Then there is a quasi-identity morphism Υ : U → U in the sense of §7.1 with Φψ1 = Φψ0 ◦ Υ. ˜ ψ is defined in (7.4) using alternative choices C˜ i , D ˜ for C i , D in (c) Suppose Φ j j ˜ ψ = Φψ ◦ Υ for some quasi-identity morphism Υ : U → U . Definition 7.1. Then Φ Sketch proof. For (a), Φψt is defined for generic t ∈ [0, 1]. We claim that Φψt is constant in t, so that Φψ0 = Φψ1 . For A > 0, consider the projection Φψt + q A Λ>0 of Φψt to Λ>0 /q A Λ>0 . Part (ii) implies that Φψt + q A Λ>0 undergoes at most finitely many changes in t ∈ [0, 1], each from a single transition in §7.2(A)– (E). We will show that Φψt + q A Λ>0 is actually unchanged by each such transition. For (A),(C),(E) this follows from the discussion in §7.2, as (7.4) counts associative Q-homology 3-spheres N weighted by Or(N )I(N ), together with part (iii), which ensures that interactions in (7.4) between pairs of associatives in (A),(C),(E) are all zero. ˜t for t ∈ (t0 , 1], and = ±1 be as in Let Nt± for t ∈ [0, 1], x ∈ X, t0 ∈ (0, 1), N §7.2(B). Then the sum (7.4) changes as t crosses t0 in two ways: ˜t (†) When t > t0 we can have terms in (7.4) from (Γ, [Nv , iv ]v∈V ) with Nv = N for some v ∈ V . This does not happen for t < t0 . ˆ [N ˆv , ˆıv ] ˆ ) in which Γ ˆ contains an edge (‡) Consider terms in (7.4) from (Γ, v∈V v w + − ˆv = Nt and N ˆw = Nt . Then the second line of (7.4) includes • − • with N a factor 12 (Nt+ × Nt− + Nt− × Nt+ ) • D. This factor (which (iii) does not require to be zero) changes by the addition of − as t increases through t0 , because of extra intersection points of Nt+ × Nt− and Nt− × Nt+ with D near (x, x) in X × X. ˆ [N ˆv , ˆıv ] ˆ ) in (‡) to trees (Γ, [Nv , iv ]v∈V ) in (†), There is a map from trees (Γ, v∈V v w ˆ with N ˆv = Nt+ and N ˆw = Nt− to a vertex v  in which we contract edges • − • in Γ ˜ in Γ with Nv = Nt . Under this map, the changes to (7.4) cancel, because we have ˜t ) = Or(Nt+ ) · Or(Nt− ) · from §7.2(B), and I(N ˜t ) = I(Nt+ )I(Nt− ) by (5.14) Or(N + − ˜t ∼ as N = Nt #Nt . Thus Φψt + q A Λ>0 is unchanged under transitions of type (B). ˜t for t ∈ (t0 , 1], and = ±1 be as Now let Nt for t ∈ [0, 1], x ∈ X, t0 ∈ (0, 1), N in §7.2(D). Then the sum (7.4) changes as t crosses t0 in two ways: ˜t (†) When t > t0 we can have terms in (7.4) from (Γ, [Nv , iv ]v∈V ) with Nv = N for some v ∈ V . This does not happen for t < t0 . ˆ [N ˆv , ˆıv ] ˆ ) in which Γ ˆ contains an edge (‡) Consider terms in (7.4) from (Γ, v∈V v w ˆv = N ˆw = Nt . Then the second line of (7.4) includes a • − • with N factor (Nt × Nt ) • D. This factor (which (iii) does not require to be zero)

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changes by the addition of −2 as t increases through t0 , because of two extra intersection points of Nt , Nt with D near (x, x) in X × X. ˆ [N ˆv , ˆıv ] ˆ ) in (‡) to (Γ, [Nv , iv ]v∈V ) in (†) , Again, there is a map from (Γ, v∈V v w ˆ with N ˆv = N ˆw = Nt to a vertex v  in Γ in which we contract edges • − • in Γ ˜ with Nv = Nt . Under this map, the changes to (7.4) cancel, because we have ˜t ) = I(Nt )2 as N ˜t ∼ ˜t ) = = Or(Nt )2 from §7.2(D), and I(N Or(N = Nt #Nt .  The factor 2 in −2 in (‡) is dealt with by the comparison between factors ˆ [N ˆv , ˆıv ]v∈Vˆ )| in (7.4). For example, in the 1/| Aut(Γ, [Nv , iv ]v∈V )| and 1/| Aut(Γ, v ˆ = • − w• we have 1/| Aut(Γ, [Nv , iv ]v∈V )| = 1 and simplest case in which Γ = • and Γ ˆ [N ˆv , ˆıv ] ˆ )| = 1 , where the 1 cancels the 2 in −2 . Thus Φψ + q A Λ>0 1/| Aut(Γ, t v∈V 2 2 is unchanged under transitions of type (D). Hence Φψt + q A Λ>0 is independent of t for all A > 0, so Φψ1 = Φψ0 , proving (a). For (b), the difference with (a) is that as (iv) does not hold, we now must allow associatives Nt in (X, ϕt , ψt ) with Cji ∩ Nt = ∅ for some i = 0, 1, 2, 3 and j. In fact, as Cji is generic and there are only countably many smooth families of 3-folds Nt , t ∈ [0, 1] in X, it is automatic that Cji ∩ Nt = ∅ for i = 0, 1, 2 for dimensional reasons, so we need only consider i = 3, and then the only possibility is that Cj3 ∩ Nt0 = {x} for some t0 ∈ (0, 1), where Nt crosses Cj3 transversely as t increases through t0 . First we consider the effect of just one such transition. So suppose that we ˜t , ˜ıt ] ∈ M(N ˜ ,α have just one family [N ˜ , ψt ) depending smoothly on t ∈ [0, 1], with ˜t = {x} for t0 ∈ (0, 1), and C 3 ∩ N ˜t = ∅ for t = t0 , and N ˜t crosses C 3 Cj˜3 ∩ N 0 j˜ j˜ transversely as t increases through t0 with intersection number = ±1, and that Cji ∩ it (Nt ) = ∅ for all i = 0, . . . , 3, j = 1, . . . , bi (X), t ∈ [0, 1] and [Nt , it ] ∈ ˜t , ˜ıt ]. M(N , α, ψt ) unless i = 3, j = j˜, t = t0 and [Nt , it ] = [N 0 0 b4 (X) i 4 Define δ = · k=1 Aj˜k ek in H4 (X; Q). Then the effect of this change on v w (7.4) is that for each labelled tree (Γ, [Nv , iv ]v∈V ) including an edge • − •, then: (∗) 12 (Nv × Nw + Nw × Nv ) • D in (7.4) increases by [Nw ] • δ as t increases ˜t , ˜ıt ] and [Nw , iw ] = [N ˜t , ˜ıt ]. through t0 if [Nv , iv ] = [N 1   ˜t ] • δ as t increases (∗∗) 2 (Nv × Nw + Nw × Nv ) • D in (7.4) increases by 2[N ˜ through t0 if [Nv , iv ] = [Nw , iw ] = [Nt , ˜ıt ]. Here • : H3 (X; Q) × H4 (X; Q) → Q is the intersection form. The reason for (∗) is ˜t crosses C 3 in X with intersection number , N ˜t × Nw (and also N ˜t × Nw ) that as N j˜ crosses Cj˜3 × Ck4 in X × X with intersection number · [Nw ] • e4k . Thus by (7.3), ˜t × Nw ) • D as t increases through t0 is the change in (N

·

b4 (X) k=1

Aij˜k [Nw ] • e4k = [Nw ] • δ.

˜t ) • D is the same. For (∗∗) we use a similar argument. The change in (Nw × N

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From (∗) and (∗∗) above we can show that   Φψ1 (θ) = v w labelled trees (Γ,[Nv ,iv ]v∈V ) S set of directed edges • → • ˜0 , ˜ in Γ with [N , i ] = [ N ı ] v v 0 for (X, ϕ0 , ψ0 )

·

 v∈V

(7.6)

· v

1 | Aut(Γ, [Nv , iv ]v∈V )|

  Or(Nv )I(Nv ) γ·[Nv ] ·q θ([Nv ]) · [Nw ] • δ | Iso([Nv , iv ])| v w  w

edges • → • in S

 1 2 (Nv

× Nw +

Nw

× Nv ) • D.

 Nv , Nw

edges • − • in Γ but not in S: are small perturbations of Nv , Nw in directions fNv , fNw

Here the labelled trees (Γ, [Nv , iv ]v∈V ) are as in (7.4) for (X, ϕ0 , ψ0 ). On the first v w line we choose a subset S of edges • − • in Γ, to each of which we assign a direction, v w ˜0 , ˜ı0 ]. For any fixed (Γ, [Nv , iv ]v∈V ), written • → •, where we must have [Nv , iv ] = [N taking the sum in (7.6) over all S is equivalent to replacing the factor 12 (Nv × Nw + Nw ×Nv )•D in (7.4) by 12 (Nv ×Nw +Nw ×Nv )•D +[Nw ]•δ for each edge (∗) (when v w ˜t ] • δ the direction • → • is fixed uniquely), and by 12 (Nv × Nw + Nw × Nv ) • D + 2[N v w w v for each edge (∗∗) (when both directions • → • and • → • are permitted), as we want. We will not construct a quasi-identity map Υ : U → U with Φψ1 = Φψ0 ◦ Υ, but we will give a first approximation. Define Υ0 : U → U by ) * ˜0 )I(N ˜0 ) Or(N ˜0 ] γ·[N ˜ θ([N0 ]) · α • δ . (7.7) Υ0 (θ) : α −→ θ(α) · exp ·q ˜0 , ˜ı0 ])| | Iso([N This is a quasi-identity map. Substitute (7.7) into (7.4) for ψ0 to give an expression for Φψ0 ◦ Υ0 . Then each term θ([Nw ]) in (7.4) is replaced by *k ) ∞  ˜0 )I(N ˜0 ) 1 Or(N ˜ ˜0 ]) · [Nw ] • δ . Υ0 (θ)([Nw ]) = θ([Nw ]) · · q γ·[N0 ] θ([N ˜0 , ˜ı0 ])| k! | Iso([N k=0 Rewrite this expression as a sum over graphs by adding k new vertices v1 , . . . , vk ˜0 and edges v•i − w• to Γ in (7.4). Then compare the result to (7.6), with Nvi = N vi w ˜0 become the directed edges v•i → w• in S. where the new edges • − • with Nvi = N What we find is that Φψ0 ◦ Υ0 agrees with the sum of all terms in (7.6) such that v w u v for each edge • → • in S, there are no other edges • − • in Γ. So Φψ0 ◦ Υ0 is a kind of leading-order approximation to Φψ1 . The author expects that there is a formula for Υ : U → U which generalizes (7.7), and yields Φψ1 = Φψ0 ◦Υ by comparison with (7.6). This formula should look like (7.7) with [· · · ] replaced by a graph sum similar to (7.6), but over labelled rooted ˜0 , and trees (Γ, [Nv , iv ]v∈V ), r with a distinguished ‘root vertex’ r ∈ V with Nr = N including some combinatorial coefficients C(Γ, r, S) ∈ Q. Equation (7.7) gives the r term when Γ = • has one vertex r and no edges. The case in which finitely many Nt cross finitely many Cj3 follows by composing the corresponding morphisms Υ for each transition in order. Then we prove the general case by reducing the target U modulo q A for A > 0, so that only finitely many transitions are relevant for any fixed A, and letting A → ∞, as in part (a). This concludes our sketch proof of (b).

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˜ be alternative choices For (c), let ψ ∈ F γ be generic, and let Cji , D and C˜ji , D ˜ in Definition 7.1, yielding superpotentials Φψ and Φψ . First suppose that there ˆ ˆ for t ∈ [0, 1] with Cˆji (0) = Cji , D(0) = D, are smooth, generic families Cˆji (t), D(t) i i ˆ ˆ ˜ ˜ ˆ Cj (1) = Cj , D(1) = D. In (7.4) replace D by D(t), and consider how the sum changes as t increases through [0, 1]. By a similar argument to (b), this happens only when Nv or Nw intersect Cˆj3 (t0 ) for some j and t0 ∈ (0, 1). Now fixing the associative Nv and deforming Cj3 over t ∈ [0, 1] so that Nv and Cj3 intersect at t = t0 , is basically the same as fixing Cj3 and deforming the associative Nv over t ∈ [0, 1] so that Nv and Cj3 intersect at t = t0 , which is what we did in (b), and it has the same effect on the sum (7.4). Hence by (b), we see ˜ ψ = Φψ ◦ Υ for some Υ : U → U as in (b) in this case. that Φ By a slightly more general argument, we can change the Cji not by smooth deformation Cji (t), t ∈ [0, 1] but by smooth bordism in X, which allows us to link ˜  any two choices Cji , C˜ji , and we can also allow any choices of D, D. 7.5. Our main conjecture. The next conjecture is the one of the main points of this paper. Conjecture 7.4. Let X be a compact, oriented 7-manifold, and fix γ in 3 HdR (X; R). Write F γ for the set of closed 4-forms ψ on X such that there exists 3 (X; R), for which (X, ϕ, ψ) is a TAa closed 3-form ϕ on X with [ϕ] = γ in HdR G2 -manifold, with the given orientation on X. Assuming Conjecture 2.16, and making some arbitrary choices, and supposing we can find a good definition for the ‘unknown multiple cover contributions’ in (7.4) to compensate for the singular behaviour in §7.2(F), Definition 7.1 gives a superpotential Φψ : U → Λ>0 for each generic ψ ∈ F γ , where U = Hom(H3 (X; Z), 1+Λ>0 ), as a smooth rigid analytic space over Λ. ˜ ψ : U → Λ>0 then We conjecture that if different arbitrary choices yield Φ ˜ Φψ = Φψ ◦ Υ for Υ : U → U a quasi-identity morphism, as in §7.1. We also conjecture that if ψ0 , ψ1 are generic elements in the same connected component of F γ , then Φψ1 = Φψ0 ◦ Υ for Υ : U → U a quasi-identity morphism. Some support for this is provided by Theorem 7.3, and its sketch proof. Conjecture 7.4 implies that any information we can extract from the superpotential Φψ , which is unchanged under reparametrizations Φψ → Φψ ◦ Υ for quasiidentity morphisms Υ : U → U, is unchanged under deformations of ψ in F γ . As a shorthand we say that such information depends only on Φψ modulo quasi-identity morphisms. Here are some examples: (i) For GWψ,α as in (7.5), let A > 0 be least such that GWψ,α = 0 for some α ∈ H3 (X; Z) with γ · α = A, or A = ∞ if GWψ,α = 0 for all α. Then A depends only on Φψ modulo quasi-identity morphisms. Also, the values of GWψ,α for any α ∈ H3 (X; Z) with γ · α = A depend only on Φψ modulo quasi-identity morphisms. Roughly, this says that the numbers of associative Q-homology spheres with least area A in X are deformation-invariant. There could exist associatives with area less than A, but their signed weighted count is zero. (ii) Whether or not Φψ has a critical point in U depends only on Φψ modulo quasi-identity morphisms. Also, the set of critical points Crit(Φψ ), as a set up to bijection rather than as a subset of U, depends only on Φψ

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modulo quasi-identity morphisms, since if Υ : U → U is a quasi-identity morphism then Υ|Crit(Φψ ◦Υ) is a bijection Crit(Φψ ◦ Υ) → Crit(Φψ ). We develop (ii) further in our discussion of G2 quantum cohomology in §7.6. For a TA-G2 -manifold (X, ϕ, ψ), the moduli spaces M(N , α, ψ) depend only on the 4-form ψ, and the superpotential Φψ depends only on ψ and the cohomology 3 (X; R). class [ϕ] = γ of ψ in HdR Conjecture 7.4 allows us to switch the focus back to the 3-form ϕ. By Proposition 2.8(b), the set of ψ compatible with a fixed good 3-form ϕ is a convex cone, and so is connected. Therefore by Conjecture 7.4, Φψ modulo quasi-identity morphisms depends only on (X, ϕ), and in fact only on ϕ up to deformations in a fixed 3 (X; R). As in Remark 2.10, we think of (X, ϕ) as the cohomology class γ ∈ HdR analogue of a symplectic manifold (Y, ω), and ψ as the analogue of an almost complex structure J on Y compatible with ω. So Φψ modulo quasi-identity morphisms is the analogue of a symplectic invariant. 7.6. G2 quantum cohomology. This section is motivated by some areas of Symplectic Geometry: quantum cohomology, as in McDuff and Salamon [66], Lagrangian Floer cohomology, as in Fukaya, Oh, Ohta and Ono [17, 18], and work of Fukaya [16] on counting J-holomorphic discs with boundary in Lagrangians in a Calabi–Yau 3-fold. The quantum cohomology QH ∗ (Y ; Λ) of a compact symplectic manifold (Y, ω) is isomorphic to the ordinary cohomology H ∗ (Y ; Λ) over a Novikov ring Λ, but it has a deformed cup product ∗ depending on the genus zero three-point Gromov– Witten invariants GWα (β1 , β2 , β3 ) of (Y, ω). If L is a compact, oriented, relatively spin Lagrangian in (Y, ω), there is a notion of bounding cochain b for L [17, 18], which is an object in the homological algebra of L satisfying an equation involving counts of J-holomorphic discs in Y with boundary in L. If a bounding cochain b exists, we say L has unobstructed Lagrangian Floer cohomology. We can form the Lagrangian Floer cohomology ring HF ∗ ((L, b), (L, b)), which is a deformed version of H ∗ (L; Λ). In contrast to quantum cohomology, we need not have HF ∗ ((L, b), (L, b)) ∼ = H ∗ (L; Λ). When (Y, ω) is a symplectic Calabi–Yau 3-fold and L ⊂ Y is a graded Lagrangian, and J a generic almost complex structure on Y compatible with ω, we can reinterpret and extend work of Fukaya [16] as follows, though Fukaya does not write things in this form. One should define a superpotential . ΦJ : U = Hom H1 (L; Z), 1 + Λ>0 −→ Λ>0 which counts J-holomorphic discs in (Y, ω) with boundary in L. This ΦJ depends on some choices, and has some wall-crossing behaviour under deformation of J, as for Φψ in §7.3–§7.5. Critical points of ΦJ correspond exactly to (equivalence classes of) bounding cochains b for L. As in §6.1, there is a strong analogy between counting J-holomorphic curves Σ in a symplectic Calabi–Yau 3-fold (Y, ω) with boundary ∂Σ in a graded Lagrangian L, and counting associative 3-folds N without boundary in a TA-G2 manifold (X, ϕ, ψ). Following this analogy, we might hope that critical points θ of Φψ should be ‘bounding cochains’ needed to define some kind of ‘G2 quantum cohomology’ QHθ∗ (X; Λ) deforming H ∗ (X; Λ), analogous to HF ∗ ((L, b), (L, b)).

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Definition 7.5. Work in the situation of §7.1–§7.3, with ψ ∈ F γ generic. Use the formula (7.5) for the superpotential Φψ . We call (X, ϕ, ψ) obstructed if Φψ has no critical points in U, and unobstructed otherwise. Suppose (X, ϕ, ψ) is unobstructed, and choose a critical point θ of Φψ . Define a Λ0 -linear map d : H 3 (X; Λ0 ) → H 4 (X; Λ0 ) by  (7.8) d(β) = GWψ,α q γ·α θ(α) · β(α) · Pd(α). α∈H3 (X;Z):γ·α>0

Here β(α) comes from the pairing H 3 (X; Λ0 ) × H3 (X; Z) → Λ0 and Pd(α) from the Poincar´e duality isomorphism Pd : H3 (X; Z) → H 4 (X; Z), and the sum in (7.8) converges in the topology on H 4 (X; Λ0 ) induced by that on Λ0 . We can interpret d as contraction with the Hessian Hessθ (Φψ ) of Φψ at θ. Now define the G2 -quantum cohomology groups QHθk (X; Λ0 ) for k  0 by ⎧ k ⎪ Λ0 ), k = 3, 4, ⎨H (X; 3 4 k QHθ (X; Λ0 ) = Ker d : H 3 (X; Λ0 ) → H 4 (X; Λ0 ) , k = 3, ⎪ 3 4 ⎩ 3 4 Coker d : H (X; Λ0 ) → H (X; Λ0 ) , k = 4. Define a product ∗ : QHθk (X; Λ0 ) × QHθl (X; Λ0 ) → QHθk+l (X; Λ0 ), written δ ∗ ∈ QHθk+l (X; Λ0 ) for δ ∈ QHθk (X; Λ0 ) and ∈ QHθl (X; Λ0 ), by: (i) If (k, l) are one of (0, 0), (0, 1), (0, 2), (0, 5), (0, 6), (0, 7), (1, 0), (1, 1), (1, 5), (1, 6), (2, 0), (2, 3), (2, 5), (3, 2), (3, 3), (5, 0), (5, 1), (5, 2), (6, 0), (6, 1), (7, 0),

(ii) (iii) (iv)

(v) (vi)

(vii)

then δ ∗ = δ ∪ , as in these cases either QHθ∗ (X; Λ0 ) = H ∗ (X; Λ0 ) in degrees k, l, k + l, or QHθ3 (X; Λ0 ) ⊆ H 3 (X; Λ0 ) for k = 3 or l = 3. If (k, l) = (0, 3) then δ ∗ = δ ∪ , where ∈ Ker d ⊆ H 3 (X; Λ0 ) implies that δ ∪ ∈ Ker d. Similarly for (k, l) = (3, 0). If (k, l) = (0, 4) then δ ∗ ( + Im d) = (δ ∪ ) + Im d, where ∈ H 4 (X; Λ0 ). Similarly for (k, l) = (4, 0). If (k, l) = (1, 2) then δ ∗ = δ ∪ . To show this is well defined we must prove that δ ∪ ∈ Ker d ⊆ H 3 (X; Λ0 ) for all δ ∈ H 1 (X; Λ0 ) and ∈ H 2 (X; Λ0 ). Now if i : N → X is an immersed associative Q-homology sphere with [N ] = α ∈ H3 (X; Z) then (δ ∪ ) · α = (i∗ (δ) ∪ i∗ ( )) · [N ] = 0, since H 1 (N ; Q) = H 2 (N ; Q) = 0 as N is a Q-homology 3-sphere, and i∗ (δ) ∈ H 1 (N ; Q), i∗ ( ) ∈ H 2 (N ; Q). Since GWψ,α counts associative Q-homology 3-spheres in class α, we have (δ ∪ ) · α = 0 if GWψ,α = 0. Hence from (7.8) we see that δ ∪ ∈ Ker d. Similarly for (k, l) = (2, 1). If (k, l) is (1,3), (2,2) or (3,1) then δ ∗ = δ ∪ + Im d. If (k, l) = (1, 4) or (2,4) then δ ∗ ( + Im d) = δ ∪ . To show this is well-defined we must show that if + Im d =  + Im d then δ ∪ = δ ∪  . As  = + dζ for ζ ∈ H 3 (X; Λ0 ), it is enough to show that δ ∪ dζ = 0. From (7.8), dζ is a linear combination of classes Pd(α) for α ∈ H3 (X; Z) with GWψ,α = 0. As in (iv), we have δ ∪ Pd(α) = 0 if δ ∈ H 1 (X; Λ0 ) or δ ∈ H 2 (X; Λ0 ), since α is represented by a Q-homology 3-sphere, so δ ∪ dζ = 0. Similarly for (k, l) = (4, 1) or (4,2). If (k, l) = (3, 4) then δ ∗ ( + Im d) = δ ∪ for δ ∈ Ker d ⊆ H 3 (X; Λ0 ) and ∈ H 4 (X; Λ0 ). As in (vi), to show this is well-defined we must show that δ ∪ dζ = 0 for ζ ∈ H 3 (X; Λ0 ). But from (7.8) we can prove

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that η ∪ dζ = ζ ∪ dη for any η, ζ ∈ H 3 (X; Λ0 ), because Hessθ (Φψ ) is a symmetric form. Thus δ ∪ dζ = 0 as dδ = 0. Similarly for (k, l) = (4, 3). (viii) If k + l > 7 then δ ∗ = 0 automatically. Since ∪ is associative and supercommutative, we see that ∗ is too. If we assume Conjecture 7.4, then G2 quantum cohomology QHθ∗ (X; Λ0 ) will be unchanged under deformations of ψ, in the same sense in which Lagrangian Floer cohomology HF ∗ ((L, b), (L, b)) is independent of J. If ψ0 , ψ1 are generic in the same connected component of F γ , Conjecture 7.4 gives Υ : U → U with Φψ1 = Φψ0 ◦ Υ. Then Υ maps critical points θ1 of Φψ1 bijectively to critical points θ0 of Φψ1 , and using the derivative dθ1 Υ of Υ at θ1 we can define a Λ0 -algebra isomorphism QHθ∗1 (X; Λ0 ) → QHθ∗0 (X; Λ0 ). There should also be a way to define an A∞ -algebra whose cohomology is QHθ∗ (X; Λ0 ), deforming the cochain cdga for H ∗ (X; Λ0 ), using similar ideas to Fukaya et al. [17, 18]. In this definition we should use the fact that we count only associative Q-homology 3-spheres N ⊂ X in the following way. Consider the 6-cycle in N × N × N , + , + , + C = (x, x , x ) : x, x ∈ N + (x , x, x ) : x, x ∈ N + (x , x , x) : x, x ∈ N . Since N is a Q-homology 3-sphere we have [C] = 0 in H6 (N × N × N ; Q), so there is a 7-cycle D on N ×N ×N with ∂D = C. The cochain-level version of multiplication ∗ should involve choosing such a 7-cycle D for each associative Q-homology sphere N in the count. The author does not know whether this G2 quantum cohomology is actually interesting. It seems likely to play some rˆ ole in M-theory, at least. 7.7. Generalizations. Here are some ways in which the picture of §7.1–§7.6 can be extended. Including a C-field. Take the field F used to define Λ in §7.1 to be F = C. Choose C ∈ H 3 (X; R)/2πH 3 (X; Z). Then we can generalize the formulae (7.4)– (7.5) defining Φψ by replacing q γ·[Nv ] by q γ·[Nv ] eiC·[Nv ] , so that (7.5) becomes  Φψ (θ) = α∈H3 (X;Z):γ·α>0 GWψ,α q γ·α eiC·α θ(α). Here as C ∈ H 3 (X; R)/2πH 3 (X; Z) and α ∈ H3 (X; Z), the product C · α lies in R/2πZ, so that eiC·α is well defined. ‘C-fields’ C of this kind are natural in the M-theory of G2 -manifolds, and have the effect of complexifying the moduli space of G2 -manifolds, with [ϕ] + iC in the complex manifold H 3 (X; C/2πiZ). Varying the cohomology class [ϕ]. So far we have worked with TA-G2 -manifolds 3 (X; R) is fixed. Here is a way to allow [ϕ] to (X, ϕ, ψ) for which the [ϕ] = γ ∈ HdR vary. Let us regard the 4-form ψ as fixed. Then Proposition 2.8(a) gives an open 3 (X; R), of cohomology classes [ϕ] of 3-forms ϕ such that convex cone KX,ψ in HdR (X, ϕ, ψ) is a TA-G2 -manifold. We can then then extend Φψ in (7.7) to a map ˆ ψ : KX,ψ × U −→ Λ>0 , Φ which maps (γ, θ) in KX,ψ × U to Φψ in (7.7) computed using [ϕ] = γ. Over F = R, we can regard KX,ψ × U as a rigid analytic space; it may be possible to glue the charts KX,ψ0 × U , KX,ψ1 × U over KX,ψ0 ∩ KX,ψ1 for different ψ0 , ψ1 , using the

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morphisms Ψ : U → U in Conjecture 7.4, to get a U-bundle over a larger open ˆ is defined. subset of H 3 (X; R), upon which a superpotential Φ Noncompact G2 -manifolds. We can consider TA-G2 -manifolds (X, ϕ, ψ) with X noncompact, if we have some control on the noncompact ends of X – some kind of convexity at infinity – which prevents associative 3-folds from escaping to infinity in X, and so changing the numbers of associatives. Counting associatives N with b1 (N ) > 0. It is tempting to try and modify (7.4) to count ‘higher genus’ associative 3-folds N with g = b1 (N ) > 0. The author does not know a way to do this in general, which is invariant under transitions of type §7.2(C). One possibility might be to try and count associatives i : N → X where N is not a Q-homology 3-sphere, but i∗ : H2 (N ; Q) → H2 (X; Q) is injective, as such N are not affected by transitions §7.2(C). 8. Remarks on counting G2 -instantons We discussed G2 -instantons on TA-G2 -manifolds (X, ϕ, ψ) in §2.4–§2.5 above. Donaldson and Segal [15, §6.2] proposed a conjectural programme to define invariants counting G2 -instantons, which would hopefully be unchanged under deformations of (ϕ, ψ), and would be analogues of Donaldson–Thomas invariants of Calabi–Yau 3-folds [48, 54]. The programme is currently under investigation by Menet, Nordstr¨ om, S´ a Earp, Walpuski, and others [68, 71, 72, 77–80]. As in [15, §6], to complete the Donaldson–Segal programme and define invariants of (X, ϕ, ψ) unchanged under deformations of ψ will require the inclusion of ‘compensation terms’ counting solutions of some equation on associative 3-folds N in X, to compensate for bubbling of G2 -instantons on associative 3-folds. So counting G2 -instantons, and counting associative 3-folds, are intimately linked. We now discuss several aspects of this programme, drawing on the ideas of §3–§7. Section 8.2 makes a proposal for how to define canonical orientations for G2 -instanton moduli spaces, based on the ideas in §3 on orienting associative moduli spaces. Section 8.4 gives two ‘thought-experiments’ describing ways in which Donaldson–Segal’s proposed invariants could change under deformations of (ϕ, ψ). Finally, §8.5 suggests a way (not yet complete) to modify the Donaldson–Segal programme to (hopefully) fix these problems. 8.1. The Donaldson–Segal programme. Suppose X is a compact 7-manifold, and (ϕ, ψ) a generic TA-G2 -structure on X. Let G be a compact Lie group, and π : P → X a principal G-bundle. Consider the moduli space M(P, ψ) of G2 -instantons on X, as in §2.4–§2.5. By analogy with Donaldson invariants of oriented 4-manifolds M [14], which count moduli spaces of instantons on M , and with Donaldson–Thomas invariants of Calabi–Yau 3-folds Y [48, 54], which can be heuristically understood as counting Hermitian–Yang–Mills connections on Y , Donaldson and Segal [15, §6] want to define invariants of (X, ϕ, ψ) by counting moduli spaces M(P, ψ). Donaldson and Segal expect [15, §4.1] that when ψ is generic M(P, ψ) will be a compact 0-manifold, that is, a finite set, and one can define an orientation on the moduli space Or : M(P, ψ) → {±1} (compare §3), though they do not give details. Then a first approximation to the invariants they want is  (8.1) DS0 (P, ψ) = [A]∈M(P,ψ) Or([A]) ∈ Z.

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They explain [15, §6.1] that DS0 (P, ψ) should in general not be unchanged under deformations of ψ, as there are index one singularities of G2 -instantons which can change the moduli spaces M(P, ψ). They expect that the typical way moduli spaces can change under deformations is as follows: Example 8.1. Let (ϕt , ψt ), t ∈ [0, 1] be a generic 1-parameter of TA-G2 structures on X. Suppose that for some t0 ∈ (0, 1) there exists a connection At on P for t ∈ [0, t0 ) which is an unobstructed G2 -instanton on (X, ϕt , ψt ), and depends smoothly on t. As t → t0 , the G2 -instanton At approaches a singular limit, in which the curvature FAt of At concentrates around a compact associative 3-fold Nt0 in (X, ϕt0 , ψt0 ). This singularity should be ‘removable’. That is, there is another principal Gbundle P  → X with a G2 -instanton connection At0 on (X, ϕt0 , ψt0 ), such that there is an isomorphism of principal G-bundles P |X\N ∼ = P  |X\N on X \ N , and up  to gauge transformations, At |X\N converges to At0 |X\N as t → t0 on any compact subset of X \ N . As t converges to t0 , the connection At near N should resemble a family of instantons with group G and charge c2 = k on the R4 normal spaces νx to N in X at x ∈ N , concentrated near 0 in νx . When G = SU(2), the second Chern classes c2 (P ), c2 (P  ) are related by c2 (P ) = c2 (P  ) + k · Pd([N ]) ∈ H 4 (X; Z). Now the moduli spaces of instantons on R4 are well understood, and can be described by the ADHM construction. Donaldson and Segal [15, §6.1] define a bundle M → N whose fibre at x ∈ N is the moduli space MG (νx , k) of instantons on νx with group G and charge k, with framing at infinity in νx depending on P  |N . Using results of Haydys, they define an equation on smooth sections s : N → M which they call the Fueter equation, which depends on A |N , and explain that the local model near N for At as t → t0 should be written in terms of a solution s of the Fueter equation. They conjecture that given a G2 -instanton (P  , A ) on (X, ϕt0 , ψt0 ), a compact associative N in (X, ϕt0 , ψt0 ), and a solution s : N → M of the Fueter equation constructed from (P  , A )|N for charge k, it should be possible to find a smooth 1parameter family of TA-G2 -manifolds (X, ϕt , ψt ), t ∈ [0, 1] including (X, ϕt0 , ψt0 ), and a smooth family of G2 -instantons (P, At ) on (X, ϕt , ψt ) for t ∈ [0, t0 ), which bubble on N as t → t0 to recover (P  , A ), s as above. This conjecture has now been proved by Walpuski [78]. When G = SU(2) and k = 1, Donaldson and Segal [15, §6.1] describe the bundle M → N and the Fueter equation for sections s : N → M more explicitly: Example 8.2. Continue in Example 8.1, but fix G = SU(2) and the charge k of instantons bubbling at N as t → t0 to be k = 1. Also suppose that the associative 3-fold N in (X, ϕt0 , ψt0 ) is unobstructed, in the sense of §2.6. The moduli space of instantons on R4 with group SU(2) and charge 1 is MSU(2) (R4 , k) ∼ = [(R4 \ {0})/{±1}] × R4 . The corresponding bundle M → N is M∼ = [(SP  \ {0})/{±1}] ×N ν. Here we choose some spin structure σ on N and write S → N for the spin bundle over N associated to σ, which has fibre H ∼ = R4 . Then SP  = (S ×N P  |N )/ SU(2)  is the spin bundle on N twisted by P |N , and SP  \ {0} is the complement of the

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zero section in SP  , so that SP  , SP  \ {0} and (SP  \ {0})/{±1} are bundles over N with fibres R4 , R4 \ {0} and (R4 \ {0})/{±1}. Dividing by {±1} means that (SP  \ {0})/{±1} is independent of the choice of spin structure σ on N . However, any section of (SP  \ {0})/{±1} lifts to a section of SP  \ {0} for SP  defined using a unique spin structure σ. Thus, sections s : N → M correspond to triples (σ, {±s1 }, s2 ) of a spin structure σ on N , a nonvanishing section s1 of the twisted spin bundle SP  → N defined using σ and P  |N , and a section s2 of ν → N . The Fueter equation on s is then equivalent to DP  ,A s1 = 0, Ds2 = 0, where DP  ,A : Γ∞ (SP  ) → Γ∞ (SP  ) is the twisted Dirac operator for (P  |N , A |N ), and D : Γ∞ (ν) → Γ∞ (ν) is as in Theorem 2.12. But by assumption N is unobstructed, so Ker D = 0, and s2 = 0. Therefore, the conclusion is that solutions s of the Fueter equation correspond to pairs (σ, s1 ), where σ is a spin structure on N , and s1 is a non-vanishing solution of the twisted Dirac equation DP  ,A s1 = 0 for the SU(2)-connection (P  |N , A |N ) on N with spin structure σ, where s1 only matters up to sign ±s1 . Donaldson and Segal’s proposal [15, §6.2] is to try to modify (8.1) to define invariants, for TA-G2 -manifolds (X, ϕ, ψ) with ψ generic:   . Or([A]) + w (P  , A ), N, k . (8.2) DS(P, ψ) = [A]∈M(P,ψ)

(P  , A ), N, k: (P  , A ) G2 -instanton on (X, ϕ, ψ) with group G, up to gauge equivalence, N = ∅ compact associative in (X, ϕ, ψ), k  1, P = P  +charge k modification along N

Here w((P  , A ), N, k) is some ‘compensation term’ which they do not define, but the crucial point is that it must be chosen so that DS(P, ψ) is unchanged under deformations of (X, ϕ, ψ) in 1-parameter families (X, ϕt , ψt ), t ∈ [0, 1]. So in Example 8.1, the first term of (8.2) changes by ±1 as t crosses t0 and [At ] disappears from M(P, ψ), and we expect w((P  , A ), N, k) for (P, A ), N, k as in Example 8.1 to change by ∓1 as t crosses t0 to compensate. When G = SU(2) and k = 1 Donaldson and Segal [15, §6.2] suggest taking w((P  , A ), N, 1) = ± 12 , where the sign is defined by using spectral flow as in §3.2. This is explained by Walpuski [79, §6.2]. Haydys and Walpuski [24, §1] give a different proposal for w((P  , A ), N, 1), which we discuss in §8.5. 8.2. Canonical orientations for moduli of G2 -instantons. As in §8.1, there are close connections between moduli spaces of G2 -instantons and of associative 3-folds in (X, ϕ, ψ). So our method in §3.2 for defining canonical orientations on associative moduli spaces M(N , α, ψ) in (X, ϕ, ψ), having chosen a flag structure F on X, might have an analogue for defining canonical orientations on G2 -instanton moduli spaces. Conjecture 8.3. Let (X, ϕ, ψ) be a compact TA-G2 -manifold and π : P → X a principal SU(2)-bundle, and write M(P, ψ) for the moduli space of irreducible G2 instanton connections A on (X, ϕ, ψ) up to gauge equivalence. We expect M(P, ψ) to be a smooth 0-manifold if ψ is generic, and an m-Kuranishi space of virtual dimension 0 in general, as for Conjectures 2.14 and 2.16. Choose a flag structure F for X, as in §3.1. Then there should be a way to define canonical orientations for the moduli spaces M(P, ψ), as manifolds or m-Kuranishi spaces, which are well behaved under deformations of (X, ϕ, ψ).

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If F, F  are flag structures on X then Proposition 3.6(b) gives a morphism

: H3 (X; Z) → {±1} satisfying (3.8). Let  : H 4 (X; Z) → {±1} correspond to under the Poincar´e duality isomorphism H3 (X; Z) ∼ = H 4 (X; Z). Then the  orientations on M(P, ψ) coming from F and F differ by a factor  ◦ c2 (P ). Here is how the author expects a proof of Conjecture 8.3 to go. We follow the method of Donaldson and Kronheimer [14, §5.4 & §7.1.6] for constructing orientations on moduli spaces M(P, g) of instanton connections on a principal SU(2)bundle P → M over a compact, oriented, generic Riemannian 4-manifold (M, g). There are three main steps in their method: (a) They define the orientation as a structure on the infinite-dimensional family B of all connections A on P , modulo gauge, not just on the finitedimensional submanifold M(P, g) ⊂ B. Here B is connected, and can be described using homotopy theory. (b) In [14, §5.4], by considering loops S 1 in B, they show that B is orientable. There are then two possible orientations on B, as B is connected. (c) In [14, §7.1.6], when c2 (P ) = k  0 in H 4 (M ; Z) ∼ = Z, they fix the orientation on B by defining it near a connection A on P which is trivial away from p1 , . . . , pk in M , and which near each pi approximates a standard SU(2)-instanton on R4 with c2 = 1, with curvature concentrated near 0. Orientations for moduli spaces M(P, ψ) of G2 -instantons on (X, ϕ, ψ) are discussed by Donaldson and Segal [15, §4.1], and in more detail by Walpuski [79, §6.1]. Walpuski does the analogues of (a),(b) above, where for (b) he shows [79, Prop. 6.3] that B is orientable for moduli spaces of G2 -instantons with gauge group SU(r) for r  2. But he does not carry out step (c), instead choosing one of the two orientations on B arbitrarily. We propose that our ideas using flag structures may be used to complete step (c). The idea would be that given a principal SU(2)-bundle P → X with c2 (P ) = β ∈ H 4 (X; Z), we would let α ∈ H3 (X; Z) correspond to β under Poincar´e duality, and choose a compact, oriented, embedded 3-submanifold N in X with [N ] = α ∈ H3 (X; Z). Here N is not required to be associative. Then we should consider a connection A on P which is trivial away from N , and near N approximates a family of small standard SU(2)-instantons with c2 = 1 on the R4 fibres of the normal bundle ν → N , as in [15, §6.1] for N associative. The orientation for B should then be determined by giving A the orientation (−1)SF(Lt :t∈[0,1]) F (N, f ), where F is the flag structure on X, and SF(Lt : t ∈ [0, 1]) is the spectral flow between an elliptic operator L0 which depends on a choice of flag f for N at t = 0, and the linearization L1 of the G2 -instanton equation at A at t = 1, where we suppose L1 is an isomorphism. 8.3. P  -flags, and canonical P  -flags. Definition 8.4. Let (X, ϕ, ψ) be a compact TA-G2 -manifold, and (P  , A ) a G2 -instanton on X with structure group SU(2), and N a compact, oriented 3dimensional submanifold in X (usually associative), and σ a spin structure on N . Then as in §8.2 we define the twisted spin bundle SP  → N and the twisted Dirac operator DP  ,A : Γ∞ (SP  ) → Γ∞ (SP  ) using σ and (P  |N , A |N ). We now repeat parts of §3.1–§3.2 with SP  → N in place of ν → N . As in Definition 3.1, let s, s ∈ Γ∞ (SP  ) be nonvanishing sections. Write 0 : N → SP 

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for the zero section, and γ : [0, 1] × N →∈ Γ∞ (SP  ) for the map γ : (t, x) → (1 − t)s(x) + ts (x). Define d(s, s ) = 0(N ) • γ([0, 1] × N ) ∈ Z . Define a P  -flag on N to be an equivalence class [s] of nonvanishing s ∈ ∞ Γ (SP  ), where s, s are equivalent if d(s, s ) = 0. Write FlagP  (N ) for the set of all P  -flags [s] on N . For [s], [s ] ∈ FlagP  (N ) we define d([s], [s ]) = d(s, s ) ∈ Z for any representatives s, s for [s], [s ]. For any [s] ∈ FlagP  (N ) and any k ∈ Z, there is a unique [s ] ∈ FlagP  (N ) with d([s], [s ]) = k, and we write [s ] = [s] + k. This gives a natural action of Z on FlagP  (N ), making FlagP  (N ) into a Z-torsor. Following Definition 3.8, let [s] be a P  -flag, and choose a representative s of unit length. There is then a unique isomorphism SP  ∼ = Λ0 T ∗ N ⊕ Λ2 T ∗ N which ∞ 0 ∗ 2 ∗ identifies s with 1 ⊕ 0 in Γ (Λ T N ⊕ Λ T N ), and identifies the symbols of DP  ,A and d ∗ + ∗ d. Thus as in (3.10) we have DP  ,A ∼ = d ∗ + ∗ d + B, for B of degree 0 as in (3.11). Define a family of first order operators At , t ∈ [0, 1] as in (3.12) by At = d ∗ + ∗ d + tB. Then A0 = d ∗ + ∗ d in (3.9), and A1 ∼ = DP  ,A under our isomorphism Λ0 T ∗ N ⊕ Λ2 T ∗ N ∼ = SP  . Thus as in Definition 3.7 we have the spectral flow SF(At : t ∈ [0, 1]) ∈ Z. P  ,A P or fN on N , called the As in Definition 3.8, there is a unique P  -flag fN canonical P  -flag of N , such that SF(At : t ∈ [0, 1]) = 0 for At : t ∈ [0, 1] conP structed using s ∈ fN . It has the property that for any P  -flag [s] for N and family At : t ∈ [0, 1] constructed from s ∈ [s] as above, we have 

P = [s] + SF(At : t ∈ [0, 1]). fN



P Canonical P  -flags fN are related to the problem of defining the weight function   w((P , A ), N, k) in (8.2) when G = SU(2) and k = 1, so that we can use Example 8.2. Suppose we are given a generic 1-parameter family of TA-G2 -manifolds (X, ϕt , ψt ), t ∈ [0, 1], and corresponding 1-parameter families (P  , At ), t ∈ [0, 1] of unobstructed G2 -instantons in (X, ϕt , ψt ), and Nt , t ∈ [0, 1] of unobstructed associative 3-folds in (X, ϕt , ψt ). Then we have a 1-parameter family of twisted Dirac operators DP  ,At for t ∈ [0, 1] on Nt . According to the Donaldson–Segal–Walpuski picture, for generic t ∈ [0, 1] we have Ker DP  ,At = 0, but for isolated t0 ∈ [0, 1] we may have Ker DP  ,At = 0, and 0 then we create or destroy a new G2 -instanton (P, At ) as t increases through t0 in [0, 1], as in Examples 8.1 and 8.2. This happens when an eigenvalue of DP  ,At passes through 0 at t = t0 , so that SF(At : t ∈ [0, 1]) jumps by 1, and so the canonical flag P fN jumps by 1 as t passes through t0 . t P Thus the canonical flag fN has the property we want of w((P  , A ), N, 1): P under deformations of (X, ϕt , ψt ), fN changes by addition of k ∈ Z exactly when P w((P  , A ), N, 1) should change by addition of k ∈ Z. Unfortunately, fN is not a number, as w((P  , A ), N, 1) should be, but a geometric structure on N .

8.4. Problems with counting G2 -instantons. Based on the ideas and results of Donaldson–Segal and Walpuski described in §8.1, and the material on P  flags in §8.3, the author expects that the following is a possible (or at least plausible) behaviour for moduli spaces of G2 -instantons and associative 3-folds under smooth deformations of TA-G2 -manifolds:

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Example 8.5. Suppose we are given a smooth family of compact TA-G2 manifolds (X, ϕt , ψt ), t ∈ [0, 1], supporting G2 -instantons and associative 3-folds as follows: (a) There is an unobstructed G2 -instanton (P  , At ) on (X, ϕt , ψt ) with structure group SU(2) for t ∈ [0, 1], depending smoothly on t. (b) For t ∈ [0, 13 ), t ∈ ( 32 , 1] there are no associatives of interest in (X, ϕt , ψt ). (c) For t ∈ ( 13 , 23 ) there are two associatives Nt+ , Nt− in (X, ϕt , ψt ), depending smoothly on t. They are unobstructed, in the same homology class, with orientations Or(Nt+ ) = 1, Or(Nt− ) = −1. (d) There are associatives N1/3 in (X, ϕ1/3 , ψ1/3 ) and N2/3 in (X, ϕ2/3 , ψ2/3 ). They are obstructed, with ON1/3 ∼ =R∼ = ON2/3 . We have Nt± → N1/3 as ± 1 2 t → 3 + and Nt → N2/3 as t → 3 − , as in §7.2(A). (e) All of Nt± , N1/3 , N2/3 are diffeomorphic to a fixed compact, oriented 3manifold N , such as N = S 3 . For simplicity we suppose H1 (N ; Z2 ) = 0, so that N has a unique spin structure. Let us now ask: how many G2 -instantons (P, At ) with structure group SU(2) are created or destroyed by bubbling a 1-instanton along Nt± from (P  , At ), as t increases over [0, 1], as described in Examples 8.1 and 8.2? Consider the oriented 4-manifold M ∼ = N × S 1 (or a twisted product) made of ± 1 2 the disjoint union of Nt , t ∈ ( 3 , 3 ) and N1/3 , N2/3 glued together in the obvious way, with its natural map M → X from the inclusions Nt± , N1/3 , N2/3 → X. On M we have a rank 4 oriented vector bundle E → M restricting to the twisted spin bundles SP  on each slice Nt± , N1/3 , N2/3 , where SP  is unique as the spin structures on Nt± , N1/3 , N2/3 ∼ = N are unique. 5The number of zeroes of a generic section of E → M , counted with signs, is k := M c2 (P  ). Suppose no G2 -instantons (P, At ) are created or destroyed over t ∈ [0, 1]. Then P P P the canonical P  -flags fN , fN do not jump, and vary continuously. There± , fN 1/3 2/3 t

± fore we can choose nonvanishing sections s± t , s1/3 , s2/3 of SP  on Nt , N1/3 , N2/3 repP P P resenting fN ± , fN1/3 , fN2/3 and varying continuously with t, and these s± t , s1/3 , s2/3 t make up a continuous, nonvanishing section of E → M , so that k = 0. In general, P k counts the jumps of fN ± as t increases over [0, 1], so we create or destroy k new t

G2 -instantons (P, At ) as t increases from 0 to 1. We expect that we can have k = 0 in Z in examples. Thus, we can have: (i) In (X, ϕ0 , ψ0 ) one G2 -instanton (P  , A0 ) and no G2 -instantons on P , where P → X is the principal SU(2)-bundle obtained from P  by gluing in a 1instanton along Nt+ , and there are no associative 3-folds of interest. (ii) In (X, ϕ1 , ψ1 ) one G2 -instanton (P  , A1 ), and k = 0 G2 -instantons on P counted with signs, and no associative 3-folds of interest. Hence, in (8.2) we have DS(P, ψ0 ) = 0 and DS(P, ψ1 ) = k = 0, so DS(P, ψ) is not deformation-invariant. If Example 8.5 is true to mathematical reality, it demonstrates a problem with the Donaldson–Segal proposal [15, §6.2] for defining invariants DS(P, ψ) in (8.2). Note that the actual choice of ‘compensation terms’ w((P  , A ), N, k) is irrelevant, since in our example there are no associatives in (X, ϕ0 , ψ0 ) or in (X, ϕ1 , ψ1 ), so the second sum in (8.2) is automatically zero. However, we can trace the failure

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to difficulties in defining w((P  , A ), N, 1) compensating for SU(2)-instantons with charge 1 bubbling along N in the way Donaldson and Segal want. We discuss a possible solution to this problem in §8.5. Here is another thought-experiment similar to Example 8.5: Example 8.6. Suppose we are given a smooth family of compact TA-G2 manifolds (X, ϕt , ψt ), t ∈ [0, 1], and a principal SU(2)-bundle P  → X with c2 (P  ) = 0 in H 4 (X; Q), supporting G2 -instantons and associatives as follows: (a) There is an unobstructed associative 3-fold Nt in (X, ϕt , ψt ) for t ∈ [0, 1], depending smoothly on t. For simplicity we suppose Nt is connected with H1 (Nt ; Z2 ) = 0, say Nt ∼ = S 3 , so that Nt has a unique spin structure. 1 2 (b) For t ∈ [0, 3 ), t ∈ ( 3 , 1] there are no G2 -instantons on P  over (X, ϕt , ψt ). − (c) For t ∈ ( 13 , 23 ) there are two gauge equivalence classes [A+ t ], [At ] of G2 instantons on P  over (X, ϕt , ψt ), depending smoothly on t. They are − unobstructed, with orientations Or([A+ t ]) = 1 and Or([At ]) = −1. (d) There are gauge equivalence classes [A1/3 ] and [A2/3 ] of G2 -instantons on P  over (X, ϕ1/3 , ψ1/3 ) and (X, ϕ2/3 , ψ2/3 ), respectively. They are both 1  obstructed, with obstruction space R. We have [A± t ] → [A1/3 ] as t → 3 + 2  and [A± t ] → [A2/3 ] as t → 3 − .

−  Consider the problem of lifting the gauge equivalence classes [A+ t ], [At ], [A1/3 ], + − [A2/3 ] to connections At , At , A1/3 , A2/3 on P depending continuously on t. As we are dealing with a loop of connections, there may be monodromy. That is, we − ± 1 2   can choose A+ t , At , A1/3 , A2/3 such that At depend smoothly on t ∈ ( 3 , 3 ), and ± + 1 2   At → A1/3 as t → 3 + , and At → A2/3 as t → 3 − . But we cannot also ensure − 2   that A− t → A2/3 as t → 3 − . Instead, we can only suppose that At → γ · A2/3 for some smooth gauge transformation γ : X → SU(2), which may induce a nontrivial map γ∗ : H3 (X; Z) → H3 (SU(2); Z) ∼ = Z. Write (γ|N )∗ : Z ∼ = H3 (N ; Z) → H3 (SU(2); Z) ∼ = Z as multiplication by k ∈ Z. We expect that we can have k = 0 in Z in examples. Let P → X be the principal SU(2)-bundle obtained from P  by gluing in a family of instantons of charge 1 along Nt . The author expects that by a similar calculation to that in Example 8.5 one can show that k G2 -instantons (P, At ) are created or destroyed by bubbling a 1-instanton along Nt from (P  , A± t ), as t increases over [0, 1], counted with signs. Thus, we can have: (i) In (X, ϕ0 , ψ0 ) there is one associative N0 , and no G2 -instantons of interest. (ii) In (X, ϕ1 , ψ1 ) there is one associative N1 , and k = 0 G2 -instantons on P , counted with signs, and no other G2 -instantons of interest. In (8.2) we have DS(P, ψ0 ) = 0 and DS(P, ψ1 ) = k = 0, so DS(P, ψ) is not deformation-invariant. There are no contributions to DS(P, ψ0 ), DS(P, ψ1 ) from N0 , N1 , as there are no G2 -instantons on P  over (X, ϕ0 , ψ0 ) or (X, ϕ1 , ψ1 ).

Again, if Example 8.6 is true to mathematical reality, it demonstrates a problem with the Donaldson–Segal proposal [15, §6.2], which we discuss in §8.5. 8.5. A suggestion for how to modify Donaldson–Segal. Examples 8.5 and 8.6 indicate that Donaldson and Segal’s proposed invariants DS(P, ψ) in (8.2) will not be deformation-invariant. However, all may not be lost. We now outline a way to modify the Donaldson–Segal programme to hopefully fix these problems.

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We summarize our main points as (i),(ii), . . . . This is not a complete proposal, just the beginnings of a possible answer. While counting G2 -instantons and counting associative 3-folds are linked, counting associative 3-folds is the more primitive problem, as one can count associatives on their own, but to count G2 -instantons with any hope of deformation-invariance, one must count associative 3-folds too. So we should really start with the problem of counting associative 3-folds. The author expects that it should only be possible to count G2 -instantons on (X, ϕ, ψ) if counting associative 3-folds on (X, ϕ, ψ) is well-behaved, by which we mean: (i) The Donaldson–Segal programme for counting G2 -instantons on a TA-G2 manifold (X, ϕ, ψ), giving an answer independent of deformations of ψ, should only work if (X, ϕ, ψ) is unobstructed in the sense of Definition 7.5. In Example 8.6, (X, ϕt , ψt ) is obstructed by the associative 3-fold Nt . The author expects that the change in invariants DS(P, ψt ) in Example 8.6 under deformations of ϕt , ψt is typical for deformations of obstructed TA-G2 -manifolds (X, ϕ, ψ). The author knows of no way to add compensation terms to restore deformationinvariance in the obstructed case. (ii) If (X, ϕ, ψ) is unobstructed then Φψ : U → Λ>0 in §7.3 has at least one critical point θ ∈ U , but this critical point may not be unique. To get deformation-invariant information from counting G2 -instantons on (X, ϕ, ψ), we should first make a choice of critical point θ of Φψ , and : whatever invariants DS(ψ, θ) we define should depend on this choice of θ. (iii) Suppose we are given a smooth 1-parameter family (X, ϕt , ψt ), t ∈ [0, 1] of TA-G2 -manifolds with [ϕt ] constant in H 3 (X; R). Then as in Conjecture 7.4, there should exist a natural quasi-identity morphism Υ : U → U with Φψ1 = Φψ0 ◦ Υ. We think of Υ as a kind of ‘parallel translation’ of associative 3-fold counting data along the family (X, ϕt , ψt ), t ∈ [0, 1]. Now Υ gives a bijection Crit(Φψ1 ) → Crit(Φψ0 ). The correct meaning of deformation-invariance for the Donaldson–Segal style invariants : : 0 , θ0 ) = DS(ψ : 1 , θ1 ) whenever θ0 ∈ DS(ψ, θ) in (ii) should be that DS(ψ Crit(Φψ0 ) and θ1 ∈ Crit(Φψ1 ) with Υ(θ1 ) = θ0 . (iv) If we follow (ii)–(iii), we generally cannot make invariants DS(P, ψ) for each principal SU(2)-bundle P → X, as in (8.2) (though see Remark 8.7 : below). Instead, we should aim to make one invariant DS(ψ, θ) in Λ>0 , as a formal power series similar to (7.4), roughly of the form   2 : (8.3) DS(ψ, θ) = DS(P, ψ) q −4π X [ϕ]∪c2 (P ) + correction terms. P → X principal SU(2)-bundle

In (i)–(iv) the author is motivated by an analogy with the Lagrangian Floer theory of Fukaya, Oh, Ohta and Ono [17, 18]. Here for a Lagrangian L in a symplectic manifold (S, ω), one needs to choose a ‘bounding cochain’ θ for L in homological algebra over a Novikov ring Λ>0 . Such θ need not exist or be unique, and we call L ‘unobstructed’ if θ exists. When (S, ω) is a symplectic Calabi–Yau 3-fold, θ corresponds to the critical point of a superpotential ΦJ : U → Λ>0 . There is a notion of ‘parallel translation’ of bounding cochains θ along smooth families Lt , t ∈ [0, 1] of Hamiltonian isotopic Lagrangians. We can now explain how to deal with Example 8.5 in our modified proposal. In Example 8.5, at least when t ∈ [0, 13 ) and t ∈ ( 23 , 1], there are no associative 3-folds

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in (X, ϕt , ψt ), so Φψt ≡ 0 and Crit(Φψt ) = U , and the extra data θt required in (ii)–(iii) is an arbitrary element of U . We could take θ0 to be the constant function 1 : H3 (X; Z) → 1 + Λ>0 , but there are many other choices. In Example 8.5 there are no associatives at t = 0 and at t = 1, so you might think that nothing changes between t = 0 and t = 1 from the point of view of counting associatives. However, the map Υ : U → U from ‘parallel translation’ along (X, ϕt , ψt ), t ∈ [0, 1] will in general not be the identity, but will depend on the (co)homology classes [ϕt ] ∈ H 3 (X; R), [Nt ] ∈ H3 (X; Z) and [M ] ∈ H4 (X; Z), ˜0 ] and δ. So if θ0 = 1, we may not have θ1 = 1. The as Υ0 in (7.7) depends on γ, [N difference in G2 -instanton counting between t = 0 and t = 1 should be compensated for by the difference between θ0 and θ1 . Our proposal for counting associative 3-folds in §7 involves counting only associative Q-homology spheres. However, in the Donaldson–Segal picture, G2 -instantons (P, A) might bubble on any compact associative 3-fold N , not just Q-homology 3-spheres, and in (8.2) we must allow N to be a general associative 3-fold. So we should explain how to bridge the gap between associative Q-homology 3-spheres, and general associative 3-folds. Haydys and Walpuski [24, §1] briefly outline a method for defining the ‘compensation terms’ w((P  , A ), N, 1) required by Donaldson and Segal, as in §8.1. They fix a line bundle L → N , and a spin structure on N with spin bundle S, and consider moduli spaces M(P,A ),N of solutions (B, Ψ) of the Seiberg–Witten type equations DB⊗A Ψ = 0, FB = μ(Ψ) on N , where B is a U(1)-connection on L with curvature FB , and Ψ : ad(P )|N → S ⊗ L is a vector bundle morphism, and DB⊗A is a twisted Dirac operator, and μ is a natural quadratic bundle map. Then M(P,A ),N has virtual dimension 0, and they wish to define w((P  , A ), N, 1) to be the virtual count [M(P,A ),N ]virt ∈ Z. We need to understand how w((P  , A ), N, 1) = [M(P,A ),N ]virt can change under deformations of (X, ϕt , ψt ), as a result of noncompactness or singularities in the moduli spaces M(P,A ),N . There are two ways in which this can happen: (A) There may be a family of solutions (Bt , Ψt ) with Ψt L2 → ∞ as t → t0 . The main result of [24] is that a rescaled limit of the (Bt , Ψt ) converges to a solution of the Fueter equation which controls bubbling of G2 -instantons along N , as in Examples 8.1–8.2. Thus, Haydys and Walpuski hope that changes in w((P  , A ), N, 1) resulting from such limits will exactly cancel changes in G2 -instanton counting, so that (8.2) is unchanged. (B) There may be a family of solutions (Bt , Ψt ) with Ψt L2 → 0 as t → t0 . While this does not cause noncompactness in M(P,A ),N , there is a problem in defining the virtual count [M(P,A ),N ]virt near solutions (B, Ψ) with Ψ = 0, as (B, 0) has stabilizer group U(1), so [M(P,A ),N ]virt may change. When Ψ = 0 the equation FB = μ(Ψ) becomes FB = 0, so (L, B) is a flat U(1)-line bundle on N . It turns out that [M(P,A ),N ]virt only changes under such transitions if the moduli space of such (L, B) has dimension 0, that is, if b1 (N ) = 0, so that N is a Q-homology 3-sphere. Our conclusion is that the Haydys–Walpuski proposal for w((P  , A ), N, 1) in (8.2) has problems for associative 3-folds N which are Q-homology 3-spheres, and these problems also involve flat U(1)-line bundles on N . Observe that this looks very similar to the programme of §7, which involves counting associative

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Q-homology 3-spheres N with weight I(N ) = |H1 (N ; Z)| in (5.12), which is the number of flat U(1)-line bundles on N . The author’s rough idea is to add some extra terms to (8.2), involving the choice of critical point θ of Φψ in (ii) above, whose changes under deformations would cancel the changes of type (B) to the Haydys–Walpuski terms, making the sum deformation-invariant. The author does not yet know how to do this. : (v) The invariant DS(ψ, θ) envisaged in (iv) should roughly be the sum of products of three kinds of terms: (a) terms counting G2 -instantons, as for  Or([A]) in (8.1); (b) Haydys–Walpuski style compensation [A]∈M(P,ψ) terms [24]; and (c) terms involving the chosen critical point θ of Φψ . This concludes our outline of modifications to the Donaldson–Segal programme. Remark 8.7. (a) From §7.1 we have been working in the ideal Λ>0 in the Novikov ring Λ in (7.1). So for instance, setting θ = 1 in (7.5) gives  GWψ,α q [ϕ]·α in Λ>0 . (8.4) Φψ (1) = α∈H3 (X;Z):[ϕ]·α>0 a

The coefficient of q in Φψ (1) is



GWψ,α , that is, it ‘counts’ associative a : 3-folds N in (X, ϕ, ψ) with area a. Similarly, the coefficient of 5 q in DS(ψ, θ) in 2 (8.3) morally ‘counts’ G2 -instantons (P, A) with energy −4π X [ϕ] ∪ c2 (P ) = a. The effect of working in Λ>0 like this is that we only get one counting invariant for each area or energy a > 0, so homology classes α with the same area, or principal bundles P with the same energy, get lumped together. α:[ϕ]·α=a

(b) If [ϕ] is generic in H 3 (X; R) then [ϕ]· : H3 (X; Z)/torsion → R is injective, so invariants in Λ>0 of the form (8.4) give an invariant for each class α in H3 (X; Z)/torsion or c2 (P ) in H 4 (X; Z)/torsion, which is not far from the system of invariants hoped for in the Donaldson–Segal proposal in §8.1. However, there is a catch. If [ϕ] is generic, and the superpotential Φψ in (7.4)– (7.5) is not identically zero, then one can show that dΦψ (θ) = 0 for all θ ∈ U, as the term in dΦψ (θ) from α ∈ H3 (X; Z) with GWψ,α = 0 and [ϕ] · α least dominates all others. So Φψ has no critical points, and (X, ϕ, ψ) is obstructed. If Φψ ≡ 0 then Φψ can only have critical points if there exist one or more pairs α1 , α2 in H3 (X; Z) with GWψ,α1 = 0, GWψ,α2 = 0, α1 = α2 and [ϕ] · α1 = [ϕ] · α2 , so that the obstructions from α1 , α2 cancel out. Then α1 − α2 lies in the kernel of [ϕ]· : H3 (X; Z)/torsion → R, and principal SU(2)-bundles P, P  → X such that c2 (P ) − c2 (P  ) lies in the subspace of H 4 (X; Q) spanned by Pd(α1 − α2 ) for all such pairs α1 , α2 contribute to the same G2 -instanton counting invariant. (c) As in (b), for a TA-G2 -manifold (X, ϕ, ψ) we have a dichotomy: either (i) Φψ ≡ 0. Then all associative 3-fold counting invariants are trivial. We can take [ϕ] generic in H 3 (X; R), and hope to define G2 -instanton counting invariants DS(α, ψ, θ) ∈ F for all α ∈ H 4 (X; Z)/torsion, depending on a choice of θ ∈ U. (ii) Φψ ≡ 0. Then we must choose a critical point θ of Φψ , which can only exist if [ϕ] lies in some proper vector subspace V of H 3 (X; R), and hope to define Donaldson–Segal style counting invariants DS(α, ψ, θ) ∈ F parametrized by α in H 4 (X; Z)/W for W = Ker([ϕ] ∪ −) ⊆ H 4 (X; Z) with rank W > 0.

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Here is an interesting special case of (i). Take X = Y × S 1 for Y a Calabi–Yau 3-fold, and initially take (ϕ, ψ) to be an S 1 -invariant TA-G2 -structure on X, e.g. the torsion-free G2 -structure induced by a Calabi–Yau structure on Y . Let N be a compact associative 3-fold in (X, ϕ, ψ). If N is S 1 -invariant then N ∼ = S 1 × Σ for some surface Σ ⊂ Y , so N is not a Q-homology 3-sphere, and it §7.3. If N is not S 1 -invariant then contributes zero to the+superpotential Φψ in , it lies in an S 1 -family eiθ · N : eiθ ∈ U(1) of associative 3-folds in X, and this family also contributes zero to Φψ , as χ(S 1 ) = 0. Thus Φψ ≡ 0, as in (i). By S 1 -localization we expect that counting G2 -instantons on (X, ϕ, ψ) gives the same answer as counting S 1 -invariant G2 -instantons on (X, ϕ, ψ), which is equivalent to counting solutions of a gauge theoretic equation on Y , essentially the ‘Donaldson–Thomas instantons’ considered by Tanaka [75]. The invariants may be an analytic version of some form of the algebro-geometric Donaldson–Thomas invariants of Y , as in Thomas [76] and Joyce and Song [47, 54].

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[42] D. Joyce, Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case, Ann. Global Anal. Geom. 26 (2004), no. 1, 1–58, DOI 10.1023/B:AGAG.0000023231.31950.cc. MR2054578 [43] D. Joyce, Special Lagrangian submanifolds with isolated conical singularities. IV. Desingularization, obstructions and families, Ann. Global Anal. Geom. 26 (2004), no. 2, 117–174, DOI 10.1023/B:AGAG.0000031067.19776.15. MR2070685 [44] D. Joyce, Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2003), no. 2, 279–347. MR2015549 [45] D. Joyce, Singularities of special Lagrangian submanifolds, pages 163–198 in S. K. Donaldson, Y. Eliashberg and M. Gromov, editors, Different Faces of Geometry, International Mathematical Series volume 3, Kluwer/Plenum, 2004. math.DG/0310460. [46] D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, vol. 12, Oxford University Press, Oxford, 2007. MR2292510 [47] D. Joyce, Generalized Donaldson-Thomas invariants, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, Surv. Differ. Geom., vol. 16, Int. Press, Somerville, MA, 2011, pp. 125–160, DOI 10.4310/SDG.2011.v16.n1.a4. MR2893678 [48] D. Joyce, An introduction to d-manifolds and derived differential geometry, pages 230–281 in L. Brambila-Paz, O. Garcia-Prada, P. Newstead and R. P. Thomas, editors, Moduli spaces, L.M.S. Lecture Notes 411, Cambridge University Press, 2014. arXiv:1206.4207. [49] D. Joyce, D-manifolds, d-orbifolds and derived differential geometry: a detailed summary, arXiv:1208.4948, 2012. [50] D. Joyce, D-manifolds and d-orbifolds: a theory of derived differential geometry, to be published by Oxford University Press, 2018. Preliminary version (2012) available at http://people.maths.ox.ac.uk/∼joyce/dmanifolds.html. [51] D. Joyce, A new definition of Kuranishi space, arXiv:1409.6908, 2014. [52] D. Joyce, Kuranishi spaces as a 2-category, arXiv:1510.07444, 2015. [53] D. Joyce, Kuranishi spaces and Symplectic Geometry, multiple volume book in progress, 2017-2027. Preliminary versions of volumes I, II available at http://people.maths.ox.ac. uk/~joyce/Kuranishi.html. [54] D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), no. 1020, iv+199, DOI 10.1090/S0065-9266-2011-00630-1. MR2951762 [55] A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003), 125–160, DOI 10.1515/crll.2003.097. MR2024648 [56] G. Lawlor, The angle criterion, Invent. Math. 95 (1989), no. 2, 437–446, DOI 10.1007/BF01393905. MR974911 [57] D. A. Lee, Connected sums of special Lagrangian submanifolds, Comm. Anal. Geom. 12 (2004), no. 3, 553–579. MR2128603 [58] Y.-I. Lee, Embedded special Lagrangian submanifolds in Calabi-Yau manifolds, Comm. Anal. Geom. 11 (2003), no. 3, 391–423, DOI 10.4310/CAG.2003.v11.n3.a1. MR2015752 [59] N. C. Leung, X. Wang, and K. Zhu, Thin instantons in G2 -manifolds and Seiberg-Witten invariants, J. Differential Geom. 95 (2013), no. 3, 419–481. MR3128991 [60] N. C. Leung, X. Wang, and K. Zhu, Instantons in G2 manifolds from J-holomorphic curves in coassociative submanifolds, Proceedings of the G¨ okova Geometry-Topology Conference 2012, Int. Press, Somerville, MA, 2013, pp. 89–110. MR3203358 [61] R. Lockhart, Fredholm, Hodge and Liouville theorems on noncompact manifolds, Trans. Amer. Math. Soc. 301 (1987), no. 1, 1–35, DOI 10.2307/2000325. MR879560 [62] R. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409–447. MR837256 [63] J. Lotay, Constructing associative 3-folds by evolution equations, Comm. Anal. Geom. 13 (2005), no. 5, 999–1037. MR2216150 [64] J. Lotay, Calibrated submanifolds of R7 and R8 with symmetries, Q. J. Math. 58 (2007), no. 1, 53–70, DOI 10.1093/qmath/hal015. MR2305050 [65] J. Lotay, Asymptotically conical associative 3-folds, Q. J. Math. 62 (2011), no. 1, 131–156, DOI 10.1093/qmath/hap036. MR2774358 [66] D. McDuff and D. Salamon, J-holomorphic curves and quantum cohomology, University Lecture Series, vol. 6, American Mathematical Society, Providence, RI, 1994. MR1286255

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[67] R. C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), no. 4, 705–747, DOI 10.4310/CAG.1998.v6.n4.a4. MR1664890 [68] G. Menet, J. Nordstr¨ om and H. N. S´ a Earp, Construction of G2 -instantons via twisted connected sums, arXiv:1510.03836, 2015. [69] J. Nordstr¨ om, Desingularizing intersecting associatives, preprint, 2012. [70] M. Ohst, Deformations of Asymptotically Cylindrical Cayley Submanifolds, arXiv:1506.00110, 2015. [71] H. N. S´ a Earp, G2 -instantons over asymptotically cylindrical manifolds, Geom. Topol. 19 (2015), no. 1, 61–111, DOI 10.2140/gt.2015.19.61. MR3318748 [72] H. N. S´ a Earp and T. Walpuski, G2 -instantons over twisted connected sums, Geom. Topol. 19 (2015), no. 3, 1263–1285, DOI 10.2140/gt.2015.19.1263. MR3352236 [73] P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z¨ urich, 2008. MR2441780 [74] D. I. Spivak, Derived smooth manifolds, Duke Math. J. 153 (2010), no. 1, 55–128, DOI 10.1215/00127094-2010-021. MR2641940 [75] Y. Tanaka, On the moduli space of Donaldson-Thomas instantons, Extracta Math. 31 (2016), no. 1, 89–107. MR3585951 [76] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR1818182 [77] T. Walpuski, G2 -instantons on generalised Kummer constructions, Geom. Topol. 17 (2013), no. 4, 2345–2388, DOI 10.2140/gt.2013.17.2345. MR3110581 [78] T. Walpuski, G2 -instantons, associative submanifolds and Fueter sections, arXiv:1205.5350, 2012. [79] T. Walpuski, Gauge theory on G2 -manifolds, PhD Thesis, Imperial College London, 2013. [80] T. Walpuski, G2 -instantons over twisted connected sums: an example, Math. Res. Lett. 23 (2016), no. 2, 529–544, DOI 10.4310/MRL.2016.v23.n2.a11. MR3512897 The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01740

Toward an algebraic Donaldson-Floer theory Jun Li

Dedicated to Simon Donaldson on the occasion of his 60th birthday Abstract. We construct the relative Donaldson polynomial invariants of a pair of a smooth divisor in a smooth surface, taking values in an operational algebraic Floer homology group. We conjecture that this pair forms an algebraic Donaldson-Floer theory.

1. introduction The relative Donaldson polynomial invariant of a four manifold N with boundary R = ∂N is a multi-linear map • Drel N : Sym H∗ (N ) −→ F H∗ (R),

taking value in the Floer homology group of R. Donaldson-Floer theory relates the Donaldson polynomial invariants DM : Sym• H∗ (M ) −→ Z of an oriented four-manifold M to the intersection pairings of the relative Donaldson polynomial invariants (1.1)

rel DM = $Drel M + , DM − %

when M is decomposed into two four-manifolds M± with boundaries along a three manifold R. This “cut and paste” technique was investigated extensively in 90’s by Braam-Donaldson, and others like Taubes, Morgan, Mrowka and Ruberman (cf. [BD1, BD2, Don1, Don2, MMR, Tau]). For more on its historical development, see [Don2]. In the inspiring paper [Don2], Donaldson envisioned an algebraic theory of Donaldson-Floer theory for algebraic surfaces. In the same paper, he demonstrated how such a theory would lead to an explicit formula of the Donaldson invariants of elliptic surfaces. Donaldson-Floer theory has since inspired later research developments in geometry and topology, beyond the subject of four-dimensional gauge theory. The theory of good degeneration of moduli spaces using the stack of expanded degenerations is one such example. This research work was partially supported by the NSF grants DMS-1564500 and DMS1601211. c 2018 American Mathematical Society

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JUN LI

The good degeneration is introduced to study moduli spaces when the underlying spaces under going a simple degeneration. A simple degeneration is an algebraic analogue of “cut and paste” in topology; it is a proper family π : X −→ C over a smooth pointed curve 0 ∈ C such that X is smooth, has smooth fiber over t = 0 ∈ C, and the central fiber X0 = X ×C 0 is a union of two smooth varieties Y+ and Y− intersecting transversally along a smooth divisor D. Let X → C be a simple degeneration of algebraic surfaces and let I be a line bundle on X. Donaldson’s vision of algebraic Donldson-Floer theory calls for the following: DF1. A family of moduli of “stable” sheaves MX/C so that for t = 0 ∈ C, MX/C,t = MX/C ×C t is the moduli of rank 2 determinant It stable sheaves on Xt . DF2. A moduli MY+ ,D of rank 2 determinant I|Y+ “relative stable” sheaves on the pair (Y+ , D) coupled with a restriction morphism r+ : MY+ ,D → MD , so that MY+ ,D with MY− ,D “reconstructs” MX/C,0 . Here MD is the moduli of rank two semistable determinant I|D bundles on D. DF3. the moduli MY± ,D defines relative Donaldson polynomial invariants DY± ,D : Sym• H∗ (Y± ) −→ H∗ (MD ), which form the Donaldson-Floer theory DXt = $DY+ ,D , DY− ,D %. The main difficulty in carrying this out, both in differential geometry and algebraic geometry, is to develop technique to deal with sheaves (on X0 ) that are not stable when when restricted to D = Y− ∩Y+ ⊂ X0 . In differential geometry, this is addressed by including the gluings of connections on Y− , Y+ , and connections on a chain of infinite cylinders R × R, subject to equivalences induced by translations of the cylinders (cf. [KM]). Inspired by Donaldson-Floer theory, and by Gieseker’s degeneration of rank two stable bundles on curves [Gie, GM], the author introduced the stack of expanded degenerations X of X → C, which is the algebraic analogue of the space of manifolds with infinite cylinder modulo R inserted. Using the stack of expanded degenerations, the author proved the algebraic analogue of Donaldson-Floer theory for GW-invariants [Li3, Li4]. We pause to describe the stack X of expanded degenerations of X/C. To begin with, we introduce the algebraic analogue of the infinite cylinder R × R. Let Δ = P(ND/Y+ ⊕ 1), where ND/Y+ is the normal bundle of D in Y+ . Because X is ∨ smooth, ND/Y+ ∼ , thus we get the same Δ when replace Y+ by Y− . Let = ND/Y − D0 , D∞ be the two distinguished sections of Δ → D. Then Δ comes with a C∗ action, fixing D0 ∪ D∞ . The space Δ◦ = Δ − D0 ∪ D∞ together with this C∗ action function as the infinite cylinder in Donaldson-Floer theory.

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The desired stack X is an Artin stack over C; its closed point over t = 0 consists of the single element [Xt ]; its closed points over 0 ∈ C are [X0 ], [X[1]0 ], [X[2]0 ], · · · , where X[n]0 = Y− ∼ Δ ∼ · · · ∼ Δ ∼ Y+ , D =D0

D∞ =D0

D∞ =D0

D∞ =D

where n many Δ are inserted, and glued along D0 and D∞ alternatively. The automorphism group of [X[n]0 ] is (C∗ )n , where the i-th copy of C∗ acts on X[n]0 via the C∗ action on the i-th copy Δ → D fixing D0 and D∞ . The logical step to realize Donaldson’s vision is to find a “stability” condition on various X[n]0 and apply the technique of expanded degenerations to construct MX/C so that elements in the central fiber MX/C,0 are “stable” sheaves on all possible X[n]0 ’s, so that (1) MX/C is proper over C, and (2) the restricton of every sheaf in MX/C,0 to singular locus of X[n]0 are stable. Unfortunately, it seems unlikely that such a stability condition exists. Building on the work of [GL], we can find a stability condition so that sheaves in the moduli MX/C,0 are locally free along the singular locus of X[n]0 . This way, we achieve DF1 and DF2 if we replace the restriction morphism by r± : MY± ,D −→ AD ,

(1.2)

where AD is the stack of rank two determinant I|D vector bundles on D. In Section 4, assuming deg I|D is odd, we will construct a numerical homology group H∗ (AD )nu of AD together with a pairing $·, ·% : H∗ (AD )nu × H2d−∗ (AD )nu −→ Q,

(1.3)

d = dim AD .

AD

be the stack of determinant I|D ⊗ ND/Y+ rank two vector bundles. We We let define the algebraic Donaldson-Floer homology to be (1.4)

HF∗ (D) = H∗ (AD )nu ⊕ H∗ (AD )nu .

Using Donaldson’s μ-map, we show that the restriction morphisms (1.2) define a multilinear map (1.5)

• Drel Y± ,D : Sym H∗ (Y± ) −→ HF∗ (AD ).

We conjecture that there are operators Rk : HF∗ (AD ) −→ HF∗ (AD ),

k ∈ Z,

so that the algebraic Donaldson-Floer theory roughly takes the following form. (For more precise statement, see Subsection 5.5.) Conjecture 1.1 (Algebraic Donldson-Floer theory). (1.6)

rel DX = $R ◦ Drel Y+ ,D , DY− ,D %.

In Section 2, we will recall the stack of expanded degenerations of a simple degeneration π : X → C; we will use this stack to construct the desired good family of stable sheaves on X relative to C, with fixed determinants. In Section 3, we will construct the moduli of relative stable sheaves on a pair (Y, D) of a smooth divisor D in a smooth surface Y . In Section 4, we will construct the numerical homology groups of AD , and construct its intersection pairing. Finally, we will construct the relative Donaldson polynomial invariants and state the conjectures. We believe that the numerical homology groups of AD can be generalized to arbitrary Artin stacks, which is under development together with Y.-H. Kiem [KL2].

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The proof of the main conjecture should be achievable by generalizing the proof of [Li4] to the case of defining a Gysin map over an Artin stacks, which will be addressed in [Li7]. Once these are fully addressed, they will provide an algebraic Donaldson-Floer theory in its simple form (assuming FNA). It will be interesting to see how much of FNA can be weakened, or entirely eliminated. This will be the subject for future research. It has been more than two decades since the seminal work of Donaldson on his vision of algebraic Donaldson-Floer theory. Since then the analogous algebrogeometric Donaldson-Floer theory for Gromov-Witten invariants and DonaldsonThomas invariants have been constructed. It is fitting that we begin to take steps to realize Donaldson’s vision of an algebraic Donaldson-Floer theory for surfaces. Such a theory will shed lights on the understanding of similar theory for Calabi-Yau threefolds (cf. [Tho]). Acknowledgments. The author thanks R. Thomas, Y.-H. Kiem and B.-S. Wu for their comments. 2. Good degeneration of moduli of stable sheaves In this section, we will use the stack of expanded degenerations to construct a good degeneration of moduli of stable sheaves on X/C. Under the favorable numerical assumption, we will show that these moduli spaces have the desired properties. This section is built on the techniques developed in [GL, Li3, Li5]. 2.1. The stack of expanded degenerations. Let π : X → C be a simple degeneration of algebraic surfaces over an affine pointed 0 ∈ C, where X0 is a union of smooth surface Y+ and Y− , intersecting transversally along a smooth connected curve D. We assume g(D) ≥ 1. In this paper, we will follow the exposition and notation developed in the survey article [Li6]. To begin with, we fix an ´etale C → A1 so that 0 ∈ C is the only point lying over 0 ∈ A1 . For A1 → A1 via z → z m , we form C m = C ×A1 A1 ; it is a curve over C via the first projection, with 0 ∈ C m the only ramification point ramified over 0 ∈ C. We let X m −→ C m be the minimal resolution of X ×C C m , with the projection the second projection to C m . The central fiber X0m (over 0 ∈ C m ) consists of m + 1 irreducible components, denoted by (2.1)

Y+ = Δ0 , Δ1 , · · · , Δm−1 , Δm = Y−

with D ∼ = Σi := Δi ∩ Δi+1 .

The X m comes with its tautological projection q : X m → X. The collection of (X → C m , q) are prototypes of expanded degenerations of X → C. The stack of expanded degeneration is the direct limit of the following finite type models X[m] → C[m]. For any m ≥ 1, we form m

C[m] = C ×A1 Am ,

Am → A1

is via (z1 , · · · , zm ) → z1 · · · zm .

We let X[m] be the small resolution of X ×C C[m] such that

(1) letting C m → C[m] be the diagonal map, then X[m] ×C[m] C m ∼ = X m;

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(2) letting τk (t) ∈ C[m] be the lifting of (· · · , t, 0, t, · · · ) ∈ Am (i.e. one 0 digit at k-th place, and all others are t), then when t specializes to 0 the singular locus of X[m]τk (t) (∼ = X0 ) specializes to Σk ⊂ X[m]0 . The pair X[m] → C[m] admits a (C∗ )m−1 -action. Let (C∗ )m−1 acts on Am via −1 (t1 , · · · , tm )σ = (σ1 t1 , σ1−1 σ2 t2 , · · · , σm−1 tm ),

σ ∈ (C∗ )m−1 .

It lifts uniquely to a (C∗ )m−1 action on X[m] → C[m], equivariant over C. We let (2.2)

Λi ⊂ X[m] ×Am (zi = 0) ⊂ X[m]

be the irreducible components of X[m] ×Am (zi = 0), where (zi = 0) ∼ = Am−1 , such that it is the proper transform of Y+ × (zi = 0) in the family X[m]. Note Λi ∩ X[m]0 = Δi ∪ · · · ∪ Δm . We define the stack X of expanded degenerations of X/C. It is a C-stack. For any C-scheme S, an S-family of expanded degenerations consists of (X → S, q), where X → S is a family of schemes and q : X → X a morphism, such that there is an open covering S = ∪Sα and C-morphisms ξα : Sα → C[nα ] so that X |S ∼ = ξ ∗ X[nα ] := X[nα ] ×C[n ] Sα , α

α

α

and that the restriction q|Sα is the composition of the first projection X |Sα → X[nα ] with the tautological projection X[nα ] → X. An arrow between (X → S, q) and (X  → S, q  ) ∈ X(S) consists of an S-isomorphism f : X → X  that commutes with the projections q : X → X and q  : X  → X. We denote by 0 ∈ C[m] the unique lifting of 0 ∈ Am . Then X[m]0 = X[m]×Am 0 is X0m = X m ×C m 0. Further X[m]0 considered as an element in X(C) has automorphism group AutX (X[m]0 ) = (C∗ )m−1 . Note that the i-th factor of (C∗ )m−1 acts trivially on X[m]0 − Δi , and acts nontrivially on Δi . 2.2. Good stable sheaves on X0m . We fix an ample line bundle H on X, and study stable rank two sheaves on X0m . We investigate possible induced ample line bundles on X0m . First, via the tautological projection q : X m → X, the pull-back q ∗ H is ample on Y± ⊂ X0m , but not on Δ[1,m−1] 1 . To get an ample line bundle on X m , we introduce divisors Δ≥k = Δk ∪ · · · ∪ Δm ⊂ X m , and introduce Q-line bundle (2.3)

H,δ := q ∗ H( δ1 Δ≥1 + δ2 Δ≥2 + · · · + δm Δm )

with δ = (δ1 , · · · , δm ) admissible. Definition 2.1. We say a sequence δ = (δ1 , · · · , δm ) admissible if δk ∈ [0, 1]∩Q and the sequence 0 = δ1 < δ2 < · · · is strictly increasing. Lemma 2.2. When > 0 is sufficiently small, for any admissible {δ· }, H,δ is ample on X m . 1

We follow the intuitive convention Δ[1,m−1] = Δ1 ∪ · · · ∪ Δm−1 .

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Proof. Since C is affine, we only need to check that H,δ |X0m is ample. For this, since is sufficiently small, and δk ∈ (0, 1), H,δ |Δ0 and H,δ |Δm are ample. To prove the lemma, we only need to check the restrictions H,δ |Δk for 1 ≤ k < m. For such k, we have H,δ |Δk = q ∗ H|Δk (− δk−1 Σk−1 + δk Σk ). Thus H,δ is ample if {δ· } is admissible. O



Fixing an ample H,δ on X m (when δ is admissible), for any coherent sheaf of -modules E, we define its Poincare polynomial to be

X0m

⊗n pE (n) = χ(E(n)) = χ(E ⊗ OX0m (H,δ )),

n sufficiently divisible.

When E is a pure dimension two sheaf, pE is a degree two polynomial in n. For f (x) = a2 x2 + a1 x + a0 , we agree l.c.f = a2 , and [f ]1 = a1 . Definition 2.3 (Slope stability). A pure dimension two coherent sheaf on X0m is H,δ -slope-stable if for any proper pure dimension two quotient sheaf E → F = 0, [pE ]1 [pF ]1 < . l.c.pE l.c.pF Definition 2.4 (Gieseker-Simpson stability). A pure dimension two coherent sheaf E on X0m is H,δ -stable if for any proper pure dimension two quotient sheaf E → F = 0, pE (n) pF (n) < , for n ' 0. l.c.pE l.c.pF We introduce the favorable numerical assumption (abbrev. FNA) that will make the above two stabilities equivalent. For convenience, for a pure dimension two sheaf E on X0 we say rank E ≺ 2 if both rank E|X± ≤ 2 and at least one of rank E|X± < 2. We let pc (n) = χ(It (n)) + χ(OXt (n)) − c, Here as always, It = I|Xt and OXt (n) =

t = 0 ∈ C.

OXt (Ht⊗n ).

Favorable Numerical Assumption (FNA). We say that (X, I, H) satisfies favorable numerical assumption if there is no pure dimension two sheaf F of OX0 modules of rank F ≺ 2 such that [pc ]1 [pF ]1 = . l.c.pF l.c.pc We have the following easy consequence. Proposition 2.5. Suppose (X, I, H) satisfies FNA. Then for any c, and an admissible sequence δ, there is an 0 > 0 so that if for some 0 < < 0 , a sheaf E of OX0m -modules of Poincare polynomial pc is H,δ -stable, then it is H ,δ -stable for any 0 <  < 0 and any admissible δ  . Proof. The proof is parallel to [Li5, Section 4].



This proposition says that the H,δ -stability of a bounded set of sheaves is independent of the choice of δ and sufficiently small > 0. Because of this, in the following whenever we say a sheaf on X0m is H,δ -stable we mean that it is H,δ -stable for an admissible δ and all sufficiently small .

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In the remainder of this paper, we fix an (X, I, H) that satisfies FNA. Definition 2.6. We say a coherent sheaf on X0m is good if it is locally free along the singular locus of X0m . We introduce types of good sheaves over X[m]0 . Let Δ → D be the standard projection; let F be a sheaf on Δ that is locally free away from a finite point set. We say F has type (0a 1b ) if for a general fiber Fξ of Δ → D over ξ ∈ D, ⊕b F|Fξ = O⊕a Fξ ⊕ OFξ (1) . Definition 2.7. Let E be a pure dimension two pure rank two sheaf on X[m]0 that is locally free away from a finite point set of X[m]0 . We say E has type O if for every Δi ⊂ Δ[1,m−1] , F|Δi has type (02 ); we say E has type I it there is an i ∈ [1, m − 1] so that F|Δi has type (01) or (11). Lemma 2.8. Assuming FNA, and let E be a pure dimension two pure rank two H,δ -stable good sheaf on X[m]0 . Then E has type O or I. In case E has type I, then there is a unique i ∈ [1, m − 1] so that F|Δi has type (01), and all other F|Δj=i has type (02 ). Proof. We begin with a simple observation. Let R be a nodal curve that is a (chain like) union of four smooth irreducible components R = R− ∪ R1 ∪ R2 ∪ R+ with R1 ∼ = R2 ∼ = P1 , and has exactly three nodes p1 = R− ∩ R1 , p2 = R1 ∩ R2 and p3 = R2 ∩ R+ . Let V be a rank two locally free sheaf on C so that for i = 1 and 2, V|Ri ∼ = ORi ⊕ ORi (1). We claim that either there is an injective OR1 ∪R2 → V, or there are two subsheaves V± → V of which the following holds: t.f. ∼ A1. V− |R− = V|R− , (V− |R1 )t.f. ∼ = O⊕2 = OR1 , (V− |R+ )t.f. = 0; R1 , (V− |R2 ) t.f. A2. V+ |R− = 0, (V+ |R1 )t.f. ∼ = V|R+ . = OR1 , (V+ |R2 )t.f. ∼ = O⊕2 R1 , (V+ |R+ ) Here the superscript “t.f.” stands for the torsion free part. Indeed, by our assumption of V|R1 and V|R2 , we have unique injective ιi : ORi (1) → V|Ri . In case (2.4)

ι1∗ (OR1 (1))|p2 = ι2∗ (OR2 (1))|p2 ⊂ V|p2 ,

then ι1∗ (OR1 (1)) ⊂ V|R1 and ι2∗ (OR2 (1)) ⊂ V|R2 patch to form a subline bundle L ⊂ V|R1 ∪R2 of degree 1 along R1 and R2 . Then OR1 ∪R2 = L(−p1 − p3 ) ⊂ V is the desired subsheaf. In case (2.4) fails, then we can find an injective ι1 : OR1 → V|R1 so that V|R1 /ι1∗ OR1 is locally free, and ι1∗ OR1 |p2 = ι2∗ (OR2 (1))|p2 ⊂ V|p2 . We then define V− ⊂ V be the subsheaf so that V− |R− = V|R− ; V− |R1 ⊂ V|R1 is the subsheaf (of V|R1 ) generated by ι1∗ OR1 and ι1∗ OR1 (1)(−p2 ); (V− |R2 )t.f. ⊂ V|R2 is the subsheaf ι2∗ OR2 (1)(−p3 ), and (V− |R+ )t.f. = 0. This sheaf satisfies condition A1. By symmetry, we obtain V+ satisfying A2. This proves the claim. We now prove the lemma. To simplifying the notation, we will rule out the case where E is a good sheaf on X[3]0 so that both E|Δ1 and E|Δ2 are of type (01). The general case can be treated similarly and will be omitted. By the observation just made, we see that one possibility is when we can find a subsheaf L ⊂ E so that L is supported on Δ1 ∪ Δ2 , and L restricted to general fibers Fξ ⊂ Δ1 and Δ2 are OFξ . In this case, we calculate [pL ]1 2 deg H|D = + O(1), l.c.pL

deg H|D

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which approaches positive ∞ when approaches 0+ . Consequently, letting F be the cokernel of L → E, for sufficiently small, [pE ]1 [pF ]1 > , l.c.pE l.c.pF impossible because E is H,δ -stable. The other possibility is when we can find two subsheaves E− and E+ ⊂ E so that E± |X± = E|X± , (E± |X∓ )t.f. = 0, and that there are sheaves R1 and R2 making the following sequence exact 0 −→ R1 ⊕ R2 −→ E− ⊕ E+ −→ E −→ 0, such that Ri is a rank one sheaf of OΔi -modules whose restriction to general fibers Fξ of Δi → D is OFξ (−1). (Compare with (A1) and (A2) above.) Because of this, l.c.pRi |=0 = [pRi ]1 |=0 . This implies pE |=0 = (pE− + pE+ )|=0 . Hence one of [pE− ]1 [pE ]1 ≥ l.c.pE− l.c.pE

or

[pE+ ]1 [pE ]1 ≥ l.c.pE+ l.c.pE

hold true. By FNA, the above two equalities can not hold simultaneously, thus one  of the strict inequality holds, violating that E is H,δ -stable. In Donaldson-Floer theory, we need to work with moduli of sheaves with fixed determinant. To this end, we will fix a line bundle I on X, and work with rank two stable sheaves on Xt of determinant Ii = I|Xt . As is known, a family of determinant It sheaves does not necessarily specialize to a sheaf on X0 of determinant I0 , even when the determinant exists. In the following, we will analyze this phenomenon in details, following [Cap]. Let E be a rank two sheaf on X m , flat over C m = C ×A1 A1 , so that E0 is a good H,δ -stable sheaf on X0m . We continue to denote by q : X m → X the tautological projection. Because E is locally free along the singular locus of X0m , det E is well-defined. Assuming det E|X m −X0m ∼ = q ∗ I|X m −X0m , then there are integers ei so that det E ∼ = q ∗ I(



ei Δi ).

Note that OX m (Δ0 + · · · + Δm ) ∼ = OX m . This leads to the following definition of line bundles on X[m] similar to I, after [Cap]. Recall that the divisor Λi ⊂ X[m] are defined in (2.2). Definition 2.9. Let I  be a line bundle on X[m]. We say I  ∼ I if there are integers e1 , · · · , em so that I ∼ = q ∗ I(e1 Λ1 + · · · + em Λm ). Given X ∈ X(S), we say a line bundle J on X has J ∼ I if for any open U ⊂ S with fU : X ×S U ∼ = X[m] ×C[m] U given by the definition of X, we can find a line bundle I  on X[m] with I  ∼ I and a line bundle L on U so that J ∼ = pr∗U l ⊗ fU∗ I  .

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2.3. Good degeneration of moduli of stable sheaves. We now construct the stack of good degenerations of stable sheaves on X/C. We agree that for any ⊗n (X , q) ∈ X(S) and a sheaf E on X , Es (n) = Ex ⊗ OXs (q ∗ H,δ ), for a choice of H,δ apparent in the context. Definition 2.10. We define a groupoid GMI,c X/C over C as follows. For any C-scheme S, we define GMI,c X/C (S) to be the triplet (E, X , q), where (X , q) ∈ X(S) and E is an S-flat sheaf of OX -modules so that for any closed s ∈ S, Es = E|Xs is a pure rank two pure dimension two good sheaf on Xs and χ(Es (n)) = pc (n); and det E ∼ I. An arrow between (E, X , q) and (E  , X  , q) ∈ GMI,c X/C (S) consists of (ρ, f ) where  ∗  ρ : X → X is an arrow in X(S), and f : E → ρ E is an isomorphism. I,c Clearly, GMI,c X/C is an Artin stack over C. For any (E, X[n]0 ) ∈ GMX/C (C), we define AutGM (E) to be the group of arrows from (E, X[n]0 ) to itself. Note that as any sheaf can be scaled by c ∈ C∗ , we have a tautological subgroup C∗ ⊂ AutGM (E). I,c Definition 2.11. We define SMI,c X/C to be the substack of GMX/C so that for I,c any C-scheme S, SMI,c X/C (S) consists of all (E, X , q) ∈ GMX/C (S) so that for any closed s ∈ S, Es is H,δ -stable and AutGM (Es )/C∗ is finite.

Because being stable is an open condition, SMI,c X/C is an open substack of I,c ∗ GMI,c X/C . Using the tautological subgroup C ⊂ AutGM (E), we define MX/C to be ∗ the rigidification of SMI,c X/C by the tautological group C ⊂ AutGM . Thus for any

E ∈ MI,c X/C (C),

AutM (E) = AutGM (E)/C∗ .

Theorem 2.12. The stack MI,c X/C is a separated, C-proper DM stack of finite type. We will prove part of this theorem in this section, and postpone the remainder to the next section. First, the separatedness is similar to the proof in [Li3, LW]. To prove that it is a DM stack, like in [Li3, LW], we first construct the coarse moduli of H,δ -stable sheaves on X[m]/C[m], then take the open substack of good H,δ stable sheaves on X[m]/C[m] that have finite AutM . Their tautological morphisms etale covering of MI,c to MI,c X/C , with varying m, form an ´ X/C . To address the properness, we begin with proving a special version of valuative criterion for MI,c X/C . Let R ⊃ k ⊃ C be a discrete valuation ring with K the field of its fractions. Given Spec R → C, we will use XR = X ×C R, and use Xk and XK to denote the closed and the generic fiber of XR . In case we have Spec R → C m , m and Xkm to denote the corresponding pullbacks of X m → C m . we use XR Lemma 2.13. Suppose we have a flat morphism Spec R → C, and an H-stable sheaf EK on XK of determinant det EK ∼ I. Then possibly after a finite base change, we can lift Spec K → C to an ´etale Spec R → C m for some m such that m the sheaf EK extends to an R-flat ER on XR so that Ek = ER |Xkm is good, and is H,δ -stable. Proof. Let XR = X ×C Spec R. Applying [GL, Thm 2.10], we can extend EK to an R-flat ER on XR so that the restricton Ek = ER |Xk is H-stable. Using the

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extension ER , possibly after a finite base change, we can lift the given Spec R → C to an ´etale Spec R → C m , and applying [GL, Lem 3.2] to extend EK to an R-flat m ER on XR . Applying [GL, Prop 3.3], we can assume that Ek is indeed good. In the end, using that (X, I, H) satisfies FNA, we conclude that Ek is H,δ -stable.  Our goal it so construct such an extension so that AutM (Ek ) is finite. In the following, we say that E|Δi is a pullback sheaf from D if there is a sheaf F on D so that, for p : Δi → D the tautological projection, E|Δi ∼ = p∗ F. Lemma 2.14. Let E be an H,δ -stable sheaf on X0m . Then AutM (E) is finite if and only if there are no Δi ⊂ Δ[1,m−1] so that E|Δi is a pullback sheaf from D. Proof. Direct check.



Proposition 2.15. Let the situation be as in Lemma 2.13. Then we can make the extension ER so that AutM (Ek ) is finite. Proof. Let ER be constructed by Lemma 2.13. In case no Ek |Δi are pullback sheaf from D (for i ∈ [1, m − 1]), then AutM (Ek ) is finite, and we are done. In case there are some i ∈ [1, m − 1] so that Ek |Δi is a pullback sheaf from D, m−1 we apply the argument in [Li3, Lem 3.9] to show that we can construct an XR , m−1 m−1 m m and a projection g : XR → XR that contracts Δi ⊂ XR a sheaf ER on XR so that g ∗ ER ∼ = ER . This way, Ek is good, H,δ -stable, and dim AutM (Ek ) < m , a flat exdim AutM (Ek ). Repeating this argument, we end up with ER on XR  (k).  tension of EK , so that (Ek , X0m ) ∈ MI,c X/C We will complete the proof of the properness in Subsection 3.4. Proof of the finite typeness part of Theorem 2.12. We show that MI,c X/C is bounded. We form the set Ξ+ = {E|Y+ | (E, X0m ) ∈ MI,c X/C (C)}. We define Ξ− similarly, with + replaced by −. Following [GL, Section 4] and [Ma], we know {c1 (F) | F ∈ Ξ± } is bounded, and {c2 (F) | F ∈ Ξ± } is bounded from below. For α = O or I, we introduce Ξ(α) = {(E, X0m ) ∈ MI,c X/C (C) | E is of type α}. (cf. type α is introduced before Definition 2.7.) We first look at Ξ(O). By [GL, Section 4], we know that for (E, X0m ) ∈ Ξ(O), c2 (E|Δi ) > 0 for all i ∈ [1, m−1]. Thus by and that for (E, X0m ) ∈ Ξ(O), mthat {c2 (F) | F ∈ Ξ± } is bounded from below, m i=0 c2 (E|Δi ) = c, we conclude that {m | (E, X0 ) ∈ Ξ(O)} is bounded. Thus Ξ(O) is bounded. We now look at Ξ(I). By Lemma 2.8, we know that for any (E, X0m ) ∈ Ξ(I), there is a unique i ∈ [1, m − 1] so that E|Δi has type (01). We let Ξ(ex) = {E|Δi | (E, X0m ) ∈ Ξ(I) and E|Δi has type (01)}. Sublemma 2.16. The set Ξ(ex) is bounded. We will prove this sublemma in Subsection 3.4. Granting the sublemma, we see that the set {c2 (F) | F ∈ Ξ(ex)} is bounded. This boundedness, combined with the argument for the boundedness of Ξ(O) shows that Ξ(I) is bounded. By Lemma  2.8, we have MI,c X/C (C)set = Ξ(O) ∪ Ξ(I), thus is bounded.

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3. Moduli of relative stable sheaves In this section, we fix a pair D ⊂ Y of a smooth connected divisor in a smooth surface, fix an ample line bundle H on Y , and fix a line bundle I on Y so that deg I|D is odd. We will construct the moduli of rank two relative stable sheaves MI,c Y,D on (D, Y ) of determinants I and second Chern class c, and constructs its associated restriction morphism rI,c,D : MI,c Y,D −→ AD ,

(3.1)

to the stack of rank two determinant I|D vector bundles on D. 3.1. Stack of expanded relative pairs. Inductively, we construct pairs D[n] ⊂ Y [n] over An , with a tautological projection q : (D[n] ⊂ Y [n]) −→ (D ⊂ Y ) as follows. We let D[0] ⊂ Y [0] be D ⊂ Y . Once (D[n] ⊂ Y [n], q) is constructed, we let Y [n + 1] be the blowing up of Y [n] × A1 along D[n] × 0 ⊂ Y [n] × A1 ; let D[n + 1] ⊂ Y [n + 1] be the proper transform of D[n] × A1 , and let Y [n + 1] → An+1 be the composite Y [n + 1] −→ Y [n] × A1 −→ An × A1 = An+1 , and let q : Y [n + 1] → Y be the composite of the tautological Y [n + 1] → Y [n] and q : Y [n] → Y . The pair D[n] ⊂ Y [n] comes with a (C∗ )n -action. The standard C∗ -action on A1 paired with the trivial action on Y lifts to a C∗ -action on Y [1] → A1 , leaving D[1] invariant. Inductively, the standard (C∗ )n -action on An lifts to a (C∗ )n -action on Y [n] → An , leaving D[n]-invariant. We call the pair D[n] ⊂ Y [n] over An with the projection q : Y [n] → Y and the given (C∗ )n -action a local model of expanded relative pairs of D ⊂ Y . Following the construction, we see that the central fiber Y [n]0 := Y [n] ×An 0 = Y ∪ Δ1 ∪ · · · ∪ Δn , where Δi and Δi+1 (we agree Δ0 = Y ) intersects transversally along Σi ∼ = D, and D[n]0 ⊂ Δn . The i-th factor of (C∗ )n acts trivially on Y [n]0 − Δi , and acts non-trivially on Δi , fixing Σi ∪ Σi+1 (we agree Σn = D[n]0 ). Let X → C be a simple degeneration as in the previous section with X0 = Y− ∪ Y+ , intersecting along the smooth D. For the model X[m] → C[m] = C ×A1 Am , if we let An × {0} × Am−n−1 ⊂ Am be the coordinate hyperplane (tn+1 = 0), then . X[m] ×Am An × {0} × Am−n−1 = Y+ [n] × Am−n−1  An × Y− [m − n − 1], where  stands for gluing via the tautological isomorphism D[n] × Am−n−1 ∼ = D × An × Am−n−1 ∼ = An × D[m − n − 1]. We define the stack Y of expanded pairs of D ⊂ Y . For any scheme S, Y(S) is the collection of all S-flat pairs D ⊂ Y together with a projection q : Y → Y so that locally it is the pullback of D[n] ⊂ Y [n] and q : Y [n] → Y for some S → An . An arrow from (D ⊂ Y, q) to (D ⊂ Y  , q  ) ∈ Y(S) is an S-isomorphism ρ : Y → Y 

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commuting with the projections q : Y → Y and q  : Y  → Y , and such that ρ|D : D ∼ = D . This way, ξ = (D[n]0 ⊂ Y [n]0 ), considered as an element in Y(C), has automorphism group AutY (ξ) = (C∗ )n . 3.2. Moduli of relative stable sheaves. In this paper, we assume that (Y, I, H) satisfies the FNA: there is no line bundle A on Y so that 2(A · H) = I · H. This way, a rank two torsion free sheaf E on Y is H-stable if and only if it is H-slope stable. For the relative surface D[n]0 ⊂ Y [n]0 , we denote by H,δ the Q-line bundle H,δ := q ∗ H( δ1 Δ≥1 + δ2 Δ≥2 + · · · + δm Δm + δm+1 D[n]0 ). For δ = {δ· } admissible as in Definition 2.1 and sufficiently small, H,δ is ample. We say a pure dimension two sheaf E on Y [n]0 is good if it is locally free along D[n]0 and the singular loci of Y [n]0 . We say E is H,δ -stable if it is H,δ -stable for an admissible δ and sufficiently small > 0. We have the following lemma on types of H,δ -stable sheaves on Y [n]0 , analogous to Lemma 2.8. Lemma 3.1. Assuming FNA, and let E be a pure dimension two pure rank two H,δ -stable good sheaf on Y [n]0 . Then E has type O. Here by E has type O we mean that for any Δi>0 ⊂ Y [n]0 , E|Δi has type O (cf. Definition 2.6). Proof. Suppose E on Y [n]0 is not of type O. Then there is a 0 < i ≤ n so that E|Δi is not of type O. We let i by the largest of such. Denoting Δ≥i = ∪k≥i Δk ⊂ Y [n]0 , then we can find a pure dimension one subsheaf L ⊂ E so that L is supported on Δ≥i , L|Δ≥i has pure rank one, and L|Δj , for j ≥ i, all have type (0). Then [χ(L(n))]1 (deg H|D + O( )) · n = , l.c.χ(L(n))

deg H|D + O( 2 ) which for > 0 sufficiently small is bigger than stable. This proves the lemma.

[pc (n)]1 l.c.pc (n) ,

violating that E is H,δ 

For any integer c, we let (3.2)

pc (m) = χ(I(m)) + χ(OY (m)) − c.

Corollary 3.2. Suppose (Y, I, H) satisfies FNA. Then for any c, and an admissible sequence δ, there is an 0 > 0 so that if for some 0 < < 0 , a sheaf E on Y [n]0 of Poincare polynomial pc is H,δ -stable, then it is H ,δ -stable for any 0 <  < 0 and any admissible δ  . Proof. The proof is parallel to [Li5, Section 4].



Like in the absolute case, we introduce the notion that a line bundle on Y [n] is similar to I. Let Λi ⊂ Y [n] be the smooth divisor lying in Y [n] ×An (zi = 0) that contains D[n] ×An (zi = 0). (Note that then Λi ∩ Y [n]0 = ∪nj=i Δi .) Definition 3.3. We say a line bundle I  on Y [n] is similar to I, denoted by I ∼ I, if there are integers e1 , · · · , en so that I  ∼ = q ∗ I ⊗ J, where 

(3.3)

J = OY [n] (e1 Λ1 + · · · + en Λn ).

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Given a (D ⊂ Y) ∈ Y(S) and a line bundle I on Y, we say I ∼ I if for any open U ⊂ S so that there is an isomorphism fU : (D, Y) ×S U ∼ = (D[n], Y [n]) ×An U given by the definition of Y, there is a line bundle I  on Y [n] similar to I and a line bundle L on U so that J ∼ = pr∗U L ⊗ pr∗Y [n] I  . We now construct the stack of relative stable sheaves on D ⊂ Y . For a sheaf E on Y [n]0 , we agree Es (m) = Es ⊗ OXs (q ∗ H ⊗m ), where q : Y [n]0 → Y as always is the tautological projection. Definition 3.4. We define the groupoid GMI,c Y,D as follows. For any scheme (S) to be the pair (E, D ⊂ Y), where (D ⊂ Y) ∈ X(S) and E S, we define GMI,c Y,D is an S-flat sheaf of OY -modules so that for any closed s ∈ S, Es = E|Ys is a pure rank two pure dimension two good sheaf on Ds ⊂ Xs such that χ(Es (n)) = pc (n), and that det E ∼ I. An arrow from (E, Y) to (E  , Y  ) ∈ GMI,c Y,D (S) consists of (ρ, f ), where ρ : Y → Y  is an arrow in Y(S), and f : E → ρ∗ E is an isomorphism. I,c The groiupoid GMI,c Y,D is an Artin stack. For any (E, Y [n]0 ) ∈ GMY,D (C), we define AutGM (E) to be the group of arrows (ρ, f ) from (E, Y [n]0 ) to itself. Note that AutGM (E) contains a tautological subgroup C∗ ≤ AutGM (E) induced by scaling E. I,c Definition 3.5. We define SMI,c Y,D be the substack of GMY,D as follows. For I,c any scheme S, SMI,c Y,D (S) consists of all (E, Y) ∈ GMY,D (S) so that for any closed s ∈ S, Es is H,δ -stable and AutGM (Es )/C∗ is finite.

Because the condition being stable and AutGM (Es )/C∗ finite is an open condiI,c I,c tion, SMI,c Y,D is an open substack of GMY,D . We define MY,D to be the rigidifica∗ tion of SMI,c Y,D by the tautological subgroup C in AutGM (E). This way, for any I,c E ∈ MY,D (C), AutM (E) = AutGM (E)/C∗ . Theorem 3.6. The stack MI,c Y,D is a separated, proper DM stack of finite type. Proof. The proof that MI,c Y,D is a DM stack, and that it is separated are parallel to the proof of Theorem 2.12, and will be omitted. We now prove that it is of finite type. For any i ≥ 0, we let (agreeing Δ0 = Y ) Ri = {E|Δi | (E, Y [n]0 ) ∈ MI,c Y,D , n ≥ i}. By Lemma 3.1, for any i ≥ 1, all F ∈ Ri are of type (02 ). It follows that det E restricted to every Δi≥1 is a pullback line bundle from D via Δi → D. In particular, det E ∼ = q ∗ I, where q : Y [n]0 → Y is the projection. ¯ we We now prove that MI,c Y,D (C) is bounded. We first show that for some n ¯ the set Ri are bounded. First, for any have Rn¯ = ∅, and that for all i ≤ n (E, Y [n]0 ) ∈ MI,c Y,D (C), as E is good, we have (cf. [GL, Lem 4.2]) (3.4)

c2 (E|Y ) + c2 (E|Δ1 ) + · · · + c2 (E|Δn ) = c.

Further, as all E|Δi≥1 are of type (02 ) and AutM (E) is finite, we have c2 (E|Δi≥1 ) > 0. Thus {c2 (F) | F ∈ R0 } is bounded from above by c. Adding that all sheaves in R0

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are e-stable for some constant e, and has determinant I, we conclude that R0 is bounded (cf. [GL, Section 4] and [Ma]). In particular, there is a c¯0 so that c2 (F) ≥ c¯0 for all F ∈ R0 . Adding ¯ = c − c¯0 , Rn¯ = ∅. c2 (E|Δi≥1 ) > 0, we conclude that for n It remains to argue that Ri≥1 are all bounded. Indeed, because R0 is bounded, the set {F|Σ1 | F ∈ R0 } is bounded. Because it contains {F|Σ1 | F ∈ R1 }, the later is bounded. Then because all sheaves in R1 have determinants isomorphic to the pullback of I|D , via Δ1 → D, and have second Chern classes bounded by c − c¯0 , we conclude that R1 is bounded. Iterating, we conclude that all Ri≤¯n are bounded. Therefore, since Ri≥¯n = ∅, we conclude that MI,c Y,D (C) is bounded. This proves the theorem except the properness part.  Remark 3.7. The proof shows that to any (E, Y [n]0 ) ∈ MI,c Y,D (C), the restriction of E to D ∼ = D[n]0 ⊂ Y [n]0 is a rank two vector bundle on D of determinant I|D . To treat the properness part of the theorem, we first verify a version of the valuative criterion. Let R ⊃ k ⊃ C be a discrete valuation ring with K the field of its fractions. Lemma 3.8. Let (EK , YK ) ∈ MI,c Y,D (K), where YK = Y × K. Then possibly after a finite base change, we can extend (EK , YK ) to (ER , YR ) ∈ MI,c Y,D (R). Proof. Since YK = Y × K is smooth. We can extend EK to a flat family of stable sheaves ER on Y × R. In case ER is locally free along D × R, we are done. If not, we apply the technique in the proof of Lemma 2.13 and Proposition 2.15 to show that possibly after a finite base change, we can extend (EK , YK ) to (ER , YR ) ∈ MI,c  Y,D (R), as desired. We will omit the details here. Corollary 3.9. Let Rc be the open substack of MI,c Y,D so that Rc (C) = I,c I,c {(E, Y ) ∈ MY,D (C)}. Let Rc be the closure of Rc in MY,D . Then Rc is proper. 3.3. Singularizing-a-sheaf. Let (E, Y [n]0 ) ∈ MI,c Y,D (C). When the traceless 2 part Ext (E, E)0 = 0, then a standard deformation technique combined with that Grothendieck’s Quot schemes are projective shows that (E, Y [n]0 ) ∈ Rc . Thus in case 2 {(E, Y [n]0 ) ∈ MI,c Y,D (C) | Ext (E, E)0 = 0} I,c is dense in MI,c Y,D , we conclude that MY,D is proper. In general, we will use the technique we call singularizing-a-sheaf. Let S be a reduced projective surface; E a pure dimension two sheaf of OS -modules, and x = {x1 , · · · , xl } ⊂ S be a reduced, length l zero-subscheme. A singularization of E along x is the kernel sheaf

E := ker{E −→ Ox } σ

for σ surjective. We say E is a general singularization along x if the σ is general. Lemma 3.10. Let E and S be as stated. Let L be a line bundle on S, and let l be an integer so that l ≥ dim Hom(E, E ⊗ L)0 . Then we can find a length l subscheme x = {x1 , · · · , xl } ⊂ S so that for a general singularization E of E along x, we have Hom(E , E ⊗ L)0 = 0.

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Proof. Let σ : E → Ox be given by the direct sum of σxi : E → kxi . We can choose x so that E is locally free near x. Let Axi = ker(σxi : E|xi → kxi ). An easy argument shows that for any u ∈ Hom(E, E ⊗ L), u lifts to Hom(E , E ⊗ L) if and only if the composite ⊂

u|x

σx

i L|xi σxi ◦ u|xi : Axi −→ E|xi −→i E ⊗ L|xi −→

are trivial for all i. For u ∈ Hom(E , E ⊗ L)0 , we can find a smooth point x ∈ S, and σx : E → kx so that σx ◦ u|x = 0. Finally, because dim Hom(E , E ⊗L)0 ≤ l, we can choose x and general E → Ox so that none of u = 0 ∈ Hom(E, E ⊗ L)0 lifts. This proves the lemma.  We need a variant of this lemma. Let (X → C, H, I) be as in the previous section. Lemma 3.11. There is an integer l depending only on (X, H, I) of which the following holds:  (1) For any (E, X[m]0 ) ∈ MI,c X/C (C), a general singularization E of E along the union x = x− ∪ x+ of two general length l subschemes x− ⊂ Y− − D and x+ ⊂ Y+ − D has Ext2X[m]0 (E , E )0 = 0.  (2) For any (E, Y [n]0 ) ∈ MI,c Y,D (C), a general singularization E of E along a general length l subscheme x ⊂ Y − D has Ext2Y [n]0 (E , E (−D[n]))0 = 0. Proof. We prove the first case. By [GL, Lem 4.3], there is a constant e, depending on (X, H) only, so that E|Y− and E|Y+ are e-stable (cf. for e-stability see [Ma]). Then by [Li2], there is an integer l depending on (X, H) (and e) so that (leting L = ωX[m]0 ) dim Hom(E|Y± , E ⊗ L|Y± )0 ≤ l. Therefore, for any (E, X[m]0 ) ∈ MI,c X/C (C), by choosing general length l zerosubschemes x± ⊂ Y± −D, and let E be a general singularization of E along x− ∪x+ , we have Hom(E |Y± , E ⊗ L|Y± )0 = 0. We now assume E has type I. Since E is of type I, by Lemma 2.8, there is an i ∈ [1, m−1] such that E|Δi has type (01), and all other E|Δj=i has type (02 ). Recall that according to (2.1), Δi intersects with the remainder of X[m]0 along Σi and Σi+1 . We normalize X[m]0 along Σi ∪ Σi+1 , to split X[m]0 into three connected schemes Y− [i − 1]0 ,

(3.5)

Δi ,

and

Y+ [m − i − 1]0 ⊂ X[m]0 .



Because E |Δi = E|Δi has type (01), we have (3.6)

Hom(E , E ⊗ L|Δi (−Σi − Σi+1 ))0 = 0.

Therefore, the canonical Hom(E ,E ⊗ L)0 → Hom(E , E ⊗ L|Y− [i−1] )0 ⊕ Hom(E , E ⊗ L|Y+ [m−i−1] )0 is injective. Since E restricted to all Δj ⊂ Y− [i − 1] and Δj  ⊂ Y+ [m − i − 1] (, Δj = Y− and Δj  = Y+ ,) have type (02 ), the vanishing (3.6) implies that the two terms on the right hand side of the above arrow vanish. Thus Hom(E , E ⊗ L)0 = 0. This proves (1).

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The proof of (2) is similar, using that E|Δi≥1 are of type (02 ). We omit the proof.  3.4. The proof of boundedness and properness. We now fill in the proofs promised earlier. Proof of the properness part in Theorem 3.6. We apply valuative criterion. Let R ⊃ k ⊆ C be a discrete valuation ring, and let (EK , Y [n]0 × K) ∈ 2 MI,c X/C (K). In case Ext (EK , EK )0 = 0, we have argued that it can be extended to (ER , YR ) ∈ MI,c Y,D (R), possibly after a finite base change. In case Ext2 (EK , EK )0 = 0, we apply Lemma 3.11 to pick a general zerosubscheme x = {x1 , · · · , xl } ⊂ Y − D, and form a general singularization EK of EK along x × K, such that Ext2 (EK , EK )0 = 0. Because there is no A so that I,c+l A · H = 12 I · H, (EK , Y [n]0 × K) ∈ MD,Y (K). As argued before, (EK , Y [n]0 × K) ∈ I,c+l Rc+l (K). Because Rc+l is proper, we can extend EK to an (ER , YR ) ∈ MD,Y (R), possibly after a finite base change. ◦ = YR − x × K, where We now construct the desired extension ER . Let YR x × K is the closure of x × K ⊂ Y [n]0 × K ⊂ YR in YR . Let Yk be the closed fiber ◦ → YR be of YR . As x ⊂ Y − D, x × K ∩ Yk lies in Y − D ⊂ Yk . We let ι : YR ◦ the inclusion. Since the complement YR − YR is codimension two and lies in the smooth part of YR , ER := ι∗ (ER |YR◦ ) is coherent, and R-flat. By our construction of EK , we know ER |YK = EK . Thus ER is an R-flat extension of EK . Finally, because ER |Yk is H,δ -slope stable, ER |Yk is H,δ -slope stable. This proves that given Spec K → MI,c Y,D , possibly after a finite base change, we can I,c  extend it to Spec R → MY,D . This proves that MI,c Y,D is proper. Proof of Sublemma 2.16. Let (E, X[m]0 ) ∈ Ξ(I), and let i ∈ [1, m − 1] be such that E|Δi has type (01). Like before (3.5), we normalize X[m]0 along Σi ∪Σi+1 , to split X[m]0 into the union of Y− [i − 1]0 , Δi , and Y+ [m − i − 1]0 . Applying Lemma 3.11, we can find an integer l depending only on (X, H) so that if we form a general singularization E of E along two general length l subschemes x− ⊂ Y− − D and x+ ⊂ Y+ − D, we will have Ext2Y− [i−1]0 (E |Y− [i−1]0 , E |Y− [i−1]0 (−Σi ))0 = 0 and Ext2Y+ [m−i−1]0 (E |Y+ [m−i−1]0 , E |Y+ [m−i−1]0 (−Σi+1 ))0 = 0.  on Since E is locally free along Σi , we can deform F− := E |Y− [i−1]0 to a sheaf F−  ∼ Y− , while deforming the pair Σi ⊂ Y− [i − 1]0 to D ⊂ Y− , so that F− |D = F− |Σi .  on Y+ , while By the same reason, we can deform F+ := E |Y+ [m−i−1]0 to a sheaf F+  deforming the pair Σi+1 ⊂ Y+ [m − i − 1]0 to D ⊂ Y+ , so that F+ |D ∼ = F+ |Σi+1 . We then glue Y− and Δi along D ∼ = Σi , and glue Δi and Y+ along Σi+1 ∼ =D   , F+ and E|Δi then glue to to form a new scheme, which is X[2]0 ; the sheaves F−   form a sheaf on X[2]0 , using the given F− |D ∼ |D ∼ = E|Σi and F+ = E|Σi+1 . We claim  that the resulting sheaf E on X[2]0 is a deformation of (E, X[m]0 ), E |Δ1 = E|Δi , and lies in MI,c+2l X/C . Indeed, the first two claims follow from our construction; the last claim follows from that being H,δ -stable is an open condition.

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Because the set of H,δ -stable sheaves V on X[2]0 of degree c1 (V) · H,δ = I · H and c2 (V) = c + 2l is a bounded set, we conclude that the set {V|Δ1 | (V, X[2]0 ) ∈ MI,c+2l  X/C } is bounded. Consequently, Ξ(ex) is bounded. Proof of the properness part in Theorem 2.12. The proof is similar to I,c I,c that of Theorem 3.6. Let Pc ⊂ MI,c X/C be the closure of MX/C ×C (C −0) in MX/C . We first prove that Pc is C-proper. Indeed, as MI,c X/C ×C (C − 0) is proper over C −0, it suffices to verify the following version of valuative criterion. Let R ⊃ k = C be a discrete valuation ring, Spec R → C flat, and (EK , XK , q) ∈ MI,c X/C (K); then up to a finite base change, we can extend EK to (ER , XR ) ∈ MI,c X/C (R). Applying Proposition 2.13, possibly after a finite base change, we can extend EK to ER on XR so that its restriction to the closed fiber Xk ∼ = X[m]0 is good and H,δ -stable. Let Ek := ER |Xk . In case no Ek |Δi are pullback sheaves from D (for i ∈ [1, m − 1]), we know that AutM (Ek ) is finite, implying (ER , XR ) ∈ MI,c X/C (R). In case there is an i ∈ [1, m − 1] so that Ek |Δi is a pullback sheaf from D, then we can apply argument in [Li3] to show that we can construct a new XR so that XR ×R K ∼ = XR ×R K,

XR ×R k = X[m − 1]0 ;

and a projection g : XR → XR that contracts Δi ⊂ XR ×R k = X[m]0 so that  ER := g∗ ER is an extension of EK in GMI,c X/C . Obviously, dim AutM (Ek ) < dim AutM (Ek ). Repeating this argument, we finally obtain an ER on XR , a flat extension of (EK , XK ), so that (ER , XR ) ∈ MI,c X/C (R). This proves that Pc is proper. In general, Let R ⊃ k = C be a discrete valuation ring, Spec R → C factor through 0 ∈ C, and (EK , XK ) ∈ MI,c X/C (K), we need to show that up to a finite base 2 change, we can extend EK to (ER , XR ) ∈ MI,c X/C (R). In case Ext (EK , EK )0 = 0, then a standard deformation argument shows that EK ∈ Pc (K). Because Pc is proper, after a finite base change of R we can find a desired extensions (ER , XR ) ∈ MI,c X/C (R). In case Ext2 (EK , EK )0 = EK of EK - 0, applying Lemma 3.11 . for a singularization 2  along a length 2l x × K ⊂ (Y− − D) ∪ (Y+ − D) × K, we have Ext (EK , EK )0 = 0, and (EK , XK ) ∈ MI,c+2l X/C (K). Thus employing the argument in the proof of Theorem 2.16, we conclude that (EK , XK ) ∈ Pc+2l , and thus (EK , XK ) can be extended, up to a finite base change, to an (ER , XR ) ∈ MI,c+2l X/C (R). From this

extension, we obtain a desired extension (ER , XR ) ∈ MI,c X/C , as in the proof of Theorem 2.16. This proves the properness part of Theorem 2.12.  4. The homology groups of the stack of vector bundles Let X → C be a simple degeneration of smooth surfaces, with central fiber X0 is a union of two smooth surfaces Y− and Y+ intersecting transversally along a smooth curve D = Y− ∩ Y+ . The Floer homology group associated to the Donaldson-Floer theory is a homology theory built upon the 3-manifold that is the unit-circle bundle ∨ . in the normal bundle ND/Y− ∼ = ND/Y + In our proposed algebraic Donaldson-Floer theory, we separate this Floer homology group into two parts: one is the numerical homology group of the stack of

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vector bundles over D; the other is a transformation of the mentioned homology group, depending on the degree deg ND/Y+ . In this section, we construct this homology group of the stack of vector bundles over D. The construction follows the ideas of Goresky-MacPherson in defining their intersection homology groups of singular spaces. This construction will be extended to more general singular Artin stacks, jointly with Y.-H. Kiem [KL2]. 4.1. Numerical homology group of a smooth Artin stack. For singular topological space, we have the usual notion of Whitney stratification (cf. [KW]). For a smooth Artin stack, its stratification is built based on the stabilizers of its closed points. Definition 4.1. A stratification of a smooth A is a collection of locally closed substacks {Ai }i∈I indexed by I such that for each i ∈ I there is an infinite group Gi so that AutA (x) ∼ = Gi for all x ∈ Ai . We say A is weakly compact if A◦ := A − ∪i∈I Ai is Deligne-Mumford and proper. We consider weakly compact smooth Artin stack exhaustible by global quotients. Let A be a smooth Artin stack, locally of finite type over C. Definition 4.2. We say A is exhaustible by global quotients if there is a sequence of smooth schemes {Wm }m≥0 of finite type acted on algebraically by groups {Gm }m≥0 , such that the quotient stacks Um = [Wm /Gm ] are dense open substacks of A, and A = ∪m Um . Let A = ∪m Um be a smooth Artin stack exhausted by open Um = [Wm /Gm ] ⊂ A, with quotient morphism ιm : Wm → Um . We let Aan be the associated smooth topological stack of A. We introduce the notion of plain cycles in A. Definition 4.3. Let V be a compact topological pseudo-manifold. A plain map f : V → Aan consists of an open covering V = ∪α Vα , a choice of Um ⊂ A, and an continuous fα : Vα → Wm , such that for any pair (α, β), there is a continuous an gαβ : Vα ∩ Vβ → Gm satisfying fα = gαβ · fβ ,2 and the cocycle condition gαβ = gβγ · gαγ for all triple (α, β, γ). Definition 4.4. A plain k-cycle (V, f ) of A is a compact oriented topological real k-dimensional pseudo-manifold V together with a plain map f : V → Aan . We define Ck (A)pl = {The Q-vector space generated by all plain k-cycles in A}. Let n = dimC A. In case we can define an intersection pairing $·, ·% : Ck (A)pl × C2n−k (A)pl −→ Q,

(4.1) we define

null-Ck (A)pl = {γ ∈ Ck (A)pl | $γ, ·% = 0}, and define Hk (A)nu = 2 By

Ck (A)pl . null-Ck (A)pl

this we mean fα (x) = gαβ (x) · fβ (x) when x ∈ Vαβ = Vα ∩ Vβ .

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Then (4.1) descends to an intersection pairing $·, ·% : Hk (A)nu × H2n−k (A)nu −→ Q.

(4.2)

Here the subscript “nu” stands for numerical. 4.2. Stratifications of stacks of bundles. Let D be a smooth genus g curve, and let I˜ ∈ Picodd (D) be an odd degree line bundle on D. We let AD be the Artin ˜ rigidified by the tautological stack of rank two vector bundles of determinants I, subgroup C∗ of automorphisms of elements in AD generated by scaling the bundles by c ∈ C∗ . Thus for any [E] ∈ AD , AutAD ([E]) = Aut(E)/C∗ . ˜ it is indeed independent of Though the stack AD is defined referenced to I, ˜ Let I˜ be another odd degree line bundle on D. Then I˜ ⊗ I˜−1 ∼ I. = L⊗2 for an L ∈ Pic(D). The correspondence E → E ⊗ L sends rank two bundles of determinants I˜ to vector bundles of determinants I˜ . This correspondence defines an isomorphism from the Artin stack of rank two vector bundles of determinants I˜ to that of determinants I˜ . Because of this, in studying the structure of AD , deg I˜ is irrelevant. In the following, we assume deg I˜ = 1. We first give a list of subspaces in AD whose partition will lead to a stratification of AD . For i ≥ 1, we let Ξi = {E ∈ AD | E has a degree i subline bundle}.3 Clearly, a further partition of Ξi will produce a stratification of AD . However, to construct a numerical homology group of AD , such Ξi are sufficient. We let Ast D ⊂ AD be the substack of stable vector bundles. It is projective, of dimension 3g − 3, and is open and dense in AD . Clearly, ∪i≥1 Ξi = AD − Ast D . We let Ξ = {Ξi }i≥1 . We form global quotients that will exhaust AD . Using Grothendieck Quotscheme we can find a sequence of smooth schemes Wm acted on algebraically by reductive groups Gm so that ιm : Wm −→ [Wm /Gm ] ⊂ AD

(4.3)

are open and exhaust AD , say

. lim dim AD − [Wm /Gm ] = −∞.

Here as usual, a point ξ ∈ AD has dim{ξ} = − dim AutAD (ξ). 4.3. Intersection pairing of plain cycles. Given a plain cycle (V, f ), we define its image in Aan D to be f (V ) := ∪α ιm (fα (Vα )) ⊂ Aan D. Accordingly, an an ι−1 m (f (V )) = ∪α Gm · fα (Vα ) ⊂ Wm .

As (V, f ) is not algebraic, the analytic closure f (V ) of f (V ) in Aan D needs some care. We define f (V ) via its preimage in Wm : an an ι−1 m (f (V )) = ∪α Gm · fα (Vα ) ⊂ Wm , an an where ∪α Gan m · fα (Vα ) is the analytic closure of ∪α Gm · fα (Vα ) in Wm . 3 Here

we only consider i ≥ 1 because all det E ∼ = I˜ have degree 1.

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Definition 4.5. Let (V, f ) and (U, h) be plain k-cycle and l-cycle respectively, of complemental dimensions k + l = 6g − 6, and both presented in [Wm /Gm ]. We say that (V, f ) and (U, h) intersect strongly-transversally if for every α, −1 st fα (Vα ) ∩ ι−1 m (h(U )) ⊂ ιm (AD ), st st an and f (V ) ∩ Ast D and h(U ) ∩ AD intersect transversally in (AD ) .

Theorem 4.6. Given two complementary dimension plain cycles (V, f ) and (U, h), both presented in [Wm /Gm ], we can find small perturbation (V, f˜) of (V, f ), ˜ of (U, h), both as plain cycles, so that (V, f˜) and (U, h) ˜ and small perturbation (U, h) intersect strongly-transversally. Further, the degree of the signed intersection ˜ ) = deg(f˜(V ) ∩ h(U ˜ ) ∩ (Ast )an ) ∈ Z deg f˜(V ) ∩ h(U D

is independent of the choices of perturbations. We define the numerical homology group of AD , granting the theorem. Let Ck (AD )pl = {The Q-vector space generated by all plain k-cycles in AD }; for any pair

. (V, f ), (U, h) ∈ Ck (AD )pl × C2n−k (AD )pl ,

n = 3g − 3,

we define (4.4)

˜ )), $(V, f ), (U, h)% = deg(f˜(V ) ∩ h(U

˜ are given in Theorem 4.6. where f˜ and h We let null-Ck (AD )pl = {(V, f ) ∈ Ck (AD )pl | $(V, f ), ·% = 0}, and define Ck (AD )pl . null-Ck (AD )pl Then every (V, f ) ∈ Ck (AD )pl has its associated class in Hk (AD )nu , and (4.4) descends to an intersection pairing Hk (AD )nu =

(4.5)

$·, ·% : Hk (AD )nu × H2n−k (AD )nu −→ Q.

4.4. Dimensions of Ξi . The remainder of this section is devoted to the proof of Theorem 4.6. To this end, we introduce some auxiliary spaces. We let Pa be the stack of pairs [L ⊂ E] of degree a subline bundles L in rank two determinants I˜ vector bundles E; arrows from [L ⊂ E] to [L ⊂ E  ] are isomorphisms ϕ : E → E  so that φ(L) = L ; rigidified by the automorphisms of E that are scaling by c ∈ C× . Thus for [L ⊂ E] ∈ Pa , 8 Hom(E/L, L), E  L ⊕ E/L; (4.6) AutPa ([L ⊂ E]) = Hom(E/L, L) × C× , E ∼ = L ⊕ E/L. The stack Pa comes with the morphism πa,1 × πa,2 × π1−a,3 : Pa −→ Pica (D) × AD × Pic1−a (D), where πa,1 × πa,2 × π1−a,3 ([L ⊂ E]) = (L, E, E/L).

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Lemma 4.7. For any L ∈ Pica (D), −1 ([L]) ≤ g − 2a − 1 dim πa,1

and

−1 dim πa,3 ([L]) ≤ g + 2a − 3.

Proof. By (4.6), −1 dim πa,1 ([L]) ≤ h1 ((E/L)∨ ⊗ L) − h0 ((E/L)∨ ⊗ L) − 1.

By Riemann-Roch, the right hand side is g − 2a − 1. This proves the first inequality. −1 −1 For the second, we use πa,3 ([L]) = π1−a,3 ([I˜ ⊗ L∨ ]).  For any B ⊂ AD and i = 1 and 3, we define - −1 . (B) ⊂ Pic(D). Pa,i (B) = πa,i πa,2 By Lemma 4.7, in case B is in general position, we have (4.7)

−1 ([L])) − dim AD . dim Pa,1 (B) ≤ dim B + (dim Pica (D) + dim πa,1

For any δ ≥ 0 and a subset B  ⊂ Pica (D), we define Λδ B  = {L(A) ∈ Pica+δ (D) | L ∈ B  , and A ⊂ D a degree δ effective divisor}. For any B  ⊂ Pica (D), we denote

- −1  . (B ) ⊂ AD . Ξa (B  ) = πa,2 πa,1

Lemma 4.8. Let (V, f ) be a plain k-cycle in AD presented in [Wm /Gm ]. Let ξ0 ∈ ι−1 m (f (V )) ⊂ Wm . Then either ιm (ξ0 ) is stable, or one of the following hold: (1) there are a ≥ 1, δ ≥ 0 such that ιm (ξ0 ) ∈ Ξa+δ (Λδ Pa,3 (f (V ) ∩ Ast D )); (2) there are a ≥ 1, δ ≥ 0 such that ιm (ξ0 ) ∈ Ξa+δ (Λδ Pa,1 (f (V ))); Proof. Let ξi ∈ ι−1 m (f (V )) be a sequence such that ξi convereges to ξ0 ∈ Wm . Let Ei (resp. E0 ) be the vector bundles associated with ξi (resp. ξ0 ). We distinguish two cases. The first is when there are infinitely many Ei that are stable. Then by passing to a subsequence, we can assume that all Ei are stable. Since Ast D is projective, we can assume that Ei converges to a stable E0 ∈ Ast D. As E0 is also a limit of the sequence Ei , we obtain a non-trivial homomorphism E0 → E0 . In case E0 is stable, E0 ∼ = E0 . In case E0 is unstable, then E0 ∈ Ξj for some j ≥ 1 and having a degree j destabilizing subbundle L0 ⊂ E0 . Because   ˜ deg L∨ 0 ⊗ I ≤ 0, the homomorphism E0 → E0 lifts to E0 → L0 . Thus for an  effective divisor A ⊂ D, say of degree δ, E0 → L0 factors through a quotient bundle homomorphism E0 → L0 (−A). Let a = j − δ = deg L0 (−A). Since E0 is stable, a ≥ 1. Then because L0 (−A) ∈ Pa,3 (f (V ) ∩ Ast D ), and because L0 ⊂ E0 is its destabilizing subbundle, we prove part (1) of the lemma. The next case is when after passing to a subsequence, all Ei are non-split and unstable. Let Li ⊂ Ei be the destabilizing subbundles. By passing to a subsequence, we can assume all deg Li = a, and the sequence Li converges to ˜ 0 ∈ Pica (D). Thus L ˜ 0 ∈ Pa,1 (f (V )). L ˜ 0 ≥ 1, L ˜ 0 → E0 ˜ 0 → E0 . As deg L This implies that we have a non-trivial L ∼ ˜ 0 (A) for an factors through the destablizing subline bundle L0 ⊂ E0 , say L0 = L effective divisor A ⊂ D of degree δ. This proves part (2) of the lemma. ˜ The last case is when after passing to a subsequence, all Ei ∼ = Li ⊕ L∨ i ⊗ I with  deg Li independent of i. This implies (2) as well. This proves the lemma.

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4.5. Strongly transversal plain cycles. It is relatively simple to perturb plain cycles. Given a plain cycle (V, f ) presented in [Wm /Gm ], via an open coverling an ∪Λ Vα = V and continues fα : Vα → Wm so that fα = gαβ · fβ , for continuous an gαβ : Vαβ → Gm satisfying the cocycle condition. To (small) perturb (V, f ), we choose open subsets Vα◦ ⊂ Vα so that the closure Vα◦ of Vα◦ in V is contained in Vα , and so that {Vα◦ }Λ covers V . We then pick one α ¯ ∈ Λ and perturb fα¯ within Vα¯◦ to f˜α¯ , namely fα¯ |Vα¯ −Vα¯◦ = f˜α¯ |Vα¯ −Vα¯◦ . We then ¯ ∈ Λ via modify all other fβ , β = α f˜β |Vβ ∩Vα¯◦ = gβ α¯ · f˜α¯ |Vβ ∩Vα¯◦

and

f˜β |Vβ −Vα¯◦ = fβ |Vβ −Vα¯◦ .

Since Vα¯◦ is compact in Vα¯ , all f˜β are continuous. Indeed, following the construction, for any pair (α, β), we have f˜β = gβα · f˜α when α = α ¯ . When α = α, ¯ the same identity holds because gαβ = gβ α¯ · gαα¯ , following from the cocycle condition (cf. Definition 4.3). Repeat this procedure for various α ¯ , we construct a perturbation of (V, f ) that is in general position. Proposition 4.9. Let (V, f ) and (U, h) be a pair of complementary dimension plain cycles in AD . Suppose (V, f ) and (U, h) are in general positions, then they intersect strongly transversally in Aan D . Namely, an f (V ) ∩ h(U ) = f (V ) ∩ h(U ) ⊂ (Ast D) ,

and they intersect transversally. Proof. We let (V, f ) and (U, h) be a pair of plain 2k-cycle and (6g − 6 − 2k)cycle in general positions. (Here for simplicity we consider even real cycles in Aan D .) We assume both are presented in [Wm /Gm ]. As both (V, f ) and (U, h) are in general an an an and h(U )∩(Ast intersect transversally in (Ast positions, f (V )∩(Ast D) D) D ) . Thus we only need to check that for any a ≥ 1, and any (Uα , hα ) of (U, g), ιm (hα (Uα )) ∩ f (V ) ∩ Ξa = ∅.

(4.8) Applying Lemma 4.8,

 f (V ) ∩ Ξa ⊂ ∪a−δ≥1 Ξa Λδ Pa−δ,3 (f (V ) ∩ Ast D)   ∪a−δ≥1 Ξa Λδ Pa−δ,1 (f (V )) .

By (4.7), we have (, recall dim = 2 dimR ,) dim Λδ Pa−δ,1 (f (V )) = dim Λδ Pa−δ,1 (f (V )) ≤ δ + (3g − 3 − k) + (2g − 2(a − δ) − 1) − (3g − 3) = 2g − k − 2a + 3δ − 1 ≤ 2g − k + a − 4. Here the last inequality follows from a − δ ≥ 1. Similarly, we have dim Λδ Pa−δ,3 (f (V ) ∩ Ast D) ≤ δ + (3g − 3 − k) + (2g − 2(1 − a + δ) − 1) − (3g − 3) = 2g − k + 2a − δ − 3.

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183

Adding that dim Pa,1 (hα (Uα )) ≤ dim hα (Uα ) ∩ ι−1 m (Ξa ) ≤ k + (2g − 2a − 1) − (3g − 3) = k − g − 2a + 2, we obtain dim Pa,1 (hα (Uα )) + dim Λδ Pa−δ,1 (f (V )) ≤ g − a − 2 < g, and dim Pa,1 (hα (Uα )) + dim Λδ Pa−δ,3 (f (V ) ∩ Ast D ) ≤ g − δ − 1 < g. Therefore, assume that both (g, U ) and (f, V ) are in general positions, and because dim Pici (D) = g, we conclude that  Pa,1 (hα (Uα )) ∩ Λδ Pa−δ,1 (f (V )) ∪ Λδ Pa−δ,3 (f (V ) ∩ Ast D ) = ∅. This proves (4.8). To prove that (g, U ) and (f, V ) are strongly-transversal, it remains to show that st f (V ) ∩ h(U ) ∩ Ast D = f (V ) ∩ h(U ) ∩ AD .

Namely, (4.9)

. f (V ) − f (V ) ∩ h(U ) ∩ Ast D = ∅.

st st Since f (V ) ∩ Ast D = f (V ) ∩ AD ∩ AD , we see that . st st dim f (V ) ∩ Ast D − f (V ) ∩ AD < dim f (V ) ∩ AD = k.

Since Ast D is projective, using that (V, f ) and (U, h) are in general position, we have (4.9). This proves the proposition.  Proof of Theorem 4.6. We sketch a proof here. The details will appear in [Li7]. By Proposition 4.9, we can perturb (V, f ) and (U, h) to general posi˜ so that (V, f˜) and (U, h) ˜ intersection strongly-transversally. tions (V, f˜) and (U, h)  ˜ Suppose (V, f ) is another small perturbation of (V, f ) so that (V, f  ) and (U, h) intersect strongly-transversally, we can find a family of small perturbation (V, fs ), s ∈ [0, 1], so that f0 = f˜, f1 = f  , and of which the following hold: (1) each (V, fs ) intersects (U, h) strongly-transversally; (2) the union ∪s∈[0,1] fs−1 (h(U )) ⊂ [0, 1] × V is a smooth real dimension one manifold with boundary f0−1 (h(U )) − f1−1 (h(U )). ˜ ) is Because V is compact, we get the signed intersection number of f˜(V ) ∩ h(U independent of the choices of small perturbations.  5. A proposed algebraic Donaldson-Floer theory In this section, we define the relative Donaldson polynomial invariants of a pair D ⊂ Y of a connected smooth divisor in a smooth algebraic surface. We assume g(D) ≥ 1. Afterwards, we state our conjectures on the algebraic Donaldson-Floer theory of a simple degeneration of surfaces.

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5.1. Virtual cycles. Let D ⊂ Y be as stated; let I be a line bundle on Y so that deg I|D is odd; let H be an ample line bundle on Y satisfying the FNA stated at the beginning of Section 3. Let π : Y −→ MI,c with D ⊂ Y Y,D , be the universal base family over the moduli space of rank two relative stable I,c ∗ sheaves on D ⊂ Y . Note that though MI,c Y,D results from rigidify the C of SMY,D , ∗ since the C only acts on the sheaves, not on the underlying base schemes, the part I,c (D ⊂ Y) in the tautological family of SMI,c Y,D descends to MY,D . We pick an ´etale atlas Uα → MI,c Y,D . For each Uα , we let (Dα ⊂ Yα ) = (D ⊂ Y) ×MI,c Uα , Y,D

and let Eα be a sheaf on Yα so that (Dα ⊂ Yα , Eα ) induce the given morphism Uα → I,c MI,c Y,D . Since elements in MY,D (C) are stable sheaves, we can assume that over Uαβ = Uα ×MI,c Uβ , with ϕα : Yαβ → Yα the projection, we have isomorphisms Y,D

∼ =

ϕ˜αβ : ϕ∗β Eβ −→ ϕ∗α Eα ,

(5.1)

saisfying the almost cocycle condition (5.2)

ϕ˜αβ ◦ ϕ˜βγ ◦ ϕ˜γα = η id : Eα ⊗OYα OYαβγ → Eα ⊗OYα OYαβγ ,

where η ∈ Γ(O∗Uαβγ ). We now look at the perfect obstruction theory of MI,c Y,D . Let πα : Yα → Uα be the projection. By [HT], Uα has a tautological perfect obstruction theory given by the Atiyah class .∨ (5.3) RHomπα (Eα , Eα )0 −→ L≥−1 Uα [−1], where the subscript “0” stands for the traceless part. By [BF, LT], we get a virtual normal cone . (5.4) Cα ⊂ Fα := h1 /h0 RHomπα (Eα , Eα )0 [1] . By (5.1) and (5.2), we see that the collection of bundle stacks Fα on Uα descend to a bundle stack F on MI,c Y,D . Further, the cone substacks Cα glue to form a cone substack C ⊂ F. We define the virtual cycle to be vir = 0!F [C] ∈ A∗ MI,c [MI,c Y,D ] Y,D ,

where 0!F is the Gysin map defined by the 0-section of F. (See [Kre].) 5.2. The μ map. We recall Donaldson’s μ map (5.5)

μ : Sym• H 2 (Y ) −→ H∗ (MI,c Y,D ).

Using GRR, the μ map can also be defined using the second Chern class of the tautological family of MI,c Y,D , should it exist. In our case, we will show that we can find a global complex of locally free sheaves on MI,c Y,D that represent RHom(Eα , Eα ) over each Yα . To this end, we first construct a canonical relative ample line bundle on Y → I,c MI,c Y,D . To each Uα → MY,D , and any closed ξ ∈ Uα , we let nξ be the integer so that ξ = (E, D[nξ ] ⊂ Y [nξ ]) ∈ MI,c Y,D (C).

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185

Following our convention, (5.6)

Y [nξ ] = Δnξ ∪ · · · ∪ Δ1 ∪ Δ0

with D[nξ ] ⊂ Δnξ .

As shown in the proof of Theorem 3.6, c2 (E|Δi ) > 0 for i > 0. We then define a line bundle Lξ on Yξ , Lξ ∼ OY , characterized by that the degree of Lξ |Δi≥1 along the fiber of Δi → D is c2 (E|Δi ) > 0. A moment of thought shows that Lξ is well-defined, and there is a line bundle L on Y so that its restriction to any closed ∼ ξ ∈ MI,c Y,D has the property L|Yξ = Lξ . I,c Because MY,D is of finite type, we can find a sufficiently large m0 so that for q : Y → Y the tautological projection, L ⊗ q ∗ H ⊗m0 is relatively ample on Y → MI,c Y,D . We next construct canonical locally free resolutions of Eα . Let πα : Yα → Uα be the projection; let Eα (m) = Eα ⊗ H⊗m . We pick sufficiently large m2 ' m1 ' 0 and form the exact sequence of sheaves φα,1

φα,0

πα∗ πα∗ (ker(φα,0 )(m2 ))(−m2 ) −→ πα∗ πα∗ (Eα (m1 ))(−m1 ) −→ Eα −→ 0. Because dim Yα /Uα = 2 and Eα are locally free along singular locus of the fibers of Yα /Uα , all terms in the above sequence, plus ker(φα,1 ), are locally free. We denote these terms by Fα,i for i = 0, 1 and 2, resulting the following exact sequence of sheaves (5.7)

φα,1

φα,0

0 −→ Fα,2 −→ Fα,1 −→ Fα,0 −→ Eα −→ 0.

Because the isomorphisms (5.1), and the almost cocycle condition (5.2), the complexes 4 ∨ 3 ∨ ∨ ⊗ Eα → Fα,1 ⊗ Eα → Fα,2 ⊗ Eα (5.8) RHom(Eα , Eα ) =q.i. Fα,0 descend to a global complex on Y, denoted by RHom(E, E). For the same reason, ∨ the collection {Fα,i ⊗ Fα,j }α descends to a locally free sheaf on Y, which we denote by Fi,j . Thus their K-classes (5.9)

RHom(Eα , Eα ) =K

2 

(−1)i+j Fi,j .

i,j=0

. -2 . i+j We define c2 RHom(E, E) = c2 Fi,j . By GRR, we have i,j=0 (−1) . 1 1 Δ(E) := c2 (E) − c21 (E) = c2 RHom(E, E) . 4 4 For ω ∈ H 2 (Y ) and  ∈ N, we define . vir ∈ H2d(c)−2 (MI,c μ(ω ⊗ ) = π∗ (Δ(E) ∪ q ∗ (ω)) ∩ [MI,c Y,D ] Y,D ), I,c where π : Y → MI,c Y,D is the projection, and d(c) = vir. dim MY,D .

5.3. Lifting pseudo-manifold cycles to AD . Following Subsection 4.2, we let AD be the stack of rank two vector bundles on D of determinants I˜ = I|D , rigidified by the standard C∗ . By Remark 3.7, restricting the (local) tautological family of MI,c Y,D to the relative divisors D[n] ⊂ Y [n], we get the restriction morphism rI,c,D : MI,c Y,D −→ AD .

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Let Pk (MI,c Y,D ) and Ck (AD )pl be the space of real k-dimensional pseudomanifold cycles in MI,c Y,D and plain cycles in AD . In this subsection, we construct a map φD : Pk (MI,c Y,D ) −→ Lk (AD ) that descends to a homomorphism of groups φD∗ : Hk (MI,c Y,D ) −→ Hk (AD )nu .

(5.10)

We begin with introducing the Grothendieck-Quot scheme over the family Y. We let p be the polynomial p(k) = χ(E(k)),

where

(E, D[n] ⊂ Y [n]) ∈ MI,c Y,D (C).

We let Quotp,I,s Y/M be the groupoid of quotient sheaves over Y/MI,c Y,D that associates to any scheme I,c S → MY,D the set of quotient sheaves σ : OYS (−ν)⊕N → F,

N = p(ν),

where ν is a sufficiently large integer to be specified later, and (DS ⊂ YS ) = (D ⊂ Y) ×MI,c S, so that (DS ⊂ YS , F) ∈ MI,c Y,D (S), and that Y,D

⊕p(ν)

h0 (σ) : πS∗ (OYS

∼ =

) −→ πS∗ (F(ν)).

By Grothendieck’s existence theorem, (5.11)

I,c Π : Quotp,I,s Y/M −→ MY,D

is a DM stack, quasi-projetive over MI,c Y,D , and is a P GL(N + 1)-stack. By [Sim], I,c since elements in MY,D are stable sheaves, by choosing ν sufficiently large, (5.11) is a P GL(N + 1)-geometric quotient. To proceed, we choose ν to be sufficiently large so that for any (E, D[n] ⊂ Y [n]) ∈ MI,c Y,D (C), the quotient homomorphism H 0 (E(ν)) ⊗ OY [n] (−ν) −→ E restricts to a quotient homomorphism τ : H 0 (E(ν)) ⊗ OD[n] (−ν) −→ E|D[n] so that h0 (τ (ν)) : H 0 (E(ν)) → H 0 (E(ν)|D[n] ) is surjective. Proposition 5.1. Let rI,c,D : MI,c Y,D → AD be the restriction morphism. Then . for any pseudo-manifold cycle (V, f ) ∈ Pk MI,c Y,D , we can find a plain cycle (V, h) ∈ Lk (AD ) so that rI,c,D ◦ f = h. Proof. One technical complication is due to that MI,c Y,D is a DM stack. By I,c using Q coefficients, homology classes of MY,D can be represented by pseudomanifold cycles with Q-coefficients. To simplify the notation, we will prove the proposition for classes lies in the largest open M ◦ ⊂ MI,c Y,D that itself is an algebraic space.

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187

We cover M ◦ by analytic open subsets Uα so that each Uα comes with a section ζα : Uα −→ Quotp,I,s Y/M I,c of the P GL(N )-bundle Quotp,I,s Y/M → MY,D .

◦ Let f : V → MI,c Y,D be a pseudo-manifold cycle so that f (V ) ⊂ M . Then V is covered by Vα = f −1 (Uα ); each fα = f |Vα lifts to

ζα ◦ fα : Vα −→ Quotp,I,s Y/M . We let M = χ(I(ν) ⊕ OD (ν)). We form the the Grothendieck-Quot scheme QuotID of all quotient sheaves OD (−ν)⊕M → F so that F are rank two and of I determinant I|D . We let QuotI,◦ D ⊂ QuotD be the open subscheme consisting of all ⊕M quotient sheaves OD (−ν) → F so that F are locally free, h1 (F(ν)) = 0 and the ⊕M 0 induced C → H (F(ν)) are isomorphisms. Then QuotI,◦ D is a P GL(M )-scheme, and [QuotI,◦ D /P GL(M )] ⊂ AD is open. We construct Rα : Uα −→ QuotI,◦ D

(5.12)

p,I,s that commutes with the restricting (to D) morphism from MI,c Y,D and QuotY/M to

AD . We continue to denote by D ⊂ Y the universal base family of MI,c Y,D . We let ˜ α ⊂ Y˜α be the pullback of D ⊂ Y to D ˜α := Quotp,I,s × I,c Uα ; U Y/M M Y,D

let α ˜ OY˜α (−ν)⊕N −→ Eα

σ ˜

˜α . We then pick a splitting be the tautological family over U (5.13)

∼ =

ηα : OY˜α (−ν)⊕N −→ OY˜α (−ν)⊕N −M ⊕ OY˜α (−ν)⊕M

˜α , the induced so that for any ξ ∈ ζα (Uα ) ⊂ U −1

ηα |ξ σ ˜ α |ξ OY˜α (−ν)⊕N |Y˜α |ξ −→ E˜α |Y˜α |ξ 0 ⊕ OY˜α (−ν)⊕M |Y˜α |ξ −→

˜ α |ξ restricting to D (5.14)

OD˜ α |ξ (−ν)⊕M −→ E˜α |D˜α |ξ

I,c is an element in QuotI,◦ D . Because MY,D is of finite type, by choosing ν sufficiently large and shrinking Uα if necessary, such ηα exists. Let Rα (ξ) be the element (5.14). This construction gives us the continuous map Rα mentioned in (5.12). We define hα = Rα ◦ fα : Vα −→ QuotI,◦ D .

By our construction, for πD : QuotI,◦ D → AD the tautological morphism, rI,c,D ◦ fα = πD ◦ hα : Vα −→ AD . We now verify that (V, h) = {(Vα , hα )} is a plain cycle.

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Let x ∈ V be represented by (Ex , Dx ⊂ Yx ) ∈ MI,c Y,D (C). In case x ∈ Vα , we have isomorphism ˜ α ⊂ Y˜α )|ζ (x) ραx : (Dx ⊂ Yx ) → (D α

and ρ˜αx : Ex ∼ = ρ∗αx (E˜α |Y˜α |ζα (x) ).

Because Ex is stable, ρ˜αx are unique up to scalars. The isomorphism ρ˜αx induces an isomorphism ∼ =

φαx : H 0 (Ex |Dx (ν)) −→ H 0 (E˜α |D˜ α |ζ

α (x)

∼ =

(ν)) −→ C⊕M ,

where the second arrow is that induced by (5.14), depending only on the choice (5.13); the first arrow is h0 (˜ ραx |D (ν)). In case x ∈ Vβ , we let gαβ (x) ∈ P GL(M ) be induced by the isomorphism ∼ =

⊕M −→ C⊕M . φαx ◦ φ−1 βx : C

(5.15)

Because (5.15) is unique up to scalars, gαβ (x) is unique. Further, by our construction, it satisfies hα (x) = gαβ (x) · hβ (x). By this construction, we see that gαβ : Vαβ → P GL(M ) are continuous; by the uniqueness, they satisfy the cocycle  condition. This proves that (V, h) = {(Vα , hα )} is a plain cycle. Corollary 5.2. The construction in Proposition 5.1 defines a homomorphism (rD,I,c )∗ : H∗ (MI,c Y,D ) −→ H∗ (AD )nu . 5.4. Relative Donaldson polynomial invariants. Let D ⊂ Y be a smooth connected curve in a smooth algebraic surface such that D2 is even and g(D) ≥ 1. Let I be a line bundle on Y so that deg I|D is odd; and let H be an ample line bundle on Y so that (Y, H) satisfies the FNA stated at the beginning of Section 3. We define the companion line line bundle of I to be I  = I(−D). Like AD , we let AD be the stack of rank two vector bundles on D of determinants I˜ = I  |D , rigidified by the standard C∗ . We define the relative Donaldson polynomial invariants of rank two sheaves on D ⊂ Y of determinant I to be . . I (5.16) DD,Y (ω ⊗ ) = ⊕c∈Z (rD,I,c )∗ μ(ω ⊗ ) ⊕c∈Z (rD,I  ,c )∗ μ(ω ⊗ ) , which is a multi-linear map Sym• H 2 (Y ) −→ HF∗ (AD ) := H∗ (AD )nu ⊕ H∗ (AD )nu . We endow HF∗ (AD ) the intersection pairing $·, ·% : HF∗ (AD ) × HF∗ (AD ) −→ Q that is the direct sum of the intersection pairings of H∗ (AD )nu and H∗ (AD )nu . 5.5. Algebraic Donaldson-Floer theory. We state the conjectural algebraic Donaldson-Floer theory. Let X → C be a simple degeneration of algebraic surfaces so that X0 = Y− ∪Y+ is a union of two smooth surface intersecting transversally along a smooth divisor D of positive genus. We let I be a line bundle on X with deg I|D odd; we pick a relative ample H on X so that (X, H, I) satisfies the FNA as stated in Subsection 2.2. We let It DX : Sym• H 2 (Xt ) −→ Q t ,Ht

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be Donaldson polynomial invariants defined via " It ⊗ DXt ,Ht (ω ) = c∈Z

I ,c ]vir t ,Ht

[MXt

189

μ(ω)⊗ ∈ Q,

MIXtt,c,Ht

is the moduli of Ht -Gieseker stable rank two determinants It second where class c sheaves on Xt ; and It ,c μ : H 2 (Xt ) −→ H 2 (MX ) t ,Ht

is Donadson’s μ-map [DK, Li1, Mor]. Because of the FNA assumption, MIXtt,c,Ht It is projective, and admits a tautological virtual cycle [MIXtt,c,Ht ]vir . Thus DX is t ,Ht It well-defined. When pg (Xt ) > 0, DXt ,Ht is a smooth invariant of the underlying oriented smooth 4-manifold of Xt . Let 0 ∈ C ◦ ⊂ C an be a disk-like (analytic) neighborhood of 0 ∈ C; let X ◦ = an X ×C an C ◦ , as an analytic space. Let t = 0 ∈ C ◦ , let ιt : Xt → X ◦ , and ι± : Y± → X ◦ be the inclusions. For ω ∈ H 2 (X ◦ , Z), we write ωt = ι∗t (ω) ∈ H 2 (Xt )

and ω± = ι∗± (ω) ∈ H 2 (Y± ).

We write ω · D to be the pairing of ω with [D] ∈ H2 (X ◦ ). Conjecture 5.3. There are operators Rk : HF∗ (AD ) −→ HF∗ (AD ),

k ∈ Z,

R0 = id, depending only on (k, g(D), deg ND/Y+ ), such that for any ω ∈ H 2 (X ◦ , Z), - I− . I+ It ⊗ ⊗ (ωt⊗ ) = $Ra DD,Y (ω− ) , DD,Y (ω+ )%, a = ω · D. DX t ,Ht − + This conjecture is inspired by [Don2], especially the discussions leading to the conclusions on [Don2, Page 123]. The author believes that the proof of this conjecture is within the reach, when FNA holds. The general case requires more work. 5.6. Added comments. The stack AD in this note is the stack of rank two vector bundles of fixed odd degree determinant line bundles, rigidified by the obvious C∗ . It can be shown that the numerical homology groups H∗ (AD )nu is canonically isomorphic to the ordinary homology groups of H∗ (Ast D ). A proof of it will be presented in [KL2]. The operators Ra are part of the Donaldson-Floer theory of the degeneration X → C. They can be constructed via the relative Donaldson polynomial invariants of (D0 ∪ D∞ , Δ). This will be address in the subsequent work [Li7]. The more challenging case is when deg I|D is even. In this case, the relevant I,c moduli spaces MI,c X/C and MY,D should be constructible along similar lines, after certain technical issues are taken care of. The more challenging part is to construct the Donaldson-Floer theory in this setting. References [BD1]

[BD2]

P. J. Braam and S. K. Donaldson, Floer’s work on instanton homology, knots and surgery, The Floer memorial volume, Progr. Math., vol. 133, Birkh¨ auser, Basel, 1995, pp. 195–256. MR1362829 P. J. Braam and S. K. Donaldson, Fukaya-Floer homology and gluing formulae for polynomial invariants, The Floer memorial volume, Progr. Math., vol. 133, Birkh¨ auser, Basel, 1995, pp. 257–281. MR1362830

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K. A. Behrend, On the de Rham cohomology of differential and algebraic stacks, Adv. Math. 198 (2005), no. 2, 583–622, DOI 10.1016/j.aim.2005.05.025. MR2183389 [BF] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88, DOI 10.1007/s002220050136. MR1437495 [Cap] L. Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994), no. 3, 589–660, DOI 10.2307/2152786. MR1254134 [Don1] S. K. Donaldson, Gluing techniques in the cohomology of moduli spaces, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 137–170. MR1215963 [Don2] S. K. Donaldson, Floer homology and algebraic geometry, Vector bundles in algebraic geometry (Durham, 1993), London Math. Soc. Lecture Note Ser., vol. 208, Cambridge Univ. Press, Cambridge, 1995, pp. 119–138, DOI 10.1017/CBO9780511569319.006. MR1338415 [DK] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR1079726 [Gie] D. Gieseker, A degeneration of the moduli space of stable bundles, J. Differential Geom. 19 (1984), no. 1, 173–206. MR739786 [GL] D. Gieseker and J. Li, Irreducibility of moduli of rank-2 vector bundles on algebraic surfaces, J. Differential Geom. 40 (1994), no. 1, 23–104. MR1285529 [GM] D. Gieseker and I. Morrison, Hilbert stability of rank-two bundles on curves, J. Differential Geom. 19 (1984), no. 1, 1–29. MR739780 [HT] D. Huybrechts and R. P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann. 346 (2010), no. 3, 545–569, DOI 10.1007/s00208-009-0397-6. MR2578562 [KL1] Y.-H. Kiem and J. Li, Vanishing of the top Chern classes of the moduli of vector bundles, J. Differential Geom. 76 (2007), no. 1, 45–115. MR2312049 [KL2] Y-H. Kiem and J. Li, in preparation. [KW] F. Kirwan and J. Woolf, An introduction to intersection homology theory, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2006. MR2207421 [Kre] A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536, DOI 10.1007/s002220050351. MR1719823 [KM] P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom. 41 (1995), no. 3, 573–734. MR1338483 [Li1] J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993), no. 2, 417–466. MR1205451 [Li2] J. Li, Kodaira dimension of moduli space of vector bundles on surfaces, Invent. Math. 115 (1994), no. 1, 1–40, DOI 10.1007/BF01231752. MR1248077 [Li3] J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR1882667 [Li4] J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR1938113 [Li5] J. Li, Moduli spaces associated to a singular variety and the moduli of bundles over universal curves, Vector bundles and representation theory (Columbia, MO, 2002), Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 57–74, DOI 10.1090/conm/322/05679. MR1987739 [Li6] J. Li, Good degenerations of moduli spaces, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 299–351. MR3184180 [Li7] J. Li, in preparation. [LT] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174, DOI 10.1090/S0894-0347-9800250-1. MR1467172 [LW] J. Li and B. Wu, Good degeneration of Quot-schemes and coherent systems, Comm. Anal. Geom. 23 (2015), no. 4, 841–921, DOI 10.4310/CAG.2015.v23.n4.a5. MR3385781 [Ma] M. Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ. 18 (1978), no. 3, 557– 614, DOI 10.1215/kjm/1250522511. MR509499 [Beh]

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J. W. Morgan, Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues, Topology 32 (1993), no. 3, 449–488, DOI 10.1016/00409383(93)90001-C. MR1231956 [MMR] J. W. Morgan, T. Mrowka, and D. Ruberman, The L2 -moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, II, International Press, Cambridge, MA, 1994. MR1287851 [Sim] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective ´ variety. II, Inst. Hautes Etudes Sci. Publ. Math. 80 (1994), 5–79 (1995). MR1320603 [Tau] C. H. Taubes, L2 moduli spaces on 4-manifolds with cylindrical ends, Monographs in Geometry and Topology, I, International Press, Cambridge, MA, 1993. MR1287854 [Tho] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR1818182 [Mor]

Department of Mathematics, Stanford University, Stanford, California Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01741

Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories Hiraku Nakajima Dedicated to Simon Donaldson Abstract. This is an introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 supersymmetric gauge theories, studied in another work of the author. This is an expanded version of an article which appeared in 第61回代数学シンポジウム報告集 (2016), written originally in Japanese.

1. Coulomb and Higgs branches – complex symplectic varieties and deformation quantization Let G be a complex reductive group and M its symplectic representation. Namely M is a vector space with a symplectic form ω, and G acts linearly on M preserving ω. Let us denote the Lie algebra of G by g. The mathematical definition of the Coulomb branch of 3d SUSY gauge theory gives a recipe for constructing a complex affine-algebraic symplectic variety1 MC ≡ MC (G, M) from (G, M): (G, M)  MC (G, M). It is motivated by research in quantum field theory in physics. It is different from known constructions of algebraic varieties, such as zero sets of polynomials, quotient spaces, moduli spaces, etc. We first construct the coordinate ring C[MC ] as a homology group with convolution product. Then we define MC as its spectrum, and study its geometric properties. As we will explain later, MC is birational to T ∗ T ∨ /W : MC ≈ T ∗ T ∨ /W = t × T ∨ /W. In physics, the right hand side is regarded as the classical description of the Coulomb branch, and MC is obtained from it after quantum correction. Here T ∨ is the dual of a maximal torus T of G, and W is the Weyl group. T ∗ T ∨ is the cotangent bundle of T ∨ , and t is the Lie algebra of T . In particular, the birational class of MC depends only on G. It is independent of the representation M. 2010 Mathematics Subject Classification. Primary 22E47; Secondary 14D20,14F43,81T13. The research of the author was supported by JSPS Kakenhi Grant Numbers 24224001, 25220701, 16H06335. 1 It has a singularity in general. It is expected that the singularity is symplectic in the sense of Beauville, but the proof is not given. c 2018 American Mathematical Society

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As we have already mentioned above, we construct a ring as a homology group with convolution product. This method has been used successfully in geometric representation theory. Since the study of representations is the main motivation there, it is usual to construct a noncommutative algebra. In fact, also for the Coulomb branch, we do get a noncommutative deformation A of MC simultaneously. Here a noncommutative deformation means a noncommutative algebra A defined over C[] such that A /A is isomorphic to the coordinate ring C[MC ] and the Poisson bracket / f˜g˜ − g˜f˜// , f˜|=0 = f, g˜|=0 = g {f, g} = / /  =0

is equal to one given by the symplectic form. We call A ≡ A (G, M) the quantized Coulomb branch. Many noncommutative algebras studied in representation theory are deformations of commutative algebras, e.g., the universal enveloping algebra U (g) of a Lie algebra g is a deformation of the symmetric algebra of g. However it is remarkable (at least to the author) that commutative algebras and their deformations appear in a systematic construction. In the first paper [Nak16b] we considered a general M, but we only constructed C[MC ] as a vector space. A definition of a product was given later in [Part II], under the assumption that M is of the form M = N ⊕ N∗ (cotangent type condition). A physical argument says that the induced homomorphism π4 (G) → π4 (Sp(M)) ∼ = {±1} must vanish in order to have a well-defined Feynman measure on the space of fields.2 We do not know whether this vanishing is required or enough to define a Coulomb branch at this moment, but the assumption M = N ⊕ N∗ is too restrictive, as more general cases have been studied in the physics literature. We will later use the notation M(G, N) when we assume M = N ⊕ N∗ after §3. There should be no fear of confusion. There is another well-known recipe for constructing a complex affine-algebraic symplectic variety from (G, M). It is the symplectic reduction M///G = μ−1 (0)//G, called the Higgs branch of the same 3d SUSY gauge theory associated with (G, M) in the physics literature. Here μ : M → g∗ is the moment map vanishing at the origin, and μ−1 (0)//G is the quotient space of μ−1 (0) by G in the sense of geometric invariant theory, namely the coordinate ring C[μ−1 (0)//G] is the space of G-invariant polynomials C[μ−1 (0)]G in the coordinate ring of μ−1 (0). When M = N ⊕ N∗ , the ring D(N) of polynomial coefficient differential operators on N gives a noncommutative deformation of M. (In order to introduce , one considers the Rees algebra associated with the filtration given by degrees of differential operators.) A noncommutative analog of the symplectic reduction has been known as a quantum symplectic reduction, which should be considered as an appropriate ‘quotient’ of D(N) of G. It gives a noncommutative deformation of MH . In representation theory, we have experienced that interesting symplectic varieties and their quantizations appear as symplectic reductions, e.g., quiver varieties and toric hyper-K¨ ahler manifolds. On the other hand, the study of Coulomb 2 This was pointed out by Witten via Braverman. It is possibly related to the existence of orientation data for the vanishing cycle considered in [Nak16b].

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branches has just started. We get symplectic varieties, whose descriptions as symplectic reductions of finite dimensional symplectic vector spaces are not known. Hence we expect the importance of Coulomb branches will increase in future. We believe that the representation theory of a quantized Coulomb branch A is easier to study than that of a quantum symplectic reduction, as it is defined as a convolution algebra, hence powerful geometric techniques (see [CG97]) can be applied. Also, the pair of Higgs and Coulomb branches of a given (G, M) is expected to be a symplectic dual pair in the sense of Braden-Licata-Proudfoot-Webster [BLPW16] in many cases. Symplectic duality expects a mysterious relation between a pair of symplectic varieties. The whole picture of symplectic duality has yet to be explored, but it at least says that it is meaningful and important to study Higgs and Coulomb branches simultaneously. It should be noted that the current framework of symplectic duality in [BLPW16] requires that both symplectic varieties have symplectic resolutions. This assumption is not satisfied in many examples of Higgs and Coulomb branches. Hence we should start to look for a more general framework of symplectic duality.

2. Physical background In §1 I have explained why the study of Coulomb branches could be interesting from a mathematical point of view. In this section I will try to explain the physical background, as far as I can. Like [Part II] this article is written so that no knowledge of physics is required for reading it, except this section. The reader does not need to understand this section, as I myself do not understand well it either. But my superficial understanding led me to find a definition given in the next section, and it is my belief that some understanding of the physics background will be necessary to achieve new results on Coulomb branches. A reader in hurry could skip this section, but it is my hope that (s)he does not. Let me emphasize that I, by no means, intend to ignore past research in physics, which strongly motivated us to obtain most of the results explained in this paper. The relevant literature can be found in [Nak16b]. In physics, like differential geometry, people use a maximal compact subgroup Gc of a complex reductive group G. Similarly we assume that M has an inner product preserved by Gc . A given pair (G, M), physicists associate a 3-dimensional supersymmetric gauge theory. It is an example of quantum field theories which are defined by path integrals of Lagrangians over the infinite dimensional space of all fields. There are two important fields, one is a connection on a principal Gc bundle P over R3 , and the other is a section of P with values in M. Other fields are spinors and sections of vector bundles associated with P . They play an important role in physics, but we ignore them as we will only give a rough understanding. Anyhow, the Lagrangian containing curvature of the connection and the differential of the section is welldefined functional, but the path integral does not have a mathematically rigorous definition. Configurations giving local minima of Lagrangian are classical solutions of motion in quantum mechanics, hence are important objects. In our situation local minimum configurations form a finite dimensional space, instead of a single

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path. It is called the moduli space of vacua. In fact, this will not be the right definition, as it gives the classical moduli space, and the actual moduli space receives corrections as we will explain below. The Lagrangian is the sum of square norms of the curvature and the derivative of the section and others. Local minima are attained when several terms vanish. We classify the moduli space of vacua by which terms vanish, and consider branches of vacua. Typical examples are the Higgs branch MH and the classical Coulomb branch. The Higgs branch is the symplectic reduction M///G explained in §1. It coincides with the hyper-K¨ahler quotient of M by Gc in differential geometry. Connections are the trivial ones, and sections are constant, hence only M remains. We do not recall the notion of hyper-K¨ ahler quotients here, so please refer to [Nak92] for example. Quiver varieties studied by the author for many years, as well as, toric hyper-K¨ ahler manifolds are examples of symplectic reductions, hence of Higgs branches. On the other hand, the classical Coulomb branch is (Tc∨ × (R3 ⊗ tc ))/W , where ∨ Tc is the dual of a maximal torus Tc of Gc , tc is the Lie algebra of Tc , and W is the Weyl group. It is the same as T ∗ T ∨ /W which appeared in §1. Sections vanish in the classical Coulomb branch, and the factor (R3 ⊗ tc ) comes from fields for which we omit the explanation here. The factor Tc∨ came from connections, but they take values in the dual torus Tc∨ and are scalars after Fourier transform in an infinite dimensional space of connections. Even this part of the physics argument is difficult to make mathematically rigorous, but we will see how T ∨ appears in §5(i) and Theorem 6.1 starting from a mathematically rigorous definition. Classical Coulomb branches and Higgs branches, and other branches of the classical moduli space of vacua contain important information of the supersymmetric gauge theory. It is an initial step to analyzing the gauge theory. One of the goal of physicists’ analysis is a description of the gauge theory as another supersymmetric quantum field theory, called a low energy effective theory, consisting ahler manifold as the target space, together with of maps from R3 with a hyper-K¨ additional fields, which we will ignore. Physicists claim that the original supersymmetric gauge theory and the low energy effective theory are equivalent as quantum field theories in low energy. For example, it implies that many quantities which physicists want to compute are the same in the two theories in low energy. The classical moduli space of vacua appears as an approximation of the target space. But it is too much hope to expect that local minima of the Lagrangian contain enough ‘quantum’ information as required by the low energy effective theory. Physicists say that the classical Coulomb branch receives quantum corrections. Namely the Coulomb branch MC is (Tc∨ ×(R3 ⊗tc ))/W only in the classical description, and the quantum description is different. It is still a hyper-K¨ ahler manifold as supersymmetry must exist also in the low energy effective theory. This part is difficult to justify directly in mathematically rigorous way. It is surprising, at least to me, that such a construction is really possible. Thus the physicists’ definition of MC is very far from mathematically rigorous unlike MH . I heard the explanation of the Coulomb branch in Witten’s series of lectures at the Newton Institute in November 1996 for the first time, but did not make it a research object for many years. Examples of Coulomb branches are familiar hyper-K¨ ahler manifolds to me, hence I had kept an interest.

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A new insight came to me when I heard Hanany’s talk in Warwick in fall 2014. Hanany explained to us there is a formula (monopole formula) computing the character of the coordinate ring C[MC ] with respect to the C× -action. The monopole formula is a sum over dominant coweights of G, and each term is a combinatorial explicit expression in a coweight. The formula passed many tests checking it reproduces the character in many known examples of Coulomb branches. After Hanany’s talk I looked for a ring whose character reproduces the monopole formula, as we can reproduce MC , at least as an affine algebraic variety, as the spectrum of the ring. Then I found a proposal in [Nak16b], which was modified in [Part II]. My path of thinking was explained in [Nak16b]. Let us recall it briefly. The starting point was [Nak16b, 1(iv),(v)]: a hypothetical topological quantum field theory obtained by a topological twist of the gauge theory produces a ring: Consider a quantum Hilbert space HS 2 for S 2 . We have a vector in Hom(HS 2 ⊗ HS 2 , HS 2 ) associated with M 3 , the 3-ball with two smaller balls removed from the interior, which produces a commutative multiplication. Then the quantum Hilbert space in question is the homology of the moduli space of solutions of the associated nonlinear PDE on S 2 , and the vector is given again by the moduli space of solutions, but on M 3 this time, whose image under the boundary value gives a homology class. This is an old idea which motivated Atiyah [Ati88] to write down axioms of topological quantum field theories based on earlier works by Donaldson, Floer, and others. I arrived at a puzzle immediately, as there is only the trivial solution for the nonlinear PDE when (Gc , M) = (SU(2), 0), as the only flat connection on S 2 is the trivial one. Since the stabilizer is nontrivial, namely SU(2), we may consider the ∗ (pt) of a point, but its spectrum is just C/ ± 1. It equivariant cohomology HSU(2) is different from the known answer in physics (i.e., the Atiyah-Hitchin manifold). I needed a correction, as a naive guess gives an immediate contradiction. I made two modifications, (a) forgetting one component of the nonlinear PDE above, corresponding to the stability condition via the Hitchin-Kobayashi correspondence, and (b) considering the sheaf of a vanishing cycle on the moduli space. The latter was motivated by recent advances in Donaldson-Thomas invariants. It will be explained in §4. In the joint work [Part II] I switched from a moduli space on ˜ = D ∪D∗ D, the gluing of two copies of the formal disk S 2 to one on a raviolo3 D D along the punctured disk D∗ . The reason was explained in [Part II, 1(i)]. Let us review it briefly. We first break 3-dimensional symmetry by choosing a time direction, thus we see the 3-dimensional space-time as a 2-dimensional movie. The 3-manifold M 3 above is a movie such that two S 2 ’s collide and become one S 2 . We suppose that one of colliding S 2 is much smaller than another like a meteor and the earth. Then we forget the time direction, and we just compare pictures before and after an incident. The incident happens at the origion of a small region D in the large S 2 , and the picture remains the same except at the origin. We ˜ This explains a difference between S 2 and D, ˜ i.e. D ˜ is a thus get an object on D. crushed S 2 in the time direction. They are the same for topological quantum field theories. Technically we take an advantage of 2-dimensional view point, as we use algebro-geometric language.

3 singular

form of ravioli, which are Italian dumplings.

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3. A mathematical definition We will review the definition of [Part II] in this section. We assume that N is a finite dimensional complex representation of a complex reductive group G. Here N may not be irreducible, nor may it be 0. The symplectic representation M is given as N ⊕ N∗ , but M does not show up in this section. Let D = Spec C[[z]] be the formal disk, D× = Spec C((z)) the formal punctured disk. We denote N((z)), N[[z]] by NK , NO respectively. Similarly let GK = G((z)), GO = G[[z]]. The affine Grassmannian GrG is the moduli space / 8 ;< / P is an algebraic G-principal bundle over D / (P, ϕ)/ isom. / ϕ : P|D× → G × D× is a trivialization of P over D× It is known that GrG has a structure of an ind-scheme as a direct limit of projective varieties. Set-theoretically, it is GrG = GK /GO . Namely we take a trivialization of P over D to regard ϕ as an element of GK , and kill the ambiguity of the choice of trivialization by taking the quotient by GO . If we further take the quotient by the left GO -action changing the trivialization ϕ, we get GO \GK /GO . It is the moduli ˜ 4. space of G-bundles over the raviolo D We then add an algebraic section s of the vector bundle P ×G N associated with the representation N to consider the moduli space T of triples (P, ϕ, s). Settheoretically, it is GK ×GO NO . Considering the Taylor expansion of s, we see that T is a direct limit of an inverse limit of finite rank vector bundles over projective schemes. We will consider homology groups of T or its closed varieties, which are rigorously defined as limits of homology groups of finite dimensional varieties. See [Part II, §2] for detail. We introduce a closed subvariety R of T by imposing the condition that ϕ(s) extends over D: R = {(P, ϕ, s) | ϕ(s) ∈ NO }/isom. Since ϕ is a trivialization over D× , ϕ(s) in general has a rational section which may have singularities at the origin. The space R is defined by requiring that the coefficients of negative powers of ϕ(s) vanish. The quotient GO \R is the moduli ˜ space of pairs of G-bundles and their sections over D. This space R is the main player of our construction. Its meaning is clearer if we consider a bigger space {(P1 , ϕ1 , s1 , P2 , ϕ2 , s2 ) ∈ T × T | ϕ1 (s1 ) = ϕ2 (s2 )}/isom. This consists of a pair of G-bundles over D, a trivialization over D× and sections of associated vector bundles such that sections are equal through trivializations. It is a fiber product T ×NK T . If we further require that (P2 , ϕ2 ) is the identity element of GrG , i.e., the point where ϕ2 extends across 0 ∈ D, we recover R. Conversely we use the action of GO on R to get T ×NK T = GK ×GO R from R. From the gauge theoretic point of view, T ×NK T parametrizes configurations of a connection and a section on D twisted at the origin 0. Namely (P1 , ϕ1 ) is before the twist, while (P2 , ϕ2 ) is after. Since the twisting happens only at the origin, they are isomorphic outside the origin. Originally we considered a connection and a section with a point singularity in 2 + 1 dimensional space-time in the 3-dimensional gauge 4 Braverman,

my collaborator, emphasizes the importance of the use of the raviolo.

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theory, but we take a 2-dimensional viewpoint by looking at two time slices, just before and after the event. Now the preparation of the space R is over, so we consider its GO -equivariant Borel-Moore homology group H∗GO (R). We define its degree so that the fundamental class of the fiber of T over the identity element of GrG has degree 0. We refer to [Part II] for the precise definition and omit it here. One can show that ∗ (pt) by using the Schubert cell H∗GO (R) vanishes in odd degree and is free over HG decomposition of the affine Grassmannian GrG . Next we introduce a convolution product ∗ : H∗GO (R) ⊗ H∗GO (R) → H∗GO (R). The rigorous definition in [Part II] is too technical to be reproduced here. Let us give a heuristic argument: We formally assume that we have an induction isomorphism H∗GK (T ×NK T ) ∼ = H∗GO (R), and T is smooth. Then using projection to the (i, j)-factor pij

T ×NK T ×NK T −−→ T ×NK T

(i, j) = (1, 2), (2, 3), (1, 3),

we define c ∗ c = p13∗ (p∗12 c ∩ p∗23 c ). This is not rigorous as we do not know how to define H∗GK (T ×NK T ), and T is not nonsingular. But we do have an alternative rigorous definition of the convolution product ∗ on H∗GO (R). We have Theorem 3.1. (H∗GO (R), ∗) is a commutative ring. The method of constructing an algebra by convolution has been used in geometric representation theory, e.g. the group ring of the Weyl group from the Steinberg variety, the universal enveloping algebra of a Kac-Moody Lie algebra from the analog of the Steinberg variety for quiver varieties, etc. But those examples give noncommutative algebras. From the general theory of convolutions, we do not get a reason why ∗ becomes commutative. An explanation of commutativity is given by recalling the geometric Satake correspondence: We consider the abelian category of GO -equivariant perverse sheaves on GrG , endow it with a tensor product via convolution product, and show that the resulting tensor category is equivalent to one of finite dimensional representations of the Langlands dual group G∨ of G. The latter category is commutative, i.e., V ⊗W ∼ = W ⊗ V , hence the former is also. A geometric explanation of this commutativity of the former is given by the Beilinson-Drinfeld one-parameter deformation of the affine Grassmannian. We can give a proof of commutativity in the above theorem using this idea [BFN17]. (In [Part II] we give another proof given by a reduction to an abelian case, where it can be shown by a direct computation.) Let us remark again that the commutativity of the product was expected if one believes that (H∗GO (R), ∗) is the quantum Hilbert space of a 3d topological field theory for S 2 . In turn, it means that there is a hidden 3d symmetry in affine Grassmannian and Beilinson-Drinfeld one-parameter deformation. This is compatible with the fact that moduli spaces of singular monopoles on R3 are identified with affine Grassmanian slices in the context of Coulomb branches of quiver gauge theories. (See in §7.)

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Now we have a commutative ring (H∗GO (R), ∗). Hence we can define the affine scheme as its spectrum: MC = Spec(H∗GO (R), ∗). We further show that (H∗GO (R), ∗) is finitely generated and integral. Hence MC is an irreducible affine variety. We also show that it is normal. A noncommutative deformation is defined as follows. We have a C× -action on the formal disk D by the loop rotation z → tz. We have induced actions on the various spaces considered above. In particular, we consider the semi-direct product GO  C× which acts on R. Hence we can consider the equivariant Borel-Moore × homology group H∗GO C (R) with respect to the larger group GO  C× , and define the convolution product as above. We thus define the quantized Coulomb branch by ×

A = (H∗GO C (R), ∗). Convolution products on affine Grassmannians and related spaces were considered earlier in [VV10, BFM05, BF08], which we use models for our definition. In [VV10], affine flag varieties instead of affine Grassmannians, and equivariant K-theory instead of equivariant Borel-Moore homology groups were used, but it is basically understood as a special case of the Coulomb branch where N is the adjoint representation. The algebra constructed there is Cherednik double affine Hecke algebra (DAHA). If we use affine Grassmannians instead of flags, we get the spherical part of the DAHA. We get the trigonometric version instead of the elliptic one if we use homology instead of K-theory. Our Coulomb branch for N = g is t × T ∨ /W . It is a remarkable example, as the Coulomb branch does not receive quantum corrections. In [BFM05, BF08], the case N = 0 was considered. The Coulomb branch is the phase space of the Toda lattice for the Langlands dual group of G, or the moduli space of solutions of Nahm’s equation on the interval. We omit further explanation. We can introduce a convolution product ∗ on the equivariant K-theory K GO (R) for R in the same way, and define the K-theoretic Coulomb branch as the specdef. GO (R), ∗). It is a general expectation that replacement of trum MK C = Spec(K homology by K-theory corresponds to a 1-dimensional higher quantum field theory compactified with S 1 . Gaiotto conjectures that MK C is isomorphic, as a complex analytic variety, to the Coulomb branch of the 4-dimensional N = 2 SUSY gauge theory on R3 ×S 1 with a generic complex structure. (See [BFN16, Remark 3.9(2)].) Here the Coulomb branch is expected to be a hyper-K¨ ahler manifold, which shares many common properties with Hitchin’s moduli spaces of solutions of the selfduality equation over a Riemann surface. Among the S 2 -family of complex two structures, two are special and other generic ones are isomorphic. 4. Not necessarily cotangent type In [Nak16b] we first made a proposal for the case when M is not necessarily of cotangent type. It was just a heuristic definition of the coordinate ring C[MC ] as a graded vector space, and a definition of the convolution product ∗ was not proposed. Nevertheless another heuristic argument yielded an idea for defining C[MC ] as H∗GO (R) (more precisely homology of the moduli space on S 2 ). We only

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have a slight advance in this direction since [Nak16b] was written. Nevertheless we believe that the original intuition is important, hence we review it in this section. The reader can safely skip this section to read other sections. 4(i). Holomorphic Chern-Simons functional. Let Σ be a compact Rie1/2 mann surface. We choose and fix a spin structure, i.e. a square root KΣ of the canonical bundle KΣ . We also fix a (C ∞ ) principal G-bundle P with a fixed reference partial connection ∂. A field consists of a pair ∂ +A : a partial connection on P . So A is a C ∞ -section of Λ0,1 ⊗(P ×G g). 1/2 Φ : a C ∞ -section of KΣ ⊗ (P ×G M). Let F be the space of all fields. There is a gauge symmetry, i.e. the complex gauge group G(P ) of all (complex) gauge transformations of P naturally acts on the space F. In fact, as we will see in examples below, we need to consider all topological types of P (classified by π1 (G)) simultaneously, but we will ignore this point. We define an analog of the holomorphic Chern-Simons functional by " 1 (4.1) CS(A, Φ) = ω((∂ + A)Φ ∧ Φ), 2 Σ where ω( ∧ ) is the tensor product of the exterior product and the symplectic form = 1/2 ω on M. Since (∂ +A)Φ is a C ∞ -section of 0,1 ⊗KΣ ⊗(P ×G M), ω((∂ +A)Φ∧Φ) = = 0,1 1,1 is a C ∞ -section of ⊗ KΣ = . Its integral is well-defined. This is invariant under the gauge symmetry G(P ). When M is of cotangent type, i.e. M = N ⊕ N∗ , we can slightly generalize the construction. Let us choose M1 , M2 to be two line bundles over Σ such that M1 ⊗ M2 = KΣ . We modify Φ as Φ1 , Φ2 : C ∞ -sections of M1 ⊗ (P ×G N) and M2 ⊗ (P ×G N∗ ) respectively. Then " (4.2) CS(A, Φ1 , Φ2 ) = $(∂ + A)Φ1 , Φ2 %. Σ

It is a complex-valued function on F. Note that F is a complex manifold, in fact a complex affine space, though it is infinite dimensional. Our holomorphic Chern-Simons functional CS is a holomorphic function on F. It is easy to see that (A, Φ) is a critical point of CS if and only if the following two equations are satisfied: (4.3)

(∂ + A)Φ = 0, μ(Φ) = 0. 1/2

The first equation means that Φ is a holomorphic section of KΣ ⊗ (P ×G M) when we regard P as a holomorphic principal bundle by ∂ + A. The second means that 1/2 Φ takes values in μ−1 (0). Therefore Φ is a holomorphic section of KΣ ⊗ (P ×G −1 −1 μ (0)), i.e. a twisted map from Σ to the quotient stack μ (0)/G. Let us denote by crit(CS) the critical locus of our holomorphic Chern-Simons functional. Since it is the critical locus of a holomorphic function on a complex manifold, we could have a sheaf ϕCS (CF ) of vanishing cycle associated with CS. This is heuristic at this stage as F is an infinite dimensional complex manifold,

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and hence it is not clear whether the usual definition of the vanishing cycle can be applied. Nevertheless it was hoped [Nak16b] that one could use an approach for usual complex Chern-Simons functional for connections on a compact Calabi-Yau 3-fold, developed by Joyce and his collaborators [BBD+15, BBBBJ15]. We thus formally define (4.4)

∗ Hc,G(P ) (crit(CS), ϕCS (CF ))

the equivariant cohomology with compact support with the sheaf of vanishing cycles ϕCS (CF ) as coefficient. The proposal in [Nak16b] was that the dual of this space (for Σ = P1 = S 2 ) has a commutative product, and to define the Coulomb branch as its spectrum. 4(ii). Derived symplectic geometry. There is an alternative approach for a construction of the perverse sheaf ϕCS (CF ) based on derived symplectic geometry [PTVV13], which I learned from Dominic Joyce during a workshop at Oxford in 2015 after [Nak16b] was written. It is an immediate consequence of results in [GR17]. Let us review it for the sake of readers. Let us first consider μ−1 (0)/G, as a derived Artin stack, as a derived fiber product (M/G) ×g∗ /G (0/G), where G acts on g∗ by the coadjoint action, and M/G → g∗ /G is the moment map. This is equipped with a 0-shifted symplectic structure. One of the main results in [PTVV13] is that the space Map(X, μ−1 (0)/G) of maps from a d-dimensional smooth and proper Calabi-Yau X to μ−1 (0)/G has a (−d)-shifted symplectic structure. In particular, for Σ an elliptic curve, (the derived version of) crit(CS)/G(P ) has a (−1)-shifted symplectic structure when 1/2 KΣ = OΣ . A modified construction for the case of twisted maps is given in [GR17]. Unlike [PTVV13], which requires genus 1, the construction of [GR17] is applicable to a compact Riemann surface Σ of any genus. Therefore (the derived version of) crit(CS)/G(P ) has a (−1)-shifted symplectic structure. There is an alternative way to define a (−1)-shifted symplectic structure, again due to [GR17]. We consider the stack of pairs ∂ + A and Φ as in (4.3), but without the equation μ(Φ) = 0. Let us denote it by SectΣ (MK 1/2 /G). Then the Σ

moment map gives a map to the stack of pairs ∂ + A and ξ, a holomorphic section of KΣ ⊗ (P ×G g∗ ). The latter is nothing but the (derived) moduli stack HiggsG (Σ) of Higgs bundles, and has a 0-shifted symplectic structure. One of main results in [GR17] says that the map SectΣ (MK 1/2 /G) → HiggsG (Σ) Σ

is a Lagrangian embedding. This result was originally observed by Gaiotto [Gai16] by a heuristic argument as in the previous subsection. There is another Lagrangian subvariety in HiggsG (Σ), the moduli stack BunG (Σ) of G-bundles on Σ. Therefore crit(CS)/G(P ) is a (derived) fiber product of two lagrangians in a 0-shifted symplectic stack, hence has a (−1)-shifted symplectic structure by [PTVV13]. Now by [BBBBJ15] the underlying Artin stack crit(CS)/G(P ), if it is oriented, has a well-defined sheaf of vanishing cycles, which is regarded as a definition of ϕCS (CF ). We do not recall the definition of an orientation here, but it is expected that its existence is guaranteed by the above condition that π4 (G) → π4 (Sp(M)) vanishes.

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4(iii). Cutting. Suppose M = N ⊕ N∗ . Then we have a C× -action on F defined by t · (A, Φ1 , Φ2 ) = (A, Φ1 , tΦ2 ). Since CS is linear in Φ2 , we have CS(t · (A, Φ1 , Φ2 )) = tCS(A, Φ1 , Φ2 ). Under this condition for finite dimensional spaces, the vanishing cycle functor was studied in [Dav13]. We hope that this result can be applied in our infinite dimensional setting, then (4.4) is isomorphic to ∗ Hc,G(P ) (RΣ , C),

where RΣ is the space of (A, Φ1 ) such that (∂ + A)Φ1 = 0, that is the space of holomorphic principal bundles (P, ∂ + A) and a holomorphic section of M1 ⊗ (P ×G N). Our space R in §3 is related to RΣ by GO \R = RD˜ /G(P ) though it is not ˜ clear whether we can take Σ = D. 5. Examples In order to illustrate that the construction in §3 is not so strange, even though we use homology groups of infinite dimensional spaces, let us give simple examples. This is based on [Part II, §4]. 5(i). Let G = C× , N = 0. This is the simplest case. Since N = 0, R is nothing but the affine Grassmannian GrG , and GrG parametrizes pairs of line bundles on D and their trivializations over D× . It is known that GrG with the reduced scheme structure is the discrete set parametrized by integers Z. (We will be interested only in homology groups of GrG and R. Hence nonreduced structures will play no role in our construction.) In fact, ϕ(z) = z n is a point corresponding to n ∈ Z. Therefore  × H∗GO (R) = H∗C (pt). n × H∗C (pt)

Note that is the polynomial ring C[w] in one variable w. Since we have a polynomial ring over each integer n, we need to calculate the product of a polynomial on m and one on n. Since we do not give the precise definition of the convolution product, we cannot perform the check, but for G = C× , the product ∗ is given by the push-forward homomorphism of the map given by tensor product ⊗

GrC× × GrC× − → GrC× . Then the product of f (w) on m and g(w) on n is f (w)g(w) on m + n. Let us denote by x the polynomial 1 on the integer n = 1. We then have ∼ C[w, x± ] = C[C × C× ]. H GO (R) = ∗

Therefore the Coulomb branch is C × C× . Since this is nothing but R3 × S 1 , the Coulomb branch does not receive the quantum correction. This is a reflection of the fact that the gauge theory is trivial in this case. Let us further consider the case when G is a torus G, and N = 0. Then GrT is a discrete space parametrized by Hom(C× , T ). Therefore  H∗TO (R) = HT∗ (pt). λ∈Hom(C× ,T )

HT∗ (pt)

is the space C[t] of polynomials on the Lie algebra t of T . On Note that the other hand, let eλ denote the fundamental class of the point λ. We have eλ ∗ eμ = eλ+μ as above. Since this can be regarded as the ring of characters of the dual T ∨ of T , the Coulomb branch is t × T ∨ = T ∗ T ∨ .

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5(ii). Let us keep G as C× , and replace the representation by the standard on N = C. As we have already explained, GrC× is a discrete set parametrized by Z, and R consists of vector spaces over integers n ∈ Z. Since the condition is that we do not get singularities by ϕ(z) = z n , we have   R= z n C[z] ∩ C[z] = z max(0,n) C[z]. n∈Z

n∈Z

 × By the Thom isomorphism for each n, we have H GO (R) ∼ = n H∗C (pt). Hence it is the same as the above example as a vector space. On the other hand, the convolution product is different. In fact, products of homology classes over n > 0 and those over n < 0 are different from above. We cannot check the assertion as we omit the definition, but the product of the fundamental classes of n = 1 and n = −1 is the image under the pushforward homomorphism for zC[z] → C[z] of the fundamental class. Since the image of this map is a codimension 1 subspace, it is nothing but the cup product of w with the fundamental class. Therefore if we denote the fundamental classes of n = 1, −1 by x, y respectively, we get xy = w. Thus H∗GO (R) ∼ = C[w, x, y]/(w = xy) ∼ = C[x, y] = C[C2 ]. Namely the Coulomb branch in this case is C2 . If we replace the representation by the 1-dimensional representation with weight N , the product xy is replaced by the image of the fundamental class under z |N | C[z] → C[z]. Therefore the coordinate ring is C[w, x, y]/(w|N | = xy). Hence the Coulomb branch is nothing but the simple singularity of type A|N |−1 . The Higgs branch N ⊕ N∗ ///C× is a single point {0} if we define it as the GIT quotient of μ−1 (0) by C× as in §1. In particular, the Higgs branch does not see the weight N . We can also consider the direct sum of N copies of the 1-dimensional representation with weight 1. The Coulomb branch is again the simple singularity of type AN −1 . The Higgs branch in this case is the closure of the minimal nilpotent orbit in sl(N ). These examples probably suggest that our definition of the Higgs branch is too naive. 6. Structures In this section we review several structures of the Coulomb branch MC . We also discuss the corresponding structures for the Higgs branch MH . They have been discussed in the physics context. A point is that they can be realized rigorously in the definition of §3. 6(i). (See [Nak16b, §4(iii)(a)] and [Part II, Remark 2.8].) H∗GO (R) is a graded algebra  by the half of the homological degree. We thus have a decomposition C[MC ] = d C[MC ]d such that C[MC ]d · C[MC ]d ⊂ C[MC ]d+d . It means that MC has a C× -action. In fact, C[MC ]d is the weight space with respect to the C× -action with weight d. In above examples, the C× -actions are of weight 1 on x, and 0 on y. Thus they are the standard action on the first factor of C × C× and C2 = C × C respectively. Remark that in general, degrees take values in integers, not necessarily nonnegative. Therefore MC may not be a cone. Here MC is a cone if C[MC ]d = 0 (d < 0), C[MC ]0 = C.

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In the physics context it is expected that the C× -action, or rather the S 1 action, extends to an SU(2)-action after a certain correction. We do not explain the correction, but it is given by a Hamiltonian torus action explained below. In particular, there will be no correction when G is semisimple. The induced SU(2)action on the two-sphere of complex structures S 2 = {aI +bJ +cK | a2 +b2 +c2 = 1} is the standard one through SU(2) → SO(3), where (I, J, K) is the hyper-K¨ ahler structure. Once we fix a complex structure I, we could see only the S 1 -action preserving I. This is the reason why we could only see the S 1 -action in the current definition, which does not realize the hyper-K¨ahler structure. For example, we have an SU(2)-action on C × C× = R3 × S 1 , once we view R3 as su(2). Our S 1 -action has the half weight. For C2 , we correct the action by a Hamiltonian S 1 -action with weights −1/2 on x, 1/2 on y. If we multiply weights by two, it becomes the restriction of the standard SU(2) = Sp(1)-action, given by the identification C2 with the quaternion field H. (It is not complex linear, hence it is different from the standard SU(2)-action on C2 . They are the left and right multiplication of quaternions respectively. Let us consider the Higgs branch MH where the SU(2)-action can be easily described. The quaternionic vector space M has an SU(2) = Sp(1)-action by multiplication of quaternions. It commutes with the G-action, hence we have an SU(2)-action on MH . It rotates the two sphere S 2 of complex structures, as it is so on M. 6(ii). (See [Part II, §3(vi)].) As for any equivariant homology group, the ∗ ∗ (pt) ∼ (pt)-lienar homomorphism group H∗GO (R) comes equipped with an HG = HG O ∗ GO HG (pt) → H∗ (R). (Remark that the convolution product c ∗ c is not naturally ∗ (pt)-linear, in fact it isn’t on the noncommutative deformation.) HG Taking the spectrum, we obtain ∗ (pt).  : MC → Spec HG

It is well-known that ∗ HG (pt) = C[g]G = C[t]W , ∗ (pt) = t/W , where t = Lie T . This is an affine space. and hence Spec HG This construction remains on the noncommutative deformation: ×

×

H∗G×C (pt) → A = H GO C (R). This is an injective algebra homomorphism. In particular, the noncommutative deformation A contains a large commutative subalgebra. Considering the specialization at  = 0, we deduce that  is Poisson commuting. Namely pull-backs of functions f , g on t/W satisfy { ∗ f,  ∗ g} = 0. We have the following Theorem 6.1 (See [Part II, §5(v)].). A generic fiber of  is T ∨ . More precisely we have the following commutative diagram, whose upper horizontal arrow is birational: / T ∗ T ∨ /W = t × T ∨ /W MCE EE mmm EE mmm m E m  EE mmthe first projection " vmmm t/W

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This is a consequence of the fixed-point localization theorem for the equivariant homology group. The localization theorem says that we have an isomorphism H TO (R) ⊗H ∗ (pt) F ∼ = H TO (RT ) ⊗H ∗ (pt) F, ∗

T

∗

T

HT∗ (pt).

where F is the quotient field of Here RT is the set of T -fixed points in R, and the isomorphism is the pushforward homomorphism of the inclusion RT → R. Combining this with the fact that H∗GO (R) is the W -invariant part of H∗TO (R), it becomes enough to compute the equivariant homology group of RT . Since RT is GrT × NT , the calculation in §5(i) shows that it is t × T ∨ . The operation ⊗HT∗ (pt) F corresponds to the restriction to the generic point of t/W . This is a standard argument which tells us that it is useful to view equivariant homology groups as families over t/W . In conclusion,  is Poisson commuting and has algebraic tori as fibers. Hence  : MC → t/W is an integrable system in the sense of Liouville, and A is its quantization. For the Higgs branch MH , we do not have a general construction of an integrable system, though we could see it in many examples. Recall that physicists say that classical Coulomb branch is (Tc∨ × (R3 ⊗ tc ))/W . This is the same as t × T ∨ /W . Hence MC is a kind of correction of the classical Coulomb branch in the sense that they are birational. But I never hear that the localization ⊗HT∗ (pt) F corresponds to going to the ‘classical limit’ in other situations. As far as I asked several physicists, an existence of the integrable system  was not known in physics context. Probably it is more natural to define a map to R3 ⊗ tc /W , as it is the ‘noncompact direction’ in the classical Coulomb branch. A generic fiber is Tc∨ . We do not have a general construction of such a map in our definition of MC , but we can directly construct it for a few examples, toric hyper-K¨ ahler manifolds, moduli spaces of singular monopoles, etc. K For a K-theoretic Coulomb branch MK C , we can define  : MC → T /W in K the same way. Recall that MC is conjecturally the Coulomb branch of a 4d SUSY gauge theory on R3 × S 1 with a generic complex structure. Though this Coulomb branch is expected to have common features as Hitchin’s moduli spaces,  is very different from Hitchin’s integrable systems: the target is not an affine space, fibers are noncompact, and it is defined for a generic complex structure. It is an important open problem to construct two special complex structures so that an analog of Hitchin’s integrable system is defined. 6(iii). (See [Nak16b, §4(iii)(c)] and [Part II, §3(v)].) It is known that the affine Grassmannian GrG is topologically a based loop group ΩG. In particular, its connected components are in bijection to the fundamental group π1 (G) of G. It is well-known that π1 (G) is a finitely generated abelian group. The homology group of R decomposes according to connected components of R, which are the same as those of GrG . This decomposition is compatible with the convolution product: let Rγ denote the connected component corresponding to γ ∈ π1 (G). Then we have H∗GO (Rγ ) ∗ H∗GO (Rγ  ) ⊂ H∗GO (Rγ+γ  ). In terms of MC = Spec H∗GO (R), this decomposition means that the Pontryagin dual π1 (G)∧ = Hom(π1 (G), C× ) of π1 (G) acts on MC . In the above examples, we have π1 (G) = π1 (C× ) = Z, and its Pontryagin dual is C× . The action is on the second factor in the first example MC = C × C× . In the second example, x has weight 1 and y has weight −1.

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Since this action extends to the noncommutative deformation H∗GO C (R), it follows that the symplectic form is preserved under the action. When G is semisimple, π1 (G) is a finite group, and its Pontryagin dual also. We obtain a torus when Hom(G, C× ) is nontrivial. Let χ ∈ Hom(G, C× ). The corresponding moment map of the C× -action via Hom(G, C× ) ∼ = Hom(C× , π1 (G)∧ ) × is given by the composition of  with dχ : g → Lie C . In particular, the action is × Hamiltonian. One can also show that the symplectic reduction of H∗GO C (R) is the Coulomb branch of the kernel of χ. See [Part II, §3(vii)(d)]. For the Higgs branch, χ ∈ Hom(G, C× ) is used to introduce a stability condition for the geometric theory quotient. Namely we can consider Proj of the ∞ invariant n −1 G,χn C[μ (0)] of semi-invariants. Here C[μ−1 (0)]G,χ = {f ∈ graded ring n=0 C[μ−1 (0)] | f (g · x) = χ(g)n f (x)}. Also we can use ζ ∈ Hom(g, Lie C× ) to perturb the defining equation as μ = ζ. 6(iv). (See [Nak16b, §5(i)] and [Part II, §3(viii)].) Suppose that N is a  containing G as a normal subgroup. The quotient representation of a larger group G  group G/G is called the flavor group in the physics literature. Let us denote it by GF .  O acts on R, we can consider the equivariant homology group H∗GO (R) Since G  O . It is a commutative ring over H ∗ (pt), with respect to the larger group G GF ∗ (pt) = hence the corresponding spectrum is a family of varieties over Spec HG F Spec C[gF ]GF . The fiber over 0 is the original MC . Namely MC has a deformation parametrized by gF //GF . Although we omit the details, we can construct (candidates) of partial resolutions of MC corresponding to cocharacters of a maximal torus TF of GF . See [Part II, §3(ix)].  On the Higgs branch MH , we have an induced action of GF = G/G. Note that the structures in this and previous subsections are swapped for MC and MH . Namely Hom(G, C× ) gives a deformation/resolution on MH and a group action on MC . On the other hand GF gives a group action on MH and a deformation/resolution on MC . 6(v). Let us consider toric hyper-K¨ahler manifolds as examples of structures of one and two subsections before. We start with an exact sequence of tori 1 → T = (C× )d−n → T = (C× )d → TF = (C× )n → 1. We take the standard representation N = Cd of T and denote its restriction to T also by N. We have MC (T, N) ∼ = C2d by the computation in §5(ii). By the construction of two subsections before, the Pontryagin dual of π1 (T) acts on C2d . This is nothing but the standard action of the dual torus T∨ of T∨ . The dual TF∨ of TF is a subtorus of T∨ , hence acts on C2d . As we explained in two subsections before, the Coulomb branch MC (T, N) for the subgroup T is nothing but the symplectic quotient C2d ///TF∨ of C2d by TF∨ . The exact sequence of dual tori 1 → TF∨ → T∨ → T ∨ → 1 identifies it as the Higgs branch for TF∨ for the representation Cd . Namely under the exchange T ↔ TF∨ , the Higgs and Coulomb branches are exchanged.

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6(vi). (See [Nak16b, §4(iii)(d)] and [BFN17, App. A]) We can extend the Hamiltonian torus action from Hom(G, C× ) to a nonabelian group action sometimes. Suppose that we have a subspace l in C[MC ] which is a Lie subalgebra with respect to the Poisson bracket { , }. For example, the space of degree 1 elements forms a Lie subalgebra as the Poisson bracket is of degree −1. We consider Hamiltonian vector fields Hf for f ∈ l, and they form a Lie subalgebra in the Lie algebra of vector fields on MC as [Hf , Hg ] = H{f,g} . Thus l acts on MC so that the transpose of the moment map is the natural homomorphism C[l∗ ] = Sym(l) → C[MC ]. In many examples l is integrated to a Lie group action. Consider the example 5(i). The symplectic form, in this example, is a standard one dw ∧ dx x . We have {x, w} = w, and Cx ⊕ Cw is a 2-dimensional Lie subalgebra. This is integrated to a C× C-action as (t, s)(x, w) = (tx, sx+w) for (t, s) ∈ C× C. This computation is not enlightening as we know the Coulomb branch explicitly. One can consider also the example 5(ii), but again it is not enlightening. λ A nontrivial example is the action of StabGQ (μ) on a slice to GrμGQ in GrGQ as the Coulomb branch of a quiver gauge theory explained in the next section. See [BFN17, App. A]. 7. Quiver gauge theories At the time of this writing, Coulomb branches of (G, N) whose Higgs branches are quiver varieties are the most studied. Let Q be a quiver with the vertex set Q0 and the edge set Q1 . For an edge h ∈ Q1 , let us denote the starting and ending vertices by o(h), i(h) respectively. Fortwo given Q0 -graded finite dimensional  complex vector spaces V = Vi , W = Wi , we set  GL(Vi ), G= i∈Q0

N=



h∈Q1

Hom(Vo(h) , Vi(h) ) ⊕



Hom(Wi , Vi ).

i∈Q0

The pair (G, N) is a quiver gauge theory. Here the G-action on N is the natural one. In physics, when Q is of type ADE, the Coulomb branch MC is identified with a moduli space of monopoles on R3 with singularities at the origin. This assertion is proved in the above mathematical definition when the monopole moduli space is replaced by its algebro-geometric analog ([BFN16]). Here the structure group of monopoles is the complex simple Lie group GQ of type Q of the adjoint type, the dimensions of Vi give the charges of monopoles, and the dimensions of Wi determine the singularity type. The definition of the algebro-geometric analog is not simple in general, but when μ = dim Wi i − dim Vi αi is dominant, it is given as follows: λ μ Consider the affine Grassmannian for GQ , and Schubert varieties GrGQ , GrGQ for  λ= dim Wi i and μ. Then the intersection of a transversal slice to GrμGQ and λ

GrGQ is MC . Under the geometric Satake correspondence, the affine Grassmannian is connected to the representation theory of the Langlands dual group G∨ Q of GQ . On the other hand, homology groups of quiver varieties have structures of representations of the Lie algebra of GQ , or of G∨ Q which is the simply-connected type. The

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symplectic duality mentioned in the introduction is (and should be) formulated so that the two constructions are related by a ‘duality’. To determine the Coulomb branches, we use the following recipe: (1) First, construct a candidate of MC . In many cases, we just take an answer given by physicists. (2) Next, construct an integrable system on the candidate, which is expected to correspond to . (3) Show that the integrable system is a flat family, and MC is normal. (4) The birational isomorphism between MC and the candidate through T ∗ T ∨ /W extends over the complement of the inverse image of a codimension 2 subset in t/W . It is a consequence of the normality that the extension outside codimension 2 guarantees the isomorphism everywhere. As we explained above, MC is birational to T ∗ T ∨ /W by an application of the localization theorem in equivariant homology groups. By a similar argument, MC can be determined at a codimension 1 subvariety by a reduction to Coulomb branches of tori and rank 1 groups. The abelian cases are determined as in §5(ii), and the rank 1 case is a hypersurface in C3 ([Part II, §6(iv)]). Therefore (4) is usually an easy step. On the other hand (3) is checked by a case-by-case argument and is usually the key point of the proof. When Q is affine type ADE, we replace monopoles by instantons. We should consider instantons on the Taub-NUT space, not on R4 in general. When μ is dominant, it is expected that moduli spaces on R4 and on the Taub-NUT space are isomorphic as complex symplectic varieties (although the hyper-K¨ ahler metrics are different). For instanton moduli spaces, either on R4 or the Taub-NUT space, the property (3) is not known. Hence we cannot prove that Coulomb branches are instanton moduli spaces in general. In fact, (3) is a delicate property. For example, nilpotent orbits are normal for type A, but not in general. On the other hand, Coulomb branches are always normal. It is known that nilpotent orbits and their intersection with Slodowy slices for classical groups appear as Higgs branches. A naive guess gives that the corresponding Coulomb branches are also, but they should not be by normality. Hanany et al find examples of Coulomb branches which are normalizations of nonnormal nilpotent orbits. For affine type A, we can use Cherkis bow varieties instead of instanton moduli spaces on the Taub-NUT space. Bow varieties are moduli spaces of solutions of Nahm’s equation, which is a nonlinear ODE. The ODE is hard to analyze, hence we rewrite bow varieties as moduli spaces of representations of a quiver with relations, and show the property (3) (see [NT17]). Thus Coulomb branches for affine quiver gauge theories of type A are all determined. 8. Quantized Coulomb branches Less is known for quantized Coulomb branches than Coulomb branches themselves. For a quiver gauge theory of finite type ADE, the quantized Coulomb branch A is isomorphic to a shifted Yangian, as proved in appendix of [BFN16]. But this was shown under the assumption that μ is dominant. General cases remain open.

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We have mentioned that the quantized Coulomb branch for N = g is the spherical DAHA. Consider the case G = GL(k) as an example of a quiver gauge theory for the Jordan quiver with V = Ck , W = 0. We generalize this case to V = Ck , W = Cr . In this case A is the spherical part of the rational Cherednik algebra associated with the wreath product Z/rZ ) Sk = (Z/rZ)k  Sk [KN16]. (The corresponding Coulomb branch is Symk (C2 /(Z/rZ)).)

References A. Braverman, M. Finkelberg, and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II, ArXiv e-prints (2016), arXiv:1601.03586 [math.RT]. ´ [Ati88] Michael Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1988), 175–186 (1989). MR1001453 [BBBBJ15] Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, and Dominic Joyce, A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications, Geom. Topol. 19 (2015), no. 3, 1287–1359. MR3352237 [BBD+15] C. Brav, V. Bussi, D. Dupont, D. Joyce, and B. Szendr˝ oi, Symmetries and stabilization for sheaves of vanishing cycles, J. Singul. 11 (2015), 85–151. With an appendix by J¨ org Sch¨ urmann. MR3353002 [BF08] Roman Bezrukavnikov and Michael Finkelberg, Equivariant Satake category and Kostant-Whittaker reduction (English, with English and Russian summaries), Mosc. Math. J. 8 (2008), no. 1, 39–72, 183. MR2422266 [BFM05] Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirkovi´ c, Equivariant homology and K-theory of affine Grassmannians and Toda lattices, Compos. Math. 141 (2005), no. 3, 746–768. MR2135527 [BFN16] A. Braverman, M. Finkelberg, and H. Nakajima, Coulomb branches of 3d N = 4 quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes), ArXiv e-prints (2016), arXiv:1604.03625 [math.RT]. [BFN17] A. Braverman, M. Finkelberg, and H. Nakajima, Ring objects in the equivariant derived Satake category arising from Coulomb branches, ArXiv e-prints (2017), arXiv:1706.02112 [math.RT]. [BLPW16] Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Quantizations of conical symplectic resolutions II: category O and symplectic duality (English, with English and French summaries), Ast´ erisque 384 (2016), 75–179. with an appendix by I. Losev. MR3594665 [CG97] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkh¨ auser Boston, Inc., Boston, MA, 1997. MR1433132 [Dav13] B. Davison, The critical CoHA of a quiver with potential, ArXiv e-prints (2013), arXiv:1311.7172 [math.AG]. [Gai16] D. Gaiotto, S-duality of boundary conditions and the Geometric Langlands program, ArXiv e-prints (2016), arXiv:1609.09030 [hep-th]. [GR17] V. Ginzburg and N. Rozenblyum, Gaiotto’s Lagrangian subvarieties via derived symplectic geometry, ArXiv e-prints (2017), arXiv:1703.08578 [math.AG]. [KN16] R. Kodera and H. Nakajima, Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras, ArXiv e-prints (2016), arXiv:1608.00875 [math.RT]. [Nak92] Hiraku Nakajima, A convergence theorem for Einstein metrics and the ALE spaces [MR1193019 (93k:53044)], Selected papers on number theory, algebraic geometry, and differential geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 160, Amer. Math. Soc., Providence, RI, 1994, pp. 79–94, DOI 10.1090/trans2/160/06. MR1308542 [Nak16a] H. Nakajima, Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, 第61回代数学シンポジウム報告集 (2016), arXiv:1612.09014 [math.RT]. [Part II]

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Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3dimensional N = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016), no. 3, 595–669. MR3565863 [NT17] Hiraku Nakajima and Yuuya Takayama, Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A, Selecta Math. (N.S.) 23 (2017), no. 4, 2553– 2633. MR3703461 [PTVV13] Tony Pantev, Bertrand To¨ en, Michel Vaqui´ e, and Gabriele Vezzosi, Shifted symplectic ´ structures, Publ. Math. Inst. Hautes Etudes Sci. 117 (2013), 271–328. MR3090262 [VV10] Michela Varagnolo and Eric Vasserot, Double affine Hecke algebras and affine flag manifolds, I, Affine flag manifolds and principal bundles, Trends Math., Birkh¨ auser/Springer Basel AG, Basel, 2010, pp. 233–289. MR3013034

[Nak16b]

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01742

An overview of knot Floer homology Peter Ozsváth and Zoltán Szabó Knot Floer homology is an invariant for knots discovered by the authors [93] and, independently, Jacob Rasmussen [107]. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology [94], changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of knot Floer homology. 1. Motivation for the construction Since the work of Simon Donaldson, gauge theory has emerged as the central tool for understanding differential topology in dimension four. Donaldson’s pioneering work from the 1980’s used the moduli space of solutions to the anti-self-dual Yang-Mills equations – or “instantons” – to construct diffeomorphism invariants of four-dimensional manifolds [14, 16]. Donaldson used these invariants to discover completely unexpected phenomena in four-dimensional topology, including a deep connection between the smooth topology of algebraic surfaces and their algebraic geometry, leading to a number of breakthroughs in the field [15, 20, 27, 35, 41, 53, 57, 114]. A corresponding invariant for three-dimensional manifolds, instanton Floer homology, was introduced by Andreas Floer. Floer’s instanton homology is the homology group of a chain complex whose generators are SU (2) representations of the fundamental group of the three-manifold Y (modulo conjugation), and whose differential counts instantons on R × Y ; see [19, 29]. Floer homology can be used as a tool for computing Donaldson’s invariants [28]. Floer formulated his instanton homology theory as a kind of infinite-dimensional Morse theory, akin to his earlier Lagrangian Floer homology, which is an invariant for a symplectic manifold equipped with a pair of Lagrangian submanifolds [30]; see also [37]. In [1], Michael Atiyah proposed a relationship between these two invariants, which is now known as the “Atiyah-Floer conjecture”. The starting point of this conjecture is a three-manifold equipped with a Heegaard splitting. The “character variety” of the Heegaard surface Σ, which is the space of representations The first author was supported by NSF grant number DMS-1405114. The second author was supported by NSF grant number DMS-1606571. c 2018 American Mathematical Society

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of π1 (Σ) into SU (2) modulo conjugation, is equipped with a pair of Lagrangian subspaces, the spaces of representations that extend over each handlebody. The Atiyah-Floer conjecture states that the Lagrangian Floer homology of these character varieties should agree with the instanton homology of the underlying threemanifold Y ; compare [116]. This statement is still a little vague: SU (2) instanton homology is defined for three-manifolds with H1 (Y ; Z) = 0, and the spaces involved on the symplectic side are singular. Nonetheless, the conjectured relationship has spurred a great deal of mathematical activity; see for example [21, 121]. In 1994, the field of four-manifold topology was revolutionized by the introduction of the Seiberg-Witten equations, a new partial differential equation coming from physics [122]. The moduli spaces of solutions to these equations could be used to construct invariants of smooth four-manifolds, just as the anti-self-dual equations are used in Donaldson’s theory. Many theorems proved earlier using Donaldson’s invariants had easier proofs and generalizations using the newly introduced Seiberg-Witten invariants [17]. The Seiberg-Witten invariants also elucidated the relationship between the differential topology of symplectic manifolds and their symplectic properties, resulting in Clifford Taubes’ celebrated proof that identified the Gromov-Witten invariants of a symplectic manifold with their Seiberg-Witten invariants [117–119]. Considerable work went into formulating a three-dimensional analogue of the Seiberg-Witten invariants. A definitive construction was given by Peter Kronheimer and Tomasz Mrowka in their monograph [54]; see also [36, 74, 78]. Heegaard Floer homology [94] grew out of our attempts to concretely understand the geometric underpinnings of Seiberg-Witten theory. A motivating problem was to find the analogue of the “Atiyah-Floer conjecture”: what Lagrangian Floer construction could be used to recapture the Seiberg-Witten invariants for three-manifolds? A clue was offered by the the following observation: the space of stationary solutions to the (suitably perturbed) Seiberg-Witten equations on R × Σ is identified the moduli space of “vortices” on Σ with some charge d. This space in turn, by early work of Taubes [115], is identified with the d-fold symmetric product of Σ, the space of unordered d-tuples of points in Σ, which we denote Symd (Σ). It was proved in [94] that Heegaard Floer homology is a well-defined threemanifold invariant, enjoying many of the properties of Seiberg-Witten theory. Although Heegaard Floer homology was designed to be isomorphic to invariants derived from the Seiberg-Witten equations, the conjectural equivalence of these two theories was verified many years after their formulation, in the work of Cagatay Kutluhan, Yi-Jen Lee, and Taubes [61]; and Vincent Colin, Paolo Ghiggini, and Ko Honda [10]. Attempts to compute Heegaard Floer homology for Dehn surgeries on knots lead naturally to a new knot invariant, knot Floer homology, discovered independently by Rasmussen [107] and by us [93]. After some discussion of Heegaard Floer homology, we will turn our attention to this knot invariant, recall some of its applications to knot theory, and then focus on recent computational advances in this theory. Acknowledgements. We would like to thank Chuck Livingston, András Stipsicz, and the referee for their many suggestions on an early draft of this paper. The work of Simon Donaldson has had a great impact on our research. Both of our PhD

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theses were based on computing Donaldson’s invariant for four-manifolds; and indeed his theory has served as an inspiration to us ever since. 2. Statement of the symplectic constructions We sketch now the construction of Heegaard Floer homology, and its corresponding knot invariant, following [93]. Before doing this, we recall some topological preliminaries. Let Σ be a surface of genus g. A complete set of attaching circles for Σ is a gtuple of pairwise disjoint, homologically linearly independent simple, closed curves. A complete set of attaching circles specifies a handlebody Uγ whose boundary is identified with Σ, so that the attaching circles bound disjoint, embedded disks in Uγ . A Heegaard splitting of a connected, closed, oriented three-manifold Y is a decomposition of Y as the union of two handlebodies, glued along their boundary. Combinatorially, a Heegaard splitting is specified by a Heegaard diagram, which consists of a triple (Σ, α, β), where Σ is an oriented surface, α = {α1 , . . . , αg } and β = {β1 , . . . , βg } are two complete sets of attaching circles for Σ. Heegaard diagrams can be thought of from the perspective of Morse theory [79,80], as follows. If Y is equipped with a self-indexing Morse function f with a unique maximum and mimum, and a gradient-like vector field v, we can let Σ be f −1 (3/2), and α is the locus of points in Σ that flow out of the index one critical points under v, and β is the locus of points in Σ that flow into the the index two critical points. We will typically work with pointed Heegaard diagrams, which consist of data H = (Σ, α, β, w), where (Σ, α, β) is a Heegaard diagram, and w ∈ Σ is an auxiliary basepoint in Σ that is disjoint from all the αi and the βj . (See Figure 2 for a somewhat complicated Heegaard diagram for S 3 , ignoring the extra basepoint labelled z.) Inside Symg (Σ), there is a pair of g-dimensional tori Tα = α1 × · · · × αg

and

T β = β1 × · · · × βg ;

e.g. Tα is the space of g-tuples of points in Σ, so that each point lies on some αi and no two points lie on the same αi . The basepoint gives rise to a real codimension two submanifold Vw ⊂ Symg (Σ), consisting of those g-tuples of points x that include the point w. The intersection points Tα ∩ Tβ are called Heegaard states for the diagram H , and they are denoted S(H ). Explicitly, if we think of the α- and β-circles as numbered by {1, . . . , g}, then Heegaard states are partitioned according to permutations σ on {1, . . . , g}. The Heegaard states of type σ correspond to points in the Cartesian product (α1 ∩ βσ(1) ) × · · · × (αg ∩ βσ(g) ). A complex structure on Σ naturally induces a complex structure on the g-fold symmetric product Symg (Σ). In fact, the g-fold symmetric product Symg (Σ) can be given a Kähler structure so that the tori Tα and Tβ are Lagrangian [103]. Versions of the Heegaard Floer homology of Y correspond to variants of Lagrangian Floer homology for Tα and Tβ in Symg (Σ), which depend on how one counts pseudoholomorphic disks which interact with the subspace Vw . Specifically, choose an almost-complex structure compatible with the symplectic structure on Symg (Σ), one can consider pseudo-holomorphic disks, as introduced by Gromov [43]. For fixed Heegaard states x and y, the pseudo-holomorphic disks

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in Symg (Σ) connecting x to y can be organized into homotopy classes of maps from the unit disk D in the complex plane to Symg (Σ), u : D → Symg (Σ), satisfying the following boundary conditions: u maps −i to x, i to y, and x + iy = z ∈ ∂D with x ≥ 0 to Tα and x ≤ 0 to Tβ . We denote the space of homotopy classes of such maps by π2 (x, y). Since w is disjoint from the αi and βj , there is a well-defined map nw : π2 (x, y) → Z which is given as the algebraic intersection number of a generic u representing φ ∈ π2 (x, y) with the oriented submanifold Vw . The moduli space of pseudo-holomorphic disks representing the homotopy class φ ∈ π2 (x, y) is denoted M(φ). This admits a natural action by R, thought of as the holomorphic automorphisms of D preserving ±i. The simplest version of Heegaard Floer homology is the homology of a chain : H ), thought of as a vector space over the field F with two elements. complex CF( Generators of this chain complex are the Heegaard states, and its differential counts : H ) is the pseudo-holomorphic disks that are disjoint from Vw ; more formally, CF( vector space generated S(H ), equipped with the differential  ∂(x) =







#

M(φ) R

y∈S {φ∈π2 (x,y)|nw (φ)=0,μ(φ)=1}

 · y.

Here, μ(φ) is the Maslov index of the homotopy class φ [37, 109]; see [64] for a very useful formulation in terms of the Heegaard diagram. As is standard in Floer theory [33,37], to make sense of the definition, the ∂-equations need to be perturbed suitably to ensure that the moduli spaces appearing above are smooth. This chain complex has a refinement CF− (H ), which is a module over the polynomial algebra F[U ] over a formal variable U , whose differential is defined by −

∂ (x) =





y∈S {φ∈π2 (x,y)|μ(φ)=1}

 #

M(φ) R

 · U nw (φ) y.

: H ). (Both complexes can in The U = 0 specialization of this chain complex is CF( fact be defined over Z coefficients; see [94].) : H )) The main theorem of [94] states that the homology of CF− (H ) (and CF( is an invariant of the underlying closed, oriented three-manifold Y represented by H. We will be concerned here with knot Floer homology [93, 107], a variant of Heegaard Floer homology. For this version, start with a doubly-pointed Heegaard diagram H = (Σ, α, β, w, z), where here the two basepoints w and z in Σ are both chosen to be disjoint from the αi and the βi for i = 1, . . . , g. This data specifies an oriented knot inside the three-manifold Y defined by the Heegaard diagram (Σ, α, β). The knot is constructed by the following procedure. Connect w to z in Σ by an arc a that is disjoint from the α-curves, and push the interior of resulting arc into the α-handlebody; similarly, connect w and z by another arc in Σ that is disjoint from the β-curves and push the interior of that into the β-handlebody to get b. The knot K is obtained as a ∪ b. It can be oriented by the convention that ∂a = z − w = −∂b. The simplest version of knot Floer homology is the homology of a chain complex H ), once again generated by Heegaard states (in a doubly-pointed Heegaard CFK(

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diagram H representing K), with differential given by     M(φ) ∂K (x) = # y. R y∈S {φ∈π2 (x,y)|nw (φ)=0=nz (φ),μ(φ)=1}

For simplicity, we hereafter restrict attention to the case where the ambient threemanifold is S 3 . Dropping the requirement that nz (φ) = 0 gives the chain complex : 3 ), which is a one-dimensional vector space. The knot whose homology is HF(S Floer complex is equipped with two gradings, the Maslov grading and the Alexander grading, induced by functions M : S(H ) → Z

and

A : S(H ) → Z

that are characterized as follows. The function M satisfies the property that if x and y are any two Heegaard states, and φ ∈ π2 (x, y) is a homotopy class of Whitney disks, then M (x) − M (y) = μ(φ) − 2nw (φ). This specifies M uniquely up to an overall additive constant. The function M : 3 ) for which the differential ∂ drops grading by induces a Z-valued grading on CF(S : 3) ∼ one; thus there is an induced grading on HF(S = F. The additive indeterminacy : 3 ) is supported in Maslov grading on M is pinned down by requiring that HF(S equal to zero. The function A satisfies the property that if x and y are any two Heegaard states, and φ ∈ π2 (x, y) is a homotopy class of Whitney disks, then A(x) − A(y) = nz (φ) − nw (φ). Once again, this specifies A up to an overall additive constant; and the differential H ) specified by the Alexander ∂K preserves the corresponding splitting of CFK( grading. H ), The Maslov and Alexander functions induce a bigrading on CFK(  d (H , s), H) = CFK CFK( d,s∈Z

d (H , s) is generated by those states x with M (x) = d and A(x) = s. where CFK The differential satisfies d (H , s) → CFK d−1 (H , s), ∂K : CFK and therefore the bigrading descends to homology  d (H , s). H) = HFK HFK( d,s∈Z

H ) has a graded Euler characteristic, which The bigraded chain complex CFK( is a Laurent polynomial in a formal variable t with integral coefficients, defined by   d (K, s)ts = d (K, s)ts . (−1)d dim CFK (−1)d dim HFK χ(CFK(K)) = d,s

d,s

This graded Euler characteristic coincides with the Alexander polynomial ΔK (t): . (2.1) χ(CFK(K)) = ΔK (t),

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+ L

− L

0 L

Figure 1. Crossing conventions in the skein relation. . where here = means that the two polynomials agree up to overall factors of t [93]. The additive indeterminacy in A is eliminated by requiring that the graded Euler characteristic is symmetric in t; i.e. Equation (2.1) holds with equality in place of . =. The information in the bigraded vector space HFK(K) is encoded in its Poincaré polynomial, a polynomial with non-negative integral coefficients in two formal variables q and t, defined by  d (K, s)q d ts . dim HFK PK (q, t) = d,s

Specializing PK to q = −1 gives the graded Euler characteristic; i.e. PK (−1, t) = ΔK (t). The construction described above is analytic in nature: the generators are combinatorial, but differentials count pseudo-holomorphic disks. Knot Floer homology has a number of different, more computationally approachable formulations. We will return to this point, but first, we outline some properties and applications of the invariant. 3. First properties We describe now some basic properties of knot Floer homology, contrasting them with corresponding properties for the Alexander polynomial. Suppose that K+ and K− are two knots with projections that differ in exactly one crossing, as shown in the first two pictures of Figure 1. Then, we can resolve the crossing to obtain a new oriented link with two components, the third picture in that figure.  − , and L  0 are three oriented links that differ as in that  +, L More generally, if L figure, we say that they form a skein triple. The Alexander polynomial for knots can be extended to oriented links, and that extension satisfies the following skein  +, L  −, L  0: relation for any skein triple L 1/2 ΔL − t−1/2 )ΔL

+ (t) − ΔL

− (t) = (t

0 (t).

This relation gives an inductive procedure for computing the Alexander polynomial. In fact, it was observed by John Conway [11] that the Alexander polynomial (for oriented links) is uniquely characterized by the above skein relation, and the normalization for the unknot U, which states that Δ U (t) is the constant polynomial 1. Extending knot Floer homology to links, the skein relation has the following analogue:  − , and L  0 are three oriented links that fit into a skein  +, L Theorem 3.1. If L triple, there is a corresponding exact triangle relating their bigraded knot Floer

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 + belong to the same component of homologies. When the two strands meeting at L  L+ , the triangle has the form  +) L HFK(

 −) L HFK(

 0) L HFK(  + belong to different components, there is a similar trianWhen the strands in L  0 ) is tensored with an appropriately graded four-dimensional L gle, except that HFK( bigraded vector space. See [93] and [86, Chapter 9] for a precise statement (with specified bigradings). The above result should be compared with Floer’s exact triangle for instanton homology; see [6, 31, 32]. The Alexander polynomial is multiplicative under connected sum. This has the following generalization to the case of knot Floer homology: 1 #K2 ) is obtained Proposition 3.2. If K1 and K2 are two knots, then HFK(K 1 ) and HFK(K 2 ); as the graded tensor product of the bigraded vector spaces HFK(K i.e. PK1 #K2 (q, t) = PK1 (q, t) · PK2 (q, t) Recall that a knot is called alternating if it has a diagram with the property that crossings alternate between over- and under-crossings as one follows the projection. By a classical theorem of Cromwell and Murasugi [12, 82], the Alexander polynomial of an alternating knot is special: its coefficients alternate in sign. This has the following analogue for knot Floer homology [90]; see also [75, 106]. Theorem 3.3. If K is an alternating knot, then the knot Floer homology for K is determined by its Alexander polynomial ΔK (t) and its signature σ(K), by the formula σ PK (q, t) = q 2 · ΔK (qt). Knot Floer homology can be given more algebraic structure. For example, there is a version which is a free chain complex over F[U ], CFK− (H ), with differential     M(φ) − ∂K (x) = # U nw (φ) y. R y∈S {φ∈π2 (x,y)|nz (φ)=0,μ(φ)=1}

Extending the Maslov and Alexander grading so that multiplication by U drops Maslov grading by 2 and Alexander grading by 1, we have that − − ∂K : CFK− d (H , s) → CFKd−1 (H , s)

− U : CFK− d (H , s) → CFKd−2 (H , s − 1).

Thus, the homology HFK− (H ) inherits the structure of a bigraded F[U ]-module. Proposition 3.4. The bigraded module HFK− (K) is finitely generated; in fact, it consists of direct summands of the form F[U ]/U m for various choices of m, and a single free summand F[U ]. The above proposition is clear from the original definitions of knot Floer homology [93, 107]; see [86, Chapter 7] for a more precise reference.

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Proposition 3.4 shows that HFK− (K) always contains U -non-torsion elements, i.e. elements ξ ∈ HFK− (K) with U m · ξ = 0 for all m. Thus, there is a natural numerical invariant of knots K, denoted τ (K) which is defined as −1 times the maximal Alexander grading of any non-torsion class ξ ∈ HFK− (K). 4. Topological applications Knot Floer homology was originally envisioned as a tool for computing the Heegaard Floer homology groups of three-manifolds obtained as surgeries on a given knot. If K is a knot in S 3 , a “surgery formula” expresses the Heegaard Floer : for three-manifolds obtained as surgeries on K in terms of another homology HF variant of knot Floer homology HFK(K) defined over the ring R = F[U, V ]/U V = 0. This knot invariant is the homology of a chain complex, CFK(H ), which is freely generated (over R) by Heegaard states, and whose differential is given by     M(φ) (4.1) ∂(x) = # · U nw (φ) V nz (φ) y. R y∈S {φ∈π2 (x,y)|μ(φ)=1}

As in the case of HFK− , the homology module inherits a bigrading: in the present case, U drops Alexander grading by one, and V raises it by one. We do not state the surgery formula here (see [98, 99]), but we do give a consequence: Theorem 4.1. [96] Suppose that K is a knot with the property that for some rational number r ∈ Q, the three-manifold Sr3 (K) obtained by Dehn surgery along K with coefficient r is a lens space. Then, all the coefficients of the Alexander polynomial are ±1 or 0; in fact, the non-zero ones alternate in sign. Thus, we can write n  (−1)k tαk ΔK (t) = k=0

where {αk }nk=0 is a decreasing sequence of integers. Moreover, when r > 0, the knot Floer homology of K is determined by this Alexander polynomial, as follows. There is a sequence of integers {mk }nk=0 determined by the formulae: m0 = 0 m2k = m2k−1 − 1 m2k+1 = m2k − 2(α2k − α2k+1 ) + 1, so that PK (q, t) =

n 

q mk tαk .

k=0

∗ (K, s) has dimension 0 or 1. In particular, for each s ∈ Z, HFK Knots that satisfy the hypothesis of Theorem 4.1 include all torus knots. Recall that any knot K ⊂ S 3 can be realized as the boundary of a compact, orientable surface F embedded in S 3 . Such a surface is called a Seifert surface for K, and the minimal genus of any Seifert surface for K is called the Seifert genus of K. It is a classical result that the degree of the Alexander polynomial gives a lower bound for the Seifert genus of a knot. This result has a sharpening for knot Floer homology, which is inspired by work of Kronheimer and Mrowka [58]:

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Theorem 4.2. [92] Knot Floer homology detects the Seifert genus g(K) of a knot K, in the sense that / ∗ (K, s) = 0}. g(K) = min{s/HFK Our original proof of the above theorem relied on many results in low-dimensional topology. To start with, Gabai’s theory of sutured manifolds equips the zero-surgery with a taut foliation [38]. A theorem of Eliashberg and Thurston [24] provides the product [−1, 1] × S03 (K) with a symplectic structure, which is suitably convex at the boundary. A theorem of Eliashberg [23] and Etnyre [25] embeds this symplectic cylinder in a closed symplectic manifold X. Donaldson’s Lefschetz pencils [18] on symplectic manifolds then provides a suitable two-handle decomposition on X for which we can prove that the Heegaard Floer four-manifold invariant is non-zero [95]. A surgery formula relating knot Floer homology with the Heegaard Floer homology of the 0-surgery then gives the required non-vanishing theorem for knot Floer homology. Juhász has an elegant proof of the above result [46] that bypasses most of the above machinery (still building on Gabai’s sutured hierarchy), using his sutured Floer homology [45]. Theorem 4.2 has the following corollary: Corollary 4.3. [92] Knot Floer homology detects the unknot, in the sense that HFK(K) has dimension one if and only if K is the unknot. The above corollary underscores how far knot Floer homology goes beyond the Alexander polynomial: there are infinitely many knots with trivial Alexander polynomial. Theorem 4.1 has a more precise statement, which expresses the sequence {αk } concretely in terms of the surgery coefficient r and the resulting lens space L(p, q) [96]. In [92], we combine this result with Corollary 4.3, to obtain the following result, first proved using Seiberg-Witten theory in our joint work with Kronheimer and Mrowka: Corollary 4.4. [60] If K ⊂ S 3 is a knot with the property that some Dehn surgery along K is homeomorphic to RP3 , then K is the unknot. See [42] for a vast generalization. A final property of knot Floer homology motivated by the Alexander polynomial is based on the classical result that the Alexander polynomial of a fibered knot is monic. This has an analogue for knot Floer homology: if K is a fibered knot with Seifert genus g = g(K), then  d (K, g) ∗ (K, g) = HFK HFK d∈Z

is one-dimensional [95]. This fact has the following remarkable converse, due to Paolo Ghiggini when g = 1 and Yi Ni when g > 1: ∗ (K, g) is one-dimensional, then K is fibered. Theorem 4.5. [39, 84] If HFK See also [46]. Ni’s theorem, combined with Theorem 4.1, immediately gives the following: Corollary 4.6. [84] If K ⊂ S 3 is a knot with the property that Sr3 (K) is a lens space, then K is fibered.

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So far, we have focused on applications on the simplest variant of knot Floer homology, HFK(K). The version HFK− (K), with its module structure over F[U ], has further applications to the unknotting number and the slice genus of a knot, which we recall here. Thinking of S 3 as a boundary of the four-ball B 4 , one can consider slice surfaces: smoothly embedded, compact, orientable surfaces in B 4 , so that F \ ∂F is mapped to B 4 \ ∂B 4 , and ∂F is mapped to K ⊂ S 3 = ∂B 4 . The slice genus of a knot K, denoted g4 (K), is the minimal genus of any slice surface for K. A knot is called a slice knot if its slice genus is 0. Clearly, the Seifert genus of K bounds the slice genus of K: g4 (K) ≤ g(K). An unknotting of K is a sequence of knots K = K0 , K1 , . . . , Kn , where Ki is obtained from Ki−1 by changing one crossing, so that Kn is the unknot. The unknotting number of K, denoted u(K), is the minimal length of any unknotting for K. An n-step unknotting for K naturally gives rise to an immersed surface in B 4 with n double points. Resolving these double-points, we can find a slice surface for K with genus n. This proves the bound g4 (K) ≤ u(K). The module structure HFK− (K), and specifically the associated integral invariant τ , gives a lower bound on the slice genus according to the following: Theorem 4.7. For any knot K ⊂ S 3 , |τ (K)| ≤ g4 (K). The above is proved in [91]; see also [107] for other similar bounds. Sucharit Sarkar gave a combinatorial proof of Theorem 4.7 from the perspective of “grid diagrams”; see [110] and [86, Chapter 8]. . Direct computation shows that for the (p, q) torus knot Tp,q , τ (Tp,q ) = (p−1)(q−1) 2 Thus, Theorem 4.7 gives another verification of following theorem of Kronheimer and Mrowka, first conjectured by Milnor [81]: Theorem 4.8. [56] For relatively prime integers p and q, the torus knot Tp,q has (p − 1)(q − 1) . u(Tp,q ) = g4 (Tp,q ) = 2 It is easy to see that the quantity appearing in the above theorem also coincides with the Seifert genus of Tp,q . Kronheimer and Mrowka’s proof of the above theorem used Donaldson invariants. A number of alternative proofs have emerged since. Rasmussen [108] gave the first combinatorial proof, using the algebraic structure on Khovanov’s knot invariants; compare [110]. There are non-orientable analogues of the slice genus, defined as follows. Consider possibly non-orientable surfaces F embedded in B 4 , meeting S 3 along K, and let γ4 (K), the non-orientable 4-genus of K denote the minimal complexity, as measured by the dimension of H1 (F ; F), for all such choices of F . For example, the torus knot T2,2n+1 bounds a n + 12 -twisted Möbius strip, so γ4 (T2,2n+1 ) = 1. Prior to 2012, the best lower bound on γ4 for any knot was 3. The situation was vastly improved by the following theorem of Joshua Batson: Theorem 4.9. [4] The non-orientable 4-genus can be arbitrarily large; for example, γ4 (T2k,2k−1 ) = k − 1. Batson’s proof goes by constructing an explicit surface with stated complexity, to give an upper bound on γ4 (T2k,2k−1 ). Next, he gives a lower bound on γ4 (T2k,2k−1 ) via a Heegaard Floer invariant associated to surgeries on the knot. An alternative proof of the above theorem is given in joint work of András Stipsicz and the authors [85], using another variant of knot Floer homology. This

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version is the homology of a chain complex CFK (H ) which, like CFK− (H ), is freely generated over F[U ] by the Heegaard states; but it is equipped with a differential     M(φ)  # ∂K x = U nw (φ)+nz (φ) y. R / y∈S {φ∈π2 (x,y)/μ(φ)=1} This complex is equipped with the single grading δ(x) = M (x) − A(x). It is  drops the δ-grading by 1, as does multiplication straightforward to check that ∂K by U . Proposition 3.4 has the following analogue: Proposition 4.10. [85, Proposition 3.5] The bigraded module HFK (K) is finitely generated; in fact, it consists of direct summands of the form F[U ]/U m for various choices of m, and a single free summand F[U ]. We can now define υ(K) to be the maximal δ-grading of any U -non-torsion element in HFK (K). Theorem 4.9 can be proved via a computation of υ(T2k,2k−1 ), combined with the following bound on the non-orientable 4-genus in terms of υ, analogous to Theorem 4.7: Theorem 4.11. [85] For any knot K ⊂ S 3 , |υ(K) − σ(K) 2 | ≤ γ4 (K). Analyzing the slice genus is a place where smooth four-dimensional topology has a clear interaction with knot theory. The slice surfaces whose genus is minimized in the definition are thought of as smoothly embedded in B 4 . Relaxing this requirement, we could ask for locally flat, topologically embedded surfaces, to obtain an analogous numerical knot invariant, called the topological slice genus, g4top (K). Correspondingly, K is said to be topologically slice if g4top (K) = 0. In a related direction, one can say that two knots K1 and K2 are concordant if there is an embedded annulus F in [1, 2] × S 3 so that F ∩ ({i} × S 3 ) is the knot Ki ; or, equivalently, if K1 #m(K2 ) is a slice knot, where here m(K) denotes the mirror of K. The connected sum operation endows this set with the structure of an Abelian group, called the smooth concordance group C. If we require the annulus to be only topologically embedded, or equivalently, if we require K1 #m(K2 ) to be only topologically slice, we obtain another group, the topological concordance group, denoted Ctop . There is a canonical homomorphism C → Ctop , whose kernel is the subgroup of topologically slice knots, CT S . Tristram [120] showed that Ctop contains a direct summand isomorphic to Z∞ ; see also [63, 70]. According to a theorem of Freedman, any knot with ΔK (t) = 1 is topologically slice. Using Donaldson’s diagonalizability theorem, Andrew Casson showed that CT S is non-trivial; see [9]. In fact, the 0-twisted Whitehead double of the trefoil, a knot for which τ (K) = 1, gives a Z-direct summand in CT S [71]. Using gauge theory, in 1995 Endo exhibited a Z∞ subgroup of CT S . In 2012, Jen Hom [44] went further, exhibiting a Z∞ direct summand of CT S by constructing infinitely many linearly independent concordance homomorphisms from CT S to Z. Her construction uses an invariant , which can be viewed as derived from knot Floer homology HFK(K) over F[U, V ]/U V . Using , she introduces an equivalence relation on the knot Floer complexes, to form a totally ordered Abelian group. The homomorphisms are then provided by the axiom of choice. In joint work with Stipsicz [101], we constructed another collection of homomorphisms to Z, using a one-parameter deformation of knot Floer homology tHFK(K). Specifically, for each rational t = pq ∈ [0, 2], there is a chain complex tCFK(K), freely generated over F[v 1/q ] by Heegaard states, whose differential now

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has the form t ∂K x=





/ y∈S {φ∈π2 (x,y)/μ(φ)=1}

 #

M(φ)



R

v tnw (φ)+(2−t)nz (φ) y.

This complex is graded by grt (x) = M (x) − tA(x), so that multiplication by v drops grading by 1. When t = 0, the complex is independent of the knot, and its homology is simply F[v]. When t = 1, the complex is CFK considered above. Define ΥK (t) to be the maximal grading of any v-non-torsion element; in particular, υ(K) = ΥK (1). Like Hom’s homomorphisms, Υ detects Z∞ direct summands of CT S [101]; see [72] for an alternative formulation of the invariant Υ and see [7, 26] for further developments. 5. Heegaard diagrams To understand knot Floer homology, it is useful to have several possible Heegaard diagrams in hand. The first Heegaard diagram, which we will call the standard diagram for a knot projection, is determined as follows. 5.1. The standard diagram for a knot projection. Fix a knot projection

D for K in R2 , together with a distinguished edge adjoining the infinite region in

the projection complement. The edge is distinguished by placing a star somewhere on the edge, as shown on the left in Figure 2. We call this data a decorated knot projection of K. To a decorated knot projection, we can associate a Heegaard diagram representing K, as follows. First, singularize the projection, so that the crossings are actually double-points. Next, take a regular neighborhood of the resulting planar graph G, to obtain a handlebody H embedded in R3 ⊂ S 3 . The regions in the complement of the graph in the plane have two distinguished regions that adjoin the marked edge, one of which is the infinite region in R2 . For each bounded region in the graph complement, there is a corresponding α-circle. In a neighborhood of each crossing, we associate a β-circle as pictured in Figure 2. Near the marking on the distinguished edge, we choose also a final meridional β-circle, again as shown in Figure 2, and place the basepoint w and z on either side of it. Note that Σ is oriented as −∂H. We recall here Kauffman’s construction of the Alexander polynomial [47].

α1

β2 β1 α3 α4 z

w

β3

α2

β4

Figure 2. Doubly-pointed Heegaard diagram for the lefthanded trefoil.

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Definition 5.1. A Kauffman state for a decorated knot projection of K is a map κ that associates to each vertex of G one of the four adjacent quadrants, subject to the following constraints: • The quadrants assigned by κ to distinct vertices are subsets of distinct bounded regions in R2 \ G. • The quadrants of the bounded region that meets the distinguished edge are not assigned by κ to any of the vertices in G. See Figure 3 for examples.

Figure 3. Kauffman states for the left-handed trefoil. Here all three of the states for this projection. Definition 5.2. Label the four quadrants about each crossing with 0, and ± 12 , according to the orientations as specified in the first line of Figure 4. The Alexander function of a Kauffman state κ, A(κ), is a sum, over each crossing, of the contribution of the quadrant occupied by the state. The Maslov function of a Kauffman state κ is obtained similarly, only now the local contributions are as specified in the second line of Figure 4. − 12 0

1 2

0

0

0

1 2

− 12

−1

1

0

0

0

0 0

0

Figure 4. Local Alexander and Maslov contributions. The first row illustrates the local Alexander contributions, and the second the local Maslov contributions of each crossing. Let S = S( D) denote the set of Kauffman states. Kauffman shows that the Alexander polynomial is computed by  (−1)M (x) tA(x) . ΔK (t) = x∈S

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(Note that Kauffman does not define M (x), which is not needed for the Alexander polynomial, only its parity.) To put Kauffman states in even more familiar territory, recall that a knot projection can be given a checkerboard coloring, coloring each region on the graph complement black and white so that the two regions meeting along each edge are colored differently. There is a planar graph, the “black graph”, whose vertices correspond to the black regions in the checkerboard coloring, and whose edges correspond to crossings in the decorated knot diagram. There is a straightforward one-to-one correspondence between maximal subtrees in the black graph and Kauffman states; see [47]. The relevance of Kauffman states to knot Floer homology is the following observation from [90]: the Heegaard states in the standard diagram for a knot projection correspond to the Kauffman states of the marked projection, via a correspondence which identifies the corresponding Maslov and Alexander functions. Although this information is insufficient to compute knot Floer homology, since the differentials count pseudo-holomorphic disks, it does give computations in many cases. For example, an elementary argument shows that for an alternating diagram, A(x)−M (x) is independent of the Kauffman state x. A little more work shows that for an alternating knot, M (x) − A(x) = σ(K) 2 . Theorem 3.3 is an immediate consequence of these considerations. Eun-Soo Lee [62] has shown that a corresponding result also holds for Khovanov homology [3, 49]. 5.2. (1, 1) diagrams. In knot theory, a knot is said to have a type (g, b) representation if there is a genus g Heegaard splitting in which the knot meets each of the two handlebodies as a union of b unknotted arcs [13]. Thus, the doubly-pointed Heegaard diagrams described above give type (g, 1) representations of knots. There is a class of knots for which the Heegaard Floer homology is particularly easy to compute, which can be represented on the torus, equipped with two basepoints; i.e. which have representations of type (1, 1). In particular, suppose that Σ is a surface of genus 1, equipped with two basepoints w and z, and two curves α and β which are isotopic (via an isotopy that crosses w and z) to two curves α and β  that meet transversely in a single intersection point. This gives a knot in S 3 . Knots with such representations include all torus knots and all 2-bridge knots (knots on which there is a height function with 4 critical points: 2 maxima and 2 minima); see [8] for a classification. For a (1, 1) knot, the Heegaard Floer homology takes place in the first symmetric product of the torus Σ, i.e. within Σ itself. Thus, the holomorphic disk counts are combinatorial; see [40, 93]. 5.3. Grid diagrams. A Heegaard diagram representing a (g, b) decomposition can be represented by a genus g Heegaard surface Σ equipped now with g+b−1 α-curves and g + b − 1 β-curves and 2b basepoints w1 , . . . , wb and z1 , . . . , zb . The α-curves are required to be pairwise disjoint, and to span a half-dimensional subspace of H1 (Σ); the β-curves are required to satisfy the same property. By our homological conditions, the surface obtained by cutting Σ along the α-curves has b connected components. Our diagrams will satisfy the following additional property: each of these connected components is required to have exactly one w-basepoint and one z-basepoint. Cutting Σ along the β-curves gives b components, and each component is required to have exactly one w-basepoint and one z-basepoint. This

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data specifies a three-manifold Y by the natural generalization of the earlier construction: attach three-dimensional two handles to [−1, 1] × Σ along {−1} × αi and {1} × βj . We are left with a three-manifold whose boundary consists of a collection of two-spheres. Fill each two-sphere with a three-ball. We can construct an oriented link in Y that meets Σ in w ∪ z, by the following construction. In each component of Σ\(α1 ∪· · ·∪αg+b−1 ), find an arc that connects the corresponding wi and zj , and push that arc into [−1, 0] × Σ, so that it meets {0} × Σ exactly at wi and zj . Find corresponding arcs in Σ \ β, and push those into [0, 1] × Σ. The two types of basepoints give one-to-one correspondences fw : π0 (Σ \ α) → π0 (Σ \ β)

and

fz : π0 (Σ \ α) → π0 (Σ \ β);

so fz−1 ◦ fw is a permutation of π0 (Σ \ α). That permutation can be written as a product of cycles; and the number of cycles in the description gives the number of components of the resulting link. Heegaard Floer homology has a generalization to this construction, as well. The ambient symplectic manifold now is Symg+b−1 (Σ), equipped with two g + b − 1dimensional tori Tα and Tβ . The chain complex CFK− (H ) now is defined over the polynomial algebra F[U1 , . . . , Ub ], with differential given by (5.1)     M(φ) nw (φ) n (φ) ∂ − (x) = # U1 w1 · · · Ub b y. R y∈S {φ∈π2 (x,y)|μ(φ)=1,nz1 (φ)=···=nzb (φ)=0}

When the multiply-pointed Heegaard diagram represents a knot K, then all of the Ui variables act the same in homology, and the resulting F[U ]-module is isomorphic to the bigraded knot Floer homology HFK− (K) described earlier; cf. [76, 97]. This observation is especially powerful for a particular class of Heegaard diagrams called grid diagrams, where Σ has genus 1, all of the α-curves are parallel (i.e. isotopic to one another), and all the β-curves are parallel. It is a classical result that every knot in S 3 has such a diagram: indeed, a projection for a knot with c crossings can be turned into a grid diagram for K with b = c + 2 α-curves and β-curves. Moreover, these diagrams are also “nice” in the sense introduced by Sucharit Sarkar. Sarkar showed that for certain Heegaard diagrams, the holomorphic disk counts appearing in the Heegaard Floer differential have an explicit, topological formulation [111]. The key result of Ciprian Manolescu, Sucharit Sarkar, and the first author in [76] states that the holomorphic disk counts appearing in Equation (5.1) for grid diagrams is a combinatorial count of certain embedded rectangles in the Heegaard torus. In [2], these techniques are used to compute the knot Floer homology groups of knots with ≤ 12 crossings. The resulting chain complex, whose generators correspond to permutations and whose differential counts embedded rectangles, can be taken as a definition for the theory rather than a computation. Invariance can be formulated and proved within the realm of grid diagrams: there is a well understood set of moves that connect any two grid diagrams representing the same knot [12, 22]. One can construct isomorphisms between the corresponding “grid homology groups”, to show that the result is a knot invariant. This is the approach taken in [77]; see also [86]. The basic setup of grid homology requires little machinery: gone are the pseudoholomorphic curves, replaced instead by embedded rectangles. This makes the material perhaps more accessible to students trying to enter the subject. The perspective offered by grids naturally points to further applications, especially to

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Legendrian knot theory [83, 102]; see also [86, Chapter 12]. Moreover, some of the topological applications have proofs purely within the framework of grid diagrams. As pointed out earlier, the slice genus bounds have a combinatorial formulation (see Theorems 4.7 and 4.8 above). Some non-orientable 4-genus bounds (see Theorem 4.9) have combinatorial proofs [85]. Working entirely in the world of grid diagrams does have some disadvantages, though. At present, many of the topological applications cannot be understood from the grid perspective. More frustratingly, the chain complexes associated to grid diagrams tend to be large and unwieldy. For a knot represented by an n × n grid diagram, the grid chain complex has n! generators. Much has been written on the topic of grid diagrams, so we refer the interested reader to the above references. We will focus instead on a different more algebraic computational approach [87, 88, 100], motivated by “bordered Floer homology” [69]. 6. Bordered preliminaries Bordered Floer homology is an invariant for three-manifolds with boundary introduced in 2008, by Robert Lipshitz, Dylan Thurston, and the first author [66,69]. This theory associates a differential graded algebra A(F ) to a surface F equipped with a parameterization. To an oriented three-manifold Y1 , equipped with an identification F ∼ = ∂Y1 , the bordered theory associates an A∞ module over this algebra, 1 ). For an oriented three-manifold Y2 whose boundary is identified denoted CFA(Y with −F, the theory associates an algebraic object, called a “type D structure” 2 ), over A(F ), which can be thought of as a kind of free differential module CFD(Y over A(F ). The module operations are defined by certain pseudo-holomorphic disks occurring in naturally adapted Heegaard diagrams that represent bordered threemanifolds. We recall here some of the formal aspects of this theory, as they serve as a motivation for some algebraic constructions for knot Floer homology which we will describe later. As a preliminary point, recall that a differential graded algebra A is a graded vector space A equipped with an associative multiplication and a differential, which are compatible by the Leibniz rule d(a · b) = (da) · b + a · (db). We suppress signs here, as we are working with coefficients in Z/2Z. Sometimes the differential and the multiplication are denoted by the more uniform notation μ1 : A → A

and

μ2 : A ⊗ A → A .

Then, the structure relations are μ1 ◦ μ1 = 0, μ2 (μ2 (a, b), c) + μ2 (a, μ2 (b, c)) = 0 (associativity), and μ1 (μ2 (a, b)) = μ2 (μ1 (a), b) + μ2 (a, μ1 (b)). Differential graded algebras have a natural generalization, A∞ algebras [48], which are graded vector spaces A equipped with a sequence of maps {μn : A⊗n → A}∞ n=1 , satisfying an infinite sequence of structure relations (generalizing the three structure relations for differential graded algebras stated above), called the A∞ relations. To state these, it is useful to think of planar trees T , with k inputs and one output. Each such tree gives rise to a map μ(T ) : A⊗k → A, where each vertex with valence d is labelled by the operation μd−1 . The A∞ relation with k inputs, states the sum

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Figure 5. The A∞ relation with four inputs. The sum of the operations associated to these trees vanishes; for example, the tree on the top left contributes μ2 (μ3 (a1 , a2 , a3 ), a4 ). of μ(T ), taken over all trees T with k inputs and exactly two internal vertices, vanishes. For example, there is a single tree with two internal vertices: it is the linear tree with two valence two vertices. So the A∞ relation in this case states that μ1 ◦ μ1 = 0. A more interesting example is shown in Figure 5. From this perspective, a differential graded algebra is an A∞ algebra with μn = 0 for all n ≥ 3. The A∞ relations can alternatively be formalized as follows. Consider the “bar complex”, the vector space ∞  Bar( A) = A⊗i , i=1

equipped with the endomorphism (6.1) ∂(a1 ⊗ · · · ⊗ an )  (a1 ⊗ · · · ⊗ ar ) ⊗ μs (ar+1 ⊗ · · · ⊗ ar+s ) ⊗ (ar+s+1 ⊗ · · · ⊗ an ). = r≥0,s>0,r+s≤n

The A∞ relation is equivalent to the condition that ∂ ◦ ∂ = 0. Over a differential graded algebra A, it is natural to consider differential graded modules N , which are equipped with a differential m1 : N → N , and an associative action m2 : N ⊗ A → N . These objects have a natural A∞ generalizations: an A∞ module N is a graded vector space equipped with a sequence of maps (6.2)

{mn : N ⊗ A⊗(n−1) → N }∞ n=1 .

Again, these are required to satisfy an A∞ relation, which is exactly as in the case for algebras, with the understanding that now all trees T have a distinguished leftmost strand (corresponding to N ), along which all vertices are labelled with mi , rather than μi which labels all other vertices. Stasheff [113] introduced A∞ algebras in his study of algebraic topology for Hspaces. They have since resurfaced in a number of settings: for example, they have taken a central role in symplectic geometry [52, 112]; they have found applications in gauge theory [5,55,60]; and of course they are also at the heart of bordered Floer homology. Although we will not need A∞ algebras in our subsequent discussions, we will be considering A∞ modules over differential graded algebras.

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Next, we recall the notation of a D structure over a differential graded algebra

A, which is another key player in the bordered theory. A type D structure is a graded vector space X, equipped with a map δ1 : X → A ⊗ X

(6.3) satisfying a structure equation

(μ2 ⊗ IdX ) ◦ (IdA ⊗δ 1 ) ◦ δ 1 + (μ1 ⊗ IdX ) ◦ δ 1 = 0. More concretely, if X has a basis {xi }ni=1 , we can write δ 1 (xi ) =

n 

ai,j ⊗ xj ,

j=1

for ai,j ∈ A. The structure relation takes the form  ai,j · aj,k = 0. dai,k + j

There is a natural pairing between A∞ modules N and type D structures X over A [69], denoted N  X, defined as follows. Iterate δ 1 to define a map δ j : X → A⊗j ⊗ X. More precisely, define δ j inductively by δ 0 = IdX , and δ j = (IdA ⊗(j−1) ⊗δ 1 ) ◦ δ j−1 for j > 0; e.g. δ 2 (x) = (IdA ⊗δ 1 ) ◦ δ 1 δ 3 (x) = (IdA ⊗A ⊗δ 1 ) ◦ (IdA ⊗δ 1 ) ◦ δ 1 . Equip the vector space N ⊗ X with the endomorphism ∞  (mj+1 ⊗ IdX ) ◦ (p ⊗ δ j (x)). D(p ⊗ x) = j=0

In general, the sum defining D may not be finite; but there are some instances where it is. For example, the module N is said to be algebraically bounded if mj = 0 for all j sufficiently large; and a type D structure X is said to be algebraically bounded if δ j = 0 for all j sufficiently large. Boundedness of either structure is sufficient to ensure finite sums in the definition of D. In cases where D is well-defined, D2 = 0; i.e. (N ⊗ X, D) is a chain complex. This chain complex is denoted N X, and it agrees with the derived tensor product of the two A∞ modules underlying N and A  X; see [69]. A key property of bordered Floer homology is a pairing theorem, which, for : ) in terms of a three-manifold Y decomposed as Y = Y1 ∪F Y2 , expresses HF(Y the above pairing between the type D and the type A structures of the pieces, : ) * H(CFA(Y 2 )). 1 )  CFD(Y HF(Y Bimodules have a natural generalization to the A∞ setting. Informally, if A1 and A2 are differential graded algebras, a type DA bimodule A1 XA2 is an object which can be viewed as a type D structure over A1 , but it also has higher operations 1 : X ⊗ A2⊗i → A1 ⊗ X, δi+1

satisfying an appropriate A∞ relation [68].

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Bimodules play the following role in the bordered theory. Recall that modules associated to a three-manifold depend on the boundary parameterization. To each mapping class φ : F → F there is a corresponding bimodule with the property that if Y1 is obtained by composing the boundary parameterization of Y1 with φ, then 1 ) is the tensor product of CFD(Y 1 ) with the associated bimodule; see [68]. CFD(Y : ) (with F coBordered Floer homology can be used to effectively compute HF(Y efficients). The key point is that the bimodules associated to mapping class group generators can be computed explicitly [67]. Thus, if we start from a Heegaard decompositon of Y , thought of as a union of two standardly bordered handlebodies, glued via an identification φ, which is expressed as a product of the mapping : ) can be obtained as an iterated tensor product, class group generators, then HF(Y where the two outermost factors are the modules associated to the standard handlebodies, and the inner factors are the bimodules associated to the mapping class group generators appearing in the factorization of φ. Conversely, Bohua Zhan [124] : ); and its has shown that this description can be taken as the definition of HF(Y topological invariance can be verified by some model computations. We will describe next an analogous bordered formulation for computing knot Floer homology; compare also [104, 123]. 7. Bordered algebras and knot invariants Bordered knot Floer homology, defined in [100] and [88], is a technique for computing knot Floer homology, which can be thought of as obtained from slicing a decorated knot projection D along horizontal slices. Specifically, cut the decorated knot projection into slices y = ti , where {ti }m i=1 is an increasing sequence of real numbers with the following properties: • the portion of the diagram with y ≤ t1 consists of a single strand with the global minimum on it • the portion with y ≥ tm consists of a single strand with the global maximum on it. • each portion of the diagram with ti ≤ y ≤ ti+1 is one of the following three standard pieces: a local maximum, a local minimum, or a crossing. To each y = ti slice of the diagram, we will associate an algebra. To each standard piece we associate a bimodule over the two algebras associated to its boundary. A chain complex computing the invariant is then obtained by tensoring together all of these bimodules. Generators for the resulting chain complex C( D) correspond to Kauffman states; and indeed generators of the intermediate bimodules correspond to certain “partial” Kauffman states. The homology of the resulting chain complex is a knot invariant. We describe these ingredients in a little more detail presently. 7.1. Partial knot diagrams. For generic t, a decorated knot projection D in the (x, y) plane meets the line y = t in 2n transverse points. We will draw our diagram so that the distinguished star is the global minimum y0 of the function y restricted to the projection. The portion of the diagram contained in the half-space in y ≥ t, for generic t > y0 , is called a upper knot diagram. Fix an upper knot diagram, and suppose that it meets the y = t slice at the 2n points {(i, t)}2n i=1 . These intersection points divide the y = t line into 2n + 1 connected components J0 = (−∞, 1), J1 = (1, 2), . . . , J2n−1 = (2n − 1, 2n), J2n =

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(2n, ∞). An idempotent state x is an n-element subset of {0, . . . , 2n}; or equivalently, we think of it as an n-element subset of the set of intervals {J0 , . . . , J2n }. An upper Kauffman state for an upper knot diagram y ≥ t is a pair (κ, x) where κ is a function that associates to each crossing in the upper knot diagram one of the four adjacent quadrants, and x is an idempotent state for the y = t slice of the diagram, subject to the following constraints: • Distinct crossings in the diagram are assigned by κ to quadrants contained in different bounded regions. A region in the knot diagram that contains a quadrant assigned by κ to some crossing is called occupied. • The unbounded region meets none of the intervals in x. • Each unoccupied bounded region contains exactly one of the intervals appearing in x on its boundary.

Figure 6. An upper Kauffman state. The black dots indicate the quadrants assigned by κ; the dark intervals on the bottom represent x. This upper knot diagram has five states. Note that any Kauffman state can be restricted to an upper knot diagram to give an upper Kauffman state. Upper Kauffman states have the following generalization. A partial knot diagram is a portion of a knot diagram contained in the (x, y) plane with t2 ≤ y ≤ t1 , so that t1 and t2 are generic. A partial Kauffman state is a triple of data (κ, x, y), where x is a collection of components in the y = t2 slice, y is a collection of components in the y = t1 slice, and κ is a map that associates to each crossing one of its four adjacent regions, subject to certain constraints. • Distinct crossings are assigned by κ to quadrants contained in distinct regions in the partial knot diagram. • If R is occupied, then y contains all the intervals in R ∩ (y = t1 ) and x contains none of the intervals in R ∩ (y = t2 ). • If R is unoccupied, then either R meets the y = t1 slice, y contains all but one of the edges of R ∩ (y = t1 ), and x contains none of the intervals in R ∩ (y = t2 ); or y contains all of the intervals in R ∩ (y = t1 ) (which now can be empty) and x contains exactly one of the intervals in the slice R ∩ (y = t2 ). Example 7.1. Consider the partial knot diagram consisting of 2n vertical lines. In this partial knot diagram, the partial Kauffman states (κ, x, y) have x = y, an arbitrary n-element subsets of {0, . . . , 2n}; and κ has no information (as there are no crossings). Example 7.2. Consider the partial knot diagram consisting of 2n vertical lines, and a single additional strand which contains a local maximum; i.e. this additional

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strand meets the bottom in two points, as in Figure 7. Assume that the maximum does not appear in the unbounded region. Then, there is a region R in the diagram that meets the top boundary in its (c−1)st interval, and it meets the bottom boundary its (c − 1)st and (c + 1)st intervals. The partial Kauffman state is then uniquely determined by x, which necessarily contains c. There are three cases; a state is said to be of type X if x∩{c−1, c, c+1} = {c−1, c}, it is of type Y if x∩{c−1, c, c+1} = {c, c + 1}, and Z if x ∩ {c − 1, c, c + 1} = {c}. See Figure 7.

Type X

Type Y

Type Z

Figure 7. Partial Kauffman states for the local maximum. We have drawn here partial Kauffman states, one of each type. Example 7.3. Consider the partial knot diagram consisting of 2n strands drawn so that the ith and (i + 1)st cross exactly once. There are four kinds of partial Kauffman states, according to which of the four regions is assigned to the crossing: N, S, E, or W. For the crossing of type N, x = y and i ∈ x; for a crossing of type S, x = y and i ∈ x; for a crossing of type W, i − 1 ∈ y, i ∈ y, i − 1 ∈ x and i ∈ x, and y \ {i − 1} = x \ {i}; for a crossing of type E, i + 1 ∈ y, i ∈ y, i + 1 ∈ x, i ∈ x, and y \ {i + 1} = x \ {i}.

Type N

Type S

Type W

Type E

Figure 8. Partial Kauffman states for crossings. We have drawn here three four Kauffman states, one of each type. 7.2. Algebras. We explain how to associate an algebra to each horizontal slice of a knot diagram. The horizontal slice can be thought of as a collection of 2n points on the real line. The portion of the knot projection above this horizontal slice gives a pairing between the 2n points. Specifically, if we slice the projection at the line y = t, so that the knot meets the line in points {(i, t)}2n i=1 , then i and j are matched if (i, t) and (j, t) are joined by an arc in the diagram contained in the portion of the diagram where y ≥ t. We denote this data by M . We will now define the corresponding algebra A(n, M ). As a preliminary step, we define an algebra B0 (m, k) associated to m points {1, . . . , m} and an integer 0 ≤ k ≤ m + 1; see [100]. The algebra is defined over the polynomial algebra F[U1 , . . . , Um ]; and it is is equipped with a set of preferred mutually orthogonal idempotents, which correspond to k element subsets of {0, . . . , m} or, equivalently, monotonically increasing functions x : {1, . . . , k} →

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{0, . . . , m} called idempotent states. Let Ix denote the idempotent corresponding to the idempotent state. As an F[U1 , . . . , Um ]-module, we have a splitting  B0 (m, k) ∼ Ix · B0 (m, k) · Iy . = x,y

Moreover, given any idempotent states x, y, the F[U1 , . . . , Um ]-module Ix · B0 (m, k)· Iy is isomorphic to F[U1 , . . . , Um ]; i.e. it is given with a preferred generator, which we denote γx,y . Thus, for any idempotent states x, y, z, we have γx,y ·γy,z = Px,y,z ·γx,z for some Px,y,z ∈ F[U1 , . . . , Um ]. Since Ix · Iy = 0 if x = y, to specify the multiplication on B0 (m, k), it suffices to specify the Px,y,z for all triples of idempotent states, which we do as follows. Each idempotent state x has a weight vector v x ∈ Zm , with components given by / vix = #{x ∈ x/x ≥ i}. nm Let Px,y,z be the monomial in F[U1 , . . . , Um ], U1n1 · · · Um , where ni is given by 1 ni = (|vix − viy | + |viy − viz | − |vix − viz |) 2  Note that B0 (m, k) is unital, with 1 = x Ix . Let Li ∈ B0 (m, k) with i ∈ {1, . . . , m} be the sum of the generators γx,y ∈ Ix · B0 (m, k) · Iy taken over all pairs of idempotent states x and y with the property that  1 if i = j vjx − vjy = 0 otherwise. Similarly, define Ri to be the sum of all the elements γy,x ∈ Iy · B0 (m, k) · Ix , where x and y run over all idempotent states as above. Let B(m, k) be the quotient algebra of B0 (m, k) by the relations Li+1 · Li = 0,

Ri · Ri+1 = 0

and Ix · Uj = 0 if x ({j − 1, j}) = ∅; i.e. if I denotes the two-sided ideal generated by Li+1 · Li , Ri · Ri+1 , and Ix · Uj as above, then B(m, k) = B0 (m, k)/I(m, k). We form the graded algebra A(n, M ) obtained by adjoining n central elements 2 = 0. Ci,j to B(2n, n), one for each {i, j} ∈ M , which satisfy the relation Ci,j We introduce a differential d on A(n, M ) which vanishes on B(2n, n) and satisfies dCi,j = Ui Uj . The algebras can be given gradings, after choosing an orientation on K; see [88]. The result now is the differential graded algebra A(n, M ). −1

Example 7.4. The algebra B(2, 1) has the following geometric description. Consider the graph with three vertices, labelled 0, 1, and 2, and two edges, one connecting 0 to 1 and another connecting 1 and 2. Think of the path as drawn horizontally, so that 1 is to the right of 0. The algebra B(2, 1) can be thought of as the quotient of the algebra of all paths in this graph, and obtained by dividing out by all paths that connect 0 and 2. The constant paths at 0, 1, and 2 correspond to the three idempotents I{0} , I{1} , and I{2} ; the first edge corresponds to L1 and R1 (whether it is oriented to the left or to the right respectively), and the second corresponds to L2 and R2 . Here, U1 = L1 · R1 + R1 · L1 and U2 = L2 · R2 + R2 · L2 . Clearly, the relation U1 U2 = 0 holds in B(2, 1).

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To construct A(1, {1, 2}), we adjoin one variable C{1,2} whose square is zero. We think of this as a differential graded algebra, but the differential is identically zero. In particular, F[U1 , U2 , C{1,2} ] . I{1} · A(1, {1, 2}) · I{1} ∼ = 2 (U1 U2 , C{1,2} ) We will view A = A(n, M ) as an algebra over the base ring of idempotents 2n+1 I( A) ∼ = F( n ) .

As such, an A∞ module over A will be a right module over I( A), and the actions mn will be multilinear over I( A): i.e. the tensor products in Equation (6.2) are taken over that ring. Similarly, a type D structure is a left module over I( A), the map δ 1 is an I( A)-module map, and the tensor product appearing in Equation (6.3) is also over that ring. Moreover, in the definition of N  X, the underlying vector space is N ⊗I(A ) X. 7.3. Bimodules. Having defined the algebra, we must associate bimodules to the various pieces. First, consider the global maximum. For the corresponding bimodule, the incoming algebra is trival; and so the bimodule is simply a type D structure. There is a single upper Kauffman state, and the corresponding generator z of the type D structure has z = I{1} · z δ z = C1,2 ⊗ z. 1

We can think of the tensor products in the construction of C( D) as an iterative procedure, starting with the above type D structure as a first step, and then successively increasing the size of the diagram covered by tensoring the type D structure in hand with the DA structure associated to the partial knot diagram immediately below it. As we will indicate below, the generators of the resulting type D structure correspond to upper Kauffman states (κ, x) for the diagram, and whose left idempotent is Ix . Thus, a key ingredient going into this definition is the type DA bimodule associated to each standard partial knot diagram. We do not describe the bimodules explicitly here; we refer the interested reader to [88]. Instead, we explain how to specify them uniquely up to homotopy equivalence. To this end, it is useful to make the following observations. Consider the dual complex for the bar complex, i.e. Cobar( A) =

∞ 

Hom( A⊗i , F),

i=1

equipped with a differential which is hom dual to ∂ as given in Equation (6.1). This is a differential graded algebra, with multiplication induced by the natural map Hom( A⊗i , F) ⊗ Hom( A⊗j , F) → Hom( A⊗(i+j) , F). A bounded A∞ module over a differential graded algebra A is the same thing as a type D structure over Cobar( A). More generally, a (bounded) DA bimodule,

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A1 X

A2

is the same thing as a type D structure over the tensor product algebra

A1 ⊗ Cobar( A2 ).

The algebra Cobar tends to be rather large; so instead, it is often convenient to find a smaller quasi-isomorphic version A . Then, up to quasi-isomorphism, a bounded A∞ module over A is equivalent to a type D structure over A . (This equivalence is called Koszul duality [65]; see also [105].) Similarly, a suitably bounded type DA bimodule A1 XA2 is uniquely determined (up to quasi-isomorphism)  by a corresponding module A1 ⊗A2 Y . Correspondingly, type D structures over A1 ⊗ A2 are called type DD bimodules over A1 and A2 . There is a handy Koszul dual algebra to A(n, M ), denoted A (n, M ), defined as follows. This is defined over the base algebra B(2n, n + 1), only now we adjoin 2n variables E1 , . . . , E2n which satisfy the following relations: Ei · b = b · Ei for all b ∈ B(2n, n + 1), Ei · Ej = Ej · Ei if i and j are not matched, and dEi = Ui . The identity map from A(n, M ) to itself can be thought of as a type DA bimodule over A(n, M ); which we can think of as the bimodule associated to the trivial diagram from Example 7.1. This is Koszul dual to the type D structure K over A(n, M ) ⊗ A (n, M ) whose generators are Ix ⊗ Iy , where x and y are complementary idempotent states; i.e. x ∪ y = {0, . . . , 2n}. The differential is specified by the element

A=

> 2n  i=1

? Li ⊗ Ri + Ri ⊗ Li + Ui ⊗ Ei

⎛ +⎝



⎞ C{i,j} ⊗ Ei , Ej  ∈ A ⊗ A ⎠ ,

{i,j}∈M

where Ei , Ej  = Ei · Ej + Ej · Ei . Specifically, δ 1 : K → ( A ⊗ A ) ⊗I(A )⊗I(A  ) K is given by δ 1 (v) = A ⊗ v 7.3.1. Crossings. We characterize the type DA bimodule of a positive crossing A2 P , where A = A (n, M ), A = A (n, M ), and M is obtained from M by 2 2 1 1 1 2 A1 composing with the transposition τ switching i and i+1. Its corresponding type DD  bimodule A2 ,A1 P is generated by partial Kauffman states with the understanding that left multiplication by Ix ⊗ I{0,...,2n}\y preserves the generator corresponding to (κ, x, y).  To describe this DD bimodule A2 ,A1 P, we introduce notational shorthand. Let N denote the sum of all the generators of P corresponding to the partial Kauffman states of type N in the sense of Example 7.3; define elements S, E, and W analogously. The differential δ 1 is specified as follows. Given any two elements X, Y ∈ {N, S, E, W}, the ( A2 ⊗ A1 ) ⊗ Y component of δ 1 (X) is a sum of terms a2 ⊗ a1 ∈ A2 ⊗ A1 , of the following types: (P-1) Rj ⊗ Lj and Lj ⊗ Rj for all j ∈ {1, . . . , 2n} \ {i, i + 1}, i.e. when X = Y. (P-2) Uj ⊗ Eτ (j) for all j = 1, . . . , 2n again with X = Y. (P-3) C{α,β} ⊗ [Eτ (α) , Eτ (β) ], for all {α, β} ∈ M2 again with X = Y.

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(P-4) Terms specified in the diagram below that connect X = Y, 1

Ri 1

1 U i+

⊗

Ri

+

⊗

L i+

N

U

i

⊗

L

i+ 1

R i+

1⊗

1⊗

Li

+

L

iL i+ 1

⊗

R

R

i

i+ 1

W

E

R

i

L ⊗

i

⊗

i+ 1

+

1

1

U

⊗

1

R i+

L

i+ 1

⊗

R

i+ 1R i

(7.1)

1

S

1 L i+

⊗

Ri

+

Ui

⊗

L i+ Li

For example, the above graph gives rise to terms (1⊗Li )⊗N+(Li ⊗1)⊗S in δ 1 (W). The negative crossing works similarly, except that Equation (7.1) is replaced by: 1

1 U i+

⊗

Li

+

1 L i+ Li

1⊗

⊗

R i+

N

U

i

⊗

R

i+ 1

1⊗

Ri

+

R

i+ 1R i

⊗

L

L

i

i+ 1

W

E

L

i

R ⊗

i

⊗

U

i+ 1

+

1

1

⊗

1 Ri

L i+

R

i+ 1

⊗

L

iL i+ 1

S

1

⊗

Ui

1

+

Li

⊗

R i+

R i+

7.3.2. Local maximum. Consider the partial knot diagram of a local maximum from Example 7.2. The type DA bimodule of this partial knot diagram A2 Ω A1 is defined over algebras A1 and A2 , and it is specified as follows. Let φc : {1, . . . , 2n} → {1, . . . , 2n + 2} be the map  j if j < c (7.2) φc (j) = j + 2 if j ≥ c. Then, (7.3)

A1 = A(n, M1 )

and

A2 = A(n + 1, φc (M1 ) ∪ {c, c + 1})

We specify this bimodule up to quasi-isomorphism by defining its dual type  DD bimodule A2 ,A1 Ω. The generators correspond to partial Kauffman states for the partial knot diagram, again with the convention that Ix ⊗ I{0,...,2n}\y preserves the generator corresponding to (κ, x, y). The differential is specified by the algebra element A ∈ A2 ⊗ A1 > 2n ?  A = (Lc Lc+1 ⊗ 1) + (Rc+1 Rc ⊗ 1) + Lφ(i) ⊗ Ri + Rφ(i) ⊗ Li + C{c,c+1} ⊗ 1 +

> 2n  i=1

? Uφ(i) ⊗ Ei

i=1

⎛

+⎝



⎞ C{φ(i),φ(j)} ⊗ Ei , Ej ⎠

{i,j}∈M

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In more detail, decomposing partial Kauffman states according to the type X, Y, and Z specified in Example 7.2, and thinking of the corresponding elements in A1 , the differential on the bimodule has terms are of the following types: (Ω-1) Rφ(j) ⊗ Lj and Lφ(j) ⊗ Rj for all j ∈ {1, . . . , 2n} \ {c − 1, c}; these connect generators of the same type. (Ω-2) Uφ(i) ⊗ Ei for i = 1, . . . , 2n (Ω-3) C{φ(i),φ(j)} ⊗ Ei , Ej  for all {i, j} ∈ M1 ; (Ω-4) C{c,c+1} ⊗ 1 (Ω-5) Terms in the diagram below connect generators of different types: Lc Lc+1 ⊗ 1

X

Y Rc+1 Rc ⊗ 1

−

c

⊗ c+ 2

L

1

Rc

−

⊗

R

1

Lc

c+ 2

⊗

−

R

1

⊗

−

L

c

Rc Lc

1

(7.4)

Z

The above description can be readily specialized to the case where the maximum appears in an unbounded region. In these cases, there is only one generator type, Z. 7.3.3. Local minimum. Turning the above example on its top, we have A2 cA1 , where A1 = A(n + 1, M1 ) and A2 = A(n, M2 ) are as follows. Let M1 be any matching that does not match c and c + 1, and let {φc (α), c}, {c + 1, φc (β)} ∈ M1 (with φc as in Equation (7.2)). Let M2 be the matching with {i, j} ∈ M2 if {φc (i), φc (j)} ∈ M1 and {α, β} ∈ M2 .  We specify this module by describing its dual type DD bimodule A2 ,A1 c . Its generators correspond to partial Kauffman states, with convention that Ix ⊗ I{0,...,2n+2}\y preserves the generator corresponding to (κ, x, y). The DD bimodule is specified by the algebra element (7.5)

⎞ ⎛ 2n  Rj ⊗ Lφ(j) + Lj ⊗ Rφ(j) + Uj ⊗ Eφ(j) ⎠ A = (1 ⊗ Lc Lc+1 ) + (1 ⊗ Rc+1 Rc ) + ⎝ j=1

+ 1 ⊗ Ec Uc+1 + Uα ⊗ Eφ(α) , Ec Ec+1 + C{α,β} ⊗ Eφ(α) , Ec Ec+1 , Eφ(β) . 7.3.4. Global minimum. When we have covered the entire diagram, save for the last piece (the global minimum), we have a type D structure C over the algebra I{1} · A(2, 1, {1, 1}) · I{1} . After dividing out by C{1,2} , what remains can be thought of as a chain complex over F[U1 , U2 ]/U1 U2 . Its homology is the invariant H(K). Dividing out the complex by U2 and taking homology gives H − (K); and dividing out by both U1 and U2 and  taking homology gives H(K). 7.4. Topological invariance. It is proved in [88] that the bigraded homology  D), H − ( D) and H( D) are invariants of the underlying oriented knot modules H(

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K represented by the diagram D. This involves checking that the homology of the chain complex is invariant under Reidemeister moves. These relations are proved locally on the level of bimodules. For example, the bimodules of a crossing satisfy the “braid relations” for any 1 ≤ i, j ≤ 2n − 1: for |i − j| > 1, these relations give quasi-isomorphisms of bimodules (7.6)

Pj  Pi * Pi  Pj ;

(where we have suppressed the algebras which come naturally from the pictures) while if j = i + 1, then, (7.7)

Pi  Pi+1  Pi * Pi+1  Pi  Pi+1 .

(again for suitably chosen algebras). Thus, we can think of these bimodules as giving a braid group action on the derived category of modules over A(n, M ); compare [51, 68, 73].  The knot invariants H(K), H − (K), and H(K) are designed to agree with their knot Floer homological analogues. One can nonetheless study them independently of holomorphic methods. For example, one can verify certain fundamental properties within the algebraic realm: relating their graded Euler characteristics with the Alexander polynomial of K, establishing a Künneth formula for connected sums, and verifying an algebraic structure result for H − (K) analogous to Proposition 3.4; see [88]. 8. Bordered knot algebras and pseudo-holomorphic curves In fact, we prove that this bordered invariant is equivalent to knot Floer homology [87]. To establish the link between the algebraic constructions and knot Floer homology, it is useful to give a pseudo-holomorphic interpretation of these structures. Upper knot diagrams can be represented by suitably decorated (partial) Heegaard diagrams. An upper Heegaard diagram is a surface Σ of genus g and 2n boundary components, labelled Z1 , . . . , Z2n , together with the following additional data: • A collection of disjoint, embedded arcs {αi }2n−1 i=1 , so that αi connects Zi to Zi+1 . • A collection of disjoint embedded closed curves {αic }gi=1 (which are also disjoint from α1 , . . . , α2n−1 ). . • A collection of embedded, mutually disjoint closed curves {βi }g+n−1 i=1 Both sets of α-and the β-circles are required to consist of homologically linearly independent curves, and the β-circles are further required to have the following combinatorial property: the surface obtained by cutting Σ along β1 , . . . , βg+n−1 , which has n connected components, is required to contain exactly two boundary circles in each component. This requirement gives a matching M on {1, . . . , 2n} (a partition into two-element subsets), where {i, j} ∈ M if Zi and Zj can be connected by a path that does not cross any βk . We sometimes abbreviate the data

H ∧ = (Σ, Z1 , . . . , Z2n , {α1 , . . . , α2n−1 }, {α1c , . . . , αgc }, {β1 , . . . , βg+n−1 }), and let M (H ∧ ) be the induced matching.

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PETER OZSVÁTH AND ZOLTÁN SZABÓ

Z1

α1

Z2

α2

Z3

α3

Z4

β

Figure 9. Upper Heegaard diagram. The diagram here is the upper Heegaard diagram for the upper planar diagram from Figure 6; the black dot represents a Heegaard state corresponding to the Kauffman state from Figure 6. An upper Heegaard state is an g + n − 1-tuple of points x, each of which is in αi ∩ βj or αic ∩ βj for various i and j, so that each circle αi contains an element in x, each βj contains an element in x, and no more than one element in x is contained on each α-arc αic . Each Heegaard state x determines a subset s(x) of {1, . . . , 2n} with cardinality n, or, equivalently, an idempotent Is(x) in A(n, M ): / s(x) = {1, . . . , 2n − 1} \ {1 ≤ i ≤ 2n − 1/x ∩ αi is non-empty}. i αi−1

Zi

αi

αi−1

αi zi ri

Figure 10. Boundary markings. On the left, we have shown a neighborhood of a boundary component Zi of Σ. To the right, we have filled in Zi , replacing it with the point zi . Fill in each boundary component Zi , to obtain a closed Riemann surface Σ, with 2n marked points zi . Extend αi into Σ to obtain a curve αi with ∂αi = zi+1 − zi , as shown in Figure 10. We will place a pair of points ri and i in a neighborhood of zi , separated by αi−1 ∪ αi . (In the special case where i = 1 and 2n, the two points ri and i are not separated by this neighborhood, as one of αi−1 or αi does not exist.) We will be working with holomorphic disks in Symg+n−1 (Σ), relative to Tβ and L0α = α1c × · · · × αgc × Symn−1 (α1 ∪ · · · ∪ α2n−1 ). Note that L0α is a singular space, with singularities contained in the locus where two points are contained on the same αi -curve. We will work away from this locus, in the subspace Lα ⊂ L0α consisting of those n−1-tuples where no two points lie on the same αi (this corresponds to the “boundary monotonicity” condition of [69]), and

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each point lies in the interior of some αi , denoted αi◦ . Clearly, Lα is disconnected; in fact ⎞ ⎛  Lα = α1c × · · · × αgc × ⎝ αt◦1 × · · · × αt◦n−1 ⎠ . {t1 ,...,tn−1 }⊂{1,...,2n−1}

The upper Heegaard states correspond to intersection points of Lα with Tβ . If x is in the component of Lα specified by {t1 , . . . , tn−1 }, then Is(x) · x = x where s(x) = {1, . . . , 2n − 1} \ {t1 , . . . , tn−1 }. Let π2 (x, y) denote the space of homotopy classes of Whitney disks as before, only now half of the boundary of the disk is mapped into L0α , and the other half into Tβ . (In fact, the disks of interest to us will have half their boundary mapped into the closure Lα ⊂ L0α of Lα .) Each φ ∈ π2 (x, y) with non-negative local multiplicities determines an algebra element b0 (φ) ∈ Ix · B0 · Iy , given by c (φ)

b0 (φ) = U1 1

c

· · · U2n2n

(φ)

· γxy ,

where ci (φ) = min(ni (φ), nri (φ)). Let X denote the vector space spanned by upper Heegaard states. Consider the map γ01 : X → B0 ⊗ X (again, where the tensor product is taken over the idempotent ring) defined by     M(φ) # γ01 (x) = · b0 (φ) · y. R / y∈S( H ) {φ∈π (x,y)/μ(φ)=1} 2 Proposition 8.1. The endomorphism γ01 satisfies the structure relation ⎞ ⎛  1 1 0 ⎝ Ui Uj ⎠ ⊗ x ∈ I ⊗ X, (μB 2 ⊗ IdX ) ◦ (IdB0 ⊗γ0 ) ◦ γ0 (x) + {i,j}∈M

where I = I(2n, n) is the ideal used in the definition of B(2n, n). Sketch of proof. In broad terms, the proof of this is the usual ∂ 2 = 0 proof in Lagrangian Floer homology: it is proved by considering one-dimensional moduli spaces of pseudo-holomorphic disks, and identifying their boundaries. In more detail, the proof rests on the following observations Observation 1. First, note that the map b0 is additive under juxtapositions, in the sense that if x, y, z ∈ Lα ∩ Tβ , φ1 ∈ π2 (x, y), and φ2 ∈ π2 (y, z) are two homotopy classes whose local multiplicities at all the i and ri are non-negative, then (8.1)

b0 (φ1 ∗ φ2 ) = b0 (φ1 ) · b0 (φ2 ).

This follows quickly from the fact that for any φ ∈ π2 (x, y), s(x)

ni (φ) − nri (φ) = vi

s(y)

− vi

,

together with the additivity of local multiplicities under juxtapositions; i.e. np (φ1 ∗ φ2 ) = np (φ1 ) + np (φ2 ) for any p ∈ Σ, and the definition of multiplication in the algebra. Observation 2. The next point is that if φ = φ1 ∗ φ2 , where φ1 ∈ π2 (x, y) and φ2 ∈ π2 (y, z) for some y ∈ Lα ∩ Tβ has an alternative decomposition φ = φ1 ∗ φ2

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242

PETER OZSVÁTH AND ZOLTÁN SZABÓ

βi−1 αi−1 x1

βi αi y1

αi+1 y2

x2

Zi

Zi+1 βi αi−1 z y

αi t

x Zi

Zi+1

Figure 11. Relations in B.

with φ1 ∈ π2 (x, y ) and φ2 ∈ π2 (y , z) with y ∈ (L0α ∩ Tβ ) \ (Lα ∩ Tβ ), then b0 (φ) ∈ I. To see why, we refer to Figure 11. At the left, the pair {x1 , x2 } represents part of an upper state x, {x1 , y2 } represents part of an upper state y, and {y1 , y2 } represents part of an upper state z. The small bigon near Zi+1 gives a term of Li+1 ⊗ y in γ01 (x); and the small bigon near Zi gives a term of Li ⊗ z in γ01 (y). Since Li+1 Li = 0, we do not need to consider the ends of the moduli space from x to z: the corresponding term is in the ideal I. Note that the alternative factorization of this moduli spaces involves {y1 , x2 } which is not in Lα ∩ Tβ . At the right is a similar picture, now with {x, t} ⊂ x, {y, t} ⊂ y, and {z, t} ⊂ z. A small bigon from x to y gives a term of y in γ01 (x). The bigon from y to z containing Zi gives a term of Ui ⊗ z in γ01 (y), which is in the ideal I. In view of the above two observations, it suffices to consider ends of moduli spaces φ ∈ π2 (x, z) for which all broken flowline decompositions φ = φ1 ∗ φ2 with φ1 ∈ π2 (x, y) and φ2 ∈ π2 (x, y) have y ∈ Lα ∩ Tβ . The usual ∂ 2 = 0 proof now shows that the number of such ends has the same parity as the number of boundary degenerations: holomorphic curves which have boundary contained entirely on L0α or Tβ . We complete the proof with two more observations: Observation 3. A homotopy class corresponding to curves with boundary contained entirely in Lα has positive coefficients at every i and mi ; thus, the associated algebra element lies in the ideal I. And finally, to keep track of the β-boundary degenerations, we have: Observation 4. There are n Maslov index 2 homotopy classes of disks ψ with boundary in Tβ , corresponding to the matchings {i, j} ∈ M , and their correspond ing algebra element is Ui · Uj ; see Figure 12. The map γ01 induces a map γ1 : X → B ⊗ X

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αi−1

αi

243

αj−1

Zi

αj

βi

Zj

Figure 12. Motivation for introducing Ci,j . satisfying the structure relation (μ2 ⊗ IdX ) ◦ (IdB ⊗γ 1 ) ◦ γ 1 =

(8.2)



Ui Uj ,

{i,j}

where here μ2 is multiplication in B. Note that since μ1 = 0 in B, the structure relation for γ 1 is nearly the type D structure relation: it would be if the right hand side were zero. Instead, this structure can be thought of as a “curved” type D structure (i.e. for an algebra with a μ0 operation); compare [50]. We can turn such an object into a type D structure over A, defining: ⎞ ⎛  δ 1 (x) = ⎝ C{i,j} ⎠ ⊗ x + γ 1 (x). {i,j}∈M

This defines the type D structure of an upper diagram. Consider the upper diagram from Figure 9. Note that we are now working in the (first symmetric product of) the two-sphere. In some sense, the type D structure is capturing the Lagrangian Floer homology of an interval with four marked points on it and the closed curve β. The type D structure has five generators corresponding to the five intersection points, which we label from left to right in the picture x1 , x2 , t, y1 , y2 ; and γ 1 is specified by the arrows in the diagram: L2 L3

L2 U1

U4

x1

x2 U3

(8.3)

R2

U2

L3

y1

t R3 U4

y2 U1

R3 R2

The verification that γ 1 satisfies the structure relation (Equation (8.2)) is straightforward, taking into account the relations in the algebra. For example, idempotents and the algebra relation ensues that U1 U2 ⊗ x1 = 0 (since we are taking tensor products over the ring of idempotents). With a little more work, one can define the A∞ module associated to a lower diagram. In this case, the higher actions count pseudo-holomorphic disks that go out to the α-boundary. The algebra actions record the sequence of walls in which

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a pseudo-holomorphic disk crosses the walls in Lα . Like in bordered Floer homology, it is clearer to express these actions in the language of Lipshitz’s cylindrical reformulation of Heegaard Floer homology [64]. A pairing theorem for recapturing knot Floer homology is then proved using a “time dilation” argument analogous to the bordered case (see [69, Chapter 9]), with a little extra attention paid now to β-boundary degenerations. Working out the the type DD bimodules for basic pieces is a fairly straightforward matter. Extending the pairing theorem to type DA bimodules then gives the following: Theorem 8.2. [87] If K ⊂ S 3 is a knot, then there are isomorphisms of bigraded modules ∼  HFK(K) ∼ HFK(K) HFK− (K) ∼ = H − (K) = H(K). = H(K) 9. Further remarks We have implemented this algorithm for calculating knot Floer homology in a computer program. We start with some comments about this computation. To effectively compute the chain complexes C( D) described above, one can start with the type D structure corresponding to the global maximum, and successively enlarge it, moving down the knot projection. The computation is significantly improved by eliminating (by passing to a homotopy equivalent complex) generators x with  ax,y ⊗ y δ 1 (x) = y

for which some ax,y is an idempotent. Another simplification is achieved by working directly with the operators γ 1 from [88, Section 13]. Recall that the generators of the complex C( D) correspond to Kauffman states; i.e. to spanning trees of the black graph. If K has n crossings, the number of Kauffman states is clearly bounded by 2n ; in fact, it appears to grow roughly like (1.7)n . With the above simplification, the size of the chain complex can be greatly reduced, provided that the knot is far from alternating. For an alternating knot,  D) vanishes; so indeed all the generators are needed. Luckily the differential in C( in the alternating case, the knot Floer homology is explicitly determined by the Alexander polynomial and the signature [90]. Our algorithm works well for computing knot Floer homology with fewer than 26 crossings. Here, most memory-intensive computations needed are for knots obtained by changing one crossing in an alternating projection. However, the program can compute the knot Floer homology groups of many knots that are much larger knots. For example, as a test example, we computed the knot Floer homology of a 91-crossing from [34]; see [100]. Throughout the above discussion, we worked in characteristic 2 to avoid signs. In fact, a knot invariant with Z coefficients can be worked out, paying a little extra care to sign conventions. This is done in [88]. One motivation is to find a knot K in S 3 whose knot Floer homology (with Z coefficients) has torsion. Despite a rather extensive search, we have not yet found such a knot. Long before the discovery of knot Floer homology, Andreas Floer proposed a construction of a knot invariant using instantons [31]. In [59], Kronheimer and Mrowka further developed this theory and also conjectured an isomorphism between (taken now with instanton knot Floer homology and the knot Floer homology HFK

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Q coefficients). Verifying this remarkable conjecture would provide a link between the fundamental group and knot Floer homology. One approach to this problem would be to give a similar algebraic method for computing instanton knot Floer homology. The knot Floer homology package described here are sufficient for many com: of surgeries on K. To understand the putations: calculating τ (K), (K), and HF − function Υ(K) and HF of surgeries on K, one needs to understand the knot Floer complex with more structure (in effect, without the U V = 0 specialization from Equation (4.1)). To study this invariant, one must work with a larger algebra; see [89]. References [1] M. Atiyah, Floer homology, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 105–108. MR1362825 [2] J. A. Baldwin and W. D. Gillam, Computations of Heegaard-Floer knot homology, J. Knot Theory Ramifications 21 (2012), no. 8, 1250075, 65, DOI 10.1142/S0218216512500757. MR2925428 [3] D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337–370, DOI 10.2140/agt.2002.2.337. MR1917056 [4] J. Batson, Nonorientable slice genus can be arbitrarily large, Math. Res. Lett. 21 (2014), no. 3, 423–436, DOI 10.4310/MRL.2014.v21.n3.a1. MR3272020 [5] J. M. Bloom, A link surgery spectral sequence in monopole Floer homology, Adv. Math. 226 (2011), no. 4, 3216–3281, DOI 10.1016/j.aim.2010.10.014. MR2764887 [6] P. J. Braam and S. K. Donaldson, Floer’s work on instanton homology, knots and surgery, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 195–256. MR1362829 [7] W. Chen. On the upsilon invariant of cable knots. arXiv:1604.04760, 2016. [8] D. H. Choi and K. H. Ko, Parametrizations of 1-bridge torus knots, J. Knot Theory Ramifications 12 (2003), no. 4, 463–491, DOI 10.1142/S0218216503002445. MR1985906 [9] T. D. Cochran and R. E. Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology 3-spheres and property P , Topology 27 (1988), no. 4, 495–512, DOI 10.1016/0040-9383(88)90028-6. MR976591 [10] V. Colin, P. Ghiggini, and K. Honda, Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA 108 (2011), no. 20, 8100–8105, DOI 10.1073/pnas.1018734108. MR2806645 [11] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, pp. 329–358. MR0258014 [12] P. R. Cromwell, Embedding knots and links in an open book. I. Basic properties, Topology Appl. 64 (1995), no. 1, 37–58, DOI 10.1016/0166-8641(94)00087-J. MR1339757 [13] H. Doll, A generalized bridge number for links in 3-manifolds, Math. Ann. 294 (1992), no. 4, 701–717, DOI 10.1007/BF01934349. MR1190452 [14] S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR710056 [15] S. K. Donaldson, Irrationality and the h-cobordism conjecture, J. Differential Geom. 26 (1987), no. 1, 141–168. MR892034 [16] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315, DOI 10.1016/0040-9383(90)90001-Z. MR1066174 [17] S. K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 45–70, DOI 10.1090/S0273-0979-96-00625-8. MR1339810 [18] S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), no. 2, 205–236. MR1802722 [19] S. K. Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, vol. 147, Cambridge University Press, Cambridge, 2002. With the assistance of M. Furuta and D. Kotschick. MR1883043

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01743

Descendents for stable pairs on 3-folds Rahul Pandharipande Dedicated to Simon Donaldson on the occasion of his 60th birthday Abstract. We survey here the construction and the basic properties of descendent invariants in the theory of stable pairs on nonsingular projective 3-folds. The main topics covered are the rationality of the generating series, the functional equation, the Gromov–Witten/Pairs correspondence for descendents, the Virasoro constraints, and the connection to the virtual fundamental class of the stable pairs moduli space in algebraic cobordism. In all of these directions, the proven results constitute only a small part of the conjectural framework. A central goal of the article is to introduce the open questions as simply and directly as possible.

Contents 0. Introduction 1. Rationality 2. Gromov–Witten/Pairs correspondence 3. Virasoro constraints 4. Virtual class in algebraic cobordism Acknowledgments References

0. Introduction 0.1. Moduli space of stable pairs. Let X be a nonsingular projective 3fold. The moduli of curves in X can be approached in several different ways.1 For an algebraic geometer, perhaps the most straightforward is the Hilbert scheme of subcurves of X. The moduli of stable pairs is closely related to the Hilbert scheme, but is geometrically much more efficient. While the definition of a stable pair takes some time to understand, the advantages of the moduli theory more than justify the effort. Definition 1. A stable pair (F, s) on X is a coherent sheaf F on X and a section s ∈ H 0 (X, F ) satisfying the following stability conditions: 2010 Mathematics Subject Classification. Primary 14N35; Secondary 14D20. The author was partially supported by SNF grant 200021-143274, ERC grant AdG-320368MCSK, SwissMAP, and the Einstein Stiftung. 1 For a discussion of the different approaches, see [42]. c 2018 American Mathematical Society

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• F is pure of dimension 1, • the section s : OX → F has cokernel of dimensional 0. Let C be the scheme-theoretic support of F . By the purity condition, all the irreducible components of C are of dimension 1 (no 0-dimensional components are permitted). By [38, Lemma 1.6], the kernel of s is the ideal sheaf of C, IC = ker(s) ⊂ OX , and C has no embedded points. A stable pair OX → F therefore defines a Cohen-Macaulay subcurve C ⊂ X via the kernel of s and a 0-dimensional subscheme2 of C via the support of the cokernel of s. To a stable pair, we associate the Euler characteristic and the class of the support C of F , χ(F ) = n ∈ Z

and [C] = β ∈ H2 (X, Z) .

For fixed n and β, there is a projective moduli space of stable pairs Pn (X, β). Unless β is an effective curve class, the moduli space Pn (X, β) is empty. A foundational treatment of the moduli space of stable pairs is presented in [38] via the results of Le Potier [16]. Just as the Hilbert scheme In (X, β) of subcurves of X of Euler characteristic n and class β is a fine moduli space with a universal quotient sequence, Pn (X, β) is a fine moduli space with a universal stable pair [38, Section 2.3]. While the Hilbert scheme In (X, β) is a moduli space of curves with free and embedded points, the moduli space of stable pairs Pn (X, β) should be viewed as a moduli space of curves with points on the curve determined by the cokernel of s. Though the additional points still play a role, Pn (X, β) is much smaller than In (X, β). If Pn (X, β) is non-empty, then Pm (X, β) is non-empty for all m > n. Stable pairs with higher Euler characteristic can be obtained by suitably twisting stable pairs with lower Euler characteristic (in other words, by adding points). On the other hand, for a fixed class β ∈ H2 (X, Z), the moduli space Pn (X, β) is empty for all sufficiently negative n. The proof exactly parallels the same result for the Hilbert scheme of curves In (X, β). 0.2. Action of the descendents. Denote the universal stable pair over X × Pn (X, β) by s OX×Pn (X,β) → F. For a stable pair (F, s) ∈ Pn (X, β), the restriction of the universal stable pair to the fiber X × (F, s) ⊂ X × Pn (X, β) s

is canonically isomorphic to OX → F . Let πX : X × Pn (X, β) → X, πP : X × Pn (X, β) → Pn (X, β) 2 When C is Gorenstein (for instance if C lies in a nonsingular surface), stable pairs supported on C are in bijection with 0-dimensional subschemes of C. More precise scheme theoretic isomorphisms of moduli spaces are proved in [40, Appendix B].

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be the projections onto the first and second factors. Since X is nonsingular and F is πP -flat, F has a finite resolution by locally free sheaves.3 Hence, the Chern character of the universal sheaf F on X × Pn (X, β) is well-defined. Definition 2. For each cohomology4 class γ ∈ H ∗ (X) and integer i ∈ Z≥0 , the action of the descendent τi (γ) is defined by ∗ τi (γ) = πP ∗ (πX (γ) · ch2+i (F) ∩ πP∗ ( · )) .

The pull-back πP∗ is well-defined in homology since πp is flat [8]. We may view the descendent action as defining a cohomology class τi (γ) ∈ H ∗ (Pn (X, β)) or as defining an endomorphism τi (γ) : H∗ (Pn (X, β)) → H∗ (Pn (X, β)) . Definition 2 is the standard method of obtaining classes on moduli spaces of sheaves via universal structures. The construction has been used previously for the cohomology of the moduli space of bundles on a curve [28], for the cycle theory of the Hilbert schemes of points of a surface [10], and in Donaldson’s famous μ map for gauge theory on 4-manifolds [6]. 0.3. Tautological classes. Let D denote the polynomial Q-algebra on the symbols { τi (γ) | i ∈ Z≥0 and γ ∈ H ∗ (X) } subject to the basic linear relations τi (λ · γ) = τi (γ + γ ) =

λτi (γ) , τi (γ) + τi ( γ) ,

for λ ∈ Q and γ, γ  ∈ H ∗ (X). The descendent action defines a Q-algebra homomorphism X : D → H ∗ (Pn (X, β)) . αn,β The most basic questions about the descendent action are to determine X )⊂D Ker(αn,β

and

X Im(αn,β ) ⊂ H ∗ (Pn (X, β)) .

Both questions are rather difficult since the space Pn (X, β) can be very complicated (with serious singularities and components of different dimensions). Few methods are available to study H ∗ (Pn (X, β)). Following the study of the cohomology of the moduli of stable curves, we define, for the moduli space of stable pairs Pn (X, β), X • Im(αn,β ) ⊂ H ∗ (Pn (X, β)) to be the algebra of tautological classes, X • Ker(αn,β ) ⊂ D to be the ideal of tautological relations since

D X = Im(αn,β ). X ) Ker(αn,β 3 Both

X and Pn (X, β) carry ample line bundles. homology and cohomology groups will be taken with Q-coefficients unless explicitly denoted otherwise. 4 All

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RAHUL PANDHARIPANDE

The basic expectation is that natural constructions yield tautological classes. For the moduli spaces of curves there is a long history of the study of tautological classes, geometric constructions, and relations, see [12, 32] for surveys. As a simple example, consider the tautological classes in the case X = P3 ,

n = 1,

β = L,

where L ∈ H2 (P3 , Z) is the class of a line. The moduli space P1 (P3 , L) is isomorphic to the Grassmannian G(2, 4). The ring homomorphism P α1,L : D → H ∗ (P1 (P3 , L)) 3

is surjective, so all classes are tautological. The tautological relations P )⊂D Ker(α1,L 3

can be determined by the Schubert calculus. Our study of descendents here follows a different line which is more accessiX ble than the full analysis of αn,β . The moduli space Pn (X, β) carries a virtual fundamental class [Pn (X, β)]vir ∈ H∗ (Pn (X, β)) obtained from the deformation theory of stable pairs. There is an associated integration map " (1) : D→Q [Pn (X,β)]vir

defined by

"

" X αn,β (D) ∩ [Pn (X, β)]vir

D= [Pn

for D ∈ D. Here,

(X,β)]vir

Pn (X,β)

" : H∗ (Pn (X, β)) → Q Pn (X,β)

is the canonical point counting map factoring through H0 (Pn (X, β)). The standard theory of descendents is a study of the integration map (1). 0.4. Deformation theory. To define a virtual fundamental class [3, 21], a 2-term deformation/obstruction theory must be found on the moduli space of stable pairs Pn (X, β). As in the case of the Hilbert scheme In (X, β), the most immediate obstruction theory of Pn (X, β) does not admit such a structure. For In (X, β), a suitable obstruction theory is obtained by viewing C ⊂ X not as a subscheme, but rather as an ideal sheaf IC with trivial determinant [7, 44]. For Pn (X, β), a suitable obstruction theory is obtained by viewing a stable pair not as sheaf with a section, but as an object [OX → F ] ∈ Db (X) in the bounded derived category of coherent sheaves on X. Denote the quasi-isomorphism equivalence class of the complex [OX → F ] in Db (X) by I • . The quasi-isomorphism class I • determines5 the stable pair [38, Proposition 1.21], and the fixed-determinant deformations of I • in Db (X) match those of the pair (F, s) to all orders [38, Theorem 2.7]. The latter property shows the scheme Pn (X, β) may be viewed as a moduli space of objects in the derived 5 The

claims require the dimension of X to be 3.

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category.6 We can then use the obstruction theory of the complex I • rather than the obstruction theory of sheaves with sections. The deformation/obstruction theory for complexes at [I • ] ∈ Pn (X, β) is governed by Ext1 (I • , I • )0

(2)

Ext2 (I • , I • )0 .

and

The obstruction theory (2) has all the formal properties 5 of the Hilbert scheme case: 2 terms, a virtual class of (complex) dimension dβ = β c1 (X), [Pn (X, β)]vir ∈ H2dβ (Pn (X, β), Z) , and a description via the χB -weighted Euler characteristics in the Calabi-Yau case [2]. 0.5. Descendent invariants. Let X be a nonsingular projective 3-fold. For nonzero β ∈ H2 (X, Z) and arbitrary γi ∈ H ∗ (X), define the stable pairs invariant with descendent insertions by " r ¨ ∂X  (3) τk1 (γ1 ) . . . τkr (γr ) = τki (γi ) . n,β

[Pn (X,β)]vir i=1

The partition function is r r / Ä ä ∂X ¨ / (4) ZP X; q / τki (γi ) = τki (γi ) qn . β

i=1

i=1

n∈Z

n,β

Since Pn (X, β) is empty for sufficiently negative n, the partition function is a Laurent series in q, r /  Ä ä / ZP X; q / τki (γi ) ∈ Q((q)) . β

i=1

The descendent invariants (3) and the associated partition functions (4) are the central topics of the paper. From the point of view of the complete tautological ring of descendent classes on Pn (X, β), the descendent invariants (3) constitute only small part of the full data. However, among many advantages, the integrals (3) are deformation invariant as X varies in families. The same can not be said of the tautological ring nor of the full cohomology H ∗ (Pn (X, β)). In addition to carrying data about the tautological classes on Pn (X, β), the descendent series are related to the enumerative geometry of curves in X. The connection is clearest for the primary fields τ0 (γ) which correspond to incidence conditions for the support curve of the stable pair with a fixed cycle Vγ ⊂ X ∗

of class γ ∈ H (X). But even for primary fields, the partition function r /  Ä ä / ZP X; q / τ0 (γi ) i=1

β

provides a virtual count and is rarely strictly enumerative. Descendents τk (D), for k ≥ 0 and D ⊂ X a divisor, can be viewed as imposing tangency conditions of the support curve of the stable pair along the divisor D. 6 The

moduli of objects in the derived category usually yields Artin stacks. The space Pn (X, β) is a rare example where the moduli of objects in the derived category has a component with coarse moduli space given by a scheme (uniformly for all 3-folds X).

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The connection of τk (D) to tangency conditions is not as close as the enumerative interpretation of primary fields — the tangency condition is just the leading term in the understanding of τk (D). The topic will be discussed in Section 2.7. 0.6. Plan of the paper. The paper starts in Section 1 with a discussion of the rationality of the descendent partition function in absolute, equivariant, and relative geometries. While the general statement is conjectural, rationality in toric and hypersurface geometries has been proven in joint work with A. Pixton in [33, 35, 37]. Examples of exact calculations of descendents are given in Section 1.4. A precise conjecture for a functional equation related to the change of variable 1 q → q is presented in Section 1.7, and a conjecture constraining the poles appears in Section 1.8. The second topic, the Gromov–Witten/Pairs correspondence for descendents, is discussed in Section 2. The descendent theory of stable maps and stable pairs on a nonsingular projective 3-fold X are conjectured to be equivalent via a universal transformation. While the correspondence is proven in joint work with A. Pixton in toric [36] and hypersurface [37] cases and several formal properties are established, a closed formula for the transformation is not known. The Gromov–Witten/Pairs correspondence has motivated much of the development of the descendent theory on the sheaf side. The first such conjectures for descendent series were made in joint work with D. Maulik, A. Okounkov, and N. Nekrasov [24, 25] in the context of the Gromov–Witten/Donaldson–Thomas correspondence7 for the partition functions associated to the Hilbert schemes In (X, β) of subcurves of X. Given the Gromov–Witten/Pairs correspondence and the well-known Virasoro constraints for descendents in Gromov–Witten theory, there must be corresponding Virasoro constraints for the descendent theory of stable pairs. For the Hilbert schemes In (X, β) of curves, descendent constraints were studied by A. Oblomkov, A. Okounkov, and myself in Princeton a decade ago [29]. In Section 3, conjectural descendent constraints for the stable pairs theory of P3 are presented (joint work with A. Oblomkov and A. Okounkov). The moduli space of stable pairs Pn (X, β) has a virtual fundamental class in homology H∗ (Pn (X, β)). By construction, the class lifts to algebraic cycles A∗ (Pn (X, β)). In a recent paper, Junliang Shen has lifted the virtual fundamental class further to algebraic cobordism Ω∗ (Pn (X, β)). Shen’s results open a new area of exploration with beautiful structure. At the moment, the methods available to explore the virtual fundamental class in cobordism all use the theory of descendents (since the Chern classes of the virtual tangent bundle of Pn (X, β) are tautological). Shen’s work is presented in Section 4. 1. Rationality 1.1. Overview. Let X be a nonsingular projective 3-fold. Our goal here is to present the conjectures governing the rationality of the partition functions of 7 A correspondence proposed in [38] between Hilbert scheme and stable pair counting (often termed DT/PT) has been well studied, especially in the Calabi-Yau case [4, 45], but is still conjectural for most 3-folds X.

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descendent invariants for the stable pairs theory of X. The most straightforward statements are for the absolute theory, but we will present the rationality claims for the equivariant and relative stable pairs theories as well. The latter two appear naturally when studying the absolute theory: most results to date involve equivariant and relative techniques. In addition to rationality, we will also discuss the functional equation and the pole constraints for the descendent partition functions. While rationality has been established in many cases, new ideas are required to prove the conjectures in full generality. The subject intertwines the Chern characters of the universal sheaves with the geometry of the virtual fundamental class. Perhaps, in the future, a point of view will emerge from which rationality is obvious. Hopefully, the functional equation will then also be clear. At present, the geometries for which the functional equation has been proven are rather few. 1.2. Absolute theory. Let X be a nonsingular projective 3-fold. The stable pairs theory for X as presented in the introduction is the absolute case. Let β ∈ H2 (X, Z) be a nonzero class, and let γi ∈ H ∗ (X). The following conjecture8 was proposed9 in [39]. Conjecture 1 (P.-Thomas, 2007). For X a nonsingular projective 3-fold, the descendent partition function ZP (X; q |

r 

τki (γi ))β

i=1

is the Laurent expansion in q of a rational function in Q(q). In the absolute case, the descendent series satisfies a dimension constraint. For γi ∈ H ei (X), the (complex) degree of the insertion τki (γi ) is e2i + ki − 1. If the sum of the degrees of the descendent insertions does not equal the virtual dimension, " dimC [Pn (X, β)]vir = c1 (X) , 6r

β

the partition function ZP (X; q | i=1 τki (γi ))β vanishes. In case X is a nonsingular projective Calabi-Yau 3-fold, the virtual dimension of Pn (X, β) is always 0 (and no nontrivial insertions are allowed). The rationality of the basic partition function ZP (X; q | 1)β was proven10 in [4, 45] by Serre duality, wall-crossing, and a weighted Euler characteristic approach to the virtual class [2]. At the moment, the proof for Calabi-Yau 3-folds does not appear to suggest an approach in the general case. 1.3. Equivariant theory. Let X be a nonsingular quasi-projective toric 3fold equipped with an action of the 3-dimensional torus T = C ∗ × C∗ × C∗ . 8 A weaker conjecture for descendent partition functions for the Hilbert scheme I (X, β) was n proposed earlier in [25]. 9 Theorems and Conjectures are dated in the text by the year of the arXiv posting. The published dates are later and can be found in the bibliography. 10 See [40] for a similar rationality argument in a restricted (simpler) setting.

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The stable pairs descendent invariants can be lifted to equivariant cohomology ∗ (X), (and defined by residues in the open case). For equivariant classes γi ∈ HT the descendent partition function is a Laurent series in q, r /  Ä äT / ZP X; q / τki (γi ) ∈ Q(s1 , s2 , s3 )((q)) , β

i=1

with coefficients in the field of fractions of ∗ HT (•) = Q[s1 , s2 , s3 ] .

The stable pair theory for such toric X is the equivariant case. A central result of [33, 35] is the following rationality property. Theorem 1 (P.-Pixton, 2012). For X a nonsingular quasi-projective toric 3fold, the descendent partition function r / Ä äT / ZP X; q / τki (γi ) β

i=1

is the Laurent expansion in q of a rational function in Q(q, s1 , s2 , s3 ). The proof of Theorem 1 uses the virtual localization formula of [14], the capped vertex11 perspective of [27], the quantum cohomology of the Hilbert scheme of points of resolutions of Ar -singularities [26, 31], and a delicate argument for pole cancellation at the vertex [33]. In the toric case, calculations can be made effectively, but the computational methods are not very efficient. When X is a nonsingular projective toric 3-fold, Theorem 1 implies Conjecture 1 for X by taking the non-equivariant limit. However, Theorem 1 is much stronger in the toric case than Conjecture 1 since the descendent insertions may exceed the virtual dimension in equivariant cohomology. In addition to the Calabi-Yau and toric cases, Conjecture 1 has been proven in [37] for complete intersections in products of projective spaces (for descendents of cohomology classes γi restricted from the ambient space — the precise statement is presented in Section 1.9). Taken together, the evidence for Conjecture 1 is rather compelling. 1.4. First examples. Let X be a nonsingular projective Calabi-Yau 3-fold, and let C⊂X be a rigid nonsingular rational curve. Let ZP (C ⊂ X; q | 1)d[C] be the contribution to the partition function ZP (X; q | 1)d[C] obtained from the moduli of stable pairs supported on C. A localization calculation which goes back to the Gromov–Witten evaluation of [11] yields (5)

ZP (C ⊂ X; q |1)d[C] =

  (−1)(μ) (μ) μd

z(μ)

i=1

(−q)mi . (1 − (−q)mi )2

11 A basic tool in the proof is the capped descendent vertex. The 1-leg capped descendent vertex is proven to be rational in [33]. The 2-leg and 3-leg capped descendent vertices are proven to be rational in [35].

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The sum here is over all (unordered) partitions of d, 

(μ)

μ = (m1 , . . . , m(μ) ) ,

mi = d ,

i=1

and z(μ) is the standard combinatorial factor 

(μ)

z(μ) =

mi · |Aut(μ)| .

i=1

The evaluation (5) played an important role in the discovery of the Gromov– Witten/Donaldson–Thomas correspondence in [24]. In example (5), only the trivial descendent insertion 1 appears. For non-trivial insertions, consider the case where X is P3 . Let p, L ∈ H∗ (P3 ) be the point and line classes in P3 respectively. Geometrically, there is unique line through two points of P3 . The corresponding partition function is also simple, (6)

ZP (P3 ; q | τ0 (p)τ0 (p))L = q + 2q 2 + q 3 .

The resulting series is not only rational, but in fact polynomial. For curve class L, the descendent invariants in (6) vanish for Euler characteristic greater than 3. In example (6), only primary fields (with descendent subscript 0) appear. An example with higher descendents is 5 1 1 q − q2 + q3 . ZP (P3 ; q | τ2 (p))L = 12 6 12 The fractions here come from the Chern character. Again, the result is a cubic polynomial. More interesting is the partition function (7)

ZP (P3 ; q | τ5 (1))L =

−2q − q 2 + 31q 3 − 31q 4 + q 5 + 2q 6 . 18(1 + q)3

The partition functions considered so far are all in the absolute case. For an equivariant descendent series, consider the T-action on P3 defined by representation weights λ0 , λ1 , λ2 , λ3 on the vector space C4 . Let 4 p 0 ∈ HT (P3 )

be the class of the T-fixed point corresponding to the weight λ0 subspace of C4 . Then, Aq − Bq 2 + Bq 3 − Aq 4 ZP (P3 ; q |τ3 (p0 ))L = (1 + q) 2 where A, B ∈ HT (•) are given by 1 1 λ0 − (λ1 + λ2 + λ3 ) , 8 24 3 9 λ0 − (λ1 + λ2 + λ3 ) . B = 8 8 The descendent insertion here has dimension 5 which exceeds the virtual dimension 2 (•). The obvious 4 of the moduli space of stable pair, so the invariants lie in HT symmetry in all of these descendent series is explained by the conjectural function equation (discussed in Section 1.7). A =

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All of the formulas discussed above are calculated by the virtual localization formula [14] for stable pairs. The T-fixed points, virtual tangent weights, and virtual normal weights are described in detail in [39]. 1.5. Example - in degree . 2. A further example in the absolute case is the degree 2 series ZP P3 ; q | τ9 (1) 2L . While a rigorous answer could be obtained, the available computer calculation here outputs a conjecture,12 Ä −q 73q 12 ZP (P3 ; q | τ9 (1))2L = 3 3 60480(1 + q) (−1 + q) − 825q 11 − 124q 10 + 5945q 9 + 779q 8 − 36020q 7 + 60224q 6 ä − 36020q 5 + 779q 4 + 5945q 3 − 124q 2 − 825q + 73 The computer calculations of Section 1.4 all provide rigorous results and could be improved to handle higher degree curves, but the code has not yet been written. 1.6. Relative theory. Let X be a nonsingular projective 3-fold containing a nonsingular divisor D⊂X. The relative case concerns the geometry X/D. While the definitions and constructions are more involved in the relative case, the basic idea is simple. The moduli space of stable pairs on X/D includes stable pairs on X which are transverse to D. The transversality condition here has two parts: (i) the section s of the stable pair has cokernel supported away from D, (ii) the equation of D is not permitted to be a zero divisor on the support of the stable pair. Conditions (i) and (ii) are not closed conditions on stable pairs on X. In a family, the support of the cokernel of s may approach D. The solution is then to change the geometry of X by bubbling off D. In fact, by appropriately bubbling X, a compact moduli space of stable pairs Pn (X/D, β) on X/D satisfying both (i) and (ii) can be obtained. The moduli space Pn (X/D, β) parameterizes stable relative pairs s : OX[k] → F

(8) 13

on the k-step

degeneration X[k].

• The algebraic variety X[k] is constructed by attaching a chain of k copies of the 3-fold P(NX/D ⊕ OD ) equipped with 0-sections and ∞-sections ι

ι

0 ∞ D −→ P(NX/D ⊕ OD ) ←− D

defined by the summands NX/D and OD respectively. The k-step degeneration X[k] is a union X ∪D P(NX/D ⊕ OD ) ∪D P(NX/D ⊕ OD ) ∪D · · · ∪D P(NX/D ⊕ OD ) , 12 The answer relies on an old program for the theory of ideal sheaves written by A. Okounkov and a newer DT/PT descendent correspondence [29]. 13 We follow the terminology of [20, 22].

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where the attachments are made along ∞-sections on the left and 0-sections on the right. The original divisor D ⊂ X is considered an ∞-section for the attachment rules. The rightmost component of X[k] carries the last ∞-section, D∞ ⊂ X[k], called the relative divisor. The k-step degeneration also admits a canonical contraction map X[k] → X

(9)

collapsing all the attached components to D ⊂ X. • The sheaf F on X[k] is of Euler characteristic χ(F ) = n and has 1-dimensional support on X[k] which pushes-down via the contraction (9) to the class β ∈ H2 (X, Z). • The following stability conditions are required for stable relative pairs: (i) F is pure with finite locally free resolution, (ii) the higher derived functors of the restriction of F to the singular14 loci of X[k] vanish, (iii) the section s has 0-dimensional cokernel supported away from the singular loci of X[k]. (iv) the pair (8) has only finitely many automorphisms covering the automorphisms of X[k]/X. The moduli space Pn (X/D, β) of stable relative pairs is a complete DeligneMumford stack equipped with a map to the Hilbert scheme of points of D via the restriction of the pair to the relative divisor, " Pn (X/D, β) → Hilb(D, [D]) . β

5

Cohomology classes on Hilb(D, β [D]) may thus be pulled-back to the moduli space Pn (X/D, β). 5 We will use the Nakajima basis of H ∗ (Hilb(D, β [D])) indexed by a partition 5 μ of β [D] labeled by cohomology classes of D. For example, the class " |μ% ∈ H ∗ (Hilb(D, [D])) , β

6 with all cohomology labels equal to the identity, is μ−1 times the Poincar´e dual i of the closure of the subvariety formed by unions of schemes of length μ1 , . . . , μ(μ) supported at (μ) distinct points of D. The stable pairs descendent invariants in the relative case are defined using the universal sheaf just as in the absolute case. The universal sheaf is defined here 14 The singular loci of X[k] , by convention, include also the relative divisor D ∞ ⊂ X[k] even though X[k] is nonsingular along D∞ as a variety. The perspective of log geometry is more natural here.

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on the universal degeneration of X/D over Pn (X/D, β). The cohomology classes γi ∈ H ∗ (X) are pulled-back to the universal degeneration via the contraction map (9). The descendent partition function with boundary conditions μ is a Laurent series in q, r /  / ä Ä / / τki (γi ) / μ ∈ Q((q)) . ZP X/D; q / β

i=1

The basic rationality statement here is parallel to the absolute and equivariant cases. Conjecture 2. For X/D a nonsingular projective relative 3-fold, the descendent partition function r / / ä Ä / / τki (γi ) / μ ∈ Q((q)) ZP X/D; q / β

i=1

is the Laurent expansion in q of a rational function in Q(q). In case X is a nonsingular quasi-projective toric 3-fold and D ⊂ X is a toric divisor, an equivariant relative stable pairs theory can be defined. The rationality conjecture then takes the form expected by combining the rationality statements in the equivariant and relative cases. Conjecture 3. For X/D a nonsingular quasi-projective relative toric 3-fold, the descendent partition function r / / äT Ä / / τki (γi ) / μ ∈ Q(s1 , s2 , s3 )(q) ZP X/D; q / β

i=1

is the Laurent expansion in q of a rational function in Q(q, s1 , s2 , s3 ). • (X) and the Nakajima basis element Of course, both γi ∈ HT " ∗ μ ∈ HT (Hilb(D, [D])) β

must be taken here in equivariant cohomology. While the full statement of Conjecture 3 remains open, a partial result follows from Theorem 1 and [33, Theorem 2] which addresses the non-equivariant limit in the projective relative toric case. Theorem 2 (P.-Pixton, 2012). For X/D a nonsingular projective relative toric 3-fold, the descendent partition function r / / ä Ä / / τki (γi ) / μ ZP X; q / i=1

β

is the Laurent expansion in q of a rational function in Q(q). As an example of a computation in closed form in the equivariant relative case, consider the geometry of the cap, C2 × P1 /C2∞ , where C2∞ ⊂ C2 × P1 is the fiber of C 2 × P1 → P1

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over ∞ ∈ P1 . The first two factors of the 3-torus T act on the C2 -factor of the cap with tangent weights −s1 and −s2 . The third factor of T acts on P1 factor of the cap with tangent weights −s3 and s3 at 0 ∈ P1 and ∞ ∈ P1 respectively. From several perspectives, the equivariant relative descendent partition function / ∂Cap ¨ / T τd (p) / (d) qn , d > 0 Zcap P ( τd (p) | (d))d = n,d

n

is the most important series in the cap geometry [34]. Here, 2 p ∈ HT (C2 × P1 )

is the class of the T-fixed point of C2 × P1 over 0 ∈ P1 , and the Nakajima basis ∗ element (d) is weighted with the identity class in HT (Hilb(C2 , d)). A central result 15 of [34] is the following calculation. Theorem 3 (P.-Pixton, 2011). We have d q d  s1 + s2  1 + (−q)i T . Zcap P ( τd (p) | (d))d = d! 2 1 − (−q)i i=1 In the above formula, the coefficient of q d , $τd (p), (d)%Hilb(C2 ,d) =

s1 + s2 , 2 · (d − 1)!

is the classical (C∗ )2 -equivariant pairing on the Hilbert scheme of points Hilb(C2 , d). The proof of Theorem 3 is a rather delicate localization calculation (using several special properties such as the a priori divisibility of the answer by s1 + s2 from the holomorphic symplectic form on Hilb(C2 , d)). The difficulty in Theorem 3 is obtaining a closed form evaluation for all d. Any particular descendent series can be calculated by the localization methods. A calculation, for example, not covered by Theorem 3 is T 2 2 (10) Zcap P ( τ2 (p) | (1))1 = (2s1 + 3s1 s2 + 2s2 )q

(1 + q 2 ) (1 + q)2

q2 . (1 + q)2 A simple closed formula for all descendents of the cap is unlikely to exist. + (6s3 (s1 + s2 ) − 2s21 − 6s1 s2 − 2s22 )

1.7. Functional equation. In case X is a nonsingular Calabi-Yau 3-fold, the descendent series viewed as a rational function in q satisfies the symmetry 1 (11) ZP (X; | 1)β = ZP (X; q | 1)β q as conjectured in [24,38] and proven in [4,45]. In fact, a functional equation for the descendent partition function is expected to hold in all cases (absolute, equivariant, and relative). For the relative case, the functional equation is given by the following formula16 [33, 34]. 15 The

formula here differs from [34] by a factor of s1 s2 since a different convention for the cohomology class p is taken. 16 The conjecture is stated in [33, 34] with a sign error: the factor of q dβ on the right side of the functional equation [33, 34] should be (−q)dβ . Then two factors of (−1)dβ multiply to 1 and yield Conjecture 4 as stated here.

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Conjecture 4 (P.-Pixton, 2012). For X/D a nonsingular projective relative 3fold, the descendent series viewed as a rational function in q satisfies the functional equation r r / / ä / ä r Ä 1 //  Ä / / / ZP X; / τki (γi ) / μ = (−1)|μ|−(μ)+ i=1 ki q −dβ ZP X; q / τki (γi ) / μ β β q i=1 i=1 where the constants are " D, |μ|=

" (μ) = length(μ) ,

dβ =

β

c1 (X) . β

The functional equation in the absolute case is obtained by specializing the divisor D ⊂ X to the empty set in Conjecture 4: r r / r Ä 1 //  Ä ä ä / ZP X; / τki (γi ) = (−1) i=1 ki q −dβ ZP X; q / τki (γi ) . β β q i=1 i=1 The functional equation in the equivariant case is conjectured to be identical, r r / r Ä 1 //  Ä äT äT / ZP X; / τki (γi ) = (−1) i=1 ki q −dβ ZP X; q / τki (γi ) . β β q i=1 i=1 Finally, in the equivariant relative case, the functional equation is expected to be same as in Conjecture 4. As an example, the descendent series for the cap evaluated in Theorem 3 satisfies the conjectured functional equation: Å ã Å d / ãT 1 q −d s1 + s2 1  1 + (−q)−i / Zcap ; τ (p) (d) = / d P q d! s1 s2 2 i=1 1 − (−q)−i d ã Å d 1 q d s1 + s2 1  (−q)i + 1 = q 2d d! s1 s2 2 i=1 (−q)i − 1 =

(−1)d−1+d cap T ZP (q; τd (p) | (d))d . q 2d

Here, the constants for the exponent of (−1) in the functional equation are |(d)|= d ,

(d) = 1 ,

dβ = 2d .

It is straightforward to check the functional equation in all the examples of Section 1.4 - 1.5. The evidence for the functional equation for descendent series is not as large as for the rationality. For the equivariant relative cap, the functional equation is proven in [34] for all descendents series ?T > r /  / cap τki (p) / (μ) ZP i=1

d

after the specialization s3 = 0. The predicted functional equation for T Zcap P ( τ2 (p) | (1))1

before the specialization s3 = 0 can be easily checked from the formula (10). The functional equation is also known to hold for special classes of descendent insertions in the nonsingular projective toric case [36] as will be discussed in Section 2.8.

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1.8. Pole constraints. Let X be a nonsingular projective 3-fold, and let β ∈ H2 (X, Z) be a nonzero class. For β to be an effective curve class, the image of β in the lattice (12)

H2 (X, Z)/torsion

must also be nonzero. Let div(β) ∈ Z>0 be the divisibility of the image of β in the lattice (12). Conjecture 5. For d = div(β), the poles in q of the rational function r / Ä ä / τki (γi ) ZP X; q / i=1

β

may occur only at q = 0 and the roots of the polynomials { 1 − (−q)m | 1 ≤ m ≤ d }. Of the above conjectures, the evidence for Conjecture 5 is the weakest. In the Calabi-Yau case with no insertions, the statement is consistent with the GopakumarVafa conjectures concerning BPS state counts. The full prediction is based on a study of the stable pairs theory of local curves where the above pole restrictions are always found. For example, the evaluation of Theorem 3 is consistent with the pole statement (even though Theorem 3 concerns the equivariant relative case). A promotion of Conjecture 5 to cover all cases also appears reasonable. 1.9. Complete intersections. Rationality results for non-toric 3-folds are proven in [37] by degeneration methods for several geometries. The simplest to state concern nonsingular complete intersections of ample divisors X ⊂ Pn1 × · · · × Pnm . Theorem 4 (P.-Pixton, 2012). Let X be a nonsingular Fano or Calabi-Yau complete intersection 3-fold in a product of projective spaces. For even classes γi ∈ H 2∗ (X), the descendent partition function r / Ä ä / τki (γi ) ZP X; q / i=1

β

is the Laurent expansion of a rational function in Q(q). By the Lefschetz hyperplane result, the even cohomology of such X is exactly the image of the restricted cohomology from the product of projective spaces. Theorem 4 does not cover the primitive cohomology in H 3 (X). Moreover, even for descendents of the even cohomology H 2∗ (X) the functional equation and pole conjectures are open. 2. Gromov–Witten/Pairs correspondence 2.1. Overview. Let X be a nonsingular projective variety. Descendent classes on the moduli spaces of stable maps M g,r (X, β) in Gromov–Witten theory, defined using cotangent lines at the marked points, have played a central role since the beginning of the subject in the early 90s. Topological recursion relations, J-functions, and Virasoro constraints all essentially concern descendents. The importance of descendents in Gromov–Witten theory was hardly a surprise: cotangent lines on the

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RAHUL PANDHARIPANDE

moduli spaces M g,r of stable curves were basic to their geometric study before Gromov–Witten theory was developed. In case X is a nonsingular projective 3-fold, descendent invariants are defined for both Gromov–Witten theory and the theory of stable pairs. The geometric constructions are rather different, but a surprising correspondence conjecturally holds: the two descendent theories are related by a universal correspondence for all nonsingular projective 3-folds. In order words, the two descendent theories contain exactly the same data. The origin of the Gromov–Witten/Pairs correspondence is found in the study of ideal sheaves in [24, 25]. Since the descendent theory of stable pairs is much better behaved, the results and conjectures take a better form for stable pairs [36, 37]. The rationality results and conjectures of Section 1 are needed for the statement of the Gromov–Witten/Pairs correspondence. Just as in Section 1, we present the absolute, equivariant, and relative cases. A more subtle discussion of diagonals is required for the relative case. 2.2. Descendents in Gromov–Witten theory. Let X be a nonsingular projective 3-fold. Gromov–Witten theory is defined via integration over the moduli space of stable maps. Let M g,r (X, β) denote the moduli space of r-pointed stable maps from connected genus g curves to X representing the class β ∈ H2 (X, Z). Let evi : M g,r (X, β) → X , Li → M g,r (X, β) denote the evaluation maps and the cotangent line bundles associated to the marked points. Let γ1 , . . . , γr ∈ H ∗ (X), and let ψi = c1 (Li ) ∈ H 2 (M g,n (X, β)) . The descendent fields, denoted by τk (γ), correspond to the classes ψik ev∗i (γ) on the moduli space of stable maps. Let ¨

τk1 (γ1 ) · · · τkr (γr )

∂ g,β

"

r 

=

[M g,r (X,β)]vir i=1

ψiki ev∗i (γi )

denote the descendent Gromov–Witten invariants. Foundational aspects of the theory are treated, for example, in [3, 21]. Let C be a possibly disconnected curve with at worst nodal singularities. The  genus of C is defined by 1 − χ(OC ). Let M g,r (X, β) denote the moduli space of maps with possibly disconnected domain curves C of genus g with no collapsed connected components. The latter condition requires each connected component of C to represent a nonzero class in H2 (X, Z). In particular, C must represent a nonzero class β. We define the descendent invariants in the disconnected case by ¨ ∂ τk1 (γ1 ) · · · τkr (γr )

g,β

" =

r  

[M g,r (X,β)]vir i=1

ψiki ev∗i (γi ).

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DESCENDENTS FOR STABLE PAIRS ON 3-FOLDS

The associated partition function is defined by17 r r /  ä ∂ Ä ¨ / (13) ZGW X; u / τki (γi ) = τki (γi ) i=1

β

g,β

i=1

g∈Z

267

u2g−2 .

Since the domain components must map nontrivially, an elementary argument shows the genus g in the sum (13) is bounded from below. 2.3. Dimension constraints. Descendents in Gromov–Witten and stable pairs theories are obtained via tautological structures over the moduli spaces 

M g,r (X, β) ,

Pn (X, β) × X

respectively. The descendents τk (γ) in both cases mix the characteristic classes of the tautological sheaves 

Li → M g,r (X, β) ,

F → Pn (X, β) × X

with the pull-back of γ ∈ H ∗ (X) via the evaluation/projective morphism. In the absolute (nonequivariant) case, the Gromov–Witten and stable pairs descendent series r r /  /  Ä Ä ä ä / / (14) ZGW X; u / τki (γi ) , ZP X; q / τki (γi ) i=1

β

β

i=1

both satisfy dimension constraints. For γi ∈ H ei (X), the (real) dimension of the descendents Gromov–Witten and stable pairs theories are 

τki (γi ) ∈ H ei +2ki (M g,r (X, β)) ,

τki (γi ) ∈ H ei +2ki −2 (Pn (X, β)) .

Since the virtual dimensions are "  c1 (TX ) + r , dimC [M g,r (X, β)]vir =

" dimC [Pn (X, β)]vir =

β

respectively, the dimension constraints " r  ei + ki = c1 (TX ) + r , 2 β i=1

c1 (TX ) β

r  ei i=1

2

" + ki − 1 =

c1 (TX ) β

exactly match. After the matching of the dimension constraints, we can further reasonably ask if there is a relationship between the Gromov–Witten and stable pairs descendent series (14). The question has two immediately puzzling features: (i) The series involve different moduli spaces and universal structures. (ii) The variables u and q of the two series are associated to different invariants (the genus and the Euler characteristic). Though the worry (i) is correct, both moduli spaces are essentially based upon the geometry of curves in X, so there is hope for a connection. The descendent correspondence proposes a precise relationship between the Gromov–Witten and stable pairs descendent series, but only after a change of variables to address (ii). 17 Our



notation follows [25, 27] and emphasizes the role of the moduli space M g,r (X, β). The degree 0 collapsed contributions will not appear anywhere in the paper.

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2.4. Descendent notation. Let X be a nonsingular projective 3-fold. Let  ), α  = ( α1 , . . . , α   > 0, α 1 ≥ . . . ≥ α   Let be a partition of size | α| and length .

 ιΔ : Δ → X   be the inclusion of the small diagonal18 in the product X  . For γ ∈ H ∗ (X), we write  ). γ · Δ = ιΔ∗ (γ) ∈ H ∗ (X  Using the K¨ unneth decomposition, we have  γ cj1 ,...,j ˆ θj1 ⊗ . . . ⊗ θj ˆ , γ·Δ=

j1 ,...,j ˆ

where {θj } is a Q-basis of H ∗ (X). We define the descendent insertion τα (γ) by  γ cj1 ,...,j ˆ τα (θj1 ) · · · τα (15) τα  ˆ−1 (θj ˆ) . (γ) =

1 −1 j1 ,...,j ˆ

Three basic examples are: • If α  = ( a1 ), then τ( a1 ) (γ) = τa1 −1 (γ) . The convention of shifting the descendent by 1 allows us to index descendent insertions by standard partitions α  and follows the notation of [36]. a2 ) and γ = 1 is the identity class, then • If α  = ( a1 ,   c1j1 ,j2 τa1 −1 (θj1 ) τa2 −1 (θj2 ) , τ( a1 , a2 ) (1) = 

j1 ,j2

where Δ = θj1 ⊗ θj2 is the standard K¨ unneth decomposition of the diagonal in X . • If γ is the class of a point, then 1 j1 ,j2 cj1 ,j2 2

τα (p) = τα 1 −1 (p) · · · τα ˆ−1 (p).

By the multilinearity of descendent insertions, formula (15) does not depend upon the basis choice {θj }. While definition (15) provides an explicit formula for the descendent insertion  τα (γ), the action of the descendent on the moduli space of stable maps M g,ˆ(X, β) is expressed geometrically by  ˆ−1 1 −1 · · · ψ α α τα · ev∗1,...,ˆ(γ · Δ) , (γ) = ψ1 ˆ where the evaluation map is 

ˆ

ev1,...,ˆ : M g,ˆ(X, β) → X  . The diagonals play a crucial role in the Gromov–Witten/Pairs correspondence for descendents — the two moduli spaces treat the diagonals differently. 18 The small diagonal Δ is the set of points of X  for which the coordinates (x , . . . , x ) are 1 ˆ all equal xi = xj .

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2.5. Correspondence matrix. A central result of [36] is the construction of  indexed by partitions α and α a universal correspondence matrix K  of positive size 19 with  K α,α  ∈ Q[i, c1 , c2 , c3 ]((u)) .  are constructed from the capped descendent vertex [36] and The elements of K satisfy two basic properties:  (i) The vanishing K α|. α,α  = 0 holds unless |α|≥ | 20  (ii) The u coefficients of K α,α  ∈ Q[i, c1 , c2 , c3 ]((u)) are homogeneous in the variables ci of degree |α|+(α) − | α|−( α) − 3((α) − 1). Via the substitution ci = ci (TX ),

(16)

 act by cup product on the cohomology of X with Q[i]((u))the matrix elements of K coefficients.  is used to define a correspondence rule The matrix K τα1 −1 (γ1 ) · · · τα −1 (γ ) → τα1 −1 (γ1 ) · · · τα −1 (γ ) .

(17)

The definition of the right side of (17) requires a sum over all set partitions P of {1, . . . , }. For such a set partition P , each element S ∈ P is a subset of {1, . . . , }. Let αS be the associated subpartition of α, and let  γS = γi . i∈S

In case all cohomology classes γj are even, we define the right side of the correspondence rule (17) by     τα (18) τα1 −1 (γ1 ) · · · τα −1 (γ ) = (KαS ,α  · γS ) . P set partition of {1,...,} S∈P  α The second sum in (18) is over all partitions α  of positive size. However, by the vanishing of property (i),  K α| , αS ,α  = 0 unless |αS |≥ | the summation index may be restricted to partitions α  of positive size bounded by |αS |. α| in the second sum in (18). The homogeneity property (ii) Suppose |αS |= | then places a strong constraint. The u coefficients of  K αS ,α  ∈ Q[i, c1 , c2 , c3 ]((u)) are homogeneous of degree α) . 3 − 2(αS ) − (

(19)

= −1. variable ci has degree i for the homogeneity.

19 Here, i2 20 The

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 For the matrix element K αS ,α  to be nonzero, the degree (19) must be nonnegative. Since the lengths of αS and α  are at least 1, nonnegativity of (19) is only possible if α) = 1 . (αS ) = ( Then, we also have αS = α  since the sizes match. The above argument shows that the descendents on the right side of (18) all correspond to partitions of size less than |α| except for the leading term obtained from the maximal set partition {1} ∪ {2} ∪ . . . ∪ {} = {1, 2, . . . , } in  parts. The leading term of the descendent correspondence, calculated in [36],  is a third basic property of K: (iii)

τα1 −1 (γ1 ) · · · τα −1 (γ ) = (iu)(α)−|α| τα1 −1 (γ1 ) · · · τα −1 (γ ) + . . . .

In case α = 1 has all parts equal to 1, then αS also has all parts equal to 1  for every S ∈ P . By property (ii), the u coefficients of K αS ,α  are homogeneous of degree 3 − (αS ) − | α|−( α), and hence vanish unless αS = α  = (1) . Therefore, if α has all parts equal to 1, the leading term is therefore the entire  formula. We obtain a fourth property of the matrix K: (iv) τ0 (γ1 ) · · · τ0 (γ ) = τ0 (γ1 ) · · · τ0 (γ ) . In the presence of odd cohomology, a natural sign must be included in formula (18). We may write set partitions P of {1, . . . , } indexing the sum on the right side of (18) as S1 ∪ . . . ∪ S|P | = {1, . . . , }. The parts Si of P are unordered, but we choose an ordering for each P . We then obtain a permutation of {1, . . . , } by moving the elements to the ordered parts Si (and respecting the original order in each group). The permutation, in turn, determines a sign σ(P ) determined by the anti-commutation of the associated odd classes. We then write     τα1 −1 (γ1 ) · · · τα −1 (γ ) = (−1)σ(P ) τα (KαSi ,α  · γS i ) . Si ∈P P set partition of {1,...,} α  The descendent τα1 −1 (γ1 ) · · · τα −1 (γ ) is easily seen to have the same commutation rules with respect to odd cohomology as τα1 −1 (γ1 ) · · · τα −1 (γ ).  in [36] expresses the coefficients explicitly in The geometric construction of K terms of the 1-legged capped descendent vertex for stable pairs and stable maps. These vertices can be computed (as a rational function in the stable pairs case and term by term in the genus parameter for stable maps). Hence, the coefficient  K α,α  ∈ Q[i, c1 , c2 , c3 ]((u)) can, in principle, be calculated term by term in u. The calculations in practice are quite difficult, and complete closed formulas are not known for all of the coefficients.

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DESCENDENTS FOR STABLE PAIRS ON 3-FOLDS

271

2.6. Absolute case. To state the descendent correspondence proposed in [36] for all nonsingular projective 3-folds X, the basic degree " c1 (X) ∈ Z dβ = β

associated to the class β ∈ H2 (X, Z) is required. Conjecture 6 (P.-Pixton (2011)). Let X be a For γi ∈ H ∗ (X), we have / Ä ä / (−q)−dβ /2 ZP X; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) β Ä dβ  = (−iu) ZGW X; u

nonsingular projective 3-fold.

/ ä / / τα1 −1 (γ1 ) · · · τα −1 (γ )

β

under the variable change −q = e . iu

Since the stable pairs side of the correspondence / Ä ä / ZP X; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) ∈ Q((q)) β

is defined as a series in q, the change of variable −q = eiu is not a priori welldefined. However, the stable pairs descendent series is predicted by Conjecture 1 to be a rational function in q. The change of variable −q = eiu is well-defined for a rational function in q by substitution. The well-posedness of Conjecture 6 therefore depends upon Conjecture 1. 2.7. Geometry of descendents. Let X be a nonsingular projective 3-fold, and let D ⊂ X be a nonsingular divisor. The Gromov–Witten descendent insertion τ1 (D) has a simple geometric leading term. Let [f : (C, p) → X] ∈ M g,1 (X, β) be a stable map. Let ev1 : M g,1 (X, β) → X be the evaluation map at the marking. The cycle ev−1 1 (D) ⊂ M g,1 (X, β) corresponds to stable maps with f (p) ∈ D. On the locus ev−1 1 (D), there is a differential (20)

df : TC,p → NX/D,f (p)

from the tangent space of C at p to the normal space of D ⊂ X at f (p) ∈ D. The differential (20) on ev−1 1 (D) vanishes on the locus where f (C) is tangent to D at p. In other words, . ∗ τ1 (D) + τ0 (D2 ) = ev−1 1 (D) −c1 (TC,p ) + ev1 (NX/D ) has the tangency cycle as a leading term. There are correction terms from the loci where p lies on a component of C contracted by f to a point of D. A parallel relationship can be pursued for τk (D) for for higher k in terms of the locus of stable maps with higher tangency along D at f (p). A full correction calculus in case X has dimension 1 (instead of 3) was found in [30]. The method

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272

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has also been successfully applied to calculate the characteristic numbers of curves in P2 for genus at most 2 in [13].21 By the Gromov–Witten/Pairs correspondence of Conjecture 6, the stable pairs descendent τk (D) has leading term on the Gromov–Witten side τk (D) = (iu)−k τk (D) + . . . . Hence, the descendents τk (D) on the stable pairs side should be viewed as essentially connected to the tangency loci associated to the divisor D ⊂ X. 2.8. Equivariant case. If X is a nonsingular quasi-projective toric 3-fold, all terms of the descendent correspondence have T-equivariant interpretations. We take the equivariant K¨ unneth decomposition in (15), and the equivariant Chern classes ci (TX ) with respect to the canonical T-action on TX in (16). The toric case is proven in [36]. Theorem 5 (P.-Pixton, 2011). Let X be a nonsingular quasi-projective toric ∗ 3-fold. For γi ∈ HT (X), we have / äT Ä / (−q)−dβ /2 ZP X; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) β / Ä äT / = (−iu)dβ ZGW X; u / τα1 −1 (γ1 ) · · · τα −1 (γ ) β

under the variable change −q = e . iu

Since the stable pairs side of the correspondence / Ä äT / ZP X; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) ∈ Q(s1 , s2 , s3 )((q)) β

is a rational function in q by Theorem 1, the change of variable −q = eiu is welldefined by substitution. When X is a nonsingular projective toric 3-fold, Theorem 5 implies Conjecture 6 for X by taking the non-equivariant limit. However, Theorem 5 is much stronger in the toric case than Conjecture 6 since the descendent insertions may exceed the virtual dimension in equivariant cohomology. In case α = (1) has all parts equal to 1, Theorem 5 specializes by property (iv) of Section 2.5 to the simpler statement / Ä äT / (21) (−q)−dβ /2 ZP X; q / τ0 (γ1 ) · · · τ0 (γ ) β / Ä äT / = (−iu)dβ ZGW X; u / τ0 (γ1 ) · · · τ0 (γ ) β

which was first proven in the context of ideal sheaves in [27]. Viewing both sides of (21) as series in u, we can complex conjugate the coefficients. Imaginary numbers only occur in −q = eiu and (−iu)dβ . After complex conjugation, we find Ä 1 // äT (−q)dβ /2 ZP X; / τ0 (γ1 ) · · · τ0 (γ ) β q / Ä äT / = (iu)dβ ZGW X; u / τ0 (γ1 ) · · · τ0 (γ ) β

21 In

higher genus, the correction calculus in P2 was too complicated to easily control.

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and thus obtain the functional equation / Ä 1 // Ä äT äT / = q −dβ ZP X; q / τ0 (γ1 ) · · · τ0 (γ ) ZP X; / τ0 (γ1 ) · · · τ0 (γ ) β β q as predicted by Conjecture 4. 2.9. Relative case. 2.9.1. Relative Gromov–Witten theory. Let X be a nonsingular projective 3fold with a nonsingular divisor D⊂X. The relative theory of stable pairs was discussed in Section 1.6. A parallel relative Gromov–Witten theory of stable maps with specified tangency along the divisor D can also be defined. In Gromov–Witten theory, relative conditions are represented by a partition 5 μ of the integer β [D], each part μi of which is marked by a cohomology class δi ∈ H ∗ (D, Z), (22)

μ = ((μ1 , δ1 ), . . . , (μ , δ )) .

The numbers μi record the multiplicities of intersection with D while the cohomol ogy labels δi record where the tangency occurs. More precisely, let M g,r (X/D, β)μ be the moduli space of stable relative maps with tangency conditions μ along D. To impose the full boundary condition, we pull-back the classes δi via the evaluation maps 

M g,r (X/D, β)μ → D

(23)

at the points of tangency. Also, the tangency points are considered to be unordered.22 Relative Gromov–Witten theory was defined before the study of stable pairs. For the foundations, including the definition of the moduli space of stable relative maps and the construction of the virtual class 



[M g,r (X/D, β)μ ] ∈ H∗ (M g,r (X/D, β)μ ) , we refer the reader to [19, 20]. 2.9.2. Diagonal classes. Definition (18) of the Gromov–Witten/Pairs correspondence in the absolute case involves the diagonal ιΔ : Δ → X s via (15). For the correspondence in the relative case, the diagonal has a more subtle definition. For the absolute geometry X, the product X s naturally parameterizes s ordered (possibly coincident) points on X. For the relative geometry X/D, the parallel object is the moduli space (X/D)s of s ordered (possibly coincident) points (p1 , . . . , ps ) ∈ X/D . 22 The evaluation maps are well-defined only after ordering the points. We define the theory first with ordered tangency points. The unordered theory is then defined by dividing by the automorphisms of the cohomology weighted partition μ.

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RAHUL PANDHARIPANDE

The points parameterized by (X/D)s are not allowed to lie on the relative divisor D. When the points approach D, the target X degenerates. The resulting moduli space (X/D)s is a nonsingular variety. Let Δrel ⊂ (X/D)s be the small diagonal where all the points pi are coincident. As a variety, Δrel is isomorphic to X. The space (X/D)s is a special case of well-known constructions in relative geometry. For example, (X/D)2 consists of 6 strata:

1• 2•

X

D

  

  

1•

1•

2•

2• D

X

D

  

X

   1• 2• D X

  

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DESCENDENTS FOR STABLE PAIRS ON 3-FOLDS

275

      1• D 2•

    

X

  

 

2• D 1•

    

X

As a variety, (X/D)2 is the blow-up of X 2 along D2 . And, Δrel ⊂ (X/D)2 is the strict transform of the standard diagonal. Select a subset S of cardinality s from the r markings of the moduli space of  maps. Just as M g,r (X, β) admits a canonical evaluation to X s via the selected 

markings, the moduli space M g,r (X/D, β)μ admits a canonical evaluation 

evS : M g,r (X/D, β)μ → (X/D)s , well-defined by the definition of a relative stable map (the markings never map to the relative divisor). The class 

ev∗S (Δrel ) ∈ H ∗ (M g,r (X/D, β)μ ) plays a crucial role in the relative descendent correspondence. By forgetting the relative structure, we obtain a projection π : (X/D)s → X s . The product contains the standard diagonal Δ ⊂ X s . However, π ∗ (Δ) = Δrel . The former has more components in the relative boundary if D = ∅.  Let 2.9.3. Relative descendent correspondence. Let α  be a partition of length .  Δrel be the cohomology class of the small diagonal in (X/D) . For a cohomology class γ of X, let  ), γ · Δ ∈ H ∗ ((X/D) rel

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276

RAHUL PANDHARIPANDE

where Δref is the small diagonal of Section 2.9.2. Define the relative descendent insertion τα (γ) by (24)

 ˆ−1 1 −1 · · · ψ α α τα · ev∗1,...,ˆ(γ · Δrel ) . (γ) = ψ1 ˆ

In case, D = ∅, definition (24) specializes to (15). Let ΩX [D] denote the locally free sheaf of differentials with logarithmic poles along D. Let TX [−D] = ΩX [D] ∨ denote the dual sheaf of tangent fields with logarithmic zeros. For the relative geometry X/D, the coefficients of the correspondence matrix  K act on the cohomology of X via the substitution ci = ci (TX [−D]) instead of the substitution ci = ci (TX ) used in the absolute case. Then, we define     τα1 −1 (γ1 ) · · · τα −1 (γ ) = τα (25) (KαS ,α  · γS ) S∈P P set partition of {1,...,l} α  as before via (24) instead of (15). Definition (25) is for even classes γi . In the presence of odd γi , a sign has to be included exactly as in the absolute case. Conjecture 7. For γi ∈ H ∗ (X), we have / / ä Ä / / (−q)−dβ /2 ZP X/D; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) / μ β / / ä Ä / / dβ +(μ)−|μ|  = (−iu) ZGW X/D; u / τa1 −1 (γ1 ) · · · τα −1 (γ ) / μ

β

under the variable change −q = eiu . The change of variables is well-defined by the rationality of Conjecture 2. A case in which Conjecture 7 is proven is when X is a nonsingular projective toric 3-fold and D ⊂ X is a toric divisor. The rationality of the stable pairs series is given by Theorem 2. The following result can be obtained by the methods of [37]. Theorem 6. For X/D a nonsingular projective relative toric 3-fold, the descendent partition function For γi ∈ H ∗ (X), we have / / ä Ä / / (−q)−dβ /2 ZP X/D; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) / μ β / / ä Ä / / = (−iu)dβ +(μ)−|μ| ZGW X/D; u / τa1 −1 (γ1 ) · · · τα −1 (γ ) / μ β

under the variable change −q = e . iu

Conjecture 7 can be lifted in a canonical way to the equivariant relative case (as in the rationality of Conjecture 3). Some equivariant relative results are proven in [37]. 2.10. Complete intersections. Let X be a Fano or Calabi-Yau complete intersection of ample divisors in a product of projective spaces, X ⊂ Pn1 × · · · × Pnm . A central result of [37] is the proof of the descendent correspondence for even classes.

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Theorem 7 (P.-Pixton, 2012). Let X be a nonsingular Fano or Calabi-Yau complete intersection 3-fold in a product of projective spaces. For even classes γi ∈ H 2∗ (X), we have / Ä ä / (−q)−dβ /2 ZP X; q /τα1 −1 (γ1 ) · · · τα −1 (γ ) β / ä Ä / dβ  = (−iu) ZGW X; u / τα1 −1 (γ1 ) · · · τα −1 (γ ) β

under the variable change −q = e . iu

Theorem 7 relies on the rationality of the stable pairs series of Theorem 4. For γi ∈ H 2∗ (X) even classes of positive degree, we obtain from Theorem 7 (under the same complete intersection hypothesis for X) the following result where only the leading term of the correspondence contributes: ? > / r s /   / −dβ /2 ZP X; q / τ0 (γi ) τkj (p) = (−q) / i=1 j=1 β ? > / r s /    / (−iu)dβ (iu)− kj ZGW X; u / τ0 (γi ) τkj (p) / i=1

j=1

β

under the variable change −q = e . Just as in the analysis of (21), the above correspondence proves the functional equation of Conjecture 4 in the case at hand. If we specialize Theorem 7 further to the case where there are no descendent insertions, we obtain Ä ä Ä ä ZP X; q = ZGW X; u iu

β

β

under the variable change −q = eiu for Calabi-Yau complete intersections in a product of projective spaces. In particular, the Gromov–Witten/Pairs correspondence hold for the famous quintic Calabi-Yau 3-fold X5 ⊂ P4 . 2.11. K3 fibrations. Let Y be a nonsingular projective toric 3-fold for which the anticanonical class KY∗ is base point free and the generic anticanonical divisor is a nonsingular projective K3 surface S. Let (26)

X ⊂ Y × P1

be a nonsingular hypersurface in the class KY∗ ⊗ KP∗1 . Using the degeneration X  Y ∪ S × P1 ∪ Y obtained by factoring a divisor of KY∗ ⊗ KP∗1 , the results of [37] yield the Gromov– Witten/Pairs correspondence for the Calabi-Yau 3-fold X.23 The hypersurface X defined by (26) is a K3-fibered Calabi-Yau 3-fold. A very natural question to ask is whether the Gromov–Witten/Pairs correspondence can be proven for all K3-fibered 3-folds. While the general case is open, results for the correspondence in fiber classes can be found in [42].24 23 The strategy here is simpler than presented in Appendix B of [42] for a particular toric 4-fold Y . 24 Parallel questions can be pursued for other surfaces. For surfaces of general type (involving the stable pairs theory of descendents), see [15].

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3. Virasoro constraints 3.1. Overview. Descendent partition functions in Gromov–Witten theory are conjectured to satisfy Virasoro constraints [9] for every target variety X. Via the Gromov–Witten/Pairs descendent correspondence, we expect parallel constraints for the descendent theory of stable pairs. An ideal path to finding the constraints for stable pairs would be to start with the explicit Virasoro constraints in Gromov– Witten theory and then apply the correspondence. However, our knowledge of the correspondence matrix is not yet sufficient for such an application. Another method is to look experimentally for relations which are of the expected shape. In a search conducted almost 10 years ago with A. Oblomkov and A. Okounkov, we found a set of such relations for the theory of ideal sheaves [29] for every nonsingular projective 3-fold X. As an example, the equations for P3 are presented here for stable pairs.25 3.2. First equations. Let X be a nonsingular projective 3-fold. The descendent insertions τ0 (1) , τ0 (D) for D ∈ H 2 (X), τ1 (1) all satisfy simple equations (parallel to the string, divisor, and dilation equations in Gromov–Witten theory): / Ä ä 6 / (i) ZP X; q / τ0 (1) · ri=1 τki (γi ) = 0, / /6 Ä Ä äβ Ä5 ä ä 6r / / (ii) ZP X; q / τ0 (D) · i=1 τki (γi ) = β D ZP X; q / ri=1 τki (γi ) , β / /6 Ä ä Ä äβ Ä ä 6 / / d d − 2β ZP X; q / ri=1 τki (γi ) . (iii) ZP X; q / τ1 (1) · ri=1 τki (γi ) = q dq β

β

All three are obtained directly from the definition of the descendent action given in Section 0.2. To prove (iii), the Hirzebruch-Riemann-Roch equation dβ ch3 (F ) = n − 2 is used for a stable pair " [F, s] ∈ Pn (X, β) , dβ = c1 (X) . β

The compatibility of (i) and (ii) with the functional equation of Conjecture 4 is trivial. While not as obvious, the differential operator dβ d q − dq 2 is also beautifully consistent with Conjecture 4. We can easily prove using (iii) that Conjecture 4 holds for r / Ä ä  / τki (γi ) ZP X; q / τ1 (1) · β

i=1

if and only if Conjecture 4 holds for r / Ä ä / ZP X; q / τki (γi ) . i=1

β

25 Since [29] is written for ideal sheaves, a DT/PT correspondence for descendents is needed to move the relations to the theory of stable pairs. Such a correspondence is also studied in [29]. I am very grateful to A. Oblomkov for his help with the formulas here.

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For example, equation (iii) yields ZP (P3 ; q | τ1 (1)τ5 (1))L =

q + 4q 2 + 17q 3 − 62q 4 + 17q 5 + 4q 6 + q 7 9(1 + q)4

when applied to (7). 3.3. Operators and constraints. A basis of the cohomology H ∗ (P3 ) is given by 1 , H , L = H2 , p = H3 where H is the hyperplane class. The divisor and dilaton equations here are / / ä Ä Ä ä / / ZP P3 ; q / τ0 (H) · D) = dZP P3 ; q / D , dL dL ã Å / / ä Ä Ä ä d / / ZP P3 ; q / τ1 (1) · D = q − 2d ZP P3 ; q / D , dL dL dq 6 where D = ri=1 τki (γi ) is an arbitrary descendent insertion. Before presenting the formulas, we introduce two conventions which simplify the notation. The first concerns descendents with negative subscripts. We define the descendent action in two negative cases: (27)

τ−2 (Hj ) = −δj,3 ,

τ−1 (Hj ) = 0 .

In particular, these all vanish except for τ−2 (p) = −1. Convention (27) is consistent with Definition 2 via the replacement ch2+i (F) → ch2+i (I[1]• ) , where I• is the universal stable pair on X × Pn (X, β). For the Virasoro constraints, the formulas are more naturally stated in terms of the Chern character subscripts (instead of including the shift by 2 in Definition 2). As a second convention, we define the insertions chi (γ) by (28)

chi (γ) = τi−2 (γ)

for all i ≥ 0. In particular, ch0 (p) acts as −1 and ch1 (Hj ) acts as 0. Let D+ be the free Q-polynomial ring with generators / ¶ © / chi (Hj ) / i ≥ 0 , j = 0, 1, 2, 3 . Via equation (28), we view D+ as an extension D ⊂ D+ of the algebra of descendents defined in Section 0.3. We define cha chb (Hj ) ∈ D+ in terms of the generators by cha chb (Hj ) =



cha (γrL )chb (γrR )

r

where the sum is indexed by the K¨ unneth decomposition  Hj · Δ = γrL ⊗ γrR ∈ H ∗ (P3 × P3 ) r

and Δ ⊂ P3 × P3 is the diagonal. Both chi (Hj ) and cha chb (Hj ) define operators on D+ by multiplication.

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RAHUL PANDHARIPANDE

To write the Virasoro relations, we will define derivations Rk : D+ → D+ for k ≥ −1 by the following action on the generators of D+ , > k ?  . j (i + j − 3 + n) chk+i (Hj ) . Rk chi (H ) = n=0

In case k = −1, the product on the right is empty and . R−1 chi (Hj ) = chi−1 (Hj ) . Definition 3. Let Lk : D+ → D+ for k ≥ −1 be the operator  L R Lk = −2 (−1)d d (a + dL − 3)! (b + dR − 3)! cha chb (H) a+b=k+2

+



a! b! cha chb (p)

a+b=k

+ Rk + (k + 1)! R−1 chk+1 (p) . The first term in the formula for Lk requires explanation. By definition, (29)

cha chb (H) = cha (p)chb (H) + cha (L)chb (L) + cha (H)chb (p)

via the three terms of the K¨ unneth decomposition of H · Δ. The notation L R

(−1)d

d

(a + dL − 3)! (b + dR − 3)! cha chb (H)

is shorthand for the sum (−1)3·1 (a + 3 − 3)! (b + 1 − 3)! cha (p)chb (H) + (−1)2·2 (a + 2 − 3)! (b + 2 − 3)! cha (L)chb (L) + (−1)1·3 (a + 1 − 3)! (b + 3 − 3)! cha (H)chb (p) . The three summands of (29) are each weighted by the factor L R

(−1)d

d

(a + dL − 3)! (b + dR − 3)!

where dL is the (complex) degree of γ L and dR is the (complex) degree of γ R with respect to the K¨ unneth summand γ L ⊗ γ R . In the second term of the formula, a! b! cha chb (p) can be expanded as a! b! cha chb (p) = a! b! cha (p)chb (p) . The summations over a and b in the first two terms in the formula for Lk require a ≥ 0 and b ≥ 0. All factorials with negative arguments vanish. For example, the formula for the first operator L−1 is L−1

= R−1 + 0! R−1 ch0 (p) .

For L0 , we have L0

= −2 · (−1)3·1 (0 + 3 − 3)! (2 + 1 − 3)! ch0 (p)ch2 (H) −2 · (−1)2·2 (1 + 2 − 3)! (1 + 2 − 3)! ch1 (L)ch1 (L) −2 · (−1)1·3 (2 + 1 − 3)! (0 + 3 − 3)! ch2 (H)ch0 (p) +ch0 (p)ch0 (p) +R0 + R−1 ch1 (p) .

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After simplification, we obtain L0 = 4ch0 (p)ch2 (H) − 2ch1 (L)ch1 (L) + ch0 (p)ch0 (p) + R0 + R−1 ch1 (p) . The operators Lk on D+ are conjectured to be the analogs for stable pairs of the Virasoro constraints for the Gromov–Witten theory of P3 . Conjecture 8 (Oblomkov-Okounkov-P.). We have ZP (P3 ; q | Lk D)dL = 0 for all k ≥ −1, for all D ∈ D+ , and for all curve classes dL. For example, for k = −1, Conjecture 8 states ZP (P3 ; q | L−1 D)dL = 0 . By the above calculation of L−1 , ZP (P3 ; q | L−1 D)dL

Ä = ZP P3 ; q Ä = ZP P3 ; q

/ ä / / (R−1 + 0! R−1 ch0 (p)) D dL / ä / / (R−1 − R−1 ) D dL

= 0, where we have also used the descendent action ch0 (p) = −1. The claim ZP (P3 ; q | L0 D)dL = 0 . is easily reduced to the divisor equation (ii) of Section 3.2 and is also true. The first nontrivial assertion of Conjecture 8 occurs for k = 1, / Ä ä / = 0, ZP (P3 ; q | L1 D)dL = ZP P3 ; q / ( − 4ch3 (H) + R1 + 2ch2 (p)R−1 ) D dL

which is at the moment unproven. For example, let D = ch3 (p) and d = 1. We obtain a prediction for descendent series for P3 , −4ZP (ch3 (H)ch3 (p))L + 12ZP (ch4 (p))L + 2ZP (ch2 (p)ch2 (p))L = 0 , which can be checked using the evaluations ZP (ch3 (H)ch3 (p))L =

ZP (τ1 (H)τ1 (p))L =

ZP (ch4 (p))L =

ZP (τ2 (p))L =

ZP (ch2 (p)ch2 (p))L =

ZP (τ0 (p)τ0 (p))L =

3 3 3 q − q2 + q3 , 4 2 4 5 1 1 q − q2 + q3 , 12 6 12 q + 2q 2 + q 3 .

3.4. The bracket. To find the Virasoro bracket, we introduce the operators  L R Lk = −2 (−1)d d (a + dL − 3)! (b + dR − 3)! cha chb (H) a+b=k+2

+



a! b! cha chb (p)

a+b=k

+Rk . We then obtain the Virasoro relations and the bracket with chk (p), [Lk , Lm ] = (m − k)Lk+m ,

[Ln , k! chk (p)] = k · (k + n)! chn+k (p).

The operators Lk are expressed in terms of Lk by: Lk = Lk + (k + 1)! L−1 chk+1 (p).

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RAHUL PANDHARIPANDE

4. Virtual class in algebraic cobordism 4.1. Overview. Let X be nonsingular projective 3-fold. From the work of J. Shen [43], the virtual fundamental class of the moduli space of stable pairs [Pn (X, β)]vir ∈ Adβ (Pn (X, β)) admits a canonical lift to the theory of algebraic cobordism26 (30)

5

[Pn (X, β)]vir ∈ Ωdβ (Pn (X, β))

where dβ = β c1 (X). Shen’s construction depends only upon the 2-term perfect obstruction theory of Pn (X, β) and is closely related to earlier work of CiocanFontantine and Kapranov [5] and Lowrey-Sch¨ urg [23]. The lift (30) leads to several natural questions. The simplest is how does the virtual class in algebraic cobordism vary with n? Let π : Pn (X, β) → • be the structure map to the point •. Then, for fixed β, we define  q n π∗ [Pn (X, β)]vir ∈ Ωdβ (•) ⊗Q Q((q)) . ZΩ P (X; q)β = n∈Z

ZΩ P (X; q)β

Is there an analogue for of the rationality and functional equation in the descendent theory of the standard virtual class? 4.2. Chern numbers. While the full data of the cobordism class (30) is difficult to analyze, the push-forward π∗ [Pn (X, β)]vir ∈ Ωdβ (•) is characterized by the virtual Chern numbers of Pn (X, β). Since Pn (X, β) has a 2-term perfect obstruction theory, there is a virtual tangent complex Tvir ∈ Db (Pn (X, β)) with Chern classes ci (Tvir ) ∈ H 2i (Pn (X, β)) . For every partition of the virtual dimension dβ , σ = (s1 , . . . , s ) ,

dβ =

 

si ,

i=1

we define an associated Chern number " σ cn,β =

 

csi (Tvir ) ∈ Z

[Pn (X,β)]vir i=1

by integration against the standard virtual class [Pn (X, β)]vir ∈ H2dβ (Pn (X, β)) . The complete collection of Chern numbers { cσn | σ ∈ Partitions(dβ ) } 26 We do not review the foundations of the theory of algebraic cobordism here. The reader can find discussions in [17, 18]. As for cohomology, we always take Q-coefficients. Shen constructs a canonical lift to algebraic cobordism [M ]vir ∈ Ω∗ (M ) of the virtual class in Chow [M ]vir ∈ A∗ (M ) obtained from a 2-term perfect obstruction theory on a quasi-projective scheme M .

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uniquely determines the algebraic cobordism class π∗ [Pn (X, β)]vir ∈ Ωdβ (•) . 4.3. Rationality and the functional equation. The rationality of the partition function ZΩ P (X; q)β is equivalent to the rationality of all the functions  ZσP (X; q)β = cσn,β q n n∈Z

for σ ∈ Partitions(dβ ). Theorem 8 (Shen 2014). The Chern class ci (Tvir ) ∈ H 2i (Pn (X, β)) can be written as a Q-linear combination of products of descendent classes ; 8 r r /   / ∗ τki (γi ) / ki ≡ 0 mod 2 , γi ∈ H (X) i=1

i=1

by a formula which is independent of n and β. Shen’s proof is geometric and constructive. Following the notation of Section 0.2, let πP : X × Pn (X, β) → Pn (X, β) be the projection and let I• ∈ Db (X × Pn (X, β)) be the universal stable pair. The class of the virtual tangent complex in K 0 (Pn (X, β)) is [−Tvir ] = [RπP ∗ RHom(I• , I• )0 ] = [RπP ∗ (I• ⊗L (I• ))∨ ] − [RπP ∗ OX×Pn (X,β) ] . The Chern character of −Tvir is then computed by the Grothendieck-RiemannRoch formula, Ä Ä ä ä (31) ch[−Tvir ] = πP ∗ ch(I• ) · ch((I• )∨ ) · Td(X) − πP ∗ Td(X) . 5 The second term of (31) is just X Td3 (X) times the identity 1 ∈ H 0 (Pn (X, β)). More interesting is the first term of (31) which can be written as Ä ä (32)

∗ ch(I• ) · ch((I• )∨ ) · Δ · Td(X) where is the projection

: X × X × Pn (X, β) → Pn (X, β) , I• and I• are the universal stable pairs pulled-back via the first and second projections X × Pn (X, β) ← X × X × Pn (X, β) → X × Pn (X, β) respectively, and Δ is the pull-back of the diagonal in X × X. Using the K¨ unneth decomposition of Δ, Shen easily writes (32) as a quadratic expression in the descendent classes — see [43, Section 3.1]. The answer is a universal formula independent of n and β. Though not explicitly remarked (nor needed) in [43], Shen’s universal formula for ch[−Tvir ] is a Q-linear combination of classes @ / A / τk1 (γ1 )τk2 (γ2 ) / k1 + k2 ≡ 0 mod 2 , γ1 , γ2 ∈ H ∗ (X)

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284

RAHUL PANDHARIPANDE

since each quadratic term appears in (32) in a form proportional to ((−1)k1 + (−1)k2 ) · τk1 (γ1 )τk2 (γ2 ) because of the universal stable pair ch(I• ) appears together with the dual ch((I• )∨ ). There are two immediate consequences of Theorem 8. If the rationality of descendent series of Conjecture 1 holds for X, then ZΩ P (X; q)β is the Laurent expansion of a rational function in Ωdβ (•) ⊗Q Q(q) . In particular, Shen’s results yield the rationality of the partition functions in algebraic cobordism in case X is a nonsingular projective toric variety (where rationality of the descendent series is proven). The second consequence concerns the functional equation. The descendents which arise in Theorem 8 have even subscript sum. Hence, if the functional equation of Conjecture 4 holds for X, then Å ã 1 Ω (33) ZP X; = q −dβ ZΩ P (X; q)β . q β The functional equation (33) should be regarded as the correct generalization to all X of the symmetry Å ã 1 = ZP (Y ; q)β ZP Y ; q β of stable pairs invariants for Calabi-Yau 3-folds Y . 4.4. An example. A geometric basis of Ω∗ (•) is given by the classes of products of projective spaces. As an example, we write the series 3 ZΩ P (P ; q)L ∈ Ω4 (•) ⊗Q Q(q)

in terms of products of projective spaces: 3 ZΩ P (P ; q)L

=

[P4 ] · f4 (q) +[P3 × P1 ] · f31 (q) +[P2 × P2 ] · f22 (q) +[P2 × P1 × P1 ] · f211 (q) +[P1 × P1 × P1 × P1 ] · f1111 (q) ,

where the rational functions27 are given by f4 (q)

=

f31 (q)

=

f22 (q)

=

f211 (q)

=

f1111 (q)

=

27 I

−4q − 40q 2 − 4q 3 ,  q 823 4 21 6 823 2 21 + 139q + q + 446q 3 + q + 139q 5 + q 4 (1 + q) 2 2 2 2

,

6q + 60q 2 + 6q 3 , . q −18 − 264q − 774q 2 − 816q 3 − 774q 4 − 264q 5 − 18q 6 , (1 + q)4 Ä 13 q + 115q + 490q 2 + 889q 3 + 1215q 4 6 (1 + q) 2 13 8 ä +889q 5 + 490q 6 + 115q 7 + q . 2

am very grateful to J. Shen for providing these formulas.

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4.5. Further directions. The study of the virtual class in algebraic cobordism of the moduli space of stable pairs Pn (X, β) is intimately connected with the study of descendents invariants. The basic reason is because the Chern classes of the virtual tangent complex are tautological classes of Pn (X, β) in the sense of Section 0.3. If another approach to the virtual class in algebraic cobordism class could be found, perhaps the implications could be reversed and results about descendent series could be proven. Acknowledgments Discussions with J. Bryan, S. Katz, D. Maulik, G. Oberdieck, A. Oblomkov, A. Okounkov, A. Pixton, J. Shen, R. Thomas, Y. Toda, and Q. Yin about stable pairs and descendent invariants have played an important role in my view of the subject. The perspective of the paper is based in part on my talk Why descendents? at the Newton institute in Cambridge in the spring of 2011, though much of the progress discussed here has happened since then. References [1] M. Aganagic, A. Klemm, M. Mari˜ no, and C. Vafa, The topological vertex, Comm. Math. Phys. 254 (2005), no. 2, 425–478. MR2117633 [2] K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338. MR2600874 [3] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR1437495 [4] T. Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), no. 4, 969–998. MR2813335 [5] I. Ciocan-Fontanine and M. Kapranov, Virtual fundamental classes via dg-manifolds, Geom. Topol. 13 (2009), no. 3, 1779–1804. MR2496057 [6] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR1079726 [7] S. K. Donaldson and R. P. Thomas, Gauge theory in higher dimensions, The geometric universe (Oxford, 1996), Oxford Univ. Press, Oxford, 1998, pp. 31–47. MR1634503 [8] S´ eminaire de G´ eom´ etrie Analytique (French), Soci´ et´ e Math´ ematique de France, Paris, 1976. ´ Tenu ` a l’Ecole Normale Sup´erieure, Paris, 1974–75; Dirig´ e par Adrien Douady et Jean-Louis Verdier; Ast´ erisque, No. 36-37. MR0424820 [9] T. Eguchi, K. Hori, and C.-S. Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B 402 (1997), no. 1-2, 71–80. MR1454328 [10] G. Ellingsrud and S. A. Strømme, Towards the Chow ring of the Hilbert scheme of P2 , J. Reine Angew. Math. 441 (1993), 33–44. MR1228610 [11] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR1728879 [12] C. Faber and R. Pandharipande, Tautological and non-tautological cohomology of the moduli space of curves, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, pp. 293–330. MR3184167 [13] T. Graber, J. Kock, and R. Pandharipande, Descendant invariants and characteristic numbers, Amer. J. Math. 124 (2002), no. 3, 611–647. MR1902891 [14] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR1666787 [15] M. Kool and R. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, arXiv:1605.02576. [16] J. Le Potier, Faisceaux semi-stables et syst` emes coh´ erents (French, with French summary), Vector bundles in algebraic geometry (Durham, 1993), London Math. Soc. Lecture Note Ser., vol. 208, Cambridge Univ. Press, Cambridge, 1995, pp. 179–239. MR1338417

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[17] M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR2286826 [18] M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63–130. MR2485880 [19] A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218. MR1839289 [20] J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR1938113 [21] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174. MR1467172 [22] J. Li and B. Wu, Good degeneration of Quot-schemes and coherent systems, Comm. Anal. Geom. 23 (2015), no. 4, 841–921. MR3385781 [23] P. E. Lowrey and T. Sch¨ urg, Derived algebraic cobordism, J. Inst. Math. Jussieu 15 (2016), no. 2, 407–443. MR3466543 [24] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263–1285. MR2264664 [25] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006), no. 5, 1286–1304. MR2264665 [26] D. Maulik and A. Oblomkov, Quantum cohomology of the Hilbert scheme of points on An resolutions, J. Amer. Math. Soc. 22 (2009), no. 4, 1055–1091. MR2525779 [27] D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande, Gromov-Witten/DonaldsonThomas correspondence for toric 3-folds, Invent. Math. 186 (2011), no. 2, 435–479. MR2845622 [28] P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337–345. MR0316452 [29] A. Oblomkov, A. Okounkov, and R. Pandharipande, in preparation. [30] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR2199225 [31] A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010), no. 3, 523–557. MR2587340 [32] R. Pandharipande, A calculus for the moduli space of curves, Proc. of Algebraic geometry Salt Lake City 2015, Proc. Sympos. Pure Math. (to appear), arXiv:1603.05151. [33] R. Pandharipande and A. Pixton, Descendents on local curves: rationality, Compos. Math. 149 (2013), no. 1, 81–124. MR3011879 [34] R. Pandharipande and A. Pixton, Descendents on local curves: stationary theory, Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨ urich, 2012, pp. 283–307. MR2987666 [35] R. Pandharipande and A. Pixton, Descendent theory for stable pairs on toric 3-folds, J. Math. Soc. Japan 65 (2013), no. 4, 1337–1372. MR3127827 [36] R. Pandharipande and A. Pixton, Gromov-Witten/pairs descendent correspondence for toric 3-folds, Geom. Topol. 18 (2014), no. 5, 2747–2821. MR3285224 [37] R. Pandharipande and A. Pixton, Gromov-Witten/Pairs correspondence for the quintic 3fold, J. Amer. Math. Soc. 30 (2017), no. 2, 389–449. MR3600040 [38] R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407–447. MR2545686 [39] R. Pandharipande and R. P. Thomas, The 3-fold vertex via stable pairs, Geom. Topol. 13 (2009), no. 4, 1835–1876. MR2497313 [40] R. Pandharipande and R. P. Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), no. 1, 267–297. MR2552254 [41] R. Pandharipande and R. P. Thomas, The Katz-Klemm-Vafa conjecture for K3 surfaces, Forum Math. Pi 4 (2016), e4, 111. MR3508473 [42] R. Pandharipande and R. P. Thomas, 13/2 ways of counting curves, Moduli spaces, London Math. Soc. Lecture Note Ser., vol. 411, Cambridge Univ. Press, Cambridge, 2014, pp. 282– 333. MR3221298 [43] J. Shen, Cobordism invariants of the moduli space of stable pairs, J. Lond. Math. Soc. (2) 94 (2016), no. 2, 427–446. MR3556447 [44] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR1818182

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[45] Y. Toda, Curve counting theories via stable objects I. DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), no. 4, 1119–1157. MR2669709 ¨rich, Ra ¨mistrasse 101, 8092 Zu ¨rich, Switzerland Departement Mathematik, ETH Zu Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01744

The Dirichlet problem for the complex homogeneous Monge-Amp` ere equation Julius Ross and David Witt Nystr¨om Abstract. We survey the Dirichlet problem for the complex Homogeneous Monge-Amp` ere Equation, both in the case of domains in Cn and the case of compact K¨ ahler manifolds parametrized by a Riemann surface with boundary. We then give a self-contained account of previous work of the authors that connects this with the Hele-Shaw flow, and give several concrete examples illustrating various phenomena that solutions to this problem can display.

1. Introduction Let X be a compact complex manifold of dimension n, Σ be a compact Riemannsurface with non-empty smooth boundary, and {ωτ }τ ∈∂Σ be a family of real (1, 1) forms or currents on X. Usually we will assume ωτ is smooth, strictly positive, and varies smoothly in τ , thus giving a smooth family of K¨ahler forms parameterized by ∂Σ. The Dirichlet problem for the complex Homogeneous Monge-Amp`ere Equation (HMAE) in this setting seeks a real (1, 1) form, or current, Ω on X × Σ satisfying (♦)

Ωn+1 = 0, Ω ≥ 0, Ω|X×{τ } = ωτ for τ ∈ ∂Σ.

It is known, under suitable hypothesis on Σ, that one can always find a solution to this equation in the sense of currents, where Ωn+1 is to be understood as the Monge-Amp`ere operator defined by Bedford-Taylor. Following Donaldson, we say that a solution is regular if Ω is smooth and the (1, 1)-forms Ω|X×{τ } are strictly positive for all τ ∈ Σ. Thus a regular solution gives a family of K¨ahler forms on X parameterized by Σ. The guiding question we will be interested in is how far a general solution to (♦) is from being regular, and whether there are conditions under which a regular solution can be guaranteed. The first part of this survey considers various instances of the HMAE, beginning with the work of Bedford-Taylor on the Dirichlet problem for domains in Cn and the pluricomplex Green function introduced by Klimek. We then turn to the setting above, which took on particular importance through work of Semmes and 2010 Mathematics Subject Classification. Primary 32W20, 35J96, 32Q15, 58J32, 76D27. c 2018 American Mathematical Society

289

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¨ JULIUS ROSS AND DAVID WITT NYSTROM

independently Donaldson, who observed that it comes up naturally as the geodesic equation in the space of K¨ahler metrics on X. In the second part we give a self-contained account of previous work of the authors that relates the HMAE when X itself is a Riemann-surface to a wellknown problem in fluid mechanics called the Hele-Shaw flow. In doing so we are able to much better understand this regularity problem, and we end with four concrete examples that show the kind of irregular behaviour that solutions to (♦) can display. In each case these will be obtained by constructing Hele-Shaw flows with particular properties. Our first example (Section 9.1) considers flows developing a “self-tangency” in which at a particular time a Hele-Shaw domain is simply connected, but has boundary that self-intersects tangentially at a point. From this we can produce a solution to the HMAE that is not twice differentiable at certain points. In fact the relation between the self-tangency and this non-differentiability is extremely explicit, and one can not only see at which points this occurs but also the directions along which the second derivative does not exist. The second example (Section 9.2) shows the Hele-Shaw flow becoming nonsimply connected, from which we can produce a solution to the HMAE admitting a definite region that does not intersect any proper harmonic disc. This obstructs the existence of a local Monge-Amp`ere foliation with proper leaves, and so this weak solution is far away from being regular. In the third example (Section 9.3) we produce a flow whose final domain is simply connected but has as boundary some (non-trivial) curve. From this we get examples of solutions to the HMAE that fail to to have the so-called “maximal rank” property. In the final example (Section 9.4) we apply work of Sakai concerning the Hele-Shaw flow for domains that have acute corners to obtain boundary data for the HMAE over the disc that is C 1,α for all α < 1 but whose weak solution is not even C 1 up to the boundary. 2. Preliminaries Throughout, D and D denote the open and closed unit disc in C respectively, × and D× and D will denote these with the origin removed. On any complex manifold X we use the convention i i (∂ − ∂), so ddc = ∂∂. dc = 2π π Given a closed real (1, 1)-form θ on a connected X, we say u : X → [−∞, ∞) is θ-plurisubharmonic (or simply θ-psh) if whenever locally θ = ddc v then u + v is plurisubharmonic. When u is upper-semicontinuous, locally integrable, and not identically −∞ we write θu := θ + ddc u, and if u is θ-psh then θu ≥ 0 in the sense of currents. The space of plurisubharmonic functions on X is denoted by Psh(X) and the θ-psh functions by Psh(X, θ). When X has complex dimension 1, being plurisubharmonic is the same as being subharmonic, and we use the more common notation Sh(X, θ) for the space of θ-subharmonic functions in this case. For general θ there might not be any θ-psh functions, but if θ is strictly positive (and thus a K¨ahler form) then there certainly are (for instance the constant

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functions). We say u is a K¨ ahler potential (with respect to θ) if it is smooth and θu strictly positive, and denote the space of such potentials by K(X, θ). Suppose u is plurisubharmonic on a domain in Cn . When u is twice differentiable its Monge-Amp`ere measure is defined as MA(u) := (ddc u)n . In general, ddc u is merely a positive current so the wedge product (ddc u)n does not immediately make sense. However, Bedford-Taylor showed that the complex Monge-Amp`ere operator can be defined for plurisubharmonic functions that are locally bounded [5]. The idea is to define the current (ddc u)m := ddc (u(ddc u)m−1 ) inductively for m ∈ {1, 2, ..., n}. Assuming (ddc u)m−1 is a positive current, it follows that it has measure coefficients, and since u is locally bounded u(ddc u)m−1 will also be a current with measure coefficients. Thus (ddc u)m := ddc (u(ddc u)m−1 ) is a well defined current and Bedford-Taylor prove that it is positive. Hence by induction MA(u) := (ddc u)n is a well defined positive (n, n)-current, i.e. a positive measure. When u is locally bounded its Monge-Amp`ere measure relative to a smooth (1, 1)-form θ is defined locally where θ = ddc v as MAθ (u) := M A(u + v). 3. The HMAE on domains in Cn 3.1. Perron-Bremermann Envelopes. Let U be a smoothly bounded domain in Cn and φ ∈ C 0 (∂U ). The Dirichlet problem for the complex Homogeneous Monge-Amp`ere Equation (HMAE) on U with boundary data φ asks for a locally bounded u ∈ Psh(U ) such that (3.1)

MA(u) = 0 on U, lim u(z) = φ(ζ) for all ζ ∈ ∂U.

z→ζ

As in the one dimensional case (i.e. when solving the Laplace equation) solutions to HMAE can be found using envelope constructions. The Perron-Bremermann envelope u is defined as (3.2)

u := sup∗ {v ∈ Psh(U ) : limsupz→ζ v(z) ≤ φ(ζ), ∀ζ ∈ ∂U },

where sup∗ means the upper-semicontinuous regularization of the supremum. A proof of the following statement can be found in [52, p18]. Theorem 3.1. Assume U is a smoothly bounded and strictly pseudoconvex domain in Cn . Then the Perron-Bremermann envelope u is the unique solution to the Dirichlet problem for the HMAE (3.1) with boundary φ. One can similarly consider the inhomogeneous Monge-Amp`ere Equation, in which one seeks a solution to M A(u) = dV where dV is a given smooth volume form. Through the work of Caffarelli, Kohn, Nirenberg, Spruck [21–23] it is known that, as long as U is strictly pseudoconvex, if φ is smooth then the solution to the inhomogeneous problem with boundary data φ is also smooth. However for the homogeneous case that we are interested in the answer is more subtle. Theorem 3.2 (Krylov). Assume U is a smoothly bounded and strictly pseudoconvex domain in Cn . If φ ∈ C ∞ (∂U ) then the solution u to the HMAE with boundary data φ lies in C 1,1 (U ). The next example shows that this regularity result is optimal.

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Example 3.3. Let U := B be the unit ball in C2 and for (z, w) ∈ ∂B let φ(z, w) := (|z|2 − 1/2)2 = (|w|2 − 1/2)2 . It can then be easily checked that the solution to the Dirichlet problem is given by u(z, w) = (max{0, |z|2 − 1/2, |w|2 − 1/2})2 . This clearly fails to be C 2 along the hypersurfaces |z|2 = 1/2 and |w|2 = 1/2. 3.2. Regular solutions, Monge-Amp` ere foliations and proper harmonic curves. Definition 3.4. We say that a solution u to Dirichlet problem (3.1) is regular if u ∈ C ∞ (U ) and if at every point of U the kernel of ddc u has complex dimension 1. Definition 3.5. Let u be a solution to (3.1). A subset of U is called a proper harmonic curve of u if it is the image of a proper holomorphic map f : Σ → U from a Riemann-surface Σ such that u ◦ f is harmonic on Σ. If u is regular, the kernel of ddc u defines a one-dimensional distribution which turns out to be integrable, and so by Frobenius Integrability Theorem yields a foliation of U whose leaves are proper harmonic curves. This is known as the associated Monge-Amp`ere foliation. Thus a way to gauge the failure of regularity of a solution is to see how far the set of proper harmonic curves is from foliating the domain. Returning to Example 3.3 one easily checks that the set of proper harmonic curves consists of the discs {(z, c) ∈ B} and {(c, w) ∈ B} for 1/2 ≤ |c|2 < 1 together with the discs {(z, cz) ∈ B} for |c| = 1. Interestingly, even though this set of discs is far from foliating the domain B their boundary circles do foliate ∂B. In particular ∂B is contained in the closure of the union of proper harmonic curves. We are not aware of any examples of solutions to the above Dirichlet problem where this is not the case. A related but different issue is that of finding local harmonic discs, i.e. nontrivial but not necessarily proper holomorphic discs along which u is harmonic. Indeed looking at Example 3.3 it is clear that through any point in B passes at least one local harmonic disc. However, an interesting construction of Sibony shows that this not always has to be the case (for the details see [52, Sect. 3.5.1] and references therein). 3.3. Pluricomplex Green functions. Another manifestation of the HMAE comes through the so-called pluricomplex Green function. Let U be a smoothly bounded strictly pseudoconvex domain in Cn and fix a point z0 ∈ U . Definition 3.6. The pluricomplex Green function of U with singularity at z0 is defined as uU,z0 := sup∗ {v ∈ Psh(U ) : v ≤ 0, νz0 (v) ≥ 1}. Above νz0 (v) denotes the Lelong number of v at the point z0 , defined by νz0 (v) = sup{t : v ≤ t ln |z − z0 |2 + O(1)} (we refer the reader to §5.1 for more on Lelong numbers). Theorem 3.7 (Demailly, Blocki). The pluricomplex Green function uU,z0 solves the HMAE on U \ {z0 } and is C 1,1 on U \ {z0 }.

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We say that the pluricomplex Green function is regular if it is C ∞ on U \ {z0 } and the kernel of ddc u one-dimensional on U \ {z0 }. Given a regular solution, the corresponding Monge-Amp`ere foliation will then consist of holomorphic curves attaching to ∂U and by the maximum principle all will pass through the point z0 . It was shown by Lempert [77] that the pluricomplex Green function is regular when the domain U is smoothly bounded and strictly convex. The discs of the corresponding Monge-Amp`ere foliation contain important information of the domain U. For z, w ∈ U , the Kobayashi distance between z and w, denoted by δK (z, w), is defined as the infimum of the Poincar´e distance between pairs x and y in D over all holomorphic discs f : D → U with f (x) = z and f (y) = w. Such a disc connecting z and w is called extremal if it realizes the Kobayashi distance. Lempert proves in [77] that when U is smoothly bounded and strictly convex, for any z, w ∈ U there exists an extremal disc (unique up to reparametrization) and that this is exactly the disc of the Monge-Amp`ere foliation of uU,z0 that passes through w. We will discuss more of Lempert’s proof in connection with the Donaldson Openness Theorem in §4.2. In contrast, Bedford-Demailly [2] give examples of smoothly bounded strictly pseudoconvex domains with a pluricomplex Green function which is not C 2 up to the boundary. It seems not to be known whether they also fail to be C 2 in the interior of the domain. 3.4. Bibliographical Remarks. The reader interested in more comprehensive surveys on this topic is referred to Berndtsson [13], Guedj [52], Guan [50], Kolodziej [68, 69], Phong-Song-Sturm [85] and Zeriahi [123]. The definition of the Perron-Bremermann envelope goes back to [83] and [19]. That this envelope is continuous when U is strictly pseudoconvex was proved by Walsh [121] who also gives examples in which this fails for more general U . That the envelope is locally C 1,1 when the domain is the unit ball was proved by Bedford-Taylor [4], where they also showed that for any smoothly bounded strictly pseudoconvex domain the solution was Lipschitz up to the boundary. The full statement of Theorem 3.2 (namely that the solution is C 1,1 all the way up to the boundary) is due to Krylov [71] (see also [52] for a detailed exposition of Krylov’s proof). Example 3.3 is due to Gamelin and Sibony (see [47] and also [52, Ex. 2.13]). The study of Monge-Amp`ere foliations goes back to the work of Bedford-Kalka [3]. Pluricomplex Green functions were introduced by Klimek [66] and independently by Zakharyuta [122]. The part of Theorem 3.7 which says that the pluricomplex Green functions solves the HMAE was first proved by Demailly [38], while the C 1,1 -regularity is due to Blocki [15]. More on the pluricomplex Green function and its applications can be found, for instance, in [17, 48, 49, 61, 62, 78]. 4. The HMAE for compact K¨ ahler manifolds 4.1. Weak and Regular Solutions. Suppose now that (X, ω) is a compact K¨ ahler manifold (without boundary) and Σ is a compact Riemann-surface with non-empty smooth boundary. Let φ ∈ C ∞ (X × ∂Σ) be chosen so φ(·, τ ) ∈ K(X, ω) for each τ ∈ ∂Σ. Letting πX : X × Σ → X be the projection, we denote by ∗ ∗ ω) the space of functions that are πX ω-plurisubharmonic on the Psh(X × Σ, πX interior of X × Σ.

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∗ Definition 4.1. We say Φ ∈ Psh(X × Σ, πX ω) ∩ C 0 (X × Σ) solves the HMAE with boundary data φ if

(4.1)

M Aπ∗ ω (Φ) = 0, lim Φ(z, ζ) = φ(ζ) for all ζ ∈ X × ∂Σ.

z→ζ

By analogy with Perron-Bremermann envelope set ∗ Φ := sup∗ {Ψ ∈ Psh(X × Σ, πX ω) : limsupz→ζ Ψ(z) ≤ φ(ζ) for ζ ∈ X × ∂Σ}

We assume henceforth that Σ is compact and carries a smooth strictly plurisubharmonic function χ such that χ|∂Σ = 0 (in fact we will mostly be concerned when Σ is either a disc or an annulus in which case this clearly holds). Using this, the following is proved almost exactly as in the local case (see, for instance [52, Ch. 1,7]). Proposition 4.2. The envelope Φ is the unique solution to (4.1). We are thus justified in calling Φ the weak solution to the HMAE with boundary data φ. The following statement (proved recently by Chu-Tosatti-Weinkove [32]) is the optimal regularity that one can expect in general. Theorem 4.3. Let φ ∈ C ∞ (X × ∂Σ) be chosen so φ(·, τ ) ∈ K(X, ω) for each τ ∈ ∂Σ. Then the weak solution Φ to the HMAE with boundary data φ lies in C 1,1 (X × Σ). Observe that by hypothesis ωτ := ω + ddc φ(·, τ ) is a K¨ahler form on X for each τ ∈ ∂Σ, and so if Φ solves (4.1) then the (1, 1)-current ∗ Ω := πX ω + ddc Φ

solves the Dirichlet problem for the HMAE with boundary data {ωτ }τ ∈∂Σ , as considered in the introduction. Following Donaldson [42] we make the following definition: Definition 4.4. We say the weak solution Φ to the HMAE (4.1) is regular if it is smooth and Φ(·, τ ) ∈ K(X, ω) for all τ ∈ Σ. Just as in the local case, a regular solution defines a foliation of X × Σ. In ∗ more detail, consider the associated form Ω := πX ω + ddc Φ. By being the weak n+1 solution to the HMAE we have Ω ≥ 0 and Ω = 0 on X × Σ. On the other hand, if Φ is regular then Ω|X×{τ } = ω + ddc Φ(·, τ ) is strictly positive for all τ ∈ Σ. Thus the kernel of Ω at each point of X × Σ is one-dimensional, and so gives a one-dimensional distribution. Since Ω is closed the distribution is integrable, and so by the Frobenius Integrability Theorem gives foliation of X × Σ. The leaves ∗ ω-harmonic along are complex since Ω is of type (1, 1) and by construction Φ is πX these leaves. As Ω|X×{τ } is strictly positive, these leaves are necessarily transverse to the fibres over Σ. If Φ is merely a weak solution then there is no reason to think such a foliation will exist. However it can still happen that there are some transverse curves along which the weak solution is harmonic. Definition 4.5. Let f : Σ → X be proper and holomorphic. We say the graph of f is a proper harmonic curve for the weak solution Φ to the HMAE if Φ ◦ f is ∗ ω-harmonic. If Σ = D is the unit disc we refer to this as a proper harmonic disc. πX

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It is in general very hard to determine whether a certain weak solution to the HMAE is regular or not. A trivial, but still sometimes useful, special case is when φ ∈ C ∞ (X × ∂Σ) is independent of the point in ∂Σ. For then Φ(z, τ ) := φ(z) for (z, τ ) ∈ X × Σ is clearly a regular solution to the HMAE, whose Monge-Amp`ere foliation is simply the horizontal slices {z} × Σ for z ∈ X. We will see in the next section that this can be used to produce non-trivial examples of regular solutions. 4.2. Donaldson’s Openness Theorem. Suppose now that Σ = D is the unit disc in C. The following theorem says that the existence of regular solutions to the HMAE persists under small perturbations of the boundary data. Theorem 4.6 (Donaldson). Suppose the weak solution to the HMAE with boundary data φ ∈ C ∞ (X × ∂D) is regular. Then for any k ≥ 2 and 0 < α ≤ 1 there is an > 0 such that if g ∈ C ∞ (X × ∂D) has C k,α -norm less than then the weak solution to the HMAE with boundary data φ + g is also regular. Very roughly, this result is obtained by casting the problem of deforming the harmonic discs of a Monge-Amp`ere foliation as an elliptic problem. Actually is it not precisely these discs that are used, but instead discs from an auxiliary construction that we now sketch. Let Θ := Θ1 + iΘ2 be a holomorphic 2-form on a complex manifold W , where Θ1 and Θ2 are real symplectic forms. A (real) submanifold V of W is said to be an LS-submanifold if it is Lagrangian with respect to Θ1 (i.e. Θ1 vanishes along W ) while being symplectic with respect to Θ2 (i.e. Θ2 restricts to a symplectic form on W ). Semmes [111] and Donaldson [42] show that given a compact K¨ ahler manifold (X, ω) there exists a holomorphic fibre bundle π : WX → X with holomorphic 2-form Θ such that K¨ahler forms in the same cohomology class as ω correspond to LS submanifolds in WX . Roughly speaking, W is constructed as follows. If ω has a local potential u on some open set U we identify WU with the (1, 0)-part of the complexified cotangent bundle  of U . If zi are local holomorphic coordinates any (1, 0)-form can be written as i ζi dzi , thus (ζi , zi ) are local holomorphic coordinates on WU and locally Θ := i dζi ∧ dzi . If V is another open set where ω has the local potential v, then over U ∩ V the transition function of the fibre bundle WX is set to be ∂(v − u). Thus there is a global section of WX , locally given by ∂u. By a simple calculation, the graph of this section is a LS-submanifold. Any K¨ ahler form cohomologous to ω comes from a K¨ahler potential φ ∈ K(X, ω), whose corresponding LS-submanifold is locally given by the graph of ∂(u + ψ). Moreover, as Donaldson shows, in [42], any closed LS-manifold in WX in the isotopy class of ∂u arises this way. Now let φ ∈ C ∞ (X × ∂D) and assume that φτ ∈ K(X, ω) for each τ ∈ ∂D. By the above, this defines a family Λτ of associated LS-submanifolds in WX . Donaldson proves the following: Proposition 4.7. There is a regular solution Φ to the HMAE with boundary data φ if and only if there is a smooth family of holomorphic discs gx : D → WX parametrized by x ∈ X such that • π(gx (0)) = x; • for each τ ∈ ∂D and each x ∈ X, gx (τ ) ∈ Λτ ;

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• for each τ ∈ D, the map x → gx (τ ) is a diffeomorphism of X. For a fixed τ ∈ D the image of the map x → gx (τ ) is the LS-submanifold associated to the K¨ ahler form ω + ddc Φ(·, τ ). Thus regular solutions to the HMAE come from these particular families of holomorphic discs. Then one can apply the deformation theory of holomorphic discs with boundary in a totally real submanifold (which is essentially an elliptic problem) to see that the existence of such a family is open as the boundary data varies, thus proving Theorem 4.6. It is interesting to note that the regularity result of Lempert for the pluricomplex Green function discussed in §3.3 is proved in a somewhat analogous manner. Recall that a holomorphic disc f : D → U with f (x) = z and f (y) = w is said to be extremal if it realizes the Kobayashi distance between z and w. Let v denote the normal vector field of ∂U pointing outward. Lempert calls a disc f stationary if it extends continuously to a map f : D → U with f (∂D) ⊆ ∂U , and if the map ∂D  ζ → [v1 (f (ζ)) : ... : vn (f (ζ))] ∈ Pn−1 extends to a holomorphic function fˆ : D → Pn−1 . Lempert proves that a stationary disc is extremal and conjectures that the converse also holds. One can interpret f being stationary as saying that the combined disc (f, fˆ) is attached to a certain totally real submanifold, and hence stationary discs persist given small perturbations of U . In particular this proves regularity for the pluricomplex Green function for domains that are small perturbations of the unit ball (thus the analogy with Donaldson’s Openness proof). To prove the result for all strictly convex domains Lempert uses a continuity argument, by establishing the required a priori estimates. 4.3. Bibliographical Remarks. In work of Mabuchi [82], Semmes [111] and Donaldson [42], the space K(X, ω) of K¨ ahler metrics cohomologous to ω is given the structure of an infinite dimensional Riemannian manifold and, somewhat amazingly, the HMAE turns out to be the geodesic equation in this space. More specifically, to find a geodesic segment joining two points φ0 , φ1 ∈ K(X, ω) requires solving the Dirichlet problem for the HMAE over X × A where A is an annulus, say A = {c0 < |τ | < c1 }, and the boundary data is taken to be φ(z, τ ) := φi (z) for |τ | = ci with i = 0, 1. Thus any smoothness properties of the weak solution to the HMAE becomes a statement about smoothness of this (weak) geodesic segment, and having a regular solution says precisely that there is a genuine (i.e. smooth) geodesic segment joining φ0 and φ1 in K(X, ω). This manifestation of the HMAE generated much interest, not least since it was observed by Donaldson [42] that the existence of a (sufficiently nice) geodesic segment joining any two points in K(X, ω) would imply uniqueness of constant scalar curvature K¨ahler metrics. We refer the reader again to [13, 50, 52, 68, 69, 85, 123] for other surveys on this topic. The statement that the weak solution to the HMAE is C 1,1 , now proved in [32], has a long history. It was proved by Chen [26] (with complements by Blocki [18]) that the weak solution has bounded Laplacian on X ×Σ and so in particular is C 1,α for any α < 1 in the interior of X × Σ. Moreover Blocki proves that if (X, ω) is assumed to have non-negative bisectional curvature then the weak solution is C 1,1 . Other works on this topic include those of Phong-Sturm [86, 88, 90], Eyssidieux– Guedj–Zeriahi [45], Demailly et al [41]. When Σ is the unit disc, C 1,1 on the interior of X × Σ has been proved by Berman [8] using a technique based on the original approach of Bedford-Taylor.

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One can more generally consider the Dirichlet problem for the HMAE on a complex manifold-with-boundary, and several of the above cited references, including [32], hold in this case as well (usually under an assumption of being weaklypseudoconcave or having Levi-flat boundary). For example, one can consider the HMAE on the total space of a (sufficiently nice) test-configuration, thus connecting K-stability with weak-geodesics (see, for instance [9, 30, 87, 99, 115] as well as the contribution by Sz´ekelyhidi in this volume). Works on the related question of the implications of the HMAE to the geometry of the space of K¨ ahler metrics include those of Arezzo-Tian [1], Berman-BoucksomGuedj-Zeriahi [11], Berndtsson-Cordero-Erausquin-Klartag-Rubinstein [14], ChenSun [29], Chen-Tian [31] and Darvas [35, 36]. Ultimately it turned out that the particular application concerning uniqueness of constant scalar curvature K¨ahler metrics cannot easily be addressed through regularity, but can resolved with just the weak solution as achieved by Berman-Berndtsson [10] (see also Chen-Li-P˘ aun [28]). Donaldson [42] gives examples of boundary data over the disc for which the weak solution is not regular, but we observe that the argument uses contradiction, and thus is non-explicit. Nevertheless, it was initially hoped that this phenomena would not hold over the annulus, and so any weak geodesic connecting two K¨ ahler potentials would be regular (and thus a geodesic in the strongest possible sense). It was not until the work of Lempert-Vivas that this was proven not to be the case. In [80] they find geodesic segments that are not C 3 up to the boundary, and later Darvas-Lempert [79] found geodesic segments that fail to be C 2 up to the boundary. In subsequent sections we will see how regularity can fail both for the HMAE over the disc and over the punctured disc. As the case for the pluricomplex Green’s function, it is currently unknown whether or not singularities can occur in the interior. 5. The Hele-Shaw Flow The rest of this paper is devoted to surveying previous work of the authors which connects the HMAE with the Hele-Shaw flow. We shall discuss two approaches to this flow, and both are useful in understanding its relation with the HMAE. First is the so-called weak Hele-Shaw flow that can be described using basic potential-theoretic constructions. The advantage of this approach is that it does not require any a priori smoothness, making it both elementary and very flexible. Second is the strong Hele-Shaw flow that is defined dynamically by describing the motion of the boundary of the flow. This necessarily requires assuming more smoothness, but has the advantage of having a physical interpretation thus making it more intuitive. Of course, the strong Hele-Shaw flow is also a weak one, and a weak Hele-Shaw flow that is also smooth will be a strong one. In this section we shall only consider the weak flow, allowing us to quickly move to the connection with the HMAE. Consideration of the strong flow will be postponed until §8. Our account is broadly self-contained, in that we include all the main features of the flow we need. That said, this represents only a tiny part of the Hele-Shaw flow theory, and the reader will find in the bibliographical remarks many references that go far beyond what is included here.

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5.1. Lelong Numbers. From now on, X will be a connected Riemann surface along with a distinguished point z0 ∈ X and ω will be a K¨ahler form on X. As X has complex dimension 1 being plurisubharmonic is the same as being subharmonic, and we let Sh(X, ω) denote the space of functions that are ω-subharmonic. Let z be a holomorphic coordinate defined near z0 . Then for ψ ∈ Sh(X, ω) the Lelong number of ψ of z0 is defined to be νz0 (ψ) := sup{c ≥ 0 : ψ ≤ c ln |z − z0 |2 + O(1)} where the inequality is to be understood as meaning there is a constant C such that ψ ≤ c ln |z − z0 |2 + C near z0 . We observe the supremum is actually attained, so if νz0 (ψ) = t then ψ ≤ t ln |z − z0 |2 + O(1). To see this, let B be a small ball centered around z0 . For any c < t the function ψ(z) − c ln |z − z0 |2 is bounded above as z tends to z0 , and lies in Sh(B \ {z0 }, ω) and thus extends to a function in Sh(B, ω) [67, Theorem 2.7.1]. On the other hand, on the boundary of the ball, ψ(z) − c ln |z − z0 |2 |∂B is bounded from above uniformly over all c < t. Thus by the maximum principle ψ(z) − c ln |z − z0 |2 is bounded above uniformly over z ∈ B and c < t. Then letting c tend to t gives ψ(z) ≤ t ln |z − z0 |2 + O(1) as claimed. The Lelong number measures the mass of the current ddc ψ at the point z0 , in that " " c (5.1) νz0 (ψ) = lim+ dd ψ = lim+ ωψ r→0

Br

r→0

Br

where Br is the ball of radius r centered at z0 [39, Theorem 2.8]. 5.2. Definitions. The basic definition on which everything else is based is the following: Definition 5.1. (Hele-Shaw Envelope) For t ∈ R let ψt := sup{ψ ∈ Sh(X, ω) : ψ ≤ 0 and νz0 (ψ) ≥ t}. We shall refer to ψt as the Hele-Shaw envelope at time t. Of course the envelope ψt depends on the background K¨ahler form ω, but this will always be clear from context. Clearly ψt ≤ 0 everywhere. Definition 5.2. (Weak Hele-Shaw Flow) For t ∈ R set Ωt := {z ∈ X : ψt (z) < 0}. We refer to Ωt as the weak Hele-Shaw domain at time t, and the collection of all such domains as the weak Hele-Shaw flow. The weak Hele-Shaw domains are generally hard to compute, unless one imposes some additional symmetry as in the following example. Example 5.3. (Radially Symmetric Case) Suppose X = C, let z0 be the origin and assume the K¨ahler form ω is radially symmetric. Then we can write ω = ddc φ for some smooth radially symmetric function φ on C, so φ(eiθ z) = φ(z) for all θ ∈ R.

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It is not hard to see that the Hele-Shaw envelopes and Hele-Shaw domains are also be radially symmetric, and we now calculate what these actually are. We assume for all t > 0 that φ satisfies the growth condition φ(z) ≥ t ln |z|2 + O(1) for |z| ' 0. It is convenient to use the variable s = − log |z|2 so our distinguished point z = 0 corresponds to s = ∞. Then we can write φ(z) = u(s) for some smooth u : R → R. By differentiating twice, one can check the condition that ω is strictly positive implies u is strictly convex, and lim

s→∞

du = 0 and ds

lim

s→−∞

du =∞ ds

(the first coming from φ(z) extending smoothly over z = 0, and the second coming from the assumed growth condition). So for t ∈ R+ there is a unique s0 ∈ R such that du |s = −t. ds 0 We let

 vt (s) :=

u(t) u(s0 ) − t(s − s0 )

for s < s0 for s ≥ s0

Then vt is the largest convex function bounded above by u with the property that vt (s) ≤ −ts + O(1) as s → ∞. We claim the Hele-Shaw envelope is given by (5.2)

ψt (z) = vt (s) − u(s)

and the weak Hele-Shaw domain is Ωt = {s > s0 } = {z : |z|2 < e−s0 }. ˜ To prove this, set ψ(z) = vt (s) − u(s) so the goal is to show ψ˜ = ψt . Observe vt ˜ being convex implies ψ ∈ Sh(C, ω) and its behaviour as s tends to infinity gives ˜ = t. Clearly ψ˜ ≤ 0, so ψ˜ ≤ ψt . For the other inequality, let ψ ∈ Sh(X, ω) νz=0 (ψ) satisfy ψ ≤ 0 and νz=0 (ψ) ≥ t. As vt is linear on {s > s0 } we have ωψ˜ = 0 on D× := {s > s0 } = {0 < |z|2 < e−s0 }. Then the difference ψ − ψ˜ is bounded as z → 0 and subharmonic on D× and thus extends to a subharmonic function on all of D [67, Theorem 2.7.1]. On the other hand ψ˜ = 0 on ∂D, and so ψ − ψ˜ ≤ 0 on ∂D. Thus by the maximum principle, ψ ≤ ψ˜ on all of D. But vt = u on the set {s ≥ s0 } so on the complement of D clearly ψ˜ = 0 ≥ ψ, and hence ψ ≤ ψ˜ ˜ and thus ψt = ψ˜ everywhere. Taking the supremum over all such ψ gives ψt ≤ ψ, as claimed. The conclusion then about the weak Hele-Shaw domain follows as this is the set on which vt is equal to u.

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5.3. Basic Properties of the Hele-Shaw Flow on compact Riemann 5 surfaces. Assume now X is compact, which in particular implies X ω is finite. It is not hard to see if ω is replaced with λω for some λ > 0 then ψt is replaced with λψλ−1 t and Ωt replaced by Ωλ−1 t . Thus without loss of generality we assume that " ω = 1. X

With this in mind we turn to some of the basic properties of the weak Hele-Shaw flow. Proposition 5.4 (Basic Properties of the weak Hele-Shaw flow in the compact case). (1) For t ≤ 0 we have ψt ≡ 0 and Ωt = ∅. (2) For t > 1 we have ψt ≡ −∞ and Ωt = X. (3) For t ∈ [0, 1] we have (a) ψt is locally bounded away from z0 . (b) ψt ∈ Sh(X, ω). (c) νz0 (ψt ) = t. (d) ωψt |Ωt = tδz0 . Our proof will use the following preliminary result. Lemma 5.5. There exists an α ∈ Sh(X, ω) ∩ C ∞ (X \ {z0 }) such (1) supX α = 0 (2) α = ln |z−z0 |2 +O(1) near z0 , so in particular νz0 (α) = 1 and (3) ω+ddc α = δz0 . Proof. Suppose z is a holomorphic coordinate on a ball B around z0 . Let ρ be a bump function identically 1 near z0 and supported in B and consider β(z) := ρ(z) log |z − z0 |2 . cohomology Then ddc β = δz0 + τ for some smooth form τ . But in Dolbeault 5 0 = [ddc β] = [δz0 ] + [τ ] = [ω] + [τ ] where the last equality uses X ω = 1 (and we are using Dolbeault cohomology of currents, which agrees with Dolbeault cohomology of smooth forms [40, IV, 6.13]). Thus τ = −ω + ddc f for some smooth function f on X, and α := β − f − C for a suitable constant C is in Sh(X, ω) ∩ C ∞ (X \ {z0 }) and satisfies conditions (1) through (3).  Remark 5.6. On P1 , with its Fubini-Study form, and coordinate z on C ⊂ P1 so z0 is the origin, we can explicitly write α = ln |z|2 − ln(1 + |z|2 ). Proof of Proposition 5.4. All of this is rather standard, and for convenience we give details. If t ≤ 0 then the constant function 0 is a candidate for the envelope defining ψt , giving (1). On5 the other 5 hand if ψ ∈ Sh(X, ω) is not identically −∞ and νz0 (ψ) ≥ t then t ≤ X ωψ = X ω = 1 by (5.1) which proves (2). So assume now t ∈ [0, 1]. Then (3a) follows as ψt is bounded from below by the function tα where α is provided by Lemma 5.5. Moreover this implies νz0 (ψt ) ≤ νz0 (tα) = t. Now let z be a holomorphic coordinate defined near z0 and consider β := sup∗ {ψ ∈ Sh(X, ω) : ψ ≤ 0 and ψ ≤ t ln |z − z0 |2 + O(1)}. Clearly β ≥ ψt and we shall show that in fact equality holds. First observe being the upper-semicontinuous regularisation of a supremum of ω-subharmonic functions, β is itself ω-subharmonic [67, Thm 2.6.1(iv)] and clearly

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β ≤ 0. We claim νz0 (β) ≥ t. The issue here is that the O(1) term in the definition of β can depend on ψ. To address this, let B be a small ball around z0 on which we can write ω = ddc ζ for some smooth function ζ. Let γ be the solution to the classical Dirichlet problem for the Laplacian (5.3)

ddc γ = 0 on B and γ|∂B = (ζ − t ln |z − z0 |2 )|∂B .

It is known [67, Theorem 2.2.6] such a γ exists, and is locally bounded on B. Then set

:= −ζ + t ln |z − z0 |2 and we claim β ≤ near z0 . To see this, suppose ψ ∈ Sh(X, ω) is such that ψ ≤ 0 and νz0 (ψ) ≥ t. Then ψ − = ψ + ζ − t ln |z − z0 |2 ∈ Sh(B \ {z0 }). On the other hand by construction (ψ − )|∂B ≤ − |∂B = γ|∂B . As z approaches 0 we have ψ ≤ t ln |z − z0 |2 + O(1) and = t ln |z − z0 |2 + O(1) so ψ − is bounded near z0 , and thus extends to a subharmonic function over all of B [67, Theorem 2.7.1]. Hence by the maximum principle ψ ≤ + γ over B. Taking the supremum over all such ψ, and then the upper semicontinuous regularisation, we deduce β ≤

near z0 as claimed. In particular β ≤ t ln |z − z0 |2 + O(1) giving νz0 (β) ≥ t. Thus β is a candidate for the envelope defining ψt , so in fact β = ψt proving items (3b) and (3c). That ψt is ω-harmonic away from z0 is proved the same way that the PerronBremermann envelope is shown to solve the HMAE. Then (3d) follows from (3c) and (5.1).  Corollary 5.7. (Openness, Connectedness) The Hele-Shaw domain Ωt is open, connected and z0 ∈ Ωt for t > 0. Proof. Openness of Ωt follows from semicontinuity of ψt , and if t > 0 then νz0 (ψ) > 0 so ψt (z0 ) = −∞ giving z0 ∈ Ωt . If Ωt were not connected then we could find a component S that does not contain z0 . Since Ωt is open, ∂S ⊂ X \ Ωt and so ψt = 0 on ∂S. As ωψt = 0 on Ωt \ {z0 } we see −ψt is subharmonic on S, so the maximum principle implies −ψt ≤ 0 on S. But this is absurd as  S ⊂ Ωt = {ψt < 0}. The next two results show the Hele-Shaw domains only depends on the value of the K¨ahler metric in a region slightly larger than that5domain. To express this precisely, suppose ω ˜ is another K¨ahler form on X, with X ω ˜ = 1, and denote by ˜ ˜ ˜. ψt and Ωt the Hele-Shaw envelopes and weak Hele-Shaw domains associated to ω Lemma 5.8 (Monotonicity). Suppose S ⊂ X is open and ω ˜ ≥ ω over S, and assume Ωt is relatively compact in S. Then ˜ t ⊂ Ωt . (5.4) ψt ≤ ψ˜t and Ω Proof. The statement is trivial if t < 0 or t > 1, so suppose t ∈ [0, 1]. From Proposition 5.4(3b) ψt ∈ Sh(X, ω), so the hypothesis implies ψ˜t ∈ Sh(S, ω ˜ ). Since Ωt is relatively compact in S we see ψt is identically zero on a neighbourhood of ˜ ≥ 0. Thus ψt ∈ Sh(X, ω ˜ ). Now X \ S, and so over this neighbourhood ω ˜ ψt = ω

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Proposition 5.4(3c) gives νz0 (ψt ) ≥ t, and so ψt is a candidate for the envelope ˜ t ⊂ Ωt  defining ψ˜t , giving ψt ≤ ψ˜t from which it follows Ω Corollary 5.9. (Locality) If ω = ω ˜ on some open S ⊂ X and Ωt is relatively ˜ t. compact in S then ψt = ψ˜t and Ωt = Ω ˜ t is ˜ t ⊂ Ωt and so Ω Proof. One application of the previous lemma tells us Ω also relatively compact in S. Then we can apply the lemma again with the roles of ω ˜ and ω reversed.  We next express the Hele-Shaw envelope in a slightly different way. Recall the function α from Lemma 5.5 that is smooth away from z0 , and satisfies ω + ddc α = δ0 and sup α = 0 and α = ln |z − z0 |2 + O(1) near z0 . X

Lemma 5.10. For t ∈ [0, 1], ψt = sup{ψ ∈ Sh(X, (1 − t)ω) : ψ ≤ −tα} + tα. Proof. The statement is trivial when t = 0, so we assume t > 0. Set u := sup{ψ ∈ Sh(X, (1 − t)ω) : ψ ≤ −tα} so the goal is to prove ψt = u+tα. Clearly ψt −tα ≤ −tα and (1−t)ω+ddc (ψt −tα) = ωψt − tδ0 ≥ 0 by Proposition 5.4(3b,d). On the other hand since νz0 (ψt ) = t we have ψt − tα is bounded near z0 . Thus ψt − tα extends over z0 to a function in Sh(X, (1 − t)ω) [67, Theorem 2.7.1] and we conclude ψt − tα ≤ u. For the other inequality, if ψ ∈ Sh(X, (1−t)ω) satisfies ψ ≤ −tα then ψ+tα ≤ 0 and tα ∈ Sh(X, tω) so by convexity ψ + tα ∈ Sh(X, ω). Moreover any such ψ is bounded above near z0 , so νz0 (ψ + tα) ≥ νz0 (tα) = t. Hence ψ + tα ≤ ψt , and taking the supremum over all such ψ gives u + tα ≤ ψt as required.  The previous Lemma casts the envelope ψt as a (translation of) the solution to an obstacle problem with obstacle −tα. A slight difference between this and the classical theory is that often the obstacle is assumed to be a smooth (or at least bounded) function, but this is easily circumvented as in the following statement. Lemma 5.11. There exists an f ∈ C ∞ (X) such that (5.5)

sup{ψ ∈ Sh(X, (1 − t)ω) : ψ ≤ −tα} = sup{ψ ∈ Sh(X, (1 − t)ω) : ψ ≤ f }.

Sketch Proof. In a small disc D around z0 on which ω = ddc ζ let v solve dd v = 0 on D and v|∂D = −tα + (1 − t)ζ|∂D . If ψ is a candidate for the envelope on the left hand side of (5.5) then by the maximum principle ψ ≤ v − (1 − t)ζ =: w on D. Now w is bounded but −tα tends to infinity near z0 , so we can find an f ∈ C ∞ (X) such that f = −tα on X \ D and f ≤ −tα on X and w ≤ f on D, and it is easy to see then that (5.5) holds for this f .  c

For more advanced information about the flow we will need some smoothness of the Hele-Shaw envelope. Note ψt will not generally be C ∞ , as can be seen in Example 5.3. However the following says this is, in some sense, the worst that can happen: Theorem 5.12 (Regularity of Hele-Shaw envelope). For t < 1 the Hele-Shaw envelope ψt is C ∞ on Ωt \ {z0 } and is C 1,1 on X \ {z0 }.

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Proof. The first statement is clear as ωψt = 0 on Ωt \ {z0 } and harmonic functions are smooth. The deeper statement is the second, which is somewhat technical and so we omit the details. When X = P1 the result we want may be reduced to known regularity of solutions of the obstacle problem for the Laplacian for domains in R2 [20] due to Cafarelli-Kinderlehrer, and the reader interested in this reduction will find details in [102, Proposition 1.1]. For general Riemann surfaces we need more machinery. For instance, there is no loss in assuming ω is integral, at which point the ψt is among the envelopes considered by [7] and [100] where the desired C 1,1 regularity is proved (strictly speaking the cited results only apply when t is rational, but the proofs given there give uniform estimates of the C 1,1 under perturbations of t and the result for all t ∈ (0, 1) then follows by approximation). We refer the reader to §5.5 for further regularity results in this direction.  Corollary 5.13. For t < 1 the boundary ∂Ωt of the weak Hele-Shaw domain has measure zero. Proof. Let u := −ψt so Ωt = {u > 0} and by the previous Theorem u is C 1,1 in a neighbourhood U of ∂Ωt . Since ωψt |U = 0, we have Δu ≥ λ > 0 on U . Fix x ∈ ∂Ωt . We first claim there is an > 0 such that for all r > 0 sufficiently small there is a y with (5.6)

u(y) ≥ r 2 and y ∈ Br (x).

To prove this, we may work locally near x and assume our distance function is the usual Euclidean one. Consider a sequence of points xn ∈ Ωt converging to x as n tends to infinity. For small r > 0 consider n sufficiently large so Br (xn ) ∩ ∂Ω is non-empty. Set v(z) := u(z) − u(xn ) − |z − xn |2 for , λ. Then v(xn ) = 0 and Δv ≥ 0 on Br (xn ) ∩ Ωt . Thus by the maximum principle applied to v on Br (xn ) ∩ Ωt we know there is a yn ∈ ∂(Br (xn ) ∩ Ωt ) with v(yn ) ≥ 0. Now ∂(Br (xn ) ∩ Ωt ) ⊂ ∂Ωt ∪ ∂Br (xn ), and if yn ∈ ∂Ωt then u(yn ) = 0, so v(yn ) < 0 which is absurd. Hence yn ∈ ∂Br (xn ), so in fact |yn − xn | = r and v(yn ) ≥ 0 becomes u(yn ) ≥ u(xn ) + r 2 . Letting n tend to infinity and taking a subsequence, we deduce there exists a y ∈ X satisfying (5.6) as claimed. We next claim there exists a c ∈ (0, 1) such that for any sufficiently small r > 0 there exists a y ∈ Br (x) and (5.7)

Bcr (y) ⊂ Ωt .

To see this let y be as in (5.6). The Lipschitz bound on ∇u near ∂Ωt , and the fact that u ≡ 0 and ∇u ≡ 0 on ∂Ωt implies that there is a bound of the form |∇u(z)| ≤ M r for dist(z, ∂Ωt ) ≤ r. Thus if |z − y| < cr we have u(z) ≥ u(y) − M cr 2 ≥ ( − M c)r 2 which is strictly positive as long as we take c < M/ . Thus Bcr (y) ⊂ Ωt proving (5.7). So, letting |A| denote the Lebesgue measure of a set A, this implies |Br (x) ∩ ∂Ωt | ≤ |Br (x)| − |Bcr (y)| = O((1 − c2 )r 2 ).

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Thus the Lebesgue density of ∂Ω at the point z satisfies |Br (x) ∩ ∂Ωt | < 1. |Br (x)|

δ(x) := lim

r→0

But the Lebesgue Density Theorem [105, Theorem 5.3.1] says δ(y) = 1 for almost all point y ∈ ∂Ω, and thus ∂Ω must have measure zero as claimed.  Corollary 5.14. For all t ∈ [0, 1) it holds that ωψt = (1 − χΩt )ω + tδz0 .

(5.8) In particular

"

(5.9)

ω = t. Ωt

Proof. Since ψt is C 1,1 , ωψt is absolutely continuous with respect to ω, thus ∂Ωt having zero measure with respect to ω means the same is true for ωψt . We thus get ωψt = χΩt ωψt + (1 − χΩt )ωψt . We have already seen ωψt = δz0 on Ωt . By definition ψt = 0 on Ωct and hence on (Ωct )◦ . As this set is open ddc ψt = 0 there, giving (1 − χΩt )ωψt = (1 − χΩt )ω. Again using that ∂Ωt has zero measure yields (1 − χΩt )ω = (1 − χΩt )ω and hence (5.8). The second statement follows from this as " " " " ω= ωψt = ((1 − χΩt )ω + tδz0 ) = X

X

X

ω + t.

Ωct

 We end this section with a final convexity property satisfied by the Hele-Shaw envelopes. Although simple, it is essential in ensuring no information is lost when we later take the Legendre transform. Lemma 5.15 (Convexity). For any given z the function t → ψt (z) is concave, decreasing and continuous in t. Proof. It is clear ψt is concave in t since if t = at1 + (1 − a)t2 where a ∈ [0, 1) and t1 , t2 ∈ [0, 1] then aψt1 + (1 − a)ψt2 ≤ ψt simply because the left hand side is clearly in Sh(X, ω), has at least Lelong number t at z0 and is bounded above by 0. That ψt (z) decreases with t is obvious, and this implies limt→s− ψt is ω-subharmonic and thus one sees lim ψt = ψs ,

t→s−

i.e. ψt is left-continuous in t. Combined with concavity this implies continuity.

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5.4. Basic properties of the Hele-Shaw flow in the plane. We will also want to discuss the weak Hele-Shaw on the plane. So suppose in this section X = C and z0 is the origin. Our K¨ahler metric ω can then be written as ω = ddc φ for some smooth function φ : C → R. We assume throughout the growth condition that for all t > 0 (5.10)

φ(z) ≥ t ln |z|2 + O(1) for |z| ' 0

So, for example, this clearly holds for the standard K¨ahler metric on C for which φ(z) = |z|2 . We are not assuming that the plane has finite area with respect to ω, and so we need to add a word as to why the basic properties of the Hele-Shaw flow from the previous section continue to hold. Given any t > 0 consider the function αt = t ln |z|2 − φ. Clearly αt ∈ Sh(C, ω) and ωαt = tδ0 and ν0 (αt ) = t. On the other hand the growth condition (5.10) implies αt is bounded as |z| tends to infinity, so subtracting a constant we may suppose αt ≤ 0. Thus we may use αt to replace the function provided by Lemma 5.5. Using this one can check the proofs of the basic properties of the Hele-Shaw envelope go through essentially unchanged and give the following. Proposition 5.16 (Basic Properties of the weak Hele-Shaw flow in the Plane). Still assuming the growth condition (5.10) holds, for all t > 0 we have (1) ψt ∈ Sh(C, ω) is locally bounded 5away from z0 (2) ν0 (ψt ) ≥ t (3) ψt ∈ C 1,1 (C \ {0}) (4) ωψt = (1 − χΩt )ω + tδ0 (5) Ωt ω = t (6) Ωt is open, connected, contains the origin and ∂Ωt has measure zero. Furthermore, analogs of the monotonicity and locality statements (Lemma 5.8 and Corollary 5.9) hold; precise statements are left to the reader. Of course one can relate the planar case and the compact case by thinking of C ⊂ P1 in the standard way. Given any large R one can find a K¨ahler form ω ˜ on P1 that agrees with ω on the ball S := {|z| < R}. Then, with an argument as in the proof of the monotonicity statement (Lemma 5.8) if one assumes the weak Hele-Shaw flow domains Ωt and Ω˜t induced by ω and ω ˜ respectively are both are relatively compact in S, then ˜ Ωt = Ωt . In this way one easily passes from statements about the Hele-Shaw on the plane to corresponding statements on P1 . 5.5. Bibliographical remarks. For a much more comprehensive survey on the Hele-Shaw flow, which also goes under the name of Laplacian-growth, the reader is referred to the book of Gustafsson-Teodorescu-Vasil‘ev [58] which also serves as a guide to the vast literature. A difference between what is written here is that we have been working on a compact Riemann surface endowed with a K¨ahler form, whereas the more classical treatment involves the complex plane, usually with the standard Euclidean structure. However this has little effect, and essentially all the fundamental results from the Hele-Shaw theory carry over without difficulty. The point of view of the Hele-Shaw flow on Riemann surfaces was taken up by Hedenmalm-Shimorin [60] and Hedenmalm-Olofsson [59], who emphasise particularly the case of simply connected Riemann surfaces, and it is from these papers

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that several of the basic properties above are taken. The case of the flow on nonsimply connected compact Riemann surfaces has been studied more recently by Skinner [114]. One can ask for more information about the structure of the boundary ∂Ωt . At least when X = C and the background metric is real analytic, it is know this boundary consists of a finite number of real simple analytic curves having a finite number of double and cusp points (see [24, 60] as well as the work of Sakai [106–110]). Constructions similar to the Hele-Shaw envelope are abundant in (pluri)potential theory (as we have seen they show up in the Perron-Bremermann envelope and pluricomplex Green function) and sometimes go under the name of “extremal envelopes” (see, for example [34, 53, 72–75, 92–95]). Lemma 5.10 casts the Hele-Shaw envelope ψt in the framework of variational inequalities and obstacle problems which is a subject in its own right (see, for instance [20, 46, 63, 84]). Perhaps the most important property of ψt we have discussed is its C 1,1 regularity (sometimes called “optimal regularity”), from which we deduced both ∂Ωt has measure zero and a formula for ωψt (in fact for this second statement, at least, one can get away with slightly less regularity). Both of these results originate with the work of Caffarelli-Kinderlehrer [20] and Caffarelli-Rivi`ere [24] who restrict attention, for the most part, to domains in Rn (although given Lemma 5.11 it may well be possible that their techniques can be used to prove Theorem 5.12). Regularity of related envelopes, especially in higher dimensions, has been taken up in many places, for instance [6, 7, 12, 33, 37, 100, 119]. The radially symmetric case from Example 5.3 can be generalised to toric manifolds, which was considered by Shiffman-Zelditch [113] and Pokorny-Singer [91]. There appear to be many different names for the domain Ωt and its complement. In [113] the analog of Ωt is called the “forbidden region”. The complement X \ Ωt is called the “equilbrium set” by Berman [7] and in the theory of variational inequalities and obstacle problems ∂Ωt sometimes goes under the name of “free boundary” and X \ Ωt goes under the name of “coincidence set” (e.g. [63, Definition 6.8]). 6. The Duality Theorem We are now ready to connect the weak Hele-Shaw flow to the HMAE. We continue with X being a compact connected Riemann-surface with distinguished 5 point z0 and background K¨ahler form ω normalised so X ω = 1. 6.1. Another HMAE. Let πX : X × D → X and πD : X × D → D be the projections. Definition 6.1. Set (6.1)   ∗  ˜ := sup Ψ ∈ Psh(X × D, πX ω) : limsupζ→ζ  Ψ(ζ) ≤ 0 for ζ ∈ X × ∂D . Φ and ν(z0 ,0) (Ψ) ≥ 1 In the above, if τ is the standard coordinate on D and z a holomorphic coordinate on X defined near z0 the Lelong number condition ν(z0 ,0) (Ψ) ≥ 1 means that for all c < 1, Ψ(z, τ ) ≤ c ln(|z − z0 |2 + |τ |2 ) + O(1) for (z, τ ) near (z0 , 0).

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˜ is analogous to the pluricomplex Green function discussed in §3.3. The Clearly Φ reason for us introducing this function is it is the weak solution for the following HMAE: ˜ lies in Psh(X × D, π ∗ ω), is locally bounded Proposition 6.2. The function Φ X away from (z0 , 0) and solves (6.2) (6.3) (6.4)

˜ = 0 on X × D \ {(z0 , 0)} ∗ ω (Φ) MAπX ˜ ≥1 ν(z ,0) (Φ) 0

˜ τ ) = 0. lim Φ(z,

|τ |→1

Furthermore (6.5)

˜ τ ) = Φ(z, ˜ eiθ τ ) for all (z, τ ) ∈ X × D and θ ∈ R. Φ(z,

Proof. We give a sketch proof. Observe first both ln |τ |2 and the function ˜ which implies it α(z) from Lemma 5.5 are candidates for the envelope defining Φ, 2 ˜ is locally bounded away from (z0 , 0) and ln |τ | ≤ Φ(z, τ ). On the other hand the ˜ ≤ 0 over X × D, maximum principle applied to the slices {z} × D shows that Φ giving (6.4) For (6.3) it is convenient to consider the blowup p : Y → X × D at the point ∗ ω + ddc Ψ ≥ 0 (z0 , 0) which has an exceptional divisor we denote by E. Suppose πX 2 2 satisfies Ψ ≤ c(ln |z − z0 | + |τ | ) + O(1) near (z0 , 0). Then E is covered by open subsets U on which E is the zero set of some holomorphic function u say, so that p∗ Ψ|U ≤ c ln |u|2 + O(1). Then similar to the proof of Proposition 5.4(3c), one can use the maximum principle to deduce in fact p∗ Ψ|U ≤ c ln |u|2 + O(1) (and thus Ψ ≤ c ln(|z − z0 |2 + |τ |2 ) + O(1)) for an O(1) term that is independent of Ψ. We leave the details to the reader. ˜ solves the claimed HMAE is as in the Perron-Bremermann The fact that Φ envelope. Finally (6.5) is a consequence of the previous statements, since if θ is ˜ ˜ eiθ τ ) is a candidate for the envelope defining Φ.  fixed then Φ(z, ˜ By (6.5) for fixed It is convenient to extend the domain of definition of Φ. −s/2 ˜ z ∈ X, the function Φ(z, e ) is independent of the imaginary part of s, and is ˜ e−s/2 ) as a convex function of s ∈ [0, ∞). subharmonic. Thus we can think of Φ(z, If we set ˜ e−s/2 ) = +∞ for s < 0 Φ(z, ˜ e−s/2 ) is a convex function for all s ∈ R. then Φ(z, Theorem 6.3 (Duality Theorem, Ross-Witt Nystr¨ om [102]). The weak so˜ τ ) to the HMAE and the Hele-Shaw envelopes ψt (z) are related by a lution Φ(z, Legendre transform. That is, ˜ τ ) − (1 − t) ln |τ |2 } (6.6) ψt (z) = inf {Φ(z, |τ |>0

and (6.7)

˜ τ ) = sup{ψt (z) + (1 − t) ln |τ |2 }. Φ(z, t

Proof. For t ∈ [0, 1] consider αt (z, τ ) := ψt (z) + (1 − t) ln |τ |2 for (z, τ ) ∈ X × D.

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∗ ∗ Clearly αt ≤ 0 and πX ω + ddc αt = πX ωψt ≥ 0. Also as νz0 (ψt ) ≥ t,

αt (z, τ ) ≤ t ln |z − z0 |2 + (1 − t) ln |τ |2 + O(1) ≤ ln(|z − z0 |2 + |τ |2 ) + O(1). ˜ giving Thus αt is a candidate for the envelope defining Φ (6.8)

˜ τ ) − (1 − t) ln |τ |2 . ψt (z) ≤ Φ(z,

˜ τ ) − (1 − t) ln |τ |2 ≥ On the other hand if t > 1 then ψt ≡ −∞ and if t < 0 then Φ(z, 2 ˜ Φ(z, τ )−ln |τ | ≥ 0 = ψt (z). Hence (6.8) holds for all t ∈ R, and taking the infimum over all |τ | > 0, ˜ τ ) − (1 − t) ln |τ |2 }. ψt (z) ≤ inf {Φ(z, |τ |>0

˜ τ ) is independent of the argument of τ , it follows For the other inequality, since Φ(z, from Kiselman’s minimum principle [65] that ˜ τ ) − (1 − t) ln |τ |2 } ψ˜t (z) := inf {Φ(z, |τ |>0

is in Sh(X, ω). We wish to show ψ˜t is a candidate for the envelope defining ψt . First, using (6.4) and letting τ → 1 gives ψ˜t ≤ 0. We claim νz0 (ψ˜t ) ≥ t. To see ˜ e−s/2 ) as a convex function in s ∈ [0, ∞). So for this, recall we are thinking of Φ(z, a fixed z ˜ e−s/2 ) + (1 − t)s}. (6.9) ψ˜t (z) = inf {Φ(z, s≥0

˜ has Lelong number at least 1 at (z0 , 0). So for any c < 1 there is a constant Now Φ C such that ˜ τ ) ≤ c ln(|z − z0 |2 + |τ |2 ) + C = c ln(|z − z0 |2 + e−s ) + C (6.10) Φ(z, for (z, τ ) near (z0 , 0). Combining with (6.9) yields (6.11) ψ˜t (z) ≤ c inf {ln(e−s + |z − z0 |2 ) + (1 − t)s} + C. s≥0

By elementary means one easily checks if t ∈ (0, 1) the infimum of ln(e−s + |z − 2 z0 | ) + (1 − t)s is attained when e−s = 1−t t |z − z0 | and at this point the right hand side of (6.11) is equal to 2

c(t ln |z − z0 |2 − (1 − t) ln(1 − t) − t ln t) + C. Hence ψ˜t (z) ≤ ct ln |z − z0 |2 + O(1) for z near z0 . Since this holds for all c < 1 we conclude νz0 (ψ˜t ) ≥ t for t ∈ (0, 1). For t = 0 one notes ln |τ |2 is a candidate for the ˜ which gives ψ˜0 = inf s≥0 {Φ(z, ˜ e−s/2 )+s} ≥ 0 and ˜ so ln |τ |2 ≤ Φ, envelope defining Φ, ˜ 0) ≤ ln |z − z0 |2 + O(1) hence in fact ψ˜0 = 0 = ψ0 . For t = 1, observe ψ˜1 (z) ≤ Φ(z, so νz0 (ψ˜t ) ≥ 1. For t < 0 then certainly νz0 (ψ˜t ) ≥ t, and thus we conclude for t ≤ 1 that ψ˜t is a candidate for the envelope defining ψt , and thus ψ˜t = ψt . Finally for t > 1 by taking s → ∞ in the definition of ψ˜t it is immediate ψ˜t ≡ −∞ and so ψ˜t = ψt for all t giving (6.6). After some rearranging, we have shown ˜ e−s/2 ) + s)}, −ψt (z) = sup{ts − (Φ(z, s∈R

˜ e−s/2 ) + s. So, the i.e. that −ψt (z) is the Legendre transform of u(s) := Φ(z, second statement follows from the first by the involution property of the Legendre transform. In fact, we can see that u(s) is convex and lower semicontinuous (since

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it is continuous on [0, ∞) and constantly −∞ on (−∞, 0)). Thus by the FenchelMoreau Theorem (see e.g. [97]) u(s) is the Legendre transform of −ψt (z) which is (6.7)).  ˜ as a solution to the HMAE over the punctured Remark 6.4. Thinking of Φ disc, we can interpret it as a weak geodesic ray in the space of K¨ahler potentials ˜ τ ) depends only on the absolute value of τ , and so K(X, ω). We have seen that Φ(z, ˜ e−s ) for s ∈ [0, ∞) give a weak using the variable s = − log |τ | the potentials Φ(z, geodesic ray in this space, starting at the potential that is identically zero when s = 0. In the limit as s → ∞, this ray ends up with a singular potential on X that puts all of its mass at the distinguished point z0 (and so it is these geodesic rays that are related through a Legendre transform to the Hele-Shaw flow). In previous work of Donaldson [43], a different free boundary problem is related, again through a Legendre transform, to the HMAE over the annulus, and thus to weak geodesic segments in K(X, ω). 6.2. Connection with the Hele-Shaw domains. So far we have related ˜ to the HMAE with the Hele-Shaw envelopes, and now we connect it the solution Φ to the weak Hele-Shaw domains. ×

Definition 6.5. Let H : X × D → R be defined by ∂ ˜ e−s/2 ) (6.12) H(z, τ ) := + Φ(z, ∂s where s := − ln |τ |2 . Here the notation means we are taking the right derivative, which by by convexity of s → H(z, e−s/2 ) always exists. Our reason for introducing this function is that it records the time at which the weak Hele-Shaw flow arrives at a given point in X. Proposition 6.6. / Ωt }. H(z, 1) + 1 = sup{t : ψt (z) = 0} = sup{t : z ∈ Proof. From (6.7) if ψt (z) = 0 then ˜ e−s/2 ) ≥ (t − 1)s Φ(z, where as always s = − ln |τ |2 , and thus by convexity H(z, 1) ≥ t − 1. For the other direction, suppose ψt (z) = a for some a < 0. Recalling for a fixed z the function t → ψt (z) is concave and decreasing in t , one sees that for t ≤ t ≤ 1 and s ≥ 0 we have ψt (z) + (t − 1)s ≤ a. On the other hand ψt ≤ 0 so if 0 ≤ t ≤ t then ψt (z) + (t − 1)s ≤ (t − 1)s. Putting this together with (6.7) gives ˜ e−s/2 ) ≤ max((t − 1)s, a) Φ(z, and so H(z, 1) ≤ t − 1, which proves the proposition.



As an application we are able to give the following statement about the movement of the boundary of the Hele-Shaw flow. By means of notation, for any S ⊂ X and r > 0 let S + Br = {z ∈ X : d(z, z  ) < r for some z  ∈ S}

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where d denotes a fixed distance function on X (for instance we could take the geodesic distance with respect to the background K¨ ahler metric determined by ω). Corollary 6.7. (1) Assume that H(·, 1) is continuous. Then the boundary of the weak HeleShaw flow is strictly increasing. That is, if z ∈ ∂Ωt for some t > 0 then z ∈ Ωt for all t > t. (2) Assume that H(·, 1) is moreover Lipschitz. Then there is a lower bound on the rate of increase of the weak Hele-Shaw flow. That is, there exist a δ > 0 such that for all 0 < t < t < 1 Ωt + Bδ(t−t ) ⊂ Ωt . Proof. We start with the first statement. Let z ∈ ∂Ωt and (zn )n∈N be a sequence of points in Ωt tending to z as n tends to infinity. Fixing n we then have zn ∈ Ωs for all s ≥ t and so H(zn , 1) + 1 ≤ t. By continuity of H(·, 1) this implies H(z, 1) + 1 ≤ t and so if t > t we must have z ∈ Ωt as desired. For the second statement, let C be the Lipschitz constant of H(·, 1), so |H(z, 1) − H(˜ z , 1)| ≤ Cd(z, z˜) for all z, z˜ ∈ X, and set δ = C −1 . Fix t > t and z ∈ Ωt + Bδ(t −t) . Then there exists z  ∈ Ωt with d(z, z  ) < δ(t − t). As z  ∈ Ωt we clearly have H(z  ) + 1 ≤ t. On the other hand if z∈ / Ωt then H(z) + 1 ≥ t giving t − t ≤ H(z) − H(z  ) ≤ Cd(z, z  ) < Cδ(t − t) = t − t which is absurd. Hence we must have z ∈ Ωt as required.



˜ lies in C 1,1 (X × Σ) then H will be Lipschitz. We Remark 6.8. Of course if Φ ˜ will see in the next section that this always holds when X = P1 , and expect that Φ should be at least C 1,α for all α < 1 when X is a general compact Riemann surface. Even in the case when X = P1 Corollary 6.7 is new (as far as we are aware). Hedenmalm-Shimorin have a similar statement [60, Proposition 3.2] but under the hypothesis Ωt is simply connected along with some regularity assumptions about ˜ and it would be interesting to ∂Ωt . The proof above rests on regularity of Φ, compare this with a proof (if one exists) that uses only one-dimensional techniques such as those from §5. 6.3. Twisting. We end this section by discussing a certain “twisting” tech˜ we have been considering nique that applies when X = P1 to show the quantity Φ can be expressed in a different way without the condition on the Lelong number. We have two motivations for wanting to do this. First, the new formulation solves the classical version of the HMAE as discussed in the introduction, and thus this twisting relates it also to the Hele-Shaw flow. Second, we can use known regularity ˜ results about this version of the HMAE to conclude regularity of Φ. The necessity of restricting to P1 is that we will make use of the existence of a global holomorphic S 1 -action. Consider P1 covered by two copies of C in the standard way with coordinates z and w = 1/z. For non-zero τ ∈ D the map ρτ : P1 → P1 given by f (z) = τ z

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is a biholomorphism fixing z0 . Restricting to those ρτ with |τ | = 1 gives a global holomorphic S 1 -action. Now ρ∗τ ω lies in the same cohomology class as ω and hence we can write ρ∗τ ω = ω + ddc φτ for 5 some smooth function φτ on X. By choosing these to be normalised by requiring φ ω = 0, the φτ are uniquely defined and X τ (6.13)

φ(z, τ ) := φτ (z) ×

is a smooth function on X × D . From now on, let Φ be the Perron-Bremermann envelope on X ×D with boundary data φ. Thus (6.14) + , ∗ Φ := sup Ψ ∈ Psh(X × D, πX ω) : limsupζ→ζ  Ψ(ζ) ≤ φ(ζ  ) for ζ  ∈ X × ∂D . ˜ and Φ is the latter is taken with respect to the So the difference between Φ “twisted” boundary data (z, τ ) → φ(z, τ ), but does not have any condition on the Lelong number at (z0 , 0). The following simple Lemma gives the explicit relationship between these two envelopes. It will be crucial later on when we wish to translate results about envelopes over the punctured disc (which connects most naturally with the Hele-Shaw flow on X) to analogous statements about envelopes over the unpunctured disc. Lemma 6.9. We have ˜ z, τ ) + φ(z, τ ) − ln |τ |2 for (z, τ ) ∈ P1 × D× . Φ(z, τ ) = Φ(τ Proof. Let β(z, τ ) := Φ(τ −1 z, τ ) − φ(τ −1 z, τ ) + ln |τ |2 . One easily checks if |τ | = 1 then β(z, τ ) = 0 and πP∗1 ω + ddc β ≥ 0 and also ν(z0 ,0) (β) ≥ 1. Hence ˜ τ ) giving one inequality, and the other is proved similarly. β(z, τ ) ≤ Φ(z,  ˜ is C 1,1 on P1 × D× . Theorem 6.10. When X = P1 the envelope Φ Proof. From the work of Chu-Tossati-Weinkove (Theorem 4.3) we have Φ is C 1,1 over X × D (we could also use the work of Blocki [18] as P1 has nonnegative bisectional curvature so [18, Theorem 1.4] applies). Thus the desired statement for ˜ follows from Lemma 6.9. Φ  Remark 6.11. It seems likely on a general compact Riemann surface that Φ also satisfies some regularity, and should be at least C 1,α for any α < 1. Our reason for saying this is Φ is describing a weak geodesic ray in the space of K¨ ahler potentials on X, and such regularity is known to hold for many related geodesic rays, such as those considered by Phong-Sturm [90]. Remark 6.12. A point c lying on the boundary ∂Ωt of the Hele-Shaw domain for t in some non-trivial interval is referred to as a stationary point. Theorem 6.10 combined with Corollary 6.7(1) imply that the Hele-Shaw flow on P1 with a smooth area form and empty initial condition never develops any stationary points (as far as we are aware this statement in the smooth case is new).

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7. Harmonic discs We return now to the case of a general compact Riemann surface X K¨ ahler form ω. The next theorem describes precisely the proper harmonic of the weak solution to the HMAE in terms of the Riemann map of those Hele-Shaw domains that are simply connected. As above consider (7.1)  ∗  ˜ := sup Ψ ∈ Psh(X × D, πX ω) : limsupζ→ζ  Ψ(ζ) ≤ 0 for ζ ∈ X × ∂D Φ and ν(z0 ,0) (Ψ) ≥ 1

with discs weak

 .

Definition 7.1. We say the graph of a holomorphic f : D → X is a proper ˜ vanishes along the graph of f away from the ˜ if π ∗ ω + ddc Φ harmonic disc for Φ X ˜ (τ ), τ ) is f ∗ ω-harmonic on D× . origin, or said another way Φ(f Theorem 7.2 (Regularity Theorem, Ross-Witt Nystr¨ om [102]). The graph of ˜ if and only if either a holomorphic f : D → X is a proper harmonic disc of Φ (1) f is the constant map f (τ ) = z0 for all τ ∈ D (where z0 is our given distinguished point in X) (2) For some t the weak Hele-Shaw domain Ωt for ω is simply connected and f : D → Ωt is a biholomorphism (i.e. a Riemann map) with f (0) = z0 . (3) f is the constant map f (τ ) = z  for all τ ∈ D, for some fixed z  ∈ X \ Ω1 . Moreover in the first case H(f (τ ), τ ) ≡ −1, in the second case H(f (τ ), τ ) ≡ t − 1 and in the third H(f (τ ), τ ) ≡ 0. Remark 7.3. More generally we would say that a proper holomorphic curve ˜ if Φ ˜ ◦ g was (πX ◦ g)∗ ω harmonic g : Σ → X × D is a proper harmonic curve of Φ −1 except at g (z0 , 0). But it is not hard to see that any such g would have to be a composition of one of the proper harmonic discs described in Theorem 7.2 with a finite cover of the unit disc, so in particular having the same image. Before the proof we need the following statement: Lemma 7.4. Fix 0 < |τ | < 1. Then H(z, τ ) = t − 1

⇐⇒

˜ τ ) = ψt (z) + (1 − t) ln |τ |2 . Φ(z,

Proof. Fix a point z ∈ X and 0 < |τ0 | < 1 and let s0 = − ln |τ0 |2 . From the Duality Theorem (6.7) ˜ τ0 ) = sup{ψt (z) + (1 − t) ln |τ0 |2 }. Φ(z, t

Now ψt (z) is continuous in t (Lemma 5.15), so for some t ˜ τ0 ) = ψt (z) + (1 − t) ln |τ0 |2 = ψt (z) − (1 − t)s0 . Φ(z, On the other hand, we certainly have ˜ e−s/2 ) ≥ ψt (z) − (1 − t)s for all s. Φ(z, ˜ e−s/2 ) at the point (s0 , Φ(z, ˜ e−s0 /2 )) is So the slope of the convex function Φ(z, equal to the slope of the linear function s → ψt (z) − (1 − t)s, which is clearly t − 1. Hence ∂ ˜ e−s/2 ) = t − 1, Φ(z, H(z, τ0 ) = + ∂s |s=s0 which is enough to prove the lemma. 

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Proof of Theorem 7.2. We shall prove if the graph of f is a proper har˜ then it is of one of the three forms in the statement of the theorem. monic disc for Φ Fix some τ0 ∈ D× and set t0 = H(f (τ0 ), τ0 ) + 1. We claim (7.2)

˜ (τ ), τ ) for all τ ∈ D. ψt0 (f (τ )) + (1 − t0 ) ln |τ |2 = Φ(f

To see this, consider ˜ (τ ), τ ) for τ ∈ D. α(τ ) := ψt0 (f (τ )) + (1 − t0 ) ln |τ |2 − Φ(f ˜ is π ∗ ω-harmonic along {(f (τ ), τ ) : τ = 0}), satisfies Then α is subharmonic (since Φ X α ≤ 0 by the Duality Theorem (6.7) and α(τ0 ) = 0 by Lemma 7.4. If f (0) = 0 ˜ is π ∗ ω-harmonic even over {(f (τ ), τ ) : τ ∈ D} and so (7.2) follows from the then Φ X maximum principle. If f (0) = 0 then by looking at the Lelong number, α extends over τ = 0 and the maximum principle still applies to give (7.2). In particular Lemma 7.4 combined with (7.2) implies H(f (τ ), τ ) ≡ t0 − 1 for all τ = 0, giving the last statement of the theorem. Suppose now that f is non-constant. We shall show f is as in case (2) of the statement, by first proving the image of f lies in Ωt0 and then proving it ˜ (τ ), τ ) is f ∗ π ∗ ω harmonic is a biholomorphism taking 0 to z0 . Observe if Φ(f X  on a neighbourhood of some τ ∈ D, then (7.2) implies ψt0 is ω-harmonic on a neighbourhood of f (τ  ). But Corollary 5.14 implies ωψt = (1 − χΩt )ω + tδz0 .

(7.3) 

so this in turn implies f (τ ) ∈ Ωt0 . ˜ (τ ), τ ) is f ∗ π ∗ ω harmonic on a neighbourhood of any nonBy hypothesis, Φ(f X   zero τ ∈ D, so f (τ ) ∈ Ωt0 for all τ  = 0. In particular Ωt0 is non-empty, so we must have t0 > 0 and so z0 ∈ Ωt0 by Corollary 5.7. If f (0) = z0 then f (0) ∈ Ωt0 . On the ˜ (τ ), τ ) is f ∗ π ∗ ω-harmonic on a neighbourhood other hand, if f (0) = z0 then Φ(f X of 0 ∈ D, giving f (0) ∈ Ωt0 . Thus in either case f (0) ∈ Ωt0 , and hence the image of f lies in Ωt0 as claimed. We next prove f is proper. To see this let τi be a sequence in D such that |τi | → 1 as i → ∞. Then by (7.2) and then (6.4) ˜ (τi ), τi ) − (1 − t0 ) ln |τi | = 0. lim ψt0 (f (τi ), τi ) = lim Φ(f

i→∞

i→∞

But Ωt0 is exhausted by the compact sets {z : ψt0 (z) ≤ −1/n} for n ∈ N and f (τi ) escapes to infinity in Ωt0 . Thus f is proper as claimed. Next we show the preimage S := f −1 (z0 ) is precisely the point 0 with multiplicity one. Given this, the fact that f is a biholomorphism with f (0) = z0 follows from a standard argument with the winding number (Lemma 7.5). Observe f (τ ) = z0 for any τ = 0, since otherwise the right hand side of (7.2) would be −∞ whereas the left hand side is finite. If f (0) = z0 then S would be empty, which is absurd by Lemma 7.5. So we conclude z0 ∈ S with some multiplicity m ≥ 1. Using (7.2) once again ˜ (τ ), τ ) ≥ ln |τ |2 . ψt0 (f (τ )) + (1 − t0 ) ln |τ |2 = Φ(f Clearly ψt0 (f (τ )) has Lelong number mt0 at 0, so the left hand side has Lelong number mt0 + (1 − t0 ) at 0. By the right hand side has Lelong number 1, giving

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mt0 + (1 − t0 ) ≤ 1, so m = 1. So in conclusion we have shown if f is not-constant then f is of the form case (2). Suppose now f ≡ z  is constant. If z  = z0 then f is as in case (1). Otherwise ˜ (τ ), τ ) is f ∗ π ∗ ω-harmonic even near τ = 0. Again (7.2) gives z  = z0 , and so Φ(f X



˜ , τ ) = ψt (z  ) + (1 − t0 ) ln |τ |2 for τ ∈ D. Φ(z 0 But this implies t0 = 1, else otherwise the right hand side takes the value −∞ at the point τ = 0, whereas the left hand side is finite. Letting τ → 1 and using (6.4) ˜ , τ ) = 0 ψ1 (z  ) = lim Φ(z τ →1



and hence z ∈ Ω1 , implying f is as in case (3). The converse, namely that each of the three listed functions, are proper harmonic discs is easier and is left to the reader.  Lemma 7.5. If f : D1 → D2 is a proper holomorphic map between two open domains in P1 then the number of preimages Np := #{f −1 (p)} (counted with multiplicity) is constant. Proof. Let γ be a smooth curve in D2 connecting two points p and q and let U be a finite union of open discs compactly supported in D1 which together cover the compact set f −1 (γ). Since the image of any boundary component of U cannot cross γ the winding numbers of the image of any such boundary component with respect to p and q must be the same. Since that winding number counts the number of preimages inside that component we get by adding up the winding numbers for  the different boundary components that Np = Nq . From this we get a description of all the proper harmonic discs for a more classical version of the HMAE, at least when X = P1 . Corollary 7.6. Let X = P1 . Then the graph of g : D → P1 is a proper harmonic disc for the weak solution to the HMAE over X × D with boundary data φ(z, τ ) from (6.13) if and only if either (1) g is the constant map g(τ ) = z0 for all τ ∈ D or (2) for some t the weak-Hele shaw domain Ωt for ω is simply connected and the map τ → τ g(τ ) is a Riemann-map from D to Ωt taking 0 to z0 or (3) g(τ ) = τ −1 z  for some fixed z  ∈ Ω1 . Proof. This is immediate from Theorem 7.2, since Lemma 6.9 implies that ˜ if and only if the graph of g(τ ) = τ −1 f (τ ) the graph of f is a harmonic disc for Φ is a harmonic disc for Φ.  Example 7.7. The above may be used to produce examples of boundary conditions for the HMAE over the (punctured) disc for which the weak solution to the HMAE is regular. For suppose X = P1 with coordinate z ⊂ C ⊂ P1 and Ωt for t ∈ (0, 1) is a smoothly varying family of simply connected domains with the property that Ωt is a symmetric disc around z = 0 with area equal to t (taken with respect to the Fubini-Study form ωF S ) for t < and t > 1 − . We will see in §9.1 that {Ωt }t∈(0,1) is the weak Hele-Shaw flow with respect to some K¨ahler form ωF S + ddc φ where φ ∈ K(X, ωF S ). Thus Theorem 7.2, the weak solution to the HMAE with boundary data determined by φ will be regular (the reader will find essentially the same example in [42]).

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We next discuss an interesting link between the Riemann map, the Hele-Shaw ˜ to the HMAE. Continue flow and the family of forms coming from the solution Φ × 1 1 to assume X = P , z0 is the origin in the chart Cz ⊂ P , and for each τ ∈ D set ˜ τ ). ωτ := ω + ddc Φ(·, Then ω1 = ω, but in general ωτ is a semipositive (1, 1)-current on X (not necessarily smooth). One can define the weak Hele-Shaw flow with respect to such ωτ in precisely the same way as the smooth case, and we denote the associated Heleτ Shaw domains by Ωω t . For r > 0 set Dr = {z ∈ C : |z| < r}. 1 Proposition 7.8. Suppose t is such that Ωω t ⊂ Cz ⊂ P is proper and simply ω connected and let ft : D → Ωt be a Riemann-map with f (0) = 0. Then for each τ ∈ D× τ Ωω t = ft (D|τ | ). σ Proof. Fix σ ∈ D× and set r := |σ|, so our aim is to show ft (Dr ) = Ωω t . iθ ˜ is invariant under (z, τ ) → (z, e τ ) (6.5) we may as well assume σ is real, so As Φ ωσ = ωr . For a function F on P1 × D and D ⊂ D we write F |D for the restriction of F ˜ to P1 × D. Then Φ| Dr is the solution to the Dirichlet problem for the HMAE with ˜ ˜ boundary data Φ(·, τ )τ ∈∂Dr and the requirement that Φ| Dr has Lelong number 1 at 1 the point (z0 , 0) ∈ Cz × Dr ⊂ P × Dr . Letting s := − ln |τ |2 consider again

∂ ˜ × Φ(z, e−s/2 ) for (z, τ ) ∈ P1 × D + ∂s which is well-defined and Lipschitz (Theorem 6.10). Clearly this is compatible with restriction, i.e. ∂ ˜ −s/2 ). H|D× (z, τ ) = + Φ| Dr (z, e r ∂s ˜ is π ∗ ω-harmonic along the graph of f and H(f (τ ), τ ) = t − 1. By Theorem 7.2, Φ X 1 Now H is also S -invariant and so this in particular implies H(z, τ ) :=

H(f (reiθ ), r) = H(f (reiθ ), reiθ ) = t − 1 for all θ ∈ R. In other words the function H(·, r) takes the value t − 1 on the boundary of f (Dr ). On the other hand Proposition 6.6 implies r H(z, r) + 1 = sup{s : z ∈ / Ωω s }

(we remark the proof of Proposition 6.6 does not require smoothness or strict r positivity assumptions of ωr ). Thus Ωω is the interior component of the curve t iθ θ → f (re ) (that is, the component containing the point z = 0), which gives r  Ωω t = f (Dr ) as claimed. 8. The Strong Hele-Shaw Flow We turn next to the strong Hele-Shaw flow. Although it is certainly possible to consider this on a general Riemann surface, for ease of exposition we shall consider only the case of the complex plane. We will, however, take the flow with respect an arbitrary area form, which generalises the classical case in which the plane is usually equipped with the standard Euclidean structure.

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316

¨ JULIUS ROSS AND DAVID WITT NYSTROM

8.1. Definitions. Let 0 < a < b < ∞ and suppose {Ωt }t∈(a,b) is a family of smoothly bounded domains in C. By this we mean given any t0 ∈ (a, b) and any point p ∈ ∂Ωt0 there exists real coordinates x, y on an open set U ⊂ C containing p such that ∂Ωt0 ∩ U = {(x, y) : y = gt0 (x)} for some smooth function gt0 . We also assume this family is smooth, by which we mean one can pick U so that gt is smooth in t for t close to t0 . As a last assumption we assume also Ωt is increasing, so Ωt ⊂ Ωt for t < t . So if n denotes the outward unit normal vector field n on ∂Ωt0 then for t close to t0 we can write ∂Ωt = {x + f (x, t)nx : x ∈ ∂Ωt0 } for some smooth function ft (x) = f (x, t) on ∂Ωt0 that is positive for t > t0 and negative for t < t0 . The normal velocity of ∂Ωt0 is defined to be dft // n. Vt0 := dt t=0 We will take the origin 0 as our distinguished point, and assume 0 ∈ Ωt for all t. For each t let pt (z) := −GΩt (z) where GΩt denotes the Green’s function for Ωt with logarithmic singularity at the origin. Thus pt = 0 on ∂Ωt and Δpt = −δ0 . The statement that pt exists and is smooth on Ωt \ {0} is classical. We also fix a smooth area form on C which we write as 1 dA κ where dA = dx ∧ dy is the standard Lebesgue measure and κ is a strictly positive real-valued smooth function on C. Definition 8.1. (Strong Hele-Shaw flow) We say {Ωt }t∈(a,b) is the strong HeleShaw flow if (8.1)

Vt = −κ∇pt on ∂Ωt for t ∈ (a, b)

where Vt is the normal velocity of ∂Ωt . When necessary to emphasise the dependence on the area form we refer to this as the strong Hele-Shaw flow with respect to the area form κ1 dA (or with respect to κ). The above has the following physical interpretation. Consider two parallel plates infinite in all directions separated by a small gap. Suppose between these two plates is some porous medium with varying permeability, and a fluid is injected into the gap through a fixed point in one of the plates at a constant rate. As the gap between the plates is small, this is essentially a two-dimensional flow that is modelled by the region Ωt that the fluid occupies at time t. We may as well assume the fluid is injected at the origin. Then the permeability of the medium is encoded by a function κ : C → R+ , so the fluid moves more freely in the areas of the plane in which κ is relatively big. The function pt models the pressure of the system, and we make some physical assumptions, namely the fluid is incompressible (meaning pt is harmonic away from the origin) and the medium itself does not exert any pressure on the system (meaning that pt is constant on the boundary, so after subtracting a constant we may as well take to be zero). The equation of motion (8.1) for the

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THE DIRICHLET PROBLEM FOR THE COMPLEX HMAE

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strong Hele-Shaw flow is then a case of Darcy’s law which describes the flow of a fluid through a porous medium. 8.2. Strong implies weak. Our next goal is to prove the strong Hele-Shaw flow is also a weak one. To do so, we start with a famous calculation due to Richardson [98]. Lemma 8.2. Suppose {Ωt }t∈(a,b) is a strictly increasing smooth family of simply connected domains in C containing the origin that satisfies (8.2)

Vt = −κ∇pt on ∂Ωt

as in (8.1). Then for any integrable subharmonic function h on Ωt , and t0 < t " dA ≥ (t − t0 )h(0). h κ Ωt \Ωt0 Proof. We compute using the Reynolds transport theorem, " " " 1 Vt ∂pt d (8.3) ds h dA = h ds = − h dt Ωt κ κ ∂n ∂Ωt ∂Ωt " " ∂h = (pt Δh − hΔpt ) dA − pt ds ≥ h(0) ∂n Ωt ∂Ωt since Δh ≥ 0 and pt = 0 on ∂Ωt and Δpt = −δ0 .



Corollary 8.3. With the assumption of the above lemma, suppose a = 0 and Ωt tends to {0} as t → 0 (i.e. given any neighbourhood U of the origin Ωt ⊂ U for t sufficiently small). Then for any integrable subharmonic function h on Ωt , " dA ≥ th(0) h (8.4) κ Ωt and equality holds if h is harmonic. In particular " dAζ (8.5) ln |z − ζ|2 / Ωt , = t ln |z|2 for z ∈ κ(ζ) Ωt " dAζ > t ln |z|2 for z ∈ Ωt . (8.6) ln |z − ζ|2 κ(ζ) Ωt Proof. Taking the limit as t0 → 0 in the above Lemma gives (8.4) The statement about harmonic functions follows as if h is harmonic then h and −h are subharmonic. Equation (8.5) follows as if z ∈ / Ωt then h(ζ) := ln |z − ζ|2 is harmonic for ζ ∈ Ωt . If z ∈ Ωt then Δ ln |z − ζ|2 = 2δz , so in Richardson’s calculation (8.3) " " d 1 h dA ≥ 2pt δz + h(0) > h(0) dt Ωt κ Ωt from which one deduces the strict inequality in (8.6).  Proposition 8.4 (Gustafsson). Suppose {Ωt }t∈(0,b) is a smooth family of strictly increasing simply connected domains that is the strong Hele-Shaw flow with respect to κ, and assume {Ωt }t∈(0,b) tends to {0} as t → 0. Then the weak HeleShaw envelope with respect to the K¨ ahler form 1 ω := dA κ

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¨ JULIUS ROSS AND DAVID WITT NYSTROM

318

is given by

"

dAζ + t ln |z|2 , κ(ζ) Ωt is the weak Hele-Shaw flow with respect to ω. ψt (z) = −

and {Ωt }t∈(0,b)

Proof. For the proof let

log |z − ζ|2

"

ψ˜t (z) := −

log |z − ζ|2 Ωt

dAζ + t ln |z|2 κ(ζ)

and write Ωw t := {z ∈ X : ψt (z) < 0} for the weak Hele-Shaw flow with respect to ω. So the goal is to prove ψ˜t = ψt and Ωw t = Ωt For large R let BR = {|z| < R} and set " dAζ for z ∈ C. φ(z) = log |z − ζ|2 κ(ζ) BR Then on BR , ddc φ = ω and

"

ωψ˜t = ddc (φ + ψ˜t ) = ddc

BR \Ωt

ln |z − ζ|2

dAζ ≥ 0. κ(ζ)

As R can be arbitrarily large this implies ψ˜t ∈ Sh(C, ω). Clearly ν0 (ψ˜t ) = t and (8.5,8.6) imply ψ˜t ≤ 0 with equality on Ωct . Thus ψ˜t is a candidate for the envelope defining the Hele-Shaw envelope, so ψ˜t ≤ ψt giving Ωw t ⊂ Ωt . Now both ψt and ψ˜t have Lelong number precisely t at the origin, the maximum principle implies ψt ≤ ψ˜t over Ωt , and so ψt = ψ˜t everywhere, and Ωt ⊂ Ωw t follows from (8.6).  8.3. Weak and Smooth implies Strong. We now show if the weak HeleShaw flow is smooth and smoothly varying, then it is in fact the strong Hele-Shaw flow. Lemma 8.5. Suppose {Ωt }t∈(0,t0 ) is a smoothly varying family of bounded increasing domains, such that for any function h that is harmonic on Ωt , " dA = th(0) (8.7) h κ Ωt Then {Ωt }t∈(0,t0 ) is the strong Hele-Shaw flow with respect to κ. Proof. This is Richardson’s calculation backwards. Let h be as in the statement. Then using the hypothesis (8.7) " " d dA Vt h(0) = = h h ds. dt Ωt κ κ ∂Ωt On the other hand, just as in (8.3) " " " ∂h ∂pt − ds = h (pt Δh − hΔpt ) dA − pt ds = h(0). ∂n ∂n ∂Ωt Ωt ∂Ωt 

 Vt ∂pt − h ds = 0 κ ∂n ∂Ωt and since this holds for all such harmonic functions we must have Vt ∂pt = on ∂Ωt κ ∂n making {Ωt }t∈(0,t0 ) the strong Hele-Shaw flow Therefore

"

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THE DIRICHLET PROBLEM FOR THE COMPLEX HMAE

319

Corollary 8.6. Suppose for some t0 the weak Hele-Shaw domains {Ωt }t∈(0,t0 ) taken respect to 1 ω := dA κ are bounded and smooth (i.e. each Ωt is smoothly bounded and varies smoothly and each Ωt is bounded for t < t0 ). Then {Ωt }t∈(0,t0 ) is the strong Hele-Shaw flow with respect to κ. Proof. Let h be harmonic on Ωt . By Proposition 5.16(4), ωψt = (1 − χΩt )ω + tδ0 , where ψt is the Hele-Shaw envelope, giving " " " dA = h hω = − hddc ψt + th(0) = th(0) κ Ωt Ωt Ωt where the last equality uses Greens formula applied to a smooth domain containing Ωt on which h is harmonic. Thus the result follows from Lemma 8.5.  8.4. Bibliographical remarks. The weak and strong point of view for the Hele-Shaw flow is a theme in the work of Gustafsson (e.g. [54–56]), and the reader interested in more is referred again to [58]. Classically this flow is considered with respect to the standard area form (Lebesgue measure), with a given initial domain Ω0 . The first problem then becomes proving short time existence of the Hele-Shaw flow, a result that goes back to Kufarev–Vinogradov [120] who prove that for a simply connected initial domain with real analytic boundary the strong Hele-Shaw flow (taken with respect to the standard Lebesgue measure) exists for some interval both forwards and backwards in time. This has then been reproved in various forms in [55, 81, 96, 118]. It is not really interesting to consider the case of empty initial condition in the classical case, as then the flow consists simply of concentric discs centered at the origin. However, if one allows a general area form, then the problem of shortterm existence of the Hele-Shaw flow with empty initial condition is non-trivial. Under the assumption that the area form is analytic and hyperbolic this short term existence is due to Hedenmalm-Shimorin [60], and when the area form is merely smooth by the authors [101]. That is, given an arbitrary smooth area form, there exists an > 0 such that the strong Hele-Shaw flow exists for 0 < t < and tends to {0} as t tends to 0. Moreover, as long as is sufficiently small, each Ωt is smoothly bounded and simply connected. The proof that we give, and the only one known at present, comes about through the connection between the Hele-Shaw flow and the Monge-Amp`ere foliation. First, using a form of Schwarz function, we interpret a simply connected Hele-Shaw domain as a holomorphic disc with boundary in a totally real submanifold (just as in Donaldson’s LS-submanifolds). This converts the short term existence problem of the Hele-Shaw flow to a problem about deforming such holomorphic discs, which is a well-known elliptic problem. The reader is referred to [101] for details. Richardson’s calculation represents an important viewpoint of the Hele-Shaw flow (see [57] for a survey). Putting h(z) = z k for k ∈ N≥1 , equation (8.4) says that for the strong Hele-Shaw flow the “complex moments” " dA zk Mk (t) := κ Ωt are constant with respect to t. This illustrates the fundamental nature of the Hele-Shaw flow. For assuming that κ is analytic and simply connected domain

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320

¨ JULIUS ROSS AND DAVID WITT NYSTROM

Ωt0 with analytic boundary, the set {Mk (t)} form local coordinates for the set of analytic perturbations of Ωt0 (that is, any nearby domain with analytic boundary is uniquely specified by its complex moments). So any such flow starting at Ωt0 can, in principle, be described by its change in complex moments. Thus the HeleShaw flow is the simplest among all possible flows, and with this viewpoint it is not surprising that it appears in so many parts of pure and applied mathematics. 9. Examples

5 We work throughout with X = P1 with K¨ ahler form ω normalised so P1 ω = 1. We consider P1 covered by two copies of C with coordinates z and w = 1/z respectively (we denote these two charts by Cz and Cw ) and let z0 be the point ˜ to the HMAE z = 0. In each case we will deduce information about the solution Φ over the punctured disc. The interested reader will easily be able to translate these to similar statements for the HMAE over the disc using Lemma 6.9. 9.1. Flows developing self-tangency. Definition 9.1. We say the Hele-Shaw for develops self-tangency at a point p ∈ Cz ⊂ P1 if there exists a t0 > 0 such that (1) Ωt is smoothly bounded, simply connected and varies smoothly for t < t0 and (2) Ωt0 is a simply connected in Cz and ∂Ωt0 is the image of a smooth locally embedded curve intersecting itself tangentially precisely at the point p (see Figure 1).

Ωt0 z0

p

Figure 1. The Hele-Shaw flow developing self-tangency I Theorem 9.2 (Ross-Witt Nystr¨ om). Suppose the Hele-Shaw flow for ω de˜ to the HMAE is not twice velops self-tangency at p. Then the weak solution Φ differentiable at the points (p, τ ) for |τ | = 1. Rather than giving a full proof we illustrate this with an instructive example. Say (x, y) are smooth coordinates centered at p, and that near p Ωt = {y < −x2 − (t0 − t)} ∪ {y > x2 + (t0 − t)} Set h(x, y) := H((x, y), 1)

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for t0 < t.

THE DIRICHLET PROBLEM FOR THE COMPLEX HMAE

where as usual H(z, τ ) :=

321

∂ ˜ Φ(z, e−s/2 ), ∂s+

and recall by Proposition 6.6 H(z, 1) + 1 = sup{t : z ∈ / Ωt }. Thus for |y| sufficiently small



h(0, y) = and from this it is clear differentiable at (p, 1).

∂h ∂y

t0 − y − 1 y > 0 t0 + y − 1 y < 0

˜ is not twice does not exist at the origin, and so Φ

Of course, for this idea have any use, we need to be able to ensure the HeleShaw can develop self-tangency. To do so we start by showing essentially any reasonable family of simply connected domains is the Hele-Shaw flow with respect to some smooth area form κ1 dA. Assume for t ∈ (a, b) that Ωt is smoothly bounded, smoothly varying simply connected and strictly increasing and each contains the origin. Take pt to be defined by pt = 0 on ∂Ωt and Δpt = −δ0 . As already mentioned, the fact pt exists and is smooth on Ωt \{0} is classical. What is also true is pt varies smoothly with t (it seems to the authors that all the known proofs of the existence of pt actually prove this stronger statement, see for instance [103, Appendix A]). Then (as observed by Berndtsson) one can reverse-engineer the defining equation for the Hele-Shaw flow to define a smooth function κ by requiring (9.1)

Vt = −κ∇pt on ∂Ωt for t ∈ (a, b).

Since {Ωt } is assumed to be strictly increasing, Vt is non-vanishing so κ is a welldefined strictly positive smooth function on some subset of C. If we further assume a = 0 and for t sufficiently small Ωt is just a disc centred at the origin with Lebesgue area t, then κ is constant on ∂Ωt for t sufficiently small, and thus extends to a smooth function across the origin. So, by construction, {Ωt }t∈(0,b) is the strong Hele-Shaw flow with respect to κ1 dA. So far we have defined a smooth κ on Ωb . Assuming that κ extends to a smooth function on Ωb , we may then extend it to a smooth function on all of P1 , giving an area form whose Hele-Shaw flow agrees with {Ωt } for t < b. We can now sketch how to use this to produce an area form whose Hele-Shaw flow develops self-tangency (see the right hand side Figure 2 and observe that in this figure have moved our distinguished point z0 to be the point −1). Fix t0 ∈ (0, 1) ˜ t be as in the figure. We assume Ω ˜ t has analytic boundary, and is and let Ω 0 0 ˜ t so z˜02 = z0 = −1. symmetric under x + iy → −x + iy. Let z˜0 := −i ∈ Ω 0 i 2 Consider the form ω ˜ := π |z| dz ∧ dz which is real analytic and strictly positive away from z = 0. Then from short time existence of the strong Hele-Shaw flow with analytic initial conditions [60, Theorem 6.2], there is a δ > 0 such that the strong ˜ t with injection point z˜0 taken respect to ω Hele-Shaw flow with initial condition Ω ˜ 0 ˜ is the pullback of the standard K¨ ahler exists for t ∈ [t0 − δ, t0 + δ]. Next observe ω ˜ t ) for t ∈ [t0 − δ, t0 ], so form ω := ddc |z|2 on C by the map f (z) = z 2 . Set Ωt := f (Ω by construction Ωt0 is self-tangent at the point p = 1. It is not hard to show that

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−1

1

˜ t0 Ω

z → z 2

Ωt0 z0

1

z˜0

Figure 2. The Hele-Shaw flow developing self-tangency II {Ωt }t∈[t0 −δ,t0 ] is the strong Hele-Shaw flow with respect to ω. We then complete this to a flow that tends to the point z0 as t tends to zero by taking Ωt0 −δ and shrinking it smoothly towards z0 . Our previous discussions show that it is possible to do so in such a way to obtain a K¨ ahler form on P1 whose Hele-Shaw flow agrees with Ωt for t ≤ t0 , and thus develops self-tangency as desired. 9.2. Multiply-connected flows. Using what has already been said, it is not hard to show there are K¨ ahler forms on P1 whose corresponding Hele-Shaw flow ceases to be simply connected at some point in time. One way to arrange this is to use the flow from the previous section that develops self-tangency at a point p at time t0 , so for a short time after t0 the domain Ωt will not be simply connected. Another way to produce such an example is to start with a K¨ ahler form that puts almost all of its mass on a given annulus A ⊂ P1 containing z0 . Physically this means the Hele-Shaw flow is modelling a fluid moving through a medium that has very high permeability on A, and low permeability outside of A. Intuitively one expects that the Hele-Shaw domains will rapidly wrap around within A before it has a chance to completely cover the bounded domain in the complement of A, thus giving a flow that at some point becomes non simply connected. This idea can be made rigorous, and we refer the reader to [102, Proposition 1.4] for details. Theorem 9.3 (Ross-Witt Nystr¨ om). Suppose ω is a K¨ ahler form on P1 and there exist two times t0 < t1 such that the weak Hele-Shaw domains Ωt with respect to ω is not simply connected for any t ∈ (t1 , t2 ). Then there exists an open set U ⊂ P1 ×D intersecting P1 ×∂D non-trivial that does not meet any proper harmonic ˜ disc of Φ. Proof. Theorem 7.2 lists all the harmonic discs, and also says the function H ˜ From this one sees that no such disc can is constant on any harmonic disc of Φ. intersect the open set U := {(z, τ ) : t1 − 1 < H(z, τ ) < t2 − 1, |τ | > 0}. Since H(z, 1) is continuous, and attains both values −1 and 0 somewhere on X, it follows from continuity that U ∩ (P1 × ∂D) is non-empty.  ˜ to the HMAE is far The point of this statement is it implies the solution Φ away from being regular, since the existence of U obstructs the possibility of a foliation of P1 × D by proper harmonic discs. It is interesting to compare with the

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example of Gamelin and Sibony, Example 3.3. There the set of proper harmonic discs did also not foliate the whole domain (which in this case was the unit ball in C2 ) but the boundaries of those discs did foliate the boundary of the domain. In our example we see even this is not the case. 9.3. Flows with simply connected final domains. Our third example concerns Hele-Shaw flows on P1 whose final domain is biholomorphic to the disc. Suppose γ is a non-trivial curve in Cw through the point w = 0 (i.e. the point z = ∞). That is, γ is image of a smooth function [0, 1] → Cw that does not intersect itself and passes through w = 0. Theorem 9.4 (Ross-Witt Nystr¨ om). Suppose the final Hele-Shaw domain of ω satisfies Ω1 = P1 \ γ. ˜ to the HMAE There there is an open subset S ⊂ P1 × D such that the solution Φ satisfies ˜ = 0 on S. πP1 ω + ddc Φ ˜ can be Said another way, we already know the rank of the form πP1 ω + ddc Φ c 2 ˜ at most 1, since (πP1 ω + dd Φ) = 0. Thus the above gives an open subset S on ˜ fails to have maximal rank. which πP1 ω + ddc Φ Proof. We shall prove the slightly weaker statement that for each τ ∈ D× ˜ τ ) vanishes on some non-empty open subset of P1 (and the the current ω + ddc Φ(·, reader is referred to [104] for the proof of the full statement). As Ω1 = P1 \ γ, and γ passes through the point w = 0, we see Ω1 is a simply connected proper subset of Cz . Consider the Riemann map f : D → Ω1 with f (0) = 0. Then by Proposition 7.8 × τ Aτ := f (D|τ | ) = Ωω 1 for τ ∈ D . In particular, Aτ is a proper subset of Cz whose complement has non-empty interior if |τ | < 1. On the other hand, for all t ∈ [0, 1] " ωτ = t. τ Ωω t

(we saw this statement Corollary 5.14 under the assumption that ωτ is a K¨ahler form, and this more general statement can be deduced using [12, Remark 1.19, Corollary 2.5]). Therefore " " ωτ = ωτ = t. Aτ

5

5

τ Ωω 1

But our normalisation is that P1 ωτ = P1 ω = 1, and so ωτ gives zero measure to  the complement of Aτ , which is precisely what we were aiming to prove. It is not hard to construct a specific example of a K¨ ahler metric on P1 for 1 which Ω1 = P \ γ for some such arc γ. To do so, let ωF S be the Fubini-Study form, so ω = ln(1 + |w|2 ) on Cw . We claim there is a φ ∈ C ∞ (P1 ) such that ω := ωF S + ddc φ > 0 and φ ≥ − ln(1 + |w|2 ) with equality precisely on γ. One can then deduce easily that Ω1 = {z : φ(w) > − ln(1 + |w|2 )} = P1 \ γ.

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¨ JULIUS ROSS AND DAVID WITT NYSTROM

To produce such a φ, assume for simplicity that γ is the interval [−1, 1] ⊂ R ⊂ Cw and let α : R → R be a non-negative smooth non-decreasing convex function with α(t) = 0 for t ≤ 1 and α(t) > 0 for t > 1. Then u(w) := α(|w|2 ) + Im(w)2 is a smooth strictly subharmonic function on Cw that vanishes precisely on γ. Using a regularised version of the maximum function, one can adjust the function

u − ln(1 + |w|2 ) for some small constant > 0 to have the correct behaviour far away from γ to ensure φ extends to a smooth function over P1 and ωF S + ddc φ > 0. The reader will find full details in [104, Section 5.4]. 9.4. Hele-Shaw flow with acute corner points. Our final example exploits work of Sakai concerning the behaviour of the Hele-Shaw flow with corner points. A point c lying on the boundary ∂Ωt of the weak Hele-Shaw domain for t in some non-trivial interval is referred to as a stationary point. Sakai proves in [110] (see also [109, Theorem 6.2]) that if ∂Ω0 contains a corner point c with angle strictly less than π/2 then c is a stationary point for the weak Hele-Shaw flow starting at Ω0 (this is to be taken as holding in the plane with its the Euclidean structure). Suppose that Ω0 ⊂ Cz ⊂ P1 is such a domain and set i ω := (1 − χΩ0 )dz ∧ dz 2 on a large ball containing Ω0 . We then extend ω to a smooth K¨ahler form outside of this ball to all of P1 . Observe that ω is absolutely continuous and semipositive, but of course not smooth. Looking back at the proofs of the Duality Theorem and it implications for the movement of the boundary of the weak Hele-Shaw flow (Corollary 6.7) it is clear that they still hold for such ω. ˜ to the HMAE Proposition 9.5. With background form ω, the weak solution Φ is not in C 1 (P1 × D). Proof. Essentially by definition, Ω0 is the weak Hele-Shaw domain at time t = 0 with respect to ω. By the result of Sakai, the corner point of Ω0 is stationary, ˜ is and thus by Corollary 6.7(1) the function H is not continuous, which means Φ 1 not C .  The implications of this can be expressed in terms of potentials. If ωF S denotes the Fubini-Study form, then (after possibly scaling ω) we can write ω = ωF S + ddc φ for some potential φ. As ω is absolutely continuous φ has bounded Laplacian, and thus lies in C 1,α for all α < 1. On the other hand combining the previous Proposition with Lemma 6.9, the weak solution Φ := sup{Ψ ∈ Psh(P1 × D, πP∗1 ωF S ) : Ψ(z, τ ) ≤ φ(τ z, τ ) for |τ | = 1} to the HMAE is not even in the class C 1 . 9.5. Final Bibliographical Remarks. The final example is new, but the first three are taken from [103], [102] and [104] respectively, and the reader will find stronger statements in these cited papers. For instance in [103] one can find an area form whose Hele-Shaw flow develops self-tangency along any given finite collection of smooth points and non-selfintersecting curve segments. Thus it is

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possible to find Dirichlet data for an HMAE that is not twice differentiable at such a prescribed set of points. And in [102] it is shown that the phenomena of having (smooth) Dirichlet data for the HMAE for which there is an open set not meeting any harmonic disc can be made to persist under small deformations of the data. Acknowledgements We wish to thank Valentino Tosatti for conversations relating to this survey, as well as the referee for helpful comments and references. References [1] C. Arezzo and G. Tian, Infinite geodesic rays in the space of K¨ ahler potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 617–630. MR2040638 [2] E. Bedford and J.-P. Demailly, Two counterexamples concerning the pluri-complex Green function in Cn , Indiana Univ. Math. J. 37 (1988), no. 4, 865–867. MR982833 [3] E. Bedford and M. Kalka, Foliations and complex Monge-Amp` ere equations, Comm. Pure Appl. Math. 30 (1977), no. 5, 543–571. MR0481107 [4] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Amp` ere equation, Invent. Math. 37 (1976), no. 1, 1–44. MR0445006 [5] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR674165 [6] R. J. Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math. 131 (2009), no. 5, 1485–1524. MR2559862 [7] R. J. Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math. 131 (2009), no. 5, 1485–1524. MR2559862 [8] R. Berman On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold arXiv:1405.6482. [9] R. J. Berman, K-polystability of Q-Fano varieties admitting K¨ ahler-Einstein metrics, Invent. Math. 203 (2016), no. 3, 973–1025. MR3461370 [10] R. J. Berman and B. Berndtsson, Convexity of the K-energy on the space of K¨ ahler metrics and uniqueness of extremal metrics, J. Amer. Math. Soc. 30 (2017), no. 4, 1165–1196. MR3671939 [11] R. J. Berman, S. Boucksom, V. Guedj, and A. Zeriahi, A variational approach to com´ plex Monge-Amp` ere equations, Publ. Math. Inst. Hautes Etudes Sci. 117 (2013), 179–245. MR3090260 [12] R. Berman and J.-P. Demailly, Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology, Progr. Math., vol. 296, Birkh¨ auser/Springer, New York, 2012, pp. 39–66. MR2884031 [13] B. Berndtsson, Convexity on the space of K¨ ahler metrics (English, with English and French summaries), Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 713–746. MR3137249 [14] B. Berndtsson, D. Cordero-Erausquin, B. Klartag, and Y. Rubinstein, Complex Legendre duality, arXiv:1608.05541. [15] Z. Blocki, The C 1,1 regularity of the pluricomplex Green function, Michigan Math. J. 47 (2000), no. 2, 211–215. MR1793621 [16] Z. Blocki, Regularity of the pluricomplex Green function with several poles, Indiana Univ. Math. J. 50 (2001), no. 1, 335–351. MR1857039 [17] Z. Blocki, The Bergman metric and the pluricomplex Green function, Trans. Amer. Math. Soc. 357 (2005), no. 7, 2613–2625. MR2139520 [18] Z. Blocki, On geodesics in the space of K¨ ahler metrics, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 3–19. MR3077245 [19] H. J. Bremermann, On a generalized Dirichlet problem for plurisubharmonic functions and ˇ pseudo-convex domains. Characterization of Silov boundaries, Trans. Amer. Math. Soc. 91 (1959), 246–276. MR0136766 [20] L. A. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Analyse Math. 37 (1980), 285–295. MR583641

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[123] A. Zeriahi, A viscosity approach to degenerate complex Monge-Amp` ere equations (English, with English and French summaries), Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 843–913. MR3137252 DPMMS, University of Cambridge, United Kingdom Email address: [email protected] Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden Email address: [email protected] Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01745

K¨ ahler-Einstein metrics G´ abor Sz´ekelyhidi Dedicated to Sir Simon Donaldson on the occasion of his 60th birthday Abstract. We survey the theory of K¨ ahler-Einstein metrics, with particular focus on the circle of ideas surrounding the Yau-Tian-Donaldson conjecture for Fano manifolds.

1. Introduction A starting point in the study of K¨ahler-Einstein metrics is K¨ ahler’s observation [104], that for Hermitian metrics satisfying what is now known as the K¨ahler condition, the Einstein equations reduce to a scalar complex Monge-Amp`ere equation. Over the many decades since, the field has grown into a very rich subject with deep connections to nonlinear PDE, geometric analysis, complex algebraic geometry, string theory, and others. The goal of this survey is to give an overview of some of these developments and in particular to showcase the diverse ideas that have been brought to bear on the problem. Let us start with K¨ ahler’s observation, and consider a Hermitian metric gj k¯ on a complex manifold M . The associated (1,1)-form, or K¨ahler form, is defined to be √ zk ω = −1gj k¯ dz j ∧ d¯ in local coordinates, and the metric g is K¨ahler if dω = 0. K¨ahler showed that in this case we can locally write the metric g in terms of a potential function φ: ∂2φ . ∂z j ∂ z¯k The Ricci curvature of g is then given by gj k¯ =

∂2 log det(g), ∂z j ∂ z¯k and so we can obtain solutions of the Einstein equation Ric = λg, by solving the scalar equation   2 ∂ φ = e−λφ . det ∂z j ∂ z¯k Under certain conditions K¨ahler potentials exist globally, not just locally. Let us suppose that M is compact. A K¨ ahler form ω on M defines a cohomology class [ω] ∈ H 2 (M ), and it is natural to consider, as Calabi [28] did, the space of all Ricj k¯ = −

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¯ K¨ ahler forms on M in a fixed cohomology class. The ∂ ∂-lemma states that any other K¨ ahler form η ∈ [ω] can be written as √ ¯ η = ω + −1∂ ∂φ for a function φ : M → R, and so the space of K¨ahler metrics in a fixed cohomology class are parametrized by scalar functions, in analogy with a conformal class in Riemannian geometry. A further important observation is that for any K¨ ahler metric g on M , its Ricci form √ zk Ric(g) = −1Rj k¯ dz j ∧ d¯ is a closed form in the first Chern class c1 (M ). Calabi [28] conjectured that conversely any representative of c1 (M ) is the Ricci form of a unique K¨ ahler metric in every K¨ahler class. This fundamental conjecture was proven by Yau [177], by solving the complex Monge-Amp`ere equation √ ¯ n = eF +c ω n (ω + −1∂ ∂φ) for φ and a constant c, given a K¨ ahler form ω and function F . Perhaps the most important case, which has had an enormous impact, is when c1 (M ) = 0. In this case Yau’s result implies that every K¨ahler class on M admits a unique Ricci flat metric. More generally, if we seek a K¨ ahler-Einstein metric ω satisfying Ric(ω) = λω, then we must have c1 (M ) = λ[ω]. In particular if λ = 0, then either c1 (M ) or −c1 (M ) must be a K¨ ahler class, and the cohomology class [ω] is uniquely determined. When c1 (M ) is negative, then the works of Yau [177] and Aubin [9] yield a K¨ahler-Einstein metric on M . It was already known by Matsushima [126], however, that when c1 (M ) is positive, i.e. M is Fano, then M can only admit a K¨ ahler-Einstein metric if its holomorphic automorphism group is reductive. Later Futaki [93] found a different obstruction stemming from the automorphism group, showing that a certain numerical invariant F (v) must vanish for all holomorphic vector fields v on M . These obstructions rule out the existence of a K¨ahler-Einstein metric on the blowup Blp P2 for instance. On the other hand, Tian [165] showed that in the case of Fano surfaces the reductivity of the automorphism group, or alternatively the vanishing of Futaki’s obstruction, is actually sufficient for the existence of a K¨ahler-Einstein metric. At this point let us digress briefly on parallel developments in the theory of holomorphic vector bundles. In algebraic geometry a basic problem is to construct moduli spaces of various objects, for instance vector bundles over a curve. It turns out that in general it is not possible to parametrize all vector bundles of a fixed topological type with a nice space, but rather we need to restrict ourselves to semistable bundles – a notion introduced by Mumford [127]. While this is a purely algebro-geometric notion, it was shown by Narasimhan-Seshadri [129], and later reproved by Donaldson [69], that stability has a differential geometric meaning: an indecomposable vector bundle over a curve is stable if and only if it admits a Hermitian metric with constant curvature. The Hitchin-Kobayashi correspondence, proved by Donaldson [70, 71] and Uhlenbeck-Yau [173], is the higher dimensional generalization of this, stating that an indecomposable vector bundle is stable if and only if it admits a Hermitian-Einstein metric. There is a particularly rich interplay between this result for complex surfaces and Donaldson theory [85] for smooth four-manifolds.

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In analogy with these results on vector bundles, Yau [178] conjectured that the existence of a K¨ahler-Einstein metric on a Fano manifold M should be related to the stability of M in a suitable sense. This conjecture was made precise by Tian [166], who generalized Futaki’s obstruction [93] to the notion of K-stability: Tian showed (see also Ding-Tian [65]) that given any C∗ -equivariant family π : X → C with generic fiber Xt ∼ = M , and Q-Fano central fiber X0 , the Futaki invariant F (X) of the induced vector field on the central fiber can be defined. The Q-Fano condition here means that X0 is a normal variety with Q-Cartier anticanonical divisor, and this assumption allows for a differential-geometric definition of F (X). Tian showed moreover that if M admits a K¨ ahler-Einstein metric, then F (X) ≥ 0 for all such families, with equality only if X is a product. This obstruction is called K-stability, and it is a far reaching generalization of Futaki’s obstruction. Indeed the latter can be viewed as a special case using only product families. The Donaldson-Uhlenbeck-Yau theorem and Yau and Tian’s conjectures on the existence of K¨ahler-Einstein metrics can be seen as two instances of a relationship between quotient constructions in symplectic and algebraic geometry, due to Kempf-Ness [106]. This is because in both settings the geometric structure we seek, a Hermitian-Einstein metric or a K¨ahler-Einstein metric, can be viewed as a zero of a moment map. This was discovered by Atiyah-Bott [8] for vector bundles over curves, and independently by Fujiki [90] and Donaldson [72] for K¨ ahler-Einstein metrics. In fact even more generally, constant scalar curvature K¨ ahler metrics, and the extremal K¨ ahler metrics introduced by Calabi [29] fit into this framework. Motivated by this, Donaldson [77] introduced a generalization of K-stability for any pair (M, L) of a projective manifold M equipped with an ample line bundle L. The definition is similar to Tian’s notion, in that we need to consider C∗ equivariant degenerations π : X → C of M , compatible with the polarization L of M . The central fiber, however, is allowed to be a singular scheme, and the corresponding numerical invariant, the Donaldson-Futaki invariant DF (X), is defined purely algebraically. In this generality we have Conjecture 1 (Yau-Tian-Donaldson). The manifold M admits a constant scalar curvature K¨ ahler metric in c1 (L), if and only if the pair (M, L) is K-stable. The conjecture can be extended [158] to characterize the existence of extremal metrics, and there are also variants for more general “twisted” equations by Dervan [62]. One direction of the conjecture is fairly well understood, namely that the existence of a constant scalar curvature metric implies K-stability (see e.g. Tian [166], Donaldson [75, 78] and Stoppa [155], Berman-Darvas-Lu [19]), however the converse in general is wide open at present. The main subject of this survey is the case when M is a Fano manifold and L = −KM , since then a constant scalar curvature metric in c1 (L) is actually K¨ ahler-Einstein. In this case Chen-Donaldson-Sun [41–44] proved the following breakthrough result. Theorem 2. A Fano manifold M admits a K¨ ahler-Einstein metric if and only if (M, −KM ) is K-stable. Our aim in this survey is not so much to describe the proof of this result, but rather to highlight the diversity of ideas that are in some way related to the Yau-Tian-Donaldson conjecture. There are several other excellent surveys on the

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subject, such as those of Thomas [162], Phong-Sturm [139] and Eyssidieux [86], with a focus on different aspects of the theory. The solution of the conjecture in the Fano case has certainly closed a chapter, but it has also set the scene for a great deal of further development, much of which is likely yet to come. Acknowledgements. It is my pleasure to thank Simon Donaldson for his advice and support over the years – it would be hard to overstate the influence that his ideas and approach to mathematics have had on my interests. I also thank Julius Ross and Valentino Tosatti for helpful comments on this survey. This work was supported in part by NSF grant DMS-1350696. 2. The moment map picture In this section we describe how the scalar curvature of a K¨ahler metric can be viewed as an infinite dimensional moment map, following Donaldson [72]. This point of view is invaluable in building intuition for the problem, and in retrospect it motivates many of the basic constructions and results that were known beforehand. We will keep the discussion at a formal level, and not delve into the precise definitions relating to infinite dimensional manifolds. Let (X, ω) denote a compact symplectic manifold, such that H 1 (X) = 0 for simplicity. Let J be the space of almost complex structures on X, compatible with ω. The space J has a natural complex structure, and each tangent space TJ J is equipped with the L2 -inner product given by the metric gJ (·, ·) = ω(·, J·). This structure turns J into an infinite dimensional K¨ ahler manifold, and the group G = Ham(X, ω) of Hamiltonian symplectomorphisms acts on J , preserving this K¨ ahler structure. We identify the Lie algebra of G with the functions C0∞ (X) with zero mean on X with respect to the volume form ω n , through the Hamiltonian construction. We further identify C0∞ (X) with its dual using the L2 inner product. The key calculation is the following. Proposition 3 (Fujiki [90], Donaldson [72]). A moment map for the action of G on J is given by μ : J → C0∞ (X) J → S − SJ , where SJ is the scalar curvature of the metric gJ whenever J is integrable, and S is its average, which is independent of J. In particular an integrable complex structure J satisfies μ(J) = 0 if and only if the K¨ahler metric gJ on X has constant scalar curvature. The precise meaning of this result is an identity relating the linearization of the scalar curvature SJ under varying the complex structure J, and the infinitesimal action of Hamiltonian symplectomorphisms on J . Indeed, let h ∈ C0∞ (M ), and let A ∈ TJ J be an infinitesimal variation of J. The variation of J by the Hamiltonian vector field vh is the Lie derivative Lvh J, and we write DSJ (A) for the variation of the scalar curvature SJ in the direction A. The content of Proposition 3 is the identity $DSJ (A), h%L2 = $JA, Lvh J%L2 , which can be checked by direct calculation. See [72, 90, 94, 167] for the details.

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Suppose for the moment that instead of the infinite dimensional group G acting on J , we had a compact group G acting on a compact K¨ahler manifold (V, ω), with moment map μ. Let us assume that ω is the curvature form of a line bundle L → V endowed with a Hermitian metric, and so V is in fact a projective manifold. Let Gc denote the complexification of G, acting on V by biholomorphisms. The Kempf-Ness theorem [106] says, in this finite dimensional situation, that a Gc -orbit contains a zero of the moment map if and only if it is polystable. One way to define polystability, that is useful in the infinite dimensional setting as well, is that an orbit Gc · p is polystable if a suitable real valued G-invariant function is proper on the orbit. More precisely we consider the function f : Gc /G → R

(2.1)

[g] → log g · pˆ, defined using a G-invariant norm on V . Here pˆ ∈ L is a non-zero lift of p, and we need a lift of the Gc -action to the total space of L in a way compatible with the choice of moment map μ. The compatibility of the lift of the action with the moment map μ can be expressed by the formula √ (2.2) dfg·p ( −1ξ) = $μ(g · p), ξ% for the variation of f , where g ∈ Gc , and ξ ∈ g. Note in particular that the critical points of f are precisely zeros of the moment map. Since the function f also turns out to be convex along geodesics in the symmetric space Gc /G, it is clear that properness of f corresponds to the existence of a zero of the moment map in the orbit Gc · p, at least if we ignore subtleties related to the possible stabilizer of p. What is less clear, however, is that to verify whether f is proper on Gc /G, it is enough to check whether f is proper along each . In fact geodesic ray in Gc /G obtained from one-parameter subgroups C∗ ⊂ Gc√ it is enough to consider only one-parameter subgroups of the form t → e −1tξ for ξ ∈ g generating a circle subgroup. For such a one-parameter subgroup we can test the properness of f by computing the limit √ −1tξ

lim f  (e

(2.3)

t→∞

· p) = $μ(q), ξ%,

√ −1tξ

where q = limt→∞ e · p. Properness of f is then equivalent to $μ(q), ξ% > 0 whenever q ∈ Gc · p. This is in essence the Hilbert-Mumford numerical criterion for stability, proved by Mumford [127], to which we refer the reader for the detailed development of this theory. Let us return to the infinite dimensional setting of the action of G on J . A first issue is that the complexification G c does not exist, but we can still try to interpret what its orbits would be if it did. Indeed in each tangent space TJ J we have a subspace spanned by elements of the form Lvh J giving the infinitesimal action of Hamiltonian vector fields, and we can simply complexify this subspace. The orbits of G c then ought to be integral submanifolds of this distribution on J . Note that ultimately we are interested in the metrics gJ determined by the pairs (ω, J), and for any diffeomorphism f the metric given by (ω, f ∗ J) is isometric to that given by ((f −1 )∗ ω, J). We can therefore switch our point of view from studying different complex structures on a symplectic manifold (X, ω) to studying different K¨ahler forms on a complex manifold (X, J), as is more standard in K¨ ahler geometry. To see what this corresponds to in terms of the complexified orbits of

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G, note that when J is integrable, √ then JLvh J = LJvh J, and at the same time we ¯ we can interpret this have the formula −LJvh ω = 2 −1∂∂h. Using the ∂ ∂-lemma as saying that in our infinite dimensional setting the role of the symmetric space ahler metrics in the K¨ ahler class [ω] (see [73]). In Gc /G is played by the space of K¨ conclusion the Kempf-Ness theorem suggests that the existence of a constant scalar curvature metric in the K¨ahler class [ω] is equivalent to stability of this class in a suitable sense. Let us see how this formal discussion motivates several of the basic constructions in the field, which were actually discovered before the moment map picture was understood: 2.1. The Mabuchi metric. In finite dimensions the metric on the symmetric space Gc /G is given by an inner product on the Lie algebra g. In the infinite dimensional setting we chose the L2 product on Hamiltonian functions, which by the above discussion correspond to variations in the K¨ahler potential. This leads to a very natural Riemannian structure on the space of K¨ahler metrics first introduced by Mabuchi [124] and later rediscovered by Semmes [147] and Donaldson [73]. For a compact K¨ ahler manifold (M, ω), Let us denote by √ H = {φ ∈ C ∞ (M ) : ω + −1∂∂φ > 0} √ the space of K¨ ahler potentials. For φ ∈ H let ωφ = ω + −1∂∂φ be the corresponding K¨ahler metric. When ω ∈ c1 (L) for an ample line bundle L, then H can also be thought of as the space of positively curved Hermitian metrics e−φ on L. Each tangent space Tφ H can be identified with C ∞ (M ), and the Mabuchi metric is defined by simply taking the L2 inner product: " f g ωφn . $f, g%φ = M

One can show that this metric turns H, at least formally, into a non-positively curved symmetric space. Of great interest is the study of geodesics in H. A calculation shows that a path φt ∈ H is a geodesic, if it satisfies the equation 1 φ¨t − |∇φ˙ t |2ωφt = 0. 2 An important observation due to Semmes and Donaldson, however, is that this geodesic equation is equivalent to a homogeneous complex Monge-Amp`ere equation. Indeed, let Aa,b = S 1 × (a, b) be a cylinder, and given a path φt ∈ H for a < t < b, define the form √ Ω = π ∗ ω + −1∂∂φt √ on the product M × Aa,b . Here π : M × Aa,b → M is the projection, and −1∂∂ involves the variables on Aa,b as well. A calculation shows that φt is a geodesic if and only if Ω is non-negative and Ωn+1 = 0, i.e. φt solves the homogeneous complex Monge-Amp`ere equation on M ×Aa,b . When a, b are finite we have geodesic segments, while if a or b is infinite, then we have geodesic rays. Since the equation Ωn+1 = 0 is degenerate elliptic, the regularity theory is very subtle. Chen [47] showed that any two potentials φ0 , φ1 ∈ H can be connected by a unique weak geodesic φt , for which Δφt is bounded, using the Laplacian on M ×Aa,b (see also Blocki [23]). This was improved to a bound on |∇2 φt | by Chu-TosattiWeinkove [50] (see also Berman [15] for a weaker result in the projective case). It

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turns out that these results are essentially optimal, since there are counterexamples to the existence of smooth geodesics, by Lempert-Vivas [110] and Darvas [53] (see also Donaldson [76] and Ross-Witt Nystr¨ om [144]). However even weak geodesics are enough for many applications, as we will describe below. 2.2. The K-energy. In the finite dimensional setting we described how the existence of a zero of the moment map is related to properness of the log-norm functional f in (2.1). In infinite dimensions this translates to the K-energy, defined by Mabuchi [123]. The formula (2.2) for the variation of the log-norm functional suggests that the K-energy K : H → R can be defined through its variation. If φt ∈ H is a path, then d K(φt ) = $S − S(ωφt ), φ˙ t %L2 (ωφt ) dt " = φ˙ t (S − S(ωφ )) ω n . t

M

φt

Mabuchi [124] showed that the K-energy is convex along smooth geodesics in H, which we now see as a general result about the log-norm functionals. It is clear from the definition that the critical points are constant scalar curvature metrics, and Mabuchi also showed that if two critical points φ0 , φ1 are connected by a smooth geodesic, then the metrics ωφ0 and ωφ1 are isometric by an automorphism of M . It was only much later that Berman-Berndtsson [17] showed that convexity holds along weak geodesics, and as an application proved a general uniqueness result along these lines. Note that uniqueness in various degrees of generality has been proven previously using other methods, see for example [13, 22, 47, 75]. In the finite dimensional setting the existence of a critical point of the log-norm functional is equivalent to its properness. Tian [166] showed that the analogous result holds for K¨ ahler-Einstein metrics, characterizing their existence in terms of properness of the K-energy in a suitable sense. See also Darvas-Rubinstein [55] for a more precise properness statement in the K¨ahler-Einstein case, in the presence of automorphisms. In the general constant scalar curvature case Berman-DarvasLu [19] showed one direction of this correspondence, namely that the existence of a cscK metric implies properness of the K-energy, as was conjectured by Tian [167]. 2.3. The Futaki invariant. A construction that predates both of the previous ones is Futaki’s obstruction [93] to the existence of a K¨ahler-Einstein metric on a Fano manifold M , analogous to the Kazdan-Warner obstruction [105] for the prescribed curvature problem on the 2-sphere. In retrospect, Futaki’s obstruction can be viewed as the first glimpse of the obstruction to K¨ ahler-Einstein metrics given by K-stability. Recall that in the finite dimensional picture, polystability of p is related to the limit (2.3) of the √ derivative of the log-norm functional along the orbit e −1tξ · pˆ of a one-parameter subgroup. The simplest example is if ξ ∈ g is in the stabilizer of p, so that the one-parameter subgroup simply acts on the line Lp . The quantity $μ(p), ξ% is then the weight of this action, and polystability requires that this weight vanishes, since otherwise the log-norm functional would not be bounded from below. The infinite dimensional analog of this weight can be defined as follows. An element ξ ∈ g in the stabilizer of a point p corresponds to a function h on M , whose Hamiltonian vector field vh preserves the complex structure of M as well, i.e. vh is

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a holomorphic Killing field on (M, ω), and vh generates a circle action on M . The corresponding weight is then " h(S − S(ω)) ω n . (2.4) F (vh ) = M

Futaki used a different, but essentially equivalent definition, and showed that F (vh ) only depends on the vector field vh , and not on the metric in the class [ω] used in the formula. In addition the invariant can be defined for any holomorphic vector field, not just those that generate circle actions. If M admits a constant scalar curvature metric in the class [ω], then it is clear from the definition that F (v) = 0 for all holomorphic vector fields v. Tian’s definition [65, 166] of K-stability is motivated by probing the properness of the K-energy along more general families of metrics φt ∈ H. From the finite dimensional picture it is most natural to consider geodesic rays, however this is technically rather difficult. Instead Tian used families of metrics on M obtained from embedding M ⊂ PN into a projective space, and then considering the restrictions of the Fubini-Study metrics σt∗ ωF S pulled back under a one-parameter family of automorphisms σt of PN . We will discuss this construction and others in more detail in Section 3. 2.4. The Ding functional. The constructions in the previous subsections apply to the general existence problem for constant scalar curvature metrics, not just K¨ahler-Einstein metrics. At the same time we will see that the K¨ahler-Einstein problem has several special features. One of these is an alternative variational description of K¨ ahler-Einstein metrics as critical points of the Ding functional D defined in [64]. Thinking of H as the space of positively curved metrics e−φ on −KM , the variation of D along a path φt is defined to be 5 " φ˙ e−φt d 1 n ˙ 5 t D(φt ) = − , φt ωφt + M dt V M e−φt M where V is the volume with respect to ω n , and we can naturally think of e−φt as defining volume forms on M . The critical points of this functional satisfy e−φ = Cωφn , so they are K¨ ahler-Einstein metrics. The Ding functional has many analogous properties to the K-energy, such as the convexity along weak geodesics proved by Berndtsson [22], but it has technical advantages over the K-energy, since defining it requires less regularity of φ. Recently Donaldson [68] gave a variation of the infinite dimensional moment map picture discussed above, in which Berndtsson’s convexity result [21] gives rise to the K¨ ahler structure on J , and the Ding functional corresponds to the log-norm functional. The weight again recovers the Futaki invariant, and the existence of a K¨ahlerEinstein metric is related to properness of D by Tian [166]. 3. K-stability In this section we survey the concept of K-stability of a Fano manifold M , or more generally a projective manifold M with an ample line bundle L, from different points of view. We first discuss the original notion for Fano manifolds, due to Tian [166], which is fairly differential geometric. A much more algebrogeometric definition for general pairs (M, L) was given by Donaldson [77]. In both of these definitions one needs to consider C∗ -equivariant degenerations of M , and

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the main difference is that in Tian’s definition the central fiber is required to be a Q-Fano variety, whereas it can be an arbitrary scheme in Donaldson’s definition. This added flexibility is needed when dealing with general polarized manifolds, but Li-Xu [118] showed that in the Fano case the two notions of K-stability are equivalent. We will now consider these two notions in more detail, along with a more analytic approach through geodesic rays in H. 3.1. Tian’s definition. The first notion of K-stability was introduced by Tian [166], in the context of Fano manifolds. As we have discussed in the previous section, Tian showed that if a Fano manifold M admits a K¨ ahler-Einstein metric, then the K-energy on the K¨ahler class c1 (M ) is proper, and K-stability can be thought of as probing this properness along certain families of metrics. Suppose that we have a C∗ -equivariant family of varieties π : X → C, with generic fiber π −1 (t) ∼ = M for t = 0. Assume in addition that the central fiber is normal, and that a power of the relative anticanonical line bundle on the regular locus extends to a relatively ample line bundle on X. In this situation we call the family X a special degeneration of M . The C∗ -action on such a special degeneration induces a C∗ -action on the central fiber X0 = π −1 (0). Using that X0 has relatively mild singularities, Tian (see also Ding-Tian [65]) showed that one can define the Futaki invariant of this C∗ -action on X0 using a differential geometric formula similar to (2.4). This is then defined to be the Futaki invariant F (X) of the special degeneration X. Note that any C∗ -action on the Fano manifold M gives rise to a product action on the trivial family X = M × C, and the Futaki invariant of this family is simply the Futaki invariant of the original C∗ -action. At the same time there are infinitely many special degenerations, even if M admits no C∗ -actions. In order to relate special degenerations to families of metrics and properness of the K-energy, note that any special degeneration π : X → C for M can be realized as a family in projective space. More precisely there is an embedding X ⊂ PN × C, such that the C∗ -action on X is induced by the action of a one-parameter subgroup σ : C∗ → SL(N + 1) on PN . Here M is embedded in PN × {1} using a basis of sections of −rKM for a suitable r > 0. We can now define a family of metrics ωt ∈ c1 (M ) by restricting the Fubini-Study metric to the non-zero fibers of X. Equivalently we can write 1 (3.1) ωt = σe∗−t ωF S |M . r Ding-Tian [65] showed that with suitable normalizing factors which we omit, d K(ωt ) = F (X), dt as suggested by (2.3) in the finite dimensional picture. Note, however, that the family ωt is usually not a geodesic ray. With these results in mind we have the following definition, due to Tian [166].

(3.2)

lim

t→∞

Definition 4. A Fano manifold M is K-stable, if F (X) ≥ 0 for all special degenerations X of M , with equality only for product degenerations. In the same paper, Tian showed that if M admits a K¨ ahler-Einstein metric then the K-energy is proper, and as a consequence he showed the following fundamental result (see also Berman [16] for the case when M admits holomorphic vector fields).

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Theorem 5. If a Fano manifold admits a K¨ ahler-Einstein metric, then it is K-stable. 3.2. The Donaldson-Futaki invariant. The central fiber of a special degeneration has fairly mild singularities, and so a differential geometric definition of the Futaki invariant was possible. On the other hand Donaldson [77] assigned an invariant, called the Donaldson-Futaki invariant, to essentially arbitrary C∗ -equivariant degenerations of a polarized manifold (M, L), by giving a purely algebro-geometric definition of the Futaki invariant for a C∗ -action on any polarized scheme (V, L). To give the definition, note that a C∗ -action λ on (V, L) induces actions on the spaces of sections H 0 (V, kL), and so in particular for each k we have a total weight wk . For large k we have expansions (3.3)

dim H 0 (V, kL) = a0 kn + a1 kn−1 + . . . wk = b0 kn+1 + b1 kn + . . . ,

and the Donaldson-Futaki invariant of λ is defined to be a1 DF (λ) = b0 − b1 . a0 When V is smooth, the equivariant Riemann-Roch formula can be used to show that this coincides with Futaki’s differential geometric definition. Such polarized schemes with C∗ -actions arise naturally as the central fibers of test-configurations. Definition 6. Let (M, L) be a polarized manifold. A test-configuration for (M, L) with exponent r is a C∗ -equivariant flat family π : (X, L) → C, such that L is relatively ample, and (π −1 (t), L|π−1 (t) ) ∼ = (M, rL), for any t = 0. In addition it is natural to require that the total space X is normal (see Li-Xu [118] and Ross-Thomas [143]). The Donaldson-Futaki invariant DF (X, L) of the test-configuration is defined to be the Donaldson-Futaki invariant of the induced C∗ -action on the central fiber. Given this definition, K-stability can be defined as follows, in analogy with Definition 4. Definition 7. A polarized manifold (M, L) is K-stable, if DF (X, L) ≥ 0 for all test-configurations for (M, L), with equality only if X ∼ = M × C. Using this definition, the following result was shown by Stoppa [155], building on work by Donaldson [78] and Arezzo-Pacard [6]. Theorem 8. Suppose that M admits a constant scalar curvature metric in c1 (L), and it has no nonzero holomorphic vector fields. Then (M, L) is K-stable. The result can be extended to the case when M has holomorphic vector fields, and also to extremal metrics (see [156], [19]). As we stated in the introduction, the Yau-Tian-Donaldson conjecture is the converse of this result, saying that if (M, L) is K-stable, then there is a constant scalar curvature metric in c1 (L). However it is likely that actually a stronger notion of stability is needed in general, in view of examples of Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman [5], that are shown to be unstable in a suitable sense by Dervan [60]. One possibility for such

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a stronger stability notion is provided by the formalism of filtrations [160, 176], while another is the concept of uniform K-stability [25, 62]. In order to compare Definitions 4 and 7, let us point out that it is fairly easy to construct interesting test-configurations, which are not special degenerations, using for instance deformation to the normal cone. This was explored in detail by Ross-Thomas [142, 143]. One can simply take any subscheme Z ⊂ M , and let X = BlZ×{0} M × C, with a suitable relatively ample line bundle L. For instance when Z is a smooth submanifold of M , then the central fiber of X will be isomorphic to BlZ M ∪ PZ , where PZ = P(NZ ⊕ C) is the projective completion of the normal bundle of Z in M , and PZ is glued along its zero section to the blowup BlZ M along its exceptional divisor. In fact Odaka [130] showed that by blowing up “flag ideals” of M × C instead of just subschemes, one can essentially recover all testconfigurations, and using this approach Odaka-Sano [133] and Dervan [61] were able to prove the K-stability of certain varieties. With this in mind, it appears that in the Fano case Definition 7 is more restrictive than Tian’s Definition 4, since test-configurations are much more general than special degenerations. It is quite remarkable then that for Fano manifolds the two notions turn out to be equivalent. This was first proven by Li-Xu [118] purely algebro-geometrically, using the minimal model program. Roughly speaking the minimal model program allowed them to modify an arbitrary test-configuration into a special degeneration, while controlling the sign of the Donaldson-Futaki invariant at each step. A more differential geometric proof also follows from ChenDonaldson-Sun’s proof [41] of the YTD-conjecture for Fano manifolds. One suggestive example is to consider a polarized toric manifold (M, L), with Delzant polytope P . It is natural in this case to only allow torus equivariant testconfigurations. The only torus equivariant test-configurations with normal central fiber are product configurations induced by a C∗ -action on M and indeed, when M is Fano, then Wang-Zhu [175] showed that the only obstruction to the existence of a K¨ ahler-Einstein metric is that given by the Futaki invariants of these C∗ actions. On the other hand, as shown by Donaldson [77], any rational piecewise linear convex function on P gives rise to a test-configuration for (M, L) and there are (non-Fano) examples where these give an obstruction to the existence of a cscK metric, not detected by product configurations. 3.3. Intersection theoretic formula. An alternative formula for the Donaldson-Futaki invariant in terms of intersection products has been very useful in more algebro-geometric developments. It was observed by Wang [174], and it is also related to the CM-polarization of Tian [136, 166]. To explain it, note that any test-configuration (X, L) can be extended trivially at infinity to obtain a C∗ -equivariant family (X, L) → P1 . The line bundle L is relatively ample, and by taking the tensor product with a line bundle pulled back from P1 we can assume that it is actually ample. A calculation shows that in terms of this family the Donaldson-Futaki invariant of a test-configuration of exponent r is (3.4)

DF (X, L) =

n n μ(M, rL) (L)n+1 + L .KX/P1 , n+1

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using the intersection product on X. Here for a polarized variety (M, L), the “slope” μ(M, L) is defined by μ(M, L) =

−KM .Ln−1 , Ln

and up to a constant multiple is it the average scalar curvature S of a K¨ ahler metric in c1 (L). This reformulation of the Donaldson-Futaki invariant has various advantages, as shown for instance in the works of Li-Xu [118], Odaka [131] and others. Here we just mention one, namely the extension of K-stability to non-algebraic K¨ahler manifolds due to Dervan-Ross [63] and Sj¨ostr¨ om Dyrefelt [150]. While the expansions (3.3) do not make sense in the absence of a line bundle, one can make sense of the intersection product (3.4) even in the K¨ ahler case. 3.4. Geodesic rays. In Section 2 we described how in the finite dimensional moment map picture, stability can be tested using geodesic rays in the symmetric space Gc /G. Donaldson [73] formulated conjectures saying that in an analogous way geodesic rays in H can be used to detect the existence of a constant scalar curvature metric. Since that time there has been enormous progress on our understanding of geodesic rays, although these conjectures are still mostly open except in the Fano case. We have seen in (3.1) that a special degeneration or a test-configuration for M gives rise to a path in the space H of K¨ahler potentials. Unless we have a product test-configuration, this path cannot be expected to be a geodesic ray in H, but rather it is a geodesic in a finite dimensional space of Bergman metrics, i.e. those obtained by restricting the Fubini-Study metric. It turns out that the relation (3.2) between the Futaki invariant of a test-configuration and the asymptotic derivative of the K-energy along the corresponding Bergman geodesic does not hold for general test-configurations. The general formula for the limit has been obtained by Paul [135] in terms of hyperdiscriminant and Chow polytopes, leading to an alternative notion of stability. To relate this to geodesic rays in H, note that a given test-configuration X for M can be realized as a family in projective spaces of arbitrarily large dimension, and in this way we obtain not one, but a whole sequence of Bergman geodesics of metrics on M from X using the formula (3.1). Phong-Sturm [140] showed that one can pass to a limit, and obtain a geodesic ray in H in a suitable weak sense, with an arbitrary initial point φ0 . One can also directly construct such a weak geodesic ray in H from the testconfiguration X in the following way (see [7], [48], [141], [16] for this in various degrees of generality). Let us denote by XΔ the family X restricted to the unit disk Δ ⊂ C. We use the “initial point” φ0 to define a metric e−φ0 on the line bundle L over ∂XΔ . The geodesic ray is then obtained by finding an S 1 -invariant metric current, and solving the homogeneous e−φ on L over XΔ , with positive curvature √ complex Monge-Amp`ere equation ( −1∂∂φ)n+1 = 0 on the interior of XΔ , in the sense of pluripotential theory. The existence of such a solution, and its regularity properties are discussed by Phong-Sturm [141]. Over the punctured disk Δ∗ the family X is biholomorphic to M × Δ∗ , and so the metric e−φ on L induces a family of metrics on L → M , which is the geodesic ray we were after.

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There has been a lot of work relating the behavior of the K-energy along such a geodesic ray to the Donaldson-Futaki invariant of the test-configuration (see e.g. [48], [141]). The first sharp result in this direction is due to Berman [16], using the Ding functional instead of the K-energy, in the case when M is Fano. He shows that along a geodesic ray φt constructed from a test-configuration (X, L) as above, we have (3.5)

DF (X, L) = lim

t→∞

d D(φt ) + q, dt

where q ≥ 0 is a rational number determined by the central fiber of the testconfiguration. An analogous formula for the K-energy, for test-configurations with smooth total space, was subsequently obtained by Sj¨ ostr¨ om Dyrefelt [150]. The relation (3.5) led Fujita [92] to study the notion of Ding stability, where the Donaldson-Futaki invariant is replaced by the asymptotic derivative of the Ding functional. As an important application, he showed that projective space has the maximal volume amongst K¨ahler-Einstein manifolds of a fixed dimension, using that K¨ahler-Einstein manifolds are stable in this sense. This direction has been taken much further in recent work of Fujita [91], Li [111], Li-Xu [116, 117], LiuXu [119], leading to new examples of K-stable varieties, as we mention in Section 5.1. A different direction was pursued by Boucksom-Hisamoto-Jonsson [25], who introduced the point of view of thinking of a test-configuration for (M, L) as a non-Archimedean metric on L. From this point of view the asymptotic derivative of the Ding functional can be seen as a non-Archimedean Ding functional. Building on these ideas, and using techniques from non-Archimedean analysis [24], as well as advances on the geometry of the space of metrics H due to Darvas [54], Berman-Boucksom-Jonsson [18] proposed a variational approach to proving Theorem 2. This approach relies far less on differential geometric methods than ChenDonaldson-Sun’s approach [41], and as such it may eventually lead to a solution of the Yau-Tian-Donaldson conjecture for singular Fano varieties. At present, however, even in the smooth case it yields a weaker result than [41] since it requires assuming uniform K-stability. 3.5. Finite dimensional approximation. In the previous section we mentioned the idea of realizing a geodesic in H as a limit of finite dimensional Bergman geodesics. This idea of approximating the geometry of H with that of larger and larger spaces of Bergman metrics is a fundamental one, going back at least to work of Tian [164] on the problem of approximating an arbitrary K¨ ahler metric ω ∈ c1 (L) on M with a sequence of metrics ωk =

1 ∗ φ ωF S . k k

Here φk : M → PNk are projective embeddings given by bases of sections of kL, for sufficiently large k. Fixing a Hermitian metric on L with curvature form ω, we have induced L2 inner products on each space of sections H 0 (kL). It is then natural to (k) (k) use an orthonormal basis {s0 , . . . , sNk } of H 0 (kL) to define the embedding φk . A calculation shows that √ 1 ∗ φk ωF S = ω + −1∂∂ log ρω,k , (3.6) k

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where the function ρω,k =

Nk 

(k)

|si |2

i=0

is called the Bergman kernel. Tian gave an asymptotic expansion of this function as k → ∞, which was later refined by Zelditch [179], Lu [120], Catlin [32] and others, showing that S(ω) −1 k + O(k−2 ), as k → ∞. 2 The lower order terms can also be computed, and are all given in terms of covariant derivatives of the curvature tensor of ω (see Lu [120]). Note that normalising constants in the coefficients depend on conventions for defining the L2 product on H 0 (kL). In addition, integrating the expansion over M recovers the HirzebruchRiemann-Roch formula. Using this expansion in (3.6) we see that ωk −ω = O(k−2 ), and so indeed we can approximate an arbitrary metric ω with Bergman metrics. Donaldson [75, 79] took this point of view further, realizing that it is advantageous to approximate the entire moment map package described in Section 2 by a sequence of analogous finite dimensional problems, a process that can be thought of as quantization [74]. A starting point is the work of Zhang [180] and Luo [122], relating the GIT stability of the Chow point of a projective submanifold V ⊂ PN under the action of SL(N +1) to the existence of a special embedding of V . Letting n denote the dimension of V , we define the matrix " √ Zj Z k ωn , M (V )jk = −1 N 2 FS |Z | V i i=0 (3.7)

ρk = 1 +

where the Zi are the homogeneous coordinates on PN . Luo shows that GIT stability is related to the existence of balanced embeddings, for which the matrix M (V ) is a multiple of the identity matrix. In fact M (V ), or rather its projection to su(N + 1) can be viewed as a moment map, and zeroes of this moment map are the balanced embeddings. Given any embedding V ⊂ PN , we can consider the restriction of the FubiniStudy metric on V , and then construct a new embedding using an L2 orthonormal basis of sections of O(1)|V as above. It turns out that V ⊂ PN is a balanced embedding if this new embedding coincides with the old one (up to the action of SU (N + 1)), i.e. if the Bergman kernel is constant. The expansion (3.7) then suggests that for a given pair (M, L), balanced embeddings using a basis of H 0 (kL) for very large k ought to be related to constant scalar curvature metrics in c1 (L). Indeed, Donaldson [75] shows that if a cscK metric exists, and M admits no holomorphic vector fields, then M also admits balanced embeddings using a basis of H 0 (kL) for sufficiently large k. Since such balanced embeddings are unique up to the action of SU (N + 1), this implies in particular that if a cscK metric exists, it is unique. Moreover M satisfies an asymptotic GIT stability condition, which implies K-semistability. By “quantizing” the K-energy as well, Donaldson [79] showed that if a cscK metric exists, and M has no holomorphic vector fields, then the K-energy is bounded below on H, generalizing Bando-Mabuchi’s result [13] in the K¨ ahler-Einstein case. Going further, using such finite dimensional approximations, Donaldson [78] showed that a destabilizing test-configuration leads to a nontrivial lower bound for the

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Calabi functional: for any metric ω ∈ c1 (L), and any test-configuration (X, L) for (M, L) we have −DF (X, L) , S(ω) − SL2 ≥ (X, L)2 where a suitable L2 -type norm on test-configurations is used for normalizing the Donaldson-Futaki invariant. This result shows, in particular, that the existence of a cscK metric implies K-semistability, even when M has holomorphic vector fields. These ideas have since been developed further in many papers. A very small sample of such developments is [45, 81, 88, 89, 112, 125, 145, 146, 148, 151]. 4. Geometric and algebraic limits Many of the ideas that we have discussed so far apply not only to the problem of K¨ ahler-Einstein metrics, but also to more general constant scalar curvature metrics. In this section this will no longer be the case, since it will be crucial that the Einstein equation controls the Ricci curvature of the metric. Our goal is to present some of the ideas involved in Chen-Donaldson-Sun’s proof of Theorem 2 [41], that is the existence part of the Yau-Tian-Donaldson Conjecture in the Fano case. Suppose that we have a Fano manifold M , and we are trying to find a metric ω ∈ c1 (M ) satisfying Ric(ω) = ω. There are several natural “continuity methods” ahler-Einstein metric. One for deforming a given initial metric ω0 ∈ c1 (M ) to a K¨ approach is to choose a reference metric α ∈ c1 (M ), and then try to find a family of metrics ωt for t ∈ [0, 1] satisfying (4.1)

Ric(ωt ) = tωt + (1 − t)α.

Using Yau’s theorem [177] we can solve this equation for t = 0, and using the implicit function theorem Aubin [10] showed that the set of t for which a solution exists is open. The remaining difficulty is then to understand what can happen to a sequence of solutions ωtk as tk → T , for some T ∈ (0, 1]. This approach was used in [56], based on the techniques of Chen-Donaldson-Sun. Chen-Donaldson-Sun [41] used a variant of this, proposed by Donaldson [80], ahler-Einstein metrics on the complement M \ D of a suitable where the ωt are K¨ smooth divisor D ⊂ M , while along D the ωt have singularities modeled on a two-dimensional cone with cone angle 2πt. In analogy with (4.1), the metrics ωt satisfy (4.2)

Ric(ωt ) = tωt + (1 − t)[D],

where [D] is the current of integration along D. Here the Ricci curvature is defined as the curvature current of a metric on −KM induced by the volume form of ωt . The advantage of this continuity path over (4.1) is that on M \ D the metric ωt is Einstein, at the expense of introducing singularities along D. At the same time in algebraic geometry it is natural to consider pairs (M, D), and metrics with cone singularities along D are useful differential geometric counterparts (see e.g. [30]). The relevant linear theory for the implicit function theorem was developed by Donaldson [82], with further refinements by Jeffres-Mazzeo-Rubinstein [102], and as in the case of (4.1) one needs to understand the limiting behavior of a sequence ωtk as tk → T . A somewhat different strategy is given by the Ricci flow, originally introduced by Hamilton [98]. On a Fano manifold M the (normalized) K¨ahler-Ricci flow is

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given by the parabolic equation ∂ ωt = ωt − Ric(ωt ), ∂t with initial metric ω0 ∈ c1 (M ). It was known for a long time that the flow exists for all time [31], and the main difficulty was understanding the behavior of ωt as t → ∞. Building on Perelman’s deep results on the Ricci flow [137] as well as on ideas in Chen-Donaldson-Sun’s proof, Chen-Wang [49] developed the necessary convergence theory for understanding this limiting behavior (see also Bamler [11]). This was then used by Chen-Sun-Wang [46] to give a proof of Theorem 2 using the Ricci flow. We now return to the continuity method (4.1), and a sequence ωtk with tk → T . If such a sequence (or a subsequence) converges to a solution ωT of (4.1), then either we can further increase t using the implicit function theorem, or T = 1 and we obtain the K¨ahler-Einstein metric that we seek. The key question is then to understand what happens when the sequence ωtk diverges. In this case we need to construct a special degeneration for M with non-positive Futaki invariant. In broad outline the strategy is the following. We first define a certain limit space Z out of the sequence (M, ωtk ) (up to choosing a subsequence), and show that this limit can be given the structure of an algebraic variety, and in addition Z can also be viewed as an algebro-geometric limit of the images of a sequence of projective embeddings φk : M → PN . For simplicity suppose that T = 1. Then the limit Z is shown to admit a possibly singular K¨ ahler-Einstein metric. If Z is biholomorphic to M , then we obtain the K¨ahler-Einstein metric that we were after, but if not, then there is a special degeneration for M with central fiber Z. This necessarily has vanishing Futaki invariant, and so M is not K-stable. The argument is a little more involved when T < 1, since then the limit Z admits a “twisted” K¨ ahler-Einstein metric, and we obtain a special degeneration with a strictly negative Futaki invariant. In the next two sections we give some more details on the construction of the limit Z, and on relating the geometric and algebraic limits. It will be helpful to consider a more general sequence (Mk , ωk ) than that obtained from the equations (4.1), even allowing the manifolds Mk to vary. Some aspects of the problem, such as the convergence theory, are easier when we consider a sequence of K¨ ahler-Einstein manifolds, however others, such as using the K-stability assumption, are easier along a continuity method such as (4.1) (see for instance Donaldson [84] for the difficulty in using a sequence of K¨ ahler-Einstein metrics). 4.1. Gromov-Hausdorff limits. In this section we will consider a sequence ahler-Einstein with volumes controlled from below, or (Mk , ωk ) that are either all K¨ they arise from (4.1) or another similar continuity method. From a geometric point of view the advantage of working with such metrics as opposed to those with just constant scalar curvature is that control of the Ricci curvature allows us to extract a Gromov-Hausdorff limit of the sequence (Mk , ωk ). This is a consequence of the Gromov compactness theorem [96] and the Bishop-Gromov volume comparison theorem [37]. For simplicity let us denote by Mk the metric space (Mk , ωk ). Up to choosing a subsequence of the Mk , the limit is a metric space (Z, d), such that we have metrics (distance functions) dk on the disjoint unions Mk  Z, extending

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those on Mk and Z, satisfying the following: for all > 0, the -neighborhoods of both Mk and Z cover all of Mk  Z for sufficiently large k. While at first (Z, d) is just a metric space, a series of works by CheegerColding [34–36] developed a detailed structure theory (see also Cheeger [33] for an exposition). The general theory applies to Riemannian manifolds with only a lower bound on the Ricci curvature, and also allows collapsing phenomena, but our sequence Mk satisfies the non-collapsing condition Vol(B(qk , 1)) > c > 0, for a constant c > 0, where qk ∈ Mk are some basepoints. This again uses the Bishop-Gromov inequality, together with Myers’s Theorem and the fact that we control the volumes of the Mk . The basic concept is that of a tangent cone. For any p ∈ Z, and a sequence rk → ∞ the Gromov compactness theorem can be applied to the sequence of pointed manifolds (Z, rk d, p) and, up to choosing a subsequence, we obtain a limit metric space Zp , called a tangent cone of Z at p. A fundamental result of Cheeger-Colding is that in our situation these tangent cones are metric cones. More precisely for each tangent cone Zp there is a length space Y of diameter at most π, such that Zp is the completion of Y × (0, ∞) using the metric . d (y1 , r1 ), (y2 , r2 ) = r12 + r22 − 2r1 r2 cos dY (y1 , y2 ). We write this space as C(Y ). A point p ∈ Z is called regular, if a tangent cone Zp is isometric to Rm (in our situation m = 2n). In fact in this case each tangent cone at p is Euclidean. This means that if we take a sufficiently small ball centered at p, and scale it up to unit size, then it will be very close to the Euclidean ball in the Gromov-Hausdorff sense. But recall that for sufficiently large k, the metric space Mk is very close to Z. This implies that we can find points pk ∈ Mk , and small balls centered at pk which, scaled to unit size, are close to Euclidean in the Gromov-Hausdorff sense. If the metrics ωk were Einstein, then results of Anderson [4] and Colding [51] would show that we can choose holomorphic coordinates centered at pk with respect to which the components of ωk are controlled in C 2,α . If all points of Z were regular, this would mean that there is a uniform scale at which we have C 2,α control of the metrics ωk . In particular we could take a limit of the K¨ahler structures (M, ωk ), and turn Z into a smooth K¨ahler manifold. Note, however, that even in this case Z may not be biholomorphic to M . When the ωk are not actually Einstein, but rather arise from (4.1), then Anderson’s result still applies at the regular points [161]. However, in general not all points of Z are regular, and instead Cheeger-Colding defined a stratification based on how close each tangent cone is to being Euclidean. More precisely, for 0 ≤ k ≤ m − 1, the set Sk is defined to be those points p ∈ Z for which no tangent cone Zp splits off an isometric factor of Rk+1 . They then showed that the Hausdorff dimension of Sk is at most k and in addition Sm−1 = Sm−2 , so Z is regular outside a codimension 2 set. A key technical step in Chen-Donaldson-Sun’s work is to improve this dimension estimate of the singular set from the Hausdorff dimension to the Minkowski dimension - i.e. to show that there is a constant K for which the singular set Sm−1 can be covered by Kr 2−m balls of radius r, for any r ∈ (0, 1). For even better estimates on the Minkowski dimension of the singular set for Einstein manifolds, see Cheeger-Naber [38, 39].

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This discussion applies also to the tangent cones Zp , as well as to further iterated tangent cones. In our situation this implies that each tangent cone Zp is a Ricci flat K¨ ahler cone, at least outside of a codimension 2√set. In addition on the ahler metric is given by −1∂∂r 2 , where r is the regular set in Zp , the Ricci flat K¨ distance from the vertex of Zp . 4.2. The partial C 0 -estimate. The theory described in the previous section gives a good understanding of the “geometric” limit of the spaces (Mk , ωk ). In this section we incorporate the holomorphic structure through the partial C 0 -estimate, which is an idea due to Tian [165] used in his work on K¨ahler-Einstein metrics on Fano surfaces. For simplicity we will assume here that Mk = M , and the sequence ωk is obtained from (4.1). Suppose that −lKM is very ample for some large l > 0. The metrics ωk define Hermitian metrics hk on −lKM , with curvature forms lωk . As in Section 3.5, we have a natural projective embedding φk : M → PN obtained by using an orthonormal basis {si } of H 0 (M, −lKM ) for the L2 inner product " $s, t%hk (lωk )n , $s, t%L2 = M

defined by hk . As before, the metric ωk is related to the pullback of the FubiniStudy metric under φk by the Bergman kernel ρωk ,l . In Section 3.5 we saw the expansion (3.7) of ρωk ,l as l → ∞ for a fixed metric ωk , whereas here the main interest is in obtaining bounds for ρωk ,l for sufficiently large l, which apply uniformly to a sequence of metrics ωk . Tian [165] conjectured that if, as in our situation, we have a positive lower bound for the Ricci curvatures of ωk , and a lower bound for the volumes, then for sufficiently large l we have a bound ρωk ,l > d > 0 from below. This is called the partial C 0 -estimate, and it is equivalent to showing that for any point p ∈ M , the bundle −lKM has a holomorphic section s satisfying sL2 = 1, and |s|2hk (p) > d. Note that if −lKM is very ample, then ρω,l is positive for any metric ω ∈ c1 (M ), and the key point is to have a uniform lower bound along the sequence of metrics ωk . For the details on how to use the partial C 0 -estimate to define the structure of an algebraic variety on Z, see Tian [165] and more generally Donaldson-Sun [67]. Here let us just note that a key consequence of the estimate is that there is a uniform bound on the derivatives of the embeddings φk : (M, ωk ) → PN , independent of k. Up to further increasing the multiple of −KM that we use, this eventually leads to the result that the Gromov-Hausdorff limit Z is homeomorphic to the algebrogeometric limit of the images φk (M ) in projective space. Let us say a few words on the proof of the partial C 0 -estimate. Fixing a point p ∈ M , for each metric ωk we must construct holomorphic sections sk of −lKM , for sufficiently large l, such that sk L2 ,hk = 1, and |sk |2hk (p) > d > 0 for some d independent of k. The simplest situation is when the geometry of each ωk is very well controlled near p at a suitable scale. For instance this would be the case if ωk did not actually depend on k (or more generally if the ωk were to converge to a smooth metric ω on M ). In this case, at a sufficiently small scale, the metrics ωk appear to be very close to the Euclidean space (Cn , ωEuc ). The trivial line bundle 2 over Cn has metric e−l|z| with curvature lωEuc , and so for sufficiently large l, there will be a holomorphic section s with |s|2 (0) = 1 and sL2 = 1 but with s decaying rapidly away from the origin. Using cutoff functions this section can be glued onto

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M to produce a smooth section sk of −lKM satisfying |sk |2hk (p) ∼ 1,

sk 2L2 ,hk ∼ 1, and ∂sk L2 ,hk ∼ 0.

Using the H¨ ormander L2 -technique, these sections can be perturbed slightly to obtain the required holomorphic sections of −lKM . This method was used by Tian [164] to find a weak form of the asymptotic expansion (3.7) of the Bergman kernel as l → ∞, for a fixed metric ω. As we have described in Section 3.5, this expansion has been very influential in the study of constant scalar curvature K¨ahler metrics. In general we do not have such good uniform control of the local geometry of the metrics ωk . The first more general situation that was understood was the case of Fano surfaces, by Tian [165]. He proved an orbifold compactness theorem for K¨ ahler-Einstein surfaces, using techniques employed in the study of Yang-Mills connections by Uhlenbeck [171, 172] (see also Anderson [3] and Bando-KasueNakajima [12]). In effect this implies that at a suitable scale, the local geometry of a non-collapsed sequence of K¨ ahler-Einstein surfaces (Mk , ωk ) is modeled either on flat C2 , or its quotient by a finite group. Using this together with the L2 technique, Tian showed the partial C 0 -estimate for such a sequence of K¨ ahler-Einstein surfaces. There was little further progress until the work of Donaldson-Sun [67], who proved the partial C 0 -estimate for non-collapsing sequences of K¨ahler-Einstein manifolds (Mk , ωk ), generalizing the result of Tian. In the proof they combined the L2 technique with the Cheeger-Colding structure theory that we described above. Very roughly the structure theory, as described above, implies that the local geometry of the (Mk , ωk ) near a point pk is modeled by Ricci flat cones C(Y ). Although Y may have singularities, the estimate on the size of the singular set ensures that we can find a cutoff function η, supported on the regular part of C(Y ), for which ∇ηL2 , 1. Taking l sufficiently large, the line bundle −lKMk with its Hermitian metric hk is modeled, at least on the regular part, by the trivial bundle over C(Y ), with trivializing section s whose norm decays exponentially fast. One can then use ηs to obtain a smooth section of −lKMk which is approximately holomorphic in an L2 -sense, and which can then be perturbed to a genuine holomorphic section sk . A gradient estimate is then used to show that |sk |2 (pk ) > d for a controlled constant d > 0, while sk L2 ∼ 1, as required. Shortly after Donaldson-Sun’s work, the partial C 0 -estimate was proven in various other settings. In their proof of the Yau-Tian-Donaldson conjecture, ChenDonaldson-Sun [43, 44] proved it for K¨ ahler-Einstein metrics with conical singularities, while Phong-Song-Sturm [138] extended Donaldson-Sun’s work to K¨ahlerRicci solitons using also some techniques of Tian-Zhang [168]. In [161] we showed that it holds along the continuity method (4.1). The ideas have also been applied to the K¨ ahler-Ricci flow, by Tian-Zhang [169] in dimensions up to 3, and ChenWang [49] in general. These results on the K¨ahler-Ricci flow also led to a proof of Tian’s general conjecture on the partial C 0 -estimate, for a non-collapsed sequence of K¨ahler manifolds with only a positive lower bound on the Ricci curvature (see Jiang [103] for dimensions up to 3, and Chen-Wang [49] in general). 4.3. The K-stability condition. In the previous sections we have described how out of a sequence (M, ωk ) along the continuity method (4.1), or from a sequence of K¨ ahler-Einstein metrics, or even from a sequence along a solution of the K¨ahlerRicci flow one can construct a limit Z, which has the structure of an algebraic

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variety. This limit variety can be singular, although it is always normal with log terminal singularities. In addition one can show that Z admits a singular K¨ ahler metric, which satisfies either an analog of the equation (4.1) with t = T (where recall that T = limk→∞ tk ), is K¨ahler-Einstein, or is a K¨ahler-Ricci soliton in the case of the Ricci flow. In this section we will explain how the K-stability assumption on M can be used to show that in the limit we obtain a K¨ahler-Einstein metric on M . In addition we discuss some of the developments regarding singular K¨ ahler-Einstein metrics, as they play an important role in the proof of the Yau-Tian-Donaldson conjecture. Let us suppose that we are working with a continuity method such as (4.1), we have T = 1, and for simplicity assume that Z is smooth. Note that Z then admits a K¨ ahler-Einstein metric, and the goal is to show that if M is K-stable, then actually Z ∼ = M . To see this, note first that Matsushima’s Theorem [126] implies that the automorphism group of Z is reductive. On the other hand, Z can also be viewed as an algebraic limit of the projective varieties φk (M ) ⊂ PN , for a sequence of embeddings φk : M → PN , and so in a suitable Hilbert scheme, Z can be viewed as an element in the closure of the GL(N + 1)-orbit of M . From this point of view it is the stabilizer of Z under the GL(N + 1) action that is reductive, and then an appropriate version of the Luna slice theorem [83, 121] implies that there is a special degeneration for M , whose central fiber is Z. Futaki’s result [93] furthermore implies that the Futaki invariant of this special degeneration must vanish. If M is K-stable, then necessarily Z ∼ ahler= M , and so M admits a K¨ Einstein metric. Suppose now that T < 1, and that for simplicity Z is smooth. Under ChenDonaldson-Sun’s continuity method (4.2) one obtains a limiting K¨ahler-Einstein metric on Z, with cone singularities along a divisor D∞ . The strategy is then to apply the argument used above, but to the pairs (M, D) and (Z, D∞ ). More precisely, one shows, using the equation on Z, that in a suitable Hilbert scheme of pairs the stabilizer of (Z, D∞ ) is reductive, and from this one constructs a testconfiguration X for (M, D) with central fiber (Z, D∞ ). The Donaldson-Futaki invariant has a natural generalization to pairs [82], called the log Futaki invariant DF (X, (1 − T )D), depending on the divisor D as well as the “angle” parameter T such that DF (X, 0D) = DF (X). It is shown in [44] that DF (X, (1 − T )D) = 0, since in fact the log Futaki invariant of any vector field on the pair (Z, (1 − ahler-Einstein metric on Z. T )D∞ ) vanishes. This uses the existence of a conical K¨ In addition DF (X, D) > 0, which corresponds to an existence result for K¨ahlerEinstein metrics with cone singularities that have small cone angles. Since the log Futaki invariant is linear in the angle parameter, it follows that DF (X) < 0. Using the continuity method (4.1) is a bit more cumbersome, because in that case instead of working with pairs of the form (M, D) that can be parametrized by a Hilbert scheme, we need to work with pairs (M, α), where α is a (1,1)-form, or even a current, and the Luna slice theorem does not apply directly in this infinite dimensional setting. In [56] we overcome this problem by an approximation argument, which we now outline. The starting point is to show that out of a sequence of solutions to (4.1) with ti → T , we obtain a limiting metric ωT on the space Z satisfying the equation Ric(ωT ) = T ωT + (1 − T )β,

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where β is a positive (1, 1)-current on Z. This can be used to show that the automorphism group of the pair (Z, β), suitably defined, is reductive, and moreover a version of the log Futaki invariant of the pair (Z, (1 − T )β) vanishes for all vector fields. The missing piece for exploiting this as above is to find a test-configuration for M with central fiber Z, under which the form α on M converges to the current β on Z, but in the end we are not able to do this. Instead, in order to apply a result such as Luna’s slice theorem, the idea is to approximate the currents α and β with currents of integration along divisors. Note first that according to Shiffman-Zelditch [149], we can write α as an average of currents of integration [M ∩ H] for hyperplane sections of M , under a suitable projective embedding M ⊂ PN . As discussed above, the geometric limit space Z is also an algebraic limit limk→∞ ρk (M ) for automorphisms ρk of PN . We can arrange that the limits ρ∞ (H) = limk→∞ ρk (H) exist for all hyperplanes H and so β is an average of the hyperplane sections [Z ∩ ρ∞ (H)] of Z. This can be used to show that a suitable tuple (Z, ρ∞ (H1 ), . . . , ρ∞ (Hl )) has the same automorphism group as (Z, β). One simple example to keep in mind is when Z is a line in P2 , and β is the Fubini-Study metric. In this case the automorphism group of (Z, β) is trivial under our definition, and so we must “mark” Z with at least 3 hyperplane sections to have a tuple with the same automorphism group. We can argue as before with these tuples, and we obtain a test-configuration X for M with central fiber Z, such that in addition the limits of the hyperplanes H1 , . . . , Hl under this test-configuration are ρ∞ (H1 ), . . . , ρ∞ (Hl ). However, since we do not know that the form α converges to β under this test-configuration, we cannot say anything about the Donaldson-Futaki invariant of X. To overcome this we choose enough hyperplane sections so that by an approximation argument the log Futaki invariant of the tuple is close to that of the pair (Z, β). It remains then to use this to show that DF (X) < 0, contradicting K-stability of M , unless we actually have Z ∼ = M and β = α, so that the continuity method can be continued past t = T . To conclude this section we make some remarks about the situation when the limit space Z is not smooth. The overall strategy remains the same, but there are substantial technical difficulties in carrying it through. To start with, one needs to make sense of the K¨ ahler-Einstein equation (or more general equations such as (4.1)) on a singular variety Z. Working on a resolution of singularities π : X → Z, the equation is equivalent to a complex Monge-Amp`ere equation √ (ω + −1∂∂φ)n = eF −φ dV, on X, where dV is a smooth volume form, however the function F need not be bounded, and the form ω is only non-negative. More precisely we have 0 ≤ eF ∈ Lp 5 n for some p > 1, and ω is also big, i.e. X ω > 0. It cannot be expected that φ is twice differentiable, and so the equation needs to be defined in a weak sense. This was done for the local theory of the complex Monge-Amp`ere operator by BedfordTaylor [14]. Further important progress in understanding such Monge-Amp`ere equations with right hand side in Lp was made by Kolodziej [109] using techniques of pluripotential theory. A detailed study of K¨ahler-Einstein metrics on singular varieties is given in Eyssidieux-Guedj-Zeriahi [87]. The lack of regularity of such a K¨ahler-Einstein metric at the singular points means that results such a Matsushima’s Theorem on the automorphism group and the vanishing of the Futaki invariant cannot be generalized in a straightforward way.

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Instead Chen-Donaldson-Sun [44] proved these results in the singular setting using the theory of geodesics and Berndtsson’s convexity and uniqueness results [22]. It is natural to expect that Theorem 2 can be extended to Q-Fano varieties, and indeed Berman [16] showed that Q-Fano varieties admitting (singular) K¨ahlerEinstein metrics are K-stable. The converse, however, is so far only known for smoothable Q-Fanos, by work of Spotti-Sun-Yao [154]. 5. Applications In this final section we briefly survey some applications of the Yau-TianDonaldson conjecture for Fano manifolds, and the techniques involved in its solution. The first is perhaps the most direct type of application, namely obtaining new examples of K¨ahler-Einstein manifolds. The second and third are more theoretical, on understanding the moduli space of Fano K¨ahler-Einstein manifolds, and the behavior of singular K¨ ahler-Einstein metrics at the singular points. 5.1. New examples. According to Theorem 2, K-stability is a necessary and sufficient condition for a Fano manifold M to admit a K¨ ahler-Einstein metric. As such it is natural to expect that the result can be used to show the existence of new K¨ahler-Einstein metrics. Unfortunately for any given Fano manifold M of dimension at least two, there are infinitely many possible special degenerations, and so at present there is no general method for testing K-stability. On the other hand there are special circumstances in which one can show that certain varieties admit a K¨ ahler-Einstein metric. In the two-dimensional case, Tian [165] used the α-invariant [163] and its refinements, to show that any Fano surface with reductive automorphism group admits a K¨ ahler-Einstein metric. More generally, calculations of the α-invariant, and related concepts such as Nadel’s multiplier ideal sheaves [128], lead to many examples of K¨ ahler-Einstein manifolds (see e.g. [40,59]). Recently Fujita [91] showed that Fano n-folds M with α-invariant α(M ) = n/(n + 1), which are borderline for Tian’s criterion [163], are still K-stable, and as such they admit K¨ahler-Einstein metrics. A different source of examples where K-stability can be checked effectively is Fano manifolds M with large automorphism groups, since in [56] we showed that it is enough to check special degenerations that are compatible with the automorphism group. For instance if M is a toric manifold, then the only torus equivariant special degenerations for M are products, and so M admits a K¨ ahler-Einstein metric whenever the Futaki invariants of all vector fields on M vanish (this result is originally due to Wang-Zhu [175], using different methods). This result has recently been generalized to reductive group compactifications by Delcroix [58] using analytic methods. Subsequently, by classifying equivariant special degenerations, Delcroix [57] generalized this result to spherical varieties, giving new examples of K¨ahler-Einstein manifolds. A more general setting is complexity-one spherical varieties [170], for instance Fano n-folds with an effective action of an (n − 1)-dimensional torus. The equivariant special degenerations of these complexity-one T-varieties were classified by Ilten-S¨ uß [101], using the theory of polyhedral divisors [2] and as an application they obtained several new examples of K-stable Fano 3-folds. Theorem 2 was generalized to the setting of K¨ahler cones in [52] and, combined with the methods of Ilten-S¨ uß [101], this lead to new examples of Ricci flat K¨ ahler

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cones. As we discussed in Section 4.1, such K¨ ahler cones arise as tangent cones of Gromov-Hausdorff limits of K¨ ahler-Einstein manifolds, but they have also been studied extensively from the point of view of Sasakian geometry [26, 95, 152]. To give a simple example, consider the affine variety Mp,q ⊂ C4 , given by the equation xy + z p + wq = 0. This has a natural action of a 2-torus, acting diagonally on ahler C4 . Using K-stability, it is shown in [52] that Mp,q admits a Ricci flat K¨ cone metric compatible with the torus action, if and only if 2p > q and 2q > p. These two conditions correspond to the calculation of a Donaldson-Futaki invariant for two torus equivariant degenerations of Mp,q , to the hypersurfaces given by the equations xy + z p = 0 and xy + wq = 0 respectively. For manifolds without such large symmetry groups, the general approach of Tian [165] for Fano surfaces has recently been revisited in the work of OdakaSpotti-Sun [134], Spotti-Sun [153] and Liu-Xu [119]. Here, given a family of Fano manifolds some elements of which are known to be K¨ ahler-Einstein, one tries to find new K¨ ahler-Einstein manifolds using a continuity method in the family. The key problem is to classify the possible Gromov-Hausdorff limits of K¨ ahlerEinstein manifolds in the family. Using this method, and combined with work of Fujita [91], Liu-Xu [119] showed that all smooth 3-dimensional Fano hypersurfaces admit K¨ ahler-Einstein metrics. Ultimately such a study can lead to a complete understanding of the K¨ ahler-Einstein moduli space and its compactification, as we will discuss in the next section. While a general algorithm for testing K-stability seems to be some ways off at present, it seems likely that these techniques will lead to many more examples. 5.2. The moduli space of K¨ ahler-Einstein manifolds. A basic question in algebraic geometry is to understand the moduli space of a certain manifold, as well as its compactifications. Perhaps the most well known example is the DeligneMumford compactification of the moduli space of hyperbolic Riemann surfaces. This compactification of the moduli space of smooth Riemann surfaces is obtained by adding certain “stable” curves with nodal singularities, and the degeneration of smooth curves to these singular ones can in turn be modeled differential geometrically by degenerating families of hyperbolic metrics. Canonically polarized manifolds are a higher dimensional generalization of hyperbolic Riemann surfaces and compact moduli spaces are obtained by adding the “KSBA-stable” varieties (see Koll´ ar-Shepherd-Barron [108] Alexeev [1], and the survey [107]). It is interesting to note that the KSBA-stable varieties were subsequently found to coincide with the K-stable varieties by work of Odaka [131], and furthermore Berman-Guenancia [20] showed that these all admit (singular) K¨ahler-Einstein metrics. To complete the analogy with hyperbolic Riemann surfaces it remains to understand the convergence of smooth K¨ ahler-Einstein metrics to the singular ones on a metric level. We emphasize that in the canonically polarized setting all smooth manifolds admit K¨ahler-Einstein metrics with negative Ricci curvature by Yau’s theorem [177], and K-stability only enters in picking out the right type of singular varieties to allow. By contrast, in the Fano case one cannot form a Hausdorff moduli space containing all the smooth manifolds, since there are non-trivial families over C that are isomorphic to a product over C∗ . For instance any non-trivial test-configuration for a Fano manifold with smooth central fiber gives such a family. One can hope,

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however, that K-stable Fano manifolds form good moduli spaces, and that a compactification is obtained by including certain singular K-stable varieties. Moreover this compact moduli space should agree with the space of K¨ ahler-Einstein Fano manifolds, compactified using the Gromov-Hausdorff topology. A first step is to understand the neighborhood of a K¨ahler-Einstein manifold in the moduli space. In [159] (see also Br¨ onnle [27]) we proved that given a K¨ ahler-Einstein manifold M , its small deformations that admit KE metrics are precisely the ones that are (poly)stable for the action of Aut(M ) on the deformation space of M . Thus a small neighborhood of M in the moduli space is modeled by a GIT quotient for the action of Aut(M ). In the case of surfaces, a global study of the moduli space of K¨ ahler-Einstein manifolds was achieved by Odaka-Spotti-Sun [134], who explicitly identified the compactified moduli space in terms of certain algebro-geometric compactifications. In higher dimensions Spotti-Sun-Yao [154] and Li-Wang-Xu [114] showed that smoothable K-stable Q-Fano varieties, i.e. the ones that we expect to appear in the compactified moduli space, all admit singular K¨ ahler-Einstein metrics, which in addition can be obtained as Gromov-Hausdorff limits of smooth KE metrics. Regarding the global structure of the moduli space, the smoothable K-stable QFano varieties are parametrized by a proper algebraic space by work of Odaka [132] and Li-Wang-Xu [114], and moreover the K-stable smooth Fanos form a quasiprojective variety by Li-Wang-Xu [115]. It remains to understand more precisely what varieties these moduli spaces parametrize in analogy with Odaka-Spotti-Sun’s work [134] on surfaces, but as we mentioned above, there are special cases where progress has been made, such as in the case of degree 4 del Pezzo manifolds by Spotti-Sun [153] and cubic threefolds by Liu-Xu [119]. 5.3. Asymptotics of singular K¨ ahler-Einstein metrics. In Section 4.3 we discussed how singular K¨ ahler-Einstein metrics play an important role in the proof of Theorem 2. Of particular interest are those singular K¨ ahler-Einstein varieties, which arise as limits of smooth K¨ahler-Einstein manifolds, in view of the importance of forming complete moduli spaces of Fano manifolds [114, 132, 134, 154], as well as Calabi-Yau manifolds [181]. We do not yet have a detailed understanding of the behavior of such singular K¨ ahler-Einstein metrics near the singular points, although much progress has been made. Such an understanding would be necessary in order to use differential geometric techniques on these singular spaces (see e.g. [30, 97] for such applications). A first step in understanding the metric behavior near the singular points is to understand the metric tangent cone, as discussed in Section 4.1. In the general Riemannian context the tangent cone may depend on the sequence of rescalings chosen to construct it, but in the setting of K¨ahler-Einstein manifolds DonaldsonSun [66] showed that the tangent cone is unique, and it has the structure of an affine variety. They also made progress towards an algebro-geometric description of the metric tangent cone, and conjectured that in general the tangent cone at a point x ∈ M of a singular K¨ahler-Einstein metric on M can be determined from the germ of the singularity (M, x) using K-stability. Note that it is not always the case that the germ of the tangent cone at the vertex is biholomorphic to the germ (M, x). For example (see [157], [99]) there is a Calabi-Yau metric on a neighborhood of the origin in the hypersurface xk0 + x21 + x22 + x23 = 0 in C4 whose tangent cone at the origin is given by C × C2 /Z2 , if k > 4.

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An important special case of this conjecture for singular Calabi-Yau varieties was proven by Hein-Sun [100], namely when the germ (M, x) is isomorphic to the germ (C, 0) of a Ricci flat K¨ ahler cone C at its vertex, and C satisfies some additional technical conditions. This uses some ideas of Li [111] and Li-Liu [113] on an alternative characterization of K-semistability based on the work of Fujita [92] we mentioned in Section 3.4. Using this, Hein-Sun showed that the singular Ricci flat metric on M near x is asymptotic, in a strong sense, to the Ricci flat metric on C. Recently, using more algebro-geometric techniques, Li-Xu [117] have also made significant progress towards resolving Donaldson-Sun’s conjecture, and they have announced a complete solution. References [1] V. Alexeev, Log canonical singularities and complete moduli of stable pairs. arXiv:9608013. [2] K. Altmann and J. Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006), no. 3, 557–607. MR2207875 [3] M. T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, J. Amer. Math. Soc. 2 (1989), no. 3, 455–490. MR999661 [4] M. T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), no. 2, 429–445. MR1074481 [5] V. Apostolov, D. M. J. Calderbank, P. Gauduchon, and C. W. Tønnesen-Friedman, Hamiltonian 2-forms in K¨ ahler geometry. III. Extremal metrics and stability, Invent. Math. 173 (2008), no. 3, 547–601. MR2425136 [6] C. Arezzo and F. Pacard, Blowing up and desingularizing constant scalar curvature K¨ ahler manifolds, Acta Math. 196 (2006), no. 2, 179–228. MR2275832 [7] C. Arezzo and G. Tian, Infinite geodesic rays in the space of K¨ ahler potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 617–630. MR2040638 [8] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR702806 ´ [9] T. Aubin, Equations du type Monge-Amp` ere sur les vari´ et´ es k¨ ahl´ eriennes compactes (French, with English summary), Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95. MR494932 [10] T. Aubin, R´ eduction du cas positif de l’´ equation de Monge-Amp` ere sur les vari´ et´ es k¨ ahl´ eriennes compactes a ` la d´ emonstration d’une in´ egalit´ e (French), J. Funct. Anal. 57 (1984), no. 2, 143–153. MR749521 [11] R. Bamler, Convergence of Ricci flows with bounded scalar curvature. arXiv:1603.05235. [12] S. Bando, A. Kasue, and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313–349. MR1001844 [13] S. Bando and T. Mabuchi, Uniqueness of Einstein K¨ ahler metrics modulo connected group actions, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 11–40. MR946233 [14] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Amp` ere equation, Invent. Math. 37 (1976), no. 1, 1–44. MR0445006 [15] R. Berman, On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold. arXiv:1405.6482. ahler-Einstein metrics, Invent. [16] R. J. Berman, K-polystability of Q-Fano varieties admitting K¨ Math. 203 (2016), no. 3, 973–1025. MR3461370 [17] R. J. Berman and B. Berndtsson, Convexity of the K-energy on the space of K¨ ahler metrics and uniqueness of extremal metrics, J. Amer. Math. Soc. 30 (2017), no. 4, 1165–1196. MR3671939 [18] R. Berman, S. Boucksom, and Jonsson, M., A variational approach to the Yau-TianDonaldson conjecture. arXiv:1509.04561. [19] R. Berman, T. Darvas, and C. H. Lu, Regularity of weak minimizers of the K-energy and applications to properness and K-stability. arXiv:1602.03114. [20] R. J. Berman and H. Guenancia, K¨ ahler-Einstein metrics on stable varieties and log canonical pairs, Geom. Funct. Anal. 24 (2014), no. 6, 1683–1730. MR3283927

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01746

Donaldson theory in non-K¨ ahlerian geometry Andrei Teleman Abstract. The Kobayashi-Hitchin correspondence gives an identification between instanton moduli spaces on complex surfaces and moduli space of polystable bundles. Proved first by Donaldson for projective surfaces, this important result has been used for explicitly describing instanton moduli spaces on certain algebraic surfaces. Complex geometry was used as a tool for solving difficult gauge theoretical problems, e.g. for computing Donaldson invariants. In a series of recent articles we have reversed the strategy and used the gauge theory to attack difficult problems in complex geometry, specifically on the existence of curves on minimal class VII surfaces with small b2 . In this article we prove a structure theorem for the instanton moduli spaces used in our approach, and we explain the consequences of this theorem.

Contents 1. Introduction. 2. An instanton moduli space on definite 4-manifolds with b1 “ 1 3. The circles of reductions 4. Curves on class VII surfaces References

1. Introduction. 1.1. Instantons and curves on class VII surfaces. A class VII surface is a (compact, connected) complex surface X with b1 pXq “ 1 and kodpXq “ ´8 [BHPV]. The classification of class VII surfaces is a longstanding, open problem in complex geometry. Important contributions on this difficult subject have been made by many prestigious mathematicians who worked on the subject, e.g. by Kodaira [Kod1]–[Kod2], Kato [Ka1]—-[Ka3], Nakamura [Na1]–[Na3], Enoki [E], Dloussky [Dl], Dloussky-Oeljeklaus-Toma [DOT], Apostolov-Dloussky [AD], Oeljeklaus-Toma [OeTo], Fujiki-Pontecorvo [FP1]–[FP2], Brunella [Br1]–[Br3]. The classification of class VII surfaces up to biholomorphism will be completed if the following conjecture is proved (see section 4.1): 2010 Mathematics Subject Classification. Primary 53C55; Secondary 53C07, 32G13. The author is indebted to Nicholas Buchdahl for careful reading the article, and for his useful remarks and suggestions. c 2018 American Mathematical Society

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Conjecture 1. Any minimal class VII surface X with b2 pXq ą 0 has b2 pXq rational curves. This conjecture shows that the main problem in the classification of class VII surfaces is the existence of curves. Note that all known minimal class VII surfaces with b2 ą 0 do contain b2 rational curves, so one comes naturally to the question: Can one prove the conjecture by defining a Gromov-Witten type invariant which counts rational curves, and then proving that, for minimal class VII surfaces, this invariant coincides with the second Betti number? This question leads naturally to the idea to use either a non-K¨ ahlerian version of the Gromov-Witten theory, or the gauge theoretical version of this theory, which is the Seiberg-Witten theory. Unfortunately up till now all attempts to use this strategy have failed. The number b2 cannot be interpreted as a Gromov-Witten invariant because, even for known surfaces, the intersection numbers of the rational curves are not constant in holomorphic families. For known class VII surfaces, the number b2 is obtained by counting the points in all Gromov-Witten moduli spaces (associated to all 2homology classes), including those with negative expected dimension. Counting the curves in a fixed homology class does not give a deformation invariant. The source of this difficulty is the “explosion of area”, a typically non-K¨ahlerian phenomenon which illustrates the non-symplectic character of the problem, and can be described briefly as follows (see [DlTe] for details). Let ď Xz “ X Ñ D p: zPD

be a holomorphic family Ť of class VII surfaces parameterised by the disk D, let χ P H2 pX , Zq, and let zPD˚ Cz “ C Ă p´1 pD˚ q be an effective divisor which is flat over the punctured disk D˚ , and such that rCz s “ χ for any z P D˚ . One can find triples pp, χ, Cq as above such that, choosing a Hermitian metric g on X , one has limzÑ0 volg pCz q “ 8, and the central fibre X0 contains no analytic cycle in the class χ. Therefore the area of the curve Cz Ă Xz “explodes” as z Ñ 0, and the Gromov-Witten moduli space associated to the limit pair pX0 , χq is empty. Taking into account this difficulty, it follows that Gromov-Witten theory cannot be easily adapted to non-K¨ahlerian geometry, so new ideas and techniques are necessary for proving the existence of curves on class VII surfaces with positive b2 . A cycle on a class VII surface X is an effective divisor D Ă X which is either an elliptic curve, or a cycle of rational curves (see section 4.1 for details). A weaker version of Conjecture 1 claims: Conjecture 2. Any minimal class VII surface X with b2 pXq ą 0 has a cycle. By the results of Nakamura [Na1], [Na3] any minimal class VII surface with a cycle is the degeneration of a 1-parameter family of blown-up primary Hopf surfaces so, if true, this conjecture will complete the classification of class VII surfaces up to deformation equivalence, so, in particular, up to diffeomorphism. In a series of articles I developed a programme to prove Conjecture 2 for class VII surfaces with small b2 , and this programme uses essentially Donaldson theory. The main object coming into the proof is the moduli space MASD pEq of projectively ASD a connections on a Hermitian 2-bundle E with c2 pEq “ 0 and detpEq “ K (the underlying differentiable line bundle of the canonical line bundle K). This moduli

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space is compact, and the space of reductions R “ MASD pEqzMASD pEq˚ is a finite a a union of circles. pEq Via the Kobayashi-Hitchin correspondence [D1], [Bu1], [LY], [LT] MASD a can be identified with a moduli space Mpst of polystable holomorphic bundles, pEq˚ with a moduli space Mst of stable holomorphic and its open subspace MASD a pst comes with a canonical involution bl0 defined by the nonbundles on X. M trivial square root l0 of rOX s in Pic0 pXq » C˚ . The fixed point set B of this involution is finite, is contained in Mst , and corresponds via the Kobayashi-Hitchin pEq˚ (see section 1.2). correspondence to a set of twisted reductions in MASD a st In relevant cases M is a smooth b2 pXq–dimensional manifold which comes with a distinguished point: the isomorphism class a “ rAs of the holomorphic bundle A defined as the essentially unique non-trivial extension of the form i

p

0 0 A ´´Ñ OX Ñ 0 . 0 Ñ K ´´Ñ

This bundle has an important property (see Proposition 4.10): if it can be written as a line bundle extension in a different way (with a kernel M ‰ i0 pKq), then X has a cycle. Our general strategy to prove Conjecture 2 has two steps: (St1) If X did not contain a cycle, then, for suitable Gauduchon metrics, Mst would contain a compact complex subspace Y of positive pure dimension with an open subspace Ya such that a P Ya Ă Y , and all the points of Ya ztau correspond to non-filtrable bundles. (St2) The existence of such a subspace Y leads to a contradiction. For b2 P t1, 2u both steps have been obtained using specific methods, which do not generalise to larger b2 (see [Te2], [Te5]). The first step was proved using explicit descriptions of the moduli spaces Mst , Mpst in the two cases; in the second step we made use essentially of the theory of Riemann surfaces (for b2 “ 1), and the theory of complex surfaces (for b2 “ 2). Recently I gave a proof of (St2) in full generality [Te6]. The proof uses a new version of the Grothendieck-Riemann-Roch theorem, which gives an equality in Bott-Chern cohomology, and is due to Bismut [Bi]. Unfortunately (St1) is still very challenging. The goal of this article is the following theorem, which solves the problem (so proves Conjecture 2) for b2 “ 3: Theorem 4.14. Let X P VIImin with 1 ď b2 pXq ď 3, and which does not contain a cycle. For suitable Gauduchon metrics on X the moduli space Mst contains a compact complex subspace Y of positive pure dimension with an open subspace Ya such that a P Ya Ă Y , and all the points of Ya ztau correspond to non-filtrable bundles. Theorem 4.14 is an easy consequence of the difficult Theorem 4.13 (see section 4.4), which is a structure theorem for the moduli space Mst , and gives a decomposition of Mst as a union of: (1) finitely many positive dimensional families of stable line bundle extensions, (2) the set B of fixed points of the canonical involution bl0 , (3) finitely many compact, positive dimensional, complex subspaces. We believe that the structure theorem Theorem 4.13 holds for any b2 ; if true, this would prove Theorem 4.14, so also Conjecture 2 in full generality. In order to avoid technical complications, we will explain the ideas and the steps of the proof assuming two simplifying assumptions: we will assume that H1 pX, Zq » Z, and

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that X does not contain curves in certain homology classes (see section 4.5). The proof of the structure theorem makes use of: ‚ A general theorem, valid for arbitrary b2 , concerning the position of the circles of reductions with respect to the connected components of the moduli space (see Theorems 3.6, 3.7 in this article). ‚ A holomorphic model theorem for the structure of the moduli space around the circles of reductions [Te7]. ‚ Results on the boundary of the Borel-Moore homology class of a pure dimensional analytic set Z Ă Mst [Te8]. ‚ Donaldson cohomology classes and their properties. Note that the moduli space used in our programme is a moduli space of PUp2q– instantons with non-trivial w2 . A natural question can be asked: can one obtain results on the existence of curves on class VII surfaces using moduli spaces of SUp2q–instantons? Moduli spaces of SUp2q–instantons on non-K¨ ahlerian surfaces, and their compactifications, have recently been investigated in [BTT1], [BTT2]. 1.2. General theory. Let pM, gq be a compact, connected, oriented Riemannian 4-manifold, and let pE, hq be a Hermitian bundle of rank 2 on M . Put D :“ detpEq, and let a be a fixed Hermitian connection on the Hermitian line bundle pD, detphqq. The fundamental object used in this article is the moduli space pEq of projectively anti-self-dual Hermitian connections on E inducing the MASD a fixed connection a on D. More precisely, denote by ApEq the affine space of Hermitian connections on pE, hq, and put L Aa pEq :“ tA P ApEq| detpAq “ au , Ba pEq :“ Aa pEq GE , AASD pEq “ tA P Aa pEq| pFA0 q` “ 0u , a where FA0 stands for the trace-free component of the curvature FA P A2 pupEqq of A, and GE :“ ΓpSUpEqq is the SUp2q–gauge group of E, i.e. the group of unitary automorphisms of E inducing the identity automorphism of D. With these notations one has ASD L pEq :“ Aa pEq GE Ă Ba pEq . MASD a Denote by PE the frame bundle of E, and by P¯E “ PE {S1 the associated principal PUp2q–bundle. The projection map q : PE Ñ P¯E is an idM -covering morphism of principal bundles compatible with the Lie group morphism Up2q Ñ PUp2q [KN]. The map ApEq “ ApPE q Ñ ApP¯E q given by A ÞÑ A¯ :“ q˚ pAq induces Ñ ApP¯E q, and A P AASD pEq if and only an isomorphism of affine spaces Aa pEq ´» a ASD ¯ ¯ if A P A pPE q. Moreover the bundle SUpEq can be identified with the bundle P¯E ˆPUp2qSUp2q associated with P¯E and the action PUp2q ˆ SUp2q Ñ SUp2q induced by the conjugation action of Up2q on SUp2q. Using these identifications we see that Remark 1.1. The map A ÞÑ A¯ induces isomorphisms ASD ¯ L ¯ L pPE q ` ˘ , MASD pEq Ñ A ˘. Ba pEq Ñ ApPE q ` ¯ a P ˆ SUp2q Γ E PUp2q Γ P¯E ˆPUp2q SUp2q In particular the spaces Ba pEq, MASD pEq are intrinsically associated with the a PUp2q–bundle P¯E , so they are independent of the parameter connection a P ApDq up to canonical isomorphisms.

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In general the canonical group morphism ` ˘ ` ˘ Γ P¯E ˆPUp2q SUp2q Ñ Γ P¯E ˆPUp2q PUp2q “ AutpP¯E q is not surjective, so MASD pEq cannot be identified with the moduli space a ASD ¯ L pPE q MASD pP¯E q :“ A AutpP¯E q of ASD connections on P¯E modulo the gauge group AutpP¯E q of this principal bundle. Putting ¯ L BpP¯E q :“ ApPE q AutpP¯ q , E and denoting by μ2 the multiplicative group t˘1u, one can see that the natural maps Ba pEq Ñ BpP¯E q , MASD pEq Ñ MASD pP¯E q a identify the spaces BpP¯E q, MASD pP¯E q with the H 1 pM, μ2 q-quotients of Ba pEq, pEq respectively [Te2]. A cohomology class ρ P H 1 pM, μ2 q acts on Ba pEq MASD a by tensor product with the flat connection aρ associated with the representation π1 pM, x0 q Ñ μ2 defined by ρ. This shows in particular that MASD pEq, endowed a with its natural H 1 pM, μ2 q-symmetry, is a finer object (contains more information) than the classical moduli space MASD pP¯E q. As in [DK, Section 4.2.2] a connection A P Aa pEq will be called irreducible if its pEq˚ stabiliser GE,A reduces to the centre μ2 of GE . We denote by Ba pEq˚ , MASD a the open subspaces of irreducible orbits in Ba pEq, MASD pEq respectively. a A connection A P Aa pEq is reducible (i.e. not irreducible) if and only if E admits an A-parallel line subbundle L. If this is the case, let bA , bK A be the induced K Hermitian connections on L, L respectively. The stabiliser GE,A is either isomorphic to SUp2q when there exists a unitary isomorphism L ÞÑ LK mapping bA onto 1 bK A , or isomorphic to S when such a unitary isomorphism does not exist. In the latter case we will say that A (and its orbit rAs) is an abelian reduction. Using the obvious isomorphism LK » D b L_ we obtain the following Remark 1.2. Let A P Aa pEq and L be an A-parallel line subbundle. Then " SUp2q if pL, bA q » pD b L_ , a b b_A q GE,A » . if pL, bA q fi pD b L_ , a b b_A q S1 If c1 pEq R 2H 2 pM, Zq then any reduction in Ba pEq is abelian. Fix a non-trivial class ρ P H 1 pM, μ2 q. An irreducible connection A P A˚a pEq will be called ρ–twisted reducible ([KM], [Te4]) if one of the following equivalent conditions is satisfied (1) rA b aρ s “ rAs. (2) The pull-back of A to the double cover defined by ρ is reducible. ¯ (3) The Euclidean 3–bundle supEq has an A-parallel line subbundle θ with w1 pθq “ ρ. A (and its orbit rAs) will be called twisted reducible (or a twisted reduction) if there exists a non-trivial class ρ P H 1 pM, μ2 q such that A is ρ–twisted reducible. Definition 1.3. A connection A P AASD pEq will be called regular if H2A “ t0u, a 2 where HA denotes the second harmonic space of the elliptic complex d

0 Ñ A0 psupEqq ´´AÑ A1 psupEqq Ñ A2` psupEqq Ñ 0

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associated with A. We shall say that MASD pEq is a regular moduli space if A is a pEq. regular for any rAs P MASD a Therefore, if H2A “ t0u, A will be called regular even when it is reducible. A connection A P AASD pEq (and its gauge class) is called locally reducible if it a is reducible on a non-empty open set. The presence of locally reducible points in the moduli space is a source of difficulties, because the standard transversality argument (see [DK, Sections 4.3.3 – 4.3.5]) does not apply at these points, so one cannot obtain regular moduli spaces using generic perturbations of the metric and the Sard-Smale theorem. Note that A is locally reducible if and only it is reducible, twisted reducible, or projectively flat, i.e. FA0 “ 0. For a connection A P AASD pEq, Chern-Weil theory gives the well-known idena tity 1 }F 0 }2 2 , (1.1) 4c2 pEq ´ c21 pEq “ 2π 2 A L which shows that Remark 1.4. With the notations above one has: pEq “ H. (1) If 4c2 pEq ´ c21 pEq ă 0 then MASD a pEq can` be identified with ˘the moduli (2) If 4c2 pEq ´ c21 pEq “ 0 then MASD a space of flat connections on P¯E modulo Γ M, P¯E ˆPUp2q SUp2q . (3) If 4c2 pEq ´ c21 pEq ą 0 then AASD pEq contains no projectively flat conneca pEq is locally reducible if and tion, in particular a connection A P AASD a only if it is either reducible or twisted reducible. The first implication is known as the topological constraint for the existence of projectively ASD connections on a given 2–bundle. 2. An instanton moduli space on definite 4-manifolds with b1 “ 1 Suppose now that b` pM q “ 0. By the first Donaldson theorem [D2] the intersection form qM of M is standard, i.e., putting b :“ b2 pM q, there exists a basis pe1 , . . . , eb q of H 2 pM, Zq{Tors with ei ¨ ej “ ´δij for 1 ď i, j ď b. Such a basis will be called standard. By the theorem of Hirzebruch-Hopf [HH] the Stiefel-Whitney class w2 pM q can be lifted to H 2 pM, Zq. Such a lift c is a characteristic element for the unimodular lattice pH 2 pM, Zq{Tors, qM q, i.e. it satisfies the congruence (2.1)

c ¨ h ” h ¨ h (mod 2) @h P H 2 pM, Zq .

Expanding the congruence class c¯ :“ c ` Tors P H 2 pM, Zq{Tors in the basis ř pe1 , . . . , eb q, we obtain a decomposition of the form c¯ “ bi“1 ai ei , and (2.1) is equivalent to the condition ai P 2Z ` 1 for 1 ď i ď b. This shows that Remark 2.1. The Stiefel-Whitney class w2 pM q has integral lifts c P H 2 pM, Zq with c2 “ ´b. For any such class there exists a standard basis pe1 , . . . , eb q of ř H 2 pM, Zq{Tors, unique up to order, such that c¯ “ bi“1 ei . We fix a lift c P H 2 pM, Zq of w2 pM q with c2 “ ´b, and let pE, hq be a Hermitian bundle of rank 2 on M with with c1 pEq “ c and c2 pEq “ 0. The invariants of the associated PUp2q–bundle are: p1 pP¯E q “ c21 pEq ´ 4c2 pEq “ ´b, w2 pP¯E q “ w2 pM q, so the isomorphism class of this bundle is intrinsically associated with M . Taking into account Remark 1.1, this also shows that the corresponding moduli space

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MASD pEq is intrinsically associated with the Riemannian 4-manifold pM, gq, up to a isomorphism. From now on we will consider the special case when b1 pM q “ 1, and b` pM q “ 0, i.e. the case when the oriented 4-manifold M has the same homological invariants as a class VII surface [BHPV]. Under these assumptions the expected dimension pEq is 2b. This moduli space plays a fundamental role in of the moduli space MASD a our strategy to prove the existence of curves on class VII surfaces using Donaldson theory. 2.1. Compactness and regularity in the Riemannian framework. We start with the first properties of this moduli space in the general framework of Riemannian 4-manifolds, and we will specialise later to class VII surfaces endowed with Gauduchon metrics. The following simple remark concerns the compactness pEq: of MASD a Remark 2.2. Let pM, gq be a Riemannian 4-manifold with b1 pM q “ 1 and negative definite intersection form, c be an integral lift of w2 pM q with c2 “ ´b2 pM q, and let pE, hq be a Hermitian bundle of rank 2 on M with with c1 pEq “ c. Suppose that b2 pM q ď 3. Then pEq “ H. (1) If c2 pEq ă 0, then MASD a pEq is compact. (2) If c2 pEq “ 0, then MASD a Proof. (1) follows using the topological constraint for the existence of projectively ASD connections (see Remark 1.4), and (2) follows from (1) using the Donaldson-Uhlenbeck compactness theorem [DK, section 4.4.1], taking into account that the lower strata of the Donaldson-Uhlenbeck compactification of M :“ pEq are empty.  MASD a On the other hand, for a large class of Riemannian 4-manifolds with b1 pM q “ 1 and negative definite intersection form, the conclusion of Remark 2.2 (in particular the compactness of M) holds without any restriction on b2 . This is a simple consequence of the Weitzenb¨ock formula for Spinc -Dirac operators which, for metrics with positive scalar curvature, allows us to strengthen the topological constraint for the existence of projectively ASD connections: Remark 2.3. Let pM, gq be a Riemannian 4-manifold with b1 pM q “ 1 and negative definite intersection form, c be an integral lift of w2 pM q with c2 “ ´b2 pM q, and let pE, hq be a Hermitian bundle of rank 2 on M with with c1 pEq “ c. Suppose that the scalar curvature sg is non-negative and strictly positive at a point. Then (1) If c2 pEq ă 0, then MASD pEq “ H. a pEq is compact. (2) If c2 pEq “ 0, then MASD a Proof. (1) For a general oriented, Riemannian 4-manifold pM, gq, let Pg be the principal SOp4q–bundle of orientation compatible orthonormal frames, τ : Q Ñ Pg be a Spinc -structure on M , Σ˘ be its spinor bundles, B be a connection on Q lifting the Levi-Civita connection, B ˘ the induced connections on Σ˘ , and b be the induced unitary connection on the Hermitian line bundle detpQq “ detpΣ˘ q. For a Hermitian connection A on E let { b,A : A0 pΣ` b Eq Ñ A0 pΣ´ b Eq D

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be the coupled Dirac operator associated with the pair pb, Aq. The index of this operator is ˘2 ` ˘ ˘ 1 `` { b,A q “ indexpD c1 pdetpQqq ` c1 pEq ` c1 pEq2 ´ 4c2 pEq ´ σpM q . 4 The Weitzenb¨ock formula for coupled Spinc -Dirac operators reads [OkTe, Proposition 2.1]: ` 1 ˘ s { b,A “ ∇˚BbA ∇BbA ` pγ b idEndpEq q p Fb idE ` FA q` ` g idΣ` bE , { ˚b,AD (2.2) D 2 4 2 ` where γ : Λ` Ñ supΣ q is the isomorphism induced by the Clifford multiplication. Now we specialise these general formulae to our case. Since c is an integral lift of w2 pM q we can choose Q such that c1 pdetpQqq “ ´c. We fix a unitary isomorj» phism detpQq ´´Ñ D_ , and we choose the Spinc p4q-connection B on Q such that ˚ _ detpBq “ j pa q. With these choices have 12 Fb idE ` FA “ FA0 for any A P Aa pEq. Therefore for any A P AASD pEq the second term on the right in (2.2) vanishes and, a { b,A q “ t0u, so indexpD { b,A q ď 0. But using our assumption on sg , we obtain kerpD { q “ ´c pEq. in our special case the index formula gives indexpDb,A 2 (2) This follows from (1) and the Donaldson-Uhlenbeck compactness theorem, as in the proof of Remark 2.2.  Furthermore, one can also prove that, for an interesting class of Riemannian metrics, our moduli space M is both compact and regular. We recall that a Riemannian metric g on a closed manifold is called of positive type if its conformal class contains a metric of positive scalar curvature. Remark 2.4. Let pM, gq be a Riemannian 4-manifold with b1 pM q “ 1 and negative definite intersection form, c be an integral lift of w2 pM q with c2 “ ´b2 pM q, and let pE, hq be a Hermitian bundle of rank 2 on M with with c1 pEq “ c, c2 “ 0. pEq is compact and Suppose that g is an ASD metric of positive type. Then MASD a regular. Proof. The vanishing of the harmonic spaces H2A follows using the Weitzenb¨ock formula for supEq-valued self-dual 2-forms [FU, (6.26) p. 111].  There are many interesting examples of ASD metrics of positive type on 4manifolds with b1 “ 1 and negative definite intersection form. Fujiki and Pontecorvo proved the existence of such metrics on special classes of class VII surfaces [FP1], [FP2]. On the other hand, our programme for proving existence of curves pEq on class VII surfaces starts with the following results: The moduli space MASD a (associated with a bundle E as in Remark 2.4) is compact on any class VII surface X endowed with a Gauduchon metric g; this moduli space is also regular if g is chosen such that degg pKX q ă 0. Moreover, on any minimal class VII surface with b2 ą 0 such metric exists. We will recall these results in section 2.2. 2.2. Compactness and regularity in the complex geometric framework. Let X be a class VII surface, i.e. a compact, connected complex surface with b1 pXq “ 1, kodpXq “ ´8 (see section 4), and let K be its canonical line bundle. For such a surface one has h1 pOX q “ 1, pg pXq “ 0, so χpOX q “ 0, and the Noether formula gives c1 pXq2 ` c2 pXq “ 0. On the other hand, since the Euler characteristic of X is b2 pXq, one obtains ´c1 pXq2 “ c2 pXq “ b2 pXq. This shows

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that the Chern class c :“ c1 pKq “ ´c1 pXq is an integral lift of w2 pXq with the property c2 “ ´b2 pXq. Let pE, hq be a Hermitian bundle of rank 2 with a fixed identification detpEq “ K, where K is the underlying differentiable line bundle of K, let a be the Chern connection of the pair pK, detphqq, and let g be a Gauduchon metric on X. The following compactness result holds in full generality [Te5]: Theorem 2.5. Let pX, gq be a class VII surface endowed with a Gauduchon metric, and let pE, hq be a Hermitian bundle of rank 2 with an identification detpEq “ K. Then pEq is empty. (1) If c2 pEq ă 0, then MASD a pEq is compact. (2) If c2 pEq “ 0, then MASD a The proof uses an argument (found by N. Buchdahl) which is similar to the one used in Remark 2.3, but makes use of the Dolbeault Dirac operator B¯A ` B¯˚ : A00 pEq ‘ A02 pEq Ñ A01 pEq A

instead of the Spinc -coupled Dirac operator. Denoting by EA the holomorphic structure on E associated with the Dolbeault operator B¯A , the condition c2 pEq ă 0 ˚ q ą 0, which implies that H 0 pEA q ‰ 0 or H 2 pEA q ‰ 0. In gives indexpB¯A ` B¯A both cases the locally free sheaf associated with EA (which will be denoted by the same symbol) admits a coherent subsheaf of rank 1, i.e. it is filtrable (see [Te2], [Te5] and section 4.2 in this article). But using a standard basis of H 2 pX, Zq{Tors associated to c “ c1 pKq (as in Remark 2.1), it is easy to prove that there exists no filtrable holomorphic 2–bundle E on X with c1 pEq “ c1 pKq and c2 pEq ă 0. For the regularity of instanton moduli spaces on complex surfaces we have the following general criterion: Proposition 2.6. Let pX, gq be a compact Gauduchon surface, pE, hq be a Hermitian bundle of rank r, and let a be a Hermitian connection of type p1, 1q on pEq, and E be the polystable holomorphic bundle pdetpEq, detphqq. Let A P AASD a associated with the Dolbeault operator B¯A . (1) If degg pKq ă 0, then H 2 pEnd0 pEqq “ 0. (2) If b1 pXq is odd, then the canonical morphism H2A Ñ H 2 pEnd0 pEqq is an isomorphism. Proof. (1) One has h2 pEnd0 pEqq “ h0 pEnd0 pEq b Kq by Serre duality. Since A is projectively ASD on pE, hq, the metric induced by h on End0 pEq is HermiteEinstein with vanishing Einstein constant. Using a Hermite-Einstein metric on K, we obtain a Hermite-Einstein metric with negative Einstein constant on End0 pEq b K, so h0 pEnd0 pEq b Kq “ 0 by a well known vanishing theorem [Ko, Theorem 1.9 p. 52]. (2) is proved by comparing the trace-free deformation elliptic complexes of E and A [Te5, Corollary 1.20].  Note that, if pX, gq is K¨ahler and A is reducible, then the canonical morphism H2A Ñ H 2 pEnd0 pEqq is not injective, because its kernel contains H0A b ωg . Therefore (2) does not hold for surfaces with b1 even. Proposition 2.7. Let X be a minimal class VII surface with b2 pXq ą 0. Then X admits Gauduchon metrics g for which degg pKq ă 0. For such a Gauduchon pEq associated with any triple pE, h, aq on X is metric the moduli space MASD a regular.

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The first statement is Lemma 2.3 in [Te5], which is a consequence of the results of [Bu2]. Using the terminology of [Te3] this statement can be reformulated as follows: for a minimal class VII surface with b2 pXq ą 0 the Bott-Chern class cBC 1 pKq cannot be pseudo-effective. The second statement follows from the first and Proposition 2.6. 3. The circles of reductions 3.1. Parameterising the reduction locus. In this section we suppose for simplicity that H1 pM, Zq » Z, which implies TorspH 2 pM, Zqq “ 0. This simplifying assumption does not play an important role in the proofs of the main results, but will allow us to avoid technical complications and emphasise the geometric ideas. Let I be the index set I :“ t1, . . . , bu, ℘pIq be its power set, and PpIq be the quotient of ℘pIq by the involution I ÞÑ I¯ :“ IzI. Therefore PpIq is the set of unordered two-term partitions of the index set I. Proposition 3.1. Let pM, gq be a Riemannian 4-manifold with H1 pM, Zq » Z, b2 pM q ą 0 and negative definite intersection form. Put b :“ b2 pM q, let c be an integral lift of w2 pM q with c2 “ ´b, and let pe1 , . . . , eb q be a standard basis of řb H 2 pM, Zq such that c “ i“1 ei . Let pE, hq be a Hermitian 2–bundle on M with c1 pEq “ c, c2 pEq “ 0. Then (1) A line bundle L on M is isomorphic to a line subbundle of E if and only ř if there exists I P ℘pIq such that c1 pLq “ eI :“ iPI ei . (2) The set of isomorphism classes of (unordered) orthogonal line bundle decompositions of E can be identified with PpIq . pEq is abelian, and (3) Any reduction in M :“ MASD a š the reduction locus R :“ MzM˚ decomposes as a disjoint union R “ λPPpIq C λ , where C λ :“ trAs P M| E has an A-parallel line subbundle L with c1 pLq P λu . (4) Let λ P PpIq. The choice of a representative I P ℘pIq gives the structure of a iH 1 pM, Rq{2πiH 1 pM, Zq-torsor on C λ , in particular C λ is a circle. Proof. (1) Let L be a line bundle which is isomorphic to a line subbundle of E. Expanding c1 pLq in the basis pe1 , . . . , eb q, and using the equation c1 pLq pc ´ c1 pLqq “ c1 pLq c1 pD b L_ q “ c2 pEq “ 0 , it follows easily that there exists I Ă I such that c1 pLq “ eI . Conversely, if c1 pLq “ eI , then L ‘ pD b L_ q » E, because the two bundles have the same Chern classes. Therefore L is isomorphic to a line subbundle of E . (2) follows from (1), and (3) follows from Remark 1.2. For (4) let L be a Hermitian line bundle with c1 pLq “ cI . Put  ( 1 Aa pLq :“ b P ApLq| Fb “ Fa ´ πiχ , 2 where χ is the harmonic representative of the de Rham class p2eI ´ cqDR . Since b` pM q “ 0, this form is ASD. Denoting by H1 pM, Rq the space of real harmonic 1-forms on M and by H1 pM, Zq Ă H1 pM, Rq the subgroup of harmonic forms representing integer cohomology classes, we see that the moduli space L Ma pLq :“ Aa pLq 8 C pM, S1 q

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has an obvious structure of an iH1 pM, Rq{2πiH1 pM, Zq-torsor, which is induced by the canonical iA1 pM q-action on ApLq. On the other hand the map u

L Ma pLq Q rbs ´´Ñ rb ‘ pa b b_ qs P C λ



is a homeomorphism.

3.2. The blown up moduli space and the orientation of the moduli space. As in the proof of Proposition 3.1, fix I P ℘pIq, and let L be a Hermitian line bundle on M with c1 pLq “ eI . For a connection b P Aa pLq, the connection bb2 b a_ P ApLb2 b Dq is ASD, so one obtains an elliptic complex over C d`b2

dbb2 ba_

ba_

0 Ñ A0 pLb2 b Dq ´´´´´´Ñ A1 pLb2 b Dq ´´´´´´Ñ A2` pLb2 b Dq Ñ 0 , pCb`b2 ba_ q b

whose index is (under our assumptions) c1 pLb2 b Dq2 “ peI ´ eI¯q2 “ ´b. It is easy to see that, since Lb2 b D is not trivial, H0 pCb`b2 ba_ q “ t0u. On the other hand, using the condition b` pM q “ 0, we have a canonical identification H2b‘pabb_ q “ H2 pCb`b2 ba_ q . Therefore Remark 3.2. Under the assumptions of Proposition 3.1, let I P ℘pIq be ¯ such that all reductions in C tI,Iu are regular. Then for any b P Aa pLq one has 1 ` dimC pH pCbb2 ba_ qq “ b. ¯

Therefore, for a regular circle of reductions C tI,Iu , the disjoint union ď H1 pCb`b2 ba_ q bPAa pLq

is a complex vector bundle of rank b over Aa pLq. The sphere bundle of this vector bundle with respect to the L2 –norm is C 8 pM, S1 q-invariant; factorising this sphere bundle by C 8 pM, S1 q, one obtains a projective bundle π L : Pa pLq Ñ Ma pLq, whose fibre Prbs over a point rbs P Ma pLq is identified with the pb ´ 1q-dimensional projective space PpH1 pCb`b2 ba_ qq. Let jL : Ma pLq Ñ Ma pD b L_ q , JL : Pa pLq Ñ Pa pD b L_ q be the diffeomorphisms defined by b ÞÑ a b b_ , α Ñ α ¯ respectively. We obtain a commutative diagram Pa pLq

(3.1)

πL ? Ma pLq uL

JL

- Pa pD b L_ q

jL

πDbL_ ? - Ma pD b L_ q

-

 uDbL_ Cλ in which the two vertical arrows are projective bundles, and all the others are diffeomorphisms. Identifying any point p P Pa pLq with its image in Pa pD b L_ q via JL , we obtain a fibre bundle π λ : P λ Ñ C λ , which depends only on the unordered ¯ and comes with an obvious uL -lifting identification pair λ “ tI, Iu, UL : Pa pLq Ñ P λ .

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We refer to [Te5, section 1.4.2] for the following Remark 3.3. Under the assumptions of Proposition 3.1, let I P ℘pIq be such ¯ that all reductions in C tI,Iu are regular. The set xASD M pEqλ :“ pMASD pEqzC λ q Y P λ a a xASD has a natural topology which makes the obvious projection M pEqλ Ñ MASD pEq a a λ ASD x continuous and proper; any sufficiently small neighbourhood of P in Ma pEqλ has a natural structure of a manifold with boundary, whose boundary is P λ . xASD Endowed with these structures, M pEqλ is called the blown-up of MASD pEq a a λ at the reduction circle C . If all reductions in the moduli space are regular we obtain xASD in a similar way the blown up M pEq of MASD pEq at the whole reduction locus; a a its underlying set is ˘ ď ˘ď` ď ˘ ` ASD xa pEq z M Cλ Pλ . λPPpIq

λPPpIq

An important property of the blown up moduli space concerns the extensibility of the Donaldson cohomology classes. Let μ be the morphism H˚ pM, Zq Ñ H 4´˚ pBa˚ pEq, Qq defined in [DK, section 5.1.2] and recall that, in general, the restriction of a class μpξq to MASD pEq˚ does not extend to MASD pEq [DK, seca a tion 5.1.4]; however it is easy to see that any such restriction does extend in a xASD canonical way to the blown up moduli space M pEq. Moreover, the pull-back a ˚ 4´˚ UL pμpξqq P H pPa pLq, Qq of such a class can be computed explicitly [Te4, Corollary 2.6], and the result is: Remark 3.4. Suppose that all reductions in MASD pEq are regular, I P ℘pIq, a and let L be a Hermitian line bundle with c1 pLq “ eI . Let rbs P Ma pLq, and Wrbs be the positive generator of H 2 pPrbs , Zq. Via the K¨ unneth decomposition H ˚ pPa pLq, Qq “ H ˚ pMa pLq, Qq b H ˚ pPrbs , Qq , and the obvious identification H1 pM, Qq ´δ» ´Ñ H 1 pMa pLq, Q, the following holds: For any pγ, uq P H1 pM, Zq ˆ H2 pM, Zq one has 1 (3.2) UL˚ pμpγqq “ δpγq b Wrbs , UL˚ pμpuqq “ xeI ´ eI¯, uy Wrbs in H ˚ pPa pLq, Qq . 2 Supposing that the whole moduli space MASD pEq is regular, the blown up a xASD moduli space M pEq will be a manifold with boundary, whose boundary is a Ť λ P . This manifold with boundary is orientable. More precisely, fixλPPpIq 1 ing an orientation O of the line H pM, Rq, we get induced orientations of the iH1 pM, Rq{2πiH1 pM, Zq-torsors Ma pLq defined above. Using the complex orientations of the fibres of π L , we obtain a canonical orientation OL of the projective fibre bundle Pa pLq. Let λ P PpIq, and I P λ. Choosing L with c1 pLq “ eI , we see that the choice of a representative I of λ gives an orientation OI of the boundary component P λ , namely the image of OL via UL . One can prove that OI is the xASD p I of M pEq, which depends only boundary orientation of a global orientation O a on O and I. This orientation coincides (up to universal sign) with the orientation associated with the lift eI ´eI¯ of w2 pXq as explained in [DK, p. 283]. In particular, for two parts I, J P ℘pIq, one has the comparison formula pJ O

pI . “ p´1q|J|´|I| O

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xASD p H . With this definition one has We will endow M pEq with the orientation O a Remark 3.5. Let I P λ, and L be a Hermitian line bundle with c1 pLq “ eI . The pull-back via UL of the boundary orientation of the boundary component xASD P λ P π0 pB M pEqq compares to OL according to the parity of |I|. a 3.3. The position of the circles of reductions in the connected components of the moduli space. In this section we will prove Theorem 3.6. Let pM, gq be a Riemannian 4-manifold with negative definite intersection form, b2 pM q ą 0 and H1 pM, Zq » Z. Put b :“ b2 pM q, let c be an integral lift of w2 pM q with c2 “ ´b. Let pE, hq be a Hermitian 2–bundle on M with pEq is compact and regular. Then c1 pEq “ c, c2 pEq “ 0, and suppose that MASD a pEq belong to the same connected component. all reduction circles in MASD a ř Proof. Let pe1 , . . . , eb q be a standard basis of H 2 pM, Zq such that c “ bi“1 ei (see Remark 2.1). Let λ0 :“ tH, Iu be the trivial two term partition of I, M0 be x0 be pEq which contains the circle C λ0 , and M the connected component of MASD a the oriented compact manifold with boundary obtained from M0 by blowing up the reduction circles which are contained in this component (see section 3.2). For every λ P PpIq put " 1 if Cλ Ă M0 nλ :“ . 0 if Cλ X M0 “ H The statement of Theorem 3.6 is equivalent to the claim nλ “ 1 @λ P PpIq .

(3.3)

Since the claim of the theorem is obvious for b “ 1 (because in this case PpIq “ tλ0 u), we assume b ě 2. Let γO be the generator of H1 pX, Zq which is compatible with the fixed orientation O of the line H1 pX, Rq, and let ph1 , . . . , hb q be the basis of H2 pX, Zq which is Poincar´e dual to pe1 , . . . , eb q (i.e. one has xei , hj y “ ´δij ). x0 is homologically trivial in M x0 , one has Since the oriented boundary B M b A ÿ ` b ˘ “ ‰E x0 “ 0 . (3.4) @pj1 , . . . , jb q P Nb with ji “ b ´ 2, μpγO q Y Y μphi qji , B M i“1

i“1

¯ P PpIq: Using Remarks 3.4, 3.5 we obtain for any λ “ tI, Iu E A ř ř b (3.5) 2b´2 μpγO q Y p Y μphi qji q, rP λ s “ p´1q|I|` iPI ji “ p´1q iPI pji `1q . i“1

This shows that the left hand side of (3.4) depends only on the Z2 -reduction of pj1 , . . . , jb q. Denote by r¨s the mod 2 reduction morphisms Z Ñ Z2 , Zb Ñ Zb2 . For an element u P Zb2 , denote by Iu P ℘pIq the associated part Iu :“ ti P I| ui “ r1su, and put 1 “ p1, . . . , 1q P Zb . With these notations we have: 

b b ÿ ÿ ˇ (  ˇ ( ji “ b ´ 2 “ rksˇ k P tN˚ ub with ki “ 2b ´ 2 “ prj ` 1sˇ j P Nb with i“1

ˇ  “ v P Zb2 ˇ |I¯v | ď b ´ 2,

i“1 b ÿ

( vi “ r0s .

i“1

To check the inclusion of the third set in the second, let v be an element of the third ř set. We show that there exists k P tN˚ ub such that rks “ v and bi“1 ki “ 2b ´ 2. We have |Iv | P 2N˚ . Choose ki :“ 2 for i P I¯v ; it suffices to note that there exists a

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family of |Iv | odd positive integers whose sum is 2b ´ 2 ´ 2|I¯v | “ |Iv | ` p|Iv | ´ 2q. The set b ÿ ˇ (  vi “ r0s v P Zb2 ˇ |I¯v | ď b ´ 2, i“1

can be identified with the set of parts  ( L Ă I| L ‰ H, |L| P 2N Ă ℘pIq via the map which ř associates to a subset L Ă I its Z2 -valued characteristic function. The Z2 -class iPI rji ` 1s of the exponent on the right hand side of (3.5) can be written as r|L X I|s, where L Ă I is the non-empty subset associated with rj ` 1s. Putting ℘ev pIq :“ tL Ă I| |L| P 2Nu we see that the system of equations (3.4) is equivalent to ÿ ev (3.6) p´1q|LXI| ntI,Iu ¯ “ 0 for L P ℘ pIqztHu . ¯ tI,IuPPpIq

Note that (3.7)

ÿ

p´1q|IXL| “ 0 @L P ℘ev pIqztHu .

¯ tI,IuPPpIq

This follows from Claim 1: For any L P ℘ev pIqztHu one has ˇ ˇ ˇ ˇ ˇ (ˇ ¯ P PpIqˇ |I X L| P 2Nuˇ “ ˇ tI, Iu ¯ P PpIqˇ |I X L| P 2N ` 1 ˇ “ 2b´2 . ˇ tI, Iu To prove the claim note that the map ¯ P ℘pLq ˆ ℘pLq ¯ , ℘pIq Q I ÞÑ pI X L, I X Lq is bijective for any subset L Ă I. On the other hand, for any non-empty set L, one has 1 |℘ev pLq| “ |℘odd pLq| “ |℘pLq| . 2 This shows that, for L P ℘ev pIqztHu, one has ˇ ˇ ˇ ¯ “ 2b´1 , ˇ I P ℘pIqˇ |I X L| P 2Nuˇ “ 1 |℘pLq| |℘pLq| 2 which proves the claim. Formula (3.7) shows that the subspace S Ă RPpIq of real solutions of the linear system (3.6) contains the line C of constant maps PpIq Ñ R. Taking into account that nλ0 “ 1 (by the definition of M0 ) we see that, in order to prove (3.3) it suffices to show that dimpSq “ 1, i.e. that there exists a linear hyperplane H Ă RPpIq such that S X H “ t0u. This follows from Claim 2: The linear system ÿ ev (3.8) p´1q|IXL| xtI,Iu ¯ “ 0 for L P ℘ pIq ¯ tI,IuPPpIq

(which is a linear system with 2b´1 equations for 2b´1 unknowns) is non-degenerate. This claim is proved noting that the matrix of the system (3.8) is a Hadamard

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matrix, i.e., a matrix with entries ˘1, and whose rows are orthogonal. Such a matrix is obviously nonsingular. In order to check the required orthogonality condition, let L, M P ℘ev pIq with L ‰ M . The inner product of the corresponding rows is ÿ ÿ p´1q|IXL| p´1q|IXM | “ p´1q|IXL|`|IXM | “ ¯ tI,IuPPpIq

“

ÿ

p´1q

¯ tI,IuPPpIq

|IXpLYM q|`|IXpLXM q|

¯ tI,IuPPpIq

ÿ

“

p´1q|IXpLYM q|´|IXpLXM q| “

¯ tI,IuPPpIq

“

ÿ

p´1q|IXppLYM qzpLXM qq| .

¯ tI,IuPPpIq

This sum vanishes by (3.7) because, since L ‰ M , we have pL Y M qzpL X M q ‰ H.  Theorem 3.6 can be generalised for general definite 4-manifolds with b1 pM q “ 1. In this more general case put Tors :“ TorspH 2 pX, Zqq; we obtain a standard řb basis pe1 , . . . , eb q in H 2 pM, Zq{Tors with c¯ “ i“1 ei , and one can prove (as in Proposition 3.1) that a Hermitian line bundle L is isomorphic with a subbundle of E if and only if there exists I P ℘pIq such that c¯1 pLq “ eI . Therefore, we obtain an identification between the set DecpEq of isomorphism classes of (unordered) orthogonal line bundle decompositions of E and the quotient of the subset (3.9)

Hc pM q :“ tu P H 2 pM, Zq| DI P ℘pIq| u ¯ “ eI u Ă H 2 pM, Zq ι

c c ´ u. Therefore the circles of reductions in MASD pEq are by the involution u ÞÑ a parameterised by classes λ P Hc pM q{xιc y. The obvious map Hc pM q Q l ÞÑ I P ℘pIq defined by the relation ¯l “ eI is a |Tors|-to-1 surjection, and it induces a |Tors|-to-1 surjection

Σ : DecpEq Ñ PpIq . The general form of Theorem 3.6 is: Theorem 3.7. Let pM, gq be a Riemannian 4-manifold with negative definite intersection form, b2 pM q ą 0 and b1 pM q “ 1. Put b :“ b2 pM q, let c be an integral lift of w2 pM q with c2 “ ´b. Let pE, hq be a Hermitian 2–bundle on M with c1 pEq “ c, c2 pEq “ 0, and suppose that MASD pEq is compact and regular. Let M0 be a a pEq which contains reductions. For any p P PpIq connected component of MASD a there exists a unique λ P Σ´1 ppq such that C λ Ă M0 . Therefore the non-empty fibres of the obvious map DecpEq Ñ π0 pMASD pEqq a are sections of Σ, so this map defines a “trivialisation” of the “bundle” DecpEq Ñ PpIq. Corollary 3.8. With the notations and under the assumptions of Theorem 3.7 the following holds: For any pair pp, p1 q P PpIq ˆ PpIq, the set 

( 1 pλ, λ1 q P Σ´1 ppq ˆ Σ´1 pp1 q| C λ , C λ belong to the same connected component

is the graph of a bijection Σ´1 ppq Ñ Σ´1 pp1 q.

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378

ANDREI TELEMAN

4. Curves on class VII surfaces 4.1. Class VII surfaces. General properties and conjectures. A class VII surface is a (compact, connected) complex surface X with b1 pXq “ 1 and kodpXq “ ´8. The second condition implies pg pXq “ 0. Therefore, using [BHPV, Theorem 2.7] one has b` pXq “ 0, so X belongs to the class of negative definite 4manifolds with b1 “ 1 considered in sections 2.1, 2.2, 3. As explained in section 2.2, the Chern class c1 pKq “ ´c1 pXq is an integral lift of w2 pXq whose square is ´b2 pXq, so Remark 2.1 applies and gives a standard basis pe1 , . . . , eb q of H 2 pX, Zq{Tors, řb unique up to order, with the property c¯1 pKq “ i“1 ei . This basis will be called the standard basis of H 2 pX, Zq{Tors. For a class VII surface X the canonical morphism H 1 pX, Zq bZ C “ H 1 pX, Cq Ñ H 1 pX, OX q is an isomorphism, so the choice of a generator χ P H 1 pX, Zq defines isomorphisms Z Ñ H 1 pX, Zq, C ´» Ñ H 1 pX, Cq “ H 1 pX, OX q. Denoting by Pic0 pXq the identity ˚ q of X, and using the standard component of the Picard group PicpXq “ H 1 pX, OX identification 1 L Pic0 pXq “ H pX, OX q , 2πiH 1 pX, Zq Ñ Pic0 pXq. The degree map associated with we obtain an isomorphism fχ : C˚ ´» any Gauduchon metric g on X is given on Pic0 pXq by the formula (4.1)

degg pfχ pζqq “ Cgχ log |ζ| ,

for a constant Cgχ P R˚ , which depends smoothly on g [LT, section 1.3]. Therefore the sign of Cgχ depends only on χ. We choose the generator χ P H 1 pX, Zq such that Cgχ ą 0; this choice gives canonical isomorphisms Z ´» Ñ H 1 pX, Zq, C ´» Ñ H 1 pX, Cq “ H 1 pX, OX q, C˚ Ñ Pic0 pXq . Let pu, tq P H 2 pX, Zq ˆ R. Using (4.1) we obtain geometric interpretations of the subsets Picu pXqt , Picu pXqąt , Picu pXqět Ă Picu pXq defined by the equation degg plq “ t, degg plq ą t, degg plq ě t respectively: the first subset is a circle, and the second (third) is punctured open (closed) disk. More generally, for a subset E Ă R we will put Picu pXqE :“ tl P Picu pXq| degg plq P Eu. Class VII surfaces with b2 “ 0 are completely classified [Te1], and the result is very simple: any class VII surface with b2 “ 0 is biholomorphic to either a Hopf surface, or to an Inoue surface. On the other hand, despite recent progress for surfaces with small b2 [Te2], [Te5], the classification of minimal class VII surfaces with b2 ą 0 remains a difficult, open problem. The class VIImin b2 ą0 of minimal class VII surfaces with b2 ą 0 contains the remarkable class of Kato surfaces. A Kato surface is a surface X P VIImin b2 ą0 which has a global spherical shell (GSS), i.e. an open submanifold U Ă X which is biholomorphic to a neighbourhood of S 3 in C2 , and such that XzU is connected [Ka1]–[Ka3], [Dl]. It is generally accepted that the Kato surfaces should be considered as the “known” surfaces in the class VIImin b2 ą0 . Indeed one has a clear construction method for all these surfaces [Ka2], and also a method to construct moduli spaces of Kato surfaces with a fixed configuration of curves [OeTo]. The following fundamental properties of Kato surfaces are easy consequences of Kato’s results:

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Proposition 4.1. Let X be a Kato surface, and b be its second Betti number. Then (1) X contains exactly b rational curves. (2) X is the degeneration of a holomorphic family of blown up primary Hopf ¯2 . surfaces, in particular X is diffeomorphic to pS 1 ˆ S 3 q#b P C Therefore all Kato surfaces with fixed second Betti number b are diffeomorphic to a very simple 4-manifold, so Kato surfaces are not very interesting from a differential topological point of view. The GSS conjecture stated by Nakamura in [Na2] claims Conjecture (the GSS conjecture). Any surface X P VIImin b2 ą0 is a Kato surface. If true this conjecture will complete the classification of class VII surfaces up to biholomorphism. An important result due to Dloussky-Oeljeklaus-Toma [DOT] states that the first property in Proposition 4.1 characterises Kato surfaces: Theorem 4.2. Any surface X P VIImin b2 ą0 with b2 pXq rational curves is a Kato surface. This theorem shows that the GSS conjecture is equivalent to Conjecture 1. Any surface X P VIImin b2 ą0 has b2 pXq rational curves. We have a weaker form of Conjecture 1 which, if true, will complete the classification of minimal class VII surfaces up to deformation equivalence. Before stating this conjecture we define: a cycle of k rational curves on a class VII surface X is a reduced effective divisor C of X such that either ř k “ 1 and C is a nodal rational curve, or k ě 2 and C can be written as a sum sPZk Cs of smooth rational curves such that either k “ 2 and C0 ¨ C1 “ 2 (two distinct intersection points), or k ě 3 and Cs ¨ Cs`r1s “ 1 for any s P Zk , Cs ¨ Ct “ 0 for t R ts ´ r1s, s, s ` r1su. More generally, a cycle on X is an effective divisor of X which is either a cycle of rational curves or a smooth elliptic curve. The possible curve configurations of Kato surfaces are [Na1]–[Na3]: Remark 4.3. Let X be a Kato surface, and D be the maximal reduced effective divisor of X. Then one of the following holds (1) D is a homologically trivial cycle of b2 pXq rational curves (generic Enoki surfaces). (2) D is the disjoint union of a homologically trivial cycle of b2 pXq rational curves and an elliptic curve E with E 2 “ ´b2 pXq (special Enoki surfaces). (3) D is a cycle of b2 pXq rational curves with D2 “ ´b2 pXq (half-Inoue surfaces). (4) D is the union of two disjoint cycle of rational curves, containing together b2 pXq curves (Inoue-Hirzebruch surfaces). (5) D is the union of a cycle of k rational curves (1 ď k ă b2 pXq) with k1 ě 1 trees of rational curves intersecting the cycle (intermediate Kato surfaces). Therefore any Kato surface X contains a cycle. The existence of a cycle is a very important property for minimal class VII surfaces because of the following important result of Nakamura ([Na1], [Na3]):

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380

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Theorem 4.4. Let X P VIImin b2 ą0 containing a cycle. Then X is a degeneration of a family of blown up primary Hopf surfaces. Therefore any surface X P VIImin b2 ą0 with a cycle belongs to the “known” deformation class of class VII surfaces, and this known deformation class contains the Kato surfaces and the blown up primary Hopf surfaces with the same second Betti number as X, the latter being generic. This result leads naturally to the following Conjecture 2. Any surface X P VIImin b2 ą0 has a cycle. If true, Conjecture 2 will complete the classification of minimal class VII surfaces up to deformation equivalence, in particular up to diffeomorphism. Conjectures 1 and 2 show that the main problem in the classification of minimal class VII surfaces with positive b2 is the existence of curves on these surfaces. Note that an important subclass of VIImin b2 ą0 has been completely classified: A min surface X P VIIb2 ą0 is called an Enoki surface if it contains a non-empty effective divisor D with D2 “ 0. If this is the case, then, by the main result of [E], X is a Kato surface, which is biholomorphic to a compactification of an affine line bundle over an elliptic curve, and the possible curve configurations are given by Remark 4.3 (1), (2). 4.2. The moduli spaces Mpst , Mst . First properties and the filtrable locus. Let pX, gq be a complex surface endowed with a Gauduchon metric g, let pE, hq be a Hermitian bundle of rank 2 on X, and let D be a holomorphic structure on the line bundle detpEq. A holomorphic structure E on E will be called Doriented if detpEq “ D. Let a be the Chern connection of the pair pD, detphqq. The non-K¨ ahlerian version of the Kobayashi-Hitchin correspondence [Bu1], [LY], [LT], [Te2] gives a homeomorphism kh pEq ´´ Ñ Mpst MASD a D pEq ,

where Mpst D pEq is the moduli space of D-oriented polystable holomorphic structures on E modulo the complex gauge group ΓpSLpEqq. The open subspace Mst D pEq Ă Mpst pEq of stable D-oriented holomorphic structures on E has a natural complex D space structure, and kh restricts to a real-analytic isomorphism ˚

kh pEq˚ ´´ ´Ñ Mst MASD a D pEq .

In general, in our non-K¨ ahlerian framework, the complex space structure of Mst D pEq pst might not extend to MD pEq. st The points of Mpst D pEq (MD pEq) correspond bijectively to isomorphism classes of polystable (respectively stable) holomorphic 2–bundles E on X with detpEq » D and c2 pEq “ c2 pEq. This remark allows us to describe these moduli spaces using the complex geometric formalism, which considers isomorphism classes of holomorphic bundles with fixed Chern classes, rather than gauge orbits of holomorphic structures on a fixed differentiable bundle. Let now pX, gq be a class VII surface endowed with a Gauduchon metric, and let pE, hq be Hermitian bundle of rank 2 on X with c2 “ 0 and detpEq “ K, where K denotes the underlying differentiable line bundle of the canonical line bundle K. The main objects used in our proofs are the moduli spaces st st Mpst :“ Mpst K pEq, M :“ MK pEq .

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Denoting by a the Chern connection of the pair pK, detphqq, these spaces are idenpEq, MASD pEq˚ via the Kobayashitified with the instanton moduli spaces MASD a a Hitchin correspondence. We know from section 2.2 that the moduli space Mpst is always compact, and that it is regular when degg pKq ă 0. We also know that, supposing b2 pXq ą 0, the moduli space Mpst contains |TorspH 2 pX, Zqq|2b2 pXq´1 circles of reductions. The complex space structure of Mst does not extend across any of these circles but, in the regular case, one has explicit holomorphic models for the corresponding ends of Mst [Te7], [Te8]. Moreover, also in the regular case, the results of section 3.3 apply, and Theorems 3.6, 3.7 give important information about the distribution of the circles of reductions in the connected components of Mpst . One of the major difficulties in the theory of holomorphic bundles on nonalgebraic surfaces is the appearance of non-filtrable bundles. We recall that a holomorphic bundle E of rank 2 on a complex surface is called filtrable if it (the associated locally free sheaf) admits a coherent subsheaf of rank 1, or, equivalently, if it admits an epimorphism onto a coherent subsheaf of rank 1. If this is the case, then there also exists an epimorphism p : E Ñ L with L torsion free of rank 1, and then kerppq is locally free, and L » L1 b IZ , where L1 is locally free of rank 1, and Z is a 0-dimensional locally complete intersection in X. If X is an algebraic surface, then any holomorphic 2–bundle E is filtrable, but this is no longer true on nonalgebraic surfaces. The appearance of non-filtrable bundles is a major difficulty in the theory of holomorphic bundles on non-algebraic surfaces; a non-filtrable bundle is stable with respect to any Gauduchon metric. On the other hand there exists no general classification method for non-filtrable bundles. The following simple result [Te5, Proposition 1.1] classifies all filtrable 2– bundles with c1 pEq “ c1 pKq and c2 pEq “ 0 on a class VII surface: Proposition 4.5. Let X be a class VII surface, and let E be a holomorphic bundle of rank 2 on X with c1 pEq “ c1 pKq and c2 pEq “ 0. Let p : E Ñ L be a sheaf epimorphism, where L is a torsion free coherent sheaf of rank 1 on X. Then (1) L and kerppq are locally free, so E becomes a line bundle extension p

0 Ñ K b L_ Ñ E ´Ñ L Ñ 0 .

(4.2)

(2) There exists I P ℘pIq such that c¯1 pLq “ eI . An extension of the form (4.2) with c¯1 pLq “ eI will be called extension of type I. The filtrable locus Φ Ă Mst decomposes as a union ď (4.3) Φ“ Mst I , IP℘pIq

where  ( st Mst I :“ rEs P M | E is the central term of an extension of type I . For I ‰ H the corresponding subset Mst I can be parameterised as follows: Let L be a Poincar´e line bundle on PicpXq ˆ X. For l P PicpXq we denote by Ll the line bundle on X defined by the restriction of L to tlu ˆ X. Therefore one has p q rLl s “ l for any l P PicpXq. Denoting by PicpXqˆX ´Ñ PicpXq, PicpXqˆX ´Ñ X 1 the two projections, and putting k :“ 2 degg pKX q we state

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382

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Proposition 4.6. Suppose that X P VIImin b2 ą0 . Let I P ℘pIqztHu, and u P eI . There exists e ą 0 such that the restriction of the coherent sheaf R1 p˚ pL _b2 b q ˚ pKqq to Picu pXqąk´e is locally free of rank |I|, and its fibre at a point l P _ Picu pXqąk´e is identified with H 1 pLl b2 b Kq via the canonical morphism. Proof. Put vg pXq :“ inftvolg pDq| D a non-empty effective divisor of Xu . By Bishop’s compactness theorem it follows that vg pXq ą 0. If T is a holomorphic line bundle on X with degg pT q ă vg pXq, then any non-trivial holomorphic section of T is nowhere vanishing, so h0 pT q ą 0 implies T » OX . _ Take e “ 12 vg pXq, and let l P Picu pXqąk . One has degg pLl b2 b Kq ă vg pXq _b2 so, taking into account that Ll b K is not trivial (because c1 pKq R 2H 2 pX, Zq), _ it follows h0 pLl b2 b Kq “ 0. On the other hand, using Serre duality we obtain _ h2 pLl b2 bKq “ h0 pLb2 l q. An important lemma proved by Nakamura [Na3, Lemma 0 1.1.3] states that, for a surface X P VIImin b2 ą0 one has h pT q “ 0 for any line bundle T with c1 pT q¨c1 pKq ă 0. Since we assumed I ‰ H, we obtain h0 pLb2 l q “ 0. Therefore _ _ _ h0 pLl b2 b Kq “ h2 pLl b2 b Kq “ 0, so h1 pLl b2 b Kq “ |I| by the Riemann-Roch theorem. The claim follows by Grauert’s locally freeness theorem.  Let Πuąk´e be the projectivisation of the holomorphic vector bundle associated with the (locally free) restriction of R1 p˚ pL _b2 b q ˚ pKqq to Picu pXqąk´e , and let Πuąk , Πuěk be the restrictions of Πuąk´e to Picu pXqąk , Picu pXqěk respectively. For _ any point l P Picu pXqąk´e and any η P H 1 pLl b2 b Kqzt0u, we denote by Eη the central term of the extension of Ll by K b L_ l defined by η. Put tΠuąk ust :“ trηs P Πuąk | Eη is stableu , and let φu : tΠuąk ust Ñ Mst be the map defined by φu prηsq :“ rEη s. This map is holomorphic, and one has ď impφu q , Mst I “ uPeI

so the map φI :“

ž uPeI

φu :

ž

tΠuąk ust Ñ Mst

uPeI

is a parametrisation of Mst I . Remark 4.7. Let I P ℘pIqztHu, and u P eI . tΠuąk ust is open in Πuąk and, for sufficiently small ε ą 0, the following holds: (1) tΠuąk ust contains the restriction Πupk,k`εq of the bundle Πuąk to the annulus Picu pXqpk,k`εq . (2) The restriction of φu to Πupk,k`εq is a holomorphic embedding, and has a continuous extension φ˜u : Πu Ñ Mpst which maps the boundrk,k`εq

ary of the manifold with boundary Πurk,k`εq onto the circle of reductions C tu,c1 pKq´uu . More precisely, for l P Picu pXqk , the fibre PpH 1 pLl b2 b Kqq Ă Πurk,k`εq is  ( mapped by φ˜u onto the singleton rpK b L_l q ‘ Ll s Ă C tu,c1 pKq´uu . The image _

Γuε :“ φu pΠupk,k`εq q

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is a smooth |I|-dimensional submanifold of Mst which is contained in Mst I , and comes with a natural structure of projective fibre bundle over Picu pXqpk,k`εq . Since the manifolds tΠuąk ust are not compact, it is very hard to describe exst plicitly the subsets impφu q, Mst I , their closures, and their Zariski closures in M . Moreover, it is not clear at all if the Zariski closure of impφu q is still |I|-dimensional. Note also that, in general φu is not necessarily injective, and for u ‰ v, the sets impφu q, impφv q are not necessarily disjoint. Now we turn to the case I “ H. The description of Mst H follows from the following Proposition 4.8. Suppose that X P VIImin b2 ą0 is not an Enoki surface. Let L be a holomorphic line bundle on X with c1 pLq P eH “ TorspH 2 pX, Zqq. Then one has h1 pL_b2 b Kq ď 1, and equality occurs if and only if Lb2 » OX . Proof. Indeed, the lemma of Nakamura used in the proof of Proposition 4.6 gives h0 pL_b2 b Kq “ 0. On the other hand, since X is not an Enoki surface, one has h2 pL_b2 b Kq “ h0 pLb2 q “ 0 except when Lb2 » OX . The claim follows by the Riemann-Roch theorem.  The set of square roots of rOX s can be identified with the subgroup H 1 pX, μ2 q of PicpXq, and has 2|Tors2 pH 2 pX, Zqq| elements, where ` ˘ Ñ H 2 pX, Zq Tors2 pH 2 pX, Zqq :“ ker H 2 pX, Zq ´2¨ [Te2]. For every u P Tors2 pH 2 pX, Zqq the component Picu pXq contains two square roots of rOX s. Therefore Proposition 4.8 states that, for any l P H 1 pX, μ2 q, one has a holomorphic 2–bundle Al , well defined up to isomorphism, which is a non-trivial extension of Ll by K b L_l . The bundle associated to l “ rOX s is just the essentially unique extension of OX by K, is called the canonical extension of X [Te3], and will be denoted by A. For any l P H 1 pX, μ2 q one has Al » A b Ll , so A is stable if and only if Al has this property. Therefore Remark 4.9. Suppose that X P VIImin b2 ą0 is not an Enoki surface. If A is st not stable, then MH “ H; if A is stable, then Mst H is the image of the map H 1 pX, μ2 q Q l ÞÑ rA b Ll s P Mst . Note that in general the map l ÞÑ rA b Ll s might not be injective. In the following section we will see that the stability of A is related to the existence of a cycle in X, so to the the validity of Conjecture 2 for X. 4.3. The canonical extension and its properties. The canonical extension (4.4)

i

p

0 0 A ´´Ñ OX Ñ 0 0 Ñ K ´´Ñ

plays an important role in our proofs. The following simple result [Te5] states that, if A can be written as a line bundle extension in a “different way”, then X has a cycle. This gives a method to prove Conjecture 2. Proposition 4.10. Suppose that the bundle A can be written as a line bundle extension (4.5)

p

i Ñ A ´Ñ L Ñ 0 0ÑT ´

with ipT q ‰ i0 pKq. Then X has a cycle.

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Proof. We sketch the proof for completeness. The composition p0 ˝ i is not trivial, because if it were, one would have ipT q Ă i0 pKq, which (taking into account that i is a bundle embedding) implies ipT q “ i0 pKq. The composition p0 ˝ i is not an isomorphism either, because if it were, the composition i ˝ pp0 ˝ iq´1 would define a right split of (4.4). Therefore impp0 ˝ iq is a proper locally free subsheaf of rank 1 of OX , so p0 ˝ i maps isomorphically T onto the ideal sheaf OX p´Cq of a nonempty effective divisor C Ă X. Restricting (4.5) to C one obtains an isomorphism KpCqC » OC , so the dualising sheaf ωC of C is trivial. When C is reduced, this implies already that C is a cycle; in the general case one can prove [Te5] that C contains a cycle.  Corollary 4.11. Let X be a class VII surface, and ( 4.4) be its canonical extension. Suppose that one of the following holds: (1) A can be written as a line bundle extension ( 4.5) with c1 pLq ‰ 0. (2) There exists a non-trivial holomorphic line bundle U with U b2 » OX such that A b U » A. Then X has a cycle. Proof. In the first case one obtains c1 pT q ‰ c1 pKq, so ipT q ‰ i0 pKq, because they are not even topologically isomorphic. In the second case one uses the isomorphism A b U » A to write A as a line bundle extension with kernel K b U fi K. In both cases Proposition 4.10 applies.  Corollary 4.12. Let pX, gq be a class VII surface endowed with a Gauduchon metric g such that degg pKq ă 0, and let ( 4.4) be its canonical extension. If A is not stable, then X has a cycle. Proof. If A is not stable, there will exist a destabilising epimorphism p : A Ñ L. We can assume that L is torsion free of rank 1. By Proposition 4.5, it follows p that L is locally free, and 0 Ñ kerppq ãÑ A ´Ñ L Ñ 0 is a line bundle extension. One has kerppq ‰ kerpp0 q because, since degg pKq ă 0, the subsheaf kerpp0 q » K does not destabilise A. Therefore X has a cycle by Proposition 4.10.  Let X P VIImin b2 ą0 . Choose a Gauduchon metric on X with degg pKX q ă 0. Such a metric always exists (see Proposition 2.7). Corollary 4.12 shows that, if X has no cycle, the subset Mst H will contain at least one point, namely the isomorphism class a :“ rAs. 4.4. A structure theorem for the moduli space. Existence of curves. Let l0 P Pic0 pXq be the non-trivial square root of rOX s, and let bl0 : Mpst Ñ Mpst be the associated involution. This involution acts as a rotation of angle π on any circle of reductions C λ Ă Mpst , so its fixed point locus B is contained in Mst . Via the Kobayashi-Hitchin correspondence B is identified with the set of ρ0 –twisted pEq˚ (see section 1.2), where ρ0 P H 1 pX, μ2 q is the image of a reductions in MASD a 1 generator of H pX, Zq in H 1 pX, Z2 q “ H 1 pX, μ2 q; the set B is finite [Te5]. The following structure theorem for the moduli space Mst gives (for suitable Gauduchon metrics) a decomposition of this space as the union of the “known” st M subspaces pMst I qI‰H , B with finitely many compact complex subspaces ˘ , Ş ` Ť Yk Ă st where Yk is of pure dimension k ą 0 and the intersection Yk is I‰H MI

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Zariski closed in Yk . The proof is based on Theorems 3.6, 3.7 obtained with gaugetheoretical methods. Theorem 4.13. Let X P VIImin b2 ą0 , and let g be a Gauduchon metric on X with degg pKq ă 0. Suppose b :“ b2 pXq ď 3. For a generic perturbation of g, the corresponding moduli space Mst decomposes as ` ď ˘ď` ď ˘ (4.6) Mst “ Mst Yk , I H‰IĂI

0ďkďb

where: (1) Yk Ă Mst is a (possibly empty) compact complex subspace of pure dimension k, and Y0 “ B, ˘ Ş`Ť st (2) For 0 ă k ď b, the intersection Yk is Zariski closed in Yk . I‰H MI We can prove now the result stated in the introduction: Theorem 4.14. Let X P VIImin b2 ą0 with b2 pXq ď 3, and which does not contain a cycle. For suitable Gauduchon metrics on X the moduli space Mst contains a compact complex subspace Y of positive pure dimension with an open subspace Ya such that a P Ya Ă Y , and all the points of Ya ztau correspond to non-filtrable bundles. Proof. Endow X with a Gauduchon metric for which the conclusion of Theorem 4.13 holds, and suppose that X does not have a cycle. Using Corollaries 4.12, 4.11 it follows that A is stable, and ` ď ˘ď a :“ rAs R Mst Y0 . I H‰IĂI

Therefore, by Theorem 4.13, there exists d with 0 ă d ď b such that a P Yd . Let Y be an irreducible component of Yd which contains a. Taking into account Theorem 4.13 (2), the decomposition (4.3) of the filtrable locus, and that Mst H is finite, it also follows that a has an open neighbourhood Ya in Y such that all the points of  Ya ztau correspond to non-filtrable bundles. Combining with the main result of [Te6] we obtain: Corollary 4.15. Any surface X P VIImin with 1 ď b2 pXq ď 3 has a cycle of curves. Proof. The existence of a compact subspace Y Ă Mst with the properties stated in Theorem 4.14 contradicts [Te6, Theorem 3.7].  4.5. The proof of the structure theorem. The cases b P t1, 2u have been studied in [Te2], [Te5]. The proof of Theorem 4.13 for b “ 3 is long and technical [Te9]. Since the goal of this article is to emphasise the geometric ideas and the role of Donaldson theory in non-K¨ahlerian geometry, we will explain here the steps and the ideas of proof under the following simplifying assumptions: (SA1) H1 pX, Zq » Z, (SA2) None of the classes ei ´ ej (1 ď i, j ď 3, i ‰ j) is represented by an irreducible curve C Ă X.

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The first simplifying assumption is equivalent to TorspH 2 pX, Zqq “ t0u; it allows us to identify the subset Hc1 pKq pXq Ă H 2 pX, Zq (see section 3) with the power set ℘pIq, and the set DecpEq of (unordered) orthogonal line bundle decompositions of E with the set PpIq of unordered two term partitions of I. Theorem 3.6 applies. Taking into account the identification Hc1 pKq pXq “ ℘pIq we will write ¯ PicI pXq, PicI pXqąk , ΠI , φI , ΓIε , C tI,Iu instead of PiceI pXq, PiceI pXqąk , ΠeI , φeI , ΓeεI , C teI ,eI¯u respectively. The role of the second simplifying assumption is cleared up by Proposition 4.16 stated after š the proof: it controls the domain of definition and the lack of injectivity of the map H‰IĂI φI . We will use the notation ℘k pIq :“ tI P ℘pIq| |I| “ ku . We explain the proof Theorem 4.13 for b “ 3 assuming the simplifying assumptions hold. The proof will make use of Proposition 4.16 and Theorem 4.17, whose proofs are purely complex geometric, and can be found in [Te9]. Proof. Step 1. Let NIpst be the connected component of Mpst containing the circle C tH,Iu , and let NI :“ NIpst X Mst . It is easy to see that NI is a connected component of Mst . Theorem 3.6 shows that NIpst contains all circles of reductions. Let Y3 be the (possibly empty) union of all connected components of Mpst which are different from NIpst . Y3 does not contain any reduction at all, so it is a compact complex manifold of dimension 3 contained in Mst , and one has ď (4.7) Mst “ NI Y3 , where Y3 is a compact 3-dimensional subspace of Mst , and NI is an irreducible 3-dimensional analytic set of Mst . Since NIpst contains all circles of reductions, it follows (by Remark 4.7 (2)) that NI contains all subsets of filtrables Mst I with . I ‰ H, in particular it contains Mst I Step 2. The map φI : ΠIąk Ñ NI is an injective holomorphic map between complex manifolds of the same dimension so, by [Kau, Proposition 46.A.1, p. 177], it is a holomorphic open embedding, in I particular its image Mst I “ impφ q is open. Using Theorem 4.17, it follows that the I complement ZI :“ NI zimpΠąk q is a hypersurface in the complex 3-fold NI , and Proposition 4.16 shows that the map ž ž φI : ΠIąk Ñ NI |I|“2

|I|“2

takes values in ZI . For I P ℘2 pIq let NI be the irreducible component of ZI which I contains Mst I “ impφ q, and let Y2 be the (possibly empty) union of the irreducible components of ZI which are different from Nt1,2u , Nt1,3u , Nt2,3u . The bidimensional analytic sets NI (I P ℘2 pIq), Y2 define 4-dimensional Borel-Moore homology classes rNI s (I P ℘2 pIq), rY2 s in Mst “ MASD pEq˚ . Identifying Mst “ MASD pEq˚ a a ASD xa pEq, we obtain 3-homology with the interior of the blown-up moduli space M ASD x classes δrNI s, δrY2 s P H3 pB Ma pEq, Zq. Since the Donaldson classes extend to xASD M pEq, we have a (4.8)

xμpγq, δrNI sy “ 0 for I P ℘2 pIq, xμpγq, δrY2 sy “ 0 ,

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where γ is a generator of H1 pX, Zq. These formulae give us accurate information about the incidence relations between the analytic sets NI , Y2 and the smooth families of extensions ΓJε (see Remark 4.7). More precisely, using [Te8, Theorem 2.2, Remark 3.8, Theorem 3.9], one can prove: 1

(a) For any I P ℘2 pIq there exists a unique I 1 P ℘1 pIq such that ΓIε Ă NI . (b) The map I ÞÑ I 1 is bijective. ¯1 ¯1 (c) NI also contains a section ΣIε of the P1 -bundle ΓIε , and this section is bl0 -invariant. (d) For any J P ℘1 pIq Y ℘2 pIq the analytic set Y2 intersects only finitely many fibres of the P|J|´1 –bundle ΓJε .

Figure 1. The irreducible component NI . A priori Y2 might be non-compact, because its closure in NIpst might intersect the reduction locus. However this situation can be avoided by perturbing the metric as follows. For a perturbation g˜ of g (in the space of Gauduchon metrics on X), ˜. Using (d) one can prove that, denote by Mst g ˜ the moduli space associated with g pst does for a generic small perturbation g˜ of g, the closure of Y2 X Mst g ˜ in Mg ˜ not intersect any circle of reductions, and is a compact complex subspace of Mst g ˜. Moreover, redefining all the objects above using g˜ instead of g, the resulting analytic set Y˜2 will coincide with this closure, so it will be compact. Therefore, replacing g by g˜ if necessary, we may suppose that Y2 is compact. Formula (4.7) and the definitions above give (4.9)

Mst “ Mst I

ď`

ď

NI

˘ď

Y2

ď

Y3 ,

IP℘2 pIq

where Yi is a (possibly empty) compact pure i-dimensional complex subspace of Mst for 2 ď i ď 3, and NI Ă Mst is a bidimensional, irreducible analytic set containing Mst I .

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Step 3. Let I P ℘2 pIq, and νI : NI Ñ NI be a normalisation of NI . The injective holomorphic map φI : ΠIąk Ñ NI has a lift fI : ΠIąk Ñ NI , which will also be injective. By [GR, Theorem, p. 166] it follows that UI :“ impfI q is open in NI , and fI induces an isomorphism ΠIąk Ñ UI . Theorem 4.17 shows that the complement ZI :“ NI zUI is an analytic set of pure dimension 1 in NI . Therefore, since νI is finite, ZI :“ Ť νI pZI q is an Ť analytic set of pure dimension 1 in NI , and one has ZI (see Fig. 1). Any irreducible component of ZI is of NI “ impφI q ZI “ Mst I one of the following types: 1

(T1) It coincides with the irreducible component NI 1 which contains impφI q. The existence of this irreducible component follows from (a), which gives 1 1 the inclusion impφI q Ă NI , and Proposition 4.16, which gives impφI q X I impφ q “ H. (T2) It coincides with the irreducible component ΣI¯1 which contains the section ¯1 ¯1 ΣIε given by (c). Note that pφI q´1 pΣI¯1 q is an bl0 –invariant section of the 1 ¯ P1 –bundle ΠIąk . (T3) Has the property: for any J P ℘1 pIq Y ℘2 pIq it intersects only finitely many fibres of the P|J|´1 –bundle ΓJε . Let YI be the Ť (possibly empty) union of irreducible components of type (T3) of ZI . Put Y1 :“ IP℘2 pIq YI . A similar argument as in the previous step shows that, replacing g by a generic perturbation of it if necessary, Y1 is compact. Formula (4.9) and the definitions above give ď` ď ˘ď ď ď ˘ď` ď (4.10) Mst “ Mst Mst pNJ Y ΣJ¯q Y1 Y2 Y3 , I I IP℘2 pIq

JP℘1 pIq

where Yi is a (possibly empty) compact pure i-dimensional complex subspace of Mst for 1 ď i ď 3, NJ Ă Mst is an irreducible, 1-dimensional analytic set containJ¯ ing Mst J , and ΣJ¯ is an irreducible, 1-dimensional analytic set containing φ pSJ¯q, ¯ ¯ where SJ¯ is a the image of an bl0 -equivariant section σJ¯ : PicJ pXqąk Ñ ΠJąk . Step 4. We claim that for any J P ℘1 pIq one has Ť (C1) NJ Ă Mst J Ť B. B. (C2) ΣJ¯ Ă Mst J¯ Indeed, let J P ℘1 pIq. We have a holomorphic map φJ : tΠJăk ust Ñ NJ , where NJ is a reduced, irreducible, pure 1-dimensional complex space. We will make use of Proposition 4.16 (4). If X contains a smooth rational curve in the class eJ ´ eJ¯ then, using the factorisation φJ “ ψ J ˝ pJ (with ψ injective and proper), it follows that φJ is surjective, so NJ “ Mst J . Note that in this case NJ contains a point of B, but this point is also an extension of type J. If X does not contain any smooth rational curve in the class eJ ´ eJ¯, then tΠJăk ust “ ΠJăk » D˚ (where D˚ Ă C is the punctured standard disk), and φJ is injective. We claim that in this case the limit bJ “

lim

φJ plq

degg plqÑ8

exists in Mst ; this limit will belong NJ , will be a fixed point of the involution bl0 , and ď ď NJ “ impφJ q tbJ u “ Mst tbJ u with bJ P B . J

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If this limit did not exist in Mst , then using the inclusion impφJ q Ă NJ and [Te8, Theorems 2.2, 3.9], it would follow that this limit exists in Mpst and belongs to a circle of reductions. But a circle of reductions cannot contain a fixed point of bl0 . Therefore in all cases we have NJ Ă Mst J Y B, and (C1) is proved. (C2) ¯ follows in a similar way by showing that the limit limdegg plqÑ8 φJ pσJ plqq exists in ΣJ¯. Using formula (4.10), the claims (C1), (C2) and putting Y0 “ B, we obtain a decomposition of the form (4.6) satisfying conditions (1). To complete the proof of the theorem it suffices to prove that for any compact complex subspace Y Ă Mst and for any I P ℘pIqztHu the intersection Mst I X Y is Zariski closed in Y . When |I| “ 1 and X contains a smooth rational curve in the class eI ´ eI¯, the claim is clear, because in this case φI is proper, so pφI q´1 pY q is finite. In all the other cases one has tΠIăk ust “ ΠIăk , φI is injective and factorises as φI “ νI ˝ fI , where νI : NI Ñ NI is the normalisation of the analytic set NI . Moreover, applying Theorem 4.17, one obtains in all cases a holomorphic map fI : NI Ñ D with impfI q “ fI´1 pD˚ q, and such that the fibres of fI over points z P D˚ correspond bijectively (via fI ) to the fibres of the projective bundle ΠIăk . Since Y is compact, and νI is proper, it follows that νI´1 pY q is a compact analytic subset of NI , so fI pνI´1 pY qq is finite. This shows that φ´1 pY q intersects only finitely many fibres of the projective bundle ΠIăk , which proves the claim.  We close the section with the two results used in the proof. The first concerns properties of the maps φI for minimal class VII surfaces with b2 “ 3; the second is a general extension theorem in complex geometry, and is of independent interest. Proposition 4.16. Let X P VIImin with b2 “ 3, and let g be a Gauduchon metric on X with degg pKq ă 0. Suppose that the simplifying assumptions (SA1) and (SA2) hold. Then (1) For any I P ℘pIq with |I| ě 2 one has tΠIąk ust “ ΠIąk . (2) The map ž ž φI : ΠIąk Ñ Mst |I|ě2

|I|ě2

is injective. st (3) If |I| ě 2 and |J| “ 1, one has Mst I X MJ “ H. (4) Suppose |J| “ 1. (a) If X does not contain any smooth rational curve in the class eJ ´ eJ¯, then tΠJąk ust “ ΠJąk and the map φJ : ΠJąk Ñ Mst is injective. (b) If the class eJ ´ eJ¯ is represented by a smooth rational curve C, then tΠJąk ust “ ΠJpk,k`cq , where c :“ volg pCq. In this case let pJ : ΠJpk,k`cq Ñ ΔJ be the double branched cover onto the quotient ΔJ of the annulus ΠJpk,k`cq “ PicJ pXqpk,k`cq by the involution rLs ÞÑ rK b L_ pCqs. Then φJ factorises as φJ “ ψ J ˝ pJ , where ψ J : ΔI Ñ Mst is holomorphic, injective and proper. In order to prove (1) and (2), note that (SA2) implies that X is not an Enoki surface and, using this fact, one can prove easily that a non-trivial extension of type I has only one line subbundle (the tautological one). Similarly, a non-trivial

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extension E of type I P ℘2 pIq has at most two line subbundles, the tautological one, and possibly a line subbundle T with Chern class eI ; if this is the case, then E becomes a non-trivial extension of type H, X has a cycle, and T does not destabilise E. Before stating the second result used in the proof of Theorem 4.13 we introduce the notations: DR :“ tz P C| |z| ă Ru , Ωrr, Rq :“ tz P C| r ď |z| ă Ru . Theorem 4.17. Let X be a reduced, normal pure n-dimensional complex space, ˚ U Ă X be an open subset (with respect to the classical topology), and g : U Ñ DR ´1 be a holomorphic map such that for any r P p0, Rq the set g pΩrr, Rqq is closed in X. Then (1) g admits a holomorphic extension f : X Ñ DR with f ´1 p0q “ XzU . (2) The complement Z :“ XzU is Zariski closed and has codimension ď 1 at any point. More precisely, Z is the union of the connected components of X which do not intersect U with a family of divisors consisting of a (possibly empty) divisor in any connected component of X which intersects U. (3) The natural map π0 pU q Ñ π0 pXq is injective. References V. Apostolov and G. Dloussky, Locally conformally symplectic structures on compact non-K¨ ahler complex surfaces, Int. Math. Res. Not. IMRN 9 (2016), 2717–2747. MR3519128 [BHPV] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR2030225 [Bi] J.-M. Bismut, Hypoelliptic Laplacian and Bott-Chern cohomology, Progress in Mathematics, vol. 305, Birkh¨ auser/Springer, Cham, 2013. A theorem of Riemann-RochGrothendieck in complex geometry. MR3099098 [Br1] M. Brunella, Locally conformally K¨ ahler metrics on Kato surfaces, Nagoya Math. J. 202 (2011), 77–81. MR2804546 [Br2] M. Brunella, A characterization of Inoue surfaces, Comment. Math. Helv. 88 (2013), no. 4, 859–874. MR3134413 [Br3] M. Brunella, A characterization of hyperbolic Kato surfaces, Publ. Mat. 58 (2014), no. 1, 251–261. MR3161518 [BTT1] N. Buchdahl, A. Teleman, M. Toma: A continuity theorem for families of sheaves on complex surfaces, arXiv:1612.09451 [math.CV]. [BTT2] N. Buchdahl, A. Teleman, M. Toma: On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces, arXiv:1701.03339 [math.CV]. [Bu1] N. P. Buchdahl, Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann. 280 (1988), no. 4, 625–648. MR939923 [Bu2] N. Buchdahl, A Nakai-Moishezon criterion for non-K¨ ahler surfaces (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 50 (2000), no. 5, 1533–1538. MR1800126 [D1] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26. MR765366 [D2] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differential Geom. 26 (1987), no. 3, 397–428. MR910015 [DK] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR1079726 [AD]

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[Dl] [DlTe] [DOT] [E] [Fi] [FU]

[FP1] [FP2]

[Gau] [GR]

[HH] [Kau]

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G. Dloussky, Structure des surfaces de Kato (French, with English summary), M´em. Soc. Math. France (N.S.) 14 (1984), ii+120. MR763959 G. Dloussky and A. Teleman, Infinite bubbling in non-K¨ ahlerian geometry, Math. Ann. 353 (2012), no. 4, 1283–1314. MR2944030 G. Dloussky, K. Oeljeklaus, and M. Toma, Class VII0 surfaces with b2 curves, Tohoku Math. J. (2) 55 (2003), no. 2, 283–309. MR1979500 ohoku Math. J. (2) 33 (1981), no. 4, I. Enoki, Surfaces of class VII0 with curves, Tˆ 453–492. MR643229 G. Fischer, Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538, Springer-Verlag, Berlin-New York, 1976. MR0430286 D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1991. MR1081321 A. Fujiki and M. Pontecorvo, Anti-self-dual bihermitian structures on Inoue surfaces, J. Differential Geom. 85 (2010), no. 1, 15–71. MR2719408 A. Fujiki and M. Pontecorvo, Twistors and bi-Hermitian surfaces of non-K¨ ahler type, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 042, 13. MR3210593 P. Gauduchon, La 1-forme de torsion d’une vari´ et´ e hermitienne compacte (French), Math. Ann. 267 (1984), no. 4, 495–518. MR742896 H. Grauert and R. Remmert, Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR755331 F. Hirzebruch and H. Hopf, Felder von Fl¨ achenelementen in 4-dimensionalen Mannigfaltigkeiten (German), Math. Ann. 136 (1958), 156–172. MR0100844 L. Kaup and B. Kaup, Holomorphic functions of several variables, De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. An introduction to the fundamental theory; With the assistance of Gottfried Barthel; Translated from the German by Michael Bridgland. MR716497 P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom. 41 (1995), no. 3, 573–734. MR1338483 M. Kato, Compact complex manifolds containing “global” spherical shells, Proc. Japan Acad. 53 (1977), no. 1, 15–16. MR0440076 M. Kato, Compact complex manifolds containing “global” spherical shells. I, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 45–84. MR578853 Ma. Kato, On a certain class of nonalgebraic non-K¨ ahler compact complex manifolds, Recent progress of algebraic geometry in Japan, North-Holland Math. Stud., vol. 73, North-Holland, Amsterdam, 1983, pp. 28–50. MR722141 S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR0152974 S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanˆ o Memorial Lectures, 5. MR909698 K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. MR0184257 K. Kodaira, On the structure of compact complex analytic surfaces. II, Amer. J. Math. 88 (1966), 682–721. MR0205280 M. L¨ ubke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR1370660 J. Li and S.-T. Yau, Hermitian-Yang-Mills connection on non-K¨ ahler manifolds, Mathematical aspects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 560–573. MR915839 I. Nakamura, On surfaces of class VII0 with curves, Invent. Math. 78 (1984), no. 3, 393–443. MR768987 I. Nakamura, Classification of non-K¨ ahler complex surfaces (Japanese), S¯ ugaku 36 (1984), no. 2, 110–124. Translated in Sugaku Expositions 2 (1989), no. 2, 209–229. MR780359

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[Na3] [OeTo] [OkTe] [Te1] [Te2] [Te3] [Te4]

[Te5] [Te6]

[Te7] [Te8] [Te9]

I. Nakamura, On surfaces of class VII0 with curves. II, Tohoku Math. J. (2) 42 (1990), no. 4, 475–516. MR1076173 K. Oeljeklaus and M. Toma, Logarithmic moduli spaces for surfaces of class VII, Math. Ann. 341 (2008), no. 2, 323–345. MR2385660 C. Okonek and A. Teleman, The coupled Seiberg-Witten equations, vortices, and moduli spaces of stable pairs, Internat. J. Math. 6 (1995), no. 6, 893–910. MR1354000 A. D. Teleman, Projectively flat surfaces and Bogomolov’s theorem on class VII0 surfaces, Internat. J. Math. 5 (1994), no. 2, 253–264. MR1266285 A. Teleman, Donaldson theory on non-K¨ ahlerian surfaces and class VII surfaces with b2 “ 1, Invent. Math. 162 (2005), no. 3, 493–521. MR2198220 A. Teleman, The pseudo-effective cone of a non-K¨ ahlerian surface and applications, Math. Ann. 335 (2006), no. 4, 965–989. MR2232025 A. Teleman, Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds, Geom. Topol. 11 (2007), 1681– 1730. MR2350464 A. Teleman, Instantons and curves on class VII surfaces, Ann. of Math. (2) 172 (2010), no. 3, 1749–1804. MR2726099 A. Teleman: A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces, to appear in Progress in Mathematics, Vol. 310 (Proceedings of the conference in honour of J. M. Bismut), arXiv:1309.0350 [math.CV]. A. Teleman, Instanton moduli spaces on non-K¨ ahlerian surfaces. Holomorphic models around the reduction loci, J. Geom. Phys. 91 (2015), 66–87. MR3327049 A. Teleman, Analytic cycles in flip passages and in instanton moduli spaces over nonK¨ ahlerian surfaces, Internat. J. Math. 27 (2016), no. 7, 1640009, 26. MR3521593 A. Teleman: Minimal class VII surfaces with b2 “ 3, in preparation.

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France Email address: [email protected]

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Proceedings of Symposia in Pure Mathematics Volume 99, 2018 http://dx.doi.org/10.1090/pspum/099/01747

Two lectures on gauge theory and Khovanov homology Edward Witten

Abstract. In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial differential equations in 4 or 5 dimensions. The second lecture is devoted to a description of the rather unusual boundary conditions by which these equations should be supplemented. An appendix describes some physical background. (Versions of these lectures have been presented at various institutions including the Simons Center at Stonybrook, the TSIMF conference center in Sanya, and also Columbia University and the University of Pennsylvania.)

1. Lecture One The first physics-based proposal concerning Khovanov homology of knots was made by Gukov, Vafa, and Schwarz [1], who suggested that vector spaces associated to knots that had been introduced a few years earlier by Ooguri and Vafa [2] were related to what appears in Khovanov homology. A number of years later, I reexpressed this type of construction in terms of gauge theory and the counting of solutions of PDE’s [3]. That is the story I will describe today. Several previous lectures are available [4, 5] (the second of these may be a better starting point) and I will take a different approach here. In any event, the goal is to construct invariants of a knot embedded in R3 (fig. 1). In the simplest version, the invariants will be obtained by simply counting, with signs, the solutions of an equation. The solutions will have an integer-valued1 topological invariant P , and if an is the “number” (counted algebraically) of solutions

2010 Mathematics Subject Classification. Primary 14D21, 57M27. Research supported in part by NSF Grant PHY-1314311. 1 To be more precise, P takes values in a Z-torsor, rather than being canonically an integer. This is related to the framing anomaly of Chern-Simons theory. See Lecture 2. c 2018 American Mathematical Society

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Figure 1. A knot embedded in R3 . with P = n, then the Jones polynomial2 of the knot will be  (1.1) J(q) = an q n . n

To get Khovanov homology, this situation is supposed to be “categorified,” that is, we want for each n to define a complex of vector spaces whose Euler characteristic is an . The only general situation that I know of in which one can naturally categorify the counting of solutions of an equation is the case that the equation whose solutions we are counting describes the critical points of some Morse function h. We will be in this framework. Our equations will be partial differential equations or PDE’s, so h will be a Morse function on an infinite-dimensional space of functions, namely the functions that appear in the PDE. The categorification will involve a middledimensional cohomology theory of the function space, analogous to Floer theory. Let us put this aside for a moment and assume we are just trying to describe the uncategorified theory, that is the Jones polynomial. The equations whose solutions I claim should be counted to define the Jones polynomial and ultimately Khovanov homology might look ad hoc if written down without an explanation of where they come from. I could have started today’s lecture by explaining the physical setup, but this might be unhelpful for some. I decided instead to try a different approach of motivating the equations by comparing to an established mathematical approach to Khovanov homology, namely symplectic Khovanov homology [6–8]. Going all the way back to the original work of Vaughn Jones [9], most approaches to the Jones polynomial define an invariant in terms of some sort of presentation of a knot, for example a projection to a plane – such as the projection used in drawing fig. 1. One defines something that is manifestly well-defined and explicitly computable once such a presentation is given. What one defines is not obviously independent of the knot presentation, but turns out to be. That step is where the magic is. And there is always some magic. 2 In approaches based on quantum field theory, the natural normalization of the Jones polynomial of a knot or link in R3 is such that the Jones polynomial of the empty link is 1. (The Jones polynomial is sometimes defined so that it equals 1 for an unknot rather than for the empty link.) We normalize the argument q of the Jones polynomial to be the instanton counting parameter, in a sense that will be explained later. With this choice, the Jones polynomial of the unknot (with standard framing) is q 1/2 + q −1/2 and in general, for a knot with zero framing, the exponents in eqn. (1.1) are half-integers.

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395

Figure 2. A knot embedded in R3 and stretched in one direction.

Figure 3. Infinite parallel strands parametrized by u, with −∞ ≤

u ≤ ∞.

An approach based on counting solutions of PDE’s has the opposite advantages and drawbacks. Topological invariance is potentially manifest (given certain generalities about elliptic PDE’s and assuming compactness is under control), but it may not be clear how to calculate. The ideal is to have manifest three- or (in the categorified case) four-dimensional symmetry together with a method of calculation. How might this be achieved? I will suggest how to guess the right equations starting from a knowledge of symplectic Khovanov homology. But in order to do this, we need to know something about a possible strategy to actually count the solutions of an equation. So I will begin by explaining what we would do if we knew which equations we want to analyze, and this will help us in guessing the equations. There is a standard strategy, applicable to the present problem, for trying to count solutions of a PDE under suitable conditions. The original version was the Atiyah-Floer conjecture concerning Floer homology of a three-manifold [10]. Adapting their approach to the present problem, the idea is to stretch a knot in one direction, say the u direction, as in fig. 2. Then one wants it to be the case that except near the ends, the solutions are independent of u. This is not automatically the case and in [11], where this strategy was followed for the present problem, it was necessary to make a perturbation to a more generic system of equations to get to a situation in which this would be true. Given this, we define a moduli space M of u-independent solutions. We can think of these as the solutions in the presence of infinite parallel strands that run in the u direction, as in fig. 3. Now as in fig. 4 consider solutions in the presence of semi-infinite strands that extend to u = +∞ or to u = −∞ but not both. Let L and Lr be the moduli

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Figure 4. Semi-infinite strands that extend to u = +∞. spaces of such solutions. Thus a point in L represents a solution in a semi-infinite situation in which the strands terminate on the left (as drawn in fig. 4). Likewise Lr parametrizes solutions in the presence of semi-infinite strands that terminate on the right. We assume that a solution in such a semi-infinite situation is independent of u for u → +∞ or u → −∞, respectively. If this is so, then L and Lr come with natural maps to M. For simplicity in our terminology, we will assume that these maps are embeddings; this amounts to assuming that each solution in the interior in fig. 2 can be extended over the left or over the right in at most one way. This assumption is not necessary but makes the explanation simpler. The solutions for a global knot like the one in fig. 2 can be understood as solutions in the middle that extend over both ends. So the global solutions are intersection points of L and Lr . The integer an that appears as a coefficient in the Jones polynomial is supposed to be the algebraic intersection number of L and Lr : (1.2)

an = L  ∩ L r .

(To be more exact, an is this intersection number computed by counting only intersections with P = n.) In this language of intersections, categorification can happen if M is in a natural way a symplectic manifold and L and Lr are Lagrangian submanifolds. Then Floer cohomology – i.e. the A-model or the Fukaya category – of M gives a framework for categorification. From the point of view of today’s lecture, the reason that all this will happen is that, even before we stretched the knot to reduce to intersections in M, the equations whose solutions we were counting are equations for critical points of some Morse function(al) h. In “symplectic Khovanov homology,” a version of such a story is developed for Khovanov homology (at least in a singly-graded version) with a very specific M. A description of this M that was proposed in [12] (and exploited in a mirror version in [13]) and which provided an important clue in my work is as follows. M can be understood as a space of Hecke modifications. Let me explain this concept. Let C be a Riemann surface and E → C a holomorphic GC bundle over C, where GC is some complex Lie group. A Hecke modification of E at a point p ∈ C is a holomorphic GC bundle E  → C with an isomorphism to E away from p: ∼ E|C\p . (1.3) ϕ : E  |C\p = For example, if GC = C∗ , the we can think of E as a holomorphic line bundle L → C. A holomorphic bundle L that is isomorphic to L away from p is (1.4)

L = L(np) = L ⊗ O(p)n

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Figure 5. A configuration of points pi ∈ R2 , at which we are going to make Hecke modifications.

for some integer n. Here n can be thought of as a weight of the Langlands-GNO dual group of C∗ , which is another copy of C∗ . The reason that I write GC , making explicit that this is the complex form of the group, is that when we do gauge theory, the gauge group will be the compact real form and I will call this simply G. In general, for any G, there is a corresponding Langlands-GNO dual group G∨ , with complexification G∨ C , such that Hecke modifications of a holomorphic GC -bundle at a point p ∈ C occur in families classified by dominant weights (or equivalently finite-dimensional representations) of G∨ C (or equivalently G∨ ). For example, if GC = GL(2, C), we can think of a GC -bundle E → C as a rank 2 complex vector bundle E → C. The Langlands-GNO dual group G∨ C is again GL(2, C), and a Hecke modification dual to the 2-dimensional representation of G∨ C is as follows. For some local decomposition E ∼ = O ⊕ O in a neighborhood of p ∈ C, one has E  ∼ = O(p) ⊕ O. The difference from the abelian case is that there is not just one Hecke modification of this type at p but a whole family of them, arising from the choice of a subbundle O of E that is going to be replaced by O(p). Because of this dependence, the Hecke modifications of this type at p form a family, parametrized by CP1 . Suppose we are given 2n points on C ∼ = R2 at which we are going to make Hecke modifications of this type of a trivial bundle rank 2 complex vector bundle E → C (fig. 5). The space of all such Hecke modifications would be a copy of (CP1 )2n , with one copy of CP1 at each point. However, there is a natural subvariety M ⊂ (CP1 )2n defined as follows. One adds a point ∞ at infinity to compactify C to CP1 , so we are now making Hecke modifications of a trivial bundle E = O ⊕ O → CP1 . A point in (CP1 )2n determines a way to perform Hecke modifications at the points p1 , p2 , . . . , p2n to make a new bundle E  . The space M is defined by requiring that E  ⊗ O(−n∞) is trivial. (If we were working in P GL(2, C) rather than GL(2, C), we would just say that E  should be trivial.) Symplectic Khovanov homology is constructed by considering intersections of Lagrangian submanifolds of the space M of multiple Hecke modifications from a trivial bundle to itself. We want to reinterpret this in terms of gauge theory PDE’s. In my work with Kapustin on gauge theory and geometric Langlands [15], an important fact was that M can be realized as a moduli space of solutions of a certain system of PDE’s. However, although M is defined in terms of bundles on

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∼ C, the PDE’s are in 3 dimensions – on R3 . As a result of this, a 2-manifold R2 = everything in the rest of the lecture will be in a dimension one more than one might expect. To describe the Jones polynomial – an invariant of knots in 3-space – we will count solutions of certain PDE’s in 4 dimensions, and the categorified version – Khovanov homology – will involve PDE’s in 5 dimensions. The 3-dimensional PDE’s that we need are known as the Bogomolny equations. They are equations, on an oriented three-dimensional Riemannian manifold W3 , for a pair A, φ, where A is a connection on a G-bundle E → W3 , and φ is a section of ad(E) → W3 (i.e. an adjoint-valued 0-form). If F = dA + A ∧ A is the curvature of A, then the Bogomolny equations are (1.5)

F = "dA φ.

(Here " is the Hodge star and dA is the gauge-covariant extension of the exterior derivative.) The Bogomolny equations have many remarkable properties and we will focus on just one aspect. We consider the Bogomolny equations on W3 = R × C with C a Riemann surface. Any connection A on a G-bundle E → C determines a holomorphic structure on E (or more exactly on its complexification): one simply writes dA = ∂ A + ∂A and uses ∂ A to define the complex structure. (In complex dimension 1, there is no integrability condition that must be obeyed by a ∂ operator.) So for any y ∈ R, by restricting E → R × C to E → {y} × C, we get a holomorphic bundle Ey → C. However, if the Bogomolny equations are satisfied, Ey is canonically independent of y. Indeed, a consequence of the Bogomolny equations is that ∂ A is independent of y up to conjugation. If we parametrize R by y, then the Bogomolny equations imply that ) * D − iφ, ∂ A = 0. (1.6) Dy Thus ∂ A is independent of y, up to a natural conjugation. The Bogomolny equations admit solutions with singularities at isolated points. To understand the basic picture, we take the three-manifold to be simply R3 , and the gauge group to be U (1). One fixes an integer n and one observes that the Bogomolny equation has an exact solution for any x0 ∈ R3 : n , F = "dφ. (1.7) φ= 2|x − x0 | I have only defined F and not the connection A whose curvature is F or the line bundle L on which A is connection. Such an L and A exist (and are essentially unique) if and only if n ∈ Z. For G = U (1), since the Bogomolny equations are linear, they have a unique solution with singularities of this type labeled by specified n1 , n2 , . . . at  integers i , F = "dφ. We specified points pi ∈ R3 (fig. 6). We simply take φ = i 2| xn−

xi |  assume that n = 0, which ensures that φ and the connection A vanish at i i infinity faster than 1/|x|. Now pick a decomposition R3 = R × R2 , where we identify R2 as C. Suppose / {y1 , . . . , yn }, that the singularities are at yi × pi , with yi ∈ R, pi ∈ C. For each y ∈ the indicated solution of the Bogomolny equations determines a holomorphic line bundle Ly → C, and upon adding a point at infinity, this naturally extends to Ly → CP1 . (Here we use the fact that A vanishes at infinity faster than 1/|x|.) Ly

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Figure 6. Points yi × pi ∈ R3 labeled by weights ni of the group U (1). is independent of y up to isomorphism as long as y is not equal to one of the yi , but even when y crosses one of the yi , Ly is constant when restricted to CP1 \pi . In crossing y = yi , Ly undergoes a Hecke modification (1.8)

Ly → Ly ⊗ O(pi )ni .

Ly is trivial for y → −∞ and for y → +∞ (again because the solution vanishes at infinity faster than 1/|x|). The solution thus describes a sequence of Hecke modifications mapping the trivial bundle to itself. We can do something similar for any simple Lie group G. (The underlying idea was introduced by ’t Hooft in the late 1970’s [14] and is important in physical applications of quantum gauge theory.) Let T be the maximal torus of G and let t be its Lie algebra. Pick a homomorphism ρ : u(1) → t. Up to a Weyl transformation, such a ρ is equivalent to a dominant weight of the dual group G∨ , so it corresponds to a representation R∨ of G∨ . We turn the singular solution (1.7) of the U (1) Bogomolny equations that we already used (more exactly, the special case of this solution with n = 1) into a singular solution for G simply by (1.9)

(A, φ) → (ρ(A), ρ(φ)).

Then we look for solutions of the Bogomolny equations for G with singularities of this type at specified points yi × pi ∈ R3 . The picture is the same as before except that now (fig. 7) the points yi × pi are labeled by homomorphisms ρi : u(1) → t, or in other words by representations Ri∨ of the dual group G∨ , rather than by integers ni . Also, now we must specify that the solution  should go to 0 at infinity faster than 1/r (for U (1), this was automatic once we set i ni = 0). Given this, such a solution describes a sequence of Hecke modifications at pi of type ρi , mapping a trivial G-bundle E → CP1 to itself. The moduli space M of solutions of the Bogomolny equations on R3 with the indicated singularities and vanishing at infinity faster than 1/r is actually a hyperK¨ ahler manifold, essentially first studied by P. Kronheimer in the 1980’s. If we pick a decomposition R3 = R × R2 , this picks one of the complex structures on the hyper-K¨ ahler manifold and in that complex structure, M is the moduli space

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Figure 7. Points in R3 labeled by homomorphisms ρi : u(1) → t, or equivalently by representations of the dual group.

Mp1 ,ρ1 ;p2 ,ρ2 ;... of all Hecke modifications of the indicated types at the indicated points, mapping a trivial bundle over CP1 to itself. This construction can be used to account for a number of properties of spaces of Hecke modifications, but for today we want to focus on the application to knot theory. The reduction to M is supposed to result from stretching a knot in one direction, so we want M to be the space of u-independent solutions of some equations, as suggested in fig. 3. We already described M via solutions of some PDE’s on R3 , so now we have to think of M as a space of u-independent solutions on R4 = R3 × R, where the second factor is parametrized by u. There actually are natural PDE’s in four dimensions that work. They play a role in the gauge theory approach to geometric Langlands [15], and are sometimes called the KW equations. They are equations for a pair A, φ where A is a connection on E → Y4 , Y4 a four-manifold, and φ is a 1-form on Y4 valued in ad(E): (1.10)

F − φ ∧ φ = "dA φ, dA " φ = 0.

In a special case Y4 = W3 × R, with A a pullback from W3 and φ = φ du (where φ is a section of ad(E) and u parametrizes the second factor in Y4 ) these equations reduce to the Bogomolny equations on W3 : (1.11)

F = "dA φ.

Therefore, the singular solution (1.9) of the Bogomolny equations that we have already studied can be lifted to a singular solution of the KW equations, but now the singularity is along a line rather than a point. Of course, the singularity is still in codimension three. We view this solution as a model that tells us what sort of codimension three singularity to look for in a more general situation. If Y4 is a 4-manifold and S ⊂ Y4 is an embedded 1-manifold, labeled by a homomorphism ρ : u(1) → t (or by a representation of G∨ ), then one can look for solutions of the KW equations with a singularity along S associated to the given choice of ρ (fig. 8). If we specialize to the case that Y4 = R3 × R, with S = ∪i Si , and Si = pi × R ⊂ 3 R × R (pi are points in R3 and R is parametrized by u) then the u-independent

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Figure 8. A four-manifold Y with an embedded 1-manifold S along which one specifies a desired singularity.

solutions of the KW equations are just the solutions of the Bogomolny equations on R3 , with the chosen singularities. So these solutions are parametrized by M; and indeed one can show that these are all solutions of the KW equations in this situation with reasonable behavior at infinity. So we have an elliptic PDE in four dimensions and we can specify in an interesting way what sort of singularity it should have on an embedded circle S ⊂ Y4 . But this sounds like a ridiculous framework for knot theory, because there is no knottedness of a 1-manifold in a 4-manifold! To resolve this point, we have to explain what is involved in categorification. Let us practice with an ordinary equation rather than a partial differential equation. Suppose that we are on a finite-dimensional compact oriented manifold N with a real vector bundle V → N with rank(V )=dimension(N ). Suppose also we are given a section s of V . We can define an integer by counting, with multiplicities (and in 5 particular with signs) the zeroes of s. This integer is the Euler class M χ(V ). In general as far as I know, there is no way to categorify the Euler class of a vector bundle. However, suppose that V = T ∗ N and that s = dh where h is a Morse function. Then the zeroes of s, which are critical points of h, have a natural “categorification” described in Morse homology. One defines a complex V with a basis vector ψp for each critical point p of h. The complex is Z-graded by assigning to ψp the “index” of the critical point p, and it has a natural differential that is defined by counting gradient flow lines between different critical points. Concretely the differential is defined by  npq ψq (1.12) dψp = q

where the sum runs over all critical points q whose Morse index exceeds by 1 that of p, and the integer npq is defined by counting flows from p to q (fig. 9). A “flow” is a solution of the gradient flow equation (1.13)

dx  = −∇h. dt

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Figure 9. A flow from one critical point to another. (To define this equation, one has to pick a Riemannian metric on the manifold N . The complex that one gets is independent of the metric up to quasi-isomorphism. One considers flows that start at p at t = −∞ and end at q at t = +∞. Such flows come in one-parameter families related by time translations and npq is the number of such families, counted algebraically.) This tells us what we need in order to be able to categorify a problem of counting solutions of the KW equations. We have to be able to write those equations as equations for a critical point of a functional Γ(A, φ): δΓ δΓ = = 0. δA δφ And the associated gradient flow equation, which will be a PDE in 5 dimensions on X5 = R × Y4 δΓ dφ δΓ dA =− , =− (1.15) dt δA dt δφ has to be elliptic, so that it will makes sense to try to count its solutions. Generically, it is not true that the KW equations on a manifold Y4 are equations for a critical point of some functional. However, this is true if Y4 = W3 ×R for some W3 . If singularities are present on an embedded 1-manifold S ⊂ Y4 then there is a further condition: The KW equations in this situation are equations for critical points of a functional if and only if S is contained in a 3-manifold W3 × p, with p a point in R. (For an explanation of “why” this is true, see the appendix.) So to make categorification possible, we have to be in the situation that leads to knot theory: S is an embedded 1-manifold in a 3-manifold W3 . Once this restriction is made, the five-dimensional flow equations exist and are indeed elliptic. (They were introduced independently in [3] and [19] and are sometimes called the HW equations.) Naively, this leads to “categorified” knot invariants for any three-manifold W3 , but to justify this claim one needs some compactness properties for solutions of the equations under consideration. I suspect that a proper proof of these compactness properties may require that the Ricci tensor of W3 is nonnegative, a very restrictive condition. (1.14)

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Figure 10. A four-manifold Y4 with boundary, with a knot placed in its boundary.

What I have described so far is supposed to correspond (for W3 = R3 , G = SO(3) and ρ corresponding to the 2-dimensional representation of G∨ = SU (2)) to “singly-graded Khovanov homology.” It is singly-graded because the only grading I have mentioned is the grading that is associated to the Morse index, or in other words to categorification. In the mathematical theory, one says that singlygraded Khovanov homology becomes trivial (it does not distinguish knots) if one “decategorifies” it and forgets the grading. In the approach I have described, this is true because in the uncategorified version, the embedded 1-manifold S is just a 1-manifold in a 4-manifold Y4 (it has no reason to be embedded in the 3-manifold W3 × p) so there is no knottedness. The physical picture makes clear where the additional “q”-grading of Khovanov homology would come from. It is supposed to come from the second Chern-class, integrated over the 4-manifold Y4 . The second Chern class is the invariant that I called P at the beginning of the lecture. But for topological reasons, the second Chern class cannot be defined in the presence of a codimension three singularity of the type I have described on an embedded one-manifold S ⊂ Y4 . (Because of the singularity, Y4 behaves as a noncompact four-manifold on which there is no topological invariant corresponding to the second Chern class of a G bundle.) And therefore the construction as I have presented it so far has no q-grading. The physical picture tells us what we have to do to get the q-grading: Y4 should be a manifold with boundary, with the knot placed in its boundary (fig. 10). The appropriate boundary condition will be the subject of Lecture Two and is such that the second Chern class can be defined. In [11], Gaiotto and I analyzed this problem (in the uncategorified situation, meaning that we counted solutions in 4 dimensions, not 5, and for the simplest case of G = SO(3)) with the aim of showing directly, without referring to the physical picture, that the Jones polynomial is  an q n (1.16) J(q) = n

where an is the number of solutions with second Chern class n. The starting point was to stretch the knot in one direction, reducing to equations in one dimension less, as in fig. 2. It turns out that the solutions in one dimension less that satisfy the boundary condition are related to a lot of interesting mathematical physics involving integrable systems, conformal field theory, geometric Langlands, and more. What

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emerges is the “vertex model” construction of the Jones polynomial; the way it emerges is somewhat along the lines of work by Bigelow [16] and Lawrence [17]. What our work added was a derivation of the vertex model from a starting point with manifest 3-dimensional symmetry. The analog of this for the categorified theory is expected to involve, in one version, a Fukaya-Seidel category with a certain superpotential. The relevant model – as well as a plausible variant that does not work – has been explored and to a considerable extent understood in [18].

2. Lecture Two In Lecture One, I explained that to define a q-grading in Khovanov homology, we have to be able to make sense of the second Chern class of a solution of the KW equations on a four-manifold Y4 , in the presence of a knot. As already explained, if the knot is represented by a codimension 1 embedded submanifold S ⊂ Y4 , this will not work, because the singularity that we want to postulate along S does not allow the definition of a second Chern class as a topological invariant. Instead, we embed the knot in the boundary of Y4 , as in fig. 10. The boundary condition that we use is subtle to describe, but has the property that the bundle is fixed on the boundary, so the second Chern class can be defined. We could actually get the q-grading for any Y4 with boundary, but to also allow categorification, we want more specifically Y4 = W × R+ , where R+ is a half-line, parametrized by y. For the Jones polynomial and Khovanov homology, we further take W = R3 . (More general choices of W are certainly also interesting, but not much is known about what to expect. See [20].) So as sketched in fig. 10, Y4 is R3 × R+ with the knot embedded in the boundary. For y → ∞, we ask for A, φ → 0. For y → 0, there is a subtle boundary condition which is one of the main points of the theory. Describing it is actually my main goal for today. This boundary condition depends on the knot K, and on the labeling of K (or of each component of a link L) by a representation R∨ of the Langlands or GNO dual group G∨ . That is the only place that K enters the setup. The desired boundary condition is an elliptic boundary condition though possibly an unexpected one. Here, “elliptic” means that although the definition may look unexpected, the resulting properties are similar to what one would get with more familiar elliptic boundary conditions such as Dirichlet or Neumann. For example, on a compact four-manifold, the linearization of the KW equation becomes a Fredholm operator and has discrete spectrum. With this boundary condition, which I will describe in some detail, the restriction to the boundary W × {y = 0} of the bundle E and connection A are specified. As a result, one can define a second Chern class " 1 Tr F ∧ F. (2.1) n= 8π 2 W ×R+ However, because the bundle is fixed on the boundary but is not trivialized, this invariant really takes values in a Z-torsor associated to framings of W and K. “Torsor” means that it is not true that n is an integer in a canonical way; rather the value of n mod Z depends only on the boundary conditions and not on the specific gauge field that satisfies them. One can “trivialize the torsor” and make n an integer by picking framings of W and K.

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To define a knot polynomial, one counts (with signs, in a standard way) the number an of solutions of the KW equations with second Chern class n, and then one defines  (2.2) J(q; K, R∨ ) = an q n . n∈Z

Compactness (not yet proved with the appropriate boundary conditions) of the solutions of the KW equations will mean that there are only finitely many terms in the sum so that this is a Laurent polynomial. As I explained in Lecture One, with this definition of the Jones polynomial, the “categorification” that leads to Khovanov homology is straightforward in principle. It arises because the KW equations can be “lifted,” in a certain sense, to certain elliptic differential equations in five dimensions, and these equations can be interpreted as gradient flow equations. But rather than say more about that today, what I want to do is to describe the boundary condition that is needed for the four (or five) dimensional equations. This boundary condition is of a possibly somewhat unfamiliar type, and understanding it is essential for making progress with this subject. The boundary conditions have been studied in [21], and were shown to be elliptic in the absence of a knot. A paper is in progress on the case with a knot [22], and I will tell a little about that case later. I will carry out this discussion in the language of the four-dimensional equations, since going to five dimensions does not change much, as was shown in [21]. The boundary condition of interest is not a simple Dirichlet or Neumann boundary condition – it is not defined by saying what fields or derivatives of fields vanish along the boundary. Rather, the boundary condition is defined by specifying a model solution of the KW equations that has a singularity along the boundary, and saying that one only wants to consider solutions of the KW equation that are asymptotic to this singular solution along the boundary. The model solution is a solution on R3 × R+ , where I will parametrize R3 by x1 , x2 , x3 and R+ by y. There is a simple exact solution with A = 0 and (2.3)

φ=

3  ti · dxi i=1

y

,

where ti are elements of the Lie algebra g of G that obey the su(2) commutation relations (2.4)

[t1 , t2 ] = t3 , and cyclic permutations.

Thus the ti are images of a standard basis of su(2) under some homomorphism ρ : su(2) → g. Every ρ leads to an interesting theory, but to get the Jones polynomial and Khovanov homology, we take ρ to be a principal embedding in the sense of Kostant. (For G = SU (2), this simply means that ρ is the identity map su(2) → g. For G = SU (N ), it means that the N -dimensional representation of g is irreducible with respect to ρ(su(2)).) The solution I have just described is what I call the Nahm pole solution, since the relevant singularity was introduced long ago by Nahm in his work on magnetic monopoles [23]. That was in the context of “Nahm’s equation,” which is an ordinary

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differential equation for three g-valued functions φ1 , φ2 , φ3 of a real variable y: dφ1 + [φ2 , φ3 ] = 0, and cyclic permutations. (2.5) dy The KW equations reduce to Nahm’s equation if we drop the dependence on x and set A = φy = 0. On R3 × R+ , the Nahm pole boundary condition just says that a solution is supposed to be asymptotic to the Nahm pole solution for y → 0. To state the Nahm pole boundary condition on M4 = W3 × R+ , for a more general 3-manifold W3 , one needs to specify some terms of O(1) in the solution (for y → 0) as well as the singular terms of order 1/y. For G = SU (2), one takes the G-bundle E on which we are solving the equations to be, when restricted to W3 × {y = 0}, the frame bundle of W3 , so that ad(E) = T W3 . Then one takes A, restricted to the boundary, to be the Levi-Civita connection on T W3 . With this choice of E, the formula (2.6)

φ=

3  ti · dxi i=1

y 

makes sense (one can think of the numerator i ti · dxi as stating the identification ad(E) ∼ = T W3 ). One can show that this choice of (A, φ) obeys the Nahm pole boundary condition up to an error of O(y), and the Nahm pole boundary condition simply says that the solution should agree with what I have described up to O(y). (One can generalize this to the case that the metric of M4 is not a product near the boundary.) Showing that the Nahm pole boundary condition is elliptic is mostly an exercise in “uniformly degenerate elliptic operators,” but one needs to know some specific facts about the KW equations. The main thing that one needs to know is that if L is the linearization of the KW equations on a half-space R4+ around the Nahm pole solution, then L as an operator between appropriate Hilbert spaces of functions on R4+ has no kernel or cokernel. Actually one can show in an elementary way that L† = −N LN −1 with an explicit matrix N , so it suffices to show that there is no kernel. Much the same argument that proves this actually proves the following statement: The only solution of the KW equations on R4+ , approaching the Nahm pole ' solution for y → 0 and also for x2 + y 2 → ∞, is the Nahm pole solution, and moreover this solution is “transverse” (in expanding around it, the operator L has zero kernel and cokernel). In terms of Khovanov homology, this means that the Khovanov homology of the empty link is of rank 1. Before trying to prove these vanishing results, I will explain a simpler vanishing result for the KW equations on a four-manifold M = M4 without boundary. This will help us know what to aim for.3 The KW equations actually have many different useful Weitzenbock formulas. I will first state some formulas that are useful if we are on a manifold without boundary. Let V = F − φ ∧ φ − "dA φ, W = dA " φ, so the KW equations are V = W = 0. Clearly then the KW equations are equivalent to the vanishing of " Tr (V ∧ "V + W ∧ "W) . (2.7) I=− M 3 The vanishing result without boundary was obtained in [15] and the case with a boundary was analyzed in [21].

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A short calculation gives   " 1 1 4 √ ij i j i j i j (2.8) I = − Fij F + Di φj D φ + Rij φ φ + [φi , φj ][φ , φ ] d x gTr 2 2 M with Rij the Ricci tensor. If Rij is non-negative, then this is a sum of non-negative terms. The condition I = 0 forces all these terms to vanish and leads to only a rather trivial class of solutions. But it is possible to say something useful even if Rij is not non-negative, because it is possible to find a family of Weitzenbock formulas. Define the selfdual and antiselfdual two-forms V + (t) = (F −φ∧φ+tdA φ)+ , V − (t) = (F −φ∧φ−t−1 dA φ)− . The equations V + (t) = V − (t) = W = 0 are a 1-parameter family of elliptic equations, parametrized by t ∈ RP1 . One finds that  −1  " t t + − 4 √ + ij − ij 2 − d x gTr V (t)V (t) + V (t)V (t) + W t + t−1 ij t + t−1 ij M " t − t−1 =I+ d4 x ijkl TrFij Fkl . 4(t + t−1 ) M In other words, the same quantity I can be written as a sum of squares in many different ways, modulo the topological invariant " 1 t − t−1 2 · 32π P, P = d4 x ijkl TrFij Fkl . (2.9) J(t) = 4(t + t−1 ) 32π 2 M Now we can deduce the following: (1) The KW equations cannot have any solutions for t = 0, ∞ except with P = 0 (if P = 0 for some solution, then by looking at the Weitzenbock formula at some value of t with J(t ) < J(t), we reach a contradiction). And (2): If the KW equations are obeyed at one value of t other than 0, ∞, then they are obeyed at all t. This is an immediate consequence of the Weitzenbock formula, once we know that P = 0. The equations then reduce to F = 0, where F = dA + A ∧ A, with A the complex connection A = A + iφ, along with dA " φ = 0. According to a theorem of Corlette [24], the solutions correspond to homomorphisms π1 (M ) → GC that are in a certain sense semi-stable. The moral of the story is that the KW equations participate in many different Weitzenbock formulas, not just one, and it is important to know all of them. However, none of the formulas that I have written down so far are useful for understanding the Nahm pole boundary condition. The reason is that if ∂M = ∅, then the preceding formulas (whose derivation involves integration by parts) will have boundary contributions if we are on a manifold with boundary, and those boundary contributions are divergent in the case of a solution with Nahm pole boundary behavior. This is inevitable because the expression that I called I in writing the Weitzenbock formula is divergent in the case of a solution with Nahm pole behavior. A formula like " (2.10) − Tr (V ∧ "V + W ∧ "W) = I + boundary correction must have a boundary correction −∞ in the case of a Nahm pole, since the left hand side is 0 and I = +∞. A Weitzenbock formula with such divergent terms is not likely to be useful. To get around this, the best we could hope for would be a Weitzenbock formula on R4+ in which I is replaced by a sum of squares of quantities whose vanishing characterizes the Nahm pole solution A = 0, φ = t · dx/y. The quantities that

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vanish4 in the Nahm pole solution are the curvature F , covariant derivatives and commutators that involve φy , namely Di φy and [φi , φy ], covariant derivatives of φb along R3 such as Da φb , and finally Wa = Dy φa + 12 εabc [φb , φc ]. (Nahm’s equation is Wa = 0.) What we need is true. Define the following sum of squares of the objects whose vanishing characterizes the Nahm pole solution: ⎛ "  1 2  I = − d4 x Tr ⎝ Fij + (Da φb )2 + (Di φy )2 2 i,j R3 ×R+ i a,b ?   2 2 + [φy , φa ] + Wa . a

a

Then there is an identity along the lines that we need: " . Tr V 2 + W 2 = I  + Δ (2.11) − M

where Δ is a certain boundary term " (2.12) Δ=−

∂R4+

Tr (φ ∧ F + . . . ) .

(I have omitted some further terms in Δ.) ' Δ is the sum of a contribution at the boundary y = 0 and on a large hemisphere x2 + y 2 >> 1. Now to get a vanishing theorem that will say that the global Nahm pole solution is the only solution on R4+ that obeys Nahm pole boundary conditions, we need to do the following. ' We have to prove that if A, φ approach the Nahm pole solution for y → 0 and for x2 + y 2 → ∞, then they approach it fast enough so that Δ = 0. Once this is established, the Weitzenbock formula will say that a KW solution that satisfies the boundary condition must have I  = 0. But I  was constructed so that it vanishes for and only for the Nahm pole solution. To find the expected behavior of a solution for y → 0 is a matter of looking at an ODE in which one ignores the x dependence, since that is nonsingular. In effect, then, we just have to look at the eigenvalues of the linearization of Nahm’s equation (or more exactly a doubled version of Nahm’s equation with A as well as φ). Half of the linearized eigenvalues are negative and half are positive. The Nahm pole boundary condition amounts to setting to 0 the coefficients of perturbations with negative eigenvalues, and allowing the positive ones. The positive eigenvalues are large enough to ensure that Δ = 0 when the Nahm pole boundary condition is obeyed. This shows that there is no contribution' to Δ from the boundary at y = 0. To show that there is no contribution to Δ at x2 + y 2 → ∞, one needs to look at the eigenvalues of the “angular” part of the operator L, which is an operator on a 3 with Nahm pole boundary conditions along the boundary. Those hemisphere S+ eigenvalues determine how fast a solution will vanish at infinity, assuming that it does vanish at infinity. Again the spectrum is such that there is no contribution to Δ. 4 In the following, indices i, j take four values corresponding to x , x , x , y, but indices a, b 1 2 3 take only three values corresponding to x1 , x2 , x3 . Also, εabc is the Levi-Civita antisymmetric tensor.

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Figure 11. A model situation in which a knot is represented by a straight line K = R in the boundary of a half-space R4+ .

This leads to the nonlinear vanishing theorem – the global Nahm pole solution is the only solution on the half-space that satisfies the boundary conditions. Much the same argument proves a linearization of the same statement: the operator L obtained by linearizing around the Nahm pole solution has trivial kernel (and hence also trivial cokernel, since L is conjugate to −L† ). Together with the machinery of uniformly degenerate elliptic operators, this leads to the ellipticity of the Nahm pole boundary condition in the absence of knots. But what are we supposed to say in the presence of knots? As already noted, the knot will be in the boundary (fig. 10). To incorporate a knot in the boundary, we introduce a refinement of the Nahm pole boundary condition, such that (A, φ) obeys the Nahm pole boundary condition at a generic boundary point away from a knot, but has some more subtle behavior near the knot. To describe what this more subtle behavior should be, we consider the case that the knot is locally a straight line R ⊂ R3 , so we work on R4+ with a knot that lives on a straight line K in the boundary (fig. 11). The idea is going to be to find a singular model solution in the presence of the knot. This solution will coincide with the Nahm pole solution near a boundary point away from K, but it will look different near a point of K. The model solution will depend on the choice of an irreducible representation R∨ of the dual group G∨ . Then a boundary condition is defined by saying that one only allows solutions of the KW equations that look like the model solution near a knot. For this to make sense, the model solution must look the same near any point of K, so we assume that the model solution is invariant under translations along K. So we reduce to equations on R2 × R+ with the knot now represented by a point p ∈ R2 (fig. 12). Once we reduce to 3 dimensions (and assume vanishing of A1 and φy in a way that can be motivated by the Weitzenbock formula) the KW equations become tractable. Pick coordinates so that x1 runs along the knot K; x2 , x3 parametrize the normal plane to K in the boundary; and y measures the distance from the

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Figure 12. Assuming translation invariance along the K direction in fig. 11, we reduce to R3+ = R2 × R+ , with the knot now represented by a point p in the boundary. boundary. Define the three operators ∂ ∂ +i + [A2 + iA3 , · ] ∂x2 ∂x3 ∂ D2 = Dy − i[φ1 , · ] = + [Ay − iφ1 , · ] ∂y D3 = [φ2 − iφ3 , · ],

D1 = D2 + iD3 =

and also the “moment map” μ = F23 − [φ2 , φ3 ] − Dy φ1 . The KW equations in this situation become (2.13)

0 = [Di , Dj ], i, j = 1, . . . , 3

along with a “moment map” condition (2.14)

μ = 0.

These equations were introduced in [15] and were called the extended Bogomolny equations. They are a sort of hybrid of three much-studied equations in the mathematics of gauge theory. If we drop D1 (by assuming that the fields are independent of x2 and x3 and that A2 = A3 = 0), we get Nahm’s equation; if we drop D2 (by assuming that the fields are independent of y and that Ay = φ1 = 0) we get Hitchin’s equation; and if we drop D3 (by setting φ2 = φ3 = 0), we get the Bogomolny equations. The full system of equations is tractable for the same reason each of those three specializations is. There are two key facts: (a) the equations [Di , Dj ] = 0 are invariant under GC -valued gauge transformations (GC is the complexification of G); (b) the combination of setting μ = 0 and dividing by G-valued gauge transformations is equivalent to forgetting the condition μ = 0 and dividing by GC -valued gauge transformations. This means that the solutions can be understood in terms of complex geometry.

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It is reasonable to expect that the model solution possesses the symmetries of the knot. So we assume that the model solution is invariant under a rotation of the boundary R2 around the point p ∈ R2 at which the knot lives, and also invariant under a scaling of R2 × R+ keeping p fixed. With these assumptions, the equations [Di , Dj ] = μ = 0 reduce to affine Toda equations which are integrable. One can find all the solutions in closed form, and the solutions that satisfy the Nahm pole boundary condition away from the knot are classified by an irreducible representation R∨ of the dual group G∨ . These solutions were found for G = SO(3), G∨ = SU (2) in [3] and more generally in [25]. How would one go about proving that the KW equations with a boundary condition defined by one of these model equations is a well-posed (elliptic) problem? Basically, modulo generalities about uniformly degenerate elliptic operators, we need to show that the operator L obtained by linearizing around one of these solutions has no kernel or cokernel. It is again sufficient to show that the kernel vanishes, since L† is conjugate to −L. Just as in the absence of a knot, we will actually find a nonlinear analog of the vanishing of the kernel of L: any solution of the KW equations on R3 × R+ (with the knot as an infinite straight line in the boundary, as before) that is asymptotic to the model solution both along the boundary and at infinity actually coincides with it. The vanishing results we want are the sort that often follow from a Weitzenbock formula. But none of the Weitzenbock formulas that we considered before are welladapted to the presence of a knot. Even the more subtle Weitzenbock formula that includes the Nahm pole singularity away from a knot " . Tr V 2 + W 2 = I  + Δ (2.15) − M

does not give any useful information, because I  (which is the sum of squares of quantities that vanish in the Nahm pole solution without a knot) is divergent in the presence of a knot so I  will be +∞ and hence Δ will be −∞ with a knot present. So we need a new Weitzenbock formula. We imitate what we did before. We find a collection of quantities Xi whose vanishing characterizes the model solution. (Some obvious Xi are real and imaginary parts of [Di , Dj ], and also μ; the others are quantities like [φi , φy ] that vanish because the model solution has A1 = φy = 0.) Then if " Tr Xi2 , (2.16) I  = − i

R3 ×R+

we have to hope that there is an identity "  (2.17) − Tr (V ∧ "V + W ∧ "W) = I  + Δ, R3 ×R+

 is a new boundary term. It turns out that there is indeed an identity like where Δ this.  = 0 in the case of a solution that obeys the KW We still need to show that Δ equations and is asymptotic near the knot to the model solution. For this, one needs to know what is the asymptotic behavior near the boundary and at infinity of a solution of the KW equations. We already know the behavior at a generic boundary point, which was used in our proof of the well-posedness of the Nahm

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Figure 13. To prove the vanishing theorem in the presence of a knot, one has to study the angular operator near a singular point on the boundary. The angular operator is defined on the indicated hemisphere.

pole boundary condition without a knot. To find the behavior near the knot and also at infinity, we now need to solve the angular part of the equation on a 2d 2 (fig. 13). hemisphere S+ Again it turns out that the eigenvalues of the angular operator are favorable,  either near p or at infinity. This fact together with so there is no contribution to Δ the relevant Weitzenbock formula imply that a solution of the KW equations on R3 × R+ that is asymptotic on the boundary and at infinity to the model solution with the knot actually coincides with that model solution. A linearized version of the same argument shows that the kernel of L vanishes, which is what we actually needed to know for ellipticity. This is the main step in showing that L is a Fredholm operator in the presence of an arbitrary knot K embedded in any three-manifold W . Some details are still needed to show that this gives an elliptic boundary condition for the nonlinear KW equations in the general case of a curved knot [22]. Appendix A. Some Physics Background In this appendix, I will briefly describe some physics background to the treatment of singly-graded Khovanov homology in Lecture One. Only a bare outline of the string/M-theory context and the framework of [3] for doubly-graded Khovanov homology will be given. I aim primarily to explain what is different for the singly-graded theory. The starting point is the existence of a six-dimensional superconformal field theory with (0, 2) supersymmetry, associated to any simply-laced Lie group G, or more precisely to its Dynkin diagram. This theory has a Spin(5) group of Rsymmetries. Making use of a subgroup F = Spin(2) ⊂ Spin(5), the theory can be topologically twisted in such a way that it can be compactified on a Riemann surface C to give a four-dimensional theory with N = 2 supersymmetry. These are the theories of class S, as studied in [26]. They have an R-symmetry group F = (Spin(3) × Spin(2))/Z2 ∼ = U(2) (the subgroup of Spin(5) that commutes with F ). Using the Spin(3) factor, a theory of class S can be topologically twisted and

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compactified on a four-manifold M in a way that preserves one supercharge Q. This gives a theory on M ×C that is topological along M and holomorphic along C. (The holomorphy along C means, for example, that the cohomology of Q acting on local operators on M × C varies homomorphically on C.) The topological-holomorphic theory on M × C still has an F = Spin(2) symmetry, because F commutes with the group F that was used in the twisting. The underlying six-dimensional (0, 2) model admits half-BPS surface operators that can be supported on any two-manifold U in six dimensions. However, when the theory is formulated on M × C as summarized in the last paragraph, if we wish to preserve the supercharge Q of the topological-holomorphic theory, the possible choices of U are quite limited. U must be of the form p × C, where p is a point in M , or Σ × q, where Σ ⊂ M is a two-manifold and q is a point in C. The reason for this is essentially that any (complete, connected) complex submanifold of C is C itself or a point q ∈ C. For constructing Khovanov homology, we take U = Σ × q. To get singly-graded Khovanov homology, we take M to be simply R4 , and C to be a cylinder R × S 1 . The six-manifold M × C is then simply R4 × R × S 1 . The supercharge Q is invariant under rotations of R4 (combined with a suitable element of Spin(3) ⊂ F) but not under more general rotations of R4 ×R. Now we use the fact that the (0, 2) model, when formulated on M  × S 1 for any five-manifold M  , with the radius of M  being much greater than that of S 1 , reduces at long distances on M  to maximally supersymmetric gauge theory, with gauge group G. In the context of the topological-holomorphic theory described above, this reduction is valid without taking any large distance limit. The resulting supersymmetric gauge theory on M  = R4 × R is infrared-free (in sharp contrast to the underlying (0, 2) model in six dimensions) and can be analyzed by classical methods. In particular, the condition for Q-invariance becomes the HW equations, which were mentioned in Lecture One. These are equations for a pair consisting of a gauge field A, and a field B on R4 × R that is an adjoint-valued section of the pullback to R4 × R of the bundle of selfdual two-forms on R4 . A surface operator in six dimensions supported on Σ × q reduces in the gauge theory description to an ’t Hooft-like surface operator supported on Σ × q  , where q ∈ R×S 1 projects to q  ∈ R. A solution of the HW equations in the presence of this surface operator is supposed to have a singularity along Σ × q  . This codimension three singularity should be modeled on the standard codimension three singularity of the Bogomolny equations, suitably embedded in the HW equations with gauge group G. To categorify the quantum knot invariants associated to a representation R∨ of a simply-laced5 compact Lie group G∨ , one studies the HW equations with gauge group G (the Langlands-GNO dual of G∨ , which in particular has the same Lie algebra as G∨ if G is simply-laced), and with the appropriate singularity along Σ × q  . The appropriate singularity is obtained, as discussed in Lecture One, by embedding a singular U (1) solution in G using the homomorphism ρ : u(1) → t ⊂ g that is dual to R∨ . The HW equations on R4 × R are compatible with the familiar codimension three singularity on a two-manifold V ⊂ R4 × R if and only if V is of the form Σ × q  with Σ ⊂ R4 , q  ∈ R. This statement is easily verified by inspection of the HW equations. The explanation that we have given here starting with the (0, 2) 5 If

G∨ is not simply-laced, one requires a refinement described in section 5.5 of [3].

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model serves to explain “why” it is true. The restriction to V ⊂ R4 × q  ⊂ R4 × R is completely essential for getting knot theory out of this construction, since the relevant topology would disappear if V were free to move in five dimensions. For example, Khovanov homology arises in the “time”-independent case Σ = Rt × K ⊂ Rt × R3 = R4 , where K is a knot in R3 and the first factor in R4 = Rt × R3 is parametrized by the “time.” If V were free to vary in R4 × R, then K would be free to vary in R3 × R (the product of the last two factors in R4 × R = Rt × R3 × R), and could be trivially unknotted. Much the same point was made in Lecture One. A more general V (not of the time-independent form Rt × K) is used to define the “morphisms” of Khovanov homology. What we have described corresponds to the singly-graded version of Khovanov homology (with only the cohomological grading and no “q”-grading). The single grading comes from the symmetry group F ∼ = Spin(2) that was maintained throughout the construction. To get doubly-graded Khovanov homology, one takes C to be not C∗ = R × S 1 but C. This is done by adding to C∗ a point q0 “at infinity.” C admits an S 1 action, leaving fixed the point q0 . In the underlying (0, 2) model, one considers a surface operator supported on U = Σ × q0 . Reduction of M × C on the orbits of S 1 leads now to a description in terms of gauge theory on M × R+ (where R+ , a half-line, is the quotient C/S 1 ). The S 1 action leads to the desired second grading. This doubly-graded version of the construction was the main subject of [3], and the details will not be repeated here. Acknowledgements I thank M. Abouzaid, C. Manolescu, and R. Mazzeo for discussions. References [1] S. Gukov, A. Schwarz, and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005), no. 1, 53–74. MR2193547 [2] H. Ooguri and C. Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), no. 3, 419–438. MR1765411 [3] E. Witten, Fivebranes and knots, Quantum Topol. 3 (2012), no. 1, 1–137. MR2852941 [4] E. Witten, Khovanov homology and gauge theory, Proceedings of the Freedman Fest, Geom. Topol. Monogr., vol. 18, Geom. Topol. Publ., Coventry, 2012, pp. 291–308. MR3084242 [5] E. Witten, “Two Lectures On The Jones Polynomial And Khovanov Homology,” arXiv:1401.6996. [6] P. Seidel and I. Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006), no. 3, 453–514. MR2254624 [7] C. Manolescu, Nilpotent slices, Hilbert schemes, and the Jones polynomial, Duke Math. J. 132 (2006), no. 2, 311–369. MR2219260 [8] M. Abouzaid and I. Smith, “Khovanov Homology From Floer Cohomology,” arXiv:1504.01230. [9] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR766964 [10] M. Atiyah, Floer homology, The Floer memorial volume, Progr. Math., vol. 133, Birkh¨ auser, Basel, 1995, pp. 105–108. MR1362825 [11] D. Gaiotto and E. Witten, Knot invariants from four-dimensional gauge theory, Adv. Theor. Math. Phys. 16 (2012), no. 3, 935–1086. MR3024278 [12] J. Kamnitzer, The Beilinson-Drinfeld Grassmannian and symplectic knot homology, Grassmannians, moduli spaces and vector bundles, Clay Math. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 2011, pp. 81–94. MR2807850 [13] S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves. I. The sl(2)-case, Duke Math. J. 142 (2008), no. 3, 511–588. MR2411561

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[14] G. ’t Hooft, On the phase transition towards permanent quark confinement, Nuclear Phys. B 138 (1978), no. 1, 1–25. MR0489444 [15] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1–236. MR2306566 [16] S. Bigelow, A homological definition of the Jones polynomial, Invariants of knots and 3manifolds (Kyoto, 2001), Geom. Topol. Monogr., vol. 4, Geom. Topol. Publ., Coventry, 2002, pp. 29–41. MR2002601 [17] R. J. Lawrence, A functorial approach to the one-variable Jones polynomial, J. Differential Geom. 37 (1993), no. 3, 689–710. MR1217166 [18] D. Galakhov and G. W. Moore, “Comments On The Two-Dimensional Landau-Ginzburg Approach,” arXiv:1607.04222. [19] A. Haydys, Fukaya-Seidel category and gauge theory, J. Symplectic Geom. 13 (2015), no. 1, 151–207. MR3338833 [20] S. Gukov, P. Putrov, and C. Vafa, “Fivebranes and 3-manifold Homology,” arXiv:1602.05302. [21] R. Mazzeo and E. Witten, The Nahm pole boundary condition, The influence of Solomon Lefschetz in geometry and topology, Contemp. Math., vol. 621, Amer. Math. Soc., Providence, RI, 2014, pp. 171–226. MR3289327 [22] R. Mazzeo and E. Witten, The Nahm pole boundary condition, The influence of Solomon Lefschetz in geometry and topology, Contemp. Math., vol. 621, Amer. Math. Soc., Providence, RI, 2014, pp. 171–226. MR3289327 [23] W. Nahm, “All Selfdual Multimonopoles for Arbitrary Gauge Groups” (NATO ASI, 1983). [24] K. Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988), no. 3, 361–382. MR965220 [25] V. Mikhaylov, On the solutions of generalized Bogomolny equations, J. High Energy Phys. 5 (2012), 112, front matter+17. MR3042947 [26] D. Gaiotto, N = 2 dualities, J. High Energy Phys. 8 (2012), 034, front matter + 57. MR3006961 School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersy 08540

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99 Vicente Mu˜ noz, Ivan Smith, and Richard P. Thomas, Editors, Modern Geometry, 2018 96 Si Li, Bong H. Lian, Wei Song, and Shing-Tung Yau, Editors, String-Math 2015, 2017 95 Izzet Coskun, Tommaso de Fernex, and Angela Gibney, Editors, Surveys on Recent Developments in Algebraic Geometry, 2017 94 Mahir Bilen Can, Editor, Algebraic Groups: Structure and Actions, 2017 93 Vincent Bouchard, Charles Doran, Stefan M´ endez-Diez, and Callum Quigley, Editors, String-Math 2014, 2016 92 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, Lie Algebras, Lie Superalgebras, Vertex Algebras and Related Topics, 2016 91 V. Sidoravicius and S. Smirnov, Editors, Probability and Statistical Physics in St. Petersburg, 2016 90 Ron Donagi, Sheldon Katz, Albrecht Klemm, and David R. Morrison, Editors, String-Math 2012, 2015 89 D. Dolgopyat, Y. Pesin, M. Pollicott, and L. Stoyanov, Editors, Hyperbolic Dynamics, Fluctuations and Large Deviations, 2015 88 Ron Donagi, Michael R. Douglas, Ljudmila Kamenova, and Martin Rocek, Editors, String-Math 2013, 2014 87 Helge Holden, Barry Simon, and Gerald Teschl, Editors, Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday, 2013 86 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, Recent Developments in Lie Algebras, Groups and Representation Theory, 2012 85 Jonathan Block, Jacques Distler, Ron Donagi, and Eric Sharpe, Editors, String-Math 2011, 2012 84 Alex H. Barnett, Carolyn S. Gordon, Peter A. Perry, and Alejandro Uribe, Editors, Spectral Geometry, 2012 83 Hisham Sati and Urs Schreiber, Editors, Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, 2011 82 Michael Usher, Editor, Low-dimensional and Symplectic Topology, 2011 81 Robert S. Doran, Greg Friedman, and Jonathan Rosenberg, Editors, Superstrings, Geometry, Topology, and C ∗ -algebras, 2010 80 D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, and M. Thaddeus, Editors, Algebraic Geometry, 2009 79 Dorina Mitrea and Marius Mitrea, Editors, Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, 2008 78 Ron Y. Donagi and Katrin Wendland, Editors, From Hodge Theory to Integrability and TQFT, 2008 77 Pavel Exner, Jonathan P. Keating, Peter Kuchment, Toshikazu Sunada, and Alexander Teplyaev, Editors, Analysis on Graphs and Its Applications, 2008 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/pspumseries/.

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PSPUM

99

˜ et al., Editors Modern Geometry • Munoz

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