Surveys in differential geometry, Vol.16, Geometry of special holonomy and related topics 1571462112, 978-1-57146-211-4

Table of contents :
Content: Gauge theory in higher dimensions, II / Simon Donaldson and Ed Segal --
An invitation to toric degenerations / Mark Gross and Bernd Siebert --
Lectures on generalized geometry / Nigel Hitchin --
Generalized Donaldson-Thomas invariants / Dominic Joyce --
Geometric structures on Riemannian manifolds / Naichung Conan Leung --
Sasaki-Einstein manifolds / James Sparks --
A survey of geometric structure in geometric analysis / Shing-Tung Yau.

Citation preview

Surveys in Differential Geometry Vol. 1:

Lectures given in 1990 edited by S.-T. Yau and H. Blaine Lawson

Vol. 2:

Lectures given in 1993 edited by C.C. Hsiung and S.-T. Yau

Vol. 3:

Lectures given in 1996 edited by C.C. Hsiung and S.-T. Yau

Vol. 4:

Integrable systems edited by Chuu Lian Terng and Karen Uhlenbeck

Vol. 5:

Differential geometry inspired by string theory edited by S.-T. Yau

Vol. 6:

Essays on Einstein manifolds edited by Claude LeBrun and McKenzie Wang

Vol. 7:

Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer edited by S.-T. Yau

Vol. 8:

Papers in honor of Calabi, Lawson, Siu, and Uhlenbeck edited by S.-T. Yau

Vol. 9:

Eigenvalues of Laplacians and other geometric operators edited by A. Grigor’yan and S-T. Yau

Vol. 10:

Essays in geometry in memory of S.-S. Chern edited by S.-T. Yau

Vol. 11:

Metric and comparison geometry edited by Jeffrey Cheeger and Karsten Grove

Vol. 12:

Geometric flows edited by Huai-Dong Cao and S.-T. Yau

Vol. 13:

Geometry, analysis, and algebraic geometry edited by Huai-Dong Cao and S.-T.Yau

Vol. 14:

Geometry of Riemann surfaces and their moduli spaces edited by Lizhen Ji, Scott A. Wolpert, and S.-T. Yau

Vol. 15:

Perspectives in mathematics and physics: Essays dedicated to Isadore Singer’s 85th birthday edited by Tomasz Mrowka and S.-T. Yau

Vol. 16:

Geometry of special holonomy and related topics edited by Naichung Conan Leung and S.-T. Yau

Volume XVI

Surveys in Differential Geometry Geometry of special holonomy and related topics

edited by Naichung Conan Leung and Shing-Tung Yau

International Press www.intlpress.com

Series Editor: Shing-Tung Yau Surveys in Differential Geometry, Vol. 16 (2011) Geometry of special holonomy and related topics Volume Editors: Naichung Conan Leung (The Chinese University of Hong Kong) Shing-Tung Yau (Harvard University) 2010 Mathematics Subject Classification. 00Bxx, 14M25, 14N35, 18Exx, 53-XX, 53C15, 53C25, 53D18, 58-XX, 70S15, 81T13.

Copyright © 2011 by International Press Somerville, Massachusetts, U.S.A. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is material in articles to which the author holds the copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author. (Copyright ownership is indicated in the notice on the first page of each article.) ISBN 978-1-57146-211-4 Printed in the United States of America. 15 14 13 12 11

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Surveys in Differential Geometry XVI

Preface

Riemannian geometry is a very rich subject in itself, having close relationships with many different branches of mathematics, and with the sciences in general. In physics, Einstein theory of general relativity taught us the importance of Einstein metrics. However, it is very difficult to find Einstein metrics on general manifolds. The most important existence result on the existence of Einstein metrics on a compact Riemannian manifold is the theorem of the second editor which says that if a manifold is K¨ ahler, with zero first Chern class, then it admits K¨ahler metrics with zero Ricci curvature. Such a manifold is called a Calabi-Yau manifold. The K¨ahlerian condition means that there is a complex structure which is compatible with the Riemannian metric up to first order. This extra structure reduces the holonomy group of the Levi-Civita connection of the Riemannian metric from the orthogonal group O (2n) to the unitary group U (n). On a Calabi-Yau manifold, the holonomy group further reduces to the special unitary group SU (n), due to the existence of a parallel holomorphic volume form. When the Riemannian manifolds has more parallel tensors, their holonomy groups will be even smaller, for instance Sp (n), G2 or Riemannian symmetric spaces. All possible Riemannian holonomy groups were classified by Berger in the 1950s. In the article “Geometric structures on Riemannian manifolds,” included in this volume, the author describes various holonomy groups and their corresponding geometries. All of them can be described in uniform manners in terms of normed division algebras and orientability. In the 1980s, physicists studying string theory found that our spacetime should be ten–dimensional. Besides the usual spacetime R3,1 , the remaining dimensions are warped in a tiny Calabi-Yau threefold. Furthermore a physical duality in string theory can be translated into a duality between the complex geometry and the symplectic geometry on different CalabiYau manifolds. Kontsevich has proposed an interpretation of the concept of Mirror symmetry, which was based on super-conformal algebra, by linking the derived category of one Calabi-Yau manifold with the Fukaya category of the mirror Calabi-Yau manifold. Kontsevich gave a talk on this subject in 2010 at the Fifth International Congress of Chinese Mathematicians v

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PREFACE

(ICCM). His proposed correspondence is now called homological mirror symmetry. Strominger, Yau, and Zaslow argued that physically mirror symmetry is a T-duality. Namely, that mirror Calabi-Yau manifolds ought to admit dual special Lagrangian torus fibrations, at least in the large complex structure limits. The base of any (singular) Lagrangian fibration has a natural (singular) affine structure. In the large complex structure limits, this affine structure is expected to break into simple pieces. For instance, a large complex structure limit of the quartic K3 surfaces is the degeneration to the union of coordinate hyperplanes in the projective three space. The corresponding limiting affine structure becomes four triangles forming the boundary of a tetrahedron. In the article “An invitation to toric degenerations,” Gross and Siebert describe a canonical construction of degenerations into a union of toric varieties. These include large complex structure limits for Calabi-Yau complete intersections in toric varieties. More importantly, in their construction, the complex geometry of the family can be read off from the tropical datum in the limit, at least in principle. This is an important step into proving the SYZ mirror symmetry conjecture. In string theory, the Calabi-Yau manifold should also be equipped with B-field. Roughly speaking, it is a harmonic two–form on the manifold. The B-field can be used to transformed the generalized geometry of the manifold. In generalized geometry, the tangent bundle T is replaced by the direct sum T ⊕ T ∗ . That is to say, going from GL(2n, R) to O(2n, 2n) as T ⊕ T ∗ has a canonical split-definite inner product. Similarly, (linear) generalized complex structure is a U (n, n) structure on T ⊕ T ∗ . Amazingly this notion includes both the complex structures and the symplectic structures as special cases. Generalized geometry has also played important roles in mirror symmetry. In the article “Lectures on generalized geometry,” Hitchin give a wonderful lecture on generalized geometry. He also gives a proof of Goto’s existence theorem for generalized K¨ ahler structures using deformation theory. In string theory, Calabi-Yau manifolds of complex dimension three are special, as they are internal manifolds in our ten–dimensional spacetimes. Recall that in low dimensional topology, for real three dimension oriented manifolds, Casson invariants count the number of flat bundles over these manifolds. These invariants can be refined to define a homology theory, called the Chern-Simons Floer homology groups, such that Casson invariants are their Euler characteristics. Donaldson and Thomas, in an earlier paper “Gauge theory in higher dimensions,” define a complex version for Casson invariants, called the Donaldson-Thomas invariants, which count the number of stable holomorphic bundles over Calabi-Yau threefolds. In the paper “Gauge theory in higher dimensions, II” within this volume, Donaldson and Segal explain how we should generalize the Chern-Simons Floer theory to the complex setting, namely a holomorphic vector bundle over the moduli space of Calabi-Yau threefolds whose rank is the Donaldson-Thomas invariants.

PREFACE

vii

Their construction uses the G2 -geometry of real seven dimensional manifolds. The natural embedding SU (3) ⊂ G2 also explains why Calabi-Yau manifolds of complex dimension three is of particular interest. Joyce and Song have developed a complete theory for Donaldson-Thomas invariants for coherent sheaves on Calabi-Yau threefolds and studied their wall-crossing properties. In “Generalized Donaldson-Thomas invariants,” Joyce has here summarized their important work. Odd-dimensional analogs of Calabi-Yau manifolds are Sasaki-Einstein manifolds. Namely, a link of a Calabi-Yau metric cone is a Sasaki-Einstein manifolds. They also play important roles in string theory and duality in physics. In “Sasaki-Einstein manifolds,” Sparks gives an exposition of the Sasaki-Einstein geometry and describes various constructions and obstructions of these metrics. We have seen that special geometry is a very rich and fascinating subject. It has an intimate relationship with physics which benefits both subjects enormously. The Editors

Surveys in Differential Geometry XVI

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Gauge theory in higher dimensions, II Simon Donaldson and Ed Segal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

An invitation to toric degenerations Mark Gross and Bernd Siebert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Lectures on generalized geometry Nigel Hitchin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generalized Donaldson–Thomas invariants Dominic Joyce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Geometric structures on Riemannian manifolds Naichung Conan Leung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Sasaki-Einstein manifolds James Sparks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A survey of geometric structure in geometric analysis Shing-Tung Yau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Surveys in Differential Geometry XVI

Gauge theory in higher dimensions, II Simon Donaldson and Ed Segal

Contents 1. 2.

Introduction Basic set-up 2.1. Exceptional holonomy 2.2. Gauge theory and submanifolds 3. Taming forms 3.1. Dimension 8 3.2. Dimension 7 3.3. Dimension 6 4. Gauge theory on tamed almost-G2 -manifolds with tubular ends 4.1. Compact 7-manifolds 4.2. 7-manifolds with tubular ends 4.3. Holomorphic bundles over moduli space 5. Finite-dimensional analogue 5.1. Morse-Novikov Theory 5.2. Discussion 6. Interaction between gauge theory and calibrated geometry 6.1. G2 -instantons and associative submanifolds 6.2. Implications 6.3. Codimension-3 theories and monopoles References

1 2 2 5 9 9 10 12 15 15 17 19 23 23 28 30 30 36 38 40

1. Introduction In this paper we follow up some of the ideas discussed in [6]. The theme of that article was the possibility of extending familiar constructions in gauge theory, associated to problems in low-dimensional topology, to higher dimensional situations, in the presence of an appropriate special geometric structure. The starting point for this was the “holomorphic Casson invariant”, counting holomorphic bundles over a Calabi-Yau 3-fold, analogous to the Casson invariant which counts flat connections over a c 2011 International Press

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S. DONALDSON AND E. SEGAL

differentiable 3-manifold. This was developed rigorously by Richard Thomas [36] in an algebro-geometric framework, and the ideas have been taken up by various authors [25]. From a differential-geometric standpoint one can make parallel discussions of two cases: one involving gauge theory and one involving submanifolds. There has been a considerable amount of work in a similar direction, involving ideas of Topological Quantum Field Theory [23], [31]. In the familiar gauge theory picture one views the Casson invariant as the Euler characteristic of the instanton Floer homology groups. Thus it is natural to hope for some analogous structure associated to a Calabi-Yau 3-fold. This was discussed in a general way in [6] but the discussion there did not pin down exactly what structure one could expect. That is the goal of the present paper. In brief, we will argue that one should hope to find a holomorphic bundle over the moduli space of Calabi-Yau 3-folds, of rank equal to the holomorphic Casson invariant (sometimes called the DT invariant) defined by Thomas. Just as in [6], many of the arguments here are tentative and speculative, since the fundamental analytical results that one would need to develop a theory properly are not yet in place. These have to do with the compactness of moduli spaces of solutions. While considerable progress has been made in this direction by Tian [38],[32], a detailed theory—in either the gauge theory or submanifold setting—seems still to be fairly distant. The issues are similar to those involved in “counting” special Lagrangian submanifolds in Calabi-Yau manifolds, which have been considered by Joyce [20] but where, again, a final theory is still lacking. The core of this article is Section 4, where we explain how to construct holomorphic bundles over Calabi-Yau moduli spaces, assuming favourable properties of a “(6 + 1)-dimensional” differential-geometric theory. The preceding Sections 2 and 3 develop background material, mostly fairly standard but introducing a point of view involving “taming forms”. In Section 5 we explain how our construction matches up with standard algebraic topology, following the familiar Floer-theory philosophy. In Section 6 we go back to discuss the central, unresolved, compactness issues. We explain the relevance of recent work of Haydys which brings in a version of the “Fueter equation”. This perhaps points the way to a unification of the gauge theory and calibrated geometry discussions and connections with the more algebrogeometric approach. We are very grateful to Richard Thomas and Dominic Joyce for many discussions of this material. The paper has been substantially revised following comments of Joyce on an earlier draft. We are also grateful to Andriy Haydys for allowing us to present part of his forthcoming work. 2. Basic set-up 2.1. Exceptional holonomy. We will begin by recalling standard material on exceptional holonomy. Some references are [19], [30]. Start

GAUGE THEORY IN HIGHER DIMENSIONS, II

3

with the positive spin representation of Spin(8) on the 8-dimensional real vector space S + . The basic fact is that this action maps onto the orthogonal group SO(S + ) = SO(8). Likewise for the negative spin representation. This is the phenomenon of “triality”: there are automorphisms of Spin(8) permuting the three representations S + , S − , R8 . In particular the stabiliser in Spin(8) of a unit spinor in S + is a copy of Spin(7) ⊂ Spin(8), which maps to Spin(7) ⊂ SO(8). A Riemannian 8-manifold X with a covariant constant unit spinor field has holonomy contained in Spin(7). In this situation we have a decomposition of the 2-forms (1)

Λ2 = Λ221 ⊕ Λ27 ,

where Λ221 corresponds to the Lie algebra of Spin(7), under the isomorphism complement. There is also a parallel Λ2 = so(8), and Λ27 is the orthogonal  4-form Ω which is equal to ( θi2 )/7 for any orthonormal basis θi of Λ27 . We can see this form in a useful explicit model. Suppose we have two copies R41 , R42 of R4 , each with spin structures. Then the positive spin space of R41 ⊕ R42 is the real part of     + 4 (2) S (R1 ) ⊗ S + (R42 ) ⊕ S − (R41 ) ⊗ S − (R42 ) . (Recall that the spin spaces in 4-dimensions are quaternionic and the complex tensor product of two quaternionic vector spaces has a natural real structure.) Fix an isomorphism Ψ between S + (R41 ) and S + (R42 ). We can regard Ψ as an element of the tensor product and we get a distinguished spinor in 8 dimensions. In other words we have a subgroup H of Spin(7) ⊂ SO(8), locally isomorphic to SU (2) × SU (2) × SU (2), consisting of automorphims of R8 which preserve the decomposition R41 ⊕ R42 and Ψ. In this picture the 4-form Ω corresponding to our distinguished spinor is (3)

dx1 dx2 dx3 dx4 + dy1 dy2 dy3 dy4 +

3 

ωi ∧ ωi .

i=1

Here xi , yi are standard co-ordinates on the two copies of R4 , ωi is a standard orthonormal basis for Λ+ (R41 ) and ωi the basis of Λ+ (R42 ) which corresponds to this under the isomorphism induced by Ψ. In fact this form Ω determines the spinor, so we could also define Spin(7) ⊂ GL(8, R) to be the stabiliser of this 4-form. The GL(8, R) orbit of Ω is a 43-dimensional submanifold A ⊂ Λ4 R8 , which can be viewed as GL(8, R)/Spin(7). On any 8-manifold we have a copy of A associated to each tangent space in the obvious way and a Spin(7) structure is equivalent to a closed 4-form which takes values in this subset. Now consider a unit spinor in S + and a unit vector in R8 . The stabiliser of the pair is the exceptional Lie group G2 , which can be regarded as a subgroup of SO(7). This means that a Riemannian product R × Y has holonomy contained in Spin(7) if and only if the holonomy of Y is contained

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in G2 ⊂ SO(7). On such a manifold Y we have a decomposition of the 2forms (4)

Λ2 = Λ214 ⊕ Λ27

where Λ214 corresponds to the Lie algebra of G2 . We have a covariant constant 3-form φ and 4-form σ, such that on the cylinder (5)

Ω = φdt + σ.

We can define the concept of a non-compact Riemannian 8 manifold with holonomy in Spin(7) and a tubular end modelled on (0, ∞) × Y , for a G2 -manifold Y . (That is, the metric differs from the product by an exponentially decaying term.) We can also consider a “neck-stretching” sequence of Spin(7)-structures on a compact manifold that degenerate to a limit which is the disjoint union of two such non-compact manifolds. That is, the metrics contain regions which are almost isometric to long finite tubes (−Ti , Ti ) × Y , where Ti → ∞. Now we can repeat the discussion, starting in 7 dimensions. Considering G2 ⊂ SO(7), the stabiliser of a unit vector is a copy of SU (3) ⊂ SO(6). A Riemannian product R × Z has a G2 -structure if and only if Z is a CalabiYau 3-fold, with holonomy in SU (3). Then we have a decomposition of the 2-forms (6)

Λ2 = Λ28 ⊕ Λ27 ,

where Λ28 corresponds to the Lie algebra of SU (3). There are covariant constant 3-forms ρ1 , ρ2 and a 2-form ω such that on the cylinder (7)

σ = ρ2 ds + ω 2 ,

φ = ωds + ρ1 .

Our notation here is that s is the co-ordinate on R. In fact either one of ρ1 , ρ2 determines the other since ρ2 = −Iρ1 , where I is the parallel complex structure. From another point of view, the complex combination θ = ρ1 + iρ2 is a holomorphic 3-form on Z. The 2-form ω lies in Λ27 , so we get a further decomposition (8)

Λ2 = Λ28 ⊕ Λ26 ⊕ ω .

Again we may consider G2 manifolds with tubular ends and neck-stretching sequences. Here we stop, although we could repeat the process to consider CalabiYau 3-folds with tubular ends, etc. This kind of neck-stretching sequence, and the converse “gluing theory” for manifolds with tubular ends, is central in the work of Kovalev [21], and many interesting new examples have been found recently by Kovalev and Nordstrom [22].

GAUGE THEORY IN HIGHER DIMENSIONS, II

5

2.2. Gauge theory and submanifolds. Next we review slightly less standard material on auxiliary differential geometric objects: submanifolds and connections. A fundamental reference for the first is the work of Harvey and Lawson [12]; a number of references for the second can be found in [6]. Start again in dimension 8. Take our model R41 ⊕ R42 above and consider the Spin(7)-orbit of the 4-plane R41 in the Grassmannian of oriented 4-planes in R8 . This is the set of Cayley 4-planes, and forms a 12-dimensional submanifold in the full Grassmannian (since the stabiliser of R41 in the 21-dimensional group Spin(7) is the 9 dimensional subgroup H). Another definition is that an oriented 4-plane Π is Cayley if the restriction of Ω to Π is the volume form. A third is that for any vector v in R8 (9)

iv (Ω)|Π = ∗Π (v  |Π ),

where ∗Π is the ∗ operator on Π induced by the metric and v  ∈ (R8 )∗ is the dual of v, again defined by the metric. Now in an 8-manifold with a Spin(7) structure we may consider Cayley submanifolds, whose tangent space at each point is Cayley. There are two fundamental properties of this condition: • Property A The condition is an elliptic PDE. As a check on this, note that the condition is given locally by 4 equations, since the set of Cayley subspaces has codimension 4 in the full Grassmannian, while 4-dimensional submanifolds near to a given one can be represented by sections of the four-dimensional normal bundle. In fact the linearisation of the condition is given by a version of the Dirac operator acting on sections of the normal bundle [26]. • Property B The submanifold is calibrated: its volume is the topological invariant given by the integral of the closed form Ω and it is a minimal submanifold, minimizing volume in its homology class. Next we move to gauge theory. We define a Spin(7)-instanton to be a connection on a bundle E, with structure group SU (l) say, whose curvature lies in Λ221 . Then we have, likewise: • Property A The equation is elliptic, modulo gauge equivalences. The linearised theory can be formulated in terms of a bundle-valued version of the elliptic complex (10)

π ◦d

7 Ω27 . Ω0 → Ω1 →

d

(Again, note the dimension check: 1 − 8 + 7 = 0.) • Property B The Yang-Mills energy is determined by the topology of the bundle. This comes from the algebraic fact that for α in Λ221 we have α ∧ α ∧ Ω = −|α|2 vol

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S. DONALDSON AND E. SEGAL

Thus for a Spin(7)-instanton:   2 |F | = − Tr(F 2 ) ∧ Ω = 8π 2 c2 (E) ∪ Ω [X]. X8

X8

It follows that Spin(7)-instantons are Yang-Mills connections. This theory has been developed in the thesis of C. Lewis [24]. Optimistically we could hope that these good properties would allow us to define invariants, modelled on the Gromov-Witten invariants of symplectic manifolds, in the submanifold case, and on the instanton invariants of 4-manifolds in the gauge theory case. In the simplest situation, where the relevant index or formal dimension is zero, one would seek to define a numerical invariant by counting solutions with appropriate signs. When the index is positive one could try to evaluate natural cohomology classes on the moduli space of solutions. There is a third property—which one might call the Floer picture—of the equations, which becomes crucial when one considers manifolds with tubular ends and neck-stretching sequences. Consider a tube I × Y , where I ⊂ R is an open interval, finite or infinite, and Y is a G2 -manifold with a 4-form σ. A connection A over I × Y yields a 1-parameter family At of connections over Y . Property C There is a locally-defined function f , on the space of connections over Y modulo gauge equivalence, such that Spin(7) instantons correspond to gradient curves of f , and (12)

d dAt 2 f (At ) = −

= − F (At ) ∧ σ 2 . dt dt

We know from analogous Floer-type theories that this is the essential property needed to control solutions over infinite tubes, and to obtain uniform control in neck-stretching sequences. The point is that one arrives in a situation where f (At ) is well-defined and the variation of f over the interval is known, so the gradient property gives bounds on   dAt 2

dt,

F (At ) ∧ σ 2 dt. dt I I To explain in a little more detail, we define a 1-form on the space of connections over Y by mapping a tangent vector δA to  (13) Tr(δA F ) ∧ σ. Y

This 1-form arises, locally in the space of connections modulo gauge equivalence, as the derivative of a function which can be written schematically as  (14) f (A) = CS(A) ∧ σ, Y

GAUGE THEORY IN HIGHER DIMENSIONS, II

7

where CS denotes the Chern-Simons form. Of course this is not really welldefined and a more precise definition is this. We choose a base point A0 and for any connection A we choose a connection A on a bundle over [0, 1] × Y with boundary values A0 and A. Then we define  2 σTrFA . (15) f (A) = [0,1]×Y

Just as in the usual Floer theory over 3-manifolds, this function is not globally well-defined, but the indeterminacy comes from the periods of σ over H4 (Y, Z). This indeterminacy is related the notion of an “adapted bundle” which we will discuss further in Section 4. But in any case we have a welldefined closed 1-form on the space of connections over Y . The Spin(7)instanton equation over the tube can be written as (16)

dAt = ∗(F (At ) ∧ σ) dt

which displays At as an integral curve of the vector field dual to this 1-form. Likewise in the submanifold set-up, we define a 1-form on the space of 3-dimensional submanifolds of Y by mapping a variation v—a vector field along a submanifold P 3 ⊂ Y 7 —to  (17) iv (σ). P

This 1-form is the derivative of a locally-defined function, determined by integrating σ over cobordisms in [0, 1] × Y , and we have an analogue of Property C above. Now we consider a connection on a bundle over Y 7 whose pull-back to the cylinder is a Spin(7)-instanton. Expressed directly over Y this condition is just that (18)

F ∧ σ = 0,

and we call the solutions G2 -instantons. In the Floer picture, these are viewed as the zeros of the 1-form on the space of connections. A 3-dimensional submanifold P ⊂ Y is called associative if R×P is Cayley. In the picture above, such a submanifold is viewed as a zero of the 1-form or critical point of the locally-defined function f . Associative submanifolds can be defined more directly by the condition that for any vector v ∈ T Y the restriction to P of the contraction iv (σ) vanishes. There is a basic algebraic model for the tangent space of Y at a point of an associative submanifold like that which we saw in 8 dimensions. We consider a 3-dimensional space R3 and a 2-dimensional complex vector space V with symmetry group SU (2). Then the tensor product of V with the spin space S = S(R3 ) has a real form R4 = (V ⊗ S)R and we have a natural isomorphism Λ2+ R4 = R3 . In other

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words, if y1 , y2 , y3 are standard co-ordinates on R3 we have a corresponding basis ω1 , ω2 , ω3 for the self-dual 2-forms on R4 . Then we have a 3-form  (19) φ= dyi ωi + dy1 dy2 dy3 , and a 4-form (20)

σ=



ωi dyj dyk + dx1 dx2 dx3 dx4 .

i,j,k cyclic

This gives a standard model for the tangent space of a G2 -manifold at a point of an associative submanifold, with yi the co-ordinates on the submanifold. A 4-dimensional submanifold Q of Y which is Cayley when regarded as a submanifold of R×Y is called co-associative. The condition can be defined more directly by saying that the restriction of the 3-form φ to Q vanishes. If we prefer, we can forget the 8-dimensional geometry and start directly in 7 dimensions, considering G2 manifolds and three kinds of differential geometric objects: associative submanifolds, coassociative submanifolds and G2 -instantons. These three conditions enjoy the same crucial Properties A,B discussed above. The equations are elliptic when set up suitably, but this is now less straightforward. At first sight, the G2 -instanton equation imposes 7 = dim Λ27 conditions on the curvature of a connection, whereas we would expect to only impose 7 − 1, taking account of gauge invariance. The explanation for this is that the curvature of any connection satisfies the Bianchi identity and the linearised problem can be formulated in terms of the elliptic complex (21)

0 → Ω0 → Ω1 → Ω6 → Ω7 → 0 d

σ∧d

d

Then the whole theory of local deformations of solutions to the G2 -instanton equations is closely analogous to the Taubes/Floer discussion of flat connections over 3-manifolds, with (21) taking the place of the de Rham complex over a 3-manifold. This theory has been developed in unpublished work of A. Tomatis and in the thesis of Henrique Sa Earp [29]. The associative condition is elliptic, with the linearisation given by a version of the Dirac operator acting on sections of the normal bundle. The co-associative condition is at first sight overdetermined, since it imposes 4 constraints on the sections of the 3-dimensional normal bundle of a 4-dimensional submanifold of Y 7 , but when set-up properly it becomes elliptic [26]. One way of doing this is to embed the discussion in that of Cayley submanifolds in the cylinder. Our third characterisation (10) of Cayley subspaces shows that if Q is any Cayley submanifold in R × Y then the R-coordinate t is a harmonic function on Q. Thus if Q is compact it must be a co-associative submanifold in some “fixed time” slice. In sum, we would optimistically hope first for a 7-dimensional theory, bearing on a compact G2 -manifold Y and yielding invariants counting associative submanifolds, G2 -instantons and co-associative submanifolds. In the

GAUGE THEORY IN HIGHER DIMENSIONS, II

9

first two cases these could be viewed as generalising the Casson invariant (since in each case we are counting the zeros of a closed 1-form). But second we could hope for a (7 + 1)-dimensional theory, assigning Floer groups to Y which should play the same role vis-` a-vis the Spin(7) discussion, for manifolds with tubular ends and neck-stretching limits, as the ordinary Floer theory does to 4-manifold invariants. Of course the second theory would be a refinement of the first, since we would view the Casson invariant as the Euler characteristic of the Floer groups. Now we repeat the discussion, dropping dimension again. The G2 instantons on R × Z 6 which are lifted up from Z are connections with (22)

F ∧ ω 2 = 0,

F ∧ ρ2 = 0.

(The conditions F ∧ρ1 = 0, F ∧ρ2 = 0 are equivalent.) From the point of view of complex geometry, the second condition is that F has type (1, 1), so the connection defines a holomorphic structure on the bundle. The other condition is the Hermitian Yang-Mills equation, and we know that the solutions correspond to “polystable” holomorphic bundles. Likewise a product R × Σ is co-associative if and only if Σ is a complex curve in Z 6 . These geometric objects in Z 6 again have the same good properties: they are defined by elliptic equations and have topological volume/energy bounds. We will take this discussion of these (6 + 1)-dimensional theories further in Section 4. One can also discuss co-associative submanifolds in a similar framework [23], [31]. The corresponding objects in 6 dimensions are the special Lagrangian submanifolds, which can be described as the critical points of a locally-defined functional [37]. The co-associative submanifolds of a tube are the gradient curves of this functional. 3. Taming forms 3.1. Dimension 8. We will now take another point of view, beginning again in 8-dimensions. Our model is Gromov’s notion of a symplectic form “taming” an almost-complex structure [11]. Let Ω0 be the standard form on R8 , as in equation (3). The convex hull of the set of negative squares −α ∧ α, for α in Λ221 is a proper cone K in Λ4 R8 so we have a dual cone of 4-forms Ω such that Ω ∧ χ > 0 for non-zero χ ∈ K. This is equivalent to saying that the quadratic form α → −α ∧ α ∧ Ω is positive definite on Λ221 . One can check that the subset of Cayley planes in Gr4 (R8 ) ⊂ Λ4 R8 lies in the cone K, so such a 4-form is also strictly positive on Cayley subspaces. Now suppose that Ω is any 4-form on an 8-manifold X which lies in the preferred subspace A at each point. We could call this an “almost Spin(7) structure”. Then Cayley submanifolds and Spin(7) instantons are defined and the equations are elliptic, just as before. However we lose the volume/energy identity for solutions. Suppose

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however we have a closed 4-form Ω which, at each point, lies in the dual cone above. Then for any Spin(7)-instanton we have a slightly weaker version of “Property B”: Property B    2 (23) |F | ≤ −C Tr(F 2 ) ∧ Ω , X

X

where the right hand side is a topological invariant. Here the constant C depends only on Ω, Ω . Similarly, we get a volume bound for Cayley submanifolds. It seems reasonable to hope that the putative theory in the case of genuine Spin(7) structures extends to this more general situation; in any case we will assume this is so for the purposes of our discussion. We call such a pair (Ω, Ω ) a tamed almost-Spin(7) structure on X. The advantage of this extension is that the notion is much more flexible. On the one hand, starting with a genuine Spin(7)-structure, it gives us scope to deform the equations, for example to achieve transversality. This is essentially the use we will make of the idea in Section 4. On the other hand, tamed almost Spin(7)-structures should be much easier to construct than Spin(7)-structures, since the condition on Ω is an open condition on a closed 4-form. 3.2. Dimension 7. We may consider dimension reductions of the theory, in this more general setting. Let V be an oriented 7-dimensional vector space and let GL+ (V ) be the linear automorphisms of positive determinant. There is an open GL+ (V )-orbit P3 ⊂ Λ3 V ∗ of “positive” forms, each of which has stabiliser isomorphic to the compact group G2 ⊂ SL(V ). Similarly there is an open orbit of positive 4-forms P4 ⊂ Λ4 V ∗ and a GL+ (V )equivariant diffeomorphism ∗ : P3 → P4 . We also denote the inverse map by ∗. The choice of notation is derived from the fact that each element φ ∈ P3 defines a natural Euclidean metric gφ on V and ∗φ is the usual Hodge dual defined by this metric. Likewise for any σ ∈ P4 . Now consider an oriented 7-manifold Y . At each point p ∈ Y we have open subsets P3,p ⊂ Λ3 T ∗ Yp , P4,p ⊂ Λ4 T ∗ Yp . We define an almost G2 structure on Y to be a 4-form σ on Y which lies in P4,p at each point p. Of course it is the same to start with a 3-form φ which lies in P3,p at each point, and an almost G2 structure defines a Riemannian metric on Y . If σ is an almost G2 -structure then the form (24)

Ω = σ + ∗σdt,

yields an almost Spin(7) structure on the cylinder R×Y . Let φ , σ  be respectively, a closed 3-form and 4-form on Y , so Ω = σ  + φ dt is closed 4-form on the cylinder. Then there is a certain open set of “taming pairs” (φ , σ  ) such that Ω is a taming form. The discussion of associative submanifolds in Y proceeds just as before. They are defined by the condition that the restriction of iv σ vanishes, for each tangent vector σ. The condition is an

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elliptic equation of Dirac type and their volume is controlled by the integral of φ over P , a topological invariant. The discussion of G2 -instantons is different in an important way. For a general σ the equation σ ∧ F = 0 for a connection is overdetermined and we do not expect any solutions. However we can consider connections over R × Y with the property that the restriction to each slice {t} × Y lies in a fixed gauge equivalence class. This corresponds to a pair (A, u) over Y where A is a connection on a bundle E and u is a section of the adjoint bundle gE . Then we have a connection A + udt on the lifted bundle over R × Y . The Spin(7)-instanton equation, expressed in 7-dimensions, becomes (25)

F ∧ σ = ∗dA u,

where ∗ is the Hodge ∗-operator of the metric gσ . This is an elliptic equation (modulo gauge equivalences) for the pair (A, u). This set-up is similar to one considered in early work of Thomas: the advantage in our present situation is that, given a taming pair (σ  , φ ), we still get a topological bound like (23) on the Yang-Mills energy. (Although σ  does not appear explicitly we need to use it in deriving the inequality.) Suppose however that σ is a closed 4-form on Y (in which case we might prefer to restrict attention to taming pairs with σ  = σ). Then the Bianchi identity implies that for any solution (A, u) as above (over a compact manifold Y ) we have dA u = 0. Thus in this case we do get a good theory of G2 -instantons, without the extra field u. The explanation is that when σ is closed we have a locally defined “Chern-Simons functional” just as before, with critical points the G2 -instantons, and the linearised theory can be expressed in terms of an elliptic complex (21). Moreover we have exactly the same gradient curve description as before. To sum up (1) If we restrict attention to tamed structures on 8-manifolds with tubular ends such that on each end the structure is defined by a closed 4-form σ on the cross section then we get a (7 + 1)dimensional differential geometric theory with the good Properties A, B  , C. So we have reasons to hope that some kind of Floer theory can be introduced into this more general and flexible, situation. The same remarks apply in the submanifold setting, for Cayley submanifolds with “associative limits”. (2) If, on the other hand, we are just interested in a compact 7-manifold Y we can study solutions (A, u) of equation (25) for any positive 4-form σ, not necessarily closed, so long as there are taming forms. This equation has properties A (with Fredholm index 0) and B  , so we expect to define a numerical invariant. This can be regarded as counting the zeros of a vector field on the space of connections modulo gauge, but the vector field is not dual to a closed 1-form. Again, the same remarks apply to aasociative submanifolds.

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Remark We note in passing that, by contrast, we expect a good theory of coassociative submanifolds for almost G2 structures where the 3-form ∗σ is closed, rather than the 4-form σ. The problem is that, for general σ the equations are overdetermined so we lose Property A. The condition d ∗ σ = 0 is the integrability condition for the overdetermined system, much as the condition dσ = 0 is the integrability condition for the G2 -instanton equations. One way of seeing this is to embed the discussion in that of Cayley submanifolds in the cylinder, as before. 3.3. Dimension 6. The discussion becomes considerably more confusing when we go down to 6 dimensions. We will need to have closed taming forms, comprising a 2-form ω  a pair of 3-forms ρ1 , ρ2 and a 4-form τ  . Thus on R2 × Z our taming form will be Ω = ω  dsdt + ρ1 dt + ρ2 ds + τ  . Then we will consider an SU (3)-structure on the tangent bundle. We recall that there is an open GL(6, R)-orbit of “positive” 3-forms in Λ3 R6 each of which determines an almost-complex structure. Thus we can take our SU (3)-structure to be specified by a 3-form ρ1 , which defines an almost complex structure and hence ρ2 = −Iρ1 and a 2-form ω which is a positive form of type (1, 1) with respect to this almost complex structure. The point to emphasise that, in this most general formulation, there are algebraic constraints on the forms ω, ρ1 but only open conditions on the taming forms. Then we get a wide variety of different extra conditions we can impose, intermediate between this most general formulation and the case of genuine Calabi-Yau structures. Start with a 7-manifold Y with tubular ends and a tamed almost G2 structure defined by a 4-form σ, not necessarily closed. Then we want to study the equation (25), for pairs (A, u) on Y . We need to have Property C: a gradient description on the ends. For this we consider a closed 3-form ρ and the closed 4-form τ on the cross-section Z and the functional  f (a, u) = CS(a) ∧ ρ + Tr(uF (a)) ∧ τ, Z

on pairs (a, u) over Z. Suppose for the moment that we take an arbitrary Riemannian metric on Z then we have a gradient equation (26)

da = ∗(F ∧ ρ + da u ∧ τ ), ds

du = ∗(F ∧ τ ) ds

On the other hand, if we can write the 4-form σ as σ = ρ2 ds+ω 2 the equation (25) becomes da du = ρ2 ∧ F + ∗da u, = ∗(F ∧ ω 2 ) ds ds So we need to arrange that these equations are the same. First we should take τ = ω 2 , that is we should suppose that ω 2 is closed. Second we should (27)

ω2 ∧

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suppose that ρ = Iρ2 for the almost complex structure defined by ρ2 , and we should use the standard metric associated to ω and this almost-complex structure. Then the equations (26) and (27) are equivalent. But we can also use ρ to define the same complex structure, and hence ρ2 . So if we start on Z with a closed positive 3-form ρ and a 2-form ω such that ω 2 is closed and ω is positive of type (1, 1) with respect to the almost-complex structure defined by by ρ we get a model for a tubular end on which equation (25) has a gradient description (Property C). We can take ρ, τ as two of our taming forms ρ1 , τ  and we also need another taming 3-form ρ = ρ2 and a taming 2-form ω  . We might want to suppose that in fact ω  = ω. (The type (1, 1) condition is the same as saying that ω ∧ ρ = 0.) In this way we obtain a (6+1)-dimensional theory with Properties A, B  , C. The stationary solutions over tubes correspond to pairs (a, u) on Z with (28)

F (a) ∧ ρ = da u ∧ ω 2

F ∧ ω 2 = 0.

The data we need over Z consists of closed forms ρ, ρ , ω but the only identity (as opposed to an open condition) that we need to impose is ρ ∧ ω = 0. We expect that we then get Floer groups associated to Z, related to the invariants of almost G2 -structures in 7-dimensions. If we want to study the 7-dimensional theory with σ closed, which fits into the (7 + 1) dimensional discussion, we are much more restricted. Then we need both ρ and Iρ to be closed 3-forms on Z, which can only occur if we have a genuine Calabi-Yau structure. In this case an integration-by-parts argument shows that for any solution of (28), da u vanishes so we are back to the equations (22). In the opposite direction, if we consider the most general set up with no particular relation between the closed taming 3-forms and the SU (3) structure, we can consider the system of equations for a triple (a, u, v) where a is a connection and u, v are sections of gE : (29)

F ∧ ρ = da u ∧ ω 2 ,

F ∧ (Iρ) = da v ∧ ω 2 ,

F ∧ ω 2 = [u, v]ω 3 .

We get a theory with Properties A, B  and we expect that counting solutions will generalise the holomorphic Casson invariant to this situation. Much of the above has a good formal interpretation. The space of connections A on a bundle E → Z has a symplectic form  δ1 , δ2 = Tr(δ1 ∧ δ2 ) ∧ ω 2 . Z

The symplectic quotient A//G by the gauge group is given by solutions of the equation F ∧ ω 2 = 0, modulo gauge equivalence. We have an induced symplectic form on A//G. Any closed 3-form ρ˜ yields a locally-defined function on the space A/G  fρ˜(a) = CS(a) ∧ ρ˜. Z

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The equation F ∧ ρ˜ = da u ∧ ω 2 is the condition defining a critical point of fρ˜, restricted to the symplectic quotient A//G ⊂ A/G. The evolution equation da ∧ ω 2 = F ∧ ρ˜, dt defines the Hamiltonian flow associated to the function fρ˜ on A//G. The difficulty in combining the (7 + 1) and (6 + 1)-dimensional theories is that we want to find another 3-form ρ such that we get the same flow as the gradient flow of fρ , and this seems to essentially restrict us to the CalabiYau case. However this restriction may not be fundamental. We expect that the (7 + 1) dimensional theory should be related, from the point of view of the 6-manifold Z, to a “Fukaya category” of Lagrangian submanifolds of A//G which would be, formally, something defined just by the symplectic structure of A//G. As usual, there is a corresponding discussion in the submanifold case. The infinite dimensional space of symplectic surfaces Σ ⊂ Z has a natural symplectic form, arising formally as a symplectic quotient [7]. Then we get locally-defined functions on this space by integrating closed 3-forms over 3-dimensional cobordisms. There is a variant of this discussion which yields deformation of the Special Lagrangian equations in a Calabi-Yau 3-fold. To explain this we set up some notation. Suppose L is a submanifold of a manifold M and ψ is a p-form on M . If the restriction of ψ to L vanishes then ψ defines a section of Ωp−1 (L, N ∗ ) where N is the normal bundle T M/T L of L in M . We will denote this section by ψN . Now consider a closed 3-form ρ and a closed 4-form τ on a 6-manifold Z. There is a locally defined functional fτ on the space of 3-dimensional submanifolds of Z, defined by integrating τ over 4-dimensional cobordisms. Let Cρ be the set of submanifolds P 3 ⊂ Z 6 such that the restriction of ρ to P vanishes. We consider the critical points of fτ restricted to Cρ . For any submanifold P ∈ Cρ the restrictions ρ|P , τ |P both vanish, the first by definition and the second for dimensional reasons, so we have well-defined bundle-valued forms ρN ∈ Ω2 (N ∗ ), τN ∈ Ω3 (N ∗ ). The Euler-Lagrange equation defining the critical points involves a function f on P (which appears as a “Lagrange multiplier”) and takes the form τN = df ∧ ρN . Thus we have a system of equations for a pair (P, f ): ρ|P = 0,

τN = df ∧ ρN ,

which are analogous to (28). (Of course we need to factor out the constant functions f .) When Z is a genuine Calabi-Yau manifold and ρ, τ are the standard forms the solutions are special Lagrangian submanifolds, as in [37], with f = 0. More generally if we write τ = ω 2 , where ω is not necessarily closed we can identify these pairs with associative submanifolds in the tube Z × R. One would expect that, for generic choices, the solutions (P, f ) are

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isolated and this deformation could be seen as removing a degeneracy in the special Lagrangian equations (which forces the latter to have solutions in moduli spaces of various dimensions, given by the first Betti number of P ). 4. Gauge theory on tamed almost-G2 -manifolds with tubular ends 4.1. Compact 7-manifolds. We will now make a slightly more detailed analysis of the (6 + 1)-dimensional theory. We will do this in the gauge theory setting, but a similar discussion applies for the submanifold case. We will also restrict attention to a case when the cross-sections of the ends are genuine Calabi-Yau manifolds. Suppose that Y is a compact 7-manifold with an almost G2 -structure defined by a 4-form σ. Let A be any connection on a bundle E → Y and form the sequence of operators (30)

Ω0 (gE ) → Ω1 (gE ) → Ω6 (gE ) → Ω7 (gE ),

as in (21). This is not in general a complex but we can make a single operator out of it in the usual way. Use the metric to identify Ωp with Ω7−p so we have (31)

DA : Ω0 ⊕ Ω1 → Ω0 ⊕ Ω1 .

The elliptic operator DA is self-adjoint if and only if σ is closed, but in any case its symbol is self-adjoint, so the index is zero. When σ is closed and A is a solution of the G2 -instanton equation we get an elliptic complex of Euler characteristic zero and we call A regular if the cohomology of this complex vanishes. This implies in particular that A is isolated in the moduli space of G2 -instantons. We will discuss briefly two more technical issues reducible conections and orientations. Suppose first that Y is manifold with holonomy equal to G2 . Then we know [19] that the harmonic 2-forms all lie in the Λ214 component in (6). This means that any complex line bundle L over Y admits a G2 -instanton connection, and in particular such a connection appears as a reducible solutions A0 on the bundle E = L ⊕ L−1 . The situation is in some respects similar to that for instantons over a 4-manifold with negative definite intersection form. The bundle gE splits as R ⊕ L2 and, at A0 , the complex (30) splits into a corresponding sum, with the interesting part given by Ω0 (L2 ) → Ω1 (L2 ) → Ω6 (L2 ) → Ω7 (L2 ). If the cohomology H 1 (L2 , A0 ) vanishes then A0 is isolated from irreducible solutions, and this is true also in families of small deformations of the G2 -structure. Thus, in this case, the irreducible solutions will not affect the enumerative discussion, counting the irreducible solutions. Since the complex has Euler characteristic 0 we expect that generically, in a family of G2 -structures, the cohomology will vanish. However we also expect that

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it will be non-trivial on some lower dimensional subset. The crucial point however is that the complex structure on L2 means that we are considering families of complex linear operators and for these the generic picture is that cohomology will appear in real codimension 2. In that case there will be no interaction between the reducible and irreducible solutions in generic 1-parameter families, which is what are relevant to our purpose. So, granted that a more detailed analysis is necessary we take this as an indication that we can ignore the potential complications from reducible solutions, and for simplicity we just ignore these reducibles in what follows. (Similar remarks apply to reductions S(U (p) × U (q)) ⊂ SU (l).) Now it seems very reasonable to assume that for generic σ all solutions are regular Even if this is not the case we could contemplate introducing further, more artificial, perturbations of the equations or ideas of “virtual cycles”, but let us assume that perturbations of σ suffice. Our basic goal, when the structure is tamed by some closed form φ is to define a number by counting the solutions with appropriate signs, and it is the issue of these signs which we take up next. As usual, we mimic the standard discussion in the Casson-Floer theory over 3-manifolds. We seek to define a “relative sign” (A, A ) ∈ {±1} for pairs of solutions with the property that

(A, A ) = (A, A ) (A , A ). This gives a way to attach signs to each solution, up to a single overall sign ambiguity. We define (A, A ) using the spectral flow of a family of operators DAt , where At is a path from A to A . Given a path, this spectral flow yields an integer and we set to be 1 or −1 as the spectral flow is even or odd. Then the essential thing is to check that this independent of the path, which is the same as saying that the spectral flow around a closed loop (in the space of connections modulo gauge equivalence) is even. Such a loop yields a connection on a bundle E over X = Y × S 1 and the spectral flow appears as the index of an elliptic operator over X. In fact this operator is just the operator apearing in the linearisation of the Spin(7)-instanton equation and can be identified simply as the Dirac operator over X, coupled to the bundle gE . Here the spin spaces in 8 dimensions are regarded as 8-dimensional real vector bundles. So this question of “orientability” in our 7-dimensional setup reduces to an algebro-topological question of showing that the index of this such a coupled Dirac operator over X is even. (More generally, if the index of all such bundles is divisible by some integer k then we expect the putative Floer theory associated to Y to be Z/k-graded.) We can use the Atiyah-Singer index theorem to express this question in terms of characteristic classes. (An explicit formula is given by Lewis in [24].) But we can also give an argument which avoids detailed calculation. Since it is odd-dimensional, the manifold Y has a nowhere vanishing vector field. This gives a reduction of the structure group of Y to SU (3) ⊂ G2 , and hence of X to SU (3) ⊂ G2 ⊂ Spin(7) ⊂ SO(8). With this reduction the spin bundles of X acquire complex structures (corresponding to V ⊕ C, V ∗ ⊕ C

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where V is the fundamental representation of SU (3). Of course we are not saying that these reductions are compatible with the differential geometric structure, but they are at the level of the symbol of the Dirac operator. Thus we can deform the coupled Dirac operator over X to a complex linear operator, and hence the (real) index is even. However we will not try to develop the theory of orientations any further here. We move on from this brief outline, which indicates how—modulo questions of compactness—we should define an integer counting the G2 instantons on a bundle E over Y . For each homology class b ∈ H3 (Y ) we consider SU (l) bundles E with c2 (E) the Poincar´e dual of b and (for simplicity) with c3 (E) = 0. There are at most a finite number of different topological types and we define an integer nb by summing the counts above over all such bundles. We expect this to be a deformation invariant, with respect to perturbations of σ. The energy bound implies that nb = 0 if [φ](b) < 0, since the moduli space is then empty. In fact nb vanishes for b outside some proper cone in the half-space {b : φ(b) > 0}. We package these numbers into a formal series  (32) fY (ψ) = nb exp(−b, ψ ), and our further hypothesis is that this converges to yield a holomorphic function of variable ψ in an open subset of H 3 (Y, C) containing the points r[φ], for large enough r. 4.2. 7-manifolds with tubular ends. Now we discuss dimension 6. If we have a Hermitian-Yang-Mills connection on a holomorphic bundle over a Calabi-Yau manifold Z 6 , as in the previous section, the linearisation of the equations (28) for pairs (a, u) yields an elliptic deformation complex of Euler characteristic zero. We call a solution regular if the cohomology vanishes, and this is just the same as saying the sheaf cohomology H ∗ (Z, gE ⊗ C) vanishes. We will assume that all solutions over Z are regular. This is a very restrictive assumption, and we will return to discuss it further below. Now suppose that Y is a non-compact manifold with tubular ends each with a Calabi-Yau cross section, and that we have an almost G2 -structure defined by a closed form σ, compatible with the product structure on the ends, up to an exponentially decaying term. If we fix a solution over each end then we have the notions of an “adapted bundle” and “adapted connection” over Y , just as in the usual 3 + 1 dimensional theory described in [8]. These are, respectively, a bundle over Y with a fixed isomorphism with the pullbacks of the chosen bundles over the ends, and a connection over Y which agrees with the model determined by the 6-dimensional solution over each end, up to an exponentially decaying term. Now fix attention on any adapted connection A over Y and form the differential operator DA as above. This is formally self-adjoint, just as before. Over an end, assuming that the connection is actually equal to the model,

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we can write DA as DA = L

d + Q, dt

where L is a skew-adjoint algebraic operator and Q is a first-order selfadjoint operator, both over Z. Under our hypothesis that Z is Calabi-Yau the composite L−1 Q is a self-adjoint first order operator over Y and the analysis of DA follows very closely that in the three-dimensional theory, as described in [8] for example. We deduce that DA : L21 → L2 is Fredholm provided that the operator Q over Z does not have a zero eigenvalue. In turn, this is the same as the hypothesis that the solution over Z is regular (since the operator Q is essentially the same as that arising in the deformation theory over Z). In the usual Floer theory we go on to consider the Fredholm index, which yields an invariant of an adapted bundle. The distinctive feature of the theory we are considering here is that the analogous index always vanishes. This is just a consequence of the fact, from the general theory, that the cokernel of DA is represented by the kernel of the formal adjoint, which is the same as DA . The consequence is that, under our restrictive hypotheses, the study of G2 -instantons on a fixed adapted bundle over Y behaves just like the compact case. We define regular solutions in just the same way, we expect that for generic σ all solutions are regular and that a count of solutions yields a deformation invariant (with respect to compactly supported variations in σ) of the adapted bundle. Now we discuss “neck stretching sequences” and gluing constructions. For simplicity consider a pair of manifolds Y1 , Y2 each with one end having the same model Z and appearing as the limit of a sequence of structures σT on a compact manifold Y . (But note that when interchanging Y1 , Y2 we have to change the sign of the 3-form on Z.) Given regular G2 -instantons on adapted bundles over Y1 , Y2 we wish to construct a glued solution over (Y, σT ) for large T . The proof follows the familiar Floer theory case closely, with one extra step. For large T we construct an approximate solution A0 with all norms of the error term σT ∧ F (A0 ) bounded by decaying exponential functions of T . Now we seek to solve the equation σT ∧ F (A0 + α) = ∗dA0 +α u, over YT , for a bundle valued 1-form α and 0-form u. We also impose the gauge fixing condition d∗A0 α = 0. Schematically, these equations can be written as DA0 s = s ∗ s + σT ∧ F (A0 ), where s is the pair (α, u), the notation s ∗ s denotes a quadratic algebraic term and DA0 is our basic elliptic operator. Mimicking the arguments in the (3 + 1)-dimensional case, we get a bound on the operator norm of the inverse of DA0 which is independent of T . Then the inverse function theorem shows that when T is large (so σT ∧ F (A0 ) is small) there is a small solution s.

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Now the final extra step is to observe that, since σT is closed, the Bianchi identity implies that in fact dA u = 0, as we have seen before. Hence A is the desired G2 -instanton. Still following the familiar pattern, we hypothesise that when T is large, all G2 -instantons over (Y, σT ) arise by this gluing construction. Given a class b ∈ H3 (Y ) we let c ∈ H2 (Z) be the image of b under the boundary map of the Mayer-Vietoris sequence of Y = Y1 ∪Y2 . This is the same as the Poincare dual on Z of the restriction of the Poincar´e dual, P DY (b), of b on Y . We use the symbol S to denote a solution over Z on a bundle with Chern class the Poincar´e dual of c. Then our hypothesis gives a gluing formula of the shape   (33) nb = n(E1 )n(E2 ), S E1 ,E2

where in the inner sum E1 , E2 run over adapted bundles with common limit S and such that the glued bundle over Y has c2 = P.D.(b). Of course we have similar formulae when we glue manifolds with more than one end. It is important to emphasise that if we work with almost G2 structures where σ is not closed we would get a different, richer theory, more like Floer theory over 3-manifolds and 4-manifolds with tubular ends. The operator DA is not self-adjoint and the index gives a non-trivial invariant of adapted bundles. Then we would get moduli spaces of different dimensions, depending on the index. However we will not pursue this further here. 4.3. Holomorphic bundles over moduli space. We have now reached the main question we wish to address in this paper. The gluing formula (33) is, in the general Floer theory framework, at the “chain level”. As we vary the Calabi-Yau structure on Z the solutions S vary and we do not have a canonical way to identify them. Further, even if choose such an identification locally the individual numbers nE1 , nE2 will change. So we seek a more invariant way of expressing the formula much as, in the ordinary Floer theory, we pass from the “chains” to the Floer homology groups. Let us for simplicity suppose that the inclusion of Z in Y induces an injection on H3 and fix a coset [b0 ] in H3 (Y )/H3 (Z). We restrict attention to classes b in this coset, which have the same image c in H2 (Z). Now we consider a series like (32)  nb+b0 exp(−b, ψ ), (34) gY (ψ) = b∈H3 (Z)

where now ψ lies in an suitable open set in H 3 (Z, C), containing the points r[θ] for large enough r. Fix the class [ω] ∈ H 2 (Z) and consider the moduli space M of pairs (I, θ) where I is a complex structure on Z which admits K¨ahler metrics in the class [ω] and θ is a nowhere-zero holomorphic 3-form. By the Torelli theorem for

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˜ Calabi-Yau manifolds this is a quotient M/Γ where Γ is the symplectic 3 ˜ mapping class group and M is immersed in H (Z, C). We have an obvious C∗ -action on M, multiplying the holomorphic form by a constant. We can also define a “norm” function by  θ ∧ θ.

(I, θ) 2 = Z

and for R > 0 we write MR for the points in M of norm greater than R. We suppose R is some fixed, suitably large, number. Then the restriction of the holomorphic function gY defines a holomorphic function on an open set in MR containing the ray lθ = {I, rθ) : r ≥ 1}. We denote this restriction also by gY . Now we make the assumption that for generic points in the moduli space M all solutions S are regular. Tentative prediction 1. Under this assumption: • There should be a holomorphic vector bundle E → MR , associated to the class c ∈ H2 (Z), and a canonical isomorphism (−1)∗ E ∼ = E ∗. • There should be an invariant gY1 which is a holomorphic section of E over a neighbourhood of the ray lθ . Likewise there should be an invariant gY2 which is a holomorphic section of (−1)∗ E over a neighbourhood of lθ . • The function gY should be the dual pairing gY1 , gY2 , formed using the isomorphism above In other words, in this situation, the structure analogous to the Floer homology of a 3-manifold is the holomorphic vector bundle E over the moduli space, and its sheaf of holomorphic sections. Remarks ˜ in which case 1. The mapping class group may not act freely on M, M is an orbifold. Then we more should strictly work with an “orbibundle” over M. 2. When we restrict the function gY from an open set in H 3 (Z, C) ˜ we could lose information. But our bundles actually extend to M over a thickening of M obtained from a quotient of such an open set and if formulated this way we get a gluing formula for the original function gY . Now we will explain the construction of the bundle. Suppose that θ, θ are two nearby Calabi-Yau structures on Z and that they are generic in the sense above, so all solutions are regular. We have a collection of critical points {S} associated to θ (with the fixed class c) and another collection S  associated to θ . Let V, V  be the complex vector space with basis elements S and S  respectively. Write S ∗ for elements of the dual basis of V ∗ . We can choose a tamed almost G2 -structure on the topological cylinder R × Z

GAUGE THEORY IN HIGHER DIMENSIONS, II

21

which is asymptotic to that defined by θ at −∞ and by θ at +∞. Fix, for the moment, closed 3-forms ψ representing cohomology classes ψ ∈ H 3 (Z) and extend these to closed 3-forms ψ on the topological cylinder, compatible with the product structure on the ends. We choose the taming form φ as the representative of Re(θ). Then for each adapted bundle E over the cylinder we have the following 1. Asymptotic limits S(E) ∈ {S}, S  (E) ∈ {S  }, at t = ±∞ respectively. 2. A real number  1 ψ(E) = − 2 ψ Tr(FA )2 , 8π Z×R where A is any adapted connection. This is independent of the choice of connection and of the representative ψ, given a choice of ψ. 3. An integer n(E) counting the number of G2 -instantons. Now we define a holomorphic function with values in Hom(V, V  )  (35) Gθ,θ (ψ) = n(E) exp(−ψ(E))S(E) ∗ ⊗ S  (E) . E

This will be defined in a neighbourhood of the ray lθ . If we change the choice of representive ψ of a class ψ to ψ + dλ we change the numbers ψ(E) to    2 ) − Tr(FS2 (E) λ. ψ(E) + Tr(FS(E) Z

This has the effect of changing the linear map Gθ,θ (ψ) to Λθ (ψ)Gθ,θ (ψ)Λθ (ψ)−1 ,

(36)

where in our bases Λθ , Λθ are diagonal matrices with entries given by the exponentials of   2 TrFS λ, TrFS2 λ, Z

Z

θ

respectively. If is another nearby generic structure and we fix the same representatives the gluing formula, extended to this situation (when we glue together two topological cylinders) yields (37)

Gθ,θ (ψ) = Gθ,θ (ψ) ◦ Gθ ,θ (ψ).

Thus to define our holomorphic bundle we decree that for each generic structure θ and choice of representatives ψ the bundle has a canonical trivialisation over some neighbourhood Uθ of lθ in the moduli space. Changing the representatives ψ changes the trivialisation by multiplying by the diagonal metric Λθ . On the the overlaps Uθ ∩ Uθ we use the maps Gθ,θ (ψ) as transition functions. The gluing formula (37) is the cocycle condition

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giving the consistency of the set of transition functions, and the isomorphism (−1)∗ (E) = E ∗ is induced by the fact that the solutions {S} of the equations defined by the 3-forms ρ, −ρ have an obvious identification. Now to define the section gY1 corresponding to Y1 we work initially in a canonical trivialisation around θ and with a choice of the representatives ψ. By our hypothesis these can be extended to closed forms over Y1 , compatible with the product structure on the end. Then for an adapted bundle E1 over Y1 and a class ψ in H3 (Z) we define ψ(E1 ) in the same manner as before. We also have a limit S(E1 ) and a number n(E1 ) Then we set g˜Y1 (ψ) =



n(E1 ) exp(−ψ(E1 )) < S(E1 ) >,

E1

a V -valued holomorphic function on a neighbourhood of lθ . The gluing formula shows first that this yields a well-defined section gY1 of E—independent of the choice of θ and of ψ—and second that gY =< gY1 , gY2 >. Remarks 1. Our bundle should have the property that its rank is equal to the DT invariant. We can think of the rank as the 0-degree component of the Chern character, and it might be that there is an extension of these ideas to a formula for all of ch E. Note that Thomas’ theory discusses a fixed Calabi-Yau manifold and there should be extensions of this which yield cohomology classes in the moduli space M, using the universal family. 2. In this Section we have fixed attention on the (6 + 1)-dimensional theory but it is natural to wonder if there is some yet higher structure, associated to the 8-dimensional geometry. Roughly speaking, we would expect this to assign Floer groups to a compact G2 manifold Y and one would like some machinery to compute these when Y = Y1 ∪Z Y2 . We make one observation in this direction. Suppose we have a vector bundle over a space B given by transition functions gα,β with respect to a cover Uα of B. Thus we have matrix entries gα,β,i,j (z) which are functions of z ∈ Uα ∩ Uβ and gα,γ,i,k (z) =



gα,β,i,j gβ,γ,j,k ,

j

on Uα ∩ Uβ ∩ Uγ . Now suppose we have a chain complex C ∗ and a chain automorphism T : C ∗ → C ∗ so we have a Lefschetz num ∗ i ber L(C , T ) = (−1) TrTi . If we have a family of such pairs, parametrised by z, the Lefschetz number becomes a function of z. ∗ So we may envisage a structure given by chain complexes Cα,β,i,j , with automorphisms, parametrised by z ∈ Uα ∩ Uβ , such that on

GAUGE THEORY IN HIGHER DIMENSIONS, II

the triple overlaps Cα,γ,i,j,k ∼



23

∗ ∗ Hom(Cα,β,i,j , Cβ,γ,j,k ),

j

where ∼ is a suitable equivalence relation, at least as strong as chain homotopy equivalence, compatible with the automorphisms. Then the Lefschetz numbers of the chain complexes give the transition functions of a holomorphic bundle. Possibly there is some structure of this kind on the moduli space M which would refine the bundle E, in the same way as the Floer homology of G2 -manifolds should refine the Casson invariant. 5. Finite-dimensional analogue 5.1. Morse-Novikov Theory. We begin with some basics. Take a manifold A, and a class [α] ∈ H 1 (A, C). This gives a local system, which we may think of either as a rank 1 complex vector bundle on A with flat connection α, or as a representation (the monodromy of the connection) ρ : π1 (A) → C∗  ρ(γ) = exp( α) γ

How do the cohomology groups H ∗ (A, ρ) of this local system behave as we vary [α]? The answer is that they form the fibres of a holomorphic sheaf E over the vector space H := H 1 (A, C). We show this as follows. An element γ ∈ π1 (A) gives a linear function on H. So we can let π1 (A) act on the ring of holomorphic functions on H by γ : f → eγ f This is an action by module automorphisms, so we have a local system on A whose fibre is the rank 1 free OH -module. The cohomology of this local system E := H ∗ (A, OH ) is then also an OH -module, i.e. a sheaf on H, and its fibre at the point [α] is the cohomology of the local system associated to [α]. We will now review Morse-Novikov theory in finite-dimensions, more specifically the reformulation due to Burghelea and Haller [2]. Let Θ be a closed real 1-form on a finite-dimensional Riemannian manifold A, and assume that Θ is “Morse”, i.e. locally the differential of a Morse function. Then the pull-back of Θ to the universal cover A˜ is globally the differential of a Morse function, and we may form the cell-complex C• given by the unstable manifolds of the zeroes of Θ on A˜ in the usual way. This cellcomplex is obviously periodic with respect the action of π1 (A). Now suppose we have a rank-1 complex local system on A given by the class of some closed complex 1-form α. Then we can use our periodic cellcomplex C• in A˜ to produce a chain-complex that in good cases will calculate

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the cohomology of this local system. What we do is twist the differential on C• by α and then take π1 (A) invariants. Assuming that Θ is in fact Morse˜ we can describe this chain-complex explicitly in Smale for the metric on A, the following way: it has a basis given by the set ZA (θ) = {p} of zeroes of Θ in A, and differential (38)

∂α p =







e

γ

α

q

i(q)=i(p)−1 γ:p→q

where i(p) denotes the index of p, and γ runs over all flow-lines of θ connecting p and q. The differential clearly varies holomorphically in α, so the homology of the chain complex is a holomorphic sheaf E on the space Ω1c of closed complex 1-forms. If we choose a section H → Ω1c (e.g. by using Hodge theory) we can pull-back and get a holomorphic sheaf on H. The fibre of this sheaf at [α] is the cohomology of the local system on A given by [α]. There are two problems with this description. Firstly, since A˜ is noncompact the cell-complex C• may not cover the whole of A˜ (indeed θ may have no zeroes at all), so we can’t guarantee that we are genuinely calculating the cohomology of the local system [α]. Secondly, the sum (37) may not converge. It is shown in [2], that neither of these issues is a problem when α = sθ for s a complex number with Re(s) > ρ, where ρ ∈ [0, ∞] is some numerical invariant depending on θ and the metric. For this range of α, the components of ∂α do converge, and the chain-complex does calculate the cohomology of [α] correctly. The invariant ρ is conjectured to be always finite, and in many examples is zero. Let us suppose something stronger: that we get convergence and the correct cohomology in an open neighbourhood of the set Re(α) = θ. This includes the set {α = (1 + iλ)θ, λ ∈ R}, where we do indeed get convergence and the right cohomology at least when ρ < 1. If this holds, for any generic α0 ∈ Ω1c we have an analytic way of constructing our sheaf E in a neighbourhood of α0 . We set Θ = Re(α0 ), then the Morse-Novikov chain-complex (38) constructed from θ will be valid in a neighbourhood of α0 , so we may define E to be its homology. Now consider the special situation in which A is a complex manifold and Θ is the real part of a holomorphic 1-form. Then the indices of all critical points are equal and, in a generic situation there are no flow lines between critical points. For any fixed α the cohomology H ∗ (A, [α]) is simply a vector space with one basis element for each zero of Θ. The sheaf on E on H 1 (A, C) is locally free and we write E also for the corresponding holomorphic vector bundle. We wish to apply this construction, formally, to the infinite-dimensional manifolds X arising from a Calabi-Yau threefold Z. Thus X is either the space of unitary connections with F ∧ ω 2 = 0, modulo gauge equivalence, or the space of symplectic surfaces Σ ⊂ Z. Assume for simplicity that we are ˜ working with genuine Calabi-Yau structures. Then given a point (I, θ) ∈ M

GAUGE THEORY IN HIGHER DIMENSIONS, II

25

we get an induced complex structure on X making it a Kahler manifold, and the Chern-Simons construction gives a holomorphic 1-form on X whose zeros correspond to stable holomorphic bundles or complex curves respectively. More generally, we have a map from H3 (Z) to H 1 (X ) and the ChernSimons construction gives a definite representative of this—a 1-form on X corresponding to a closed 3-form on Z. ˜ to Allowing the complex structure to vary, we get a map from M 1 H (X ; C), which is in fact a holomorphic embedding. So we should get a ˜ as the restriction of the twisted cohomology sheaf E, assuming sheaf over M of course that we had made sense of the latter. Then everything is invariant under the mapping class group Γ so we can descend to the moduli space M. It should now be fairly clear how our (conjectural) construction fits into this picture. Although we do not have any reason to believe that there is a cohomology sheaf over the whole of H 3 (Z, C) we can make sense of ˜ An individual fibre, over a generic point, this over a neighbourhood of M. is rather uninteresting—having an almost-canonical basis. The interesting structure appears in the way these are fitted together into a vector bundle. To understand this we go back to our finite-dimensional situation and consider Morse theory on the universal cover. Let θ again be a closed real 1-form on a finite-dimensional Riemannian manifold A, and (by abuse of notation) let θ also denote its pull-back to the ˜ Assume that θ is Morse-Smale, then we want to consider universal cover A. ˜ the resulting Morse complex for A. Choose lifts p˜ of each zero p ∈ A of θ, so the zeros of θ on A˜ are  ZA˜ (θ) = {γ(˜ p), γ ∈ π1 (A)} p

This is a basis for the Morse complex. The set of flow-lines is π1 (A)-invariant, γ −1 φ to be the (signed) number of so when i(q) = i(p) − 1 we may define Npq flow-lines between γ(˜ p) and φ(˜ q ), this depends only the product γ −1 φ in π1 (A). Then the differential in the Morse complex is   γ −1 φ d(γ(˜ p)) = Npq φ(˜ q) i(q)=i(p)−1 φ∈π1 (A)

Since Morse homology is actually isomorphic to singular homology, it doesn’t change as we vary θ and the metric. Floer observed in [10] that we can give an a priori proof of this fact, without reference to singular homology, by counting flow-lines in families. Suppose we have a suitably generic path (θt , gt ) of (π1 (A)-invariant) closed 1-forms and metrics, where (θ0 , g0 ) and (θ1 , g1 ) are Morse-Smale. Then using some standard function we can define a 1-form on A˜ × [0, 1] which has zeros  {θ0 = 0} × {0} {θ1 = 0} × {1}

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S. DONALDSON AND E. SEGAL

−1 φ

γ Let Mpq

be the (signed) number of flow-lines γ(˜ p) × {0} −→ φ(˜ q ) × {1}

where i(p) = i(q), and define a linear map Ψ between the Morse complexes for (θ0 , g0 ) and (θ1 , g1 ) by   γ −1 φ Mpq φ(˜ q) (39) Ψ(γ(˜ p)) = i(q)=i(p) φ∈π1 (A)

Then it can be shown that Ψ is a map of chain complexes, and that it is functorial with respect to composition of paths. Furthermore, if we have a homotopy between two such paths, then by a similar construction there is an induced homotopy between the corresponding chain maps. It follows that Ψ is a homotopy equivalence. (Here we are assuming, in the finite-dimensional ˜ case, that we do not run into problems due to the noncompactness of A.) Now suppose that θ0 = Re(α0 ) and θ1 = Re(α1 ) for some complex 1forms α0 and α1 . We can use the above discussion to understand how the Morse-Novikov complexes for (α0 , g0 ) and (α1 , g1 ) are related. To do this, we need the following construction. Let (θ0 , g0 ) and (θ1 , g1 ) be any two Morse-Smale pairs. Suppose, we have a π1 (A)-invariant linear map



T : ZA˜ (θ0 ) R → ZA˜ (θ1 ) R of the vector spaces underlying the corresponding Morse complexes. Then  −1 φ γ Tpq φ(˜ q) T : γ(˜ p) → q,φ γ for some set of real numbers {Tpq }. Examples are the Morse differential d when (θ0 , g0 ) = (θ1 , g1 ), and the map Ψ (39) when (θ0 , g0 ) = (θ1 , g1 ). Then for any closed complex 1-form α on A we may formally define a map

Tˆ : ZA (θ0 ) C → ZA (θ1 ) C by setting Tˆα (p) =



γ(˜ q)

γ Tpq e

p ˜

α

q

q,γ

though of course this may not converge. It is easy to check that this process is (formally) a homomorphism, i.e. if we have three Morse-Smale pairs (θ0 , g0 ), (θ1 , g1 ), (θ2 , g2 ) and maps

ZA˜ (θ0 )

R

U

T → ZA˜ (θ1 ) R → ZA˜ (θ2 ) R

GAUGE THEORY IN HIGHER DIMENSIONS, II

27

then (40)

 ˆα = T Tˆα U Uα

Also applying this to the Morse differential we get

γ(˜   q) γ Npq e p˜ α q dˆα (p) = i(q)=i(p)−1

γ

which is just the Morse-Novikov differential ∂α . Now take two generic pairs (α0 , g0 ), (α1 , g1 ) of closed complex 1-forms and metrics on A, and choose a generic path between them in the space of closed complex 1-forms and metrics. Take the real parts of all the 1-forms to get the same data in the space of closed real 1-forms and metrics. We get two Morse complexes with differentials d0 and d1 , and a chain map Ψ between them. Assume there is some neighbourhood U ⊂ Ω1c , containing α0 ˆ α converge for all α ∈ U. Then over and α1 , such that the maps dˆ0α , dˆ1α and Ψ U we have two Morse-Novikov complexes of holomorphic vector bundles on ˆ is U, given by the differentials dˆ0 and dˆ1 . However, by (40) we know that Ψ a holomorphic chain map between the two complexes. If we have two such paths and a homotopy between them, we get two chain maps Ψ1 and Ψ2 between the Morse complexes, and chain-homotopy Ξ. Then, assuming everything converges in U, we get two holomorphic chain ˆ 1 and Ψ ˆ 2 between our Morse-Novikov complexes on U and a holomormaps Ψ ˆ between them. Similarly, assuming that all the necessary phic homotopy Ξ maps converge in U, we can show that our two Morse-Novikov complexes are homotopy equivalent. In the model situation we are considering, the complex 1-forms on A that are actually holomorphic, so at generic points the Morse-Novikov chain ˆ are complexes have no differential. This means the homotopy equivalences Ψ 1 2 ˆ and Ψ ˆ are homotopic maps then they just isomorphisms, and that if Ψ are actually identical. We now have a picture of how our bundle E on H 1 (A, C) is built up. Suppose we have a simply-connected region U ⊂ M consisting of only two “chambers” U1 and U2 in which the Morse-Smale condition holds, separated by a wall on which it fails. Firstly, take two points x, y ∈ U1 , and a path between them. The corresponding map Ψ counts flow-lines in the family of 1-forms and metrics given by the path, but since the Morse-Smale condition holds everywhere along the path we will just see a single flow-line from any zero of αx to the corresponding zero of αy , so Ψ is just the identity map. This means that E should be trivial over each region U1 and U2 . However, if we have a path between x ∈ U1 and y ∈ U2 then the family will contain additional flow-lines arising from the “unexpected” flow-lines that occur on the wall, so the map Ψ will be non-trivial. Passing to the holomorphic ˆ that we should use to patch E when version, we get the transition function Ψ we cross the wall.

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We can formulate this construction a little more precisely. Pick any x ∈ U1 and y ∈ U2 and any path between them. The corresponding map Ψxy is independent of the choice of path since U is simply-connected. Assume that ˆ xy converges in all of U. Let Vx , Vy be the trivial bundles on U with fibres Ψ ZA (αx ) C and ZA (αy ) C respectively. Take another copy of each of these bundles, and denote them Vxy := Vx and Vyx := Vy . Define Φ

EU = coker(Vxy ⊕ Vyx −→ Vx ⊕ Vy ) where

 Φ=

ˆ xy id Ψ ˆ −1 id Ψ xy



Then over U1 we have EU ∼ = Vx by projecting onto the first factor, and over U2 we have EU ∼ = Vy by projecting onto the second factor, but across ˆ xy . This is the wall these isomorphisms differ by the transition function Ψ precisely the vector bundle that we wanted. In fact this picture of a finite wall-and-chamber decomposition is misleading. Since there are infinitely many homotopy classes in which nonMorse-Smale flow-lines can appear, we actually expect the set of walls to be dense in the parameter space H 1 (A, C). However, this construction adapts easily. Let U again be simply connected, but have possibly infinitely many walls in it. For every point z ∈ U that doesn’t lie on a wall we let Vz be the trivial bundle on U spanned by ZA (αz ), and for every distinct ordered pair x, y of such points we take a copy Vxy := Vx . For every pair x, y we get a map Ψxy of Morse complexes, and again this is independent of the choice of ˆ xy converges in all of U for all x, y ∈ U. path between x and y. Assume that Ψ Define Φ Vxy −→ Vz ) EU = coker( x,y

z

where the map Φ has components ˆ xy : Vxy → Vz δxz id + δyz Ψ At any generic point z we have EU ∼ = Vz by projecting onto that factor, so in particular EU is a finite rank vector bundle. 5.2. Discussion. In our infinite-dimensional situation we interpret Floer’s time dependent vector fields, used in constructing the chain homotopy, as almost-G2 structures on topological cylinders. The count of flow lines for the time dependent vector field gives the count of G2 -instantons, and the Novikov series handle the passage to the universal cover. Then it should be clear how our definition of the holomorphic bundle over M matches up with the finite-dimensional discussion. To sum up, we interpret the holomorphic bundle E over M as the “middle-dimensional” cohomology of the infinite

GAUGE THEORY IN HIGHER DIMENSIONS, II

29

dimensional space X , with coefficients in the local system over X defined by classes in H 3 (Z, C). When Floer first introduced his theory he gave a different treatment of the parameter-dependence of his homology which was more explicit but technically more complicated. Later, he introduced the “time-dependent vector field” trick which gave a simpler approach, but the original method had the advantage of giving a more precise description of the chain homotopies. In a similar way we can give a more explicit description of the co-efficients which define the transition function of our holomorphic bundle. This is closer to the discussion in [6] and in [27]. The “Novikov” aspect of the theory is confusing here, so let us begin by considering a finite dimensional complex manifold A˜ and a proper holomorphic map f : A˜ → C with a finite number of critical values z1 , . . . , zN ∈ C, where we suppose each zi corresponds to a unique ˜ The gradient curves of the function Re(f ) on A˜ map critical point Ai ∈ A. under f to line segments with fixed imaginary part. So if, as will generically be the case, the imaginary parts of the zi are all different there can be no gradient curves joining the critical points. However we are free to rotate the picture, multiplying f by any complex number of modulus 1. Given a pair zi , zj we can choose this phase so that we are allowed to have gradient curves joining Ai , Aj , mapping to the line segment zi zj in C, which generically will not contain any other zk . The count of such gradient curves gives a number Nij . This is the intersection number of the vanishing cycles of f at the two points, when the fibre f −1 (t) is transported along the segment zi zj so that the two vanishing cycles can be viewed as homology classes in the same space. Now suppose we have a generic 1-parameter family of such situations. Then in the family a third point zk may move across the segment zi zj . When this happens the number Nij changes by ±Nik Nkj . Another way of expressing this is that we change the homotopy class of the path in C \ {z1 , . . . , zN } used to identify the fibres near zi and zj . The parallel transport along the two paths differs by monodromy around zk : a Dehn twist in the vanishing cycle associated to zk . Now given a generic f we let Vf be the vector space with basis symbols Ai associated to the critical points. As we move in a generic 1-parameter family from f0 to f1 say the Ai move continuously so we have a naive, and rather trivial, isomorphism between Vf0 and Vf1 (depending on the path). We modify this in the following way. If in the family the critical point zj moves across the ray {z : Im(z) = Im(zi ), Re(z) > Re(zi )} then we map the basis element Ai , before the crossing, to Ai ± Nij Aj , after the crossing. Now replace the hypothesis that f is proper and has a finite number of critical points by the situation we had before, where A˜ is the universal cover of some A, so there are a finite number of critical points up to the action of H1 (A). The same procedure gives a more explicit recipe for the local trivialisation of the twisted cohomology bundle, except now we need to keep track of the relative homotopy class of paths between critical points, which brings in the Novikov series. Translated into our infinite dimensional picture

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this gives a way to describe the transition functions of our holomorphic bundle, using G2 -instantons on genuine tubes R × Z, exploiting the freedom to multiply the 3-form θ by a phase. With all of this discussion in place, we return to discuss our main hypothesis, that for generic points in the moduli space M all solutions are regular. This seems to be a serious restriction, not often satisfied in practice. However there is an obvious strategy for removing this restriction. We work with suitable generic perturbations of the Calabi-Yau structure, involving triples ω, ρ, ρ with ω ∧ ρ = 0. It is very reasonable to expect that for generic perturbations of this kind all solutions are regular. But as we explained in Section 3 we have then to give up the assumption that σ is closed, so we get nonzero Fredholm indices for adapted bundles. But this just means that, in the finite-dimensional analogue, we need to compute twisted cohomology using a 1-form with zeros of different indices so we can have a nontrivial chain complex. What one would expect is that it is possible to define a collection of sheaves over the moduli space MR which can be interpreted as the different twisted cohomology groups of A//G. But we will not go into this further here. Finally, we point out that, as explained by Burghelea and Haller, in the finite dimensional situation the twisted cohomology groups can be computed by a variant of Witten’s complex, and described using differential forms satisfying Witten’s deformation of the Hodge-Laplace equation. It could well be that the structure we are discussing in this article arise this way in Quantum Field Theory, and are perhaps well-known there. 6. Interaction between gauge theory and calibrated geometry 6.1. G2 -instantons and associative submanifolds. So far we have ignored the fundamental problem of compactness of moduli spaces. In this section we attempt to explore this and to get a glimpse of what modifications are required to take account of the problem. Suppose that σi is a sequence of G2 -structures on a fixed 7-manifold Y with limit σ. Suppose we have corresponding sequence of G2 -instantons on a fixed SU (l) bundle. Then, according to Tian, after taking a subsequence the connections converge off a set P of Haussdorf dimension at most 3 and each three-dimensional component satisfies the associative condition, in a generalised sense. The sequence of 4-forms Tr(FA2 i ) allows us to attach a multiplicity to each 3-dimensional component. Let us suppose that in fact P is a connected smooth associative submanifold. Roughly speaking, we expect that if the multiplicity is k then transverse to P the connections are modelled on instantons on R4 with c2 = k, and the behaviour as i → ∞ mimics the familiar bubbling of instantons over 4-manifolds. We expect also that the singularity at P in the limiting connection is removable, so that limiting connection extends to a smooth G2 -instanton over Y [32]. In this section we discuss the converse question. Let σ be a G2 structure on Y and A be a

GAUGE THEORY IN HIGHER DIMENSIONS, II

31

G2 -instanton on a bundle E over Y . Let k be a positive integer and P be an associative submanifold in Y . When does the triple (A, P, k) appear as the limit of smooth G2 -instantons with respect to a sequence of deformations σi of σ? In this subsection we will explain that there is a natural candidate criterion for “bubbling” question. In particular when the gauge group is SU (2) and when k = 1 we will argue that this occurs if for some spin structure on P the coupled Dirac operator on E|P , defined by the restriction of the connection A, has a nontrivial kernel. Several authors have considered related problems, mostly emphasising the similar question involving Cayley submanifolds and Spin(7)-instantons in 8-dimensions. Lewis proved an existence theorem for Spin(7)-instantons using a gluing construction, choosing a Cayley submanifold with very special properties [24]. Brendle considered the general question of existence of Yang-Mills connections [5], and Spin(7)-instantons in particular [4], but restricting attention to the case when (the analogue of) the integer k above is 1. The construction we want to explain here is due to Haydys [14],[15], and related to ideas of Pidstrigatch [28] and Taubes [35]. We refer to the paper of Haydys [15] for a more complete account, and a discussion of various other interesting related matters. To begin, suppose that V is a quaternionic manifold, with a multiplication map μ : T V × H → T V.

(41)

Then there is an elliptic “Fueter equation” for maps f : R3 → V which is (42)

I

∂f ∂f ∂f +J +K = 0. ∂y1 ∂y2 ∂y3

In the case when V is H this is just the Dirac equation for a spinor field. (There is a similar equation for maps from R4 to V , but we will emphasise the 3-dimensional version.) Now suppose that there is an action of SU (2) on V permuting the I, J, K. More precisely, this means that μ in (41) is an SU (2)-equivariant map, for the induced action of SU (2) on T V and the standard action by automorphisms of the quaternions. Let P be an oriented Riemannian 3-manifold with a spin structure and F r → P the corresponding principal SU (2) bundle. Then we can form the associated bundle (43)

V = F r ×SU (2) V.

For each point y ∈ P there is an obvious way to make R ⊕ T Py into an algebra Hy , isomorphic to H but not canonically so. Our hypotheses imply that there is a natural Hy structure on the tangent bundle of the fibre V y . Thus there is a Fueter equation for sections of V ,  ei ∇i s = 0, i

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where ei is any orthonormal frame in T Py and ∇i denotes the “vertical part” of the derivative of the section, defined using the horizontal subspace induced from the Levi-Civita connection. Slightly more generally still, suppose that G is another Lie group and there is an action of G × SU (2) on V , where now G preserves the quaternionic structure. Let Q → P be a G-bundle with connection, so we have a principal G × SU (2) bundle Q ×P F r over Y . Then we can form an associated fibre bundle (Q ×P F r) ×G×SU (2) V, and, as Haydys observed [14], there is an intrinsic Fueter equation for sections of this bundle over P . With this background in place, we return to consider an associative submanifold P ⊂ Y . To explain the basic idea, we begin with the model case when P, Y are flat, so we can take local co-ordinates y1 , y2 , y3 , x1 , x2 , x3 , x4 in which  dyi dyj ωk , σ = dx1 dx2 dx3 dx4 + i,j,k cyclic

Λ+ R4 .

For > 0 let σ be −2 times the where ωi is a standard basis for pull-back of σ under the map which multiplies the xa co-ordinates by . Thus  dyi dyj ωk . σ = 2 dx1 dx2 dx3 dx4 +  We have a limit σ ∗ = dyi dyj ωk which is not a positive form, but we can still consider the equation F ∧ σ ∗ = 0, which we refer to as the σ ∗ -instanton equation. Let N → P be the normal bundle of P ⊂ Y . The G2 -structure induces a bundle isomorphism Λ+ N → T P , which one finds is covariant constant with respect to the standard induced connections on T P, N . Fix a spin structure on P . Then we get another complex vector bundle U → P , with a connection and structure group SU (2), such that N is canonically identified with the real part of U ⊗C S. Using the connection on N we get a canonical 3-form σ ∗ on the total space of N . Let exp be the exponential map from a neighbourhood of the zero-section in N to Y , let exp (ξ) = exp( ξ) and let σ be the 4-form −2 exp∗ (σ). Then one can see that the limit of σ as tends to zero is σ ∗ . We can define σ ∗ -instantons: connections on bundles over the total space N , as above and it clearly reasonable to expect that these are the blow-up limits of sequences of connections developing a singularity along P . Now we want to bring in a G2 -instanton connection A on another SU (l) bundle E over Y . This defines a connection A|P on the restriction E|P . Let N∞ be the 4-sphere bundle over P obtained by adjoining a section at infinity to the R4 -bundle N . W define a σ ∗ -instanton on N with asymptotic limit A|P to be given by a connection A on a bundle E → N∞ which satisfies the σ ∗ -instanton equations on the dense subset N ⊂ N∞ and such that the restriction of A to the infinity section is equivalent to A|P . Note that this data defines an integer Chern class k, given by the restriction of E to any

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4-sphere fibre. We expect that a triple (A, P, k) can only occur as the limit of a sequence of G2 -instantons, for nearby structures, if there is a solution of the σ ∗ -instanton equation with Chern class k and asymptotic limit A|P . Let M = Mk,l be the moduli space of “framed” SU (l) instantons of Chern class k over S 4 = R4 ∪ {∞}. By a framing we mean a trivialisation of the fibre over ∞. We write R4 as the real part of S + ⊗ S − where S + , S − are the spin spaces. Then there is a natural action of the group SU (2) × SU (2) × SU (l) on M , where the two SU (2) factors act on S + , S − , and SU (l) acts on the framing at infinity. There is a quaternionic structure on M which is preserved by SU (l) and the second copy of SU (2) and permuted by the first copy of SU (2). So we are in the situation above, with G = SU (2) × SU (l). Given our SU (l)-bundle E|P and our SU (2)-bundle U → P we form a G-bundle as the fibre product and then we get a bundle M → Y with fibre M and a Fueter equation for sections of M . Theorem 1. (Haydys [15]) There is a one-to-one correspondence between solutions of the σ ∗ instanton equation with Chern class k and asymptotic limit A|P and sections of the bundle M over P which satisfy the Fueter equation. This can be thought of a variant of the “adiabatic limit” for Spin(7)instantons over products discussed in [6]. In the case when k = 1 the Fueter equation appears in [4] as a “balancing condition”. It is natural to expect that the equation in [5] associated to general Yang-Mills solutions can be interpreted as the equation defining a harmonic section of the bundle M , as in [16]. We use the connection on N to split the tangent space of N into horizontal and vertical subspaces, isomorphic to T P and (U ⊗ S)R respectively. With respect to this splitting, F (A) ∧ σ ∗ has two components, say F1 , F2 where F1 takes values in the normal bundle of P and F2 in the tangent bundle (both lifted to N and tensored with the bundle of Lie algebras). The σ ∗ -instanton equation thus splits into two separate conditions. We show that there is a one-to-one correspondence between • connections A over N∞ isomorphic to A over the section at infinity and satisfying F2 = 0 • smooth sections of M → Y . Then we show that the further condition F1 = 0 is equivalent to the Fueter equation. The condition F2 = 0 just asserts that the restriction of A to each fibre of N is an anti-self-dual connection. There is a tautological bundle over M × S 4 which is equivariant for an action of SU (2) × SU (2) × SU (l) and which has a fixed trivialisation over M × {∞}. On this bundle we have a standard connection which restricts tautologically to the S 4 slices and which is compatible with the trivialisation over M × {∞}. Using the group action ˜ over the pull-back π ∗ (M ) of M to N∞ , which we construct a bundle E

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can also be viewed as a bundle over P with fibre S 4 × M . The connections ˜ A section s of M on T P, U and E induce a natural connection A0 on E. ∗ induces a section s of π (M ) and we have a connection As = s∗ (A0 ) on ˜ over N∞ . The stated properties of the connection on the a bundle s∗ (E) universal bundle imply that As is isomorphic to A over the section at infinity and satisfies the condition F2 = 0, simply because the connection on the universal bundle is anti-self-dual on each R4 slice in M × R4 . Conversely, it is a straightforward formal exercise to check that all such connections arise in this way. The remaining task is to match up the curvature condition F1 = 0 with the Fueter equation. For this we should recall some more of the theory of instantons over R4 . We can regard R ⊕ Λ+ as an algebra H. Of course, as before, H is isomorphic to the quaternions but we prefer not to fix such an isomorphism. Then H acts naturally by multiplication on itself and also acts on R4 . Let A be a finite energy instanton over R4 . We have a defomation d

d+

A A 1 0 + complex Ω0 → Ω1 → Ω+ and an elliptic operator d∗A ⊕ d+ A :Ω →Ω ⊕Ω . The tangent space of the moduli space M at A can be identified with the L2 solutions a of (d∗A ⊕ d+ A )a = 0 [33]. The crucial points are

• d∗A ⊕ d+ A commutes with the H action induced by the actions on R ⊕ Λ+ and R4 . Thus we get an action of H on the tangent space of M , which is just the quaternionic structure mentioned before. • The component of the curvature of the connection A on the universal bundle in T ∗ M ⊗ T ∗ R4 = Hom(T M, T ∗ R4 ) is the tautological map given by evaluating a ∈ T M at a point in R4 . In particular this commutes with the action of H.

Now we work at a fixed point in P and fix an orthonormal basis ei for the tangent space of P at this point. Identifying the fibre of N at this point with R4 we get a basis ωi of Λ+ . Suppose we have a section s of M → P . With respect to the given connections this has a covariant derivative, with three components a1 , a2 , a3 ∈ T M corresponding to the tangent vectors ei . The Fueter equation is ωi (ai ) = 0. By the second observation above this implies that for each point x ∈ R4  ωi (Fai , ) = 0, where (Fai , ) is the bundle-valued 1-form on R4 obtained by pairing the curvature of the universal bundle with ai ∈ T M . Unravelling the definitions one sees that the left hand side of this equation is precisely the component F1 of the curvature. Thus a solution of the Fueter equation does yield a solution of the σ ∗ -instanton equation and the only remaining thing is to see that there are no other solutions. Suppose we have any solution A of the equation F2 = 0, isomorphic to A over the infinity section. Restriction to the fibres of N defines a section s of M and A must agree with As in the fibre direction. Thus the only

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possibility is that A = As + Φ where Φ is a bundle-valued 1-form vanishing in the fibre direction. If we fix a point in P and a basis ei as above then Φ has three components φi which are sections of the adjoint bundle over R4 . If ai are the derivatives of s, as above, then the curvature condition F1 = 0 becomes (44)

ωi (ai + dA φi ) = 0.

The hypothesis on the connection over the infinity section is equivalent to the condition that |φi (x)| → 0 as x ∈ R4 tends to infinity. Thus what we need to show is that in this situation all the φi vanish. Now write Φ = φi ωi , a bundle-valued self-dual 2-form over R4 . The equation (44) is equivalent to  d∗A Φ = ωi (ai ). + ∗ Since d+ A ωi (ai ) = 0 we have the identity dA dA Φ = 0. The Weitzenbock formula in this situation tells us that  ∗ (d∗A dA φi ) ωi . d+ A dA Φ = i

(This uses the fact that A is an anti-self-dual connection.) So we deduce that d∗A dA φi = 0 and then the maximum principle implies that φ vanishes, since |φi | tends to zero at infinity. This completes the proof of the theorem. There is a standard map from M to R4 which takes a connection to the centre of mass of its curvature density |F |2 and the derivative of this map is H-linear. In fact M is a product M  × R4 , where M  is the “centred” moduli space. It follows that there is a bundle map from M to N which takes solutions of the Fueter equation for sections of M to sections of the corresponding equation for sections of N . The latter is just the linear Dirac equation appearing in the theory of deformations of the associative submanifold P . We assume P is “regular” so this equation has no non-zero solution. This means that we can replace the instanton moduli space M by the centred moduli space M  throughout the discussion above. Let us consider the case when l = 2 and k = 1. Then, up to translation and dilation of R4 and gauge equivalence, there is just one instanton which is the standard connection on the negative spin bundle over S 4 . The framed moduli space can naturally be written as M  = (S + \ {0})/ ± 1, where S + is the spin space, and this is compatible with the quaternionic structure. (The quotient by ±1 comes from the centre of SU (2).) Tracing through the definitions we find that   M  = (S + ⊗ E)R \ 0 / ± 1. where S + is the spin bundle over P and 0 denotes the zero-section. Note that the bundle M  does not depend on the choice of a spin structure on Y but if we have a section of M  there is a unique choice of spin structure

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for which this lifts to a section of (S + ⊗ E)R . Making this choice, we see that solutions of the Fueter equation correspond (up to ±1) to solutions of the Dirac equation for sections of E-valued spinors over P , using the Dirac operator coupled to A. Thus, in sum we expect that the condition that a pair P, A appear as the limit of G2 -instantons, with multiplicity k = 1, is that there is a nontrivial harmonic spinor for the restriction A|P , for some choice of spin structure on P . (Note that we are assuming here that the harmonic spinor does not vanish, but this should be true generically for dimensional reasons. A case when the harmonic spinor vanishes somewhere would require further analysis.) We could carry out the same discussion for complex curves in a CalabiYau 3-fold and Hermitian-Yang-Mills connections (solutions of (22)), or embed this in the situation above by taking the product with a circle. But in this 6-dimensional case there is an alternative, algebro-geometric, point of view, which leads to the same conclusion. Let Σ be a smooth curve in a Calabi-Yau 3-fold Z0 . Suppose π : Z → Δ is a deformation of Z0 , where Δ is a disc in C. Thus Σ can be viewed as a curve in the central fibre π −1 (0) ⊂ Z. Blow up this curve to get π ˜ : Z˜ → Δ. Then π ˜ −1 (0) = V ∪D Z˜0 , where Z˜0 is the blow-up of Z0 along Σ, D ⊂ Z˜0 is the exceptional divisor, which is a CP1 bundle over Σ, and V is a CP2 bundle over Σ which contains ˜ restriction to a copy of D. If we have a holomorphic bundle E over Z, fibres gives a family of bundles Et over the deformations Zt = π −1 (t) for non-zero t but if the restriction of E to the CP2 fibres in V is non-trivial this family will not extend to give a bundle over Z0 . One expects this to give the algebro-geometric description of a family of Hermitian-Yang-Mills connections developing a singularity along Σ. The analogue of the connection A in the discussion above is furnished by the restriction of E to Z˜0 ⊂ π ˜ −1 (0), which we assume to be the lift of a bundle E over Z0 . The algebro-geometric analogue of the question we have discussed above is to ask: given Σ, E, k when is there a bundle over π ˜ −1 (0) which is isomorphic to the pull-back of E over Z˜0 and has c2 = k on the CP2 fibres in V . It is a straightforward algebraic geometry exercise to show that when k = 1 this occurs precisely 1/2 when there is a non-vanishing holomorphic section of E ⊗ KΣ over Σ, for 1/2 some choice of spin structure KΣ . 6.2. Implications. The Dirac operator on E-valued spinors over P is naturally a real operator and we expect to encounter a zero eigenvalue in real codimension 1. Thus it seems likely that a naive count of G2 -instantons will not yield an invariant. What one would expect is needed is a count which includes triples (A, P, k) of a connection on a different bundle and an associative submanifold P , thought of as having multiplicity k. We should count these with some weight W (A, P, k). For example with bundles of rank

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l = 2 and when k = 1 we would need some way to determine the weight W so that in a generic 1-parameter family it changes by ±1 when one eigenvalue of the Dirac operator crosses 0 (and a prerequisite for understanding the signs would be to develop a theory of orientations and signs in the “naive” theory, which we have not discussed). This is much the same issue involved in regularising the “dimension” in the ordinary Floer theory, as we discussed in Section 4.1. Formally W could be given by the ±1/2 where the sign is determined by the parity of the “number of negative eigenvalues” of the coupled Dirac operator on P . It seems possible that this can be done, using the theory of spectral flow. For larger values of k new issues arise, since the contribution from P involves an essentially non-linear problem. We should still expect to encounter solutions in real codimension 1. The reason is that there is a dilation action on M and hence on M which preserves the quaternionic structure. Thus a single solution of the Fueter equation generates a 1-dimensional family, by dilation. The linearisation of the Fueter equation has Fredholm index 0 but this dilation action implies that, in a family, we expect to encounter solutions in real codimension 1, just as for the linear Dirac operator. Given any section s of M , let sˆ denote the vertical vector field defined by the infinitesimal dilation action, and let D(s) be the expression appearing in the Fueter equation, which is also a vertical vector field. Then we have a nonlinear eigenvalue equation D(s) = λˆ s, for sections s, generalising the eigenvalue equation for the Dirac operator. (In [13], Section 3.4, Haydys develops a more general theory of these eigenvalue equations, in terms of a “Swann bundle”.) Of course we have a solution of the Fueter equation just when there is a zero eigenvalue. So it seems that one needs an extension of the theory of spectral flow which would enable one to define the weight W (A, P, k) by a regularisation of the “number of negative eigenvalues” for this nonlinear problem. If one seeks, more ambitiously, to construct a Floer theory in 7-dimensions then it seems likely that one would have to assign a Floer group (or, perhaps better, chain complex) to (A, P, k), giving the contribution to the overall Floer homology (and with Euler characteristic W (A, P, k)). This may be related to recent work of Hohloch, Noetal and Salamon [17]. We could think of a “completed” space of connections with a point having a neighborhood modelled on the product of a Hilbert space with a cone over a space L, where L is the space of sections of M , modulo dilation. In a finite-dimensional analogue the contribution of this point in a Morse theory description of the homology of the total space will involve the theory of the Conley index; the homology of L and various subsets. It is possible that there is a “Floer-analogue” of this which can be formulated in terms of the solutions of the eigenvalue equation, and “flow lines” between them.

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Even leaving aside this complication in 7-dimensions, of bubbling along an associative submanifold, it seems likely that the naive count of solutions in 6 dimensions is not the right thing to consider for the purposes of developing a 6 + 1 dimensional theory, including the conjectural holomorphic bundle E we have discussed. Consider a nonsingular G2 instanton over a topological cylinder R × Z. Restriction to slices {t} × Z gives a family of connections over Z. One can imagine a case when as t → ∞ these connections develop a singularity along Σ ⊂ Z. This phenomenon suggests that one would have to take account of pairs (E, Σ) in Z in order to have the correct gluing identities. This fits in with the fact that the Thomas’ algebro-geometric approach to the holomorphic Casson invariant includes contributions from sheaves, not just bundles. There seems to be a lot of scope for work relating the algebro-geometric and differential-geometric points of view. 6.3. Codimension-3 theories and monopoles. This subsection is rather more speculative. Given a noncompact Riemannian 3-manifold B with an end of a suitable kind one can study the Bogomolny monopole equation F (A) = ∗dA Φ for a connection A and section Φ of the adjoint bundle. We will just consider the case of structure group SU (2). Solutions correspond to translation-invariant instantons on R × B. One also imposes asymptotic conditions on the ends of B, the most important being that |Φ| → r −1 at infinity, where r > 0 is fixed. The most familiar case is when B = R3 and then one can reduce to the case when r = 1 by scaling the metric. But in general r will be a genuine parameter and it will not be possible to identify solutions for different values of r. The references [3], [9], [18], [1], and many others, give more details about this monopole theory. Now suppose we have a noncompact G2 -manifold Y . We can study the analogous equation (25) on Y with the asymptotic condition |u| → r−1 at infinity. Let us imagine that, for manifolds Y with an end of a suitable kind, we can find a set-up which leads to a Fredholm problem and to invariants, which would be numbers in the case of index zero. Then we could study the behaviour of solutions as the parameter r varies, in particular as r → 0. We can find plausible models for this based on compact co-associative submanifolds Q ⊂ Y , in much the same way as we modelled the blow-up behaviour around associative submanifolds. For simplicity consider first the flat case, so we have standard co-ordinates xa on Q and yi normal to Q. The equations (25) can be written very schematically as (45)

∇y u = Fyy + Fxx

∇x u = Fxy .

If we change variables, replacing yi by ryi and u by r−1 u, then take the limit as r tends to zero, we get a limiting equations (46)

∇y u = Fyy

∇x u = Fxy .

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The first of these equations is the Bogomolony monopole equation on each R3 slice normal to Q and the second equation is the Fueter equation for the resulting map from Q to the moduli space of monopoles on R3 . To say this more systematically and generally, recall that for each integer k > 0 we have a moduli space Monk of “centred” monopoles of charge k on R3 . (The charge is a topological invariant given by the degree of φ over a large sphere, in any trivialisation of the bundle.) This moduli space is a hyperkahler manifold of real dimension 4(k − 1) and the rotations of R3 act on Monk , permuting the complex structures. Given any compact coassociative submanifold Q ⊂ Y we can form a bundle Monk → Q with fibre Monk much as before and there is a Fueter equation for sections of Monk . We interpret solutions of this Fueter equation as possible asymptotic limits for sequences of solutions of (25) over Y as the parameter r tends to 0. These solutions are localised around Q and should be very close to reducible away from Q, with the structure group reduced to S 1 ⊂ SU (2) by the “Higgs field” u (which would be almost covariant constant away from Q). We could make exactly the same discussion for a noncompact Calabi-Yau manifold Z 6 , and we will now switch our focus to this case as it is simpler. We suppose the elementary topology of the set-up allows us to associate a class in H3 (Z) to our problem, analogous to the monopole charge in 3-dimensions. Then for each class κ ∈ H3 (Z) we expect to have a numerical invariant nκ , counting solutions of (22). Our model for the asymptotic behaviour involves a special Lagrangian submanifold P ⊂ Z. For each k we form a monopole bundle over P and we have a Fueter equation for sections. This has index 0 and we imagine we can define a number w(k, P ) by counting the solutions. Then we could hope to express the number nκ as some kind of count of special Lagrangian submanifolds P , weighted by the numbers w(k, P ). The simplest guess is a formula of the shape (47)



nκ = κ=





w(ki , Pi ).

ki [Pi ] i

This may be a rather crude approximation to the truth of the matter, because we have not discussed what happens when two special Lagrangians intersect. But in any case we could hope that there is some way of computing nκ from data localised around special Lagrangian submanifolds. This picture, if it stands up to closer scrutiny, is rather similar to Taubes’ relation between the Seiberg-Witten and Gromov invariants of a symplectic four-manifold W [34]. The moduli space of vortices on R2 would play the role in that case that the moduli space of monopoles does above. For a given “charge” k the moduli space of vortices is just the k-fold symmetric product of R2 [18]. When the ambient space is actually a complex surface we arrive in the realm of ordinary algebraic geometry. Given a curve Σ ⊂ W a section of the appropriate “vortex bundle” corresponds to an infinitesimal deformation of the order k formal neighbourhood of Σ, as a subscheme of W .

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All of this discussion assumes that we can indeed find the correct set-up to define numerical invariants nκ . The point we want to emphasise is that, if this can be done, one might hope that these are easier to define than the counts of special Lagrangians. Thus, by analogy, the Seiberg-Witten invariants of a 4-manifold are much easier to define than the Gromov invariants. Then we could take (47) as the definition, or a guide to the definition, of an invariant counting special Lagrangian submanifolds. References [1] Atiyah, M.F. and Hitchin, N.J. The geometry and dynamics of magnetic monopoles Princeton U.P. (1988). [2] Burghelea, D. and Haller, S. On the topology and analysis of a closed 1-form, I (Novikov’s theory revisited) In:Essays on geometry and relatd topics, Vol. 1 133–175 Monogr. Enseign. Math. 38 Geneva (2001). [3] Braam, P.J. Magnetic monopoles on 3-manifolds Jour. Differential Geometry 30 425– 464 (1989). [4] Brendle, S. Complex anti-self-dual instantons and Cayley submanifolds arxiv DG/0302094. [5] Brendle, S. On the construction of solutions to the Yang-Mills equations in higher dimensions arxiv DG/0302093. [6] Donaldson, S.K. and Thomas, R.P. Gauge Theory in higher dimensions In: The Geometric Universe, Huggett et al. Eds. Oxford U.P. (1998). [7] Donaldson, S.K. Moment maps and diffeomorphisms Asian J. Math. 3 1–15 (1999). [8] Donaldson, S.K. Floer homology groups in Yang-Mills theory Cambridge U.P. (2000). [9] Floer, A. Monopoles on asymptotically flat manifolds In: The Floer Memorial Volume, Birkhauser 3–41 (1995). [10] Floer, A. An instanton invariant for 3-manifolds Commun. Math. Phys. 118 215–240 (1989). [11] Gromov, M. Pseudoholomorphic curves in symplectic geometry Inventiones Math. 82 307–347 (1985). [12] Harvey, R. and Lawson, H. B. Calibrated geometries Acta Math. 148 47–157 (1982). [13] Haydys, A. Generalised Seiberg-Witten equations and hyperkahler geometry Thesis, Gottingen (2006). [14] Haydys, A. Nonlinear Dirac operators and quaternionic analysis Commun. Math. Phys. 281 251–286 (2008). [15] Haydys, A. Gauge theory, calibrated geometry and harmonic spinors arxiv 0902.3738. [16] Hong Y.-J., Harmonic maps into the moduli spaces of flat connections Ann. Global Analysis and Geometry 17 441–473 (1999). [17] Hohloch, S. Noetal, G. and Salamon, D. Hypercontact structures and Floer homology Geometry and Topology 13 2543–2617 (2009). [18] Jaffe, A. and Taubes, C.H. Vortices and monopoles Birkhauser, Boston. [19] Joyce, D. Riemannian holonomy groups and calibrated geometry In: Calabi-Yau manifolds and related geometries 1–68, Universitext, Springer (2003). [20] Joyce, D. On counting special Lagrangian homology spheres In:Topology and Geometry:Commemorating SISTAG Berrick et al Eds. Contemporary Meth. 314 125–151 Amer. Math. Soc. (1999). [21] Kovalev, A.G. Twisted connected sums and special Riemannian holonomy J. Reine Angew. Math. 565 125–160 (2003). om, J. Asymptotically cylindrical 7-manifolds with holo[22] Kovalev, A.G. and Nordstr¨ nomy G2 , with applications to compact irreducible G2 -manifolds Annals Global Analysis and Geom. 38 221–257 (2010).

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[23] Leung, N.C. Topological Quantum Field Theory for Calabi-Yau 3-folds and G2 -manifolds Adv. Theor. Math. Phys. 6 575–591 (2002). [24] Lewis, C. Spin(7) instantons Oxford University D. Phil thesis (1998). [25] Maulik, D. , Nekrasov, N. Okounkov, A. and Pandharipande, R. Gromov-Witten theory and Donaldson-Thomas theory, I Compos. Math 142 1263–1285 (2006). [26] Mclean, R. Deformations of calibrated submanifolds Commun. Analysis and Geometry 6 707–745 (1998). [27] Oancea, A. and Salamon, D. Lefschetz fibrations, intersection numbers and representations of the framed braid group Preprint arxiv 0708.2051. [28] Pidstrigatch, V. Hyperkahler manifolds and the Seiberg-Witten equations (English trans.) Proc Steklov Inst. Math. 246 249–262 (2004). [29] S´ a Earp, H. N. Instantons on G2 -manifolds Imperial College, London Ph. D Thesis (2009). [30] Salamon, S. M. Riemannian geometry and holonomy groups Pitman Rs. Notes Math. 201 (1989). [31] Salur,S. Deformations of asymptotically cylindrical coassociative submanifolds with moving boundary DG/0601420. [32] Tao, T and Tian, G. A singularity removal theorem for Yang-Mills fields in higher dimensions J. Amer. Math. Soc. 17 557–593 (2004). [33] Taubes, C.H. Stability in Yang-Mills theories Commun. Math. Phys. 91 235–263 (1983). [34] Taubes, C.H. SW ⇒ Gr: from the Seiberg-Witten equation to pseudoholomorphic curves Jour. Amer. Math. Soc. 9 845–918 (1996). [35] Taubes, C. H. Nonlinear generalisations of a 3-manifold’s Dirac operator In: Trends in Math. Phys. AMS/IP Studies Adv. Math. 13 475–486 (1998). [36] Thomas, R.P. A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on K3 fibrations Jour. Differential Geometry 54 367–438 (2000). [37] Thomas, R.P. Moment maps, monodromy and mirror manifolds In Symplectic geometry and mirror symmetry, Seoul 2000 World Scientific 467–498 (2001). [38] Tian, G. Gauge theory and calibrated geometry Ann. Math. 151 193–268 (2000). Imperial College London E-mail address: [email protected]

Surveys in Differential Geometry XVI

An invitation to toric degenerations Mark Gross and Bernd Siebert

Contents Introduction 1. Purely toric constructions 2. Introducing singular affine structures 3. Examples without scattering 4. Scattering 5. Three-dimensional examples References

43 44 50 54 63 72 77

Introduction In [GrSi2] we gave a canonical construction of degenerating families of varieties with effective anticanonical bundle. The central fibre X of such a degeneration is a union of toric varieties, glued pairwise torically along toric prime divisors. In particular, the notion of toric strata makes sense on the central fiber. A somewhat complementary feature of our degeneration is their toroidal nature near the 0-dimensional toric strata of X; near these points the degeneration is locally analytically or in the ´etale topology given by a monomial on an affine toric variety. Thus in this local model the central fiber is a reduced toric divisor. A degeneration with these two properties is called a toric degeneration. The name is probably not well-chosen as it suggests a global toric nature, which is not the case as we will emphasize below. A good example to think of is a degeneration of a quartic surface in P3 to the union of the coordinate hyperplanes. More generally, any Calabi-Yau complete intersection in a toric variety has toric degenerations [Gr2]. Thus the notion of toric degeneration is a very versatile one, conjecturally giving all deformation classes of Calabi-Yau varieties with maximally unipotent boundary points. Our construction has a number of remarkable features. It generalizes the construction of a polarized toric variety from an integral polyhedron (momentum polyhedron) This work was partially supported by NSF grants 0505325 and 0805328. c 2011 International Press

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to non-toric situations in a highly non-trivial, but canonical fashion. It works order by order, each step being controlled by integral affine (“tropical”) geometry. This has the striking consequence that any complex geometry feature of the degeneration that is determined on a finite order deformation of the central fibre can, at least in principle, be read off tropically. An important ingredient in this algorithm is the scattering construction, introduced by Kontsevich and Soibelman in a rigid-analytic setup in dimension two [KoSo]. The purpose of these notes is to provide an extended introduction to [GrSi2]. The emphasis is on highlighting some features of the construction by going through examples explicitly. To avoid repeating ourselves, we will introduce most concepts in an ad hoc fashion and refer to [GrSi2] for the general case and more technical definitions. 1. Purely toric constructions 1.1. Toric varieties from polyhedra. To start with let us recall the algebraicgeometric construction of a toric variety from a convex integral polyhedron σ ⊆ MR , the intersection of finitely many closed halfspaces. To keep track of functorial behaviour we work in MR := M ⊗Z R for some free abelian group M  Zn of rank n. If σ is bounded it is the convex hull in MR of finitely many points in M ⊆ MR . In any case, for each face τ ⊆ σ we have the cone generated by σ relative to τ :    Kτ σ := R≥0 · (σ − τ ) = m ∈ MR  ∃m0 ∈ τ, m1 ∈ σ, λ ∈ R≥0 : m = λ · (m1 − m0 ) . These cones are finite rational polyhedral, that is, there exist u1 , . . . , us ∈ M with Kτ σ = C(u1 , . . . , us ) := R≥0 · u1 + · · · + R≥0 · us . Note also that Kτ σ ∩ (−Kτ σ) = Tτ = τ − τ , the tangent space to τ . So Kτ σ is strictly convex if and only if τ is a vertex. The integral points of Kτ σ define a subring C[Kτ σ ∩ M ] of the Laurent polynomial ring C[M ]  C[z1±1 , . . . , zn±n ] by restricting the exponents to integral points of Kτ σ. For m ∈ Kτ σ ∩ M we write z m for the corresponding monomial. The invertible elements of this ring are precisely the monomials cz m with m ∈ Λτ := Tτ ∩ M and c ∈ C \ {0}. Example 1.1. Let σ ⊆ R2 be the triangle with vertices v1 = (0, 0), v2 = (1, 0), v3 = (0, a) with a ∈ N \ {0}. Then   Kv1 σ = C (1, 0), (0, 1)   Kv2 σ = C (−1, a), (−1, 0)   Kv3 σ = C (0, −1), (1, −a) For i = 1, 3 the monoid of integral points Kvi σ ∩ Z2 ⊆ Z2 is freely generated by the primitive generators (1, 0), (0, 1) and (0, −1), (1, −a) of the extremal rays. In other words, N2 −→ Kv1 σ ∩ Z2 , (α, β) −→ α · (1, 0) + β · (0, 1) = (α, β)

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and N2 −→ Kv3 σ ∩ Z2 ,

(α, β) −→ α · (0, −1) + β · (1, −a) = (β, −α − βa)

are isomorphisms of additive monoids. This shows C[Kvi σ ∩ Z2 ]  C[x, y],

i = 1, 3,

as abstract rings. For i = 2 the integral generators (−1, a), (−1, 0) of the extremal  −1 rays of Kv2 σ generate a proper sublattice of Z2 of index a = det −1 a 0 . Thus (−1, 0), (−1, a) also do not suffice to generate Kv2 σ ∩ Z2 as a monoid, for a > 1. It is not hard to show that a minimal set of generators of Kv2 σ rather consists of the a + 1 elements (−1, 0), (−1, 1), . . . , (−1, a). A good way to view C[Kv2 σ ∩ Z2 ] is as the ring of invariants of C[x, y] under the diagonal action of Z/a by a-th roots of unity ζ ∈ C, ζ a = 1: x −→ ζ · x,

y −→ ζ · y.

Under this identification z (−1,i) ∈ C[Kv2 σ ∩ Z2 ] corresponds to the invariant monomial xi y a−i . The remaining rings associated to higher dimensional faces of σ are  dim τ = 1 C[x, y ±1 ], C[Kτ σ]  ±1 ±1 C[x , y ], τ = σ.  As the example indicates, rings of the form C[Kτ σ ∩ M ] (toric rings) can be difficult to describe in terms of generators and relations. To obtain examples that can be easily written down in classical projective algebraic geometry, in this paper we therefore almost exclusively restrict ourselves to polyhedra σ with Kv σ ∩ M  Nn as a monoid, for any vertex v ∈ σ. If m1 , . . . , ms are the generators of the extremal rays of Kv σ, a necessary and sufficient condition for this to be true is s = n and det(m1 , . . . , ms ) = 1. Now given a convex integral polyhedron σ ⊆ MR , with dim σ = n for simplicity, and a face τ ⊆ σ we obtain the affine toric variety   Uτ := Spec C[Kτ σ ∩ M ] .   Since C[Kτ σ ∩ M ] ⊆ C[M ] any Uτ contains the algebraic torus Uσ = Spec C[M ]  Gnm . More generally, if τ ⊆ τ  then C[Kτ σ∩M ] is canonically a subring of C[Kτ  σ∩M ], and hence we have an open embedding Uτ  −→ Uτ . These open embeddings are mutually compatible. Hence the Uτ glue to a scheme Xσ of dimension dim σ. In other words, there are open embeddings Uτ → Xσ inducing the morphisms Uτ  → Uτ for all τ ⊆ τ  ⊆ σ. The multiplication action on Uσ = Gnm extends to Xσ . Hence Xσ is a toric variety. Note that according to this definition

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toric varieties have a distinguished closed point, the unit of Gnm . Moreover, for faces τ  ⊆ τ ⊆ σ the ring epimorphism  z m , m ∈ Kτ  τ m C[Kτ  σ ∩ M ] −→ C[Kτ  τ ∩ M ], z −→ 0, otherwise, induces a closed embedding ιτ : Xτ → Xσ with image disjoint from Gnm ⊆ Xσ unless τ = σ. The images of the various Xτ are called toric strata of Xσ , the image of Uσ = Gnm the big cell. If τ, τ  ⊆ σ are two faces it holds ιτ (Xτ ) ∩ ιτ  (Xτ  ) = ιτ ∩τ  (Xτ ∩τ  ). Here we make the convention X∅ := ∅. Hence the face lattice of σ readily records the intersection pattern of the toric strata of Xσ . In particular, the facets (codimension one faces) of σ are in one-to-one correspondence with the toric prime divisors, the irreducible Weil divisors that are invariant under the torus action. Example 1.2. For σ = conv{(0, 0), (1, 0), (0, a)} from Example 1.1 we claim that projective plane P(1, a, 1). Recall that P(1, a, 1) is the quotient of Xσ isthe weighted  3 A \ (0, 0, 0) by the action of Gm that on closed points is given by λ · (x0 , x1 , x2 ) = (λx0 , λa x1 , λx2 ),

λ ∈ C∗ .

In fact, on A3 \ V (x0 ) the ring of invariants of the action is C[x, y] with x = x1 /xa0 , y = x2 /x0 and similarly on A3 \ V (x2 ). On the other hand, on A3 \ V (x1 ) the ring of invariants is generated by xi0 xa−i 2 /x1 , i = 1, . . . , a. This is the ring of invariants of the diagonal Z/a-action on A2 . Hence P(1, a, 1) has an affine open covering with spectra of the rings C[Kvi σ ∩ Z2 ] discussed in Example 1.1. The gluing morphisms between these open sets are the same as given by toric geometry. By construction the scheme Xσ depends only on the cones Kτ σ, hence only on the normal fan of σ with elements the dual cones (Kτ σ)∨ ⊆ MR∗ . More generally, toric varieties are constructed from fans. In particular, integrality and boundedness of σ can be weakened to rationality of the cones Kτ σ. Those toric varieties coming from integral polyhedra (bounded or not) are endowed with a toric ample line bundle. In fact, defining the cone over σ   C(σ) := cl R≥0 · (σ × {1}) ⊆ MR × R, the ring C[C(σ) ∩ (M × Z)] is graded by deg z (m,h) := h ∈ N. Taking the closure cl here is important in the unbounded case. It adds the asymptotic cone lima→0 a · σ to MR × {0}. It is then not hard to see that one has a canonical isomorphism   Xσ  Proj C[C(σ) ∩ (M × Z)] . Although C[C(σ) ∩ (M × Z)] is not in general generated in degree 1, integrality of the vertices of σ implies that the sheaf O(1) on the right-hand side is nevertheless locally free. This yields the toric ample line bundle mentioned above.

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1.2. Toric degenerations of toric varieties. Now let us see how certain unbounded polyhedra naturally lead to toric degenerations with general fibre a toric variety. Let σ ˜ ⊆ MR × R be an (n + 1)-dimensional convex integral polyhedron that is closed under positive translations in the last coordinate:   σ ˜=σ ˜ + 0 × R≥0 . Let q : MR × R → MR be the projection and σ := q(˜ σ ). Then the non-vertical part of ∂ σ ˜ is the graph of a piecewise affine function ϕ:σ→R with rational slopes. The domains of affine linearity of ϕ define a decomposition P of σ into convex polyhedra. In terms of this data, σ ˜ is the upper convex hull of the graph of ϕ:    σ ˜ = (m, h) ∈ MR × R  h ≥ ϕ(m) . Thus σ ˜ is equivalent to a polyhedral decomposition P of the convex integral polyhedron σ together with a function ϕ on σ that is piecewise affine and strictly convex with respect to P and takes integral values at the vertices of P. Now Xσ˜ is an (n + 1)-dimensional toric variety that comes with a toric morphism π : Xσ˜ → A1 . In fact, each of the rings C[Kτ˜ σ ˜ ∩ (M × Z)] is naturally a C[t]-algebra by letting t = z (0,1) , and the gluing morphisms are homomorphisms of C[t]-algebras. The preimage of the closed point 0 ∈ A1 is set-theoretically the union of toric prime divisors of Xσ˜ that map non-dominantly to A1 . It is reduced if and only if ϕ has integral slopes, that is, takes integral values at all integral points, not just the vertices. To see this let v˜ = (v, ϕ(v)) ∈ σ ˜ be a vertex and ϕv a piecewise linear function on MR which agrees with ϕ(v + ·) − ϕ(v) close to 0. In other words, the graph of ϕv is the boundary of the tangent cone    ˜ = (m, h) ∈ MR × R  h ≥ ϕv (m) Kv˜ σ of σ ˜ at v˜. A C-basis for C[Kv˜ σ ˜ ∩ (M × Z)]/(t) is given by z (m,h) ,

ϕv (m) ≤ h < ϕv (m) + 1,

and z (m,h) is nilpotent modulo (t) if and only if ϕv (m) < h, that is, if ϕv (m) is not integral. Assume now that ϕ(m) ∈ Z for all m ∈ σ ∩ M . Then C[Kv˜ σ ˜ ∩ (M × Z)]/(t) has one monomial generator z (m,ϕv (m)) for any m ∈ Kv σ ∩ M , and the relations are    z (m+m ,ϕv (m+m )) , ∃τ ∈ P : v ∈ τ and m, m ∈ Kv τ   (m,ϕv (m)) (m ,ϕv (m )) z ·z = 0, otherwise.

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Figure 1.1. The polyhedron σ ˜ (left) and the normal fan of σ ˜ (right). In other words, π −1 (0) is the scheme-theoretic sum (fibred coproduct) of the n-dimensional toric varieties Xτ˜  Xq(˜τ ) with τ˜ ⊂ ∂ σ ˜ projecting bijectively onto some τ ∈ P [n] . These are precisely the toric prime divisors of Xσ˜ mapping non-dominantly to A1 . As in [GrSi2] P [k] denotes the set of k-dimensional cells of the polyhedral complex P. To understand general fibres π −1 (t), t = 0, we localize at t. This has the effect of removing the lower boundary of σ ˜ , that is, of going over to σ ˜ + (0 × R) = σ × R. Thus π −1 (A1 \ {0}) = Xσ×R = Xσ × (A1 \ {0}). Thus each general fibre π −1 (t), t = 0, is canonically isomorphic to Xσ . Note, however, that these isomorphisms degenerate as t approaches 0. Example 1.3. Here is a degeneration of P1 to two copies of P1 featuring an   [1] Al−1 -singularity in the total space. Let σ = [0, a + 1], P = [0, a], [a, a + 1] , ϕ(0) = ϕ(a) = 0, ϕ(a + 1) = l as in Figure 1.1. The slopes of ϕ are 0 and l on the two 1-cells. The boundary of σ ˜ has two non-vertical components. Each gives one of the two irreducible components of π −1 (0). Their point of intersection is the 0-dimensional toric stratum defined by the vertex v˜ = (a, 0) of σ ˜ . The monoid Kv˜ σ ˜ ∩ Z2 has generators m1 = (−1, 0), m2 = (1, l), m3 = (0, 1) with relation m1 + m2 = l · m3 . Hence, C[Kv˜ σ ˜ ∩ Z2 ]  C[z1 , z2 , t]/(z1 z2 − tl ), with zi = z mi for i = 1, 2 and t = z (0,1) defining π. This is a local model of a smoothing of a nodal singularity with an Al−1 -singularity in the total space. Thus the changes of slope of ϕ at the non-maximal cells of P determine the singularities of the total space. Another way to understand the total space is from the normal fan of σ ˜ . It can be obtained by subdividing the fan of A1 ×P1 by the ray through (−l, 1). This corresponds to a weighted blow up at one of the zero-dimensional toric strata, leading to another P1 over 0 ∈ A1 and the Al−1 -singularity. Note also that the length a of the interval is completely irrelevant to the complex geometry; it only changes the polarization which we did not care about at this point. 

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Example 1.4. For a two-dimensional example consider the convex hull   σ = conv (1, 0), (0, 1), (−1, 0), (0, −1) with the polyhedral decomposition into 4 standard 2-simplices shown in the figure below. At (x, y) ∈ σ the function ϕ is given by 0, x, y and x+y, respectively, depending on the maximal cell containing (x, y) as shown in the figure. Thus the slope of ϕ changes by 1 along each interior 1-cell of P. The associated degeneration has as central fibre a union of 4 copies of P2 , glued pairwise along the coordinate lines as indicated by P. Thus the singular locus is a union of 4 projective lines, joined in one point Xv˜ , where v˜ = (0, 0, 0) ∈ σ ˜ . The monoid Kv˜ σ ˜ is generated by m1 = (1, 0, 1), m2 = (0, 1, 1), m3 = (−1, 0, 0), m4 = (0, −1, 0) fulfilling the single relation m1 + m3 = m2 + m4 . Thus C[Kv˜ σ ˜ ∩ Z3 ]  C[z1 , z2 , z3 , z4 ]/(z1 z3 − z2 z4 ), with C[t]-algebra structure defined by t = z (0,0,1) = z1 z3 = z2 z4 . This shows that the total space Xσ˜ has a singular point isomorphic to the origin in the affine cone over a smooth quadric, while the central fibre is a product of two normal crossing singularities. The general fibre is isomorphic to X  always, which here is a  σ1, as 1 maps the fan of P1 × P1 toric Z/2-quotient of P1 × P1 . In fact, −1 1 to the normal fan of σ. Restricted to the big cell G2m ⊂ P1 × P1 , this map is given by C[u±1 , v ±1 ] −→ C[x±1 , y ±1 ],

u −→ xy, v −→ x−1 y.

The subring of C[x±1 , y ±1 ] generated by x±1 y ±1 is the invariant ring for the involution (x, y) → (−x, −y), and this involution extends to P1 × P1 . Note that Xσ has 4 isolated quotient singularities. These correspond to the vertices of σ. It is also instructive to write down this degeneration embedded projectively. As explained at the end of §1.1 we have to take the integral points of the cone C(˜ σ ) ⊆ R4 over σ ˜ as generators of a graded C[t]-algebra, with the degree given by the projection σ ) ∩ Z4 ] is generated as a C[t]-algebra to the last coordinate and t = z (0,0,1,0) . Now C[C(˜ by the monomials X = z (1,0,1,1) , Y = z (−1,0,0,1) , Z = z (0,1,1,1) , W = z (0,−1,0,1) , U = z (0,0,0,1) . Note that these generators are in one-to-one correspondence with the integral points of σ. The relations are XY − tU 2 ,

ZW − tU 2 .

This exhibits Xσ˜ as the intersection of two quadrics in P4A1 = A1 × P4 .



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2. Introducing singular affine structures From a birational classification point of view toric varieties are boring as they are all rational. In particular, it is impossible to construct degenerations of non-rational varieties directly by the method of §1.2. The idea in [GrSi2] is that one can get a much larger and more interesting class of degenerations by gluing toric pieces in a non-toric fashion. The central fibre is still represented by a cell complex P of integral polyhedra, but the integral affine embedding of the cell complex into Rn exists only locally near each vertex. In other words, the underlying topological space of P is an integral affine manifold B, with singularities on a cell complex Δ ⊆ B of real codimension 2 that is a retract of |P [n−1] | \ |P [0] |. We then construct the total space X of the degeneration order by order, by gluing torically constructed non-reduced varieties, “thickenings” of toric varieties so to speak, in a non-toric fashion. Thus our starting data are integral cell-complexes with compatible integral affine charts near the vertices. We call these integral tropical manifolds ([GrSi2], Definition 1.2) because they arise naturally as the bounded parts of the embedded tropical varieties associated to the degeneration. 2.1. Degenerations of hypersurfaces. A hypersurface X ⊆ Pn+1 of degree d ≤ n+2 can be degenerated to a union of d coordinate hyperplanes simply by deforming the defining equation. For example, for n = 2 let f ∈ C[X0 , . . . , X3 ] be a general homogeneous polynomial of degree d ≤ 4. Then X0 . . . Xd−1 + tf = 0 defines a family π : Y → A1 with π −1 (0) a union of d coordinate hyperplanes. This is not a semistable family because Y is not smooth at the intersection of V (f ) with  the singular locus of V (X0 · . . . · Xd−1 ). The latter consists of d2 projective lines V (Xi , Xj ), 0 ≤ i < j ≤ d − 1. Since f is general the intersection of V (f ) with any of these projective lines consists of d reduced points with two nonzero coordinate entries each, that is, not equal to the 4 special points [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]. Near any of these points Y → A1 is locally analytically given by the projection of the three-dimensional A1 -singularity V (xy − wt) ⊆ A4 to the t-coordinate. Note that for w = 0 this is a product of a semistable degeneration of a curve with A1 \ {0}, but this fails at w = 0, which contains the singular point of Y . While the local model of this degeneration is still toric, the singular points of Y are general points of one-dimensional toric strata of π −1 (0). Hence this is a very different degeneration than the torically constructed ones in §1.2. Our first aim is to obtain this local degeneration naturally from a tropical manifold. This was the starting point for [GrSi2] in March 2004. 2.2. A singular affine manifold. There is a famous two-dimensional singular integral affine manifold in the theory of integrable systems, called the focus-focus

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Figure 2.1. Various charts I–III for the focus-focus singularity: On the left, the shaded area is a fundamental domain for B. In the two other figures, the dashed vertical lines indicate the parts of to be removed; the horizontal curves are all straight affine lines. singularity [Wi], which is the model for our singular affine manifolds B at general points of Δ, the codimension 2 singular locus of the affine structure. An (integral) affine structure on a topological manifold is an atlas with transition functions in the (integral) affine linear group. Parallel transport of tangent vectors is well-defined on such manifolds. In fact, an affine manifold comes naturally with a flat, torsion-free, but usually non-metric connection. The focus-focus singularity is (the germ at the origin P of) R2 with an integral affine structure away from P  such  that parallel transport counterclockwise around P gives the transformation 11 01 . This affine manifold B can be constructed by gluing R2 \ (R≥0 × {0}) with the given affine structure to itself by the integral affine transformation (x, y) −→ (x, x + y),

for x ≥ 0, −x < y < 0,

see Chart I in Figure 2.1. Note that ∂y is an invariant tangent vector, and indeed the projection (x, y) → x on R2 \ (R≥0 × {0}) to the first coordinate descends to a continuous map B → R that is integral affine away from P ∈ B. The preimage of the origin is a line through P . Alternatively, B can be constructed from the two charts R2 \ ({0} × R≥0 ) and 2 R \ ({0} × R≤0 ), each covering B minus one half of , via      (x, y), x 0, see Charts II and III in Figure 2.1. Note that we do not know how to continue any non-vertical affine line across P . 2.3. The basic example. The focus-focus singularity admits a polyhedral decomposition by decomposing along the invariant line . This decomposition has two maximal cells σ± , with preimages R≥0 × R and R≤0 × R in any of the charts of Figure 2.1. We can also define a strictly convex, integral affine function ϕ : B → R by letting ϕ = 0 on the left maximal cell σ− and ϕ(x, y) = x on the right maximal cell σ+ . Again, this is independent of the chosen chart, and it takes integral values at any integral point of B.

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Now let us try to treat this situation with a singularity in the same way as we did in §1.2. We can still describe the central fibre as before: X0 = Xσ− Xρ Xσ+ ,       Xσ− = Spec C[x, w±1 ] , Xσ+ = Spec C[y, w±1 ] , Xρ = Spec C[w±1 ] , where ρ = σ− ∩ σ+ , x = z (−1,0) , y = z (1,0) , w = z (0,1) . By writing this it is important to fix one affine realization of σ± , say the Chart II in Figure 2.1. Intrinsically, the exponent (−1, 0) defining x is an integral tangent vector on σ− , while the exponent (1, 0) defining y is an integral tangent vector on σ+ . For w we can take either maximal cell, because the exponent (0, 1) just represents the global integral vector field ∂y defined above. Continuing to work in Chart II the procedure of §1.2 yields the deformation V (xy − t) of V (xy) ⊆ A2 × (A1 \ {0}). But in Chart III the tangent vector (1, 0) of σ+ changes to (1, 1). Since z (1,1) = wy the same procedure applied in this chart therefore yields V (xy − wt) for the deformation. Because w is invertible V (xy − t) and V (xy −wt) are isomorphic as schemes over C[t], but not as deformation with the given embedding of X0 . Of course, this phenomenon is just due to the affine monodromy around P . As expected, the singularities of the affine structure lead to inconsistencies in the naive approach using toric geometry. k+1 . In other The starting point to overcome this problem is to work modulo  t k+1  words, we want to construct a k-th order deformation Xk → Spec C[t]/(t ) . The topological spaces of X0 and Xk are the same, so we only have to deal with the structure sheaf. We now use toric geometry merely to define the correct non-reduced versions (k-th order thickenings) of the toric strata. We then glue two maximaldimensional strata intersecting in codimension one by a non-toric automorphism of the common codimension one stratum, but the choice of this automorphism is different depending on which chart we use. In the present example, the thickenings of the toric strata suggested by toric geometry are given by the rings   Rσk − = Sk [x1 , y1 , w±1 ]/ x1 y1 − t, y1k+1 ,   Rσk + = Sk [x2 , y2 , w±1 ]/ x2 y2 − t, xk+1 , 2   k+1 k Rρ,σ , = Sk [x1 , y1 , w±1 ]/ x1 y1 − t, xk+1 1 , y1 −   k+1 k Rρ,σ , = Sk [x2 , y2 , w±1 ]/ x2 y2 − t, xk+1 2 , y2 + with Sk = C[t]/(tk+1 ). Here we distinguish between x and y as monomials on σ− and on σ+ . Thus y1 = z (1,0,1) in either chart, while y2 = z (1,0,1) in Chart II and y2 = z (1,1,1) in Chart III. These monomials only depend on the affine structure on σ± and hence have an intrinsic meaning. For the thickening of the ρ-stratum, however, we obtain two rings, depending on which maximal cell σ± the monomials live on. For the monomial w we don’t need to make any choices because it corresponds to a globally defined vector field. In any case, we have two natural Sk -algebra epimorphisms k , q− : Rσk − −→ Rρ,σ −

k q+ : Rσk + −→ Rρ,σ , +

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exhibiting the thickened ρ-stratum as closed subscheme of the σ− - and σ+ -stratum, respectively. k k Affine geometry suggests two isomorphisms hII , hIII : Rρ,σ → Rρ,σ , depending − + on which chart we use to go from σ− to σ+ : hII : x1 −→ x2 ,

y1 −→ y2 ,

hIII : x1 −→ wx2 , y1 −→ w

−1

w −→ w y2 , w −→ w

Now comes the point: To remedy the inconsistency caused by this, let α ∈ C∗ and compose hII with the automorphism gII : x2 −→ (α + w) · x2 , y2 −→ (α + w)−1 y2 , w −→ w  k  . For Chart III we compose with of the localization Rρ,σ + α+w gIII : x2 −→ (1 + αw−1 ) · x2 ,

y2 −→ (1 + αw−1 )−1 y2 ,

w −→ w.

We then indeed obtain     gII ◦ hII (x1 ) = (α + w) · x2 = (1 + αw−1 ) · wx2 = gIII ◦ hIII (x1 ),     gII ◦ hII (y1 ) = (α + w)−1 · y2 = (1 + αw−1 )−1 · w−1 y2 = gIII ◦ hIII (y1 ). The result of gluing Spec Rσk − and Spec Rσk + along the codimension one strata via this k isomorphism is given by the fibre product Rσk − ×(Rρ,σ k )α+w Rσ+ . In this fibre product +

k the homomorphism Rσk + → (Rρ,σ ) is the composition of q+ with localization, + α+w k k while Rσ− → (Rρ,σ+ )α+w composes q− and the localization homomorphism with gII ◦ hII = gIII ◦ hIII . It can be shown ([GrSi2], Lemma 2.34) that generators for this fibre product as an Sk [w±1 ]-algebra are     x = x1 , (α + w)x2 , y = (α + w)y1 , y2

with the single relation (coming from x1 y1 = t = x2 y2 ) (2.1)

xy − (α + w)t = 0.

This is the k-th order neighbourhood of the surface degeneration discussed in §2.1, with the A1 -singularity at x = y = 0 and w = −α. 2.4. General treatment in codimension one. The discussion in §2.3 generalizes to arbitrary dimension n as follows. Let ρ be a codimension one cell of P, Xσ± and Xρ with separating the maximal cellsσ− , σ+ .This gives three toric varieties  the respective big cells Spec C[Λσ± ]  Gnm and Spec C[Λρ ]  Gn−1 m . For an integral polyhedron τ we use the notation Λτ for the lattice of integral tangent vector fields on τ . The convex integral piecewise affine function ϕ changes slope along ρ by some integer k > 0. We then obtain the same rings and gluing morphisms as in §2.3, except that w has to be replaced by n − 1 coordinates w1 , . . . , wn−1 for Gn−1 m ⊆ Xρ .

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There are affine charts on the interiors of σ± and near each vertex of ρ. Each vertex k k v ∈ ρ suggests an isomorphism hv : Rρ,σ → Rρ,σ . For a different vertex v  ∈ ρ we − + m m have hv (z ) = hv (z ) for m ∈ Λρ because these tangent vectors are globally defined on σ− ∪σ+ . But for general m ∈ Λσ− , parallel transport Πv to σ+ through v  is related to parallel transport Πv through v as follows: (2.2)

Πv (m) = Πv (m) + m, dˇρ  · mρv v .

Here mρv v is some element of Λρ , viewed canonically as a subset of Λσ+ , and dˇρ is ∗  the primitive generator of Λv /Λρ  Z that evaluates positively on tangent vectors pointing from ρ into σ+ . This monodromy formula implies  ρ m,dˇρ · hv (z m ). hv (z m ) = z mv v ρ

Taking x = z m , y = z −m in the chart at v with m, dˇρ  = 1 yields hv (x) = z mv v ·hv (x), ρ hv (y) = z −mv v · hv (y). Thus to define the gluing invariantly we can compose hv with k an automorphism of Rρ,σ of the form + −dˇρ ,m

gv : z m −→ fρ,v

· zm

for functions fρ,v ∈ C[Λρ ], indexed by vertices v of ρ and fulfilling the change of vertex formula (2.3)

ρ

fρ,v = z mv v · fρ,v .

In the example in §2.3 we can think of v as lying on the upper half of , v  on the ρ lower half and fρ,v = 1 + αw−1 , fρ,v = α + w. Then mρv v = (0, 1), z mv v = w, and the change of vertex formula reads α + w = w · (1 + αw−1 ). In the general case the result of gluing is the hypersurface xy − fρ,v (w1 , . . . , wn−1 )tk = 0. 3. Examples without scattering In Section 2 we worked in an affine (in the algebraic geometry sense) neighbourhood of a singular point of the total space of the degeneration. We now want to look at projective examples in two dimensions, with one or several singular points. To this end we work with bounded tropical manifolds and use the toric procedure from §1.2 for the treatment away from the singularities. 3.1. Overview of the general procedure. To obtain degenerations of complete varieties we need to start from a bounded tropical manifold (B, P), where B denotes the underlying singular integral affine manifold and P is the decomposition into integral polyhedra, together with an (in general multi-valued) integral piecewise affine function ϕ. The tropical manifold determines readily the prospective  central fibre X0 = σ∈Pmax Xσ from the maximal cells, glued pairwise torically in codimension one. The most general approach also allows one to compose the gluing

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in codimension one with a toric automorphism, but as this is a rather straightforward generalization that only complicates the formulas we do not include it here. In the notation of [GrSi2] this means taking se = 1 for any inclusion e : ω → τ of cells ω, τ ∈ P. To obtain local models near the singular points we need to choose, for each ρ ∈ [n−1] P and vertex v ∈ ρ, a regular function fρ,v on an open set in Xρ ⊆ X0 . The open set is the affine neighbourhood Xτ Uv := X0 \ τ ∈P, v∈τ

of the corresponding point Xv ∈ X0 . As explained in §2.4, consistency of the gluing dictates the change of vertex formula (2.3) that relates fρ,v and fρ,v for vertices v, v  ∈ ρ. There is also a compatibility condition for the fρ,v for fixed v and all ρ containing a codimension-2 cell that makes sure everything stays toric ´etale locally at general points of the toric strata. We will explain this condition when we need it later on (see the discussion after Example 4.1). Next, the piecewise affine function ϕ determines local toric models for the degeneration, defined by the toric procedure from §1.2. Explicitly, if v ∈ P is a vertex and ϕv is a piecewise linear representative of ϕ introduced in §1.2 then the local model near the zero-dimensional toric stratum Xv is        Spec C[P ] −→ Spec C[t] , P = (m, h) ∈ M × Z  h ≥ ϕv (m) . Here t = z (0,1) makes C[P ] into a C[t]-algebra. Following the discussion in Section 2 we now work to some finite t-order k, that is modulo tk+1 , and decompose according to toric strata. This gives toric local models that depend on k, a toric affine open neighbourhood Uω ⊆ X0 of Int(Xω ), a toric stratum Xτ intersecting Uω and a maximal-dimensional reference cell σ ⊇ τ . In analogy with Uv for ω ∈ P we take Xτ . Uω := X0 \ τ ∈P, ω⊆τ

Thus we need σ ∈ Pmax and cells ω, τ ∈ P with ω ⊆ τ ⊆ σ. The last condition is equivalent to Uω ∩Xτ = ∅. The corresponding ring, localized at all the gluing functions k k fρ,v with ρ ⊇ τ , is denoted Rω→τ,σ , see [GrSi2], §2.1 for details. Thus  Spec Rω→τ,σ is a k-th order non-reduced version (“thickening”) of (Xτ ∩ Uω ) \ ρ⊇τ V (fρ,v ), v ∈ ω arbitrary. Note that for v, v  ∈ τ the gluing functions fρ,v and fρ,v differ by a monomial that is invertible on Uω , so V (fρ,v ) ∩ Uω = V (fρ,v ) ∩ Uω .

k The relation ideal defining Rω→τ,σ in (the localization at ρ⊇τ fρ,v of) C[P ] is generated by all monomials z (m,h) , (m, h) ∈ Λτ ⊕ Z, that have τ -order at least k + 1. Here one defines the τ -order as the maximum, over all σ  ∈ Pmax containing τ , of the orderof vanishing of z (m,h) on the big cell of Xσ , viewed canonically as a subset of  Spec C[P ] . Now even in examples with few cells this procedure requires the gluing of many affine schemes. We can, however, sometimes use the following shortcut. The integral points of B provide a basis of sections of a natural very ample line bundle on the

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central fibre X0 , hence providing a closed embedding X0 → PN −1 , N the number of integral points of B. In sufficiently simple examples the gluing morphisms defining the deformation of X0 readily homogenize to describe the k-th order deformation of k+1 ). The toric nature of this X0 as a subspace of PN Ok , Ok = Spec Sk , Sk = C[t]/(t construction makes sure that this procedure gives the right local models. This is how we compute most of the examples in this paper. 3.2. One singular point I: Two triangles. For our first series of examples we fit the basic example from §2.3 into a projective degeneration. This amounts to finding a polyhedral decomposition of a bounded neighbourhood of the singular point P of the focus-focus singularity in Figure 2.1. One of the easiest ways to do this is to  take two standard triangles (integrally affine isomorphic to conv (0, 0), (1, 0), (0, 1) ), one on each half-plane σ± , and with P at the center of the common edge. In particular, P should not be integral, but rather half-integral. Thus let us first shift our Charts I–III by (0, 1/2) so that the singular point P is at (0, 1/2). Now in Chart II take     σ1 = conv (0, 0), (−1, 0), (0, 1) , σ2 = conv (0, 0), (1, 0), (0, 1) as maximal cells for the polyhedral decomposition, see Figure 3.1. This is one of the two choices making the boundary locally convex. In fact, we can always bring σ1 to the suggested form by applying a transvection a1 01 on Chart I for some a ∈ Z. This descends to an integral affine  transformation of the focus-focus singularity. Then σ2 = conv (0, 0), (1, b), (0, 1) for some b ∈ Z. Convexity at (0, 0) implies b ≥ 0, while convexity at (0, 1) implies b ≤ 1 (check in Chart III). Local convexity of ∂B is indispensible to define local toric models for the deformation. Both choices are abstractly isomorphic, b = 0 making the lower boundary a line, b = 1 the upper one. There are 4 integral points on this tropical manifold B1 , the vertices of P. We interpret these as homogeneous coordinates in a P3 = Proj C[X, Y, Z, W ] . The central fibre X0 is then given by the hypersurface XY = 0, because X and Y are the only vertices not contained in the same cell. In fact, if P consists of standard simplices, X0 can be defined by the Stanley-Reisner ideal [Ho],[St] of the simplicial complex given by P. This is clear from the definitions. Now the gluing computation in §2.3 readily homogenizes. In fact, viewing the computation in Chart II as being homogenized with respect to Z, the relation (2.1) becomes  XY W −t α+ = 0. ZZ Z

Figure 3.1. The two charts defining the tropical manifold B1 .

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Figure 3.2. Torically constructed degeneration of P1 × P1 Clearing denominators yields the family of quadric hypersurfaces XY − tZ(αZ + W ) = 0. The analogous computation in the other chart gives the same result. Hence we obtain a projective degeneration with general fibre isomorphic to P1 × P1 . The total space has another A1 -singularity of a toric nature at X = Y = Z = 0. This singularity is due to the fact that ∂B1 is not straight at (0, 1). It is interesting to compare this deformation to the torically constructed one for a diagonally subdivided unit square with ϕ changing slope by 1 along the diagonal. As shown in Figure 3.2 we again have 4 homogeneous generators X, Y, Z, W and the relation XY − tZW = 0. The change of coordinates W → αZ + W shows that this family is isomorphic to the previous one, the only difference being that now both A1 -singularities of the total space are at 0-dimensional toric strata of P3 . Thus in a sense, by introducing the singular point in the interior we have straightened the boundary of the momentum polytope at the vertex labelled Z and moved one of the two A1 -singularities in the total space to a non-toric position. The fact that the introduction of a singular point leads to an isomorphic family is a rather special phenomenon due to the large symmetry in this example. 3.3. One singular point II: Blow-up of P2 . A similar tropical manifold, which we denote B2 , leads to a degeneration of the blow-up of P2 (Figure 3.3). Note that the boundary of B2 is straight at both vertices labelled U and W . Again we take ϕ

Figure 3.3. Three charts defining the tropical manifold B2 .

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Figure 3.4. Torically constructed degeneration of P2 blown up once to change slope by 1 along the only interior edge, with  vertices labelled U and W . 4 There are 5 integral points, so we work in PA1 = Proj C[t][X, Y, Z, U, W ] . Now σ2 is not a standard simplex. Thus we have a relation that does not receive any corrections from monodromy, namely W Y − U Z = 0. The gluing computations at the vertices labelled U and W homogenize to the two equations XY − t(αU + W )U = 0, XZ − t(αU + W )W = 0. For U W = 0 these are related by the substitution Z = W Y /U . The central fibre X0 has two irreducible components P2 and P1 ×P1 , glued along a P1 . The total space has only one singular point, the A1 -singularity at [0, 0, 0, 1, −α] ∈ (X0 )sing = P1 . The general fibre of this degeneration π : X → A1 contains the (−1)-curve E = V (X, αU +W, αY + Z) whose contraction yields a P2 . This is a well-known example: The 3 relations form the 2 × 2-minors of the matrix

 U W X , Y Z t(αU + W ) which for t = 1 describe the cubic scroll, see e.g. [Ha], Example 7.24. Again, this example has a toric analogue (Figure 3.4). The relations are W Y − U Z = 0,

XY − tU 2 = 0,

XZ − tU W = 0.

In fact, the substitution U → αU + W , Y → αY + Z yields the non-toric ideal above. 3.4. Propagation. Once we start changing the gluing of components somewhere we are forced to change at other places as well to keep consistency. Thus in a sense the gluing functions propagate. Example 3.1. Consider the non-compact example of a tropical manifold with 4 maximal cells σ1 , . . . , σ4 shown in Figure 3.5. If we take the piecewise affine function ϕ with ϕ(1, 0) = 1, ϕ(0, 1) = 1, ϕ(−1, 0) = 0, ϕ(0, −1) = 0, the toric local model at v is C[t] −→ C[x, y, z, w]/(xy − zw),

t −→ xy = zw.

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Figure 3.5. A tropical manifold forcing propagation of the gluing function. The figure on the right shows the tangent vectors belonging to the generators of the toric model. In terms of integral tangent vectors at v the generators are x = z (1,0,1) ,

y = z (−1,0,0) ,

z = z (0,1,1) ,

w = z (0,−1,0) .

Now let us try to glue together the k-th order neighbourhood of X0 following the general procedure of §2.4 with gluing function fρ3 ,v = 1 + y and all other fρi ,v = 1. We k defining the thickenings of the irreducible components Xσi , 4·2 = have 4 rings Rv→σ i ,σi k k 8 rings for the one-dimensional toric strata Rv→ρ , Rv→ρ (i taken modulo 4), i ,σi i ,σi−1 and 4 rings, identified mutually via an affine chart at v,   k = C[x, y, z, w, t]/ (xy − zw, xy − t) + I Rv→v,σ k i for the thickening of the zero-dimensional toric stratum Xv . Here Ik = (x, z)k+1 + (x, w)k+1 + (y, z)k+1 + (y, w)k+1 is the ideal generated by monomials that are divisible by tk+1 at the generic point of Xσi for some i. For the toric local model we want to take the inverse limit of all these rings with respect to the homomorphisms that we have introduced between them. These are of two kinds. First, for τ  ⊆ τ we have k k Rv→τ,σ → Rv→τ  ,σ defining the inclusion of toric strata with reference cell σi . Second, i i k k there are the change of reference cell isomorphisms Rv→τ,σ → Rv→τ,σ . This requires i i±1 compatibility of the compositions. In the present case this comes down to checking the following. Let θvk be the composition k k k k k Rv→v,σ −→ Rv→v,σ −→ Rv→v,σ −→ Rv→v,σ −→ Rv→v,σ 1 2 3 4 1

of changing the reference cell σi counterclockwise around the origin by crossing the ρi . The compatibility condition is θvk = id. But crossing ρ1 , ρ2 and ρ4 yields the identity. Thus θvk equals the change of reference cell isomorphism from crossing ρ3 , which is x → x,

y → y,

z → (1 + y)z,

w → (1 + y)−1 w.

Recall that we pick up a negative power of fρ3 ,v = 1 + y if the tangent vector points from ρ3 into σ3 , while monomials with exponents tangent to ρ3 are left invariant.

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Note also that because y is nilpotent now, 1 + y is invertible, so this is a well-defined k automorphism of Rv→v,σ . In any case, we see θvk = id as soon as k ≥ 2. 1 To fix this it is clear how to proceed: Just take fρ1 ,v = 1 + y as well. Then by the sign conventions, crossing ρ1 gives the inverse of what we had for ρ3 . Then θvk = id as k for any k (and any i).  automorphism of Rv→v,σ i Remark 3.2. There is an important observation to be made here. On Xρ3 the monomial y is nonzero, that is, of order zero with respect to t. But once we move to a general point of Xρ1 we have y = x−1 t because of the relation xy = t. Thus the t-order increases. And this is not an accident, but happens whenever we follow a monomial z (m,h) , (m, h) ∈ Λv × Z in direction −m ([GrSi2], Proposition 2.6). More precisely, if ϕ differs by aρ ∈ Λ⊥ ρ along a codimension one cell ρ then a monomial with tangent vector m passing through ρ changes t-order by −m, aρ . This implies that even in examples where the invariant line emanating from a singular point is unbounded, the gluing functions always converge t-adically. Said differently, for the construction of a k-th order deformation of X0 , we need to propagate the contribution of each singular point only through finitely many cells.  Of course, there is no reason for the propagation to stay in the 1-skeleton of P. The gluing functions then separate from the (n − 1)-skeleton of P and move into the interiors of maximal cells. Example 3.3. Let us modify Example 3.1 as shown in Figure 3.6 with ϕ(−1, 0) = 0, ϕ(0, −1) = 0, ϕ(1, 1) = 1. The gluing functions are fρ2 ,v = 1+x and all other fρi ,v = 1. The toric local model at v is now the normal crossing degeneration C[t] −→ C[x, y, z],

t −→ xyz

with x = z (−1,0,0) , y = z (0,−1,0) , z = z (1,1,1) . Following the gluing isomorphisms around v leads to the automorphism θvk : x → x, of the ring

y → (1 + x)−1 y,

z → (1 + x)z

  k = C[x, y, z, t]/ xyz − t, xk+1 , y k+1 , z k+1 . Rv→v,σ i

Figure 3.6. Propagation of the gluing function into a maximal cell. The figure on the right shows the tangent vectors belonging to the generators of the toric model.

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For k > 0 this is again not the identity, and we can’t fix this by changing any  of the gluing functions. But since 1 + x is invertible on the whole thickening Spec C[x, y, z, t]/  (xyz − t, xk+1 ) of Xσ1 , the inverse of θvk can be viewed as an automorphism of Xσ1 that propagates along the ray p = R≥0 · (1, 0) emanating from the codimension two cell v. We then change the gluing procedure by taking one copy of the thickening of Xσ1 for each connected component of σ1 \ p. Passing through p means applying the attached automorphism. The thickenings of the codimension one strata Xρ1 and Xρ3 are viewed as closed subschemes of the copies labelled by the connected component of σ1 \ p containing ρ1 and ρ3 , respectively. Inserting p makes the gluing consistent to all orders. The result is isomorphic to   Spec C[x, y, z, t]/(xyz − (1 + x)t) .  Apart from the gluing functions we have now introduced another object, automorphisms propagating along rays. The higher-dimensional generalization of rays are walls ([GrSi2], Definition 2.20): They are one-codimensional polyhedral subsets of some maximal cell σ, emanating from a two-codimensional polyhedral subset q into the interior of σ, and extending all the way through σ along some integral tangent vector −mp, that is, p = σ ∩ (q − R≥0 · mp). The attached automorphism is of the form m,np (m,h)  ·z , z (m,h) −→ 1 + cpz (mp ,hp ) where hp > 0 is the t-order of the attached monomial and cp ∈ C. We will discuss how to obtain these walls systematically in the context of scattering in Section 4. 3.5. Several singular points: The non-interacting case. For several singular points P1 , . . . , Pl ∈ B the monodromy invariant direction determines an affine line μ emanating from each Pμ . As long as these coincide or do not intersect, the construction presented so far works. Example 3.4. Here is an example with two singular points having parallel invariant lines 1 , 2 , see Figure 3.7. We will see that, depending on a choice of parameter,   it leads to a degeneration either of the Hirzebruch surface F2 = P OP1 (−2) ⊕ OP1 or of P1 × P1 . In this case we have two codimension one strata along which we glue. Taking ϕ to change slope by 1 along each interior edge again, the homogenization of the gluing along codimension one cells gives the polynomials XV − t(U + R)U, XS − t(R + U )R, U Y − t(V + λS)V, RY − t(λS + V )S. Here λ ∈ C∗ is a parameter that we can not get rid of by automorphism. Note that λ determines the relative position of the zero loci of the gluing functions when compared in Xσ2 = P1 × P1 . There is one more toric relation RV − SU   = 0 from σ2 . The corresponding subscheme of P5A1 = ProjC[t] C[t][X, Y, R, S, U, V ] has one extra component V (R, S, U, V ). In fact, saturating with respect to any of R, S, U, V gives another relation XY − t2 (U + R)(V + λS) = 0. This relation can also be deduced directly from following the line connecting the vertices X and Y in the chart shown in Figure 3.7 on the right.

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Figure 3.7. Two charts for a tropical manifold with two parallel invariant lines 1 , 2 .

Figure 3.8. A tropical manifold for a degeneration of P2 blown up twice. The invariant lines emanating from the two singular points agree. These six polynomials are the 2 × 2 minors of the matrix

 X t(V + λS) U R t(U + R) Y V S For fixed t = 0, this is a scroll in P5 whose ruling is given by the lines whose equations are linear combinations of the two rows of the above matrix. The image of this scroll under the rational map κ : Xt → P3 given by [x, y, u, v, r, s] → [x0 , x1 , x2 , x3 ] = [u + r, y, v, s] has equation X0 X1 − t(X2 + X3 )(X2 + λX3 ) = 0 in P3 . For λ = 1 this is a smooth quadric and κ induces an isomorphism Xt  P1 × P1 . For λ = 1 the quadric X0 X1 − t(X2 + X3 )2 = 0 has an A1 -singularity, κ contracts the (−2)-curve given by Y = V + S = U + R = X = 0, and Xt  F2 .

Example 3.5. Figure 3.8 features two singularities with the same invariant line. The general fibre of the corresponding degeneration is a P2 blown up in two different points. We don’t bother to write down the homogeneous equations. This example is easy to generalize to any number of blown up points, which then gives our first genuinely non-toric examples. Many more examples can be obtained from [Sy], which contains a toolkit for the construction of two-dimensional affine manifolds with focus-focus singularities. The symplectic 4-manifold constructed in this reference from such a singular affine manifold is a symplectic model for the general fibre of our degeneration.

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Figure 4.1. A tropical manifold leading to inconsistent gluing. The figure on the right shows the tangent vectors belonging to the generators of the toric model. 4. Scattering If the invariant lines emanating from singular points of a two-dimensional affine structure intersect, the gluing construction becomes inconsistent again. 4.1. First example of scattering. We study the modification of Example 3.1 shown in Figure 4.1, with gluing functions fρ1 ,v = 1 + y = 1 + x−1 t, fρ2 ,v = 1 + w = 1 + z −1 t, fρ3 ,v = 1 + y, fρ4 ,v = 1 + w. k Following the gluing isomorphisms of Rv→v,σ by going counterclockwise around v, i with starting cell σ1 say, yields

x −→ (1 + w)x   −→ 1 + (1 + y)−1 w x   −1  w (1 + w)−1 x −→ 1 + 1 + (1 + w)y    −1 −1 −→ 1 + 1 + (1 + (1 + y)w)y (1 + y)w 1 + (1 + y)w x   −1  −1 (1 + y)w 1 + (1 + y)w x = 1 + (1 + y) + (1 + y)wy   = 1 + (1 + wy)−1 w (1 + w + wy)−1 x   = (1 + wy) + w (1 + wy)−1 (1 + w + wy)−1 x = (1 + wy)−1 x, and similarly, y −→ (1 + wy)y,

z −→ (1 + wy)z,

w −→ (1 + wy)−1 w.

Thus we can again achieve θvk = id for all k by inserting the ray p = R≥0 · (1, 1) with attached function 1 + wy = 1 + t2 x−1 z −1 ! 4.2. A projective example. Of course, this example can also be made projective, say as in Figure 4.2. The ideal defining the degeneration is generated by the two polynomials XY − t(U + W )U, ZW − t(U + Y )U.

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Figure 4.2. Chart for the tropical manifold B3 . The dashed lines denote cuts. These relations can be most simply derived from gluing k-th order standard thickenings of Xσ3 , Xσ4 and of Xσ2 , Xσ3 , respectively, and let k tend to infinity. This procedure does not seem to depend on p. In fact, in this example, the insertion of p merely assures that the same relations are obtained with any other pairs of neighbouring maximal cells. As an abstract deformation this is isomorphic to the toric one with ideal (XY − 2 tU , ZW − tY 2 ) presented in Example 1.4. 4.3. Systematic procedure for the insertion of walls. The long computation for the insertion of the ray p in Example 4.1 looks like quite an accident. But there is a systematic procedure to insert rays (or walls in higher dimension) to achieve consistency to any finite order. The number of rays to be inserted becomes arbitrarily large with increasing k, however, essentially with the only exception the example discussed in §4.1. This procedure originates from the two-dimensional situation of [KoSo]. Of course, due to the non-discrete nature of the discriminant locus the higher dimensional situation is much more involved. Systematically we proceed as follows. By induction one arrives at a collection of walls and modifications of the gluing functions, a structure (in its architectural meaning) S as we call it. A structure consists of walls and so-called slabs. A slab b is a polyhedral subset of some ρ ∈ P [n−1] together with a collection fb,v of higher order corrections of the gluing functions fρ,v , one for each vertex v ∈ ρ that is contained in a connected component of ρ \ Δ intersecting b. These fulfill the same change of k vertex formula (2.3) as the fρ,v , but interpreted in the rings Rρ→ρ,σ for any σ ∈ Pmax containing ρ. We need slabs because a wall hitting a codimension one cell ρ may lead to different corrections on both sides of the wall. Technically we refine P to a polyhedral decomposition PS in such a way that the walls lie in the (n − 1)-skeleton of PS . Then the underlying subsets of the slabs are exactly the (n − 1)-cells of PS contained in the (n − 1)-skeleton of P. For consistency of the gluing construction it suffices to follow loops around the codimension two cells of PS , the joints of the structure (or rather of PS ). This is shown in [GrSi2], Lemma 2.30; the precise definition of consistency is in Definition 2.28. The fact that the gluing construction for a structure consistent to order k gives a k-th order toric degeneration occupies §2.6 in [GrSi2]. Thus consistency is really a codimension two feature. For the scattering computation at a joint j we can therefore work over the ring S = C[[t]][Λj] of Laurent

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polynomials with coefficients in C[[t]] and exponents in Λj, the lattice of integral tangent vectors along j. In fact, if θj is the composition of the gluing morphisms following a small loop around such a joint, then θj is the identity on any monomial z (m,h) with m tangent to j. By making the discriminant locus Δ sufficiently non-rational or by deforming it a little, we can assume the intersection of Δ with j to be of codimension one in j. Let p ∈ j \ Δ be a general point and denote by σj ∈ P the smallest cell containing j. Note that σj has dimension n − 2, n − 1 or n. The computations take place in an affine chart at a vertex v ∈ σj in the same connected component of σj \ Δ as p. In an affine chart at v decompose the lattice of integral tangent vectors by choosing a complement Λ  Z2 to Λj ⊆ Λv Λv = Λj ⊕ Λ. While the procedure essentially is always the same, the setup for the scattering differs somewhat depending on dim σj ∈ {n − 2, n − 1, n}. For being explicit let us assume dim σj = n − 2, which is is the most difficult case; it is also the situation for the initial scattering of codimension one cells of P intersecting in a codimension two cell j = σj of P. At a later stage (k > 0) the inclusion j ⊆ σj may be strict and there may be additional walls running into j from scatterings at other joints. The scattering procedure is a computation in the rings Rσk j →σj ,σ using induction on k. However, these rings are not practical for computations with a computer algebra system, and they would require the introduction of log structures in Example 4.1. Log structures are necessary to make things work globally in the end, but on the level of this survey it does not seem appropriate to get into these kinds of technicalities. We therefore present the scattering computation in a traditional polynomial ring. For any order k  ≥ 0 the composition of gluing morphisms around j, starting at a   reference cell σ ∈ Pmax , defines an automorphism θjk of the ring Rσk j →σj ,σ . This ring

is a finite Sf -algebra, where f = ρ⊇σj fρ,v . Let ϕ be the restriction to {0} × Λ  Z2 of a non-negative representative of ϕ. By changing the representative by an affine function we can assume ϕ(0, 0) = ϕ(−1, 0) = 0,

0 ≤ ϕ(0, 1) ≤ ϕ(1, 1),

which by convexity and integrality of ϕ implies ϕ(1, 1) ≥ 1. 

Letting x := z (−1,0,0) , y := z (0,−1,0) , z := z (1,1,1) this exhibits Rσk j →σj ,σ as a quotient of an Sf -subalgebra of   (4.1) Rk := Sf [x, y, z]/ (xyz − t) + (x, y, z)k+1 , for some k ≥ k  . The smallest possible k is the minimum of the degrees a + b + c for xa y b z c having σj-order (introduced in §3.1) larger than k  . Conversely, any compu tation in Rk can be obtained from a computation in Rσk j →σj ,σ for k   0. Thus the result of the scattering procedure is the same with both sorts of rings.

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Initially the only scattering happens at codimension two cells τ ∈ P. The starting data are the gluing functions fρ,v for all codimension one cells ρ ⊇ τ where v ∈ τ is some vertex selecting an affine chart for the computation. Consistency for k = 1 already requires a compatibility condition for the restrictions fρ,v |Xτ that we can see by studying the following local example. Example 4.1. To consider a local scattering situation for k = 1 take j = τ = × {0} ⊆ Rn as an affine subspace and let ρi = τ × R≥0 mi , i = 1, . . . , l, be affine half-hyperplanes emanating from τ in directions mi ∈ Z2 = Λ in the normal lattice Zn /Λτ = Λ. (Strictly speaking this is not a legal situation because τ does not have a vertex, but this is irrelevant for the present discussion.) We assume the ρi labelled cyclically around τ , that is, ρi separates two maximal cell σi−1 , σi containing τ , for any i ∈ Z/lZ. Let ni = (ai , bi ) be a primitive normal vector to Λρi that evaluates non-negatively on σi . Then passing from σi−1 to σi gives the automorphism

Rn−2

i ,m θi1 : z m −→ fρ−n · zm i

of the ring R1 = Sf [x, y, z]/(x, y, z)2 from (4.1). Because in this local example there is no vertex we drop the reference vertex from the notation for the gluing functions fρi . The loop around the joint τ gives the automorphism θτ1 = θk1 ◦ . . . θ11 . Because we work modulo (x, y, z)2 the effect of applying θki to a monomial is multiplication with a power of the restriction fρi |Xτ ∈ Sf . We obtain    ai i ,bi ),(−1,0) f θτ1 (x) = fρ−(a | · x = · x, ρ i Xτ i i

and analogously, θτ1 (y) =



fρi |Xτ

i

bi

· y,

θτ1 (z) =

i



fρi |Xτ

−ai −bi

· z.

i

Thus consistency for k = 1 requires the following multiplicative condition for the slab functions   b a (4.2) fρi |Xτ i = 1 and fρi |Xτ i = 1.  i

i

Example 4.1 shows that apart from the change of vertex formula (2.3), slab functions need to fulfill a number of algebraic equations to achieve consistency for k = 1. More precisely, each τ ∈ P [n−2] yields two multiplicative equations as in (4.2) for the restrictions fρ,v |Xτ of the slab functions for those ρ ∈ P [n−1] that contain τ , see Equation (1.10) in [GrSi2] for the general form. Note that by the change of vertex formula the relations obtained from different vertices v, v  ∈ τ are equivalent. For those readers familiar with [GrSi1], what this shows is that specifying the gluing functions fρ,v for k = 1 is essentially the same as specifying a log structure on X0 (cf. [GrSi1], Theorem 3.22). Thus the log structures of [GrSi1] on X0 can be viewed as “initial conditions” for our construction. To explain the next steps (k > 1) let us look at the following typical scattering situation in dimension 3. Let the reference vertex be v = 0 ∈ R3 and let the codimension

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two cell τ be contained in the coordinate axis R · (0, 0, 1). We have three codimension one cells ρ1 , ρ2 , ρ3 ⊇ τ with Kτ ρi /Tτ spanned by (−1, 0, 0), (−1, −3, 0) and (2, 3, 0), respectively. For each i there is a unique slab bi ⊆ ρi containing x. The piecewise affine function ϕ is uniquely defined by ϕ|τ = 0 and ϕ(−1, 0, 0) = ϕ(−1, −3, 0) = 0,

ϕ(2, 3, 0) = 3.

Following the maximal cells σ1 , σ2 , σ3 containing τ counterclockwise, with σ1 containing ρ1 and ρ2 , we have ϕ(a, b) = 0, 3a − b and b for (a, b) ∈ σ1 , σ2 , σ3 , respectively. In particular, ϕ(2, 3, 0) = 3 is the smallest possible value making ϕ integral and with ϕ|σ1 = 0. The exponents of our monomials take values in    P := (m, h) ∈ Λ × Z  h ≥ ϕ(m) . As slab functions take fb1 ,v = 1 + z (0,0,1,0) + 2z (−1,0,0,1) , fb2 ,v = 1 + z (0,0,1,0) − z (−1,−3,1,0) , fb3 ,v = 1 + z (0,0,1,0) + 5z (2,3,0,3) , where the exponents lie in Λ × Z, Λ = Λv . Thus in z (a,b,c,h) the first two coordinates (a, b) determine the direction in Λ/Λτ , the third coordinate c is for the toric parameter on Xτ and h − ϕ(a, b) determines the divisibility by t. We also assume there is one wall p containing τ coming from some earlier step, say with Kτ p/Tτ spanned by (−1, 1, 0) and with associated function fp = 1 + 7z (−1,1,0,2) . Our scattering problem can be visualized in the two-dimensional normal space to τ as in Figure 4.3 on the left. For the explicit scattering computation we now go over to the Sf -algebra Rk from (4.1). Here Sf = C[[t]][w, w−1 ]1+w with w = z (0,0,1,0) , and x = z (−1,0,0,0) ,

y = z (0,−1,0,0) ,

z = z (1,1,0,1) .

Note that xyz = t defines the t-algebra structure, and a monomial xa y b z c wd represents an element of P if and only if   ϕ(c − a, c − b, d) = max 0, c − b, 3(c − a) − (c − b) ≥ c. With these notations our input data is fb1 ,v = 1 + w + 2tx, fb2 ,v = 1 + w − wxy 3 , fb3 ,v = 1 + w + 5xz 3 , fp = 1 + 7tx2 z.

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Figure 4.3. A scattering procedure: Starting data (left) and result at order 10. The labels are explained in the text.

It is now easy to compute the composition θvk by a computer algebra system. Seeing the various terms come up at higher order and getting rid of them by insertion of walls is very instructive. We therefore encourage the reader to do that and verify the following computations. Note that as we have seen in §4.1, the naive insertion procedure leads to huge expressions quickly, so a computation in Q[x, y, z, w] would be too slow. It is therefore important to expand in a Taylor series to order k in each of x, y and z after each application of an automorphism associated to a slab or wall. Note also that to compute θvk as an automorphism of Rτk→τ,σ1 we have to observe the difference between the τ -order of a monomial xa y b z c and its degree a + b + c. For example, both x2 y and x5 y 5 z 5 = t5 have τ -order 5, but x2 y is already visible modulo (x, y, z)4 while x5 y 5 z 5 requires working modulo (x, y, z)16 . The computation in Rk gives consistency up to k = 3. At degree 4 one has to insert three walls, denoted 4a, 4b, 4c in Figure 4.3 on the right. The associated functions are 1 + 5(1 + w)2 xz 3 ,

1 + 2(1 + w)2 x2 yz,

1 − (1 + w)2 wxy 3 .

This is just the part fbi − (1 + w) of the slab functions that do not cancel, with an appropriate power of 1 + w that we will explain shortly. Similarly, at degree 6 the incoming wall p starts contributing, forcing its continuation by a wall in direction (1, −1) with function 1 + 7(1 + w)5 tx2 z (6a in Figure 4.3 on the right). Then everything goes well up to degree 8, where we find interactions of two of such terms, namely some multiples of x3 yz 4 = tx2 z 3 , x3 y 4 z = tx2 y 3 , x2 y 3 z 3 = t2 yz. These monomials have directions (1, 3), (−2, −3), (1, 0), so have to scatter away in the directions (−1, −3), (2, 3), (−1, 0) occupied by the slabs. Since we do not want to allow walls to lie above slabs the only possibility is to change the slab functions. Here is the change

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that makes things consistent to degree 9: fb 1 ,v = 1 + w + 2tx − 15(1 + w)2 t2 wyz, fb 2 ,v = 1 + w − wxy 3 + 30(1 + w)2 tx2 z 3 , fb3 ,v = 1 + w + 5xz 3 − 6(1 + w)2 wtx2 y 3 . Let us go through degree 10 in detail. The composition modulo (x, y, z)11 yields  700x4 yz 5 14x5 y 2 z 3 x −→ 1 − − − 56(1 + w)2 wx4 y 4 z 2 · x 1+w 1+w  175x4 yz 5 28x5 y 2 z 3 − + 56(1 + w)2 wx4 y 4 z 2 · y y −→ 1 + 1+w 1+w  525x4 yz 5 42x5 y 2 z 3 z −→ 1 + + · z. 1+w 1+w The occurring monomials are x4 yz 5 = tx3 z 4 , x5 y 2 z 3 = t2 x3 z, x4 y 4 z 2 = t2 x2 y 2 , which have directions (1, 4), (−2, 1) and (−1, −1), respectively. (The fact that the last one points in the same direction as z explains why it does not show up in the last line: It belongs to a wall acting trivially on z.) Thus we want to insert walls in directions (−1, −4), (2, −1) and (1, 1) (10a, 10b and 10c in Figure 4.3) with functions 1 + atx3 z 4 ,

1 + bt2 x3 z,

1 + ct2 x2 y 2 ,

for some a, b, c ∈ C[w]. We now explain how to determine the coefficients a, b, c. Since they come in a product with monomials of degree 10, they do not interact with any monomial of non-zero degree. Hence the influence to the composition can be computed from a scattering diagram with only the wall in question and the three slabs bi with functions fbi = 1 + w, i = 1, 2, 3. Here is the computation of the composition for the wall 10a: 4 b1 b2 10a  x −→ x −→ (1 + w)3 x −→ 1 + atx3 z 4 · (1 + w)3 x  b3  −→ 1 + 4at(1 + w)−5 x3 z 4 · x. 4  Here we used 1 + at(1 + w)−5 x3 z 4 = 1 + 4at(1 + w)−5 x3 z 4 modulo (x, y, z)10 . The second term has to cancel with −700(1 + w)−1 tx3 z 4 , which leads to a = 175(1 + w)4 . Said differently, we just have to commute the wall automorphism past all the automorphisms of slabs in counterclockwise direction up to the reference cell. For the wall labelled 10a there is only one such slab, b3 , with normal vector (−3, 2), and the yields monomial is x3 z 4 = z (1,4,0,4)  . The application of the associated automorphism   the power − (−3, 2), (1, 4) = −5 of 1 + w. The coefficient 4 = − (4, −1), (−1, 0) is picked up from passing x past the wall in question. By the form of the automorphisms, looking at θvk (y) or θvk (z) gives the same result. This is a consistency check. The other coefficients can be computed analogously, giving 1 + 175(1 + w)4 tx3 z 4 ,

1 + 14(1 + w)7 t2 x3 z,

1 − 56(1 + w)4 t2 wx2 y 2

for the functions associated to the walls 10a, 10b and 10c, respectively.

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Continuing in this fashion we can insert new walls essentially uniquely to make the gluing consistent to any finite τ -order. Remark 4.2. 1) There is one miracle here. Namely, while the scattering computation works only after localization at 1 + w, no negative powers of 1 + w ever have to be inserted. This is indispensable because walls and slabs can only carry regular functions. For general slab functions and slab configurations this might indeed not be true, see the discussion in §4.4.1 of [GrSi2]. So we need a condition, and the condition we found to make it work ([GrSi2], Definition 1.25,ii) morally says that the deformation problem has a unique solution locally in codimension 2 up to isomorphism. This is a natural condition. Nevertheless, the proof that no denominators occur in §4.4 of [GrSi2] is the most technical part of the story. We refer to [GrSi2] for a more complete discussion. 2) For dim σj ∈ {n − 1, n} the algorithm is the same, but there are higher degree monomials with vanishing σj-order. In the (n − 1)-dimensional case a perturbation argument as in §4.4 suffices to show that one can achieve consistency without changing the slab functions and without introducing denominators, see [GrSi2] §4.3. In the n-dimensional case this is not an issue because there are no slabs and hence there are no denominators and all monomials have positive order anyway. This last case is essentially the situation that was treated by Kontsevich and Soibelman in [KoSo]. In essence, our approach trades the difficult gradient flow arguments in [KoSo] for algebraic arguments for the dim σj = n − 1 case, while the dim σj = n − 2 case is only non-trivial for n ≥ 3 and is the most difficult issue. 4.4. The perturbation idea. Sometimes a simple geometric perturbation of the starting data allows one to simplify a scattering situation. In [GrSi2], §4.2 we formalize this in the notion of infinitesimal scattering diagrams. Here we just want to illustrate the concept by one simple example. Example 4.3. Consider the two-dimensional example of a local scattering situation with xyz = t shown in Figure 4.4. By formally perturbing the slabs we arrive at Figure 4.5 on the left. The functions are the slab functions on the various polyhedral pieces. While this is not a legal polyhedral decomposition, we can still insert walls inductively to make the diagram consistent to any finite order. The result is shown

Figure 4.4. Scattering of three non-trivial slabs.

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Figure 4.5. Perturbed scattering diagram.

Figure 4.6. Result of collapsing Figure 4.5, right.

Figure 4.7. A tropical manifold for a degeneration of cubic surfaces. in Figure 4.5 on the right. Note that this diagram is entirely determined by the most simple scattering situation of §4.1 via affine isomorphisms. Now the point of this discussion is that we can collapse this diagram again by superimposing walls and slabs with the same direction if necessary. For the present situation the result is shown in Figure 4.6. This configuration of slabs and walls is consistent to all orders.  We can also make this example projective. The smallest polarization produces a degeneration of cubic surfaces as follows. Example 4.4. Consider the tropical manifold depicted in Figure 4.7, with three unit triangles as maximal cells and one focus-focus singularity on each of the three interior edges. The piecewise affine function is defined by the values 0, 0, 0, 1 on the 4 vertices U, X, Y, Z, respectively. The vertices also define homogeneous coordinates

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written by the same symbols. This exhibits X0 as the union of hyperplanes V (XY Z) ⊆ P3 . Dehomogenizing with respect to U gives the affine coordinates x0 = X/U , y0 = Y /U , z0 = Z/U , and in these coordinates we define the slab functions by fρ1 ,U = 1 + x0 ,

fρ2 ,U = 1 + y0 ,

fρ3 ,U = 1 + z0 .

Let us compute an affine chart for the deformation. To this end we look at Figure 4.6 and perform a computation similar to the basic gluing computation in §2.3. As reference chamber choose the upper left quadrant in Figure 4.6. In this chamber x = X/U is the uncorrected toric coordinate x0 , but y = Y /U and z = Z/U receive corrections from crossing walls and slabs. For z we need to cross the wall with function 1 + y0 , while for y there is the wall with function 1 + z0 and the slab with function (1 + x0 )(1 + y0 z0 ). The complete list of relations therefore is x0 y0 z0 = t x = x0 z = (1 + y0 )z0 y = (1 + x0 + z0 + y0 z0 + x0 y0 z0 )y0 . Eliminating x0 , y0 , z0 exhibits an affine chart for the deformation as the hypersurface   xyz = t (1 + t) + x + y + z . in A4 with coordinates x, y, z, t. Homogenizing yields the degeneration of singular cubic surfaces   XY Z = t (1 + t)U 3 + (X + Y + Z)U 2 . The general fibre has 3 ordinary double points at [1, 0, 0, 0], [0, 1, 0, 0] and [0, 0, 1, 0]; these come from the vertices X, Y, Z on the boundary of B.  5. Three-dimensional examples So far we have essentially considered two-dimensional examples, the only exception being the three-dimensional sample scattering computation in §4.3. We already observed in this example one complication in higher dimensions, the potential presence of poles in the scattering procedure, see Remark 4.2,(1) for a discussion how this is handled. We also observed in this example that the scattering procedure generally requires higher order corrections to the slab functions, and the slab functions themselves propagate. In two dimensions ad hoc solutions can be used to do this. In higher dimensions this propagation is more complicated and we need a homological argument ([GrSi2], §3.5). Another, more fundamental difference in higher dimensions is the fact that the codimension two intersection loci of walls and slabs, the joints, are higher dimensional. Thus in the automorphism θj associated to a loop around a joint j there may be monomials tangent to j. These can not be removed by inserting walls. It nevertheless turns out that once we have run the homological argument, this only happens for joints j contained in a codimension two cell σj of P, and the remainρ ing terms are either undirectional (pure t-powers, tl ) or of the form tk z mvv /fρ,v

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Figure 5.1. The tropical manifold for a degeneration of KP2 (left) and its projection to the plane (right). ([GrSi2], Proposition 3.23). Here mρvv is defined by affine monodromy as in (2.2). In other words, these terms arise from pure t-powers on some other affine chart. The final step in our algorithm uses a normalization procedure to get rid of these terms also ([GrSi2], §3.6). The normalization essentially says that the logarithms of the slab functions do not contain pure t-powers. Remarkably, this step not only makes sure that no obstructions arise in the deformation process, but also makes t a canonical parameter in the sense of mirror symmetry, traditionally denoted q. Examples featuring all phenomena in this process are very complicated to run through explicitly. We therefore content ourselves with one non-compact example treating a degeneration of the total space of KP2 , the so-called “local P2 ” from the mirror symmetry literature [ChKlYa], and its mirror. Example 5.1. Consider the tropical manifold B shown in Figure 5.1. There is a total of six maximal cells σ0 , . . . , σ5 with only σ0 bounded, a prism. The maximal cells adjacent to the sides of the prism are σ1 , σ2 , σ3 as shown in the figure. The remaining σ4 , σ5 are adjacent to the bottom and top of σ0 , respectively. The affine structure is then completely determined by   σ0 = conv (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1) , by the directions (2, −1, ±1), (−1, 2, ±1), (−1, −1, ±1) of the unbounded edges in a chart on the upper (plus sign) and lower (minus sign) half, and by requiring the monodromy around the discriminant locus (the dashed lines in the figure) to be primitive and positive. Note that (0, 0, 1) is an invariant tangent vector defining an affine projection as suggested in Figure 5.1 on the right. As piecewise affine function ϕ we take a minimal one changing slope by 1 along each codimension one cell and vanishing on σ0 . In particular, on the six unbounded edges ϕ takes the value 1 at the first integral point different from a vertex. The nontrivial slabs (fb = 1) are the six vertical cells of codimension one. If we denote by s the monomial with direction (0, 0, −1) and with σ0 -order 0 and by s its inverse, the gluing functions are fb = 1 + s (upper half),

fb = 1 + s (lower half).

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Figure 5.2. The generators of the affine patches (left: bottom, right: top). Note that by (4.2) and the change of vertex formula (2.3) any one slab function determines all the others. This structure is already consistent along the three vertical edges of the prism. To get consistency everywhere it only remains to insert six vertical walls in σ4 and σ5 each. These extend the six vertical codimension one cells to infinity. There is no scattering because the monomials carried by the walls and slabs all point in the same direction (0, 0, 1), the invariant tangent vector, and hence the corresponding automorphisms mutually commute. To study this example it is not advisable to write down the homogeneous coordinate ring. In fact, because this is a non-compact example it does not suffice to take the six vertices X, Y, Z, U, V, W as generators of a homogeneous coordinate ring. Rather we need a number of generators of degree 0 defined by tangent vectors in the unbounded directions. Due to the non-simplicial nature of the polyhedra this leads to a long list of generators and an even longer list of relations. Instead we use the construction via gluing affine patches from [GrSi2]. There are six vertices, and correspondingly we have a cover of the degeneration by six affine open sets. Figure 5.2 shows our choices of generators, where the arrows should be thought of as tangent to edges. Thus in a chart at U the generating monomials are v = z (1,0,0,0) , z = z (0,1,0,0) , r = z (−1,−1,1,1) , s = z (0,0,−1,0) , t = z (0,0,0,1) , with single relation rvzs = t, a semi-stable (normal crossings) degeneration. Because of the symmetry of the example the situation is analogous in the five other charts. Now we have to adjust these local models by the slab functions. A computation analogous to §2.3 gives rvzs = (1 + s)t (at U ),

pwxs = (1 + s)t (at V ),

quys = (1 + s)t (at W ),











r vzs = (1 + s )t (at X),





p wxs = (1 + s )t (at Y ),

q uys = (1 + s )t (at Z).

Here the variables take reference to any maximal cell containing them. Thus x, y, z, u, v, w can all be thought of as monomials on σ0 , while for example r is a monomial

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on σ2 or σ3 . The patching between these charts is given by affine geometry in the relevant maximal cells. In particular, variables with the same name are all identified, and we have relations such as x = v −1 ,

u = xw−1 ,

s = s−1 ,

q = pw3

etc.

As a consistency check one can verify compatibility of the relations with the gluing. For example, going from the chart at U to the chart at V means the substitution v → x−1 ,

z → x−1 w,

r → x3 p,

s → s.

This maps the relation rvzs = (1 + s)t to pwxs = (1 + s)t, as expected. Similarly, to go from U to X means substituting v → v,

z → z,

r → r s ,

s → (s )−1

into rvzs = (1 + s)t, leading to r vzs = (1 + s )t. At this point we have written down a degeneration π : X → A1 = Spec C[t] with X covered by six affine open sets. We claim that a general fibre Xt is an open subset of the total space KP2 of the canonical bundle of P2 . To this end fix t ∈ C \ {0} and define a projection κ : Xt −→ P2 by viewing the triples X, Y, Z or U, V, W as homogeneous coordinates on P2 . Thus set-theoretically the restriction of κ to the chart at U is (r, v, z, s) −→ [1, v, z]. It is straightforward to check compatibility with the patching. For example, in the intersection with the chart at V we find κ(p, w, x, s) = [x, 1, w] = [v −1 , 1, v −1 z] = [1, v, z]. Analogous computations show compatibility on the ring level. The fibre of κ over a closed point of P2 , say [1, v, z], is the hypersurface rvzs − (1 + s)t = 0 in A2 = Spec C[r, s] . Note that Xt is disjoint from s = 0 or from s = 0 since t = 0, so it suffices to work in one chart only. If vz = 0 this is a hyperbola, hence isomorphic to A1 \ {0}. On the other hand, if vz = 0 we must have s = −1 and r has no restrictions, so this is an A1 . The global meaning of this comes by observing that the fibre coordinates p, q, r transform dually to the sections dx ∧ dw, dy ∧ du and dz ∧ dv of KP2 . For example, since y = z −1 v and u = z −1 ,    qdy ∧ du = q −z −2 vdz + z −1 dv) ∧ −z −2 dz = qz −3 dz ∧ dv = rdz ∧ dv. Above we computed the fibre of κ over [1, v, z] to be given by (rvz − t)s = t. For given r, v, z this has a solution s as long as rvz − t = 0. Thus rvz − t = 0 describes the hypersurface locally that is being removed from KP2 to obtain Xt . Globally we are removing the graph of a rational section of KP2 with poles along the toric divisor V (XYZ). 

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Figure 5.3. The tropical manifold for the mirror of Example 5.1 (left) and the central triangle containing the discriminant locus (right). Example 5.2. An example with somewhat complementary features to Example 5.1 is provided by the mirror. Following the general recipe of [GrSi1] the tropical manifold together with the (multi-valued) function ϕ is obtained by a discrete version of the Legendre transform ([GrSi1], Construction 1.15). By this construction the polyhedral decompositions of a tropical manifold and its mirror are combinatorially dual to each other. As this is not primarily a paper about mirror symmetry we do not explain this construction, but only state the result. As shown in Figure 5.3 the tropical manifold is a double tetrahedron glued from six standard simplices. We have five exterior vertices labelled X, Y, Z, U, V and one interior vertex W . In an affine chart at W the five emanating edges point in the directions (0, 0, ±1), (1, 0, 0), (0, 1, 0), (−1, −1, 0). The affine structure is then completely determined by requiring the monodromy around the edges of the discriminant locus (dashed in the figure) to be primitive and positive. The function ϕ is single-valued and can be taken to take value 0 on X, Y, U, W and value 1 on Z and V . This again has the property to change slope by 1 along each cell of codimension one. There are three non-trivial slabs, the three horizontal triangles containing the discriminant locus. Up to automorphisms there is only one set of gluing functions possible at order 0, namely, in affine coordinates x = X/W , y = Y /W , z = Z/W at W : fρ1 ,W = 1 + x + y,

fρ2 ,W = 1 + y + z,

fρ3 ,W = 1 + z + x.

The expressions at the other vertices follow from this by the change of vertex formula (2.3). To make this structure consistent to all orders only requires propagating the slab functions to the neighbouring slabs. This leads to fb,W = 1 + x + y + z, for any of the three slabs b = ρi . The approach by homogeneous coordinates works well again in this case. We get the toric relation XY Z = tW 3 ,

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and the homogenization of the gluing relation uv = (1 + x + y + z)t gives U V = t2 (X + Y + Z + W )W. As one checks in local coordinates this is a complete set of relations. For t =0 the  projective variety Xt is a conic bundle over Proj C[X, Y, Z, W ]/(XY Z − tW 3 ) with singular fibers over (X + Y + Z + W )W = 0. This base space of the conic bundle is the quotient of P2 by the Z/3-action ξ · [x0 , x1 , x2 ] = [ξx0 , ξ 2 x1 , x2 ], for ξ a primitive third root of unity. The suggestion in the literature for the mirror to KP2 is to take the family of non-complete Calabi-Yau varieties defined by uv = 1 + x + y + tx−1 y −1 in C2 ×(C∗ )2 [ChKlYa]. Here u, v are the coordinates on C2 . This is exactly the open subset of our family fibering over the big cell of the weighted projective space. See [Gr1], §4 for a discussion how this fits with the SYZ picture of local mirror symmetry. As written this family does not come correctly parametrized for the purpose of mirror symmetry. Rather, a period integral defines a new parameter q related to t by the so-called mirror map. It is one striking feature of our approach that this mirror map comes up naturally via the normalization condition. The present example is too local to illustrate the need for doing this, but as mentioned at the beginning of this section, generally our algorithm requires the logarithm of the slab functions to not contain any pure t-powers,  see l [GrSi2], §3.6. In the present example this means adding a power series g = l≥0 al t to fb,W with the property that k   (−1)k+1   x + y + z + g(xyz) ∈ C[[x, y, z]] log fb,W + g = k k≥1

does not contain any monomials (xyz)l = tl . This condition determines the coefficients ak inductively: g(t) = −2t + 5t2 − 32t3 + 286t4 − 3038t5 + · · · It follows from the period computations in [GbZa] that the modified family   XY Z = tW 3 , U V = t2 X + Y + Z + (1 + g(t))W W, is then indeed written in canonical coordinates, that is, the mirror map becomes trivial.  References [ChKlYa] [GbZa]

T.M. Chiang, A. Klemm, S.-T. Yau, E. Zaslow: Local mirror symmetry: calculations and interpretations, Adv. Theor. Math. Phys. 3 (1999), 495–565. T. Graber, E. Zaslow: Open-string Gromov-Witten invariants: calculations and a mirror “theorem”. in “Orbifolds in mathematics and physics (Madison, WI, 2001)”, 107–121, Contemp. Math., 310, Amer. Math. Soc. 2002.

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[Gr1] [Gr2] [GrSi1] [GrSi2] [Ha] [Ho]

[KoSo]

[St] [Sy]

[Wi]

M. Gross: Examples of special Lagrangian fibrations, in: Symplectic geometry and mirror symmetry (Seoul, 2000), 81–109, World Sci. Publ. 2001. M. Gross: Toric Degenerations and Batyrev-Borisov Duality, Math. Ann. 333 (2005), 645–688. M. Gross, B. Siebert: Mirror symmetry via logarithmic degeneration data I, J. Differential Geom. 72 (2006), 169–338. M. Gross, B. Siebert: From real affine to complex geometry, to appear in Annals of Math. J. Harris: Algebraic geometry, Springer 1992. M. Hochster: Cohen-Macaulay rings, combinatorics and simplicial complexes, in: Ring theory II, B.R. McDonald, R.A. Morris (eds.), Lecture Notes in Pure and Appl. Math. 26, M. Dekker 1977. M. Kontsevich, Y. Soibelman: Affine structures and non-Archimedean analytic spaces, in: The unity of mathematics (P. Etingof, V. Retakh, I.M. Singer, eds.), 321–385, Progr. Math. 244, Birkh¨ auser 2006. R. Stanley: Combinatorics and commutative algebra, Second ed., Birkh¨ auser 1996. M. Symington: Four dimensions from two in symplectic topology, in: Topology and geometry of manifolds (Athens, GA, 2001), 153–208, Proc. Sympos. Pure Math. 71, Amer. Math. Soc. 2003. J. Williamson: On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58 (1936), 141–163.

UCSD Mathematics, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA E-mail address: [email protected] ¨ t Hamburg, Bundesstraße 55, 20146 Hamburg, Department Mathematik, Universita Germany E-mail address: [email protected]

Surveys in Differential Geometry XVI

Lectures on generalized geometry Nigel Hitchin

Preface These notes are based on six lectures given in March 2010 at the Institute of Mathematical Sciences in the Chinese University of Hong Kong as part of the JCAS Lecture Series. They were mainly targeted at graduate students. They are not intended to be a comprehensive treatment of the subject of generalized geometry, but instead I have attempted to present the general features and to focus on a few topics which I have found particularly interesting and which I hope the reader will too. The relatively new material consists of an account of Goto’s existence theorem for generalized K¨ahler structures, examples of generalized holomorphic bundles and the B-field action on their moduli spaces. Since the publication of the first paper on the subject [11], there have been many articles written within both the mathematical and theoretical physics communities, and the reader should be warned that different authors have different conventions (or occasionally this author too!). For other accounts of generalized geometry, I should direct the reader to the papers and surveys (e.g. [2], [8]) of my former students Marco Gualtieri and Gil Cavalcanti who have developed many aspects of the theory. I would like to thank the IMS for its hospitality and for its invitation to give these lectures, and Marco Gualtieri for useful conversations during the preparation of these notes. Contents Preface 1. The Courant bracket, B-fields and metrics 1.1. Linear algebra preliminaries 1.2. The Courant bracket 1.3. Riemannian geometry

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Subject classification: Primary 53D18. c 2011 International Press

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2. Spinors, twists and skew torsion 2.1. Spinors 2.2. Twisted structures 2.3. Skew torsion 3. Generalized complex manifolds 3.1. Generalized complex structures 3.2. Symmetries and twisting ¯ 3.3. The ∂-operator 4. Generalized K¨ ahler manifolds 4.1. Bihermitian metrics 4.2. Goto’s deformation theorem 4.3. Deformation of the bihermitian structure 5. Generalized holomorphic bundles 5.1. Basic features 5.2. Co-Higgs bundles 5.3. Holomorphic Poisson modules 6. Holomorphic bundles and the B-field action 6.1. The B-field action 6.2. Spectral covers 6.3. Twisted bundles and gerbes References

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1. The Courant bracket, B-fields and metrics 1.1. Linear algebra preliminaries. Generalized geometry is based on two premises – the first is to replace the tangent bundle T of a manifold M by T ⊕ T ∗ , and the second to replace the Lie bracket on sections of T by the Courant bracket. The idea then is to use one’s experience of differential geometry and by analogy to define and develop the generalized version. Depending on the object, this may or may not be a fertile process, but the intriguing fact is that, by drawing on the intuition of a mathematician, one may often obtain this way a topic which is also of interest to the theoretical physicist. We begin with the natural linear algebra structure of the generalized tangent bundle T ⊕ T ∗ . If X denotes a tangent vector and ξ a cotangent vector then we write X + ξ as a typical element of a fibre (T ⊕ T ∗ )x . There is a natural indefinite inner product defined by (X + ξ, X + ξ) = iX ξ

(= ξ, X = ξ(X))

using the interior product iX , or equivalently the natural pairing ξ, X or the evaluation ξ(X) of ξ ∈ Tx∗ on X. This is to be thought of as replacing the notion of a Riemannian metric, even though on an n-manifold it has signature (n, n).

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In block-diagonal form, a skew-adjoint transformation of T ⊕ T ∗ at a point can be written as   A β . B −At Here A is just an endomorphism of T and B : T → T ∗ to be skew-adjoint must satisfy (B(X1 + ξ1 ), X2 + ξ2 ) = (B(X1 ), X2 ) = −(B(X2 ), X1 ) so that B is a skew-symmetric form, or equivalently B ∈ Λ2 T ∗ , and its action is X + ξ → iX B. Since 2  0 0 =0 B 0 exponentiating gives (1)

exp B(X + ξ) = X + ξ + iX B

This B-field action will be fundamental, yielding extra transformations in generalized geometries. It represents a breaking of symmetry in some sense since the bivector β ∈ Λ2 T plays a lesser role. 1.2. The Courant bracket. We described above the pointwise structure of the generalized tangent bundle. Now we consider the substitute for the Lie bracket [X, Y ] of two vector fields. This is the Courant bracket which appears in the literature in two different formats – here we adopt the original skew-symmetric one: Definition 1. The Courant bracket of two sections X + ξ, Y + η of T ⊕ T ∗ is defined by 1 [X + ξ, Y + η] = [X, Y ] + LX η − LY ξ − d(iX η − iY ξ). 2 This has the important property that it commutes with the B-field action of a closed 2-form B: Proposition 1. Let B be a closed 2-form, then exp B([X + ξ, Y + η]) = [exp B(X + ξ), exp B(Y + η)]. Proof. We need to prove that [X + ξ + iX B, Y + η + iY B] = [X + ξ, Y + η] + i[X,Y ] B and we shall make use of the Cartan formula for the Lie derivative of a differential form α: LX α = d(iX α) + iX dα. First expand [X + ξ + iX B, Y + η + iY B] = [X + ξ, Y + η] + LX iY B − LY iX B 1 − d(iX iY B − iY iX B). 2

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The last two terms give d(iY iX B) = LY iX B − iY d(iX B) by the Cartan formula, and so yield [X + ξ + iX B, Y + η + iY B] = [X + ξ, Y + η] + LX iY B − iY d(iX B) = [X + ξ, Y + η] + i[X,Y ] B + iY LX B − iY d(iX B) = [X + ξ, Y + η] + i[X,Y ] B + iY iX dB by the Cartan formula again. So if dB = 0 the bracket is preserved.



The inner product and Courant bracket naturally defined above are clearly invariant under the induced action of a diffeomorphism of the manifold M . However, we now see that a global closed differential 2-form B will also act, preserving both the inner product and bracket. This means an overall action of the semi-direct product of closed 2-forms with diffeomorphisms Ω2 (M )cl  Diff(M ). This is a key feature of generalized geometry – we have to consider B-field transformations as well as diffeomorphisms. The Lie algebra of the group Ω2 (M )cl  Diff(M ) consists of sections X + B of T ⊕ Λ2 T ∗ where B is closed. If we take B = −dξ, then the Lie algebra action on Y + η is (X − dξ)(Y + η) = LX (Y + η) − iY dξ = [X, Y ] + LX η − LY ξ + d(iY ξ). It is then easy to see that we can reinterpret the Courant bracket as the skew-symmetrization of this: 1 [X + ξ, Y + η] = ((X − dξ)(Y + η) − (Y − dη)(X + ξ)). 2 However, although the Courant bracket is derived this way from a Lie algebra action, it is not itself a bracket of any Lie algebra – the Jacobi identity fails. More precisely we have (writing u = X+ξ, v = Y +η, w = Z+ζ) Proposition 2. 1 [[u, v], w] + [[v, w], u] + [[w, u], v] = d(([u, v], w) + ([v, w], u) + ([w, u], v)) 3 Proof. If u = X + ξ, let u ˜ = X − dξ be the corresponding element in the Lie algebra of Ω2 (M )cl  Diff(M ). We shall temporarily write uv for the action of u ˜ on v (this is also called the Dorfman “bracket” of u and v) so that the Courant bracket is (uv − vu)/2. We first show that (2)

u(vw) = (uv)w + v(uw).

To see this note that u(vw) − v(uw) = u ˜v˜(w) − v˜u ˜(w) = [˜ u, v˜](w) since u ˜, v˜ are Lie algebra actions, and the bracket here is just the commutator. But

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(uv)w is the Lie algebra action of uv = u ˜v = [X, Y ] + LX η − iY dξ which acts as [X, Y ] − d(LX η − iY dξ) = [X, Y ] − LX dη + LY dξ using the Cartan formula and d2 = 0. This however is just the bracket [˜ u, v˜] in the Lie algebra of Ω2 (M )cl  Diff(M ). To prove the Proposition we note now that the symmetrization (uv + vu)/2 is 1 1 (LX η − iY dξ + LY ξ − iX dη) = d(iX η + iY ξ) = d(u, v) 2 2 while we have already seen that the skew-symmetrization (uv − vu)/2 is equal to [u, v]. So we rewrite the left hand side of the expression in the Proposition as one quarter of (uv − vu)w − w(uv − vu) + (vw − wv)u − u(vw − wv) + (wu − uw)v − v(wu − uw) Using (2) we sum these to get (−1) times the sum r of the right-hand column. If  is the sum of the left-hand column then this means  + r = −r. But then −r = 3(+r) is the sum of terms like (uv −vu)w +w(uv −vu) = 4([u, v], w). The formula follows directly.  There are two more characteristic properties of the Courant bracket which are easily verified: (3)

[u, f v] = f [u, v] + (Xf )v − (u, v)df

where f is a smooth function, and as usual u = X + ξ, and (4)

X(v, w) = ([u, v] + d(u, v), w) + (v, [u, w] + d(u, w)).

1.3. Riemannian geometry. The fact that we introduced the inner product on T ⊕ T ∗ as the analogue of the Riemannian metric does not mean that Riemannian geometry is excluded from this area – we just have to treat it in a different way. We describe a metric g as a map g : T → T ∗ and consider its graph V ⊂ T ⊕ T ∗ . This is the set of pairs (X, gX) or in local coordinates (and the summation convention, which we shall use throughout) the span of ∂ + gij dxj . ∂xi The subbundle V has an orthogonal complement V ⊥ consisting of elements of the form X − gX. The inner product on T ⊕ T ∗ restricted to X + gX ∈ V is iX gX = g(X, X) which is positive definite and restricted to V ⊥ we get the negative definite −g(X, X). So T ⊕T ∗ with its signature (n, n) inner product can also be written as the orthogonal sum V ⊕ V ⊥ . Equivalently we have reduced the structure group of T ⊕ T ∗ from SO(n, n) to S(O(n) × O(n)).

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The nondegeneracy of g means that g : T → T ∗ is an isomorphism so that the projection from V ⊂ T ⊕ T ∗ to either factor is an isomorphism. This means we can lift vector fields or 1-forms to sections of V . Let us call X + the lift of a vector field X to V and X − its lift to V ⊥ , i.e. X ± = X ±gX. We also have the orthogonal projection πV : T ⊕ T ∗ → V and then 1 1 πV (X) = πV (X + gX + X − gX) = X + . 2 2 We can use these lifts and projections together with the Courant bracket to give a convenient way of working out the Levi-Civita connection of g. First we show: Proposition 3. Let v be a section of V and X a vector field, then ∇X v = πV [X − , v] defines a connection on V which preserves the inner product induced from T ⊕ T ∗. Proof. Write v = Y + η, then observe that ∇f X v = πV [f X − , v] = πV (f [X − , v] − (Y f )X − + (v, X − )df ) using Property (3) of the Courant bracket. But V and V ⊥ are orthogonal so πV X − = 0 = (v, X − ) and hence ∇f X v = f ∇X v. Now using the same property we have ∇X f v = πV (f [X − , v] + (Xf )v − (v, X − )df ) = f ∇X v + (Xf )v since (v, X − ) = 0 and πV v = v. These two properties define a connection. To show compatibility with the inner product take v, w sections of V , then (∇X v, w) + (v, ∇X w) = (πV [X − , v], w) + (v, πV [X − , w]) = ([X − , v], w) + (v, [X − , w]) since πV is the orthogonal projection onto V and v, w are sections of V . Now use Property (4) of the Courant bracket to see that X(v, w) = ([X − , v] + d(X − , v), w) + (v, [X − , w] + d(X − , w)). But (X − , v) = 0 = (X − , w) and we get X(v, w) = (∇X v, w) + (v, ∇X w) as required.  Using the isomorphism of V with T (or T ∗ ) we can directly use this to find a connection on the tangent bundle. Directly, we take coordinates xi and then from the definition of the connection, the covariant derivative of ∂/∂xj + in the direction ∂/∂xi is   ∂ ∂ πV − gik dxk , + gj dx . ∂xi ∂xj

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Expanding the Courant bracket gives ∂gj ∂gj ∂gij ∂(−gik ) 1 ∂gik dx − dxk − d(gji − (−gij )) = dx + dxk − dxk . ∂xi ∂xj 2 ∂xi ∂xj ∂xk Projecting on V we get ∂ 1 ) (dxk +g k 2 ∂x



∂gjk ∂gij ∂gik + − ∂xi ∂xj ∂xk



1 = g k 2



∂gjk ∂gij ∂gik + − ∂xi ∂xj ∂xk



∂+ ∂x

which is the usual formula for the Christoffel symbols of the Levi-Civita connection. Example: Here is another computation – the so-called Bianchi IX type metrics (using the terminology for example in [5]). These are four-dimensional metrics with an SU (2) action with generic orbit three-dimensional and in the diagonal form g = (abc)2 dt2 + a2 σ12 + b2 σ22 + c2 σ32 where a, b, c are functions of t and σi are basic left-invariant forms on the group, where dσ1 = −σ2 ∧ σ3 etc. If Xi are the dual vector fields then [X1 , X2 ] = X3 and LX1 σ2 = σ3 etc. Because of the even-handed treatment of forms and vector fields in generalized geometry, it is as easy to work out covariant derivatives of 1-forms as vector fields. Here we shall find the connection matrix for the orthonormal basis of 1-forms e0 = abc dt, e1 = aσ1 , e2 = bσ2 , e3 = cσ3 . By symmetry it is enough to work out derivatives with respect to X1 and ∂/∂t. First we take X1 , so that X1− = X1 − a2 σ1 . For the covariant derivative of e0 consider the Courant bracket [X1 − a2 σ1 ,

∂ + (abc)2 dt] = 2aa σ1 . ∂t

−1 2 But e+ 0 = (abc) (∂/∂t+(abc) dt) and using Property (3) of the bracket and the orthogonality of X1− , e+ 0 we have

[X1− , e+ 0]=

2a 2a σ1 = e1 bc abc

Projecting on V and using πV e1 = e+ 1 /2, we have (5)

∇X1 e0 =

a e1 . abc

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For the 1-form e1 note that, since LX1 σ1 = 0 1 [X1− , X1+ ] = [X1 − a2 σ1 , X1 + a2 σ1 ] = − d(a2 + a2 ) = −2aa dt. 2 But e1 = a−1 X1+ , and again using Property (3) and the orthogonality of   X1− , X1+ we have [X1− , e+ 1 ] = −2a dt. Projecting onto V gives πV (−2a dt) =   −2 −(a dt + a (abc) ∂/∂t). So a e0 . abc (Note that with (5) this checks with the fact that the connection preserves the metric.) −1 + For e+ 2 = b X2 we have ∇X1 e1 = −

(6)

[X1− , X2+ ] = [X1 − a2 σ1 , X2 + b2 σ2 ] = [X1 , X2 ] + LX1 b2 σ2 + LX2 a2 σ1 − 0 = X3 + (b2 − a2 )σ3 −1 2 2 and so [X1− , e+ 2 ] = b (X3 + (b − a )σ3 ). Projecting onto V ,

1 −1 1 −1 2 2 2 −2 πV [X1− , e+ 2 ] = b (X3 + c σ3 ) + b (b − a )(σ3 + c X3 ) 2 2 so that (7)

1 2 (c + b2 − a2 )e3 . 2bc Now we covariantly differentiate with respect to t.     ∂ ∂ + 2 2 −1 ∂ − (abc) dt, e0 = − (abc) dt, (abc) + (abc)dt ∂t ∂t ∂t (abc) ∂ 1 + (abc) dt + (abc) dt − d(2(abc)) =− (abc)2 ∂t 2   (abc) ∂ 2 = − + (abc) dt (abc)2 ∂t ∇X1 e2 =

so projecting onto V gives ∇ ∂ e0 = 0.

(8)

∂t

and finally (similar to the first case above)   ∂ − (abc)2 dt, X1 + a2 σ1 = 2aa σ1 − 0 − 0 ∂t so that



   ∂ ∂ + 2 2 −1 2 − (abc) dt, e1 = − (abc) dt, a (X1 + a σ1 ) ∂t ∂t a = 2a σ1 − 2 (X1 + a2 σ1 ) a

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and projecting onto V we get ∇ ∂ e1 =

(9)

∂t

a e1 . a

The point to make here is that the somewhat mysterious Courant bracket can be used as a tool for automatically computing covariant derivatives in ordinary Riemannian geometry. 2. Spinors, twists and skew torsion 2.1. Spinors. In generalized geometry, the role of differential forms is changed. They become a Clifford module for the Clifford algebra generated by T ⊕ T ∗ with its indefinite inner product. Recall that, given a vector space W with an inner product ( , ) the Clifford algebra Cl(W ) is generated by 1 and W with the relations x2 = (x, x)1 (in positive definite signature the usual sign is −1 but this is the most convenient for our case). Consider an exterior differential form ϕ ∈ Λ∗ T ∗ and define the action of X + ξ ∈ T ⊕ T ∗ on ϕ by (X + ξ) · ϕ = iX ϕ + ξ ∧ ϕ then (X + ξ)2 · ϕ = iX (ξ ∧ ϕ) + ξ ∧ iX ϕ = iX ξϕ = (X + ξ, X + ξ)ϕ and so Λ∗ T ∗ is a module for the Clifford algebra. We have already remarked that we can regard T ⊕T ∗ as having structure group SO(n, n) and if the manifold is oriented this lifts to Spin(n, n). The exterior algebra is almost the basic spin representation of Spin(n, n), but not quite. The Clifford algebra has an anti-involution – any element is a sum of products x1 x2 . . . xk of generators xi ∈ W and x1 x2 . . . xk → xk xk−1 . . . x1 defines the anti-involution. It represents a “transpose” map a → at arising from an invariant bilinear form on the basic spin module. In our case the spin representation is strictly speaking S = Λ∗ T ∗ ⊗ (Λn T ∗ )−1/2 . Another way of saying this is that there is an invariant bilinear form on Λ∗ T ∗ with values in the line bundle Λn T ∗ . Because of its appearance in another context it is known as the Mukai pairing. Concretely, given ϕ1 , ϕ2 ∈ Λ∗ T ∗ , the pairing is  n−2j ϕ1 , ϕ2  = (−1)j (ϕ2j + ϕ12j+1 ∧ ϕ2n−2j−1 ) 1 ∧ ϕ2 j

where the superscript p denotes the p-form component of the form.

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The Lie algebra of the spin group (which is the Lie algebra of SO(n, n)) sits inside the Clifford algebra as the subspace {a ∈ Cl(W ) : [a, W ] ⊆ W and a = −at } where the commutator is taken in the Clifford algebra. Consider a 2-form B ∈  Λ2 T ∗ . The Clifford action of a 1-form ξ is exterior multiplication, so B · ϕ = bij ξi · ξj · ϕ = bij ξi ∧ ξj ∧ ϕ defines an action on spinors. Moreover, being skew-symmetric in ξi it satisfies B t = −B. Now take X +ξ ∈ W = T ⊕ T ∗ and the commutator [B, X + ξ] in the Clifford algebra: B ∧ (iX + ξ∧)ϕ − (iX + ξ∧)B∧ = B ∧ iX ϕ − iX (B ∧ ϕ) = −iX B ∧ ϕ. So this action preserves T ⊕ T ∗ and so defines an element in the Lie algebra of SO(n, n). But the Lie algebra action of B ∈ Λ2 T ∗ on T ⊕ T ∗ was X + ξ → iX B, so we see from the above formula that the action of a B-field on spinors is given by the exponentiation of −B in the exterior algebra : ϕ → e−B∧ ϕ. One may easily check, for example, that the Mukai pairing is invariant under the action: e−B ϕ1 , e−B ϕ2  = ϕ1 , ϕ2 . This action, together with the natural diffeomorphism action on forms, gives a combined action of the group Ω2 (M )cl Diff(M ) and a corresponding action of its Lie algebra. We earlier considered the map u → u ˜ given by X + ξ → X − dξ and on any bundle associated to T ⊕ T ∗ by a representation of SO(n, n) we have an action of u ˜. We regard this now as a “Lie derivative” Lu in the direction of a section u of T ⊕ T ∗ . In the spin representation there is a “Cartan formula” for this: Proposition 4. The Lie derivative of a form ϕ by a section u of T ⊕T ∗ is given by Lu ϕ = d(u · ϕ) + u · dϕ. Proof. d(X + ξ) · ϕ + (X + ξ) · dϕ = diX ϕ + d(ξ ∧ ϕ) + iX dϕ + ξ ∧ dϕ = LX ϕ + dξ ∧ ϕ using the usual Cartan formula and the fact that B = −dξ acts as −B = dξ.  In fact replacing the exterior product by the Clifford product is a common feature of generalized geometry whenever we deal with forms. The Lie derivative acting on sections of T ⊕ T ∗ is the Lie algebra action we observed in the first lecture so (10)

Lu v − Lv u = 2[u, v]

where [u, v] is the Courant bracket.

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2.2. Twisted structures. We now want to consider a twisted version of T ⊕ T ∗ . Suppose we have a nice covering of the manifold M by open sets Uα and we give ourselves a closed 2-form Bαβ = −Bβα on each two-fold intersection Uα ∩ Uβ . We can use the action of Bαβ to identify T ⊕ T ∗ on Uα with T ⊕ T ∗ on Uβ over the intersection. This will be compatible over threefold intersections if (11)

Bαβ + Bβγ + Bγα = 0

on Uα ∩ Uβ ∩ Uγ . We have seen in the first lecture that the action of a closed 2-form on T ⊕ T ∗ preserves both the inner product and the Courant bracket, so by the above identifications this way we construct a rank 2n vector bundle E over M with an inner product and a bracket operation on sections. And since the B-field action is trivial on T ∗ ⊂ T ⊕ T ∗ , the vector bundle is an extension: 0 → T ∗ → E → T → 0. π

Such an object is called an exact Courant algebroid. It can be abstractly characterized by the Properties (3) and (4) of the Courant bracket, where the vector field X is πu, together with the Jacobi-type formula in Proposition 2. The relation (11) says that we have a 1-cocycle for the sheaf Ω2cl of closed 2-forms on M . There is an exact sequence of sheaves 0 → Ω2cl → Ω2 → Ω3cl → 0 d

and since Ω2 is a flabby sheaf, we have H 1 (M, Ω2cl ) ∼ = H 0 (M, Ω3cl )/dH 0 (M, Ω2 ) = Ω3cl /dΩ2 = H 3 (M, R) so that such a structure has a characteristic degree 3 cohomology class. Example: The theory of gerbes fits into the twisted picture quite readily. Very briefly, a U (1) gerbe can be defined by a 2-cocycle with values on the sheaf of C ∞ circle-valued functions – so it is given by functions gαβγ on threefold intersections satisfying a coboundary condition. In the exact sequence of sheaves of C ∞ functions exp 2πi

1 → Z → R → U (1) → 1 the 2-cocycle defines a class in H 3 (M, Z). If we think of the analogue for line bundles, we have the transition functions gαβ and then a connection on the line bundle is given by 1-forms Aα on open sets such that Aβ − Aα = (g −1 dg)αβ on twofold intersections (where we identify the Lie algebra of the circle with R).

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A connective structure on a gerbe is similarly a collection of 1-forms Aαβ such that Aαβ + Aβγ + Aγα = (g −1 dg)αβγ on threefold intersections. Clearly Bαβ = dAαβ defines a Courant algebroid, and its characteristic class is the image of the integral cohomology class in H 3 (M, R). An example of this is a Hermitian structure on a holomorphic gerbe on a complex manifold (defined by a cocycle of holomorphic functions hαβγ with values in C∗ ). A Hermitian structure on this is a choice of a cochain kαβ of positive functions with |hαβγ | = kαβ kβγ kγα . Then Aαβ = (h−1 dc h)αβ defines ¯ a connective structure. Here dc = I −1 dI = −i(∂ − ∂). The bundle E has an orthogonal structure and so an associated spinor bundle S. By the definition of E, S is obtained by identifying Λ∗ T ∗ over Uα with Λ∗ T ∗ over Uβ by ϕ → e−Bαβ ϕ.

A global section of S is then given by local forms ϕα , ϕβ such that ϕα = e−Bαβ ϕβ on Uα ∩ Uβ . Since Bαβ is closed and even, dϕα = d(e−Bαβ ϕβ ) = −dBαβ ∧ e−Bαβ ϕβ + e−Bαβ dϕβ = e−Bαβ dϕβ and so we have a well-defined operator d : C ∞ (S ev ) → C ∞ (S od ). The Z2 -graded cohomology of this is the twisted cohomology. There is a more familiar way of writing this if we consider the inclusion of sheaves Ω2cl ⊂ Ω2 . Since Ω2 is a flabby sheaf, the cohomology class of Bαβ is trivial here and we can find 2-forms Fα such that on Uα ∩ Uβ Fβ − Fα = Bαβ . Since Bαβ is closed dFβ = dFα is the restriction of a global closed 3-form H which represents the characteristic class in H 3 (M, R). But then e−Fα ϕα = e−Fβ eBαβ ϕα = e−Fβ ϕβ defines a global exterior form ψ. Furthermore dψ = d(e−Fα ϕα ) = −H ∧ ψ + e−Fα dϕα . Thus the operator d above defined on S is equivalent to the operator d + H : Ωev → Ωod on exterior forms. Example: For gerbes the full analogue of a connection is a connective structure together with a curving, which is precisely a choice of 2-form Fα

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such that Fβ − Fα = dAαβ . In this case the 3-form H such that H/2π has integral periods is the curvature. Remark: Instead of thinking in cohomological terms about writing a cocycle of closed 2-forms Bαβ as a coboundary Fβ −Fα in the sheaf of all 2-forms, there is a more geometric interpretation of this choice which can be quite convenient. The B-field action of Fα gives an isomorphism of T ⊕ T ∗ with itself over Uα and the relation Fβ − Fα = Bαβ says that this extends to an isomorphism E∼ = T ⊕ T ∗. More concretely, X over Uα is mapped to X + iX Fα ∈ E and defines a splitting (in fact an isotropic splitting) of the extension 0 → T ∗ → E → T → 0. With the same coboundary data, we identified the spinor bundle S with the exterior algebra bundle and now we note that (X + iX Fα ) · e−Fα ϕα = −e−Fα iX ϕα so the two are compatible. We have a choice – either consider E, S with their standard local models of T ⊕ T ∗ and Λ∗ T ∗ , or make the splitting and give a global isomorphism. The cost is that we replace the ordinary exterior derivative by d + H and, as can be seen from the proof of Proposition 1, replace the standard Courant bracket by the twisted version (12)

[X + ξ, Y + η] + iY iX H.

2.3. Skew torsion. If we replace T ⊕ T ∗ by its twisted version E we may ask how to incorporate a Riemannian metric as we did in the first lecture. Here is the definition: Definition 2. A generalized metric is a subbundle V ⊂ E of rank n on which the induced inner product is positive definite. Since the inner product on T ∗ ⊂ E is zero and is positive definite on V , V ∩ T ∗ = 0 and so in a local isomorphism E ∼ = T ⊕ T ∗ , V is the graph of a ∗ map hα : T → T . So, splitting into symmetric and skew symmetric parts hα = gα + Fα . On the twofold intersection hα (X) = hβ (X) + iX Bαβ . Thus hα (X)(X) = hβ (X)(X) = g(X, X) for a well-defined Riemannian metric g, but Fα = Fβ + Bαβ . Associated with a generalized metric we thus obtain a natural splitting. Now the definition of a connection in Proposition 3 makes perfectly good sense in the twisted case. To see what we get, let us redo the calculation

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using local coordinates. In this case V is defined locally by X + gX + iX Fα and V ⊥ by X − gX + iX Fα (we are just transforming the V and V ⊥ of the metric g by the orthogonal transformation of the local B-field Fα .) So the appropriate Courant bracket is 

∂ ∂ − gik dxk + Fik dxk , + gj dx + Fj dx ∂xi ∂xj



and this gives the terms for the Levi-Civita connection plus a term1 ∂Fj ∂Fik 1 dx − dxk − d(Fji − Fij ) = ∂xi ∂xj 2



∂Fj ∂Fi ∂Fij − + ∂xi ∂xj ∂x

 dx .

(We could also have used the twisted bracketas in (12).) Writing the skew bilinear form Fα as i 0 and T itself to be stable for stability of the co-Higgs bundle.

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We refer the reader to [16], [15] for more results about co-Higgs bundles, but here we shall give some examples on projective spaces slightly more interesting than those above. Example: Consider M = CPm = P(W ). We denote by O(m) the line bundle of degree m on P(W ) and by V (m) the tensor product V ⊗ O(m). The tangent bundle fits into the Euler sequence of holomorphic bundles 0 → O → W (1) → T → 0 from which we obtain W ∼ = H 0 (CPm , T (−1)). We also see that Λm T ∼ = O(m + 1) and hence T∗ ∼ = Λm−1 T ⊗ Λm T ∗ ∼ = Λm−1 T (−(m + 1)). This means that T ⊗ T ∗ (1) ∼ = T ⊗ Λm−1 T (−m) = T (−1) ⊗ Λm−1 (T (−1)) and from the (m + 1)-dimensional space of sections of T (−1) we can construct by tensor and exterior product many sections, not just scalars, of T ⊗ T ∗ (1). Take one, ψ, and a section w of T (−1). Then set φ = ψ ⊗ w ∈ H 0 (CPm , End T ⊗ T ). By construction, φ2 = [ψ, ψ] ⊗ w ∧ w = 0, and the tangent bundle itself is stable so this gives plenty of examples of co-Higgs bundles on projective space. The simplest concrete example, where we can write down the moduli space, is the case of the bundle V = O ⊕ O(−1) on CP1 . Since Λ2 T = 0, there is no integrability condition on φ in one dimension. Here the tangent bundle is O(2) and so we must have   a b φ= c −a where a, b, c are sections of O(2), O(3), O(1) respectively. Since the degree and rank of V are coprime, there are no semi-stable bundles which means, in this one-dimensional case, that the moduli space is smooth. We first define a canonical six-dimensional complex manifold. We denote by p : T CP1 → CP1 the projection and η ∈ H 0 (T CP1 , p∗ T ) the tautological section. This is a section of p∗ O(2). Now define M = {(x, s) ∈ T CP1 × H 0 (CP1 , O(4)) : η 2 (x) = s(p(x))}. Proposition 11. M is naturally isomorphic to the moduli space of stable rank 2 trace zero co-Higgs bundles of degree −1 on CP1 .

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Proof. 1. First note that any vector bundle on CP1 is a sum of line bundles by the Birkhoff-Grothendieck theorem. If the decomposition is O(m) ⊕ O(−1 − m), then a, b, c are sections of O(2), O(2m + 3), O(1 − 2m). If c is zero, then O(m) is φ-invariant, so by stability m < −1/2. If m = −1, then by changing the order of the subbundles we are in the same situation. If m ≤ −2 then b is a section of a line bundle of negative degree and so vanishes – then the invariant subbundle O(−1 − m) contradicts stability. Thus the vector bundle in this moduli space is always V = O ⊕ O(−1). 2. Since c is a non-zero section of O(1) it vanishes at a distinguished point z = z0 . Then a(z0 ) is a point x in the total space of O(2) = T CP1 . It is well-defined because an automorphism of O ⊕ O(−1) is defined by   A B 0 C where A, B, C are sections of O, O(1), O and where c = 0 the action on a is trivial. 3. The determinant det φ is a section s of O(4), and at z = z0 , c vanishes so we have det φ(z0 ) = −a(z0 )2 , so set p = −det φ. This defines a map from the moduli space to M. 4. In the reverse direction, choose an affine parameter z on CP1 such that the point x maps to z = 0 and write p(z) = a20 + zb(z) where b(z) is a cubic polynomial. Then η 2 (0) = a20 so η(0) = ±a0 and   η(0) b(z) z −η(0) is a representative Higgs field. Note that M is a fibration of elliptic curves y 2 = c0 + c1 z + · · · + c4 z 4 over the five-dimensional vector space of coefficients c0 , . . . , c4 . We shall see this again when we consider the B-field action in the next lecture.  Remark: We saw in Section 3.3 that the ∂¯J -complex for an ordinary complex structure was defined by ∂¯

Ω0,q (Λp T ) → Ω0,q+1 (Λp T ). ¯ complex is defined by ∂¯ ± φ where For a co-Higgs bundle (V, φ) the D ∂¯ : Ω0,q (V ⊗ Λp T ) → Ω0,q+1 (V ⊗ Λp T ) and

φ : Ω0,q (V ⊗ Λp T ) → Ω0,q (V ⊗ Λp+1 T ). Note that the total degree p + (q + 1) = (p + 1) + q is preserved. It is easy ¯ complex is the hypercohomology of the to see that the cohomology of the D complex of sheaves φ

· · · → O(V ⊗ Λp T ) → O(V ⊗ Λp+1 T ) → · · ·

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5.3. Holomorphic Poisson modules. We observed that a holomorphic Poisson module is a holomorphic vector bundle V with a first order holomorphic linear differential operator ¯ : O(V ) → O(V ⊗ T ) D whose symbol is 1 ⊗ σ : V ⊗ T ∗ → V ⊗ T . Relative to a local holomorphic ¯ is defined by a “connection matrix” A of vector fields: basis si of V , D ¯ i = sj ⊗ Aji . Ds ¯ is a flat holoWhen σ is non-degenerate it identifies T with T ∗ and then D morphic connection. Example: If X = σ(df ) is the Hamiltonian vector field of f then the Lie derivative LX acts on tensors but the action in general involves the second derivative of f . However for the canonical line bundle K = Λn T ∗ we have LX (dz1 ∧ · · · ∧ dzn ) =

∂Xi (dz1 ∧ · · · ∧ dzn ) ∂zi

and, since σ ij is skew-symmetric, ∂Xi ∂ = ∂zi ∂zi



∂f σ ∂zj ij

 =

∂σ ij ∂f ∂zi ∂zj

which involves only the first derivative of f . Thus ¯ df  {f, s} = LX s = Ds, defines a first order operator. The second condition for a Poisson module follows from the integrability of the Poisson structure: since σ(df ) = X, σ(dg) = Y implies σ(d{f, g}) = [X, Y ], it follows that {{f, g}, s} = L[X,Y ] s = [LX , LY ]s = {f, {g, s}} − {g, {f, s}}. This clearly holds for any power K m . ¯ : O(V ) → O(V ⊗T ) is globRemark: A holomorphic first-order operator D ally defined as a vector bundle homomorphism α : J 1 (V ) → V ⊗ T where J 1 (V ) is the bundle of holomorphic 1-jets of sections of V . It is an extension 0 → V ⊗ T ∗ → J 1 (V ) → V → 0

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and its extension class in H 1 (M, End V ⊗T ∗ ), the Atiyah class, is the obstruction to splitting the sequence holomorphically. When V is a line bundle, and M is K¨ahler, this is the first Chern class in H 1,1 . ¯ is the homomorphism α restricted to V ⊗T ∗ ⊂ J 1 (V ), The symbol σ of D ¯ so the existence of D means that σ ∈ H 0 (M, Hom(V ⊗ T ∗ , V ⊗ T ) lifts to a class α ∈ H 0 (M, Hom(J 1 (V ), V ⊗T ). In the long exact cohomology sequence of the extension, this means that the map σ : H 1 (M, End V ⊗ T ∗ ) → H 1 (M, End V ⊗ T ) applied to the Atiyah class is zero. In the case of a line bundle this is the cup product we encountered in Section 4.3 applied to the first Chern class, so in particular we see that the existence of a Poisson module structure on the canonical bundle means that the image of c1 (T ) in H 1 (M, T ) is zero. So, as in Section 4.3, if c1 (T ) is represented by a K¨ahler form, Goto’s theorem, to first order, keeps the complex structures I+ , I− in the same diffeomorphism class. Just because the Lie derivative of a Hamiltonian vector field does not make the tangent bundle a Poisson module does not mean that it can’t be one. If we take two vector fields X1 , X2 on CP2 then σ = X1 ∧ X2 defines a Poisson structure. It is holomorphic symplectic where σ is non-zero, which is away from a cubic curve C – the curve where X1 ∧ X2 = 0 i.e. where X1 and X2 become linearly dependent. Here we can step back and view CP2 as a generalized complex manifold: away from C, σ −1 defines a holomorphic section of the canonical bundle K which we can write as a closed complex 2-form B + iω. The generalized complex structure here is a symplectic structure ω transformed by the B-field B. But on such a structure, a generalized holomorphic bundle is a flat vector bundle. Now X1 and X2 are linearly independent away from C so we can try ¯ on T by making them covariant constant i.e. DX ¯ 1 = DX ¯ 2= and define D 0. Then we need to show that this extends as a holomorphic differential ¯ 2 = 0 condition is already satisfied on an open set and so operator – the D holds everywhere. Take a local holomorphic basis ∂/∂z1 , ∂/∂z2 for T in a neighbourhood of a point of C, and then ∂ Xi = Pji ∂zj so (17)

σ = X1 ∧ X2 = det P

∂ ∂ ∧ . ∂z1 ∂z2

¯ relative to this basis is given by a matrix A of A“connection matrix” for D vector fields such that ¯ i = D(Pji ∂ ) = σ(dPji ) ∂ + Pji Akj ∂ 0 = DX ∂zi ∂zj ∂zk

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which has solution A = −σ(dP )P −1 = −σ(dP ) From (17) this is

 adj P

adj P . det P

∂P ∂ ∂P ∂ − ∂z2 ∂z1 ∂z1 ∂z2



¯ is well-defined. which is holomorphic and so D We can extend this argument to other rank 2 vector bundles V with Λ2 V ∼ = K ∗ , so long as they have two sections s1 , s2 to replace the vector fields X1 , X2 . A generic vector field on CP2 has three zeros: suppose V has a section s with k simple zeros x1 , . . . , xk (and then the second Chern class c2 (V ) = k). Then s defines an exact sequence of sheaves 0 → O → O(V ) → K ∗ ⊗ I → 0 s

where I is the ideal sheaf of the k points. If H 0 (CP2 , K ∗ ⊗ I) = 0, in other words if there is a cubic curve C passing through the k points, then from the exact cohomology sequence (and using H 1 (CP2 , O) = 0) we can find a second section s2 of V , and if s1 = s, s1 ∧ s2 vanishes on the curve C, which defines a holomorphic Poisson structure. The Serre construction provides a means of constructing such bundles (see for example [3] Section 10.2.2). Away from the k points we have an extension of line bundles 0 → O → O(V ) → K ∗ → 0 s

which is described by a Dolbeault representative α ∈ Ω0,1 (K) = Ω2,1 . It extends to an extension as above if it has a singularity at each of the points of the form 1 dz1 ∧ dz2 ∧ (¯ z2 d¯ z1 − z¯1 d¯ z2 ). 4r4 ¯ =  λi δx = β a linear combination of delta funcIn distributional terms ∂α i i tions of the points. Such a sum defines a class in H 2 (M, K). Since H 2 (M, K) is dual to 0 H (M, O) ∼ = C, this class is determined by evaluating it on the function 1. But  λi β, 1 = i

¯ = β for α. so if the λi sum to zero the class is zero and one can solve ∂α Thus, given a cubic curve and a collection of k points with non-zero scalars λi whose sum is zero, we obtain a rank 2 Poisson module with c2 (V ) = k.

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Remark: The Serre construction can also be used to generate co-Higgs bundles on CP2 with nilpotent Higgs field φ. This time we require Λ2 V ∼ = ¯ = β for a distribution defining a class in O(1) and we want to solve ∂α H 2 (CP2 , O(−1)) which is dual to H 0 (CP2 , O(−2)) = 0. Hence there is no constraint on the λi s. We obtain an extension s

π

0 → O → O(V ) → O(1) ⊗ I → 0. Choosing a section w of T (−1), v → s(wπ(v)) defines φ ∈ H 0 (CP2 , End V ⊗ T ) whose kernel and image lie in the trivial rank one subsheaf. 6. Holomorphic bundles and the B-field action 6.1. The B-field action. On a complex manifold we can pull back a holomorphic vector bundle by a holomorphic diffeomorphism to get a new one, but in generalized geometry we have learned that the group Ω2 (M )cl  Diff(M ) replaces the group of diffeomorphisms and in particular that a closed real (1, 1)-form B preserves the generalized complex structure determined by an ordinary complex structure. We shall study next the effect of this action on generalized holomorphic bundles. Recall that in this case a generalized holomorphic bundle is defined by an operator   ∂s ∂ ¯ Ds = + A¯i s d¯ zi + φj s ∂ z¯i ∂zj where (V, φ) is a co-Higgs bundle. The transform of this by B is then the operator   ∂s ∂ j ¯ Ds = + A¯i + φ Bj¯i s d¯ zi + φj s ∂ z¯i ∂zj More invariantly we write iφ B ∈ Ω0,1 (End V ) for the contraction of the Higgs field φ ∈ H 0 (M, End V ⊗ T ) with B ∈ Ω0,1 (T ∗ ) and then we have a new holomorphic structure ∂¯B = ∂¯ + iφ B on the C ∞ bundle V . But the condition φ2 = 0 means that iφ B commutes with φ and so φ is still holomorphic with respect to this new structure: ∂¯B φ = 0. Remark: Note that if U ⊂ V is a holomorphic subbundle with respect to ∂¯ then if it is also φ-invariant, it is holomorphic with respect to ∂¯B . So stability is preserved by the B-field action. ¯ for θ ∈ Ω1,0 and define ψ = iφ θ, a section of End V . Now suppose B = ∂θ i ¯ ¯ = φj B ¯. Then In coordinates ψ = φ θi which implies (∂ψ) i ji ¯ ¯ = [φk θk , φj B ¯] = [φk , φj ]θk B ¯ = 0 [ψ, ∂ψ] i ji ji

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since φj and φk commute. This means (unusually for a non-abelian gauge theory) that if we exponentiate to an automorphism of the bundle V we have ¯ + iφ Bψ. exp(−ψ)∂¯ exp ψ = ∂ψ Thus if B1 and B2 represent the same Dolbeault cohomology class in H 1 (M, T ∗ ), the two actions are related by an automorphism. Hence H 1 (M, T ∗ ) acts on the moduli space of stable co-Higgs bundles. Example: If we take the canonical Higgs bundle V = O ⊕ T and φ(λ, X) = (X, 0) as in Section 5.2, then for [B] ∈ H 1 (M, T ∗ ) the structure ∂¯B defines a non-trivial extension 0→O→V →T →0 which still has a canonical Higgs field. We shall investigate this action in more detail next. 6.2. Spectral covers. In the case of Higgs bundles, Simpson reinterpreted in [18] a Higgs sheaf on M in terms of a sheaf on P(O ⊕ T ∗ ) whose support is disjoint from the divisor at infinity. This can be adapted immediately replacing T ∗ by T . In standard local coordinates yi , zj on T M given by the vector field y i ∂/∂zi , we pull back the rank k bundle V under the projection p : T M → M and define an action of y i by φi . Since [φi , φj ] = 0 this defines a module structure over the commutative ring of functions polynomial in the fibre directions. More concretely, suppose in a neighbourhood of a point some linear combination of the φi , say φ1 , has distinct eigenvalues. Then since by the Cayley-Hamilton theorem φ1 satisfies its characteristic equation, on the support of the sheaf y1 is an eigenvalue of φ1 , and the kernel of φ1 − y1 defines a line bundle U ⊂ p∗ V . Since all φi commute with φ1 , L is an eigenspace for φi with eigenvalue y i . If the m characteristic equations of φi are generic, they define an m-dimensional submanifold S of T M which is an m-fold covering of M under p. There will be points at which φ1 has coincident eigenvalues, but under suitable genericity conditions S will still be smooth with a line bundle L. The action of φ on U is φ|U = φi |U

∂ ∂ = yi ∂zi ∂zi

which is the tautological section of p∗ T on the total space of T . The one-dimensional case of this, where M = CP1 , was much studied from the point of view of integrable systems before its important application to the moduli space of Higgs bundles, and the co-Higgs situation on CP1 is a particular case described, for example, in [15]. In this case, where T = O(2), φ is a holomorphic section of End V (2) and its characteristic equation is det(η − φ) = η k + a1 η k−1 + · · · + ak = 0

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where ai is a section of O(2i) on CP1 . Interpreting η = yd/dz as the tautological section of p∗ O(2), this is the vanishing of a section of p∗ O(2k) on the algebraic surface T CP1 and it defines a spectral curve S which, by the adjunction formula, has genus g = (k − 1)2 and is a branched covering of CP1 of degree k. We reconstruct a co-Higgs bundle by taking the direct image p∗ L of a line bundle L on S. For any open set U ⊆ CP1 , by definition H 0 (U, p∗ L) = H 0 (p−1 (U ), L). The sheaf p∗ L defines a rank k vector bundle and the direct image of multiplication by the tautological section η : H 0 (p−1 (U ), L) → H 0 (p−1 (U ), L(2)) defines a Higgs field φ. (Note that the line bundle L is not quite the same as the eigenspace bundle U . The direct image gives a canonical evaluation map p∗ V → L so that L∗ ⊂ p∗ V ∗ is the eigenspace bundle of the dual endomorphism φt .) The degree of the line bundle L and that of the vector bundle V are easily related – the direct image definition implies that H 0 (S, L ⊗ p∗ O(n)) ∼ = H 0 (CP1 , p∗ L(n)) and taking n large, these are given by the Riemann-Roch formula. The result is deg V = deg L + k − k 2 . Examples: 1. Consider the example of V = O ⊕ O(−1) in the previous lecture. Here k = 2 and deg V = −1, so deg L = 1. The curve S has genus (k − 1)2 = 1 and is an elliptic curve (y 2 = c0 + c1 z + · · · + c4 z 4 ). The line bundle L has degree one and hence has a unique section which vanishes at a single point, which is η(0) in our description of the moduli space. 2. If deg L = g − 1 = k 2 − 2k then deg V = −k, so V (1) has degree zero. Now a vector bundle E on CP1 is trivial if it has degree zero and H 0 (CP1 , E(−1)) = 0, so V (1) is trivial if 0 = H 0 (CP1 , V ) = H 0 (S, L), which is if the divisor class of L does not lie on the theta divisor of S. In this case a co-Higgs bundle consists of a k × k matrix whose entries are sections of O(2). Now consider the B-field action from the point of view of the spectral cover. Since φ is only changed by conjugation, the spectral cover is unchanged – it is only the holomorphic structure on the line bundle which can change. But the change in the holomorphic structure on V was ∂¯ → ∂¯ + iφ B

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and on U ⊂ V φ acts via the tautological section η of p∗ T M , so we are changing the holomorphic structure of U by ∂¯ → ∂¯ + iη B. In other words, we have [B] ∈ H 1 (M, T ∗ ) which we pull back to p∗ [B] ∈ H 1 (T M, p∗ T ) then contract with η ∈ H 0 (T M, p∗ T ) to get the class ηp∗ [B] ∈ H 1 (T M, O). Exponentiating to H 1 (T M, O∗ ) defines a line bundle LB . Restricting to S the B-field action is U → U ⊗ LB . Let us look at this action in the two examples above. Since H 1,1 (CP1 ) is one-dimensional a real closed (1, 1) form is cohomologous to a multiple of idz ∧ d¯ z (1 + z z¯)2 This form integrates to 2π over CP1 . Pulling back to T CP1 and contracting with yd/dz we obtain the class in H 1 (S, O) represented by iyd¯ z . (1 + z z¯)2 1. In the first example, S is an elliptic curve and a point of the moduli space is defined by a point x on this curve, so tensoring with a line bundle LB is a translation. The non-vanishing 1-form dz/y is equal to du in the uniformization and then two points x, x are related by a translation u → u + a if x dz =a y x modulo periods. On the other hand our class in H 1 (S, O) pairs with dz/y ∈ H 0 (S, K) by integration: idz ∧ d¯ z = 4π ¯)2 S (1 + z z so this determines the translation. 2. In the second example, since the bundle is trivial we may write the Higgs field in End Ck ⊗ H 0 (CP1 , O(2)) as a matrix with entries quadratic polynomials in z. Write it thus: φ = (T1 + iT2 ) + 2iT3 z + (T1 − iT2 )z 2 . Then, as derived in [15], tensoring by LB is integrating to time t = 1 the system of nonlinear differential equations called Nahm’s equations. dT1 = [T2 , T3 ], dt

dT2 = [T3 , T1 ], dt

dT3 = [T1 , T2 ]. dt

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These equations arise in the study of non-abelian monopoles and are dimensional reductions of the self-dual Yang-Mills equations. From these examples it is clear that the B-field action can be highly non-trivial. What it also shows is that the action on the moduli space can be quite badly behaved, for the Nahm flow could be an irrational flow on the Jacobian of the spectral curve. 6.3. Twisted bundles and gerbes. Now suppose we replace the generalized complex structure on T ⊕ T ∗ by a twisted version on the bundle E defined by a 1-cocycle Bαβ of closed real (1, 1)-forms. What is a generalized holomorphic bundle now? The general definition is the same – a vector ¯ but we want to understand it in bundle V with a differential operator D more concrete terms. If we think of E as obtained by patching together copies of T ⊕ T ∗ then over each open set Uα , V has the structure of a co-Higgs bundle – a holomorphic structure Aα and a Higgs field φα . On the intersection Uα ∩ Uβ these are related by the B-field action of Bαβ : (18)

(Aβ )¯i = (Aα )¯i + φj (Bαβ )j¯i ,

(φβ )j = (φα )j .

Consider first the case of V = L a line bundle. Then, because End V is holomorphically trivial for all of the local holomorphic structures, φ is a global holomorphic vector field X. So consider the (0, 1) form Aαβ = iX Bαβ . ¯ αβ = 0 and X is holomorphic so that The (1, 1) form Bαβ is closed so ∂B ¯ ¯ ∂Aαβ = 0. Locally write Aαβ = ∂fαβ , then, since Bαβ is a cocycle, on threefold intersections fαβ + fβγ + fγα is holomorphic. Write gαβγ = exp 2πi(fαβ + fβγ + fγα ) then this defines a holomorphic gerbe. ¯ But the local holomorphic structure on L is defined by a ∂-closed form ¯ Aα , so writing Aα = ∂hα we have from (18) that kαβ = fαβ + hα − hβ is holomorphic and moreover gαβγ = exp 2πi(kαβ + kβγ + kγα ). This is a holomorphic trivialization of the gerbe, or as is sometimes said, a line bundle over the gerbe. The ratio of any two trivializations (i.e. writing gαβγ as a coboundary) is a cocycle which defines the transition functions for a holomorphic line bundle. In the untwisted case a generalized holomorphic bundle was just a line bundle and a vector field; here any two differ by such an object. In more invariant terms we have taken the class in H 2 (M, T ∗ ) defined by the (1, 2) component of the 3-form H, and contracted with the vector field X ∈ H 0 (M, T ) to get a class in H 2 (M, O). Exponentiating gives us an element in H 2 (M, O∗ ) which is the equivalence class of the holomorphic

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gerbe defined by gαβγ . The existence of a trivialization of the gerbe is the statement that this class is zero. Now consider the general case: over each Uα we can consider the spectral cover in T M . This is defined by characteristic polynomials of components of φ. The C ∞ transition functions for the vector bundle V conjugate φ and so leave these polynomials invariant. It follows that the local spectral covers fit together into a global spectral cover S ⊂ T M . The eigenspace bundle U however, only has local holomorphic structures. But φ acts on U via the tautological section η of p∗ T M , and so we are in a parallel situation to the one we just considered: a gerbe on T M defined by the cocycle Aαβ = iη p∗ Bαβ . In the untwisted case, a co-Higgs bundle was determined by a line bundle on the spectral cover, in this case it is a trivialization of the gerbe. The language of gerbes is convenient to describe things on the spectral cover, but a C ∞ bundle V with local holomorphic structures is not readily adaptable to conventional algebraic geometric language on M itself. As far ¯ but it is still useful to as generalized geometry is concerned we have D, rephrase the structure in more conventional language. For that purpose, we can split the extension E and work with T ⊕ T ∗ and the Courant bracket twisted with a 3-form H. ¯ = ∂¯A − H 1,2 ± φ where The generalized Dolbeault complex is now D ∂¯A : Ω0,q (V ⊗ Λp T ) → Ω0,q+1 (V ⊗ Λp T ), the 3–form H 1,2 acts by contraction in the (1, 0) entry, H 1,2 : Ω0,q (V ⊗ Λp T ) → Ω0,q+2 (V ⊗ Λp−1 T ) and the Higgs field acts like this φ : Ω0,q (V ⊗ Λp T ) → Ω0,q (V ⊗ Λp+1 T ). ¯ 2 = 0 now becomes The condition D 2 = iφ H, ∂¯A

∂¯A φ = 0,

φ2 = 0.

This shape of structure has appeared in the literature. For example, replacing V by End V (and thereby getting a complex which governs the deformation theory of a generalized holomorphic bundle), we obtain a curved differential graded algebra – an algebra with derivation where d2 a = [c, a] and dc = 0. This is an identifiable concept, but nevertheless, packaged in the language of generalized geometry it becomes quite natural. References [1] J-M.Bismut, A local index theorem for non-K¨ ahler manifolds Math. Ann. 284 (1989) 681–699.

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[2] G.Cavalcanti, Introduction to generalized complex geometry (Lecture notes for the mini course given at the 26th Coloquio Brasileiro de Matematica, IMPA, Brazil) http://www.staff.science.uu.nl/ caval101/homepage/Research.html [3] S.K.Donaldson & P.B.Kronheimer, The geometry of four-manifolds, Oxford University Press, New York, 1990. [4] S.J.Gates, C.M.Hull and M.Roˇcek, Twisted multiplets and new supersymmetric nonlinear σ-models, Nuclear Phys. B 248 (1984) 157–186. [5] G.W.Gibbons and C.N.Pope, The positive action conjecture and asymptotically Euclidean metrics in quantum gravity, Comm. Math. Phys. 66 (1979) 267–290. [6] R.Goto, Deformations of generalized K¨ ahler structures and bihermitian structures, arXiv:0910.1651 [7] M.Gualtieri, Branes on Poisson varieties, in The Many Facets of Geometry (O.GarciaPrada, J-P.Bourguignon and S.Salamon (eds.)) 368–394, Oxford Univ. Press, Oxford, (2010). [8] M.Gualtieri, Generalized complex geometry, Annals of Mathematics 174 (2011) 75–123. [9] M.Gualtieri, Generalized K¨ ahler geometry, arXiv:math/1007.3485. [10] N.J.Hitchin, Riemann surfaces and integrable systems. Notes by Justin Sawon, in Oxf. Grad. Texts Math. 4 Integrable systems (Oxford, 1997) 11–52, Oxford Univ. Press, New York, (1999). [11] N.J.Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003) 281–308. [12] N.J.Hitchin, Generalized Geometry – an introduction, in Handbook of PseudoRiemannian Geometry and Supersymmetry, V. Cortes (ed.) IRMA Lectures in Mathematics and Theoretical Physics 16 185–208, European Mathematical Society (2010). [13] N.J.Hitchin, Instantons, Poisson structures and generalized K¨ ahler geometry, Comm. Math. Phys. 265 (2006), 131–164. [14] N.J.Hitchin, Bihermitian metrics on Del Pezzo surfaces, Journal of Symplectic Geometry 5 (2007) 1–7. [15] N.J.Hitchin, Generalized holomorphic bundles and the B-field action, J. Geom. Phys. 61 (2011) 352–362. [16] S.Rayan, Co-Higgs bundles on P 1 , arXiv:1010.2526. ´ [17] C.T.Simpson, Higgs bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992), 5–95. [18] C.T.Simpson, Moduli of representations of the fundamental group of a smooth pro´ jective variety. II, Inst. Hautes Etudes Sci. Publ. Math 80 (1994), 5–79. Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK E-mail address: [email protected]

Surveys in Differential Geometry XVI

Generalized Donaldson–Thomas invariants Dominic Joyce

Abstract. This is a survey of the book [16] with Yinan Song. Donaldson–Thomas invariants DT α (τ ) ∈ Z ‘count’ τ -(semi)stable coherent sheaves with Chern character α on a Calabi–Yau 3-fold X. They are unchanged under deformations of X. The conventional definition works only for classes α with no strictly τ -semistable α sheaves. Behrend showed  αthat DT (τ )can be written as a weighted Euler characteristic χ Mst (τ ), νMαst (τ ) of the stable moduli scheme Mα we call the ‘Behrend st (τ ) by a constructible function νMα st (τ ) function’. We discuss generalized Donaldson–Thomas invariants ¯ α (τ ) ∈ Q. These are defined for all classes α, and are equal DT to DT α (τ ) when it is defined. They are unchanged under deformations of X, and transform according to a known wall-crossing formula under change of stability condition τ . We conjecture that they ˆ α (τ ) ∈ Z can be written in terms of integral BPS invariants DT when the stability condition τ is ‘generic’. We extend the theory to abelian categories mod-CQ/I of representations of a quiver Q with relations I coming from a superpotential W on Q, and connect our ideas with Szendr˝ oi’s noncommutative Donaldson–Thomas invariants, and work by Reineke and others on invariants counting quiver representations. The book [16] has significant overlap with a recent, independent paper of Kontsevich and Soibelman [18].

1. Introduction This is a survey of the book [16] by the author and Yinan Song. Let X be a Calabi–Yau 3-fold over C, and OX (1) a very ample line bundle on X. Our definition of Calabi–Yau 3-fold requires X to be projective, with H 1 (OX ) = 0. Write coh(X) for the abelian category of coherent sheaves on X, and K(X) for the numerical Grothendieck group of coh(X). Let τ denote Gieseker stability of coherent sheaves w.r.t. OX (1). If E is a coherent c 2011 International Press

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sheaf on X then [E] ∈ K(X) is in effect the Chern character ch(E) of E in H even (X; Q). For α ∈ K(X) we can form the coarse moduli schemes Mαss (τ ), Mαst (τ ) of τ -(semi)stable sheaves E with [E] = α. Then Mαss (τ ) is a projective C-scheme whose points correspond to S-equivalence classes of τ -semistable sheaves, and Mαst (τ ) is an open subscheme of Mαss (τ ) whose points correspond to isomorphism classes of τ -stable sheaves. For Chern characters α with Mαss (τ ) = Mαst (τ ), following Donaldson and Thomas [4, §3], Thomas [33] constructed a symmetric obstruction theory on Mαst (τ ) and defined the Donaldson–Thomas invariant to be the virtual class  DT α (τ ) = [Mα (τ )]vir 1 ∈ Z, st

an integer which ‘counts’ τ -semistable sheaves in class α. Thomas’ main result [33, §3] is that DT α (τ ) is unchanged under deformations of the underlying Calabi–Yau 3-fold X. Later, Behrend [1] showed that Donaldson– Thomas invariants can be written as a weighted Euler characteristic   DT α (τ ) = χ Mαst (τ ), νMαst (τ ) , where νMαst (τ ) is the Behrend function, a constructible function on Mαst (τ ) depending only on Mαst (τ ) as a C-scheme. Conventional Donaldson–Thomas invariants DT α (τ ) are only defined for classes α with Mαss (τ ) = Mαst (τ ), that is, when there are no strictly τ -semistable sheaves. Also, although DT α (τ ) depends on the stability condition τ , that is, on the choice of very ample line bundle OX (1) on X, this dependence was not understood until now. The main goal of [16] is to address these two issues. For a Calabi–Yau 3-fold X over C we will define generalized Donaldson– ¯ α (τ ) ∈ Q for all α ∈ K(X), which ‘count’ Thomas invariants DT τ -semistable sheaves in class α. These have the following important properties: ¯ α (τ ) ∈ Q is unchanged by deformations of the Calabi–Yau • DT 3-fold X. ¯ α (τ ) lies in Z and equals the conven• If Mαss (τ ) = Mαst (τ ) then DT tional Donaldson–Thomas invariant DT α (τ ) defined by Thomas [33]. • If Mαss (τ ) = Mαst (τ ) then conventional Donaldson–Thomas invariants DT α (τ ) are not defined for class α. Our generalized invariant ¯ α (τ ) may lie in Q because strictly semistable sheaves E make DT ¯ α (τ ). For ‘generic’ τ (complicated) Q-valued contributions to DT α ¯ we have a conjecture that writes the DT (τ ) in terms of other, ˆ α (τ ). integer-valued invariants DT • If τ, τ˜ are two stability conditions on coh(X), there is an explicit ¯ α (˜ change of stability condition formula giving DT τ ) in terms of the β ¯ DT (τ ).

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These invariants are a continuation of the author’s programme [9–15]. We begin in §2 with some background material on constructible functions and stack functions on Artin stacks, taken from [9,10]. Then §3 summarizes ideas from [11–14] on Euler-characteristic type invariants J α (τ ) counting sheaves on Calabi–Yau 3-folds and their wall-crossing under change of stability condition, and facts on Donaldson–Thomas invariants from Thomas [33] and Behrend [1]. Section 4 summarizes [16, §5–§6], and is the heart of the paper. Let X be a Calabi–Yau 3-fold, and M the moduli stack of coherent sheaves on X. Write χ ¯ : K(X) × K(X) → Z for the Euler form of coh(X). We will explain that the Behrend function νM of M satisfies two important identities 

¯ 1 ],[E2 ]) νM(E1 )νM(E2 ), νM(E1 ⊕ E2 ) = (−1)χ([E  νM(F )dχ −  νM(F  )dχ 1 1

[λ]∈P(Ext (E2 ,E1 )): λ ⇔ 0→E1 →F →E2 →0  = dim Ext1 (E2 , E1 )

−

[λ ]∈P(Ext (E1 ,E2 )): λ ⇔ 0→E2 →F  →E1 →0  dim Ext1 (E1 , E2 ) νM(E1

⊕ E2 ).

˜ ˜ : SFind (M) → L(X), where We use these to define a Lie algebra morphism Ψ al ind SFal (M) is a special Lie subalgebra of the Ringel–Hall algebra SFal (M) of ˜ X, a large algebra with a universal construction, and L(X) is a much smaller α ˜ explicit Lie algebra, the Q-vector space with basis λ for α ∈ K(X), and Lie bracket ¯ ˜α, λ ˜ β ] = (−1)χ(α,β) ˜ α+β . [λ χ(α, ¯ β)λ If τ is Gieseker stability in coh(X) and α ∈ K(X), we define an element in SFind al (M) which ‘counts’ τ -semistable sheaves in class α in a special ¯ α (τ ) ∈ Q by way. We define the generalized Donaldson–Thomas invariant DT   ˜α. ˜ ¯α (τ ) = −DT ¯ α (τ )λ Ψ ¯α (τ )

By results in [14], the ¯α (τ ) transform according to a universal transformation law in the Lie algebra SFind al (M) under change of stability condition. α ˜ α transform according to the same law ˜ ¯ Applying Ψ shows that −DT (τ )λ ˜ in L(X). This yields a wall-crossing formula for two stability conditions τ, τ˜ on coh(X): ¯ α (˜ DT τ) =   (1)



iso. κ:I→C(X): connected, classes  κ(i)=α simplyof finite i∈I connected sets I digraphs Γ, vertices I

(−1)|I|−1 V (I, Γ, κ; τ, τ˜) · 1

·(−1) 2

 i,j∈I

|χ(κ(i),κ(j))| ¯

·

 

i∈I

¯ κ(i) (τ ) DT

χ(κ(i), ¯ κ(j)), i

j

edges • → • in Γ

where V (I, Γ, κ; τ, τ˜) ∈ Q are combinatorial coefficients, and there are only finitely many nonzero terms.

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¯ α (τ ) is unchanged under deformations of X, we introTo prove that DT duce auxiliary invariants P I α,n (τ  ) ∈ Z counting ‘stable pairs’ s : O(−n) → E, for n  0 and E ∈ coh(X) τ -semistable in class α ∈ K(X). The moduli  space Mα,n stp (τ ) of such stable pairs is a proper fine moduli C-scheme with a symmetric obstruction theory, so by the same proof as for Donaldson–  Thomas invariants [33], the virtual count P I α,n (τ  ) of Mα,n stp (τ ) is deformation-invariant. By a wall-crossing proof similar to that for (1) we find that 

l (−1)l  ¯ X (−n)]−α1 −···−αi−1 ,αi ) (−1)χ([O l! i=1 α1 ,...,αl ∈C(X),  

l1: α1 +···+αl =α, ¯ αi (τ ) . χ ¯ [OX (−n)]−α1 −· · ·−αi−1 , αi DT

P I α,n (τ  ) =

τ (αi )=τ (α), all i

Using deformation-invariance of the P I α,n (τ  ) and induction on rank α we ¯ α (τ ) is deformation-invariant. find that DT ¯ α (τ ) lie in Q rather than Z. So Examples show that in general the DT ¯ α (τ ) in terms of it is an interesting question whether we can rewrite the DT some system of Z-valued invariants, just as Q-valued Gromov–Witten invariants of Calabi–Yau 3-folds are (conjecturally) written in terms of Z-valued ˆ α (τ ) for Gopakumar–Vafa invariants [7]. We define new BPS invariants DT α ∈ C(X) to satisfy  1 ˆ α/m ¯ α (τ ) = DT DT (τ ), m2 m1, m|α

ˆ α (τ ) ∈ Z for all α if the stability condition τ is and we conjecture that DT ‘generic’. Evidence for this conjecture is given in [16, §6.1–§6.5 & §7.6]. Section 5 summarizes [16, §7], which develops an analogue of Donaldson– Thomas theory for representations of quivers with relations coming from a superpotential. This provides a kind of toy model for Donaldson–Thomas invariants using only polynomials and finite-dimensional algebra, and is a source of many simple, explicit examples. Counting invariants for quivers with superpotential have been studied by Nakajima, Reineke, Szendr˝ oi and other authors for some years [5,24–27,29,30,32], under the general name of ‘noncommutative Donaldson–Thomas invariants’. Curiously, the invariants studied so far are the analogues of our pair invariants P I α,n (τ  ), and the ˆ α (τ ) seem to have received no attention, although ¯ α (τ ), DT analogues of DT they appear to the author to be more fundamental. A recent paper by Kontsevich and Soibelman [18], summarized in [19], has considerable overlap with both [16] and the already published [9–15]. The two were completed largely independently, and the first versions of [16, 18] appeared on the arXiv within a few days of each other. Kontsevich and Soibelman are far more ambitious than us, working in triangulated categories rather than abelian categories, over general fields K rather than C, and with general motivic invariants rather than the Euler characteristic. But for this

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reason, almost every major result in [18] depends explicitly or implicitly on conjectures. The author would like to acknowledge the contribution of [18] ˆ α (τ ) and integrality in §4.4 below, and to the material on to the ideas on DT quivers with superpotential in §5. The relationship between [16] and [18] is discussed in detail in [16, §1.6]. Acknowledgements. The author would like to thank Tom Bridgeland, Richard Thomas, Bal´ azs Szendr˝ oi, and his co-author Yinan Song. This research was supported by EPSRC grant EP/D077990/1. 2. Constructible functions and stack functions We begin with some background material on Artin stacks, constructible functions, and stack functions, drawn from [9, 10]. We restrict to the field K = C. 2.1. Artin stacks and constructible functions. Artin stacks are a class of geometric spaces, generalizing schemes and algebraic spaces. For a good introduction to Artin stacks see G´omez [6], and for a thorough treatment see Laumon and Moret-Bailly [20]. We work throughout over the field C. We make the convention that all Artin stacks in this paper are locally of finite type, with affine geometric stabilizers, that is, all stabilizer groups IsoF(x) are affine algebraic C-groups, and substacks are locally closed. Artin C-stacks form a 2-category. That is, we have objects which are C-stacks F, G, and also two kinds of morphisms, 1-morphisms φ, ψ : F → G between C-stacks, and 2-morphisms A : φ → ψ between 1-morphisms. Definition 2.1. Let F be a C-stack. Write F(C) for the set of 2-isomorphism classes [x] of 1-morphisms x : Spec C → F. Elements of F(C) are called C-points of F. If φ : F → G is a 1-morphism then composition with φ induces a map of sets φ∗ : F(C) → G(C). For a 1-morphism x : Spec C → F, the stabilizer group IsoF(x) is the group of 2-morphisms A : x → x. When F is an Artin C-stack, IsoF(x) is an algebraic C-group, which we assume is affine. If φ : F → G is a 1-morphism,   composition induces a morphism of C-groups φ∗ : IsoF([x]) → IsoG φ∗ ([x]) , for [x] ∈ F(C). We discuss constructible functions on C-stacks, following [9]. Definition  2.2. Let F be an Artin C-stack. We call C ⊆ F(C) constructible if C = i∈I Fi (C), where {Fi : i ∈ I} is a finite collection of finite type Artin C-substacks Fi of F. We call S ⊆ F(C) locally constructible if S ∩ C is constructible for all constructible C ⊆ F(C). A function f : F(C) → Q is called constructible if f (F(C)) is finite and f −1 (c) is a constructible set in F(C) for each c ∈ f (F(C)) \ {0}. A function f : F(C) → Q is called locally constructible if f · δC is constructible for all constructible C ⊆ F(C), where δC is the characteristic function of C. Write CF(F) and LCF(F) for the

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Q-vector spaces of Q-valued constructible and locally constructible functions on F. Following [9, §4–§5] we define pushforwards and pullbacks of constructible functions along 1-morphisms. Definition 2.3. Let F, G be Artin C-stacks and φ : F → G a representable 1-morphism. For f ∈ CF(F), define CFstk (φ)f : G(C) → Q by   CFstk (φ)f (y) = χ F ×φ,G,y Spec C, πF∗ (f ) for y ∈ G(C), where F×φ,G,y Spec C is a C-scheme (or algebraic space) as φ is representable, and χ(· · · ) is the Euler characteristic of this C-scheme weighted by πF∗ (f ). Then CFstk (φ) : CF(F) → CF(G) is a Q-linear map called the stack pushforward. Let θ : F → G be a finite type 1-morphism. The pullback θ∗ : CF(G) → CF(F) is given by θ∗ (f ) = f ◦ θ∗ . It is a Q-linear map. Here [9, §4–§5] are some properties of these. Theorem 2.4. Let E, F, G, H be Artin C-stacks and β : F → G, γ : G → H be 1-morphisms. Then CFstk (γ ◦ β) = CFstk (γ) ◦ CFstk (β) : CF(F) → CF(H),

(2)

(γ ◦ β)∗ = β ∗ ◦ γ ∗ : CF(H) → CF(F),

(3)

supposing β, γ representable in (2), and of finite type in (3). If E 

η

ψ

θ

F

/G

φ



/H

is a Cartesian square with η, φ representable and θ, ψ of finite type, then the following commutes:

CF(E) O

/ CF(G) O

CFstk (η)

ψ∗

θ∗

CF(F)

CFstk (φ)

/ CF(H).

2.2. Stack functions. Stack functions are a universal generalization of constructible functions introduced in [10, §3]. Here [10, Def. 3.1] is the basic definition. Definition 2.5. Let F be an Artin C-stack. Consider pairs (R, ρ), where R is a finite type Artin C-stack and ρ : R → F is a representable 1-morphism. We call two pairs (R, ρ), (R , ρ ) equivalent if there exists a 1-isomorphism ι : R → R such that ρ ◦ι and ρ are 2-isomorphic 1-morphisms R → F. Write [(R, ρ)] for the equivalence class of (R, ρ). If (R, ρ) is such a pair and S is a closed C-substack of R then (S, ρ|S), (R \ S, ρ|R\S) are pairs of the same kind. Define SF(F) to be the Q-vector space generated by equivalence classes [(R, ρ)] as above, with for each closed C-substack S of R a relation (4)

[(R, ρ)] = [(S, ρ|S)] + [(R \ S, ρ|R\S)].

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Elements of SF(F) will be called stack functions. We relate CF(F) and SF(F). Definition 2.6. Let n F be an Artin C-stack and C ⊆ F(C) be constructible. Then C = i=1 Ri (C), for R1 , . . . , Rn finite type C-substacks of F. Let ρi : Ri → F be the inclusion 1-morphism. Then [(Ri , ρi )] ∈ SF(F). Define δ¯C = ni=1 [(Ri , ρi )] ∈ SF(F). We think of this as the analogue of the characteristic function ιF : CF(F) →  δC ∈ CF(F) of¯ C. Define a Q-linear map stk SF(F) by ιF(f ) = 0=c∈f (F(C)) c · δf −1 (c) . Define Q-linear πF : SF(F) → CF(F) by n  n stk πFstk i=1 ci [(Ri , ρi )] = i=1 ci CF (ρi )1Ri , where 1Ri is the function 1 ∈ CF(Ri ). Then πFstk ◦ιF is the identity on CF(F). The operations on constructible functions in §2.1 extend to stack functions. Definition 2.7. Let φ : F→G be a representable 1-morphism of Artin C-stacks. Define the pushforward φ∗ : SF(F)→SF(G) by  m (5) φ∗ : m i=1 ci [(Ri , ρi )] −→ i=1 ci [(Ri , φ ◦ ρi )]. Let φ : F→G be of finite type. Define the pullback φ∗ : SF(G)→SF(F) by  m (6) φ∗ : m i=1 ci [(Ri , ρi )] −→ i=1 ci [(Ri ×ρi ,G,φ F, πF)]. The tensor product ⊗ : SF(F) × SF(G) → SF(F × G) is  n   m (7) i=1 ci [(Ri , ρi )] ⊗ j=1 dj [(Sj , σj )] = i,j ci dj [(Ri ×Sj , ρi ×σj )]. Here [10, Th. 3.5] is the analogue of Theorem 2.4. Theorem 2.8. Let E, F, G, H be Artin C-stacks and β : F → G, γ : G → H be 1-morphisms. Then (γ ◦β)∗ =γ∗ ◦β∗ : SF(F)→SF(H),

(γ ◦β)∗=β ∗ ◦γ ∗ : SF(H)→SF(F),

for β, γ representable in the first equation, and of finite type in the second. If E

/G η

ψ  θ φ  /H F

is a Cartesian square with θ, ψ of finite type and η, φ representable, then the following commutes:

SF(E) O

θ∗

SF(F)

η∗

/ SF(G) O ∗

φ∗

/ SF(H).

ψ

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In [10, §3] we relate pushforwards and pullbacks of stack and constructible functions using ιF, πFstk . Theorem 2.9. Let φ : F → G be a 1-morphism of Artin C-stacks. Then (a) φ∗ ◦ιG =ιF ◦φ∗ : CF(G)→SF(F) if φ is of finite type; stk ◦ φ = CFstk (φ) ◦ π stk : SF(F) → CF(G) if φ is representable; (b) πG ∗ F and stk : SF(G) → CF(F) if φ is of finite type. (c) πFstk ◦ φ∗ = φ∗ ◦ πG We define some projections Πvi n : SF(F) → SF(F), [10, §5]. Definition 2.10. For any Artin C-stack F we will define linear maps Πvi : n SF(F) → SF(F) for n  0. Now SF(F) is generated by [(R, ρ)] with R 1-isomorphic to a quotient [X/G], for X a quasiprojective C-variety and G a special algebraic C-group, with maximal torus T G . Let S(T G ) be the set of subsets of T G defined by Boolean operations upon closed C-subgroups L of T G . Define a measure dμn : S(T G ) → Z to be additive upon disjoint unions of sets in S(T G ), and to satisfy dμn (L) = 1 if dim L = n and dμn (L) = 0 if dim L = 0 for all algebraic C-subgroups L of T G . Define   Πvi n [(R, ρ)] =  (8) 

|{w ∈ W (G, T G ) : w · t = t}|  {t} [X /CG ({t})], ρ ◦ ι{t} dμn . G |W (G, T )| t∈T G Here X {t} is the subscheme of X fixed by t, and CG ({t}) is the centralizer of t in G, and ι{t} : [X {t} /CG ({t})] → [X/G] is the obvious 1-morphism. The integrand in (8), regarded as a function of t ∈ T G , is a constructible function taking only finitely many values. The level sets of the function lie in S(T G ), so they are measurable w.r.t. dμn , and the integral is well-defined. In [10, §5] we show (8) induces a unique linear map Πvi n : SF(F) → SF(F). Here [10, §5] are some properties of the Πvi n. Theorem 2.11. In the situation above, we have: 2 vi vi vi vi (i) (Πvi n ) = Πn , so that Πn is a projection, and Πm ◦ Πn = 0 for m = n.  (ii) For all f ∈ SF(F) we have f = n0 Πvi n (f ), where the sum makes vi sense as Πn (f ) = 0 for n  0. (iii) If φ : F → G is a 1-morphism of Artin C-stacks then Πvi n ◦ φ∗ = vi φ∗ ◦ Πn : SF(F) → SF(G). n vi vi (iv) If f ∈ SF(F), g ∈ SF(G) then Πvi n (f ⊗g) = m=0 Πm (f )⊗Πn−m (g). Roughly speaking, Πvi n projects [(R, ρ)] ∈ SF(F) to [(Rn , ρ)], where Rn is the substack of points r ∈ R(C) whose stabilizer groups IsoR(r) have rank n.

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¯ 2.3. Stack function spaces SF(F, χ, Q). We will also need another ¯ family of spaces SF(F, χ, Q), from [10, §5–§6]. Definition 2.12. Let F be an Artin C-stack. Consider pairs (R, ρ), where R is a finite type Artin C-stack and ρ : R → F is a representable ¯ 1-morphism, with equivalence as in Definition 2.5. Define SF(F, χ, Q) to be the Q-vector space generated by equivalence classes [(R, ρ)], with the following relations: (i) Given [(R, ρ)] as above and S a closed C-substack of R we have [(R, ρ)] = [(S, ρ|S)] + [(R \ S, ρ|R\S)], as in (4). (ii) Let R be a finite type Artin C-stack, U a quasiprojective C-variety, πR : R×U → R the natural projection, and ρ : R → F a 1-morphism. Then [(R × U, ρ ◦ πR)] = χ([U ])[(R, ρ)]. Here χ(U ) ∈ Z is the Euler characteristic of U . It is a motivic invariant of C-schemes, that is, χ(U ) = χ(V )+χ(U \V ) for V ⊂ U closed. (iii) Given [(R, ρ)] as above and a 1-isomorphism R ∼ = [X/G] for X a quasiprojective C-variety and G a very special algebraic C-group acting on X with maximal torus T G , we have [(R, ρ)] =



Q∈Q(G,T G ) F (G, T



[X/Q], ρ ◦ ιQ ,



G , Q)

where ιQ :[X/Q]→R ∼ =[X/G] is the natural projection 1-morphism. Here Q(G, T G ) is a certain finite set of C-subgroups of T G , and F (G, T G , ¯ χ,Q : Q) ∈ Q are a system of rational coefficients defined in [10, §6.2]. Define Π F   ¯ ¯ χ,Q : c [(R , ρ )] →

c [(R , ρ )]. Define SF(F) → SF(F, χ, Q) by Π i i i i i∈I i i∈I i F pushforwards φ∗ , pullbacks φ∗ , tensor products ⊗ and projections Πvi n on ¯ the spaces SF(∗, χ, Q) as in §2.2. The important point is that (5)–(8) are ¯ compatible with the relations defining SF(∗, χ, Q), or they would not be well¯ defined. The analogues of Theorems 2.8, 2.9 and 2.11 hold for SF(∗, χ, Q). Here [10, §5–§6] is a useful way to represent these spaces. It means that ¯ by working in SF(F, χ, Q), we can treat all stabilizer groups as if they are abelian. ¯ Proposition 2.13. SF(F, χ, Q) is spanned over Q by [(U ×[Spec C/ T ], ρ)], for U a quasiprojective C-variety and T an algebraic C-group isomorphic to Gkm × K for k  0 and K finite abelian. Moreover   Πvi n [(U × [Spec C/T ], ρ)] =

[(U × [Spec C/T ], ρ)], 0,

dim T = n, otherwise.

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3. Background material on Calabi–Yau 3-folds We now summarize some facts on Donaldson–Thomas invariants and other sheaf-counting invariants on Calabi–Yau 3-folds prior to our book [16]. Sections 3.1–3.3 review material from the author’s series of papers [11–14], and §3.4 explains results on Donaldson–Thomas theory from Thomas [33] and Behrend [1]. For simplicity we restrict to Calabi–Yau 3-folds and to the field K = C, although much of [1, 11–14, 33] works in greater generality. 3.1. The Ringel–Hall algebra of a Calabi–Yau 3-fold. We will use the following notation for the rest of the paper. Definition 3.1. A Calabi–Yau 3-fold is a smooth projective 3-fold X over C, with trivial canonical bundle KX . In §4 we will also assume that H 1 (OX ) = 0. The Grothendieck group K0 (X) of coh(X) is the abelian group generated by all isomorphism classes [E] of objects E in coh(X), with the relations [E] + [G] = [F ] for each short exact sequence 0 → E → F → G → 0. The Euler form χ ¯ : K0 (X) × K0 (X) → Z is a biadditive map satisfying    (9) χ ¯ [E], [F ] = i0 (−1)i dim Exti (E, F ) for all E, F ∈ coh(X). As X is a Calabi–Yau 3-fold, Serre duality gives Exti (F, E) ∼ = Ext3−i (E, F )∗ , so dim Exti (F, E) = dim Ext3−i (E, F ) for all E, F ∈ coh(X). Therefore χ ¯ is also given by     χ ¯ [E], [F ] = dim Hom(E, F ) − dim Ext1 (E, F ) (10)   − dim Hom(F, E) − dim Ext1 (F, E) . Thus the Euler form χ ¯ on K0 (X) is antisymmetric. The numerical Grothendieck group K(X) is the quotient of K0 (X) by the kernel of χ. ¯ Then χ ¯ on K0 (X) descends to a nondegenerate, biadditive Euler form χ ¯ : K(X) × K(X) → Z. Define the ‘positive cone’ C(X) in K(X) to be   C(X) = [E] ∈ K(X) : 0 ∼  E ∈ coh(X) ⊂ K(X). = Write M for the moduli stack of objects in coh(X). It is an Artin C-stack, locally of finite type. Points of M(C) correspond to isomorphism classes [E] of objects E in coh(X), and the stabilizer group IsoM([E]) in M is isomorphic as an algebraic C-group to the automorphism group Aut(E). For α ∈ C(X), write Mα for the substack of objects E ∈ coh(X) in class α in K(X). It is an open and closed C-substack of M. Write Exact for the moduli stack of short exact sequences 0 → E1 → E2 → E3 → 0 in coh(X). It is an Artin C-stack, locally of finite type. For j = 1, 2, 3 write πj : Exact → M for the 1-morphism projecting 0 → E1 → E2 → E3 → 0 to Ej . Then π2 is representable, and π1 × π3 : Exact → M × M is of finite type. In [12] we define Ringel–Hall algebras, using stack functions.

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¯ Definition 3.2. Define bilinear operations ∗ on SF(M), SF(M, χ, Q) by   f ∗ g = (π2 )∗ (π1 × π3 )∗ (f ⊗ g) , using pushforwards, pullbacks and tensor products in Definition 2.7. They are well-defined as π2 is representable, and π1 × π3 is of finite type. By [12, Th. 5.2], whose proof uses Theorem 2.8, this * is associative, and makes ¯ SF(M), SF(M, χ, Q) into noncommutative Q-algebras, called Ringel–Hall algebras, with identity δ¯[0] , where [0] ∈ M is the zero object. The projection ¯ χ,Q : SF(M) → SF(M, ¯ Π χ, Q) is an algebra morphism. M As these algebras are inconveniently large for some purposes, in [12, ¯ al (M, χ, Q) using the algebra Def. 5.5] we define subalgebras SFal (M), SF structure on stabilizer groups in M. Suppose [(R, ρ)] is a generator of SF(M). Let r ∈ R(C) with ρ∗ (r) = [E] ∈ M(C), for some E ∈ coh(X). Then ρ induces a morphism of stabilizer C-groups ρ∗ : IsoR(r) → IsoM([E]) ∼ = Aut(E). As ρ is representable this is injective, and induces an isomorphism of IsoR(r) with a C-subgroup of Aut(E). Now Aut(E) = End(E)× is the C-group of invertible elements in a finite-dimensional C-algebra End(E) = Hom(E, E). We say that [(R, ρ)] has algebra if whenever r ∈  stabilizers  R(C) with ρ∗ (r) = [E], the C-subgroup ρ∗ IsoR(r) in Aut(E) is the Cgroup A× of invertible elements in a C-subalgebra A in End(E). Write ¯ al (M, χ, Q) for the subspaces of SF(M), SF(M, ¯ SFal (M), SF χ, Q) spanned over Q by [(R, ρ)] with algebra stabilizers. Then [12, Prop. 5.7] shows that ¯ al (M, χ, Q) are subalgebras of the Ringel–Hall algebras SF(M), SFal (M), SF ¯ SF(M, χ, Q). ¯ al (M, χ, Q) are closed under Now [12, Cor. 5.10] shows that SFal (M), SF vi ¯ the operators Πn on SF(M), SF(M, χ, Q) defined in §2.2. In [12, Def. 5.14] ¯ ind we define SFind al (M), SFal (M, χ, Q) to be the subspaces of f in SFal (M) ¯ ind (M, χ, Q) ¯ al (M, χ, Q) with Πvi (f ) = f . We think of SFind (M), SF and SF 1 al al as stack functions ‘supported on virtual indecomposables’. ¯ ind In [12, Th. 5.18] we show that SFind al (M), SFal (M, χ, Q) are closed ¯ al (M, χ, Q). Thus, under the Lie bracket [f, g] = f ∗ g − g ∗ f on SFal (M), SF ind ind ¯ ¯ al (M, χ, Q). SFal (M), SFal (M, χ, Q) are Lie subalgebras of SFal (M), SF As in [12, Cor. 5.11], Proposition 2.13 simplifies to give: ¯ al (M, χ, Q) is spanned over Q by elements of the Proposition 3.3. SF k form [(U × [Spec C/Gm ], ρ)] with algebra stabilizers, for U a quasiprojec¯ ind (M, χ, Q) is spanned over Q by [(U × tive C-variety and k  0. Also SF al [Spec C/Gm ], ρ)] with algebra stabilizers, for U a quasiprojective C-variety. All the above except (10) works for X an arbitrary smooth projective Cscheme, but our next result uses the Calabi–Yau 3-fold assumption on X in an essential way. We follow [12, §6.5–§6.6], but use the notation of [16, §3.4].

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Definition 3.4. Define an explicit Lie algebra L(X) over Q to be the Q-vector space with basis of symbols λα for α ∈ K(X), with Lie bracket (11)

[λα , λβ ] = χ(α, ¯ β)λα+β

for α, β ∈ K(X). As χ ¯ is antisymmetric, (11) satisfies the Jacobi identity and makes L(X) into an infinite-dimensional Lie algebra over Q. ¯ ind (M, χ, Q) → L(X) by Define a Q-linear map Ψχ,Q : SF al  χ,Q (12) Ψ (f ) = α∈K(X) γ α λα , ¯ ind (M, χ, Q) is where γ α ∈ Q is defined as follows. Proposition 3.3 says SF al spanned by elements [(U × [Spec C/Gm ], ρ)]. We may write  (13) f |Mα = ni=1 δi [(Ui × [Spec C/Gm ], ρi )], where δi ∈ Q and Ui is a quasiprojective C-variety. We set  γ α = ni=1 δi χ(Ui ). This is independent of the choices in (13). Now define Ψ : SFind al (M) → L(X) χ,Q χ,Q ¯ ◦ ΠM . by Ψ = Ψ In [12, Th. 6.12], using equation (10), we prove: χ,Q : SF ¯ ind (M, χ, Q) → L(X) Theorem 3.5. Ψ : SFind al (M) → L(X) and Ψ al are Lie algebra morphisms.

3.2. Stability conditions on coh(X) and invariants J α(τ ). Next we discuss material in [13] on stability conditions. We continue to use the notation of §3.1, with X a Calabi–Yau 3-fold. Definition 3.6. Suppose (T, ) is a totally ordered set, and τ : C(X) → T a map. We call (τ, T, ) a stability condition on coh(X) if whenever α, β, γ ∈ C(X) with β = α + γ then either τ (α) < τ (β) < τ (γ), or τ (α) > τ (β) > τ (γ), or τ (α) = τ (β) = τ (γ). We call (τ, T, ) a weak stability condition on coh(X) if whenever α, β, γ ∈ C(X) with β = α + γ then either τ (α)  τ (β)  τ (γ), or τ (α)  τ (β)  τ (γ). For such (τ, T, ), we call a nonzero sheaf E in coh(X) (i) τ -stable if for all S ⊂ E with S ∼  0, E we have τ ([S]) < τ ([E/S]); = and (ii) τ -semistable if for all S ⊂ E with S ∼  0, E we have τ ([S])  = τ ([E/S]). For α ∈ C(X), write Mαss (τ ), Mαst (τ ) for the moduli stacks of τ -(semi) stable E ∈ A with class [E] = α in K(X). They are open C-substacks of Mα . We call (τ, T, ) permissible if: (a) coh(X) is τ -artinian, that is, there exist no infinite chains of subobjects · · ·E2 E1 E0 = X in A and τ ([En+1 ])τ ([En /En+1 ]) for all n; and (b) Mαss (τ ) is a finite type substack of Mα for all α ∈ C(X).

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Here are two important examples: Example 3.7. Define G to be the set of monic rational polynomials in t of degree at most 3:   G = p(t) = td + ad−1 td−1 + · · · + a0 : d = 0, 1, 2, 3, a0 , . . . , ad−1 ∈ Q . Define a total order ‘’ on G by p  p for p, p ∈ G if either (a) deg p > deg p , or (b) deg p = deg p and p(t)  p (t) for all t  0. We write p < q if p  q and p = q. Fix a very ample line bundle OX (1) on X. For E ∈ coh(X), the Hilbert 0 polynomial PE is the unique polynomial in Q[t]  such that PE  (n) = dim H (E(n)) for all n  0. Equivalently, PE (n) = χ ¯ [OX (−n)], [E] for all n ∈ Z. Thus, PE depends only on the class α ∈ K(X) of E, and we may write Pα instead of PE . Define τ : C(X) → G by τ (α) = Pα /rα , where Pα is the Hilbert polynomial of α, and rα is the (positive) leading coefficient of Pα . Then (τ, G, ) is a permissible stability condition on coh(X) [13, Ex. 4.16], called Gieseker stability. Gieseker stability is studied in [8, §1.2]. Write Mαss (τ ), Mαst (τ ) for the coarse moduli schemes of τ -(semi)stable sheaves E with class [E] = α in K(X). By [8, Th. 4.3.4], Mαss (τ ) is a projective C-scheme whose C-points correspond to S-equivalence classes of Gieseker semistable sheaves in class α, and Mαst (τ ) is an open C-subscheme whose C-points correspond to isomorphism classes of Gieseker stable sheaves in class α. Example 3.8. In the situation of Example 3.7, define   M = p(t) = td + ad−1 td−1 : d = 0, 1, 2, 3, ad−1 ∈ Q, a−1 = 0 ⊂ G and restrict the total order  on G to M . Define μ : C(X) → M by μ(α) = td + ad−1 td−1 when τ (α) = Pα /rα = td + ad−1 td−1 + · · · + a0 , that is, μ(α) is the truncation of the polynomial τ (α) in Example 3.7 at its second term. Then as in [13, Ex. 4.17], (μ, M, ) is a permissible weak stability condition on coh(X). It is called μ-stability, and is studied in [8, §1.6]. α (τ ),  In [13, §8] we define interesting stack functions δ¯ss ¯α (τ ) in SFal (M).

Definition 3.9. Let (τ, T, ) be a permissible weak stability condition α (τ ) = δ ¯Mα (τ ) in SFal (M) for α ∈ C(X). on coh(X). Define stack functions δ¯ss ss α (τ ) is the characteristic function, in the sense of Definition 2.6, of That is, δ¯ss the moduli substack Mαss (τ ) of τ -semistable sheaves in M. In [13, Def. 8.1] we define elements ¯α (τ ) in SFal (M) by  (−1)n−1 ¯α1 α2 αn (14) ¯α (τ ) = (τ ) ∗ · · · ∗ δ¯ss (τ ), δss (τ ) ∗ δ¯ss n n1, α1 ,...,αn ∈C(X): α1 +···+αn =α, τ (αi )=τ (α), all i

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where ∗ is the Ringel–Hall multiplication in SFal (M). Then [13, Th. 8.2] proves α (15) δ¯ss (τ ) =



1 α1 ¯ (τ ) ∗ ¯α2 (τ ) ∗ · · · ∗ ¯αn (τ ). n!

n1, α1 ,...,αn ∈C(X): α1 +···+αn =α, τ (αi )=τ (α), all i

There are only finitely many nonzero terms in (14)–(15). Equations (14) and (15) are inverse, so that knowing the ¯α (τ ) is equivα (τ ). If Mα (τ ) = Mα (τ ) then  α (τ ). The alent to knowing the δ¯ss ¯α (τ ) = δ¯ss ss st α α α ¯ difference between ¯ (τ ) and δss (τ ) is that ¯ (τ ) ‘counts’ strictly semistable sheaves in a special, complicated way. Here [13, Th. 8.7] is an important α (τ ). The proof is highly property of the ¯α (τ ), which does not hold for the δ¯ss nontrivial, using the full power of the configurations formalism of [11–14]. Theorem 3.10. ¯α (τ ) lies in the Lie subalgebra SFind al (M) in SFal (M). In [14, §6.6] we define invariants J α (τ ) ∈ Q for all α ∈ C(X) by (16)

  Ψ ¯α (τ ) = J α (τ )λα .

This is valid by Theorem 3.10. These J α (τ ) are rational numbers ‘counting’ τ -semistable sheaves E in class α. When Mαss (τ ) = Mαst (τ ) we have (17)

  J α (τ ) = χ Mαst (τ ) ,

that is, J α (τ ) is the Euler characteristic of the moduli space Mαst (τ ). In the notation of §3.4, this is not weighted by the Behrend function νMαst (τ ) , and so is not the Donaldson–Thomas invariant DT α (τ ). As in [16, Ex. 6.9], the J α (τ ) are in general not unchanged under deformations of X.

3.3. Changing stability conditions and algebra identities. In α (τ ),  [14] we prove transformation laws for the δ¯ss ¯α (τ ) under change of stability condition. These involve combinatorial coefficients S(∗; τ, τ˜) ∈ Z and U (∗; τ, τ˜) ∈ Q defined in [14, §4.1]. Definition 3.11. Let (τ, T, ),(˜ τ , T˜, ) be weak stability conditions on coh(X). Let n  1 and α1 , . . . , αn ∈ C(X). If for all i = 1, . . . , n − 1 we have either (a) τ (αi )  τ (αi+1 ) and τ˜(α1 + · · · + αi ) > τ˜(αi+1 + · · · + αn ) or (b) τ (αi ) > τ (αi+1 ) and τ˜(α1 + · · · + αi )  τ˜(αi+1 + · · · + αn ),

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then define S(α1 , . . . , αn ; τ, τ˜) = (−1)r , where r is the number of i = 1, . . . , n − 1 satisfying (a). Otherwise define S(α1 , . . . , αn ; τ, τ˜) = 0. Now define U (α1 , . . . , αn ; τ, τ˜) =  (−1)l−1 l S(βbi−1 +1 , βbi−1 +2 , . . . , βbi ; τ, τ˜) · i=1 l m  1lmn, 0=a0 0. Harmonic maps behave like harmonic one forms. The Bochner formula can be generalized to Δ |df |2 =

1 |∇df |2 + RcM (df, df ) − RmN (df, df, df, df ) . 2

Using this formula, Eells-Sampson [41] showed that if N is negatively curved, but not necessary compact, then any homotopy class of map f : M → N admits a unique harmonic map representative, which also minimizes the harmonic energy. If, moreover RcM > 0, then there is no nontrivial harmonic map at all. These Bochner type arguments for harmonic maps were refined by Siu, Yau and others to prove various superrigidity type theorems for locally symmetric spaces. We note that holomorphic maps between K¨ ahler manifolds are harmonic. When M is one dimensional, i.e. a circle S 1 , then f (M ) is a geodesic in N and the parametrization f has constant speed. In general, harmonic maps and minimal submanifolds are quite different objects as the former one depends on the choice of gM . Sigma model. When M is two dimensional, then the harmonic energy E (f ) is unchanged if we scale gM to eu gM with u any smooth function on M , i.e. E (f ) depends only on the conformal structure on M . The sigma model, or σ-model, in physics considers E (f ) as the action functional on the space of maps, or bosonic fields, M ap (M, N ) and this gives a conformal field theory. When we couple it with fermionic fields, i.e. spinors of M twisted by f ∗ TN , there is a supersymmetry (abbrev. N = 1 SUSY) between bosons and fermions. When N is a K¨ahler (resp. hyperk¨ ahler) manifold, then we can add more fermionic fields and obtain a N = 2 (resp. N = 4) SUSY σ-model. In the K¨ ahler case, we have     2 2 2 ¯  . ∂f E (f ) = |df | = |∂f | + M

On the other hand,



∗

M



2



|∂f | −

f ω= M

M

M

M

 2 ¯  . ∂f

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Thus



∗

E (f ) =



M



 2 ¯  . ∂f

f ω+2 M

Note that M f ∗ ω = [ω] (f (M )) depends only on the homotopy class of the map f . Thus in afixed homotopy class, holomorphic maps have the least harmonic energy M f ∗ ω. This continues to hold true for any symplectic manifold M with a compatible metric. Similar situations also happen for gauge theory over an oriented Riemannian four manifold (section 4.1). Witten [134] studied twisted versions of N = 2 σ-models and defined two different models of TFT, called the A-model and B-model. The A-model describe the quantum symplectic geometry of N and Gromov-Witten invariants (section 5.4) are partition functions in this model. The B-model describe the complex geometry of N and there are no instanton effects. When N is a Calabi-Yau manifold, the mirror symmetry conjecture says that there should be another Calabi-Yau manifold N  such that the A-model and the B-model on N and N  got interchanged (section 6.3).

4. Oriented four manifolds Oriented manifolds of dimension two are special as they are one dimensional complex manifolds, i.e. Riemann surfaces. This is because SO (2) ∼ = U (1). Similarly, oriented manifolds of dimension four are one dimensional quaternionic manifolds because SO (4) = Sp (1) Sp (1). As Λ2 R4 ∼ = sp (1) ⊕ sp (1) ∼ = so (4), given any oriented Riemannian four manifold M , we have a decomposition of its two forms into self-dual (SD) and anti-self-dual (ASD) components, Λ2 (M ) = Λ2+ (M ) ⊕ Λ2− (M ) . Indeed this corresponds to the eigenspace decomposition for the Hodge star operator ∗ on the space of two forms. There is a corresponding decomposition for harmonic two forms, via Hodge theory we have 2 2 H 2 (M ) = H+ (M ) ⊕ H− (M ) .

This decomposition depends only on the conformal structure of M . Note that   φ∧φ=± |φ|2 M

M

2 (M ). Thus the intersection product q 2 for φ ∈ H± M on H (M ) is positive 2 (M ) (resp. H 2 (M )) The signature τ (M ) = (resp. negative) definite on H+ − 2 2 dim H+ (M ) − dim H− (M ) is a topological invariant of M and it equals to the characteristic number p1 (M ) /3 by the Hirzebruch signature formula.

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4.1. Gauge theory in dimension four. Donaldson theory. First we note that the Yang-Mills energy functional  |FA |2 M

on an oriented Riemannian four manifold (M, g) depends only on the conformal class of the metric g. As curvature tensors are matrix-valued two forms, for Hermitian connections DA on any complex vector bundle E over an oriented Riemannian four manifold M , we have FA = FA+ + FA− with

1 (FA ± ∗FA ) ∈ Ω2± (M, ad (E)) , 2 the (anti-)self-dual (abbrev. SD/ASD) components of FA . Notice that     + 2  − 2 2   F  |FA | = FA + A FA± :=

M

M

M

and when c1 (E) = 0, we also have  −1 T r F A ∧ FA c2 (E) = 2 8π M    + 2  − 2 −1 1   F  . = 2 FA + 2 8π M 8π M A Thus the Yang-Mills energy functional satisfies,    + 2 2 2 F  . |FA | = 8π c2 (E) + 2 A M

M

Therefore, when c2 (E) ≥ 0, the absolute minimal for the Yang-Mills energy is realized by ASD connections, i.e. FA+ = 0, also called instantons. For a line bundle over R3,1 , the ASD-equation FA+ = 0 is the Maxwell equation for electromagnetism. In general, when E is a line bundle, this linear equation is well-understood via Hodge theory. The situation is much more complicated when the rank r of E is higher as the moduli space of solutions is not compact. When r equals two, using foundational works of Uhlenbeck, Taubes and others, Donaldson [38] studied the intersection theory on this moduli space and defined new and powerful invariants for the differentiable structures on four manifolds. When M is a K¨ ahler surface, then the instanton equation is the same as the Hermitian Yang-Mills equation with zero slope. Donaldson [36] showed that this latter equation is solvable precisely when E is a polystable holomorphic vector bundle over M . In higher dimensions, this result is proved by Uhlenbeck-Yau [125].

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Witten [132] described the Donaldson invariants as partition functions of a topological field theory (TFT) with N = 2 supersymmetries. This TFT links the 4-dimensional Donaldson theory and the 3-dimensional ChernSimons Floer theory and 2-dimensional gauge theory as explained by Atiyah [3]. Seiberg-Witten theory. After much work from both mathematicians and physicists, Witten [132] found that the Donaldson theory should be equivalent to a much simpler theory, the so-called Seiberg-Witten (SW) theory. In this theory, DA is a connection of a line bundle L. The SW-equation is FA+ = σ (φ) DA φ = 0. Here φ ∈ SL+ is a positive spinor field twisted by L. The operator DA is the    ∗ is a natural isomorphism. twisted Dirac operator and σ : ad SL+ → Λ2+ TM The Weitzenbock formula R ∗ ∗ DA DA φ = D A DA φ + φ + FA+ · φ 4 implies that the SW-equation has no nontrivial solution if M has positive scalar curvature. It can also be used to show that the moduli space of solutions to the SW-equation is compact. This makes the SW theory much simpler than the Donaldson theory. The compactness of the moduli space of SW equation still holds true even if we allow the spinor field φ to have eigencomponents by Leung-Xu [100] using an eigenvalue estimate by VafaWitten [126]. SW theory has many important applications in different branches of four dimensional theory, including (i) differential topology, for instance the Thom conjecture [78], (ii) K¨ ahler geometry, (iii) Riemannian geometry, for instance the study of Einstein metric [80][87][90] and (iv) symplectic geometry, for instance SW=GW by Taubes and classification of symplectic four manifolds with b+ = 1 [104][118]. Vafa-Witten theory. Vafa and Witten also studied a N = 4 TFT on four manifolds in [127] and derived the Vafa-Witten equation FA+ + [B, B] + [C, B] = 0

∗ B + DA C = 0, DA

where B ∈ Ω2+ (M, ad (E)) and C ∈ Ω0 (M, ad (E)). The S-duality in physics predicts that the generating function for the Euler characteristics of these moduli spaces is a modular form. More recently, Vafa-Witten introduced another N = 4 TFT [127] and conjectured that S-duality for this theory would give a physical explanation of the Langlands program. 4.2. Riemannian geometry in dimension four. Recall that the ∗ . With Riemannian curvature tensor Rm is a self-adjoint operator on Λ2 TM

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respect to the decomposition of two forms into SD and ASD components, we have +

W + R/12 Rc0 Rm = . Rc0 W − + R/12 Here R is the scalar curvature, Rc0 is the tracefree part of the Ricci curvature and W = W + + W − is the Weyl curvature (section 12) and it depends only on the conformal structure on M. When W = 0, M is conformally flat. M is called an ASD (resp. SD) manifold if W + = 0 (resp. W − = 0). Note that if we reverse the orientation of M , then W + and W − get interchanged. The 2 (M ) − dim H 2 (M ) can be expressed as signature τ (M ) = dim H+ −   + 2  − 2 1 W  − W  . τ (M ) = 12π 2 M When M is an Einstein manifold, the Euler characteristic χ (M ) can be computed as follows,   + 2  − 2 1 W  + W  + R2 /24. χ (M ) = 2 8π M In particular, we obtain the Hitchin inequality 2 τ (M ) ≤ χ (M ) , 3 where the equality sign holds if and only if W − = R = 0. When M is K¨ahler, this inequality can be rewritten as c21 (M ) ≤ 4c2 (M ). In the Kahler-Einstein case, it can be sharpened to c21 (M ) ≤ 3c2 (M ) , the Miyaoka-Yau inequality. The equality sign holds if and only if the universal cover of M is the complex ball BC2 = SU (2, 1) /U (2). Note that Yau showed that every K¨ ahler manifold with c1 (M ) = 0 or < 0 admits a K¨ahlerEinstein metric. Thus this inequality is an important tool in the study of the classification problem of complex algebraic surfaces. Lebrun [80] generalized the Miyaoka-Yau inequality to the real case with the help of the Seiberg-Witten invariants. If the Seiberg-Witten invariant is nonzero with respect to the reversed orientation on a K¨ahler-Einstein surface, then the index τ (M ) is nonnegative, and it is zero if and only if the universal cover of 1 = (SU (1, 1) /U (1))2 . M is covered by the product of complex disks BC1 ×BC 2 1 1 Recall that BC and BC × BC are the only Hermitian symmetric spaces of noncompact type in this dimension [87]. There is also a non-Hermitian symmetric space, namely the hyperbolic ball B 4 , for which one can also find a characterization using Chern number inequality by using the non-Abelian Seiberg-Witten equation [90].

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Twistor transform. We can associate to every oriented Riemannian four manifold M a 6-dimensional manifold Z, called the twistor space of M , which is equipped with an almost complex structure JZ . It is the total space of a fiber bundle π S2 → Z → M with fiber over x ∈ M being the set of all linear orthogonal complex structures on Tx M , which is a copy of SO (4) /U (2)  S 2 . For instance, when M = S 4 , we have Z = CP3 and the fibration is given by sending a complex line in C4 to the quaternionic line in H2 that it spans as S 4 = HP1 . The necessary and sufficient condition for (Z, JZ ) to be a complex manifold is W + = 0 on M . Every K¨ ahler surface with zero scalar curvature is an ASD 4-manifold. Taubes [117] showed that after taking connected sum with sufficiently many copies of CP2 , every M admits an ASD metric. Atiyah, Hitchin and Singer [4] showed that if M is an ASD 4-manifold, then the conformal geometry of M can be described in terms of the complex geometry of Z, called the twistor transformation. We will discuss a particular interesting class of ASD four manifolds, namely the K3 surfaces, in section 6.4. 5. K¨ ahler geometry A K¨ ahler manifold is a Riemannian manifold M with a compatible complex structure. Recall that M is a complex manifold if it has a covering by complex coordinate charts with holomorphic transition functions. In the linear setting, a complex vector space is equivalent to a real vector space V with a R-linear homomorphism J : V → V satisfying J 2 = −1. It is compatible with an inner product g on V if g is Hermitian, i.e. for any u, v ∈ V we have g (Ju, Jv) = g (u, v) . In this case, we have a non-degenerate two form ω ∈ Λ2 V ∗ defined by ω (u, v) = g (Ju, v) . Thus V is a symplectic vector space. Indeed any two of these structures g, J and ω determines the third one, i.e. the intersection of any two of the three subgroups O (2n), GL (n, C) and Sp (2n, R) in GL (2n, R) is always the unitary group U (n). On a manifold M , an almost complex structure is a linear complex structure Jx on every tangent space Tx M . M is called a K¨ ahler manifold if g is Hermitian and J is parallel, ∇J = 0. Equivalently a K¨ ahler manifold is a Riemannian manifold with holonomy U (n). A K¨ahler manifold is always a complex manifold by the theorem of Newlander and Nirenberg which states that an almost complex structure is integrable, i.e. coming from the linearization of a complex structure on M if and only if the Nijenhuis tensor N ∈ Ω2 (M, T M ) vanishes, where 4N (u, v) = [u, v] + J [Ju, v] + J [u, Jv] − [Ju, Jv]

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for any u, v ∈ Tx M and for any x ∈ M . In particular, if J is parallel with respect to some torsion free connection ∇, say the Levi-Civita connection, then M is a complex manifold. An important observation is that if M is already a Hermitian complex manifold, then the K¨ ahlerian condition ∇J = 0, or equivalently ∇ω = 0 ∈ Ω2 (M, TM ), can be reduced to the closedness of the K¨ ahler form, dω = 0 ∈ 3 Ω (M ). This implies that every complex submanifold of a K¨ ahler manifold is always K¨ ahler. In particular, every projective algebraic manifold M in CPN is K¨ahler, thus providing abundant examples of K¨ ahler manifolds. Kodaira embedding theorem gives a necessary and sufficient condition for a compact K¨ ahler manifold M to be projective, namely the K¨ ahler class [ω] ∈ H 2 (M, R) should be defined over Z. As a result, K¨ahler geometry and complex algebraic geometry are intimately related to each other. 5.1. K¨ ahler geometry—complex aspects. Dolbeault cohomology and Hodge (p, q)-decomposition. A linear complex structure on a real vector space V can be rephrased as a decomposition V ⊗ C = V 1,0 ⊕ V 0,1 satisfying V 0,1 = V 1,0 . Indeed V 1,0 and V 0,1 are the ±i eigenspaces of J. Taking tensor powers, we have k  V ⊗C= V p,q with V q,p = V p,q . p

q

p+q=k

V 1,0 ⊗ V 0,1 . We have a corresponding decompositions for Here V p,q = differential forms on any almost complex manifold M ,  Ωk (M, C) = Ωp,q (M ) with Ωq,p (M ) = Ωp,q (M ) . p+q=k

The exterior differentiation d : Ωk (M, C) → Ωk+1 (M, C) decomposes ¯ with ∂ : Ωp,q (M ) → Ωp+1,q (M ) and N : Ωp,q (M ) → into d = ∂ + N + ∂¯ + N p+2,q−1 Ω (M ) being the tensor product with the Nijenhuis tensor. M being a complex manifold, i.e. N = 0, is equivalent to ∂¯2 = 0 : Ωp,q (M ) → Ωp,q+2 (M ) . That is, we have an elliptic complex ∂¯ ∂¯ ∂¯ 0 → Ω0,0 (M ) → Ω0,1 (M ) → · · · → Ω0,n (M ) → 0 with ∂¯2 = 0,

called the Dolbeault complex. The corresponding cohomology group is called the Dolbeault cohomology, denoted    Ker ∂¯     H∂p,q (M ) = . ¯ Im ∂¯  p,q Ω

Its dimensions numbers of M.

hp,q

(M ) =

dim H∂p,q ¯

(M )

(M )’s are called the (p, q)-Hodge

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Similarly, on any holomorphic vector bundle E over M , there is a twisted ∂¯ operator, ∂¯E : Ωp,q (M, E) → Ωp,q+1 (M, E)

2 with ∂¯E = 0,

and the corresponding cohomology groups are denoted as H∂p,q ¯ (M, E). p,q 0,q p ∗ ∼ There is a canonical identification H∂¯ (M, E) = H∂¯ (M, Λ TM ⊗ E) and ∗ ⊗ E). Analogous to the Poincar´ we simply write it as H q (M, Λp TM e duality, the Serre duality gives the isomorphism ∗ H p,q (M, E) ∼ = H n−p,n−q (M, E ∗ ) .

In particular,

∗ H q (M, E) ∼ = H n−q (M, KM ⊗ E ∗ ) .

If we endow both M and E with Hermitian metrics, then we can define the twisted Laplacian operator ∗ ∗ ¯ + ∂¯E ∂E , Δ∂¯E = ∂¯E ∂¯E ∗ is the adjoint of ∂ ¯E and the Hodge theorem of representing cohowhere ∂¯E mology classes by harmonic forms has a direct generalization here which gives an isomorphism   ∼ H∂p,q ¯ (M, E) = Ker Δ∂¯E |Ωp,q (M,E) .

When M is a K¨ahler manifold, one can prove that Δ = 2Δ∂¯ and this has many important consequences. For instance, we have the following Hodge (p, q)-decomposition,  q,p p,q H∂p,q H k (M, C) = ¯ (M ) with H∂¯ (M ) = H∂¯ (M ) . p+q=k

In particular, every odd degree Betti number of a compact K¨ ahler manifold is even, b2l+1 (M ) ∈ 2Z. This collection of subspaces H p,q (M )’s in H k (M, Z) ⊗ C is called the Hodge structure of M . It depends only on the complex structure on M , but not on the choice of K¨ ahler metrics and it carries important informations about the complex structure. Hard Lefschetz action. Recall that ω (u, v) = g (Ju, v) defines a symplectic structure on M . In particular, [ω]l = 0 ∈ H 2l (M, R) for any 0 ≤ l ≤ n. Let L : Ωk (M ) → Ωk+2 (M ) L (φ) = φ ∧ ω and Λ be its adjoint operator. Then H = [L, Λ] : Ωk (M ) → Ωk (M ) is a multiple of the identity operator, H = (n − k). The relationships [H, L] = 2L,

[H, Λ] = −2Λ and

[L, Λ] = H

defines an sl (2, R)-action on Ω∗ (M ). The closedness of ω implies that L can be descended to H ∗ (M, R) and the K¨ ahler property of M , i.e. ω is

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parallel, implies that both L and Λ commute with Δ and therefore the above sl (2, R)-action can be descended to H ∗ (M, R), called the Hard Lefschetz action. This action allows us to recover H ∗ (M, R) from Ker (Λ), called the primitive cohomology and to reduce the Hodge structure on H ∗ (M, C) to the primitive cohomology and define a polarized Hodge structure. K¨ ahler identities. First order K¨ ahler identities give relationships between zeroth order operators L, Λ, H and first order operators ∂, ∂, ∂ ∗ , ∗ ∂ . They are ! ∗ [L, ∂ ∗ ] = i∂, L, ∂ = −i∂,   ∗ [Λ, ∂] = i∂ , Λ, ∂ = −i∂ ∗ , and all other brackets are zero. Indeed these formulas hold true even when J is only an almost complex structure, namely M is a symplectic manifold with a compatible almost complex structure J, sometimes called an almost K¨ ahler structure. Second order K¨ ahler identities are Δ = 2Δ∂ = 2Δ∂ , ¯ ∂ ∗ and ∂¯∗ . We have seen earlier that and Δ commutes with L, Λ, H, ∂, ∂, the identity Δ = 2Δ∂ is used to define the Hodge structure on H ∗ (M, C). Recall that the identities among zeroth order operators L, Λ, H define a sl (2, R)-action on Ω∗ (M ) and there is a canonical identification between Lie algebras sl (2, R) ∼ = su (1, 1). Indeed all the above identities together define ∗ an action on Ω (M, C) by the super Lie algebra su (1, 1)sup = su (1, 1) ⊕ C1,1 ⊕ R ∗

with C1,1 spanned by ∂, ∂, ∂ ∗ , ∂ and R spanned by Δ. Indeed this super Lie algebra action encompasses the hard Lefschetz action and all the K¨ ahler identities. It has natural generalizations to manifolds defined over other normed division algebras (section 10). Deformation of complex structures. A complex structure on M can be ¯ characterized by the ∂-operator ∂¯ : Ωp,q (M ) → Ωp,q+1 (M )  2 satisfying the integrability condition ∂¯ = 0. A different complex structure is then given by ∂¯ + φ with φ ∈ Ω0,1 (M, TM ) . The integrability condition becomes ¯ + [φ, φ] = 0. ∂φ For a smooth family of complex structures ∂¯ + φ (t), φ (t) = tφ1 + t2 φ2 + t3 φ3 + · · ·

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we have ¯ =0 ∂φ 1 ¯ ∂φ2 = − [φ1 , φ1 ] and so on. The first equation says that infinitesimal deformations of complex struc  0,1 ¯ tures are parametrized by Ω (M, TM ) ∩ Ker ∂ . Up to diffeomorphisms, they are parametrized by the cohomology group H 1 (M, TM ). When H 2 (M, TM ) = 0, the second equation for φ2 can always be solved and the same is true for all other φj ’s. Using the Hodge theory for any given φ1 , we can obtain a convergent power series solution φ (t) for small t and therefore we ¯ This have an honest family of deformations of the complex structure ∂. means that the moduli space of complex structures is smooth and with tan¯ gent space H 1 (M, TM ) at ∂. 2 Even when H (M, TM ) = 0, the integrability condition can sometimes be solved for any given φ1 . For example, this is the case for Calabi-Yau manifolds. In general there is a Kurinishi (nonlinear) map κ defined on a neighborhood of the origin, κ : H 1 (M, TM ) → H 2 (M, TM ) , ¯ such that the moduli space of complex structures  on M near ∂ is given by 3 −1 κ (0). The map satisfies κ (φ) = [φ, φ] + O |φ| . The space H 2 (M, TM ) is called the obstruction space for the deformations. For a holomorphic bundle E over a fixed complex manifold M , infinitesimal deformations of E are parametrized by H 1 (M, End (E)) and the obstruction space is given by H 2 (M, End (E)). There are also analogous spaces for deformations of flat bundles. For a holomorphic map f : C → M , the space of infinitesimal  deforma- 0 tions (resp. obstruction space) of f with fixed M is given by H C, Nf (C)/M   (resp. H 1 C, Nf (C)/M ) We remark that the deformation theory only deal with the local structure of moduli spaces. Construction of a global moduli space as a complex variety, or projective variety, is a different matter, which is often dealt with via the geometric invariant theory. Subvarieties and coherent sheaves. We remark that on any K¨ ahler mank 2k ifold M , the form ω /k! ∈ Ω (M ) is always a calibration form and those submanifolds calibrated by it are precisely complex submanifolds in M . This follows from the Wirtinger formula. As a corollary, complex submanifolds S in K¨ ahler manifolds are volume minimizers and they define nontrivial cohomology classes because S ω k /k! is the volume of S, P D [S] ∈ H∂p,p (M ) ∩ H 2p (M, Z) , where p = n − k is the complex codimension of S. This continues to hold true even when S is singular, namely a subvariety in M . The famous Hodge

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conjecture asks that whether every class in H∂p,p (M ) ∩ H 2p (M, R) is represented a Q-linear combination of subvarieties in a projective manifold M . The Chern classes cp (E) for any holomorphic vector bundle also lie in H∂p,p (M )∩H 2p (M, Z) and this continues to hold true for any coherent sheaf, which allows E to be singular. Quasi-isomorphism classes of complexes of coherent sheaves form a derived category Db (M ). It contains much information about M , for instance, it determines M completely when c1 (M ) is either positive or negative [15]. 5.2. K¨ ahler geometry—Riemannian aspects. Hermitian Yang-Mills metrics. Given any Hermitian metric hE on a holomorphic vector bundle E over a complex manifold M , there is a unique Hermitian connection DE satisfying DE hE = 0 and (DE )0,1 = ∂ E : Ω0 (M, E) → Ω0,1 (M, E) . In terms of any local holomorphic frame ei ’s on E, we have DE = d + h−1 ∂h, and   FE = ∂ h−1 ∂h ∈ Ω1,1 (M, ad (E)) ,

  where h = hij with hij = hE (ei , ej ). This implies that c1 (E) =

 i i  [T rE (FE )] = ∂∂ log det h 2π 2π

and Chern classes are of type (p, p), i.e. cp (E) ∈ H p,p (M ) ∩ H 2p (M, Z). For holomorphic bundles over a K¨ ahler manifold, the Yang-Mills equa∗ tion DE FE = 0 is equivalent to DE (ΛFE ) = 0 because of the K¨ahler identity Λ, ∂ = −i∂ ∗ . Thus an eigenbundle in E for the bundle endomorphism ΛFE is a holomorphic (parallel) subbundle. Thus, unless E is reducible, the Yang-Mills equation reduces to a first order differential equation, called the Hermitian-Yang-Mills equation, ΛFE = μE I, where the constant μE is called the slope of E and it is given by2  1 ω n−1 , μE = c1 (E) ∧ r M (n − 1)! where r is the rank of E. We remark that when M is a K¨ahler surface, then the equations FE0,2 = ΛFE = 0 is the same as the ASD equation for unitary connections on the four manifold M . This is because Λ2+ ∼ = Rω ⊕ Re Λ0,2 ⊕ Im Λ0,2 . 2For simplicity, we have normalized the volume of M to be one.

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A Hermitian-Yang-Mills connection is not just a critical point for the  2 Yang-Mills functional |FE | , it is an absolute minimizer! This can be seen from the following equality, derived from the Chern-Weil theory, 2 n   2 n       FE − 1 T r (FE ) I  ω = ΛFE − 1 T r (ΛFE ) I  ω     n! r n! r M M    4π 2 (r − 1) 2r 2 − c1 (E) − c2 (E) r r−1 M ω n−2 . × (n − 2)! As a corollary, if a holomorphic bundle E admits a Hermitian-Yang-Mills connection, then it must satisfy the following Chern number inequality,   2r 2 n−2 c1 (E) ω ≤ c2 (E) ω n−2 . r − 1 M M Furthermore, if the equality sign holds, then FE = 1r T r (FE ) I, i.e. a projectively flat connection and thus DE corresponds to a homomorphism ρ : π 1 (M ) → PU (r). A necessary condition for the existence of Hermitian-Yang-Mills connection is E being a Mumford polystable bundle, i.e. a direct sum of Mumford stable bundle. Recall that E is Mumford stable if every nontrivial coherent subsheaf S in E satisfies the following slope inequality, μS < μE . The celebrated theorem of Donaldson [36], Uhlenbeck and Yau [125] says that the converse is also true. Namely a holomorphic bundle E admits a Hermitian Yang-Mills connection if and only if E is a Mumford polystable bundle. Mumford stability was introduced to construct projective moduli space of holomorphic bundles over Riemann surfaces via geometric invariant theory (GIT). The correct notion of stability in higher dimensions is the Gieseker stability which replaces the slope inequality by the normalized Hilbert polynomial inequality,     1 1 χ M, S ⊗ L⊗k < χ M, E ⊗ L⊗k rk (S) rk (E) for sufficiently  L is a line bundle over M with c1 (L) represented  large k. Here by ω and χ M, E ⊗ L⊗k is given by n       i q ⊗k q ⊗k χ M, E ⊗ L (−1) dim H M, E ⊗ L T rE e 2π FE +kω T dM , = = q=0

M

by the Riemann-Roch formula and the Chern-Weil theory. Notice that the dominating terms for large k is given by the slope. In [88] the author showed

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that Gieseker polystability is a necessary and sufficient condition for the existence of a bounded solution to the following equation,  i [2n] χ M, E ⊗ L⊗k  ω n F +kω e 2π E T dM = IE , rk (E) n! for sufficiently large k on any sufficiently smooth holomorphic bundle E. The relationship with the symplectic geometry will be discussed in section 5.3. Curvature for K¨ ahler metrics. When h = g is a Hermitian metric on the tangent bundle E = TM , then the K¨ ahlerian condition for g is equivalent to DE being torsion free. Yet another equivalent definition of g being a K¨ ahler metric is the existence of holomorphic normal coordinate, i.e. given any point p0 ∈ M, there exists a local holomorphic coordinate z j = xj + iy j ’s such that for any nearby point p, we have   gij (p) = δ ij + O |p − p0 |2 . The symmetries for the curvature tensor Rm of a K¨ahler metric are richer than the Riemannian  case. First we note that the inclusion u (n) ⊂ o (2n) corresponds to V 1,1 R ⊂ Λ2 VR for any Hermitian vector space V . This implies that Rmi¯jk¯l is Hermitian symmetric with respect to i¯j and also to k¯l for any complex coordinates z j = xj + iy j ’s. Second the Ricci tensor  1,1  ∗ ∼ Rc being Hermitian symmetric means that Rc ∈ u (n) = Λ TM R and the corresponding (1, 1)-form Rc (J·, ·) will again be denoted as Rci¯j . Explicitly we have ¯

Rci¯j = g kl Rmi¯jk¯l ∂ ∂ = i j log det (gk¯l ) . ∂z ∂ z¯ In particular, the first Chern class of M (modulo torsion) is represented by the Ricci form, c1 (M )R =

i [Rc] . 2π ¯

The scalar curvature is given by R = g ij Rci¯j = Δ log det (gk¯l ). Recall in the Riemannian case that when RcM ≥ 0, harmonic one forms are parallel and they must be zero if RcM > 0. In the K¨ ahler case, holomorphic p-forms are harmonic. When M has RcM ≥ 0, they are parallel and zero if RcM > 0, i.e. M is Fano. If Rm = λI as an endomorphism of u (n), then the universal cover of M must be either the complex projective space CPn , the complex ball or Cn according to λ being positive, negative or zero. These are called complex space forms. If Rc = λω, i.e. M is a K¨ ahler-Einstein manifold, then first of all c1 (M )R must be represented by a multiple of some K¨ahler form ω 0 . Unless c1 (M )R = ∗ or its inverse is positive and 0, the canonical line bundle KM = Λn TM

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therefore M is a projective manifold by the Kodaira’s embedding theorem. If we write Rc (ω 0 ) = λω 0 + i∂∂f and ω = ω 0 + i∂∂φ for some smooth functions f and φ on M with average one, then the K¨ahler-Einstein equation is reduced to the following fully nonlinear second order elliptic equation, called the complex Monge-Amp`ere equation,

  ∂2φ det gi¯j + i j = e−λφ+f det gi¯j . ∂z ∂ z¯ When c1 (M )R = 0, Yau [137] solved this equation and proved that there is a unique metric in every K¨ ahler class with zero Ricci curvature, Rc = 0. In fact, Yau’s theorem also solved a conjecture of Calabi which says that every (1, 1)-form representing c1 (M )R can be represented uniquely as the Ricci form of a metric in any K¨ ahler class on any compact K¨ahler manifold M . As a corollary of Yau’s theorem, c1 (M )R = 0 implies that c2 (M ) [ω]n−2 ≥ 0, for any K¨ ahler class [ω] and the equality sign holds if and only if the universal cover of M is Cn . Zero Ricci curvature means that the canonical line bundle ∗ is a flat line bundle. Suppose that c (M ) = 0 as an integral KM = Λn TM 1 class, then KM is indeed trivial and its covariant constant section defines a holomorphic volume form on M . Thus the holonomy group of M is reduced to SU (n) and such a manifold is called a Calabi-Yau manifold and plays a very important role in string theory. When c1 (M )R < 0, Aubin and Yau solved the Monge-Amp`ere equation, thus proving that there is a unique K¨ ahler-Einstein metric in the K¨ ahler class −c1 (M )R . The Chern number inequality becomes 2 (n + 1) (M ) . (−1)n c2 (M ) cn−2 1 n When the equality sign holds, then the universal cover of M must be the complex hyperbolic ball. When c1 (M )R > 0, M is called a Fano manifold and there are nontrivial obstructions to the existence of K¨ ahler-Einstein metrics, for instance the Futaki invariant. Yau conjectured that there should be a notion of stability which relates to the existence of such a canonical metric. This problem has been studied by many mathematicians including Donaldson, Mabuchi, Phong, Tian and others. Donaldson showed that this relationship between stability and the existence of canonical metrics should continue to hold true for constant scalar curvature K¨ ahler metrics, i.e. R = const. (−1)n cn1 (M ) ≤

5.3. K¨ ahler geometry—symplectic aspects. Recall that the K¨ ahler form ω (u, v) = g (Ju, v) defines a closed non-degenerate two form, i.e. a symplectic form. On the classical level, symplectic geometry is a much more linear theory than the complex geometry (section 5.3). On the quantum level, it becomes a very rich and challenging subject which includes

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the theories of Gromov-Witten invariants and the Fukaya-Floer category. The mirror symmetry conjecture says roughly that complex geometry and quantum symplectic geometry should be equivalent to each other. Basic symplectic geometry. Let us start by reviewing some aspects of the classical symplectic geometry. First the Darboux lemma says that every symplectic manifold is locally standard which is R2n with ω0 =

n  j=1

dxj ∧ dyj .

We can view this as the cotangent bundle T ∗ Rn of Rn with coordinates xj ’s and dual coordinates yj ’s along fibers. In fact ω 0 defines a canonical symplectic structure on the cotangent bundle M = T ∗ X of any manifold X. A vector field v on M preserves the symplectic form ω if and only if ιv ω is a closed one form. If ιv ω = df is exact, then v is called a Hamiltonian vector field. Moser’s lemma says that if ω t is an one parameter family of symplectic forms in M representing the same cohomology class, then all these (M, ω t )’s are symplectomorphic to each other. Thus the moduli space of symplectic structures on M is locally isomorphic to H 2 (M, R). One can also include B-fields B on M and such that the moduli space of symplectic structures with B-fields B +iω is a complex space which is locally isomorphic to H 2 (M, C). This concept is originated in string theory and important in mirror symmetry. The natural class of submanifolds in (M, ω) consists of Lagrangian submanifolds L which are n-dimensional submanifolds in M satisfying ω|L = 0. For instance the zero section and every fiber in the cotangent bundle T ∗ X are Lagrangian submanifolds. In general, if we regard a section L in T ∗ X as the graph of an one form φ ∈ Ω1 (X), then L is a Lagrangian submanifold if and only if φ is a closed form, i.e. dφ = 0. Indeed a neighborhood of any Lagrangian submanifold L is always symplectically equivalent to the cotangent bundle T ∗ L and therefore the moduli space of Lagrangian submanifolds L in M modulo Hamiltonian equivalences is locally given by H 1 (L, R). If we consider the moduli space of A-cycles (L, DE ), i.e. L is a Lagrangian submanifold in M and DE is a flat U (1)-connection over L, then it is again a complex manifold and locally isomorphic to H 1 (L, C). Lagrangian fibrations. A Lagrangian fibration π : M → B is called a (singular) real polarization on M , or an integrable system. It plays an important role in geometric quantization of the symplectic manifold M . Another standard way to obtain geometric quantization is to equip M with a complex structure such that ω is the corresponding K¨ ahler form, this is called a complex polarization. Away from singular fibers, we have a surjective bundle homomorphism T M → π ∗ T B over M and the kernel is called the vertical tangent bundle Tvert M . For Lagrangian fibration, we have Tvert M ∼ = π ∗ T ∗ B.

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Thus we have n commuting vector fields along fibers which are linearly independent at every point. This determines canonical affine structures on smooth fibers. When π is proper, i.e. fibers are compact, then fibers are tori Tx∗ B/Λx . This gives a lattice subbundle Λ in T ∗ B over B outside the discriminant locus Disc (π). As a result, the base B\Disc (π) also has a GL (n, Z) × Zn -affine structure. Toric varieties PΔ are examples of symplectic manifolds with Lagrangian fibrations in which the fibers are orbits of an Hamiltonian torus action (see below) and the base is a convex polytope Δ. The simplest compact toric varieties are certainly complex projective spaces CPn+1 . Hamiltonian action and symplectic reduction. Suppose (M, ω) is a symplectic manifold with a Hamiltonian action by a compact Lie group G with moment map μ : M → (LieG)∗ . We recall that a moment map μ is a G-equivariant map such that for any v ∈ Lie (G) and v # the vector field on M that v generates, then v # is the Hamiltonian vector field for the function x → μ (x) (v), i.e. ιv# ω = d (μ (v)) . If a 2n-dimensional symplectic manifold M has an effective Hamiltonian T k -action, then k is at most n. When k = n, the moment map μ : M → Rn is a Lagrangian fibration on M and the image is convex polytope Δ in Rn . Such a M is called a toric variety and its geometry is completely dictated by the polytope Δ. For instance, when Δ is the standard simplex in Rn , M is the complex projective space CPn with the toric action is induced from n T n ⊂ (C× ) ⊂ Pn . When M has a Hamiltonian G-action, there is a procedure to divide out the symmetry to produce another (possibly singular) symplectic manifold M//G = μ−1 (0) /G called the symplectic quotient or symplectic reduction. We can also replace 0 by other coadjoint orbit. We assume that (M, ω) is a K¨ahler manifold with ω being defined over Z, thus [ω] = c1 (L) for a positive line bundle L. Suppose that the complexification GC of G acts holomorphically on M and the action can be lifted to L. We can apply the Geometric Invariant Theory, developed by Mumford, to construct a quotient space M/GC within the category of algebraic varieties. In this construction, one needs to remove unstable points in M to ensure that the quotient space is Hausdorff. Kempf-Ness showed that this complex algebraic approach and the symplectic approach of taking quotient are equivalent to each other, M/GC ∼ = M//G.

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This identification is particularly fruitful in many infinite dimensional settings as a guiding principle. Symplectic geometry and gauge theory. Given a Hermitian complex vector bundle E over a symplectic manifold (M, ω), the space A (E) of unitary connections on E has a natural symplectic form Ω defined as follows: The tangent space of A (E) at any connection DA can be identified as Ω1 (M, ad (E)). Given any tangent vectors B and C, we define  ω n−1 . Ω (DA ) (B, C) = T rE B ∧ C ∧ (n − 1)! M The action on A (E) by the group G (E) of gauge transformations of E preserves Ω and its moment map is given by μ : A (E) → Ω2n (M, ad (E)) μ (DA ) = FE ∧

ω n−1 . (n − 1)!

When M is a Riemann surface, both Ω and μ are independent of the symplectic form on M and the symplectic quotient A (E) //G (E) = {DA : FA = 0} /G (E) , is the moduli space of flat connections on E, i.e. Hom (π 1 (M ) , U (r)) /U (r) . There are natural generalizations of this to other compact Lie groups. The infinite dimensional analog of the identification between GIT quotient and symplectic quotient suggests that every polystable holomorphic bundle over M admits a unitary flat connection. This was proved by Narishima-Seshadra. When M is a K¨ ahler manifold and E is a holomorphic vector bundle over it, we can restrict our attention to the subset Ahol (E) consisting of those connections DA satisfying FA0,2 = 0, namely (DA )0,1 defines a holomorphic structure on E. In this case, the suggested isomorphism between Ahol (E) //G (E) and Ahol (E) /G C (E) is the theorem of Donaldson [36] and Uhlenbeck-Yau [125] which says that every Mumford polystable holomorphic bundle admits a unique Hermitian-Yang-Mills connection. However, in algebraic geometry, the correct stability condition is the Gieseker stability defined using the Hilbert polynomial,    ! i ⊗k χ M, E ⊗ L T rE e 2π FA +kωIE ∧ T dM = M

The analog of the equivalence between GIT quotient and the symplectic quotient was established in [88] for the existence of solutions to the following equations for large k, !(n,n) χ M, E ⊗ L⊗k  ω n i F +kωI E ∧ T dM = e 2π A IE . r n!

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This equation is also a moment map equation for the G (E)-action on Ahol (E) with respect to the following nonconstant symplectic form Ωk [89]:  ! i T rE e 2π FA +kωIE ∧ B ∧ C ∧ T dM . Ωk (DA ) (B, C) = sym

M

The Gieseker stability is an asymptotic stability as k goes to infinity. For each finite k, Donaldson studied the finite dimensional GIT/symplectic quotients equivalences and conjectured that Gieseker stable bundles should admit balanced metrics for large k’s and they converge to the HermitianYang-Mills metrics if E is also Mumford polystable. This problem was solved by Wang [129]. Space of K¨ ahler forms. Donaldson and Semmes showed that the space of K¨ ahler metrics in a fixed K¨ ahler class on M is an infinite dimensional symplectic manifold with a Hamiltonian action by Dif f (M ). Furthermore, the moment map can be identified with the scalar curvature of a K¨ahler metric. Thus the GIT/symplectic quotients equivalences should relate the existence of constant scalar curvature K¨ ahler metrics with GIT stability of the manifold M . In the next section, we discuss symplectic geometry on the quantum level. 5.4. Gromov-Witten theory. Gromov-Witten invariants. Given a complex structure J on M , we study complex submanifolds C in M , namely Tx C is J-linear in Tx M for any x ∈ C. This notion continues to make sense for any almost complex structure and such submanifolds C are called J-pseudo holomorphic submanifolds, or simply J-holomorphic submanifolds. Recall that an almost complex structure is simply a complex structure on the tangent bundle TM . If we choose a generic almost complex structure on M , then it admits no J-holomorphic submanifolds C of dimC C ≥ 2, even locally, as the Cauchy-Riemann equation is an over-determined system of differential equations. However, there are always many J-holomorphic curves, at least locally. In order to count the number of such curves, it turns out that it is better to use the parametrized version, namely we consider the moduli space Mcurve (M ) of J-holomorphic maps from genus g Riemannian surfaces Σ to M , f : Σ → M. When M is equipped with a compatible symplectic structure ω, then Mcurve (M ) has a natural compactification Mcurve (M ) by stable maps where the domain Riemann surface Σ is allowed to have nodal singular points and f is required to have only finite number of automorphisms. Recall that every J-holomorphic curve is calibrated by ω, in particular, f (Σ) always represents a nontrivial homology class [f (Σ)] ∈ H2 (M, Z) \ {0}.   Infinitesimal deformations of f are parametrized by H 0 Σ, Nf (Σ)/M   (section 5.1). When H 1 Σ, Nf (Σ)/M = 0, then f always has unobstructed

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deformations. In this case, Mcurve (M ) is smooth and its dimension is determined by the Riemann-Roch formula,  curve (M ) = f ∗ c1 (M ) + (g − 1) (3 − n) . dimC M Σ

In order to count the number of curves in M , we impose conditions to cut down the dimension of Mcurve (M ). For instance, we can require f (Σ) to pass through specific points in M . These define the Gromov-Witten invariants, or simply GW-invariants. These are invariants for the deformation class of symplectic forms on M . In particular, it is independent of the choice of compatible almost complex structures on M . Even when Mcurve (M ) is singular, there is a theory using virtual fundamental class to define these invariants. GW-invariants can be interpreted as the partition functions for the A-model TFT in the N = 2 SUSY σ-model on the K¨ ahler manifold M . In general, GW-invariants are difficult to compute. When dimR M = 4, Taubes showed that GW-invariants with no constraints are equivalent to the SW-invariants. This result has far-reaching consequences in four dimensional symplectic geometry. When the symplectic manifold (M, ω) has a lot symmetries, for instance a toric variety, then GW-invariants can be computed in many instances via Bott localization, at least in the genus zero case. This method can be generalized to complex hypersurfaces of small degrees in Fano toric varieties as well. It was initiated by Kontsevich, motivated from the mirror symmetric conjecture for Calabi-Yau manifolds. The mirror theorem which computes the genus zero GW-invariants for quintic CY threefolds in terms of the variation of Hodge structures of its mirror manifold was proven by developing this approach by Givental [47] and Liu-Lian-Yau [105]. There are other approaches in determining GW-invariants, for example the YauZaslow argument for the number of rational curves on K3 surfaces (section 6.4) and various physical methods coming from dualities in string theory and M-theory. The genus zero GW-invariants can be used to deform the cup product structure on the cohomology ring H ∗ (M ) and results in the quantum cohomology ring QH ∗ (M ) for the symplectic manifold (M, ω). It is originated from the (closed) string theory. QH ∗ (M ) can be formally interpreted as the middle dimensional cohomology ring of the free loop space LM via the Witten-Morse theory for the symplectic area functional A : LM → R  γ1 A (γ) = ω. γ0

Here γ t is an one-parameter family of loops in M connecting γ = γ 1 to a γ background γ 0 . Here γ 1 means the integration over the two dimensional 0 " surface γ t spanned by the path of loops. Critical points of A (γ) are t∈[0,1]

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constant loops in M , thus M can be viewed as the critical set of A inside LM . After choosing a compatible metric on M , gradient flow lines in LM correspond to J-holomorphic cylinders in M . Floer theory of Lagrangian intersections. In open string theory, the boundaries of a string, namely a path γ (t) in M , lies on Lagrangian submanifolds Li in M , i.e. γ (0) ∈ L0 and γ (1) ∈ L1 . Similarly, we consider the symplectic area functional A on the space LL0 →L1 M of all such paths. Critical points of A are constant paths and therefore correspond to intersection points in L0 ∩ L1 and gradient flow lines of A are holomorphic strips between Li ’s joining two intersection points. Floer and others developed the Witten-Morse theory in this setting and defined Floer cohomology M (L , L ) for Lagrangian intersections. Fukaya and others [45] groups HFLagr 0 1 extended these structures and defined the Fukaya category F uk (M ), which is conjecturally dual to the derived category of coherent sheaves under mirror symmetry. 6. Calabi-Yau geometry 6.1. Calabi-Yau manifolds. A K¨ ahler manifold M is called a CalabiYau manifold if it admits a parallel holomorphic volume form Ω ∈ Ωn,0 (M ), i.e. ∇Ω = 0 ¯ = vM , (2i)−n (−1)n(n+1)/2 Ω ∧ Ω where vM = ω n /n! is the Riemannian volume form and n is the complex dimension of M . Equivalently, M is a Riemannian manifold with holonomy group inside SU (n). If M is compact, then Ω being parallel can be replaced ¯ = 0. The complex volume form Ω defines a Cby holomorphicity, i.e. ∂Ω orientation on M and it fits into the unified description of geometries of special holonomy in terms of normed division algebras (see section 10). Recall that a volume form vM on M defines a symplectic form on KΣ (M ) = M apemb (Σ, M ) /Dif f (Σ) where dim Σ = dim M − 2. One can also use the holomorphic volume form Ω to define a holomorphic symplectic C (M ) where dim Σ = n−2. Furthermore form on the isotropic knot space KΣ R the special Lagrangian geometry of M can be interpreted as the complex C (M ) (section 9.4). symplectic geometry of KΣ Since the Ricci curvature of a K¨ ahler metric is given by ∂2 log det (gk¯l ) , ∂z i ∂ z¯j any Calabi-Yau manifold has zero Ricci curvature. In particular, its canonical line bundle KM is trivial and c1 (M ) = 0. By Yau’s celebrated theorem [137], any compact K¨ ahler manifold M with c1 (M ) = 0 admits a unique Calabi-Yau metric in every K¨ ahler class. Thus it is easy to identify which K¨ ahler manifold admits a Calabi-Yau metric. Nevertheless, we still do not Rci¯j = −

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know how to write down explicitly any nontrivial Calabi-Yau metric on any compact Calabi-Yau manifold. For instance, any smooth projective hypersurface of degree n + 2 in CPn+1 is a Calabi-Yau manifold. More generally, a hypersurface in a toric variety XΔ representing the class c1 (XΔ ) is a (possibly singular) Calabi-Yau variety if and only if Δ is a reflexive polytope, i.e. both Δ and its polar dual polytope  are integral. This construction can be easily generalized to complete intersections in toric varieties and produces many examples of Calabi-Yau manifolds. Even though we do not know any explicit Calabi-Yau metric on compact manifolds, there are many such examples on noncompact manifolds, including the total space of the canonical line bundle KCPn−1 of CPn−1 and the cotangent bundle TS∗n of the sphere. These metrics are found by utilizing the symmetries of these spaces to reduce the Monge-Amp`ere equation to an ODE. Every holomorphic p-form in H p,0 (M ) is parallel since RcM = 0 for a Calabi-Yau manifold M . The existence of such forms can usually be used to reduce the holonomy group of M to a smaller subgroup. Using the deRham decomposition for holonomy groups, up to a finite cover, M is a product of irreducible factors and they are (i) complex tori Cn /Γ, (ii) irreducible hyperk¨ ahler manifolds, i.e. hol = Sp (n/2) or (iii) strict Calabi-Yau manifolds, i.e. hol = SU (n). Only   complex tori have nontrivial holomorphic one forms and hp,0 (Cn /Γ) = np . Hyperk¨ ahler manifolds admit holomorphic 2l,0 2l+1,0 symplectic forms, indeed h = 1 and h = 0 for irreducible hyperk¨ ahler manifolds. We will discuss more about the hyperk¨ ahler geometry in section 11.1. When hol = SU (n), the only nontrivial holomorphic p-form is the holomorphic volume form and therefore hp,0 = 0 for 1 ≤ p ≤ n − 1. By the Lefschetz hyperplane theorem, any Calabi-Yau hypersurface, or complete intersection, in a Fano toric variety is a strict Calabi-Yau manifold provided that n ≥ 2. Since KM is trivial for a Calabi-Yau manifold, we have   H q (M, TM ) ∼ = H q M, Λn−1 T ∗ ∼ = H n−1,q (M ) . M

In particular, the spaces of infinitesimal deformations and obstructions are given by H n−1,1 (M ) and H n−1,2 (M ) respectively. Tian [122] and Todorov [124] proved that the Kuranishi map κ : H 1 (M, TM ) → H 2 (M, TM ) is zero. Thus the moduli space of complex structures on M is always smooth and of dimension hn−1,1 (M ). This moduli space has many nice properties, especially in complex dimension three (section 7). 6.2. Special Lagrangian geometry. Special Lagrangian submanifolds. For K¨ ahler manifolds, complex submanifolds are calibrated and therefore they are absolute minimizers for the volume functional. For Calabi-Yau manifolds, there is a natural class of Lagrangian submanifolds which are calibrated. Harvey and Lawson [59]   found that for any given phase angle θ, the differential form Re eiθ Ω ∈

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Ωn (M ) is a calibrated form. Furthermore a submanifold L in M is cali iθ  brated by Re e Ω if and only if   iθ ω|L = 0 and Im e Ω |L = 0. Such a L is called a special Lagrangian submanifold of phase θ. We remark that for any Lagrangian submanifold L in M , the volume form vL for the induced metric satisfies vL = eiθ(x) Ω|L for some function θ (x) : L → R/2πZ. It is the Hamiltonian function for the mean curvature vector field on L, i.e. ιH ω = dθ (x) . This implies that the mean curvature flow preserves the Lagrangian property and they satisfy the PDE ∂θ = Δθ. ∂t This is a non-linear equation as the Laplacian is defined with respect to the induced metric on L which is changing in time. Lagrangian mean curvature flow in Calabi-Yau manifolds enjoys many nice properties and the same is true for hyperlagrangian mean curvature flow in hyperk¨ ahler manifolds (section 10.3). Unless specified otherwise, we assume that θ = 0 for the calibrating form. When M = Cn , a Lagrangian graph Graph (dφ) over Rn is special if and only if φ (x) satisfies

∂2φ Im det δ jk + i j k = 0. ∂x ∂x Examples of special Lagrangian submanifolds include the real locus of a Calabi-Yau manifold, complex Lagrangian submanifolds in a hyperk¨ ahler manifold (section 11.1). Many explicit examples of special Lagrangian submanifolds had been constructed in the noncompact setting by imposing symmetries to reduce the equation to an ODE. There are also various compact examples constructed using singular perturbation method to resolve singular special Lagrangians. C (M ) is an infiIn section 9.5, we show that the isotropic knot space KΣ nite manifold with an H-structure such that (i) every special Lagrangian submanifold L of phase 0 in M determines a J-complex Lagrangian subC (L) in KC (M ) and (ii) certain special Lagrangian submanifold manifold KΣ Σ C (M ). Thus L of phase π/2 in M determines a J-holomorphic curve in KΣ the Calabi-Yau geometry can be interpreted as the hyperk¨ ahler geometry in C (M ). the isotropic knot space KΣ

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Moduli of Special Lagrangian submanifolds and A-branes. Recall that given any Lagrangian submanifold L in M , its neighborhood is isomorphic symplectically to T ∗ L and nearby Lagrangian submanifolds are graphs of closed one forms on L. When L is special, then its infinitesimal deformations are parametrized by harmonic one forms on L. McLean [108] proved that there is no obstruction to extend any infinitesimal deformation to a honest one, thus the moduli space MSLag (M ) of special Lagrangian submanifolds L in M is smooth with tangent space H 1 (L, R). Hitchin [61] defined analogs of the (multi-valued) Abel-Jacobi map, p : MSLag (M ) → H 1 (L0 , R) and p : MSLag (M ) → H n−1 (L0 , R)  immersion of MSLag (M ) into T ∗ and1 showed that 1(p, p ) is a Lagrangian H (L0 , R) ∼ = H (L0 , R) × H n−1 (L0 , R). To define p, we consider a path of special Lagrangian submanifolds Lt together with a loop γ t in each Lt , then p (L1 ) ∈ H 1 (L0 , R) ∼ = Hom (H1 (L0 , R) , R) is given by  γ1 p (L1 ) (γ 0 ) = ω. γ0

p

Similarly, is defined using Im Ω in place of ω. Note that the  moduli space of flat U (1)-connections over L is naturally H 1 L, S 1 = iH 1 (L, R) /H 1 (L, Z). Thus we could include flat U (1)connections over L to complexify the moduli space of special Lagrangian submanifolds to the moduli space of A-branes MA−brane (M ). It has a natural symplectic structure with a Lagrangian fibration,   H 1 L, S 1 → MA−brane (M ) → MSLag (M ) . Hitchin described MA−brane (M ) using the symplectic reduction method as follows: Fix a n-dimensional volume manifold (L, vL ), then the mapping space M ap (L, M ) has a natural Dif f (L, vL )-invariant symplectic form ω M ap given by  ω M ap (f ) (X, Y ) =

L

ω (X, Y ) vL ,

where X, Y ∈ T[f ] M ap (L, M ) ∼ = Γ (L, f ∗ TM ). When H 1 (L) = 0, this action is Hamiltonian and the moment map μ is given by μ (f ) = [α] ∈ Ω1 (L) /dΩ0 (L) with f ∗ ω = dα. Thus M ap (L, M ) //Dif f (L, vL ) = μ−1 (0) /Dif f (L, vL ) Lag ∼ = M[L] (M ) ,

those components containing f (L) in the moduli space of Lagrangian submanifolds in M .

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We can also restrict our attention to the complex submanifold M apΩ (L, M ) in M ap (L, M ) consisting of those f ’s satisfying f ∗ Ω = vL . Then SLag M apΩ (L, M ) //Dif f (L, vL ) ∼ = M[L] (M ) .

However, this is a discrete set as special Lagrangian submanifolds L with H 1 (L) = 0 are rigid. In general, when H 1 (L) = 0, the above construction has a natural generalization with the symplectic quotient isomorphic to (M ). MA−brane [L] Analogous to representing every deRham cohomology class by a unique harmonic form, we expect that most Hamiltonian equivalent classes of Lagrangian submanifolds in Calabi-Yau manifolds admits a unique special Lagrangian representative. The uniqueness question was studied by Thomas and Yau [120]. The existence part is a hard analytic problem. Schoen and Wolfson studied it using a variational approach while Smoczyk, M.T. Wang and others studied it using the mean curvature flow. Notice that the mean curvature flow preserves the class of Lagrangian submanifolds and its stationary points are given by special Lagrangian submanifolds inside CalabiYau manifolds. Special Lagrangian fibration is an important ingredient in the SYZ proposal to explain the mirror symmetry phenomenons. Since we do not know how to write down the Calabi-Yau metrics, it is in general very difficult to find such fibrations, with the exception of complex Lagrangian fibrations on hyperk¨ ahler manifolds. In the next section, we will explain how we attempt to construct special Lagrangian fibrations on Calabi-Yau hypersurfaces in CPn+1 . 6.3. Mirror symmetry. Mirror symmetry is a duality transformation which interchanges symplectic geometry and complex geometry between mirror Calabi-Yau manifolds (see e.g. [30]). Physical origin, a brief encounter. Mirror symmetry is originated from the physical studies of the superstring theory. The spacetime in superstring theory has dimension ten. In order to reduce to our usual four dimensional spacetime, we need to compactify six dimensions. Furthermore, this six dimensional internal space X must be a compact Calabi-Yau threefold, possibly coupled with a E8 × E8 -bundle over it depending on the types of string theory under considerations. There are particular topological sectors of this string theory, called the A-model and the B-model. From a mathematical point of view, they correspond to the symplectic geometry and the complex geometry of X. Motivated from physical considerations, Greene and Plesser predicted that there should be a conjugate theory in which the A-model and B-model switch to each other, at least near the large complex structure limit (abbrev. LCSL). The corresponding compactified Calabi-Yau threefold Y is called

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the mirror manifold to X. ←−−−−−−−−−−−→ A-model on X B-model on Y mirror symmetry (symplectic geometry) (complex geometry) The simplest Calabi-Yau threefold is the zero locus of a degree five homogeneous polynomial f in CP4 , the quintic Calabi-Yau threefold. For example if we take f (z0 , z1 , . . . , z4 ) = z05 + z15 + · · · + z45 + ψ (z0 z1 · · · z4 ) then X = {f = 0} is a smooth Calabi-Yau threefold, called the Fermat Calabi-Yau threefold, provided that ψ is any complex number not equal to one. Candelas et al [29] did a highly nontrivial calculation of this equivalence for the Fermat Calabi-Yau threefolds and showed physically that the number of rational curves of any degree in X can be read off explicitly from the periods of Y . This is an astonishing discovery as it relates two very different but equally important subjects in algebraic geometry, namely the enumerative geometry of X and the variation of complex structures of Y . Mirror of A-cycles and B-cycles. In 1994 Kontsevich [76] proposed a more precise conjecture on this duality between symplectic and complex geometries, called the homological mirror symmetry (HMS): If X and Y are mirror manifolds to each other, then the Fukaya-Floer category of Lagrangian intersections in X is equivalent to the bounded derived category of coherent sheaves on Y . HMS conjecture works for Calabi-Yau manifolds of any dimension. For K3 surfaces, this conjecture was verified by Seidel [115]. There are also generalizations of this duality for Fano manifolds and general type manifolds. The mirror symmetry conjecture predicts that (special) Lagrangian submanifolds should behave like (Hermitian Yang-Mills) holomorphic vector bundles, modulo quantum effects. Thomas and Yau [120] formulated a very interesting conjecture on the existence of special Lagrangian submanifolds which is the mirror of the theorem of Donaldson, Uhlenbeck and Yau on the existence of Hermitian Yang-Mills connections. SYZ proposal. Strominger, Yau and Zaslow proposed a resolution of mirror symmetry in their groundbreaking paper [116]. They conjectured that (i) mirror Calabi-Yau manifolds X and Y should admit special Lagrangian torus fibrations with sections in the large volume/complex structure limit; ∗ T dual ←−−−tori −→ T ↓ ↓ X Y ↓ ↓ B B∗

(ii) they are dual torus fibrations to each other; (iii) a fiberwise FourierMukai transformation along fibers interchanges the symplectic (resp.

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complex) geometry on X with the complex (resp. symplectic) geometry on Y . It roughly says that the mysterious duality is simply a Fourier transformation! The quantum corrections coming from holomorphic curves are higher Fourier modes. A brief reasoning behind SYZ is as follows: From physical considerations, B-branes are complex submanifolds, or more generally derived equivalent classes of complexes of coherent sheaves, and A-branes are special Lagrangian submanifolds coupled with unitary flat bundles. As mirror symmetry should identify the complex geometry of Y with the symplectic geometry of X, their moduli spaces of branes should be identified as well, at least at LCSL where quantum corrections had been suppressed. Since any space Y is always the moduli space of points which are complex submanifolds, Y should also be the moduli space of certain A-branes in X. Furthermore the underlying Lagrangian submanifolds L of these A-branes should cover X everywhere once, just like what points in Y did. Since deformations of L are parametrized by H 1 (L, R), we must have dim H 1 (L, R) = n and X should admit a special Lagrangian torus fibration π

T → X → B. When we consider the complex submanifold which is Y itself, the moduli space is a single point and the corresponding A-brane in X would be a rigid  special Lagrangian in X. Since Y [Y ]∪[pt] = 1, this rigid special Lagrangian submanifold should be a section to the above special Lagrangian fibration on X. Next, given any torus fiber T in X, its dual torus T ∗ parametrizes flat U (1)-bundles over T , namely A-branes in X with support T . Under mirror symmetry, this T ∗ also parametrizes corresponding B-branes in Y, which are points in Y . Thus T ∗ is a subspace in Y and therefore Y also has a torus fibration by such T ∗ ’s. T ∗ → Y → B∗. π

One can further argue that these two are dual special Lagrangian torus fibrations. Besides giving dual fibrations on mirror manifolds X and Y , we have a transformation between special Lagrangian fibers in X with zero dimensional complex submanifolds in Y , namely points. This is a special case of a fiberwise Fourier-Mukai transformation. For more general special Lagrangian submanifolds in X, say a section to the above fibration, then the intersection point of it with any fiber T would determine a flat U (1) connection on T ∗ because (T ∗ )∗ = T . By patching them for various fibers T , we obtain a U (1) connection on the whole manifold Y . One expects that this determines a holomorphic line bundle on Y which is the mirror to the section

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in X. This was verified in [101] in the semiflat case. We call this transformation between the symplectic geometry of X and the complex geometry of Y the SYZ mirror transformation. The SYZ transformation was generalized to the mirror symmetry for local Calabi-Yau manifolds by Leung-Vafa [97]. In [66], Hori-Vafa gave a physical proof of the mirror symmetry using the SYZ proposal. Special Lagrangian fibrations. It is difficult to find special Lagrangian fibrations on compact Calabi-Yau manifolds as we do not know their metrics well. For noncompact Calabi-Yau manifolds, there are examples with explicit Calabi-Yau metrics. Most of these examples also admit explicit special Lagrangian fibrations. For instance, π : C3 → R3   π (z1 , z2 , z3 ) = |z1 |2 − |z2 |2 , |z2 |2 − |z3 |2 , Im z1 z2 z3 is a special Lagrangian fibration on C3 with generic fibers T 2 × R topologically. In the following, we explain how we expect special Lagrangian fibrations should appear for hypersurfaces. Suppose that M is a degree n + 2 CY hypersurface in CPn+1 M = {f (z0 , z1 , . . . , zn+1 ) = 0} . The most singular one is given by the union of coordinate hyperplanes, namely M∞ = {z0 z1 · · · zn+1 = 0} ⊂ CPn+1 . For the family of Calabi-Yau manifolds Mt defined by f (z0 , z1 , . . . , zn+1 ) + t · z0 z1 · · · zn+1 = 0, the limit as t goes to infinity is called the large complex structure limit (abbrev. LCSL). It can be characterized in terms of the period and it is also called the maximal unipotent monodromy limit [29]. Notice that the n smooth part M∞ \Sing (M∞ ) is a union of (C× ) = T n ×Rn and one expects that these T n -fibration can be perturbed and extended to give a special Lagrangian fibration on Mt for t large. Without the special condition, this approach was carried out by Gross, Mikhalkin, Ruan and Zharkov. However the question of whether one can make the Lagrangian fibrations on X special is a much more delicate question as we do not understand the behavior of the Calabi-Yau metrics, whose existences are asserted by the celebrated theorem of Yau [137]. This approach can be generalized to Calabi-Yau hypersurfaces X in any Fano toric variety PΔ . Furthermore, their mirror manifolds Y are Calabi-Yau hypersurfaces in another Fano toric variety P∇ whose defining polytope is the polar dual to Δ. Thus we can see that the Lagrangian fibration structures on X and Y should be given by dual tori, at least away from singular fibers.

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The situation is quite different for Calabi-Yau twofolds, namely K3 surfaces, or more generally for hyperk¨ ahler manifolds. In this case, the CalabiYau metric on X is K¨ahler with respect to three complex structures I, J and K. When X admits a J-holomorphic Lagrangian fibration, then this fibration is a special Lagrangian fibration with respect to the K¨ ahler metric ω I , as well as ω K . Furthermore, SYZ also predicts that mirror symmetry is merely a twistor rotation from I to K in this case. For K3 surfaces, there are plenty of elliptic fibrations and they are automatically complex Lagrangian fibrations because of their low dimension. Furthermore Gross-Wilson [57] described the Calabi-Yau metrics for generic elliptic K3 surfaces by using the singular perturbation method. They used model metrics which were constructed by Greene, Shapere, Vafa and Yau [51] away from singular fibers and by Ooguri-Vafa [111] near singular fibers. Recall that the base space B of any compact Lagrangian fibration π

T → X → B, admits a canonical integral affine structure, possibly with singularities. This affine structure will dictate the Calabi-Yau geometry at the large complex structure limit. Outside the preimage of the singular set of B, the total space X is given by the quotient of the cotangent bundle T ∗ B by a lattice subbundle symplectically. In order to understand the A-model on X, we need to be able to describe rational curves and holomorphic disks on X in terms of the affine structure on B. There has been much progress on this by the work of Fukaya, Kontsevich-Soibelman, Siebert-Gross and others. Here tropical geometry plays an important role. The tangent bundle T B of any affine manifold B admits a canonical complex structure away from its singularities. Kontsevich and Soibelman [77], Gross and Siebert [55] described how to deform this complex structure at the large complex structure limit to nearby complex structures. Recall that the physical calculations of Candelas et al [29] showed that the variation of their Hodge structures should determine the Gromov-Witten invariants of rational curves of the mirror manifold. This important formula was later proven by Givental, Lian, Liu and Yau via a clever computation of Gromov-Witten invariants using localization method. The above program will eventually give a mathematical explanation of this phenomenon. Explicit SYZ mirror transformation. Leung-Yau-Zaslow [101], [94] used the SYZ transformation to verify various correspondences between symplectic geometry and complex geometry between semi-flat Calabi-Yau manifolds. In this situation, there is no quantum corrections from instantons, namely rational curves or holomorphic disks. To include quantum corrections in the SYZ transformation for Calabi-Yau manifolds is a much more difficult problem. However in the Fano case, there are recent results on

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applying the SYZ transformation with quantum corrections by Auroux [8], Chan-Leung [31][32]. In order to understand the mirror transformation, we look at the simplest example, namely Cn . If we view Cn as a tangent bundle over B ∼ = Rn , M = TB then it has a canonical complex structure with holomorphic volume form ΩM = dz 1 ∧ dz 2 ∧ · · · ∧ dz n . If we view Cn as a cotangent bundle of B ∼ = Rn , W = T ∗B then it has a canonical symplectic form  ωW = dxj ∧ dyj . Every linear subspace P in B determines a complex subspace T P in M ∗ as well as a Lagrangian subspace NP/B in W . They are fiberwise dual to each other with respect to the natural projections of T B and T ∗ B to B. Any Lagrangian section L in W is the graph of a closed one form η on B. Let DA = d + iα be a flat U (1)-connection over L, then ∂¯E = ∂¯ + iα + β defines a new holomorphic structure on the topologically trivial complex line bundle over M . These are the simplest Fourier-Mukai transformations, or the mirror transformations. Any Riemannian metric g on B ∼ = Rn induces compatible Riemannian metric gM and gW on (co)tangent bundles M and W . Suppose g is a Hessian metric, i.e. there is a convex function φ : B → R such that gij =

∂2φ . ∂xi ∂xj

Then gM is a CY metric on M = T B if and only if φ satisfies the real MongeAmp`ere equation 2

∂ φ det = 1. ∂xi ∂xj On the other hand, gW is a CY metric on W = T ∗ B if and only if the Legendre transformations ψ of φ satisfies the real Monge-Amp`ere equation

2 ∂ ψ = 1. det ∂xi ∂xj We recall that the Legendre transformation of a convex function φ : B → R is another convex function ψ : B ∗ → R satisfying  φ+ψ= xj xj

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where the dual affine spaces B and B ∗ are identified under the map B → B∗   ∂φ xj x1 , . . . , xn = j . ∂x From above discussions, it is not surprising that the mirror symmetry transformation should be a fiberwise FM transform coupled with a Legendre transform along base directions. In simple situations, namely semi-flat CY manifolds [101][94], or Fano toric manifolds [32], explicit mirror transformations can be constructed to explain various dualities between complex and symplectic geometries. However, the general situation is still far from having a complete understanding, despite recent progress by Gross-Siebert [56]. 6.4. K3 surfaces. K3 surfaces as one dimension H-manifolds. In this section, we study two dimensional Calabi-Yau manifolds M in greater details. Since SU (2) = Sp (1), a Calabi-Yau surface is the same as an one H-dimensional hyperk¨ ahler manifold (section 10). This is similar to the fact that every oriented surface is a complex curve because of the isomorphism SO (2) = U (1). The volume form on a Riemann surface is always a (K¨ ahler) symplectic form. Similarly, the holomorphic volume form ΩJ ∈ Ω2,0 (M ) on a Calabi-Yau surface is always a holomorphic symplectic form. We decompose ΩJ into real and imaginary parts, ΩJ = ω I − iω K and we define I and K by ω I (u, v) = g (Iu, v)

and ω K (u, v) = g (Ku, v) .

Then just like the original complex structure J on M , both I and K are orthogonal complex structures on M with K¨ ahler forms ω I and ω K respectively. Furthermore they satisfy the Hamilton relation I 2 = J 2 = K 2 = IJK = −id. This gives an explicit description of the H-structure on any Calabi-Yau surface. See section 11.1 for more discussions on hyperk¨ahler manifolds. There are only two classes of compact Calabi-Yau surfaces, namely complex tori C2 /Λ and K3 surfaces. A K3 surface is a simply connected K¨ahler surface M with c1 (M ) = 0. Siu and Todorov proved that the K¨ ahlerian property of M is automatic. The 2 cohomology group H (M, Z) together with the quadratic form qM given by the intersection product is isomorphic to





0 1 0 1 0 1 LK3 = (−E8 ) ⊕ (−E8 ) ⊕ ⊕ ⊕ 1 0 1 0 1 0

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where E8 is the Cartan matrix for the exceptional Lie group E8 . In particular, qM has signature (3, 19). Every smooth quartic surface in CP3 is a K3 surface. Also, given any complex torus A = C2 /Λ, then A/Z2 has 16 ordinary double points. By blowing them up, we obtain a K3 surface. A particular nice class of K3 surfaces M are elliptic K3 surfaces with sections. Moduli of K3 surfaces. The moduli space of complex structures on K3 surfaces can be described in terms of the period map τ . Given a K3 surface M with holomorphic symplectic form ΩJ , the span of [Re ΩJ ] and [Im ΩJ ] in H 2 (M, R) defines an element, called the period τ , in the period domain D, D = SO+ (3, 19)/SO(2) × SO(1, 19). Here we have fixed an isomorphism between H 2 (M, Z) with a fixed lattice LK3 . It is a nontrivial fact that the period map is a global isomorphism, called the global Torelli theorem for K3 surfaces. If we restrict to K3 surfaces which are projective, then the moduli space of marked projective K3 surfaces is SO+ (2, k)/SO(2) × SO(k), with 1 ≤ k ≤ 18 depending on the rank of H 2 (M, Z) ∩ H 1,1 (M ). This is a Hermitian symmetric space of type IV. Note that every K3 surface can be deformed to a projective surface. Every Einstein metric g on M is automatically K¨ ahler by observations of Hitchin and Todorov. Therefore it determines a S 2 -family of complex structures on M . The analog of the period map for Einstein metrics on M associates to each Einstein metric g the span of [ω I ], [ω J ] and [ω K ] in 2 (M ) the space of self-dual harmonic two H 2 (M, R), which coincides with H+ forms on M . Thus the moduli space of marked (orbifold) Einstein metrics on M with unit volume is SO+ (3, 19)/SO(3) × SO(19). If we allow the volume of M to vary, then this adds a R+ -factor to the above moduli space. From physical motivations, we also consider B-fields which are elements in H 2 (M, U (1)). Then this extended moduli space [2] is isomorphic to SO+ (3, 19) SO+ (4, 20) × R+ × H 2 (M, R)  . SO (3) × SO (19) SO (4) × SO (20) The above isomorphism is defined as follows: The lattice of the total cohomology is







0 1 0 1 0 1 0 1 ∗ H (X, Z)  (−E8 ) ⊕ (−E8 ) ⊕ ⊕ ⊕ ⊕ 1 0 1 0 1 0 1 0

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and it has signature (4, 20). Given any (H+ , v, B) we associate to it a space∗ like 4-dimensional space in H R) spanned by x − q (x, B) [M ] with x ∈  (M,  1 H+ , together with 1 + B + v − 2 q (B) [M ]. There are conjectural dualities between the geometry, or more precisely the physics, of K3 surfaces and flat tori T d coupled with flat E8 ×E8 -bundles for small d. Recall that the moduli space of flat tori is SO+ (d, d)/SO(d) × SO(d). When d equals 2 (resp. 3 and 4), the corresponding structures on K3 surfaces for this duality are algebraic elliptic K3 surfaces with sections (resp. Einstein metrics and Einstein metrics with B-fields). Bundles over K3 surfaces. For any vector bundle E over a K3 surface M with a fixed complex structure J, the integrability condition FA0,2 = 0 for a connection DA to define a holomorphic structure on E is equivalent to FA ∧ ΩJ = 0 where ΩJ = ω I − iω K . Note that FA0,2 = FA ∧ ω J = 0 is identical to the ASD equation FA+ = 0 and this system of equations is equivalent to FA0,2 = 0 with respect to I, J and K, i.e. a tri-holomorphic bundle. This is also the same as the following system of equations, FA ∧ ω I = FA ∧ ω J = FA ∧ ω K = 0. Recall that FA ∧ω J is the moment map for a gauge group action on the space of unitary connections on E. Therefore, the moduli space MASD of ASD connections, or polystable bundles, on M can be regarded as a hyperk¨ ahler quotient A (E) ///G (E). In particular, it is a hyperk¨ ahler manifold (section 11), but not necessarily compact. Using the algebraic geometry approach, Mukai [109] showed that the moduli space of polystable coherent sheaves on M admits a canonical holomorphic symplectic form. Examples of such include the Hilbert scheme of n-points in M and the universal compactified Jacobians of curves in M . Holomorphic curves vs special Lagrangians in K3. Given any real surface C in M , it is a J-holomorphic curve if and only if ΩJ |C = 0, i.e. C is a J-complex Lagrangian submanifold in (M, ΩJ ). In particular, the normal bundle of C is isomorphic to the cotangent bundle of C. For instance, every rational curve in M is a (−2)-curve in the sense that the degree of the normal bundle is −2. Since ΩJ = ω I − iω K , C is a Lagrangian submanifold in M with respect to both the symplectic forms ω I and ω K . Of course, this is also equivalent to ω K |C = 0 and Im ΩK |C = 0, i.e. C is a special Lagrangian submanifold in the Calabi-Yau surface (M, K, ω K , ΩK ). Therefore, if (M, J) has an elliptic fibration with section, then the same fibration is a special Lagrangian fibration with section on (M, K, ω K , ΩK ) . In an elliptic fibration with section on a K3 surface M , the section S is always a smooth rational curve, i.e. CP1 . In the generic case, singular fibers

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are nodal rational curves with one node. The total number of singular fibers is 24. This is because the Euler characteristic e (M ) = 24 and smooth elliptic fibers have zero Euler characteristic. Yau-Zaslow formula. Motivated from physical considerations, YauZaslow [138] found an amazing formula for the number of rational curves on K3 surfaces in terms of a quasi-modular form. This conjectural formula was generalized by G¨ ottsche [49] to arbitrary genus (and arbitrary projective surface): Suppose C is a holomorphic curve in M representing a cohomology class [C] with q (C) = 2d − 2 and its divisibility, or index, as r. If C is a smooth curve, then d is equal to the genus of C and also to the dimension of the linear system of C. If we denote the number of genus g curves in X representing [C] as Ng (d, r). Then the Gottsche-Yau-Zaslow formula says that ⎛ ⎛ ⎞ ⎞g ∞    # 1 24 d k−1 Ng (d, r) q = ⎝ k⎝ d⎠ q ⎠ 1 − qd d≥0 k=1 d≥1 d|k

g d q = G2 (q) . dq Δ (q) where G2 (q) is the Eisenstein series. Notice that a generic K¨ahler K3 surface has no curve at all. Using the family Gromov-Witten invariants for the twistor family of complex structures on M , Bryan-Leung [19][20] gives a well-defined definition of Ng (d, f ). When [C] is a primitive class, i.e. r = 1, this formula was proved in [19] and when g = 0, i.e. the original YZ conjecture, it was proved by Klemm-Maulik-Pandharipande-Scheidegger [74]. 7. Calabi-Yau 3-folds Calabi-Yau manifolds M of complex dimension 3 have particularly rich geometry. It is related to the fact that M × S 1 is a G2 -manifold and their geometries are governed by the largest normed division algebra, namely the octonion O (see section 10). Physically this is also the most important dimension in superstring theory. 7.1. Moduli of CY threefolds. When M is a Calabi-Yau threefold with a fixed holomorphic volume form Ω, there are natural cubic forms ∗ ) and H 1 (M, T ), called the A-Yukawa coupling Y defined on H 1 (M, TM M A and the B-Yukawa coupling YB defined as follows: Recall that every element ∗ ) ∼ H 1,1 (M ) can be represented by an (1, 1)-form, then φ ∈ H 1 (M, TM = 3 $

∗ H 1 (M, TM )→C  YA (φ1 , φ2 , φ3 ) = φ1 ∧ φ2 ∧ φ3 .

YA :

M

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If we represent η i ∈ H 1 (M, TM ) by an element in Ω0,1 (M, TM ), then wedging them together and contracting with Ω gives (η 1 ∧ η 2 ∧ η 3 )Ω ∈ Ω0,3 (M ). Then YB :

3 $



YB (η 1 , η 2 , η 3 ) =

H 1 (M, TM ) → C

M

Ω ∧ (η 1 ∧ η 2 ∧ η 3 )Ω.

The moduli space Mcpx of complex structures on Calabi-Yau 3-folds is a projective special K¨ ahler manifold, as first observed by Bryant-Griffiths [25] (see also [43]). If we include the choice of a holomorphic volume form %cpx is a C× -bundle over Mcpx and on M , then this extended moduli space M it is a special K¨ahler manifold. Let us recall the definition of special K¨ ahler manifold. For any K¨ ahler manifold, its K¨ ahler form is locally determined by a real valued function φ, called the K¨ ahler potential, i.e. ω = i∂∂φ (z, z¯), while for special K¨ ahler manifold, it is determined by a holomorphic function F (z), called the prepotential, which satisfies

∂F i 1 φ = Im z¯ , 2 ∂z i for suitable holomorphic coordinates z’s. From an intrinsic point of view, a K¨ ahler manifold (M, ω) with a torsion free flat symplectic connection ∇ is a special K¨ahler manifold if ∇ ∧ J = 0. Such a manifold has a local holomorphic function F, called the prepotential, which governs the special K¨ ahler geometry. In terms of a suitable dual holomorphic Darboux coordinates z’s and w’s, we have ∂F wj = j , ∂z for all j. When X is a special K¨ahler manifold, its cotangent bundle M = T ∗ X admits a natural hyperk¨ ahler structure. %cpx , we note that the tangent space at [M ] ∈ To define this structure on M % Mcpx (resp. [(M, Ω)] ∈ Mcpx ) is naturally identified with H 1 (M, TM ) ∼ = H 2,1 (M ) (resp. H 3,0 (M ) ⊕ H 2,1 (M )). Notice that the Poincar´e pairing on the complex vector space H 3 (M, C) defines a holomorphic symplectic form and such that H 3,0 (M ) ⊕ H 2,1 (M ) is a complex Lagrangian subspace in it. We choose a symplectic basis Ai ’s, Bi ’s for the middle homology group H3 (M, Z) modulo torsion, i.e. Ai ∩ Aj = 0 = Bi ∩ Bj Then

and Ai ∩ Bj = δ ij .

 i

z =

Ai

Ω

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%cpx . Similarly defines a local holomorphic coordinate system on M  wi = Ω Bi

define another such coordinate and it is related to z i ’s by a holomorphic   Legendre transformation, i.e. there exists a local holomorphic function F z i such that ∂F wi = i . ∂z %cpx . This gives the prepotential function on M i %cpx , one has In terms of this coordinate z ’s on M YB = ∇3 F. %cpx . Such structures determine a so-called Frobenius manifold structure on M In general, the cotangent bundle of any special K¨ ahler manifold admits a canonical hyperk¨ ahler structure. In our situation, the fibers of the cotangent %cpx can be identified as the universal covering of the intermediate bundle of M Jacobian J (M ),   J (M ) = H 3 (M, Z) \H 3 (M, C) / H 3,0 (M ) ⊕ H 2,1 (M ) . %cpx is the universal intermediate Jacobian J univ Therefore H 3 (M, Z) \T ∗ M for Calabi-Yau threefolds. It admits a hyperk¨ ahler structure with a complex Lagrangian fibration over the extended moduli space of Calabi-Yau threefolds, %cpx . π : J univ → M We remark that the second fundamental form of a Lagrangian  submanifold L is always a symmetric cubic tensor in Γ L, Sym3 TL∗ . For a Lagrangian fibration, if we integrate the second fundamental forms for fibers, then we obtain a cubic tensor on the base of the fibration. For our complex %cpx , this cubic tensor is nothing but the Lagrangian fibration J univ → M B-Yukawa coupling YB . 7.2. Curves and surfaces in Calabi-Yau threefolds. A real codimension two submanifold S in M is a complex surface if and only if Ω|S = 0. Given any complex surface S in M , its normal bundle NS/M equals to the canonical line bundle KS of S. When S is a del Pezzo surface, for instance CP2 , NS/M is negative and therefore S can be contracted to a point. We call KS a local Calabi-Yau manifold. When S is a Calabi-Yau surface, i.e. a K3 surface or an Abelian surface, then NS/M is trivial and S can be deformed and gives a complex surface fibration structure on M . Similarly when C is an elliptic curve, it often happens that C is a fiber of an elliptic fibration on M . There has been much studies on elliptic Calabi-Yau threefolds, including the geometry of its K¨ahler cone by Wilson, stable bundles over them by Donagi, Friedman, Morgan and others.

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Flops, smoothing and extremal transitions. When C is a smooth rational curve in M , then NC/M ∼ = O (−d) ⊕ O (d − 2) for some integer d. In the generic situation with NC/M ∼ = O (−1) ⊕ O (−1) negative, C is called an (−1, −1)-curve, then C can be contracted to a rational double point, i.e. locally given by   2 z1 + z22 + z32 + z42 = 0 ⊂ C4 . There is another small resolution of this rational double point and gives another (−1, −1)-curve C  as the exceptional set. This birational transformation is called a simple flop. Suppose M  is obtained by flopping C in M , the cohomology rings of M and M  are different, in particular the topology of M changes under flops. Nevertheless, if we consider the quantum product by including contributions from genus zero GW invariants to the classical cup product, then the quantum cohomology ring of M and M  are the same [86]. In general, we expect that the quantum geometries, or the string theories, of M and M  are the same. We could also consider the smoothing of the rational double point by deforming the complex structure to  2  z1 + z22 + z32 + z42 = t ⊂ C4 , for small non-zero t. Topologically, this process replaces the singular point   by a (Lagrangian) S 3 , which is given by z12 + z22 + z32 + z42 = t ∩ R4 when t > 0. This is called a vanishing cycle for the isolated singular point. Indeed this is the mirror to the blowing up process which replaces the singular point by a (holomorphic) S 2 ∼ = CP1 . The process of contracting a (−1, −1)curve followed by a smoothing is called an extremal transition. The mirror symmetry conjecture has predicted several surprising consequences for such transitions, including a formula relating open GW invariants on O (−1) ⊕ O (−1) over S 2 with the Chern-Simons theory on S 3 . Thus the rational double point can be resolved in three different ways. One by smoothing (with a S 3 vanishing cycle) and two by blowing up (with a S 2 exceptional cycle). When we view these from the G2 perspectives, all three look the same and they are related by a triality symmetry (section 8.5). GW-invariants and multiple cover formula for CY3 . For any Calabi-Yau 3-fold M , the moduli space of genus g holomorphic curves C in M always has expected dimension zero. Thus we denote the GW-invariant which counts the number of genus g curves in M representing a class β ∈ H2 (M, Z) as Ngβ (M ). Notice that GW-invariants count the number of stable maps from genus g curves to M and every curve C ⊂ M representing β ∈ H2 (M, Z) will contribute to the GW-invariant for the class dβ because of a degree d stable map f : Σ → M could multiply cover its image f (Σ) = C. Such contributions to the GW-invariants should be subtracted in order to count the number of

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curves in M . The multiple cover formulas tell us how each curve in the class β contributes to the GW-invariant for the class dβ. In the genus zero case, the contribution is d−3 when C is an (−1, −1)-curve. There are also multiple cover formula for other contractible curves [17]. Superpotential for curves. Dimension being three plays a special role here as holomorphic curves are critical points of the following Chern-Simons type functional Φ among real surfaces Σ in M ,  Φ (Σ) = Ω B

Ω3,0 (M )

is the holomorphic volume form on M and B is a real where Ω ∈ three dimensional singular chain in M with ∂B = Σ − Σ0 for any fixed background surface Σ0 in M . %univ Thus the universal moduli space M curve of curves in Calabi-Yau 3-folds %cpx . We recall the universal is expected to be a generic finite cover of M Abel-Jacobi map univ %univ . AJ : M curve → J To explain it, we fix a background C0 and for any holomorphic curve C in M homologous to C0 we define AJ (C) ∈ H 3 (M, Z) \H 3 (M, C) /(H 3,0 (M ) ⊕ H 2,1 (M )) by evaluating it on any cohomology class η ∈ H 3 (M, C) as follows,  η, AJ (C) [η] = B

where B is any singular chain in M with ∂B = C − C0 . The map AJ defines a complex Lagrangian subspace in J univ [121]. Locally, it is given by the %cpx , where graph of an exact holomorphic one form dΦ on M %univ Φ:M curve → C  Ω. Φ (M, C) = B

This is called the superpotential in the physics literature. Gopakumar-Vafa conjecture. A holomorphic curve C ⊂ M coupled with a holomorphic line bundle L over C is called a BPS state in string theory. The moduli space MBP S (M ) of such pairs (C, L) carries a forgetful map to the moduli space of curve in M , π : MBP S (M ) → Mcurve (M ) . One would like to find a good compactification of MBP S (M ). For K3 surfaces, Yau and Zaslow [138] studied this space and derived an amazing formula for the generating functions of the number of rational curves in K3 surfaces, the Yau-Zaslow formula. From physical considerations, Gopaku-  mar and Vafa [48] conjectured that the cohomology group H ∗ MBP S (M ) should admits an sl (2, R) × sl (2, R)-action and suitable combinations of

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its multiplicities are called the BPS number denoted nβg (M ) ∈ Z and they determine Gromov-Witten invariants of M of every genus explicitly by the following formula,  g≥0 β∈H2 (M,Z)

Ngβ

(M ) u

2g−2 β

q =



nβg

g≥0 β∈H2 (M,Z)

1 ku 2g−2 kβ 2 sin (M ) q k 2 k>0

where Ngβ (M ) is the GW-invariant and nβg (M ) is the BPS number. 7.3. Donaldson-Thomas bundles over Calabi-Yau threefolds. Holomorphic bundles E over a Calabi-Yau threefold M is another type of BPS states in string theory. Recall that a connection DA defines a holomorphic structure on E if and only if (FA )0,2 = 0. Similar to the special feature of flatness FA = 0 on three manifolds, (FA )0,2 = 0 is the Euler-Lagrange equation for the holomorphic Chern-Simons functional CShol on Calabi-Yau threefolds M ,

 2 CShol (DA ) = Ω ∧ T r A ∧ dA + A ∧ A ∧ A , 3 M where DA = DA0 + A for some fixed background connection DA0 . Equivalently,  CShol (DA ) = Ω ∧ T rFA2¯ , M ×[0,1]

with DA¯ a connection over M × [0, 1] extending DA0 and DA on its boundaries. This is similar to the functional Φ for real surfaces in M . We can %univ → J univ on therefore have an analog of the Abel-Jacobi map AJbdl : M bdl %univ of holomorphic bundle over Calabi-Yau the universal moduli space M bdl 3-folds and it is an Lagrangian with superpotential given by %univ → C Φbdl : M bdl  Ω ∧ T rFA2¯ . Φ (M, DA ) = M ×[0,1]

Indeed, this is a special case of the special geometry of cycles in G2 -manifolds [73] (section 8). For Calabi-Yau threefolds M , using techniques from algebraic geometry, Donaldson and Thomas [40] define a count DTM for the number of stable holomorphic bundles over M which is invariant under deformations of complex structures on M . Indeed one can allow E to be singular as well, namely a coherent sheaf on M . For example every holomorphic curve C in M defines an ideal sheaf IC on M which is automatically stable and the corresponding

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Donaldson-Thomas invariants DTM are conjectured by Maulik-NekrasovOkounkov-Pandharipande [107] to be related to GW-invariants by ZDT,β (q) = ZGW,β (u) ZDT,0 (q)

 iu + q = 0. Here Z n after a change of coordinate e (q) = DT,β n In,β q and  ZGW,β (u) = g Ng,β u2g−2 . It is proven that ZDT,0 (q) is given by the  MacMahon function M (q) = n≥1 (1 − q n )−n . Recently Pandharipande and Thomas [113] defined a related invariant by counting stable pairs OM → OC in the derived category Db (M ) instead of IC and this has the effect of getting rid of the contributions of embedded points in C to the DT-invariants. They are called the PT-invariants and denote ZP T,β (u). They conjectured the following relationship ZDT,β (q) = ZP T,β (u) ZDT,0 (q) Pandharipande and Thomas formally defined BPS count nβg in terms of PT-invariants for irreducible curve classes. 7.4. Special Lagrangian submanifolds in CY3 . An A-brane (L, DA ) in a Calabi-Yau manifold M consists of a Lagrangian submanifold L in M together with a flat U (1)-connection over L. In complex dimension three, A-branes are critical points of the following Chern-Simons type functional  L1

L0

(ω + FA )2

where (Lt , DAt ) is a smooth path of A-branes connecting a background (L0 , DA0 ) to (L1 , DA1 ) = (L, DA ). The moduli space of A-branes has a natural symplectic structure and admits action by the group of gauge transformation G =  a Hamiltonian  C ∞ M, S 1 of the flat line bundle over L. The corresponding moment map μ is given by μ (L, DA ) = Im Ω|L . Therefore the symplectic quotient is the moduli space of special Lagrangian submanifolds in M [119]. Because of these special properties in dimension three, one might expect to be able to count the number of special Lagrangian spheres in Calabi-Yau threefolds, as studied by Joyce [72] and other people. 7.5. Mirror symmetry for Calabi-Yau threefolds. General mirror symmetry conjectures are expected to hold true for Calabi-Yau manifolds of any dimension. The conjecture was originated from the studies of string theory in which the Calabi-Yau manifolds always have complex dimension

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three. Indeed there are many mysterious dualities for the geometry/physics of Calabi-Yau threefolds, which are not shared by Calabi-Yau manifolds of higher dimensions. A reason that Calabi-Yau threefold geometry is much richer is due to the fact that SU (3) ⊂ G2 = Aut (O). As a matter of fact, many of these dualities comes from dualities/triality in G2 -geometry/physics conjecturally. We have explain that the complex geometry and the symplectic geometry for Calabi-Yau threefolds are expected to be unchanged under flops and (mirror) transformed to each other under extremal transitions. This includes the large N duality for Chern-Simons theory of knots in S 3 . Nekrasov argued physically that the large N Chern-Simons theory on S 3 can also be described in terms of the geometry of the moduli spaces of ASD connections on R4 . Very roughly speaking, the reason is as S 3 shrinks to a point in its cotangent bundle and creates a conical singularity, the physics of the ten dimensional spacetime T ∗ S 3 × R4 can be approximated by the physics of the base manifold R4 . Atiyah-Witten [7] observed that both the O (−1, −1) bundle over S 2 where ordinary flops take place and the cotangent bundle over S 3 where extremal transitions take place are quotients of the spinor bundle over S 3 , a seven dimensional G2 -manifold, by S 1 in different ways. These different ways are connected by G2 triality flops. Thus when lifting to the G2 -geometry, flops and extremal transitions are the same operation. Only when we quotient it by S 1 to obtain Calabi-Yau manifolds, then we break the symmetry. We expect that G2 -flops on S 3 will preserve both the associative geometry and the coassociative geometry on G2 -manifolds. The closed on M × R3,1 is conjecturally dual to the M 1 string  theory 3,1 theory on S × M × R . This led to the conjecture of Gopakumar-Vafa [48] and eventually the many developments of counting curves in Calabi-Yau threefolds, including the Donaldson-Thomas invariants, the PandharipandeThomas invariants and their relationships with the Gopakumar-Vafa BPSinvariants. For the open string theory on X × R3,1 with the boundaries of string lying on a special Lagrangian submanifold L in M , the conjectural M-theory dual is a G2 -manifold X which admits a S 1 -fibration to M whose fibers over L degenerate to points. For M-theory on more general G2 -manifolds X, the conjectural duals are Calabi-Yau threefolds M coupled with E8 ×E8 -bundles, as studied by Gukov-Yau-Zaslow [58]. Another link between Calabi-Yau threefolds and G2 -manifolds comes from the topological M-theory [33] in which the Hitchin volume functional on the space of three forms is being quantized. The natural projectively flat connection over the parameter space can be interpreted as the BCOV anomaly equation [12] for Calabi-Yau threefolds. The BCOV theory was originally arisen from studies of the Kodaira-Spencer theory of gravity and their mirror symmetry between A- and B-models. The BCOV mirror conjecture is a powerful tool to compute higher genus Gromov-Witten invariants

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for Calabi-Yau threefolds as their generating function is the partition function in A-model of this theory. 8. G2 -geometry 8.1. G2 -manifolds. A seven dimensional Riemannian manifold (M, g) is called a G2 -manifold if it has a parallel vector cross product (VCP) on its tangent spaces. Recall that a linear homomorphism J : Rm → Rm is an orthogonal complex structure if and only if it satisfies (i) Ju⊥u and (ii) |Ju| = |u| for any vector u. The only constraint is m being an even integer. Now we generalize it to a bilinear homomorphism × : Rm ⊗ Rm → Rm by requiring (i) u × v perpendicular to both u and v and (ii) |u × v| equals the area of the parallelogram spanned by u and v. Obviously the standard vector product, or sometimes called the cross product, in R3 is an example of such. One way to express this product is by identifying R3 with Im H and defining u × v = Im (u · v) with u · v the product of quaternions. The same formula also defines a VCP on the imaginary octonions R7 ∼ = Im O. In fact these are all VCPs [50]. The octonion product can also be recovered from the VCP, and thus the group of isometries of R7 preserving × equals the group of algebra automorphisms of O, which is the exceptional Lie group G2 . Therefore G2 manifolds do have holonomy groups G2 . Since SU (3) ⊆ G2 , the product of any Calabi-Yau threefold with R is always a G2 -manifold. Like CY manifolds, G2 -manifold always Einstein manifolds, i.e. Rc = 0. The analog of the K¨ ahler form ω (u, v) = g (Ju, v) is the G2 -form Ω ∈ Ω3 (M ) given by Ω (u, v, w) = g (u × v, w) . When M = X × R with X being a Calabi-Yau threefold with K¨ ahler form ω X and holomorphic volume form ΩX , then we have Ω = Re ΩX − ω X ∧ dt. Ω is a three form analog of a symplectic form as for any x ∈ M , Ωx ∈ Λ3 Tx∗ M lies in an open (positive) GL (7, R)-orbit. Unlike a symplectic form, the metric g is determined by Ω, g (u, v) vM = ιu Ω ∧ ιv Ω ∧ Ω. Furthermore, ∇Ω = 0 can be reduced to ΔΩ = 0. Thus a G2 -manifold can be characterized as a seven dimensional manifold processing a harmonic positive three form Ω. Despite such simplifications, there is no general existence result for compact G2 -manifolds, analogous to Yau’s theorem for Calabi-Yau manifolds. Using singular perturbation method to resolve singularities of T 7 /Γ for certain finite group Γ-action, Joyce [70] constructed the first nontrivial compact examples. Explicit noncompact examples were constructed earlier by Bryant,

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Salamon and others by imposing symmetries to reduce the G2-condition to an ODE. This included total spaces of the spinor bundle S S 3 over S 3 , the     self-dual two-form bundles Λ2+ S 4 and Λ2+ CP2 . Analogous to the (p, q)-decomposition for K¨ ahler manifolds, in the G2 setting, we have Ω1 (M ) = Ω17 Ω2 (M ) = Ω27 ⊕ Ω214 Ω3 (M ) = Ω31 ⊕ Ω37 ⊕ Ω327 , and Ωkd ∼ by the Hodge star operator ∗. These components correspond = Ω7−k d to the irreducible decomposition of G2 acting on Λ∗ R7 . Here the subscript indicate the dimension of these irreducible components inside Λ∗ R7 . We have g ∧Ω Ω (i) Λ31 is spanned by Ω, (ii) Λ17 ∼ = Λ27 ∼ = Λ37 via T ∗ → T → Λ2 T ∗ and T ∗ → ∗ Λ4 T ∗ → Λ3 T ∗ , (iii) Λ214 corresponds to g2 ⊂ so (7) ∼ = Λ2 R7 and (iv) Λ327 ∼ = 2 7 Sym0 R corresponds to deformations of the G2 -metrics with fixed volume. 8.2. Moduli of G2 -manifolds. Given any G2 -manifold M with G2 form Ω, we define a G2 -Yukawa coupling as follows 3 $

H 3 (M ) → R    ˜ ∧φ ˜ ∧φ ˜ Ω YΩ (φ1 , φ2 , φ3 ) = ∗Ω ∧ φ 1 2 3 YΩ :

˜ ∈ Ω1 (M, TM ) is defined by where for any φi ∈ Ω3 (M ), φ i ιv1 ∧v2 φi = ιv1 ∧v2 ∧φ˜ Ω ∈ Ω1 (M ) i

for any vector v1 and v2 . It satisfies YΩ (Ω, Ω, Ω) = YΩ (Ω) =



2

|Ω| =

 Ω ∧ ∗Ω.

When M = X × S 1 , then YΩ is a combination of YA and YB for the CalabiYau threefold X. Unless a compact G2 -manifold is a metric product M = X × S 1 up to a finite cover, the holonomy group is the full G2 . Via Bochner argument, we have H 1 (M ) = 0 and therefore H7k (M ) = 0 for any k. Thus H 3 (M ) ∼ = 3 (M ) parametrizes infinitesimal deformations of G -metrics H13 (M ) ⊕ H27 2 on M . If we fix a marking, i.e. an isomorphism from H3 (M, Z) /T or to a fixed ˜ G of marked G2 -structure on M is locally lattice, then the moduli space M 2 3 isomorphic to H (M ) and given by ˜ G → H 3 (M ) p:M 2 (M, Ω) → [Ω] .

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˜ G consisting of those If we restrict our attention to MG2 ⊂ M 2 G2 -metrics with unit total volume, then up to a positive constant, the WeilPetersson metric is equal to YΩ (Ω, −, −), or ∇2 YΩ (Ω). In particular, the metric is of Hessian type. This is the real analog to the special K¨ahler structure on the moduli space of complex structures on Calabi-Yau threefolds. Such a geometry is captured by the cubic form ∇3 YΩ (Ω), which is nothing but the G2 -Yukawa coupling YΩ itself. Similarly to the universal intermediate Jacobian in the Calabi-Yau threefolds situation, we construct the bundle ˜G JG2 → M 2   3 1 whose fiber over (M, Ω) is H M, S [73]. JG2 has a natural K¨ ahler structure in such a way that the above map is a Lagrangian fibration. 8.3. (Co-)associative geometry. (Co-)associative submanifolds. The analog of the complex geometry defined by J (resp. symplectic geometry defined by ω) is the associative geometry defined by × (resp. coassociative geometry defined by Ω) on the G2 -manifold M [84]. Since Rm admits a VCP only when m = 3 or 7, ×-invariant submanifolds in M must have dimension three and they are called associative submanifolds, or instantons. They are calibrated by Ω and therefore absolute minimal submanifolds in M . Infinitesimal deformations of associative submanifolds are parametrized by twisted harmonic spinors and their deformations could be obstructed. The G2 -analog of Lagrangian submanifolds are coassociative submanifolds in M , which are four dimensional submanifolds C with Ω|C = 0. They are calibrated by ∗Ω, like special Lagrangian submanifolds. The normal bundle NC/M is isomorphic to the bundle Λ2+ (C) of self-dual two forms on C. Infinitesimal deformations of C as coassociative submanifolds are 2 (C) and their deformaparametrized by harmonic self-dual two forms H+ tions are always unobstructed. In particular, their moduli space Mcoass (M ) is smooth. Coassociative submanifolds are natural boundary conditions for the free boundary value problem of associative submanifolds with nonempty boundaries. In the next section, we will explain that the theory of associative submanifolds and coassociative submanifolds in G2 -manifolds are special examples of instantons and branes in manifolds with VCP. We could also view a G2 -manifold M , or more precisely M × S 1 , as an O-oriented O-manifold (see section 10). From this point of view, associative (resp. coassociative) submanifolds are H-Lagrangian submanifolds of type I (resp. type II) in M . When M = X ×S 1 , a submanifold of the form Σ×S 1 (resp. L×{p}) is an associative submanifold in M if and only if Σ is a holomorphic curve (resp. L is a special Lagrangian with phase zero) in the Calabi-Yau threefold X. Similarly, a submanifold of the form L × S 1 (resp. C × {p}) is a coassociative

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submanifold in M if and only if L is a special Lagrangian with phase π/2 (resp. C is a complex surface) in X. Analogous to the roles that holomorphic disks play in the intersection theory of Lagrangian submanifolds in symplectic manifolds, one should consider associative submanifolds bounding the intersections of coassociative submanifolds in the G2 -setting. This theory is also closely related to the Seiberg-Witten theory of the coassociative submanifold C [99]. G2 -TFT. It is natural to couple (co-)associative submanifolds C with U (r)-connections on E over C. We consider the following Chern-Simons type functional CSG2 : M ap (C, M ) × A (C) → R  T r exp (f ∗ (Ω + ∗Ω) + FA ) , CSG2 (f1 , DA1 ) = [0,1]×C

where f : [0, 1] × C → M with f (t, x) = ft (x) is an one parameter family of maps joining f1 with a background map f0 . This map is invariant under the natural action of the connected component of the large gauge group G˜ consisting of g˜ : E → E E ↓ M

g˜

→

E ↓ g → M

covering a diffeomorphism g of M and linear isomorphisms along fibers. It fits into the following fiber bundle with fiber the usual group of gauge transformations, G → G˜ → Dif f (C) When dim C = 3, CSG2 (f1 , DA1 ) =



1 ∗Ω + T rFA2 . 2 [0,1]×C

Critical points are (f1 , DA1 )’s with f1 (C) as associative submanifold in M and DA1 a flat connection over f1 (C). Thus the moduli space of critical points, {dCSG2 = 0} /G˜ = MA−cycle (M ) is the moduli space of associative submanifolds coupled with flat bundles, called associative cycles, or simply A-cycles in M . When dim C = 4,  CSG2 (f1 , DA1 ) = T rΩ ∧ FA . [0,1]×C

Suppose that DA1 is a U (1)-connection, then the Euler-Lagrange equations is Ω|f1 (C) = 0 and FA+1 = 0.

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Thus the moduli space of critical points, {dCSG2 = 0} /G˜ = MC−cycle (M ) is the moduli space of coassociative submanifolds coupled with ASD bundles, called coassociative cycles, or simply C-cycles in M . Formally, the Witten-Morse theory for CSG2 defines topological field theories, called A-TFT and C-TFT for G2 -manifolds [91][39]. Moduli of A- and C-cycles. The moduli spaces MA−cycle (M ) and MC−cycle (M ) process natural three forms, four forms and cubic tensors, analogous to the G2 -forms and Yukawa couplings. When r = 1, the forgetting maps to the moduli spaces of associative and coassociative submanifolds, MA−cycle (M ) → Massoc (M ) and MC−cycle (M ) → Mcoass (M ) behave somewhat similar to an associative and coassociative fibrations respectively [84]. We consider the universal moduli spaces of A- and C-cycles, i.e. & MA−cycle = MA−cycle (M ) and ˜G M ∈M 2

MC−cycle =

&

MC−cycle (M ) .

˜G M ∈M 2

There are analogs of the Abel-Jacobi maps to the universal G2 -intermediate Jacobian, AJG2 : MA−cycle → JG2 and AJG2 : MC−cycle → JG2 , whose images give Lagrangian subspaces in the K¨ ahler manifold JG2 . Recall ˜ G . Locally, these Lagrangian that JG2 has a Lagrangian fibration over M 2 ˜ G and these functions are subspaces are graphs of exact one forms on M 2 G2 -superpotentials as in the physics literatures. 8.4. G2 -Donaldson-Thomas bundles. Donaldson and Thomas generalized the ASD connections over oriented four manifolds and introduced the DT-connections on vector bundles E over manifolds with holonomy SU (3), G2 and Spin (7). In the G2 -setting, the DT-equation for the curvature FA is ∗FA + FA ∧ Ω = 0, or equivalently, FA ∧ (∗Ω) = 0 ∈ Ω6 (M, ad (E)) .

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DT-connections are absolute minimizers of the Yang-Mills functional We can also rewrite the DT-equation as



|FA |2 .

FA ∈ Ω214 (M, ad (E)) . Recall that g2 ⊂ so (7) corresponds to the component Λ214 ⊂ Λ2 . Globally ∗ ⊂ Λ2 T ∗ . Thus the we have the infinitesimal frame bundle g2 (M ) ∼ = Λ214 TM M above equation is the same as FA ∈ g2 (M ) ⊗ ad (E) . In this form, the G2 -DT connections are special O-connections over M from the point of view of geometry over normed division algebra (section 10). We could consider the previous Chern-Simons type functionals to this setting, CSG2 : A (M ) → R  T r exp (f ∗ (Ω + ∗Ω) + FA ) CSG2 (DA1 ) = [0,1]×M

  1 2 1 4 T r ∗Ω ∧ FA + FA . = 2 4! [0,1]×M 

The Euler-Lagrange equation is a perturbation of the G2 -DT equation, FA ∧ ∗Ω +

1 3 F = 0. 3! A

The original G2 -DT equation is the Euler-Lagrangian equation for the func tional [0,1]×M ∗Ω ∧ T rFA2 . The moduli space MDT (M ) of G2 -DT connections over M has many properties similar to those for moduli of (co-)associative cycles. For instance, it has a natural three form, four form and a cubic tensor, similar to the G2 forms and Yukawa couplings. 8.5. G2 -dualities, trialities and M-theory. The mystery of the nonassociativity of O is hidden in the triality and we expect that the full symmetry for G2 -geometry is a mirror triality. GYZ duality. For mirror symmetry, mirror Calabi-Yau threefolds X and Y admit dual special Lagrangian T 3 -fibrations. Furthermore, the mirror transformation between the complex geometry and the symplectic geometry is a certain generalization of the Fourier-Mukai transformation along these T 3 fibers. Gukov-Yau-Zaslow [58] argues physically that if we couple X with a stable E8 × E8 -bundle, then its string theory is dual to the M-theory on a G2 -manifold M with a coassociative K3-fibration. On the fiber level, this is the duality that we mentioned in section 6.4. However, it is not clear how to transform the Calabi-Yau geometry on X to the G2 -geometry on M . One is also curious whether this is part of a triality symmetry among X, Y and M .

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Triality flops. When we resolve a rational double point singularity in a Calabi-Yau threefold, we could either smooth it by deforming its complex structure, thus producing a vanishing cycle S 3 , or we could blow it up, i.e. deforming its symplectic structure, and producing an exceptional cycle S 2 . The mirror symmetry for this extremal transition has produced many exciting mathematics. Recall that there are two ways to blowup and they are related by a simple flop. It is expected that the quantum symplectic geometry is unchanged under a simple flop. Atiyah-Witten [7] discovered that the extremal transition from S 2 to S 3 and the flop between S 2 ’s can be naturally  unified from the G2 -perspectives. 3 3 Namely the G2 -singular cone C S × S admits three different resolutions, each of them is the spinor bundle SS 3 over S 3 and they are related by triality G2 -flops. Only when we quotient by S 1 , three different resolutions give T ∗ S 3 , OS 2 (−1, −1), OS 2 (−1, −1) and they are related by extremal transitions and flop. Hence the dualities for Calabi-Yau threefolds become more symmetric from the G2 -perspectives. It is natural to ask how the various G2 -TFTs behave under triality G2 -flops. For instance, it is interesting to ask whether the skein relation for knot invariants is a consequence of the triality for the associative geometry under a G2 -flop. In M-theory, physicists study the membrane theory on an eleven dimensional spacetime R3,1 × M with M a G2 -manifold. M-theory reduces to the superstring theory when M is the product of a Calabi-Yau threefold with S 1 whose size goes to zero. As we mentioned above, there are many duality transformations in M-theory. We hope that they can be combined together to form a mirror triality. We remark that there is a similar story for eight dimensional Spin (7)manifolds, which is related to F-theory in physics. However we will not further discuss it here. 9. Geometry of vector cross products A K¨ ahler structure on a Riemannian manifold (M, g) is a parallel Hermitian complex structure J : Tx M → Tx M . J being a linear Hermitian complex structure is equivalent to Ju ⊥ u

and

|Ju| = |u| .

Similarly, a G2 -structure is a parallel product structure × : Tx M ⊗ Tx M → Tx M satisfying similar defining properties. Their corresponding symplectic 2-form ω and G2 3-form Ω play important roles in these geometries. Another example is the standard volume form on R3 gives the vector product, or cross product, u × v on R3 . Such product structures coming from volume forms and K¨ ahler forms have particularly nice properties and they are examples of real vector cross products (abbrev. R-VCP) [85]. VCPs are classified and

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the remaining two are G2 -VCP and Spin (7)-VCP. Manifolds with these structures play important roles in string theory, M-theory and F-theory. For C-VCP, there are only two of such and they come from holomorphic volume forms and holomorphic symplectic forms. The corresponding K¨ ahler manifolds are Calabi-Yau manifolds and hyperk¨ ahler manifolds. Thus RVCP and C-VCP give a nice uniform description of special holonomies. 9.1. VCP manifolds. Riemannian manifolds with VCP. The classical vector product, or cross product, on R3 satisfies the property that u×v is perpendicular to both u and v and with length equal to the area of the parallelogram spanned by u and v. The only such structure in higher dimension is the G2 -structure. A real vector cross product is such a product structure but we allow the number r of variables to be different from two. When r = 1, this is a Hermitian complex structure. We define an r-fold vector cross product (R-VCP or simply VCP) on a Euclidean space Rm to be a skew-symmetric multi-linear form × : Rm ⊗ · · · ⊗ Rm → Rm which satisfies (i) (u1 × · · · × ur ) ⊥ui

for all i

(ii) |u1 × · · · × ur | = u1 ∧ · · · ∧ ur . Here · is the norm on r-forms. Similar to the K¨ ahler form and the G2 -form, we define the VCP-form Ω ∈ Ωr+1 (Rm ) as follow Ω (u1 , . . . , ur , ur+1 ) = u1 × · · · × ur , ur+1  . The condition (i) for × ensures that Ω is skew-symmetric and condition (ii) is equivalent to |ιu1 ιu2 . . . ιur Ω| = 1 for any orthonormal vectors ui ’s. It turns out that there are very few possibilities of such and they are classified by Gray [50]: Any VCP-form must be one of the following: (1) volume form, (2) symplectic form, (3) G2 -form and (4) Spin (7)-form. G2 and Spin (7) VCPs live in 7 and 8 dimensional spaces respectively and they are both directly linked to the octonion O. In terms of R7  Im O and R8  O and octonion multiplications, we have, in the G2 -case, u × v = Im uv ΩG2 = dx123 − dx167 + dx145 + dx257 + dx246 − dx356 + dx347 ,

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and in the Spin (7)-case, u × v × w = (u (¯ v w) − w (¯ v u)) /2    ΩSpin(7) = −dx1234 − dx5678 − dx21 + dx34 dx65 + dx78       − dx31 + dx42 dx75 + dx86 − dx41 + dx23 dx85 + dx67 . Here dxijk means dxi ∧ dxj ∧ dxk and so on. Note that G2 and Spin (7) are the symmetry groups of the corresponding VCP-forms in R7 and R8 respectively. A Riemannian manifold (M, g) with a VCP (resp. closed VCP and parallel VCP ) [85] means that it has VCP × on each of its tangent space and Ω is a smooth form (resp. closed form and parallel form). For simplicity, we usually call a Riemannian manifold with a parallel VCP simply a VCPmanifold. 9.2. Instantons and branes. The closedness of Ω implies that Ω ∈ Ωr+1 (M ) is a calibrating form. A (r + 1)-dimensional submanifold C in M is calibrated by Ω if and only if it is preserved by × and they are called instantons. For instance, when M is a K¨ ahler manifold, instantons are holomorphic curves; when M is a G2 -manifold, instantons are associative submanifolds; when M is a Spin (7)-manifold, instantons are Cayley submanifolds. The following characterization of an instanton is useful in studying the deformation theory of instantons: First we denote the symmetry group of × as G ⊂ O (m) and its Lie algebra as g ⊂ o (m) ∼ = Λ2 V ∗ . Over a VCPmanifold M , by putting g for each tangent space V = Tx M together, we ∗ . We define a homomorphism have a subbundle gM ⊂ Λ2 TM τ : Λr+1 V → Λ2 V as the composition of the VCP of the last r components follows by the wedge product, i.e. τ (u0 ∧ u1 ∧ · · · ∧ ur ) = u0 ∧ (u1 × · · · × ur ) . The image of τ lies inside g⊥ , the orthogonal complement of the Lie algebra of G. Over M , we have   ⊥ τ ∈ Ωr+1 M, gM and C is a instanton in M if and only if   ⊥ τ |C = 0 ∈ Ωr+1 C, gM . Using this, one can show that infinitesimal deformations of an instanton in M are parametrized by twisted harmonic spinors on C. The well-behaved free boundary condition for holomorphic curves C in a K¨ ahler manifold C is by requiring the boundary ∂C to lie inside a Lagrangian

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submanifold L ⊂ M . This notion generalizes to VCP-geometry. A submanifold L in M is called a brane if (i) Ω|L = 0 and (ii) dim L = (n + r − 1) /2. Indeed this is the largest possible dimension among submanifolds L satisfying Ω|L = 0. Lagrangian submanifolds in K¨ ahler manifolds, hypersurfaces in volume manifolds and coassociative submanifolds in G2 -manifolds are examples of branes. However, brane does not exist in any Spin (7)-manifold. Suppose L is a brane in M , VCP determines a homomorphism from Λr TL to the normal bundle NL/M . Taking the adjoint of it, NL/M becomes a subbundle of Λr TL∗ which is spanned by VCP-forms of degree r on L. Infinitesimal deformations of a brane are parametrized by such degree r closed VCP-forms on L and they all have unobstructed deformations. In particular, the moduli space of branes in M is always smooth. Thus the theory of VCP gives a unified approach to describe the geometries of various important geometric objects in K¨ ahler geometry, G2 geometry and Spin (7)-geometry. The following table summarizes all these structures. VCP form

Volume form Sympl. form

G2 -form

Instanton

Domain

Assoc. submfd. Cayley submfd.

Brane

Hypersurface Lagr. submfd. Coass. submfd. n/a

Holo. curve

Spin (7)-form

9.3. Symplectic geometry on higher dimensional knot spaces. Symplectic forms are 1-fold VCP forms. Indeed the geometry of r-fold VCP on M can be described in terms of the symplectic geometry of the infinite dimensional knot space KΣ (M ) [83]. Here KΣ (M ) is the set of all submanifolds in M which are diffeomorphic to a fixed (r − 1)-dimensional space Σ, that is KΣ (M ) = M apemb (Σ, M ) /Dif f (Σ). Here M ap (Σ, M ) is the space of all embeddings from a fixed (r − 1)-dimensional manifold Σ to M . Via transgression, any (r + 1)-form Ω on M defines a 2 -form ω K on KΣ (M ), i.e.  ω K (f ) (X, Y ) = ιf∗ (X)∧f∗ (Y ) Ω. Σ

It is clear that dω K = 0 if and only if dΩ = 0. The Riemannian metric on M defines one on KΣ (M ). One can show that Ω is a r-VCP form on M if and only if ω K is an 1-VCP form on KΣ (M ), in particular a symplectic form. Furthermore, a submanifold L in M is a brane if and only if KΣ (L) is a Lagrangian submanifold in KΣ (M ) . Given a holomorphic curve D in KΣ (M ), it is a family of submanifolds in M , i.e. there is a fiber bundle Σ → C → D and a map C → M , then C is an instanton in M provided that the curve satisfies a normality condition. Thus the geometry of VCP on M is nothing but the symplectic geometry on KΣ (M ). For instance, one would expect to have intersection theory of branes defined using instantons,

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similar to the intersection theory of Lagrangian submanifolds defined using J-holomorphic discs. This interpretation of VCP geometry in terms of symplectic geometry on knot spaces is particularly fruitful for C-VCP. 9.4. C-VCP geometry. K¨ ahler manifolds with C-VCP. We can similarly define a r-fold complex vector cross product (C-VCP) on a Hermitian vector space Cm [85]. More precisely, a C-VCP form Ω ∈ Ωr+1,0 (Cm ) must satisfy |ιu1 ιu2 . . . ιur Ω| = 1 for any orthonormal complex vector ui ’s. There are only two C-VCPs, namely it must be either (i) a complex volume form or (ii) a complex symplectic form. Suppose (M, ω) is K¨ahler manifold with a holomorphic form Ω defining a C-VCP, then Ω is automatically parallel. As a result, a K¨ ahler manifold with a holomorphic C-VCP must be either (i) a Calabi-Yau manifold or (ii) a hyperk¨ ahler manifold. Instantons. Since Ω is a complex valued differential form, we  have  a S 1 -family of calibrating forms. Namely for any real number θ, Re eiθ Ω is a calibrating form and those submanifolds C it calibrates are called instantons with phase θ. Instantons can be characterized as those (r + 1)-dimensional submanifolds C in M satisfying   Im eiθ Ω |C = ω|C = 0. When M is a Calabi-Yau manifold, then C is a special Lagrangian submanifold with phase θ. When M is a hyperk¨ahler manifold, then C is a Jθ -holomorphic curve where Jθ = (cos θ) I + (sin θ) K. N-branes and D-branes. There are two types of boundary conditions for instantons: Neumann boundary condition and Dirichlet boundary condition. The corresponding branes are called N-branes and D-branes. A N-brane is a real (n + r − 1)-dimensional submanifold L in M satisfying Ω|L = 0. A D-brane is a middle dimensional submanifold L in M satisfying   ω|L = Re eiθ Ω |L = 0. As in the real case, N-branes are biggest dimension submanifolds where the restriction of Ω vanishes. This forces L to be a complex submanifold in M . When M is a Calabi-Yau manifold, a N-brane (resp. D-brane) is a complex hypersurface (resp. special Lagrangian submanifold with phase θ +π/2) in M . Even though complex hypersurfaces and special Lagrangian submanifolds seems to be very different from each other, both of them correspond to ˆ Σ (M ) of M complex Lagrangian submanifolds in the isotropic knot space K

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(as will be explained below). The difference is they are complex Lagrangian ˆ Σ (M ). with respect to different complex structures in the twistor family of K When M is a hyperk¨ ahler manifold, a N-brane (resp. D-brane) is a J-complex Lagrangian submanifold (resp. Jθ+π/2 -complex Lagrangian submanifold) in M . The deformation theory of instantons with boundaries lying in N-branes and D-branes was studied by Schoen’s school. The following table summarizes these structures. C-VCP mfd.

Calabi-Yau mfd.

Hyperk¨ ahler mfd.

Instanton

SLag. w/ θ = 0

J-holomorphic curve

D-brane

Cpx. hypersurface

J-cpx. Lagr. submfd.

N-brane

SLag. w/ θ = π/2

I-cpx. Lagr. submfd.

9.5. Hyperk¨ ahler geometry on isotropic knot spaces of CY. Recall that the R-VCP geometry on M can be interpreted as the symplectic geometry on KΣ (M ), the space of submanifolds in M , i.e. KΣ (M ) = M apemb (Σ, M ) /Dif f (Σ) . We will simply write M ap (Σ, M ) for M apemb (Σ, M ). For C-VCP we should complexify this picture and consider the quotient of M ap (Σ, M ) by the complexification of Dif f (Σ), which does not exist! So one tries to replace the complex quotient by the symplectic quotient: Suppose (M, g, J, ω) is a K¨ ahler manifold with a C-VCP form Ω ∈ Ωr+1,0 (M ) . In order to induce a symplectic form ω M ap on M ap (Σ, M ) from ω on M , we need to fix a background volume form ν Σ on Σ, then  (f ∗ ω) (u, v) ν Σ ω M ap (f ) (u, v) = Σ

for any f ∈ M ap (Σ, M ) and any u, v ∈ Tf M ap (Σ, M ) = Γ (Σ, f ∗ TM ). Since we have fixed a background volume form on Σ, only volume preserving diffeomorphisms of Σ preserves ω M ap . The moment map for the Dif f (Σ, ν Σ )-action on M ap (Σ, M ) is μ : M ap (Σ, M ) → Ω1 (Σ) /dΩ0 (Σ) μ (f ) = [α] where f ∗ ω = dα (section 6.2). In particular, μ−1 (0) consists of isotropic embeddings of Σ into M . However, the symplectic quotient M ap (Σ, M ) //Dif f (Σ, ν Σ ) = μ−1 (0) /Dif f (Σ, ν Σ )

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is too big as we should take the symplectic quotient by the larger group Dif f (Σ) . In [83], a natural coisotropic foliation D on μ−1 (0) is constructed and the quotient space ˆ Σ (M ) = μ−1 (0) /Dif f (Σ) K is a natural substitute for the nonexisting symplectic quotient M ap (Σ, M ) // ˆ Σ (M ) admits Dif f (Σ), called the isotropic knot space. Furthermore, K almost complex structures I, J and K satisfying the Hamilton relation and a natural 1-fold C-VCP structure. However, in this infinite dimensional setˆ Σ (M ) is a holomorphic symplectic manifold, not ting, we only know that K hyperk¨ ahler. Thus we have constructed an infinite dimensional holomorphic symˆ Σ (M ) from any Calabi-Yau manifold M . Analogous to plectic manifold K ˆ Σ (M ), where Jθ = the real situation, a normal Jθ -holomorphic curve in K (cos θ) I + (sin θ) K, corresponds to an instanton, i.e. a special Lagrangian submanifold with phase θ in M . More interestingly, (i) if L is a complex hypersurface (i.e. a N-brane) in ˆ Σ (L) is a J-complex Lagrangian submanifold in K ˆ Σ (M ) and (ii) M, then K if L is a special Lagrangian submanifold with phase θ + π/2 (i.e. a D-brane ˆ Σ (L) is a Jθ+π/2 -complex Lagrangian submanifold with phase θ), then K ˆ Σ (L) also defines a complex Lagrangian in ˆ Σ (M ). The reason that K in K ˆ Σ (M ) is because it is automatically sitting inside μ−1 (0). K For quaternionic-K¨ ahler manifolds, i.e. Hol (M, g) ⊆ Sp (n) Sp (1), there is no well-defined H-VCP structure because of the noncommutativity of H. In summary, we have explained the source of reduction to every nonsymmetric Riemannian holonomy groups as M carrying different A-VCP with A ∈ {R, C, H}. none

r=n−1

r=1

r=2

r=3

R-VCP

O (n)

SO (n)

U (n)

G2

Spin (7)

C-VCP

U (n)

SU (n)

Sp (n)

H-VCP

Sp (n) Sp (1) 10. Geometry over normed division algebras

10.1. Manifolds over normed algebras. Normed division algebras. We have seen that Riemannian manifolds (M, g) with special holonomy groups have various algebraic structures on their tangent bundles T M . It is not surprising that such structures are related to normed algebras A. We recall that a real algebra A with a norm · is a normed algebra if it satisfies the obvious compatibility condition: ab = a b

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for any a, b ∈ A. Note that a normed algebra is always a division algebra. It is a classical fact that R, C, H and O form the complete list of normed division algebras. In this section, we will see that different holonomy groups correspond to manifolds being defined over various A and whether it is A-oriented or not. Each tangent space of a Riemannian manifold (M, g) certainly has a norm. If it also has an A-module structure, then it will reduce the holonomy group of M . For instance, a Riemannian manifold with a parallel C-module structure on its tangent spaces is a K¨ ahler manifold. We can give a unified way to describe all Riemannian holonomy groups and their geometries from this point of view, even including Riemannian symmetric spaces (see section 13). There are many advantages to this. It helps us to discover new links between different kinds of geometries and obtain new results. For example the hard Lefschetz sl (2, R)-action for compact K¨ahler manifolds can be naturally generalized to all holonomy groups in [96] and the results of mean curvature flow for Lagrangian submanifolds in Calabi-Yau manifolds was generalized to hyperlagrangian submanifolds in the H case in [98]. We could even include conformal geometry in the same arena with the help of Jordan algebras (section 12). The first issue is O-module does not make sense as O is not an associative algebra. Using Jordan algebras, one can resolve this problem as long as the O-dimension is not more than three. In fact, we need to use this and its extension via the magic square to describe exceptional symmetric spaces in terms of two normed division algebras. For the time being, we will restrict to the one dimensional case, i.e. O itself. A-manifolds and A-orientations. Now we define a Riemannian Amanifold M to be a Riemannian manifold with its holonomy group lies inside the group GA (n) of twisted isomorphisms of An , which is defined as follows: Suppose V is a normed linear A-module of rank n. A R-linear isometry φ of V is called a twisted isomorphism if there exists θ ∈ SO (A) such that φ (vx) = φ (v) θ (x) for any v ∈ V and x ∈ A. When A is R or C, θ is necessarily the identity. The reason of introducing θ is because a H-structure is given by the twistor S 2 -family of complex structures, rather than a particular choice of the triple (I, J, K). We have the symmetry group GH (n) is Sp (n) Sp (1). In the O case, GO (1) is the exceptional holonomy group Spin (7). Next we introduce the notion of an A-orientation for Riemannian A-manifolds. Orientability requires the notion of determinant. In the complex and quaternionic cases, detA corresponds to the projection to the second component in U (n) = SU (n) × U (1) /Zn and Sp (n) Sp (1) respectively. In the octonion case, it is simply the octonion product because we only consider the one dimensional case. Therefore a twisted isomorphism g ∈ GA (n) is called special if detA (g) fixes 1 ∈ A. That is g is an element in the isotropic

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subgroup of 1 in GA (n), which we denote HA (n) and the corresponding manifolds are called special A-manifolds. They are given explicitly in the following table: A

R

C

H

O

GA (n)

O (n)

U (n)

Sp (n) Sp (1)

Spin (7)

HA (n)

SO (n)

SU (n)

Sp (n)

G2

Their corresponding Riemannian geometries are as follows. A

R

C

H

O

GA (n)

Riemannian

K¨ ahler

Quaternionic-K¨ ahler

Spin (7)

HA (n)

Volume

Calabi-Yau

Hyperk¨ ahler

G2

In the above definition, a G2 -manifold is a real eight dimensional manifold. However its universal cover is a Riemannian product manifold M × R with M a seven dimensional G2 -manifold as defined previously. This is because G2 = Aut (O) fixes R ⊂ O and therefore sits inside SO (7). suA (1, 1)sup -action. When M is a K¨ahler manifold, the hard Lefschetz theorem says that there is a natural sl2 (R)-action on differential forms Ω∗ (M ) and the cohomology group H ∗ (M, R). Notice that sl2 (R) ∼ = su (1, 1). Indeed there is a super Lie algebra su (1, 1)sup = su (1, 1) ⊕ C1,1 ⊕ R which acts on the space of differential forms Ω∗ (M ) extending the hard ¯ Lefschetz action. Here C1,1 acts by the first order differential operators ∂, ∂, ∗ ∗ ¯ ∂ and ∂ and R acts by the second order differential operator, the Laplacian Δ. Having such an action encompass (i) the hard Lefschetz action, (ii) first order K¨ ahler identities and (iii) second order K¨ ahler identities. Suppose M is only an oriented Riemannian manifold, then Ω∗ (M ) admits an so (1, 1)sup = so (1, 1) ⊕ R1,1 ⊕ R where R1,1 acts via d and d∗ and R acts via Δ. When M is a hyperk¨ahler manifold, there is a similar action of sp (1, 1)sup = sp (1, 1) ⊕ H1,1 ⊕ R ahler identities. on Ω∗ (M ) encompassing all the hyperk¨ The existence of such actions can be explained via the geometry over normed algebras point of view. The basic reason is the exterior algebra Λ∗ (An ) can be naturally identified with the spinor representation for the vector space An,n . Such actions have natural generalization to all (A-oriented) A-manifolds [93].

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10.2. Gauge theory over (special) A-manifolds. When a Riemannian manifold has special holonomy, there is usually a special kind of connections on bundles over them, for instance holomorphic bundles over K¨ ahler manifolds and Donaldson-Thomas bundles over G2 - and Spin (7)-manifolds. Using normed division algebras to describe different holonomy groups gives us a particular nice unified way to describe all these different types of connections. If M is a (special) A-manifold, then the Lie algebra of its holonomy group is (hA (n) ⊂)gA (n) ⊂ so (m), where m and n are dimensions of M over R and A respectively. This determines a subbundle in the bundle of ∗ , denoted 2-forms Λ2 TM ∗ hA (TM ) ⊂ gA (TM ) ⊂ Λ2 TM .

Recall that the curvature FA of a connection DA on a bundle E is a ad (E)∗ ⊗ad (E) . On a (special) A-manifold M , it is natuvalued 2-form, FA ∈ Λ2 TM ral to require FA to lie inside Γ (M, gA (TM ) ⊗ ad (E)) (resp. Γ(M, hA (TM ) ⊗ ad (E))) and we call such a connection a (special) A-connection. Most of them can be identified with well-known Yang-Mills connections as indicated in the following table. A-connections C FE0,2 = FE2,0 = 0 (Holomorphic bundles) H F ∈ gH (TM ) ⊗ ad (E) (B-bundles) O ∗FE + Θ ∧ FE = 0

Special A-connections FE0,2 = FE2,0 = ΛF = 0 (Hermitian Yang-Mills bdls.) 0,2 FI0,2 = FJ0,2 = FK =0

(ASD or hyperholomorphic bdls.) F ∧Θ=0

(Spin (7)-Donaldson-Thomas bdls.) (G2 -Donaldson-Thomas bdls.) 10.3. A-submanifolds and (special) Lagrangian submanifolds. A-submanifolds in A-manifolds. Complex submanifolds form an important class of submanifolds in K¨ ahler manifolds. However there are very few quaternionic submanifolds in a H-manifold as they are always totally geodesic submanifolds. And we have no O-submanifold as G2 - and Spin (7)manifolds already have O-dimension one. However, the geometry of (special) Lagrangian submanifolds in (special) A-manifolds is very rich. (Special) A/2-Lagrangian submanifolds. When a symplectic vector space (V, ω) has a compatible complex structure J, a Lagrangian subspace L in V gives an orthogonal decomposition V = L ⊕ JL. When V is an A-module, it has several compatible complex structures J1 , . . . , J2r−1 where r = dimR A/2 and therefore V has several symplectic structure ω 1 , . . . , ω 2r−1 . For example,

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when V = Hn , a subspace which is both ω I - and ω K -Lagrangian is automatically J-linear and therefore a J-complex Lagrangian subspace. In general, if L ⊂ An is Lagrangian with respect to half of the symplectic forms, say ω 1 , . . . , ω r , then it is Jl -linear for the remaining complex structures. In particular, it is a (A/2)-module. Here we denote C/2 = R, H/2 = C and O/2 = H. We call such a L a (A/2)-Lagrangian subspace in V. Globally on M , this defines the notion of (A/2)-Lagrangian submanifolds in any Riemannian A-manifold. There is also a corresponding notion of orientability for such L, i.e. phase angles are constant on L, which defines the class of special (A/2)-Lagrangian submanifold and all of these are calibrated submanifolds. To describe the phase angle concretely, we recall the Grassmannian of (A/2)-Lagrangian subspaces in Cn (resp. Hn and O) is U (n) /SO (n) (resp. Sp (n) Sp (1) /U (n) U (1) and Spin (7) /Sp (1)3 ). There is phase angle map to S 1 (resp. S 2 / ± 1 and S 4 ) and the fiber over θ consists of special (A/2)-Lagrangian subspaces of phase angle θ. Thus giving any (A/2)-Lagrangian submanifold L, there is a phase angle map θ : L → Sd where d = dim A/2 and in the H-case it is well-defined up to the antipodal map. L is special if this is the constant map. We identify all the them in the following table. A

A 2 -Lagrangian

C

Lagrangian submfd.

special Lagr. submfd.

H

hyperlagrangian submfd.

complex Lagrangian submfd.

O

Cayley submfd.

(co-)associative submfd.

submfd.

special

A 2 -Lagr.

submfd.

When L is a Lagrangian submanifold in a Calabi-Yau manifold M , the phase angle function θ (x) satisfies ιH ω = dθ (x) , where H is the mean curvature vector of L. This implies that Lagrangian submanifolds in M , or more generally in a K¨ ahler-Einstein manifold, are invariant under the mean curvature flow. And the phase angle θ (x) satisfies the non-linear PDE ∂θ = Δθ ∂t where Δ is the Laplacian operator with respect to the induced metric on L which is changing it t. It does not develop type I singularity. Furthermore, if L is minimal, then it must be calibrated, namely a special Lagrangian submanifold.

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Indeed, the same is true for A/2-Lagrangian submanifolds in any special A-manifold in general. When A = O, Cayley submanifold is already calibrated and therefore the other interesting situation is for hyperlagrangian submanifolds L in a hyperk¨ ahler manifold M [98]. In this case, the phase angle function θ (x) : L → S 2 / ± 1 has the following nice property: The target S 2 is the twistor family of complex structures on M and for any x ∈ L, the tangent space Tx L ⊂ Tx M is complex linear with respect to the complex structure Jθ(x) . We also have ¯ ιH ΩJθ(x) = ∂θ, and θ (x) satisfies the harmonic map flow. These can be used to show that the mean curvature flow preserves the hyperlagrangian property and type I singularity does not appear, just as what happens in the Calabi-Yau case. 11. Quaternion geometry When we go from R to C and then to H, their geometries become more and more rigid. For instance, every higher dimensional H-manifold is Einstein. Also every complex submanifold in a K¨ ahler manifold has zero mean curvature, in fact an absolute volume minimizer, and every quaternionic submanifold has zero second fundamental form, i.e. a totally geodesic submanifold. This is because the second fundamental form must be skew-Hermitian with respect to I and J (of course K as well). Using IJ = K, it becomes Hermitian with respect to K and therefore it must be zero. In general, every quaternionic map from Hm to Hn must be affine H-linear. Twistor spaces. Recall that a Riemannian H-manifold M has its holonomy group in Sp (n) Sp (1). It is also called a quaternionic-K¨ ahler manifold. If we also assume H-orientability, then its holonomy group is inside Sp (n) and it is a hyperk¨ ahler manifold. Every tangent space Tx M is modeled on Hn , thus it has three complex structures I, J and K. Indeed, any element in the unit sphere S 2 in the three dimensional vector space spanned by I, J and K is a complex structure on Tx M . They form a S 2 -fiber bundle S 2 → Z → M. The total space is called the twistor space. In fact, this is the associated bundle for the Sp (n) Sp (1)-frame bundle over M via the adjoint action of the Sp (1)-factor on S 2 ⊂ Im H. Equivalently, the inclusion Sp (n) Sp (1) ⊂ SO (4n) defines a subbundle in the bundle of two forms, Λ2sp(n) ⊕ Λ2sp(1) ⊂ Λ2 , and Z is the unit sphere bundle inside Λ2sp(1) . When M is of dimension four, the above inclusion is the decomposition of any two form into ASD and SD components, Λ2− ⊕ Λ2+ = Λ2 .

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The tangent space of Z at any (x, Jx ) in Z splits into vertical and horizontal components via the Riemannian metric on M . Since each fiber is S 2 , the vertical component has a natural complex structure and Jx is a complex structure on the horizontal component, thus Z has a natural almost complex structure. In fact, it is integrable as long as n ≥ 2, namely the twistor space is always a complex manifold. Moreover every S 2 fiber is a holomorphic curve in Z. When n = 1, this requires W + = 0. Furthermore, the H-geometry on M can be described in terms of the complex geometry on Z, the so-called twistor transformation. 11.1. Hyperk¨ ahler geometry. On a hyperk¨ ahler manifold M , Horientibility guarantees that each I, J and K can be consistently defined over the whole manifold M . In particular Z = M × S 2 , but non-holomorphically. Indeed, the projection to S 2 is the S 2 -family of complex structures on M , called the twistor family. The three complex structures satisfies the Hamilton relation, I 2 = J 2 = K 2 = IJK = −id. Given any J in the twistor family, (M, g, J) is a Calabi-Yau manifold which corresponds to the embedding Sp (n) ⊂ SU (2n), analogous to U (n) ⊂ SO (2n) in the complex case. The corresponding K¨ ahler form will be denoted as ω J . The parallel form ΩJ = ω I − iω k ∈ Ω2,0 J (M ) defines a holomorphic symplectic structure on (M, J). The J-holomorphic volume form for the Calabi-Yau structure on (M, J) is simply the top exterior power of ΩJ . Conversely, if a compact K¨ahler manifold (M, J) admits a holomorphic ∗ as holomorphic vecsymplectic form ΩJ , then TM is isomorphic to TM tor bundles and therefore all its odd Chern classes vanish. In particular, c1 (M ) = 0 and it admits a Ricci flat K¨ ahler metric g by Yau’s theorem. ΩJ is a parallel form with respect to g and both Re ΩJ and Im ΩJ define K¨ ahler structures on M which make M into a hyperk¨ ahler manifold. Examples of hyperk¨ ahler manifolds. Using this method, Beauville constructed higher dimensional hyperk¨ ahler manifolds from four dimensional ones S, i.e. K3 surfaces and Abelian surfaces, by resolving the singularities of symmetric powers of S, called the Hilbert schemes of points in S. Instead of considering moduli space of points in S, Mukai [109] showed that the moduli space M of stable coherent sheaves on S has a canonical holomorphic symplectic form. When there is no strictly semi-stable sheaf, then M is compact and therefore a hyperk¨ ahler manifold. O’Grady [110] studied the resolutions of the compactification of M and constructed two compact hyperk¨ ahler manifolds which are not birational to Beauville’s examples. Recall that the cotangent bundle of any complex manifold X admits a canonical holomorphic symplectic form but we cannot use Yau’s theorem to assert that T ∗ X is hyperk¨ ahler as it is noncompact. Nevertheless,

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Calabi [28] has constructed explicit hyperk¨ ahler metric on T ∗ Pn . In fact, the cotangent bundle of any Hermitian symmetric space of compact type is hyperk¨ ahler [14]. Hyperk¨ ahler manifolds also arises from the geometry of Calabi-Yau threefolds Y . The infinite dimensional space of isotropic knots in Y carries a natural H-structure [83]. Moreover, complex surfaces or special Lagrangian submanifolds in Y define complex Lagrangian submanifolds in it. As we mentioned in section 7.1, if we enlarge the moduli space of complex structures on Calabi-Yau threefolds by including a choice of the holomorphic volume forms, then the universal intermediate Jacobian over it is a hyperk¨ ahler manifold. Moreover, under the Abel-Jacobi map, the universal moduli spaces of holomorphic curves define a complex Lagrangian subvariety in it. The underlying structures on this enlarged moduli space of Calabi-Yau manifolds is the special K¨ ahler structure [43]. Moduli of hyperk¨ ahler manifolds. The moduli spaces of hyperk¨ ahler manifolds have many nice properties. First since M is always a CalabiYau manifold, the moduli space of complex structures on M is smooth, by the Tian-Todorov lemma. Every complex deformation Mt of M is again a hyperk¨ ahler manifold and the corresponding holomorphic symplectic form Ωt satisfies the following important property: q0 (Ωt ) = 0 and q0 (Re Ωt ) > 0, Here q0 : H 2 (M, C) → C    n n−1 ¯ n−1 2 n−1 ¯ n ¯ n−1 φ, Ω Ω Ω φ Ωn Ω φ + (1 − n) Ω q0 (φ) = 2 is called the Beauville-Bogomolov quadratic form. It is a quadratic form of signature (3, b2 − 3) on H 2 (M, R). The local Torelli theorem says that the period map τ : T → D = SO+ (3, b2 − 3) /SO (2) SO (1, b2 − 3) which sends a marked hyperk¨ ahler manifold Mt to the real two plane in 2 H (M, R) spanned by Re Ωt and Im Ωt is a local isomorphism. Complex Lagrangian submanifolds. As we mentioned before, any quaternionic submanifold in a hyperk¨ ahler manifold is totally geodesic. Thus there are very few of such. On the other hand, the geometry of complex Lagrangian submanifolds is rather rich. Bryant also studied other types of submanifolds in hyperk¨ ahler manifolds [26][21]. A complex submanifold L of middle dimension in a holomorphic symplectic manifold M is called a complex Lagrangian submanifold if the restriction of ΩJ to M is zero. Any middle dimensional complex submanifold L in M with c1 (L) positive is always a complex Lagrangian submanifold because there is no nontrivial holomorphic two form on L by Bochner arguments. For instance any CPn in M is a complex Lagrangian submanifold. As in the real case, the cotangent

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bundle of any complex manifold X is always a holomorphic symplectic manifold with the zero section X being a complex Lagrangian submanifold in it. Hitchin showed that when M is a hyperk¨ahler manifold and L is a middle dimensional submanifold in M , then the vanishing of ΩJ on L already implies that L is a complex submanifold. On the other hand, a complex submanifold L in M being a complex Lagrangian can be characterized by the property that the cohomology class [L] it represents is Ω-primitive [17]. Furthermore, the rational cobordism class of L is completely determined by [L] ∈ H 2n (M, Z). Since complex Lagrangian submanifolds are examples of special Lagrangian submanifolds, the moduli space McxLag (M ) of complex Lagrangian submanifolds in M is smooth. Hitchin [62] defined an analog of the Abel-Jacobi map τ : McxLag (M ) → H 1 (L0 , C) by integrating Ω along a path of 1-cycles between any complex Lagrangian submanifold L and a fixed one L0 . Since L0 is a K¨ ahler manifold, H 1 (L0 , R) is a complex vector space. Hitchin showed that τ is a local embedding with image a complex Lagrangian subspace in H 1 (L0 , C)  T ∗ H 1 (L0 , R) . On ahler metric. The the cotangent space T ∗ H 1 , there is a canonical pseudo K¨ cxLag restriction of it to M (M ) is positive definite and of Hessian type. This ahler manifold. implies that McxLag (M ) is a special K¨ From a physical perspective, the σ-model with a hyperk¨ ahler manifold M as a target has N = 4 SUSY. If we require the domain Riemann surfaces to have boundaries and their images lie inside a fix complex Lagrangian submanifold L ⊂ M , then only half of the SUSY can be preserved. Hence we have a N = 2 SUSY theory. Moduli spaces of such theories are generally known (physically) to have special K¨ ahler geometry. In our case here, it is simply the moduli space of complex Lagrangian submanifolds in M (possible with M varying). Therefore we expect such a moduli space is special K¨ ahler. Complex Lagrangian fibrations. There are many examples of hyperk¨ahler manifolds that admit complex Lagrangian fibrations. For instance, the moduli space of curves coupled with line bundles in any K3 surface S has a complex Lagrangian fibration with each fiber consisting of sheaves which support on a fixed curve in S. This moduli space plays an important role in the original derivation of the Yau-Zaslow formula (section 6.4). Every elliptic fibration on a K3 surface S is a complex Lagrangian fibration because of the low dimension. This induces complex Lagrangian fibrations on the Hilbert schemes of points in S. Matsushita [106] proved that every holomorphic fibration on an irreducible projective hyperk¨ ahler manifold is a complex Lagrangian fibration provided that the base is normal. The base must be a Fano variety with the same Hodge number as CPn (or simply write Pn ). A folklore conjecture says that the base must always be Pn . Note that Huybrechts proved that

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given any compact hyperk¨ ahler manifold M , a given complex structure is projective if and only if there exists a holomorphic line bundle L over M such that q0 (c1 (L)) > 0 where q0 is the Beauville-Bogomolov quadratic form. For a complex Lagrangian fibration on a projective hyperk¨ ahler manifold, every smooth fiber is an Abelian variety. We call such a fibration an algebraically integrable system. In this case, the base has a natural special K¨ ahler structure away from the discriminant locus. Furthermore, the second fundamental forms for the fibers define a symmetric cubic tensor on the base which characterizes this fibration, as studied by Donagi-Markman [34][35]. Hyperk¨ ahler flops. There are explicit examples of hyperk¨ ahler metrics on cotangent bundles of any compact Hermitian symmetric space, for instance the complex Grassmannian GrC (r, n) or Pn . Using the fact that the cotangent bundles of a vector space V and its dual space V ∗ are both equal to V × V ∗ , there is a canonical identification between T ∗ Pn and T ∗ (Pn )∗ outside their zero sections Pn and Pn∗ respectively. This surgery operator can be done on any holomorphic symplectic manifold M of complex dimension 2n. Namely if M contains a complex (Lagrangian) submanifold Pn , then we can replace it by Pn∗ and obtain a new holomorphic symplectic manifold M  . This is called a hyperk¨ ahler flop. n  Since M \P and M \Pn∗ are isomorphic, every complex Lagrangian L in M determines one in M  , called the Legendre transform of L, denoted L∨ . Note that their intersections with Pn and Pn∗ are dual varieties. The usual Pl¨ ucker formula for dual varieties was generalized in [92] and give L1 · L2 +

n∗ ∨ n∗ (L1 · Pn ) (L2 · Pn ) (L∨ ∨ ∨ 1 · P ) (L2 · P ) = L . · L + 1 2 (−1)n+1 (n + 1) (−1)n+1 (n + 1)

We recall the classical Plucker formula for an algebraic curve C ⊂ P2 and its dual curve C ∨ ⊂ P2∗ , d∨ = d (d − 1) − 2δ − 3κ, where d (resp. d∨ ) is the degree of C (resp. C ∨ ) and δ (resp. κ) is the number of double points (resp. cusp) of C. Thus this gives a duality transformation between the categories of coherent sheaves on complex Lagrangians submanifolds in M and its hyperk¨ ahler flop M  . When the H-dimension of M is two, Hu and Yau [68] showed that hyperk¨ ahler flops are the only birational transformations among hyperk¨ ahler manifolds. Note that birational transformations are isomorphisms in Hdimension one. For higher dimensional hyperk¨ ahler manifolds, Mukai [109] studied a hyperk¨ ahler flop along a family of Pk in M when the total space Z of the family is a complex coisotropic submanifold in M of complex codimension k. This is also called a Mukai elementary modification. There is also a natural generalization of the Legendre transformation between categories of complex Lagrangians in this setting [92].

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More general birational transformations can be constructed when the above coisotropic family of Pk develops singularities. Then the model birational transformation between T ∗ Pn and T ∗ Pn∗ will be replaced by Springer correspondences, for instance between cotangent bundles of complex Grassmannians T ∗ GrC (r, Cn ) and T ∗ GrC (r, Cn∗ ) . This stratified version of the Mukai elementary transformation is constructed by Markman. 11.2. Quaternionic-K¨ ahler geometry. Oriented four manifolds are quaternionic-K¨ ahler manifolds of H-dimension one, because Sp (1) Sp (1) = SO (4) . We usually referred to quaternionic-K¨ ahler manifolds M as those with holonomy group equals Sp (n) Sp (1) instead of a subgroup of it. When n ≥ 2, quaternionic-K¨ ahler manifolds have two very special properties: (1) the Ricci curvature of M equals g or −g and they are called positive and negative respectively. Namely they are Einstein manifolds. (2) The canonical almost complex structure on the twistor space Z is always integrable, i.e. Z becomes a complex manifold. As a result, we usually require oriented four manifolds to have these properties to be called quaternionic-K¨ ahler, namely they are ASD Einstein four manifolds. The list of compact Riemannian symmetric spaces which are quaternionic-K¨ ahler is Sp (n) U (n) O (n) G2 , , , , Sp (n − 1) Sp (1) U (n − 2) U (2) O (n − 4) O (4) Sp (1) Sp (1) ×

E6 E7 E8 F4 , , , . Sp (3) Sp (1) SU (6) Sp (1) Spin (12) Sp (1) E7 Sp (1)

Notice that for each simply connected compact Lie group, there is exactly one compact quaternionic-K¨ ahler symmetric space associated to it. It has been a folklore conjecture that these are the only quaternionicK¨ ahler manifolds in the positive case and this has recently been proven by Kobayashi. 12. Conformal geometry Two Riemannian metrics g and g  on M are called conformally equivalent if there is a smooth function u on M such that g  = e2u g. A conformal structure on M is an equivalence class of Riemannian metrics g under conformal equivalences. For example, the standard metric on S n \ {p} is conformally equivalent to Rn and we called such a metric conformally flat. The following combination of the curvature tensor depends only on the conformal class of g,

1 R R W = Rm − Rc − g ◦ g − g ◦ g, n−2 n 2n (n − 1) and W is called the Weyl curvature tensor. When dim M ≥ 4, g is locally conformally flat if and only if W = 0. When dim M = 2, M is a Riemann

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surface and conformal structures  the same as complex structures on M .  are In particular, the group Conf S 2 of conformal transformations of S 2 is infinite dimensional. However, when n = dim M ≥ 3, we have Conf (S n ) = SO (n + 1, 1) . This group is generated by SO (n) together with inversions. There are many important topics in conformal geometry, for instances (i) the Yamabe problem, solved by Schoen, says that there is always a constant scalar curvature metric any conformal class of metrics; (ii) the  within 2 harmonic energy E (f ) = Σ |df | for a map f : (Σ, gΣ ) → (M, gM ) is invariant under conformal change of gΣ when dim Σ = 2. This fact is essential to the two dimensional conformal field theory (abbrev. 2d CFT) in physics (section 3.7); (iii) the Yang-Mills energy functional YM (DA ) = M |FA |2 for a connection DA on a unitary bundle Cr → E → (M, gM ) is invariant under conformal changes of gM when dim M = 4. This fact is important for the Donaldson theory (section 4.1). Other conformally invariant quantities include the Willmore functional in dimension four and the conformal volume of Li-Yau. In this section, we will only explain that conformal geometry can be interpreted as an extension of real, complex, quaternion and octonion geometry with the help of Jordan algebras [79]. Jordan algebras. Due to the nonassociativity of the octonions, we do not have a well-defined notion of its modules On . Nonetheless, when n ≤ 3 there is a way to define the symmetry group of the non-existing object On , by considering the space of self-adjoint operators. Recall on Rn , the space of self-adjoint operators is simply the space of symmetric n × n matrices, denoted by Sn (R). The symmetrization of ordinary matrix multiplication A ◦ B = (AB + BA) /2 makes Sn (R) into a formally real Jordan algebra. Namely it is an algebra over R whose multiplication ◦ is commutative and power associative (that is, (a ◦ a) ◦ a = a ◦ (a ◦ a)), together with a1 ◦ a1 + · · · + an ◦ an = 0 ⇒ a1 = · · · = an = 0. The same product also makes the space Sn (A) of Hermitian symmetric matrices with entries in A ∈ {R, C, H} into a Jordan algebra. When n = 3, an analog of the product can still be defined for A = O, making S3 (O) into an exceptional Jordan algebra (see e.g. [9]) even though O lacks of associativity. Inside Sn (A) we may collect all rank one projections, which are matrices p with p◦p = p and tr p = 1, to form the projective space APn−1 . For instance, even though one has problem to define O3 , each rank one projection operator in S3 (O) can be interpreted as an octonion line in O3 , and the space of them forms the octonion projective plane OP2 , which can be identified as the symmetric space F4 /Spin (9).

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Since Sn (A) is the space of self-adjoint operators on An , it should share the same automorphism group HA (n) as An . This is indeed true in the classical cases when A ∈ {R, C, H} and continues to have such an interpretation in the exceptional case A = O. The following gives a complete list of simple formally real Jordan algebras and their automorphism groups: A

R

C

H

O

Rm

Sn (A)

Sn (R)

Sn (C)

Sn (H)

S3 (O)

S2 (Rm )  Rm ⊕ R1,1

APn−1

RPn−1

CPn−1

HPn−1

OP2

AP1 = Sm

HA (n)

SO (n)

SU (n)

Sp (n)

F4

SO (m + 1)

Amazingly there is one more item in the list of Jordan algebras besides those coming from normed division algebras, namely the spin factor S2 (Rm )  Rm ⊕ R1,1 . It consists of 2 × 2 matrices of the form ⎛ ⎞

v a−b v ↔ ⎝b⎠ v a+b a where v ∈ Rm and a, b ∈ R, and we set v · w = v t w for v, w ∈ Rm to carry out matrix multiplication. The embedded projective space is ⎫ ⎧⎛ ⎞ ⎬ ⎨ v ⎝ b ⎠ : v 2 + b2 = 1 ∼ = Sm . ⎩ 1 4⎭ 2

Notice that the automorphism group SO (m + 1) of Sm is contained as a maximal compact subgroup in the non-compact group Conf(Sm ) = SO (m + 1, 1). A natural question arises: For A ∈ {R, C, H, O}, is there a symmetry group of APn−1 which gives an analog to the conformal symmetry SO (m + 1, 1) of Sm ? To answer this question, one identifies Sm as the conformal boundary of the hyperbolic ball B m+1 := {M ∈ S2 (Rm ) : detM = 1} ∼ = SO(m + 1, 1)/SO(m + 1) on which SO (m + 1, 1) acts as isometries. Under this identification, one has Conf(Sm ) ∼ = Isom(B m+1 ) = SO (m + 1, 1) which preserves collinearity in the sense that Conf(Sm ) maps circles to circles in Sm . Now for A ∈ {R, C, H}, if we collect the symmetries of APn−1 which are linear but not necessarily isometries, we obtain the group SL (n, A). Analogously APn−1 can be identified as a part of the conformal boundary of {M ∈ Sn (A) : detM = 1} ∼ = SL (n, A) /SU (n, A) on which SL (n, A) acts as isometries. We get the answer for A ∈ {R, C, H}: SL (n, A) can be regarded as the

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conformal symmetry of APn−1 , which plays the same role as SO (m + 1, 1) acting on Sm . In general, let’s denote these symmetry groups as NA (n) which are listed below. Notice that HA (n) sits inside NA (n) as a maximal compact subgroup. A

R

C

H

O

Rm

NA (n)

SL (n, R)

SL (n, C)

SL (n, H)

E6−26

SO (m + 1, 1)

HA (n)

SO (n)

SU (n)

Sp (n)

F4

SO (m + 1)

Here E6−26 is a split Lie group of type E6 and its maximum compact subgroup is F4 . We observe that when m = 1, 2, 4 and 8, NRm (2) = SL (2, A) with A being real, complex, quaternion and octonion respectively. Hence, sl (2, R) = so (2, 1), sl (2, C) = so (3, 1),  sl (2, H) = so (5, 1), sl (2, O) = so (9, 1). In general we have sl (2, A) = so A ⊕ R1,1 [9]. The above point of view integrates conformal geometry with real, complex, quaternionic and octonionic geometries. 13. Geometry of Riemannian symmetric spaces 13.1. Riemannian symmetric spaces. A Riemannian manifold (M , g) is called a Riemannian symmetric space, or simply a symmetric space, if every point x ∈ M has an involutive isometry σ x ∈ Dif f (M, g) satisfying σ x (x) = x and dσ x = −id on Tx M . Recall that involutive means σ 2x = id. A typical example is the Grassmannian GrR (r, n) of r-dimensional linear subspaces in Rn . Suppose x represents a subspace P0 , then σ x (P ) is the reflection of P along P0 with respect to the orthogonal decomposition Rn = P0 ⊕ P0⊥ . Similarly, if Rn is endowed with a complex (resp. symplectic) structure, then the Grassmannian of complex (resp. Lagrangian) subspaces in Rn is also a symmetric space for the same reason. We will explain in this section that all compact Riemannian symmetric spaces are basically of these types. A noncompact example of a symmetric space is the Grassmannian GrR+ (r, n) of spacelike subspaces, namely it is the set of r-dimensional subspaces P in Rr,n−r such that restriction of the signature (r, n − r) inner  2   2  product ·, · = rj=1 dxj − nk=r+1 dxk to P is positive definite. Note that GrR+ (r, n) is naturally an open subset in GrR (r, n) and they are called the compact/noncompact dual to each other. Similarly, the Grassmannian of spacelike complex (resp. Lagrangian) subspaces in Cr,n−r is a noncompact symmetric space, denoted as GrC+ (r, n) (resp. LGrC+ (n) when n = 2r). The curvature tensor of a symmetric space is always covariant constant, i.e. ∇Rm = 0. This is because dσ x acts as (−1) on Tx M and ∇Rm is an odd degree tensor. Conversely, the universal cover of any Riemannian manifold

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M with ∇Rm = 0 is a symmetric space and we call such a M a locally symmetric space. A symmetric space M has a large group of isometries Dif f (M, g) which acts transitively on M . That is, for any p, q ∈ M , there is an isometry σ satisfying σ (p) = q. Indeed σ is the involutive symmetry σ x for the midpoint x of the geodesic segment joining p and q. If we denote G = Dif f (M, g) and K as the isotropy subgroup fixing a point x ∈ M , then we have M∼ = G/K. In particular, dσ x (x) = −id on Tx M = g/k, where g and k are the Lie algebras of G and K respectively. Indeed there is a Lie algebra homomorphism σ : g → g satisfying σ 2 = id and the eigenspace decomposition of σ is given by g = k⊕p with eigenvalues 1 and −1 on k and p respectively. Equivalently, g has a decomposition g = k ⊕ p as vector spaces satisfying [k, k] ⊂ k, [k, p] ⊂ p and

[p, p] ⊂ k.

Note that p = Tx M . Furthermore, up to covering, M is determined by such a structure (g, σ), called an involutive Lie algebra. The Riemannian structure on M is reflected by the existence of a positive inner product on g which is invariant under σ and adg k and this is called an orthogonal structure on (g, σ). The curvature of M can be expressed in terms of the Lie algebra structure of g, namely for any X, Y ∈ p = Tx M , we have Rm (X, Y ) X, Y M = − [X, [X, Y ]] , Y g 1 Rc (X, Y ) = −T rp (adX ◦ adY ) = − X, Y g . 2 Orthogonal involutive Lie algebras and therefore Riemannian symmetric spaces has been completely classified by Cartan and we will describe them as Grassmannians. We recall from section 3 that holonomy groups of Riemannian manifolds are classified by Berger into two types: the first type are manifolds defined over a normed algebra A and with or without A-orientation; the second type are locally symmetric spaces. Suppose M = G/K is an irreducible symmetric space, then it is compact (resp. noncompact) if and only if its curvature is positive (resp. negative), with the only exception that the universal cover of M is Rn in which case its curvature is zero. As the curvature of M can be expressed in terms of the Killing form of g, M is compact if and only if G is a compact Lie group and M is noncompact if and only if K is a maximal compact subgroup in a noncompact Lie group G. Symmetric spaces always come in pairs (Mc , Mn ) with Mc = Gc /Kc (resp. Mn = Gn /Kn ) is a compact (resp. noncompact) symmetric space and the Lie algebras have the same complexification gc ⊗ C ∼ = gn ⊗ C,

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and under this isomorphism, kc ∼ = kn

and pc ∼ = ipn .

Mc and Mn are called compact/noncompact dual to each other. In many cases, Mn is naturally an open subset in Mc . For instance O (r, n − r) O (n) ⊂ O (r) O (n − r) O (r) O (n − r) or equivalently, GrR+ (r, n) ⊂ GrR (r, n) as the open subset consisting of those r-dimensional subspaces which are spacelike. Every compact Lie group G is an example of symmetric spaces when we view it as a quotient of G × G by G under the diagonal action. The Lie group O (n) (resp. U (n) and Sp (n)) can be viewed as the Grassmannian of maximal isotropic subspaces in Rn,n (resp. Cn,n and Hn,n ). Recall that a maximal isotropic subspace is a half dimensional linear subspace with the restriction of the signature (n, n) inner product to it being identically zero. When a Lie group G is noncompact, it also determines a symmetric space M = G/K where K is a maximal compact subgroup in G. Indeed every noncompact symmetric space is of this form. For example, when G is the complexification K C of a compact Lie group K, then K C /K is the noncompact dual to K. 13.2. Jordan algebras and magic square. Every compact Lie group is a compact symmetric space. Up to finite covers, they are determined by their Lie algebras and they are classified into An = su (n + 1), Bn = so (2n + 1), Cn = sp (n), Dn = so (2n), E6 , E7 , E8 , F4 and G2 . Note that classical Lie algebras, i.e. An , Bn , Cn and Dn , consist of infinitesimal symmetries of inner product spaces Rn , Cn and Hn . Recall that G2 can also be interpreted as the symmetry group of O. The remaining ones can be interpreted as the symmetries of (A ⊗ B)3 with both A and B normed division algebras. However, this magic square approach of Tits [123] and Freudenthal [44] is more subtle (see [9] for a beautiful account). They show that   gn (A, B) = DerSn (A) ⊕ Sn (A)T r=0 ⊗ Im B ⊕ Der(B) admits a natural Lie algebra structure. Here Sn (A) is the Jordan algebra of A-Hermitian n × n matrices (n = 3 if A or B is the octonion). On the one hand, going from A to 2A, for instance from R to C, corresponds to enlarging the symmetry of a linear space V to its complexification V ⊕ V = V ⊗ C with one added complex structure. We can also identify this as the tangent bundle T V of V . On the other hand, going from B to 2B corresponds to enlarging the symmetry of V to its symplectification V ⊕ V ∗ = T ∗ V . The

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corresponding Lie groups Gn (A, B) (up to finite cover and center) are given in the following table. B\A R C H O

R SO (n) SU (n) Sp (n) F4

C SU (n) SU (n)2 SU (2n) E6

H Sp (n) SU (2n) SO (4n) E7

O F4 E6 E7 E8

This is called the magic square because of the symmetry of interchanging A and B, just like mirror symmetry! Loosely speaking, Gn (A, B) is the group of symmetries of (A ⊗ B)n . The set of all real linear subspaces in (A ⊗ B)n which are complex linear with respect to all complex structures coming from A and B will be called (A, B)-Grassmannian, or simply Grassmannian, and it is denoted as GrAB (r, n), or symbolically as {(A ⊗ B)r ⊂ (A ⊗ B)n } and they are given in the following table.

A\B R C H O

R

C

H

O

O(n) U(n) Sp(n) F4 O(k)O(n − k) U(k)U(n − k) Sp(k)Sp(n − k) Spin(9) 2 U(n) U(n) U(2n) E6 U(k)U(n − k) U(k)2 U(n − k)2 U(2k)U(2n − 2k) Spin(10)U(1) Sp(n) U(2n) O(4n) E7 Sp(k)Sp(n − k) U(2k)U(2n − 2k) O(4k)O(4n − 4k) Spin(12)Sp(1) F4 E6 E7 E8 Spin(9) Spin(10)U(1) Spin(12)Sp(1) SO(16)

Every GrAB (r, n) is a compact Riemannian symmetric space G/K with G = Gn (A, B) and K equals to Gr (A, B) Gn−r (A, B), possibly up to U (1) or Sp (1) factors. Next we consider real linear subspaces in (A ⊗ B)n which are complex linear with respect to all complex structures coming from A and B except the last J from A and instead we require it to be Lagrangian with respect to ω J . The set of all such subspaces is called Lagrangian(A, B)-Grassmannian, or simply Lagrangian Grassmannian, and it is denoted as LGrAB (n), or  n A ⊂ (A ⊗ B)n and they are given in the following symbolically as 2 ⊗B table.

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B\A

R

C

H

O

R

n/a

n/a

n/a

n/a

C

U (n) O (n) Sp (n) U (n) F4 Sp (3) Sp (1)

SU (n)2 SU (n) U (2n) U (n)2 E6 SU (6) Sp (1)

SU (2n) Sp (n) SO (4n) U (2n) E7 Spin (12) Sp (1)

E6 F4 E7 E6 U (1) E8 E7 Sp (1)

H O

Each LGrAB (n) is a compact Riemannian symmetric space G/K with G = A  Gn (A, B) and K equals to Gn 2 , B , possibly up to U (1) or Sp (1) factors. This is because the orthogonal complement of a Lagrangian subspace is naturally its dual space. Note that the symmetric space G2 /SO (4) = G2 /Sp (1)2 is the Lagrangian Grassmannian LGrOR (1) . With a little more work, SO (2n) /U (n) can be described in a similar fashion [69]. In fact we have given the list of all symmetric spaces of compact type up to covering with the exceptions of E6 /Sp (4) and E7 /SU (8). They can be realized as follows: We consider real linear subspaces in (A ⊗ B)3 which are complex linear with respect to all complex structures coming from A and B except the last complex structures JA from A and JB from B and instead we require it to be Lagrangian with respect to both ω JA and ω JB . The set of all such subspaces is called double Lagrangian (A, B)-Grassmannian, or simply double Lagrangian and

it is denoted as LLGrAB , or sym  Grassmannian,  3 A B 4 2 bolically as Λ 2 ⊗ 2 ⊂ (A ⊗ B) and they are given in the following table. B\A

R

C

R

n/a

n/a 2

C

n/a

H

n/a

O

n/a

SU (3) SO (4) SU (6) SU (4) E6 Sp (4)

H

O

n/a

n/a

SU (6) SU (4) SO (12) U (6) E7 SU (8)

E6 Sp (4) E7 SU (8) E8 SO (16)

Each LLGrAB is a compact  A B  Riemannian symmetric space G/K with G = G3 (A, B) and K = G4 2 , 2 ! A reason for having dimension four appearing in K is roughly as follows: If we consider the decomposition of two forms on R4 into self-dual and anti-self-dual forms, Λ2 R4 = Λ2+ R4 ⊕ Λ2− R4 , then

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each Λ2± R4 is a Lagrangian subspace in a copy of C3 in the decomposition C3 ⊗ C = C3 ⊕ C3 . Thus SU (3)2 /SO (4) = [SU (3) /SO (3)]2 and similarly we have SU (6) /SU (4) = SU (6) /SO (6). We should warn our readers that much care is always needed whenever A or B equals O [69]. These give a simple description of all locally symmetric spaces of compact type up to covering as Grassmannians of complex/Lagrangian subspaces in (A ⊗ B)n except compact Lie groups G. In fact compact Lie groups Gn (A, B) can also be described as the Grassmannians of maximal isotropic subspaces in T ∗ (A ⊗ B)n with respect to the canonical symmetric bilinear form of type (m, m). These structures are related to the generalized complex structures as introduced by Hitchin. ˆ of noncompact type? In many cases How about symmetric spaces M ˆ ˆ ⊂ M . The simplest M admits an open embedding into its compact dual M example is GrR+ (r, n) =

O (r, n − r) O (n) ⊂ = GrR (r, n) O (r) O (n − r) O (r) O (n − r)

where GrR+ (r, n) is the Grassmannian of spacelike linear subspaces of dimension r in Rr,n−r . Indeed this holds true in general for Hermitian symmetric spaces as well. 13.3. Hermitian and quaternionic symmetric spaces. Hermitian symmetric spaces. A Riemannian symmetric space G/K is called a Hermitian symmetric space if it admits a G-invariant complex structure. This happens precisely when K has a U (1)-factor, possibly up to a finite cover. Classical irreducible compact Hermitian symmetric spaces are classified into (i) Type I(p,q) SU (p + q) /S (U (p) U (q)), (ii) Type II(n) Spin (2n) /U (n), (iii) Type III(n) Sp (n) /U (n) and (iv) Type IV(n) Spin (n + 1) /Spin (n) U (1). There are natural embeddings Gn /K ⊂ p ⊂ Gc /K where the first embedding realizes Gn /K as a bounded domain in CN , called the Harish-Chandra embedding and Gn /K ⊂ Gc /K is called the Borel embedding. An example of these embeddings is   GrC+ (r, n) ⊂ Hom Cr , Cn−r ⊂ GrC (r, n) . Conversely, every bounded domain D in CN has a natural K¨ ahler metric, called the Bergman metric. If it is also symmetric in the sense that every point x ∈ D is fixed by a biholomorphic involution as an isolated fixed point, then this involution must preserve the Bergman metric. Thus noncompact Hermitian symmetric spaces are the same as bounded symmetric domains.

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The bounded symmetric domain Sp (2n, R) /U (n) can be transformed to the Siegel upper half space Sn+ (R)+iSn (R) by the Cayley transformation, where Sn+ (R) is the set of positive definite symmetric matrices of rank n. It is well-known that Sp (2n, R) /U (n) is the universal cover of the moduli space of n-dimensional Abelian varieties, or equivalently the moduli space of weight one polarized Hodge structures. Indeed every noncompact Hermitian symmetric space can be described as the moduli space of polarized Hodge structures of certain types by the work of Deligne. Recall that Sn (R) is a Jordan algebra and every Jordan algebra is of the form Sn (A), with A ∈ {R, C, H, O, Rm }. When A = O, we have n = 3 and S3 (O) is the exceptional Jordan algebra; when A = Rm , we have n = 2 and S2 (Rm ) ∼ = Rm,1 is the spin factor. Every Sn+ (A) + iSn (A) is a noncompact Hermitian symmetric space and they form the complete list of tube domains. We have A

R

Sn+ (A) + iSn (A)

C

H

O

Rm

Sp (2n, R) U (n, n) O∗ (4n) E7,3 O (m + 2, 2) 2 U (n) U (2n) E6 U (1) O (m + 2) O (2) U (n)

In fact, they are simply the Grassmannians of spacelike (C ⊗ A)-Lagrangians in (H ⊗ A)n , S + (A) + iSn (A) ∼ = LGr+ (n) . n

HA

+ We also remark that Sn+ (A) ∼ (n) . = LGrCA Quaternionic symmetric spaces. A symmetric space M = G/K is called a quaternionic symmetric spaces if it is also a quaternionic-K¨ ahler manifold, namely its holonomy group lies inside Sp (n) Sp (1) with 4n = dimR M . Similar to the characterization of Hermitian symmetric spaces, G/K is a quaternionic symmetric space if and only if K contains a Sp (1)-factor, possibly up to a finite cover. This Sp (1)-factor always corresponds to the longest root of G. As a matter of fact, there is exactly one compact quaternionic symmetric space for each simple Lie group, up to finite covers. Classical compact quaternionic symmetric spaces are

Sp (n + 1) U (n + 2) , Sp (n) Sp (1) U (2) U (2)

and

O (n + 4) . O (n) O (4)

Note that Sp (n + 1) /Sp (n) Sp (1) ∼ = HPn is the set of H-lines in Hn+1 . Recall from the magic square that when n = 2m, su (2m) should be regarded as the set of infinitesimal symmetries of (C ⊗ H)m . Thus U (n + 2) /U (n) U (2) ∼ = (C ⊗ H) Pm . Similarly, O (n + 4) /O (n) O (4) ∼ = (H ⊗ H) Pm with n = 4m. We can also include the exceptional symmetric space (O ⊗ H) P2 ∼ = E7 /SO (12) O (4) in this list of quaternionic symmetric spaces of the form (A ⊗ H) Pm for some normed division algebra A.

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Exceptional compact quaternionic symmetric spaces are E6 E7 E8 G2 F4 , , , and . Sp (3) Sp (1) SU (6) U (2) SO (12) O (4) E7 Sp (1) Sp (1)2 The first four in the above list correspond to the sets of H-Lagrangians in (O ⊗ A)3 with A = R, C, H and O respectively, i.e. LGrOA (3). And the last one is simply the set of H-Lagrangians in O, i.e. LGH (1). 14. Conclusions In this article, we summarized certain aspects of geometric structures on Riemannian manifolds M . From a topological perspective, we discussed bundles and submanifolds, leading to K-theory and cohomology theory. When M is equipped with a Riemannian metric, we look for canonical structures by minimizing the energy of a connection and the volume of a submanifold. The Euler-Lagrange equations are the Yang-Mills equations and the minimal surface equations. They are both system of second order partial differential equations. Most spaces we encounter in nature have more geometric structures. That is why symmetry is such an important subject in physics. Spaces with the largest amount of symmetries are Riemannian symmetric spaces. They are model spaces in geometry as well as many important moduli spaces. A compatible complex structure J on M gives a symplectic structure ω via ω (u, v) = g (Ju, v). This makes M into a K¨ ahler manifold. K¨ ahler geometry is a very rich subject as it includes the complex algebraic geometry for projective manifolds. K¨ ahlerian is also the necessary and sufficient condition for the existence of N = 2 supersymmetry for σ-models in physics. J defines the complex geometry on M while ω defines the symplectic geometry on M . These two geometries are conjecturally dual to each other by the mirror symmetry conjecture, provided that M has a C-orientation, namely a Calabi-Yau manifold. Berger classified all possible reduced holonomy groups and the author described them in terms of normed algebras A ∈ {R, C, H, O} and Aorientability. This means that geometric structures on (M, g) do always come from, possibly more than one, J or equivalently ω. This approach also gives a unified description of various canonical Yang-Mills connections and calibrated submanifolds in M . As a result, we found new results and relationships among these geometries. These canonical connections and submanifolds are not just critical points of the energy/volume functionals, they are absolute minimizers. Furthermore they are governed by first order PDEs and they define, at least formally, topological invariants of M via various topological field theories. The O-geometry has the richest structure and its corresponding geometries have holonomy groups Spin (7) ⊃ G2 ⊃ SU (3) in dimension 8, 7 and 6 respectively. The lack of associativity for O is rescued by the triality as we

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have seen it reflected in G2 -geometry. The octonion structure is also equivalent to the vector cross product, which is a natural generalization of the complex structure, thus we have seen that G2 -geometry resembles the K¨ahler geometry in many ways. The H-Lagrangian geometry in the O-geometry brings the beautiful low dimensional geometry and physics in dimension 4, 3 and 2 into an integral part of the O-geometry. For instance the author expects that the skein relation for knots in R3 and the Morgan-MrowkaSzab´ o formula for the SW-invariants are consequences of the G2 -triality. The duality and triality transformations among different geometries have proven to be a powerful tools to uncover many structures and producing amazing formulas. Many of these formulas have been verified by computational methods. The SYZ approach to explain the mirror symmetry has sparked many exciting developments in mathematics, as well as in physics. We expect to continue to see many more of such developments in the coming years until we fully understand the SYZ transformations.

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Surveys in Differential Geometry XVI

Sasaki-Einstein manifolds James Sparks

A Sasaki-Einstein manifold is a Riemannian manifold (S, g) that is both Sasakian and Einstein. Sasakian geometry is the odd-dimensional cousin of K¨ ahler geometry. Indeed, just as K¨ ahler geometry is the natural intersection of complex, symplectic, and Riemannian geometry, so Sasakian geometry is the natural intersection of CR, contact, and Riemannian geometry. Perhaps the most straightforward definition is the following:  a Riemannian manifold (S, g)  is Sasakian if and only if its metric cone C(S) = R>0 × S, g¯ = dr2 + r2 g is K¨ ahler. In particular, (S, g) has odd dimension 2n−1, where n is the complex dimension of the K¨ ahler cone. A metric g is Einstein if Ricg = λg for some constant λ. It turns out that a Sasakian manifold can be Einstein only for λ = 2(n − 1), so that g has positive Ricci curvature. Assuming, as we shall do throughout, that (S, g) is complete, it follows from Myers’ Theorem that S is compact with finite fundamental group. Moreover, a simple calculation shows that a Sasakian metric g is Einstein with Ricg = 2(n − 1)g if and only if the cone metric g¯ is Ricci-flat, Ricg¯ = 0. It immediately follows that for a Sasaki-Einstein g ) ⊂ SU (n). manifold the restricted holonomy group of the cone Hol0 (¯ The canonical example of a Sasaki-Einstein manifold is the odd dimensional sphere S 2n−1 , equipped with its standard Einstein metric. In this case the K¨ ahler cone is Cn \ {0}, equipped with its flat metric. A Sasakian manifold (S, g) inherits a number of geometric structures from the K¨ ahler structure of its cone. In particular, an important role is played by the Reeb vector field. This may be defined as ξ = J(r∂r ), where J denotes the integrable complex structure of the K¨ahler cone. The restriction of ξ to S = {r = 1} = {1} × S ⊂ C(S) is a unit length Killing vector field, and its orbits thus define a one-dimensional foliation of S called the Reeb foliation. There is then a classification of Sasakian manifolds, and hence 1991 Mathematics Subject Classification. Primary 53C25, 53C15. Key words and phrases. Sasakian geometry, Sasaki-Einstein manifolds. The author is supported by a Royal Society University Research Fellowship. c 2011 International Press

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also Sasaki-Einstein manifolds, according to the global properties of this foliation. If all the orbits of ξ are compact, and hence circles, then ξ integrates to a locally free isometric action of U (1) on (S, g). If this action is free, then the Sasakian manifold is said to be regular; otherwise it is said to be quasi-regular. On the other hand, if ξ has a non-compact orbit the Sasakian manifold is said to be irregular. For the purposes of this introduction it is convenient to focus on the case of dimension 5 (n = 3), since this is the lowest non-trivial dimension. Prior to the 21st century, the only known examples of Sasaki-Einstein 5-manifolds were regular. As we shall explain, a regular Sasaki-Einstein manifold is the total space of a principal U (1) bundle over a K¨ ahler-Einstein manifold of positive Ricci curvature. The classification of such Fano K¨ ahlerEinstein surfaces due to Tian-Yau then leads to a classification of all regular Sasaki-Einstein 5-manifolds. Passing to their simply-connected covers, these are connected sums S 5 #k(S 2 × S 3 ) where k = 0, 1, 3, 4, 5, 6, 7, 8. For each of k = 0, 1, 3, 4 there is a unique such regular Sasaki-Einstein structure, while for 5 ≤ k ≤ 8 there are continuous families of complex dimension 2(k − 4). However, before 2001 it was not known whether quasi-regular Sasaki-Einstein 5-manifolds existed, and indeed there was even a conjecture that irregular Sasaki-Einstein manifolds do not exist. The progress over the last decade has been dramatic. Again, focusing on dimension 5, it is now known that there exist Sasaki-Einstein structures on #k(S 2 × S 3 ) for all values of k. These include infinitely many toric Sasaki-Einstein metrics, meaning that the torus T3 acts isometrically on the Sasakian structure, for every value of k. Indeed, for k = 1 these metrics are known completely explicitly, giving countably infinite families of quasi-regular and irregular Sasaki-Einstein structures on S 2 × S 3 . The list of Sasaki-Einstein structures on #k(S 2 × S 3 ) also includes examples with the smallest possible isometry group, namely the T ∼ = U (1) generated by a quasi-regular Reeb vector field. In particular, there are known to exist infinitely many such structures for every k, except k = 1, 2, and these often come in continuous families. Moreover, S 5 itself admits at least 80 inequivalent quasi-regular Sasaki-Einstein structures. Again, some of these come in continuous families, the largest known having complex dimension 5. There are also quasi-regular Sasaki-Einstein metrics on 5-manifolds which are not connected sums of S 2 × S 3 , including infinitely many rational homology 5-spheres, as well as infinitely many connected sums of these. Similar abundant results hold also in higher dimensions. In particular, all 28 oriented diffeomorphism classes on S 7 admit Sasaki-Einstein metrics. In this article I will review these developments, as well as other results not mentioned above. It is important to mention those topics that will not be covered. As in K¨ ahler geometry, one can define extremal Sasakian metrics and Sasaki-Ricci solitons. Both of these generalize the notion of a SasakiEinstein manifold in different directions. Perhaps the main reason for not discussing these topics, other than reasons of time and space, is that so far

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there are not any obvious applications of these geometries to supergravity theories and string theory, which is the author’s main interest. The article is arranged as follows. Section 1 is a review of all the necessary background in Sasakian geometry; section 2 covers regular SasakiEinstein manifolds; section 3 describes the construction of quasi-regular Sasaki-Einstein structures, focusing in particular on links of weighted homogeneous hypersurface singularities; in section 4 we describe what is known about explicit constructions of Sasaki-Einstein manifolds; section 5 covers toric Sasakian geometry and the classification of toric Sasaki-Einstein manifolds; section 6 describes obstructions; and section 7 concludes with some open problems. Acknowledgements. I would like to thank Charles Boyer for comments on the first version of this article. 1. Sasakian geometry 1.1. Sasakian basics. We begin with an introduction to Sasakian geometry. Many of the results here are elementary and follow almost immediately from the definitions. The reader is referred to [14, 48, 75] or the recent monograph [21] for detailed proofs. Definition 1.1. A compact Riemannian manifold (S, g) is Sasakian if  and only if its metric cone C(S) = R>0 × S, g¯ = dr2 + r2 g is K¨ahler. It follows that S has odd dimension 2n − 1, where n denotes the complex dimension of the K¨ ahler cone. Notice that the Sasakian manifold (S, g) is naturally isometrically embedded into the cone via the inclusion S = {r = 1} = {1} × S ⊂ C(S). We shall often regard S as embedded into C(S) this way. There is also a canonical projection p : C(S) → S which forgets the r coordinate. Being K¨ ahler, the cone (C(S), g¯) is equipped with an integrable complex structure J and a K¨ ahler 2-form ω, both of which are parallel ¯ of g¯. The K¨ahler structure of with respect to the Levi-Civita connection ∇ (C(S), g¯), combined with its cone structure, induces the Sasakian structure on S = {1} × S ⊂ C(S). The following equations are useful in proving many of the formulae that follow: (1.1)

¯ r∂ (r∂r ) = r∂r , ∇ ¯ r∂ X = ∇ ¯ X (r∂r ) = X, ∇ r r ¯ X Y = ∇X Y − g(X, Y )r∂r . ∇

Here X and Y denote vector fields on S, appropriately interpreted also as vector fields on C(S), and ∇ is the Levi-Civita connection of g, . The canonical vector field r∂r is known as the homothetic or Euler vector field. Using the relations (1.1), together with the fact that J is parallel, ¯ = 0, one easily shows that r∂r is real holomorphic, Lr∂ J = 0. It is then ∇J r

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natural to define the characteristic vector field (1.2)

ξ = J (r∂r ) .

Again, elementary calculations show that ξ is real holomorphic and also Killing, Lξ g¯ = 0. Moreover, ξ is clearly tangent to surfaces of constant r and has square length g¯(ξ, ξ) = r 2 . We may similarly define the 1-form (1.3) η = dc log r = i(∂¯ − ∂) log r, where as usual dc = J ◦ d denotes the composition of exterior derivative with the action of J on 1-forms, and ∂, ∂¯ are the usual Dolbeault operators, with ¯ It follows straightforwardly from the definition that d = ∂ + ∂. (1.4)

η(ξ) = 1,

iξ dη = 0.

Here we have introduced the interior contraction: if α is a (p + 1)-form and X a vector field then iX α is the p-form defined via iX α(X1 , . . . , Xp ) = α(X, X1 , . . . , Xp ). Moreover, it is also clear that 1 1 g¯ (J(r∂r ), X) = 2 g¯(ξ, X). 2 r r Using this last formula one can show that the K¨ ahler 2-form on C(S) is (1.5)

(1.6)

η(X) =

1 ¯ 2 1 . ω = d(r2 η) = i∂ ∂r 2 2

The function 12 r2 is hence a global K¨ahler potential for the cone metric. The 1-form η restricts to a 1-form η |S on S ⊂ C(S). One checks from Lr∂r η = 0 that in fact η = p∗ (η |S ). In a standard abuse of notation, we shall then not distinguish between the 1-form η on the cone and its restriction to the Sasakian manifold η |S . Similar remarks apply to the Reeb vector field ξ: by the above comments this is tangent to S, where it defines a unit length Killing vector field, so g(ξ, ξ) = 1 and Lξ g = 0. Notice from (1.5) that η(X) = g(ξ, X) holds for all vector fields X on S. Since the K¨ ahler 2-form ω is in particular symplectic, it follows from (1.6) that the top degree form η ∧ (dη)n−1 on S is nowhere zero; that is, it is a volume form on S. By definition, this  makes η a contact 1-form on S. Indeed, the open symplectic manifold C(S) = R>0 × S, ω = 12 d(r2 η) is called the symplectization of the contact manifold (S, η). The relations η(ξ) = 1, iξ dη = 0 from (1.4) imply that ξ is the unique Reeb vector field for this contact structure. We shall hence also refer to ξ as the Reeb vector field of the Sasakian structure. The contact subbundle D ⊂ T S is defined as D = ker η. The tangent bundle of S then splits as (1.7)

T S = D ⊕ Lξ ,

where Lξ denotes the line tangent to ξ. This splitting is easily seen to be orthogonal with respect to the Sasakian metric g.

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Next define a section Φ of End(T S) via Φ |D = J |D , Φ |Lξ = 0. Using = −1 and that the cone metric g¯ is Hermitian one shows that Φ2 = −1 + η ⊗ ξ,

(1.8)

g (Φ(X), Φ(Y )) = g(X, Y ) − η(X)η(Y ),

(1.9)

where X, Y are any vector fields on S. In fact a triple (η, ξ, Φ), with η a contact 1-form with Reeb vector field ξ and Φ a section of End(T S) satisfying (1.8), is known as an almost contact structure. This implies that (D, JD ≡ Φ |D ) defines an almost CR structure. Of course, this has been induced by embedding S as a real hypersurface {r = 1} in a complex manifold, so this almost CR structure is of hypersurface type (the bundle D has rank one less than that of T S) and is also integrable. The Levi form may be taken to be 1 ¯ 2 ahler form of the cone (1.6). Since this is positive, 2 i∂∂r , which is the K¨ by definition we have a strictly pseudo-convex CR structure. The second equation (1.9) then says that g |D is a Hermitian metric on D. Indeed, an almost contact structure (η, ξ, Φ) together with a metric g satisfying (1.9) is known as a metric contact structure. Sasakian manifolds are thus special types of metric contact structures, which is how Sasaki originally introduced them [91]. Since 1 g(X, Y ) = dη(X, Φ(Y )) + η(X)η(Y ), 2 we see that 12 dη |D is the fundamental 2-form associated to g |D . The contact subbundle D is symplectic with respect to this 2-form. The tensor Φ may also be defined via Φ(X) = ∇X ξ. This follows from the last equation in (1.1), together with the calculation ¯ Xξ = ∇ ¯ X (J (r∂r )) = J(X). Then a further calculation gives ∇ (1.10)

(∇X Φ) Y = g(ξ, Y )X − g(X, Y )ξ,

where X and Y are any vector fields on S. This leads to the following equivalent definitions of a Sasakian manifold [14], the first of which is perhaps the closest to the original definition of Sasaki [91]. Proposition 1.2. Let (S, g) be a Riemannian manifold, with ∇ the LeviCivita connection of g and R(X, Y ) the Riemann curvature tensor. Then the following are equivalent: (1) There exists a Killing vector field ξ of unit length so that the tensor field Φ(X) = ∇X ξ satisfies (1.10) for any pair of vector fields X, Y on S.

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(2) There exists a Killing vector field ξ of unit length so that the Riemann curvature satisfies R(X, ξ)Y = g(ξ, Y )X − g(X, Y )ξ, for any pair of vector fields X, Y on S. ahler. (3) The metric cone (C(S), g¯) = (R>0 × S, dr2 + r2 g) over S is K¨ The equivalence of (1) and (2) follows from an elementary calculation relating (∇X Φ)Y to R(X, ξ)Y . We have already sketched the proof that (3) implies (1). To show that (1) implies (3) one defines an almost complex structure J on C(S) via J (r∂r ) = ξ,

J(X) = Φ(X) − η(X)r∂r ,

where X is a vector field on S, appropriately interpreted as a vector field on C(S), and η(X) = g(ξ, X). It is then straightforward to check from the definitions that the cone is indeed K¨ ahler. We may think of a Sasakian manifold as the collection S = (S, g, η, ξ, Φ). As mentioned, this is a special type of metric contact structure. 1.2. The Reeb foliation. The Reeb vector field ξ has unit length on (S, g) and in particular is nowhere zero. Its integral curves are geodesics, and the corresponding foliation Fξ is called the Reeb foliation. Notice that, due to the orthogonal splitting (1.7), the contact subbundle D is the normal bundle to Fξ . The leaf space is clearly identical to that of the complex vector field ξ − iJ(ξ) = ξ + ir∂r on the cone C(S). Since this complex vector field is holomorphic, the Reeb foliation thus naturally inherits a transverse holomorphic structure. In fact the leaf space also inherits a K¨ ahler metric, giving rise to a transversely K¨ ahler foliation, as we now describe. Introduce a foliation chart {Uα } on S, where each Uα is of the form Uα = I ×Vα with I ⊂ R an open interval and Vα ⊂ Cn−1 open. We may introduce coordinates (x, z1 , . . . , zn−1 ) on Uα , where ξ = ∂x and z1 , . . . , zn−1 are complex coordinates on Vα . The fact that the cone is complex implies that the transition functions between the Vα are holomorphic. More precisely, if Uβ has corresponding coordinates (y, w1 , . . . , wn−1 ), with Uα ∩ Uβ = ∅, then ∂zi = 0, ∂ w¯j

∂zi = 0. ∂y

Recall that the contact subbundle D is equipped with the almost complex structure JD , so that on D ⊗ C we may define the ±i eigenspaces of JD as the (1, 0) and (0, 1) vectors, respectively. Then in the above foliation chart Uα , (D ⊗ C)(1,0) is spanned by ∂zi − η (∂zi ) ξ.

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Since ξ is a Killing vector field, and so preserves the metric g, it follows that g |D gives a well-defined Hermitian metric gαT on the patch Vα by restricting to a slice {x = constant}. Moreover, (1.4) implies that       dη ∂zi − η (∂zi ) ξ, ∂z¯j − η ∂z¯j ξ = dη ∂zi , ∂z¯j . The fundamental 2-form ωαT for the Hermtian metric gαT in the patch Vα is hence equal to the restriction of 12 dη to a slice {x = constant}. Thus ωαT is closed, and the transverse metric gαT is K¨ ahler. Indeed, in the chart Uα notice that we may write η = dx + i

n−1 

∂zi Kα dzi − i

i=1

n−1 

∂z¯i Kα d¯ zi ,

i=1

where Kα is a function on Uα with ∂x Kα = 0. The local function Kα is a K¨ ahler potential for the transverse K¨ ahler structure in the chart Uα , as observed in [56]. Such a structure is called a transversely K¨ ahler foliation. T T We denote the collection of transverse metrics by g = {gα }. Although g T so defined is really a collection of metrics in each coordinate chart, notice that we may identify it with the global tensor field on S defined via 1 g T (X, Y ) = dη(X, Φ(Y )). 2 That is, the restriction of g T to a slice {x = constant} in the patch Uα is equal ahler form ω T may be defined globally as to gαT . Similarly, the transverse K¨ 1 2 dη. The basic forms and basic cohomology of the Reeb foliation Fξ play an important role (for further background, the reader might consult [104]): Definition 1.3. A p-form α on S is called basic if iξ α = 0,

Lξ α = 0.

We denote by ΛpB the sheaf of germs of basic p-forms and ΩpB the set of global sections of ΛpB . If α is a basic form then it is easy to see that dα is also basic. We may thus define dB = d |Ω∗B , so that dB : ΩpB → Ωp+1 B . The corresponding complex ∗ (F ) (Ω∗B , dB ) is called the basic de Rham complex, and its cohomology HB ξ the basic cohomology. Let Uα and Uβ be coordinate patches as above, with coordinates adapted to the Reeb foliation (x, z1 , . . . , zn−1 ) and (y, w1 , . . . , wn−1 ), respectively. Then a form of Hodge type (p, q) on Uα α = αi1 ···ip ¯j1 ···¯jq dzi1 ∧ · · · ∧ dzip ∧ d¯ z¯j1 ∧ · · · ∧ d¯ z¯jq , is also of type (p, q) with respect to the wi coordinates on Uβ . Moreover, if α is basic then αi1 ···ip¯j1 ···¯jq is independent of x. There are hence globally

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well-defined Dolbeault operators p+1,q , ∂B : Ωp,q B → ΩB p,q p,q+1 ∂¯B : ΩB → ΩB .

Both are nilpotent of degree 2, so that one can define the basic Dolbeault p,∗ complex (Ωp,∗ , ∂¯B ) and corresponding cohomology groups HB (Fξ ). These invariants of the Reeb foliation are important invariants of the Sasakian manifold. Clearly dB = ∂B + ∂¯B , and we may similarly define the operator dcB = i(∂¯B − ∂B ). In each local leaf space Vα we may define the Ricci forms as ρTα = −i∂ ∂¯ log det gαT . These are the (1, 1)-forms associated to the Ricci tensors RicTα of gαT via the complex structure, in the usual way. Via pull-back to Uα under the projection Uα → Vα that identifies points on the same Reeb orbit, one sees that these patch together to give global tensors ρT and RicT on S, just as was the case for the transverse K¨ahler form and metric. In particular, the transverse Ricci form ρT is a global basic 2-form of Hodge type (1, 1) which is closed under dB . The corresponding basic cohomology 1,1 B class [ρT /2π] is denoted by cB 1 (Fξ ) ∈ HB (Fξ ), or simply c1 , and is called the basic first Chern class of Fξ . Again, this is an important invariant of B B B the Sasakian structure. We say that cB 1 > 0 or c1 < 0 if c1 or −c1 is represented by a transverse K¨ ahler form, respectively. In particular, the Sasakian structure will be called transverse Fano if cB 1 > 0. Thus far we have considered a fixed Sasakian manifold S = (S, g, η, ξ, Φ). It will be important later to understand how one can deform such a structure to another Sasakian structure on the same manifold S. Since a Sasakian manifold and its corresponding K¨ ahler cone have several geometric structures, one can fix some of these whilst deforming others. An important class of such deformations, which are analogous to deformations in K¨ ahler geometry, is summarized by the following result: Proposition 1.4. Fix a Sasakian manifold S = (S, g, η, ξ, Φ). Then any other Sasakian structure on S with the same Reeb vector field ξ, the same holomorphic structure on the cone C(S) = R>0 × S, and the same transversely holomorphic structure of the Reeb foliation Fξ is related to the original structure via the deformed contact form η  = η + dcB φ, where φ is a smooth basic function that is sufficiently small. Proof. The proof is straightforward. We fix the holomorphic structure on the cone, but replace r by r = r exp φ, where 12 (r )2 will be the new global K¨ ahler potential for the cone metric. Since J and ξ are held fixed, from (1.2) we have r ∂r = r∂r , which implies that φ is a function on S. The ¯  2 . Since new metric on C(S) will be g¯ (X, Y ) = ω  (X, JY ) where ω  = 12 i∂ ∂r r∂r is holomorphic, it is clear that g¯ is homogeneous degree 2. Moreover, one easily checks that the necessary condition Lξ r = 0, which means that φ

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is basic, implies that g¯ is also a cone. Then η  = dc log r = η + dcB φ. For small enough ϕ, η  ∧ (dη  )n−1 is still a volume form on S  = {r = 1} ∼ = S, or in other words ω  is still non-degenerate on the cone, and similarly g¯ will be a non-degenerate metric. Thus we have defined a new K¨ahler cone, and hence new Sasakian structure. This deformation is precisely a deformation of the transverse K¨ ahler 1,1 metric, holding fixed its basic cohomology class in HB (Fξ ). Indeed, clearly   T 1 ω = ω T + dB dcB φ. 2 In fact the converse is also true by the transverse ∂ ∂¯ lemma proven in [43], which is thus an equivalent way to characterize these transverse K¨ ahler B deformations. Notice that the basic first Chern class c1 (Fξ ) is invariant under such deformations, although the contact subbundle D will change.  Later we shall also consider deforming the Reeb vector field ξ whilst holding the holomorphic structure of the cone fixed. 1.3. Regularity. There is a classification of Sasakian manifolds according to the global properties of the Reeb foliation Fξ . If the orbits of the Reeb vector field ξ are all closed, and hence circles, then ξ integrates to an isometric U (1) action on (S, g). Since ξ is nowhere zero this action is locally free; that is, the isotropy group of every point in S is finite. If the U (1) action is in fact free then the Sasakian structure is said to be regular. Otherwise, it is said to be quasi-regular. If the orbits of ξ are not all closed the Sasakian structure is said to be irregular. In this case the closure of the 1-parameter subgroup of the isometry group of (S, g) is isomorphic to a torus Tk , for some positive integer k called the rank of the Sasakian structure. In particular, irregular Sasakian manifolds have at least a T2 isometry. In the regular or quasi-regular case, the leaf space Z = S/Fξ = S/U (1) has the structure of a compact manifold or orbifold, respectively. In the latter case the orbifold singularities of Z descend from the points in S with non-trivial isotropy subgroups. Notice that, being finite subgroups of U (1), these will all be isomorphic to cyclic groups. The transverse K¨ ahler structure described above then pushes down to a K¨ ahler structure on Z, so that Z is a compact complex manifold or orbifold equipped with a K¨ ahler metric h. Digression on orbifolds. The reader will not need to know much about orbifolds in order to follow this article. However, we briefly digress here to sketch some basics, referring to [14] or [21] for a much more detailed account in the current context. Just as a manifold M is a topological space that is locally modelled on Rk , so an orbifold is a topological space locally modelled on Rk /Γ, where Γ is a finite group of diffeomorphisms. The local Euclidean charts {Ui , ϕi } ˜i , Γi , ϕi }. Here of a manifold are replaced with local uniformizing systems {U

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˜i is an open subset of Rk containing the origin; Γi is, without loss of U ˜i → Ui generality, a finite subgroup of O(k) acting effectively on Rk ; and ϕi : U is a continuous map onto the open set Ui ⊂ M such that ϕi ◦ γ = ϕi for ˜i /Γi → Ui is a homeomorphism. These all γ ∈ Γi and the induced map U charts are then glued together in an appropriate way. The groups Γi are called the local uniformizing groups. The least common multiple of the orders of the local uniformizing groups Γi , when it is defined, is called the order of M and denoted ord(M ). In particular, M is a manifold if and only if ord(M ) = 1. One can similarly define complex orbifolds, where one may take the Γi ⊂ U (k) acting on Ck . If x ∈ M is point and p = ϕ−1 i (x) then the conjugacy class of the isotropy ˜i . One denotes this subgroup Γp ⊂ Γi depends only on x, not on the chart U Γx , so that the non-singular points of M are those for which Γx is trivial. The set of such points is dense in M . In the case at hand, where M is realized as the leaf space Z of a quasi-regular Reeb foliation, Γx is the same as the leaf holonomy group of the leaf x. For orbifolds the notion of fibre bundle is modified to that of a fibre orbibundle. These consist of bundles over the local uniformizing neighbour˜i that patch together in an appropriate way. In particular, part of hoods U the data specifying an orbibundle with structure group G are group homomorphisms hi ∈ Hom(Γi , G). The local uniformizing systems of an orbifold ˜i → U ˜j is a diffeomorare glued together with the property that if φji : U phism into its image then for each γi ∈ Γi there is a unique γj ∈ Γj such that φji ◦ γi = γj ◦ φji . The patching condition is then that if Bi is a fibre bundle ˜i there should exist a corresponding bundle map φ∗ : Bj | ˜ → Bi over U ij φji (Ui ) such that hi (γi ) ◦ φ∗ij = φ∗ij ◦ hj (γj ). Of course, by choosing an appropriate ˜i ×F where F is the fibre refinement of the cover we may assume that Bi = U on which G acts. The total space is then itself an orbifold in which the Bi may form the local uniformizing neighbourhoods. The group Γi acts on Bi ˜i × F to (γ −1 pi , f hi (γ)), where γ ∈ Γi . Thus the local by sending (pi , f ) ∈ U uniformizing groups of the total space may be taken to be subgroups of the Γi . In particular, when F = G is a Lie group so that we have a principal G orbibundle, then the image hi (Γi ) acts freely on the fibre. Thus provided the group homomorphisms hi inject into the structure group G, the total space will in fact be a smooth manifold. This will be important in what follows. The final orbinotion we need is that of orbifold cohomology, introduced by Haefliger [58]. One may define the orbibundle P of orthonormal frames over a Riemannian orbifold (M, g) in the usual way. This is a principal O(n) orbibundle, and the discussion in the previous paragraph implies that the total space P is in fact a smooth manifold. One can then introduce the classifying space BM of the orbifold in an obvious way by defining BM = (EO(n) × P )/O(n), where EO(n) denotes the universal O(n) bundle and the action of O(n) is diagonal. One then defines the orbifold homology,

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cohomology and homotopy groups as those of BM , respectively. In particu∗ (M, Z) = H ∗ (BM, Z), lar, the orbifold cohomology groups are denoted Horb and these reduce to the usual cohomology groups of M when M is a manifold. The projection BM → M has generic fibre the contractible space ∗ (M, R) → H ∗ (M, R). EO(n), and this then induces an isomorphism Horb Typically integral classes map to rational classes under the natural map ∼ ∗ (M, Z) → H ∗ (M, R) → Horb H ∗ (M, R). orb Returning to Sasakian geometry, in the regular or quasi-regular case the leaf space Z = S/Fξ = S/U (1) is a manifold or orbifold, respectively. The Gysin sequence for the corresponding U (1) (orbi)bundle then implies that the projection map π : S → Z gives rise to a ring isomorphism π ∗ : ∗ (F ), thus relating the cohomology of the leaf space Z to the H ∗ (Z, R) ∼ = HB ξ basic cohomology of the foliation. We may now state the following result [15]: Theorem 1.5. Let S be a compact regular or quasi-regular Sasakian manifold. Then the space of leaves of the Reeb foliation Fξ is a compact K¨ ahler manifold or orbifold (Z, h, ωZ , JZ ), respectively. The corresponding projection π : (S, g) → (Z, h), is a (orbifold) Riemannian submersion, with fibres being totally geodesic circles. Moreover, the cohomology class [ωZ ] is proportional to an integral 2 (Z, Z). class in the (orbifold) cohomology group Horb In either the regular or quasi-regular case, ωZ is a closed 2-form on Z which thus defines a cohomology class [ωZ ] ∈ H 2 (Z, R). In the regular case, the projection π defines a principal U (1) bundle, and ωZ is proportional to the curvature 2-form of a unitary connection on this bundle. Thus [ωZ ] is proportional to a class in the image of the natural map H 2 (Z, Z) → H 2 (Z, R), since the curvature represents 2πc1 where c1 denotes the first Chern class of the principal U (1) bundle. In the quasi-regular case, the projection π is instead a principal U (1) orbibundle, with ωZ again proportional to a 2 (Z, Z) classifies isocurvature 2-form. The orbifold cohomology group Horb morphism classes of principal U (1) orbibundles over an orbifold Z, just as in the regular manifold case the first Chern class in H 2 (Z, Z) classifies principal U (1) bundles. The K¨ ahler form ωZ then defines a cohomology class [ωZ ] ∈ H 2 (Z, R) which is proportional to a class in the image of the natural 2 (Z, Z) → H 2 (Z, R) → H 2 (Z, R). map Horb orb A K¨ ahler manifold or orbifold whose K¨ ahler class is proportional to an integral cohomology class in this way is called a Hodge orbifold. There is no restriction on this constant of proportionality in Sasakian geometry: it may be changed via the D-homothetic transformation defined in the next section.

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The converse is also true [15]: Theorem 1.6. Let (Z, h) be a compact Hodge orbifold. Let π : S → Z be a principal U (1) orbibundle over Z whose first Chern class is an integral class defined by [ωZ ], and let η be a 1-form on S with dη = 2π ∗ ωZ (η is then proportional to a connection 1-form). Then (S, π ∗ h + η ⊗ η) is a Sasakian orbifold. Furthermore, if all the local uniformizing groups inject into the structure group U (1) (the hi ∈ Hom(Γi , U (1)) are all injective), then the total space S is a smooth manifold. We close this subsection by noting that (1.11)

1 iξ ω = − dr2 , 2

where recall that ω is the K¨ahler form on the cone (C(S), g¯). Thus 12 r2 is precisely the Hamiltonian function for the Reeb vector field ξ. In the regular/quasi-regular case the K¨ ahler manifold/orbifold (Z, h, ωZ , JZ ) may then be viewed as the K¨ ahler reduction of the K¨ ahler cone with respect to the corresponding Reeb U (1) action. 1.4. The Einstein condition. We begin with the following more general definition: Definition 1.7. A Sasakian manifold S = (S, g, η, ξ, Φ) is said to be η-Sasaki-Einstein if there are constants λ and ν such that Ricg = λg + νη ⊗ η. An important fact is that λ + ν = 2(n − 1). This follows from the second condition in Proposition 1.2, which implies that for a Sasakian manifold Ricg (ξ, ξ) = 2(n − 1). In particular, Sasaki-Einstein manifolds, with ν = 0, necessarily have λ = 2(n − 1). Definition 1.8. A Sasaki-Einstein manifold is a Sasakian manifold (S, g) with Ricg = 2(n − 1)g. It is easy to see that the η-Sasaki-Einstein condition is equivalent to the transverse K¨ahler metric being Einstein, so that RicT = κg T for some constant κ. To see the equivalence one notes that (1.12)

˜ Y˜ ) = RicT (X, Y ) − 2g T (X, Y ), Ricg (X,

˜ Y˜ are lifts where X, Y are vector fields on the local leaf spaces {Vα } and X, T T to D. Then Ric = κg together with (1.12) implies that Ricg = (κ − 2)g + (2n − κ)η ⊗ η.

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Given a Sasakian manifold S one can check that for a constant a > 0 the rescaling (1.13)

g  = ag + (a2 − a)η ⊗ η,

η  = aη,

1 ξ  = ξ, a

Φ = Φ,

gives a Sasakian manifold (S, g  , η  , ξ  , Φ ) with the same holomorphic structure on C(S), but with r = ra . This is known as a D-homothetic transformation [97]. Using the above formulae, together with the fact that the Ricci tensor is invariant under scaling the metric by a positive constant, it is then straightforward to show that if S = (S, g, η, ξ, Φ) is η-Sasaki-Einstein with constant λ > −2, then a D-homothetic transformation with a = (λ + 2)/2n gives a Sasaki-Einstein manifold. Thus any Sasakian structure which is transversely K¨ahler-Einstein with κ > 0 may be transformed via this scaling to a Sasaki-Einstein structure. The Sasaki-Einstein case may be summarized by the following: Proposition 1.9. Let (S, g) be a Sasakian manifold of dimension 2n−1. Then the following are equivalent (1) (S, g) is Sasaki-Einstein with Ricg = 2(n − 1)g. (2) The K¨ ahler cone (C(S), g¯) is Ricci-flat, Ricg¯ = 0. (3) The transverse K¨ ahler structure to the Reeb foliation Fξ is K¨ ahlerEinstein with RicT = 2ng T . g) ⊂ It immediately follows that the restricted holonomy group Hol0 (¯ SU (n). Notice that a Sasaki-Einstein 3-manifold has a universal covering space which is isometric to the standard round sphere, so the first interesting dimension is n = 3, or equivalently real dimension dim S = 5. 1,1 Since ρT represents 2πcB 1 (Fξ ) ∈ HB (Fξ ), clearly a necessary condition for a Sasakian manifold to admit a transverse K¨ ahler deformation to a Sasaki-Einstein structure, in the sense of Proposition 1.4, is that cB 1 = B c1 (Fξ ) > 0. Indeed, we have the following result, formalized in [48]: Proposition 1.10. The following necessary conditions for a Sasakian manifold S to admit a deformation of the transverse K¨ ahler structure to a Sasaki-Einstein metric are equivalent: 1,1 (1) cB 1 = a[dη] ∈ HB (Fξ ) for some positive constant a. 2 (2) cB 1 > 0 and c1 (D) = 0 ∈ H (S, R). (3) For some positive integer  > 0, the th power of the canonical line  bundle KC(S) admits a nowhere vanishing holomorphic section Ω with Lξ Ω = inΩ.

As described in [106], the space X = C(S)∪{r = 0}, obtained by adding the cone point at {r = 0} to C(S) ∼ = R>0 × S, can be made into a complex analytic space in a unique way. In fact it is simple to see that X is Stein, and

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the point o = {r = 0} ∈ X is an isolated singularity. Then (3) above implies that, by definition, X is -Gorenstein: Definition 1.11. An analytic space X with an isolated singularity o ∈  X is said to be -Gorenstein if KX\{o} is trivial. In particular, if  = 1 one says that X is Gorenstein. Proof. (Proposition 1.10) The equivalence of (1) and (2) follows immediately from the long exact sequence [104] relating the basic cohomology of the foliation Fξ to the cohomology of S (see [48]). The Ricci form ρ of the cone (C(S), g¯) is related to the transverse Ricci form, by an elementary calculation, via ρ = ρT − ndη. Here we are regarding ρT as a global basic 2-form on S, pulled back to the cone C(S). If condition (1) holds then by the above comments there is a D-homothetic transformation so that [ρT ] = 1,1 n[dη] ∈ HB (Fξ ). It now follows from the transverse ∂ ∂¯ lemma [43] that ¯ (f is there is a smooth function f on C(S) with r∂r f = ξf = 0 and ρ = i∂ ∂f f n the pull-back of a basic function on S). But now e ω /n! defines a flat metric on KC(S) , where recall that ω is the K¨ahler form for g¯. There is thus a multi˜ of KC(S) such that Ω = Ω ˜ ⊗ is a global holomorphic section valued section Ω  of KC(S) , for some positive integer  > 0, with Ω = 1. Using the fact that f is invariant under r∂r and that ω is homogeneous degree 2, the equality n in ¯˜ = ef ω , n(n−1)/2 ˜ Ω ∧ Ω (−1) 2n n!  implies that Lr∂r Ω = nΩ. 1.5. 3-Sasakian manifolds. In dimensions of the form n = 2p, so dim S = 4p−1, there exists a special class of Sasaki-Einstein manifolds called 3-Sasakian manifolds: Definition 1.12. A Riemannian manifold (S, g) is 3-Sasakian if and only if its metric cone C(S) = R>0 × S, g¯ = dr2 + r2 g is hyperK¨ahler. This implies that the cone has complex dimension n = 2p, or real dimension 4p, and that the holonomy group Hol(¯ g ) ⊂ Sp(p) ⊂ SU (2p). Thus 3-Sasakian manifolds are automatically Sasaki-Einstein. The hyperK¨ ahler structure on the cone descends to a 3-Sasakian structure on the base of the cone (S, g). In particular, the triplet of complex structures gives rise to a triplet of Reeb vector fields (ξ1 , ξ2 , ξ3 ) whose Lie brackets give a copy of the Lie algebra su(2). There is then a corresponding 3-dimensional foliation, whose leaf space is a quaternionic K¨ ahler manifold or orbifold. This extra structure means that 3-Sasakian geometry is rather more constrained, and it is somewhat more straightforward to construct examples. In particular, rich infinite classes of examples were produced in the 1990s via a quotient construction (essentially the hyperK¨ ahler quotient). A review of those developments was given in a previous article in this journal series [14], with a

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more recent account appearing in [19]. We note that the first non-trivial dimension for a 3-Sasakian manifold is dim S = 7, and also that 3-Sasakian manifolds are automatically regular or quasi-regular as Sasaki-Einstein manifolds (indeed, the first quasi-regular Sasaki-Einstein manifolds constructed were 3-Sasakian 7-manifolds). We will therefore not discuss 3-Sasakian geometry any further in this article, but focus instead on the construction of Sasaki-Einstein manifolds that are not 3-Sasakian. 1.6. Killing spinors. For applications to supergravity theories one wants a slightly stronger definition of Sasaki-Einstein manifold than we have given above. This is related to the following: Definition 1.13. Let (S, g) be a complete Riemannian spin manifold. Denote the spin bundle by S S and let ψ be a smooth section of S S. Then ψ is said to be a Killing spinor if for some constant α (1.14)

∇Y ψ = αY · ψ,

for every vector field Y , where ∇ denotes the spin connection of g and Y · ψ is Clifford multiplication of Y on ψ. One says that ψ is imaginary if α ∈ Im(C∗ ), parallel if α = 0, or real if α ∈ Re(C∗ ). It is a simple exercise to show that the existence of such a Killing spinor implies that g is Einstein with constant λ = 4(m − 1)α2 , where m = dim S. In particular, the existence of a real Killing spinor implies that (S, g) is a compact Einstein manifold with positive Ricci curvature. The relation to Sasaki-Einstein geometry is given by the following result of [6]: Theorem 1.14. A complete simply-connected Sasaki-Einstein manifold admits at least 2 linearly independent real Killing spinors with α = + 12 , − 12 for n = 2p − 1 and α = + 12 , + 12 for n = 2p, respectively. Conversely, a complete Riemannian spin manifold admitting such Killing spinors in these dimensions is Sasaki-Einstein with Hol(¯ g ) ⊂ SU (n). Notice that in both cases Hol(¯ g ) ⊂ SU (n), so that in particular a simplyconnected Sasaki-Einstein manifold is indeed spin. Moreover, in this case  = 1 in Proposition 1.10 so that the singularity X = C(S)∪{r = 0} is Gorenstein. Indeed, a real Killing spinor on (S, g) lifts to a parallel spinor on (C(S), g¯) [6], and from this parallel spinor one can construct a nowhere zero holomorphic (n, 0)-form by “squaring” it. We refer the reader to [75] for details. When a Sasaki-Einstein manifold is not simply-connected the existence of Killing spinors is more subtle. An instructive example is S 5 , equipped with its standard metric. Here X = C3 is equipped with its flat K¨ ahler metric. 3 Denoting standard complex coordinates on C by (z1 , z2 , z3 ) we may consider the quotient S 5 /Zq , where Zq acts by sending (z1 , z2 , z3 ) → (ζz1 , ζz2 , ζz3 ) with ζ a primitive qth root of unity. For q = 2 this is the antipodal map,

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giving RP5 which is not even a spin manifold. In fact of all these quotients only S 5 and S 5 /Z3 admit Killing spinors. For applications to supergravity theories, a Sasaki-Einstein manifold is in fact defined to satisfy this stronger requirement that it admits Killing spinors. Of course, since π1 (S) is finite by Myers’ Theorem [80], one may always lift to a simply-connected cover, where Theorem 1.14 implies that the two definitions coincide. We shall thus not generally emphasize this distinction. The reader might wonder what happens to Theorem 1.14 when the number of linearly independent Killing spinors is not 2. For simplicity we focus on the simply-connected case. When n = 2p − 1, the existence of 1 Killing spinor in fact implies the existence of 2 with opposite sign of α, so that (S, g) is Sasaki-Einstein. If there are more than 2, or at least 2 with the same sign of α, then (S, g) is the round sphere. When n = 4, so that dim S = 7, it is possible for a Riemannian spin 7-manifold (S, g) to admit a single real Killing spinor, in which case (S, g) is said to be a weak G2 holonomy manifold ; the metric cone then has holonomy contained in the group Spin(7) ⊂ SO(8). In all other dimensions of the form n = 2p, the existence of 1 Killing spinor again implies the existence of 2, implying (S, g) is Sasaki-Einstein. A simplyconnected 3-Sasakian manifold has 3 linearly independent Killing spinors, all with α = + 12 . If there are more than 3, or at least 2 with opposite sign of α, then again (S, g) is necessarily the round sphere. For further details, and a list of references, the reader is referred to [14]. 2. Regular Sasaki-Einstein manifolds 2.1. Fano K¨ ahler-Einstein manifolds. Theorem 1.5, together with Proposition 1.9, implies that any regular Sasaki-Einstein manifold is the total space of a principal U (1) bundle over a K¨ ahler-Einstein manifold (Z, h). On the other hand, Theorem 1.6 implies that the converse is also true. In fact this construction of Einstein metrics on the total spaces of principal U (1) bundles over K¨ ahler-Einstein manifolds is in the very early paper of Kobayashi [62]. Theorem 2.1. A complete regular Sasaki-Einstein manifold (S, g) of dimension (2n − 1) is the total space of a principal U (1) bundle over a compact K¨ ahler-Einstein manifold (Z, h, ωZ ) with positive Ricci curvature Rich = 2nh, which is the leaf space of the Reeb foliation Fξ . If S is simplyconnected then this U (1) bundle has first Chern class −[nωZ /πI(Z)] = −c1 (Z)/I(Z), where I(Z) ∈ Z>0 is the Fano index of Z. Conversely, if (Z, h, ωZ ) is a complete simply-connected K¨ ahler-Einstein manifold with positive Ricci curvature Rich = 2nh, then let π : S → Z be the principal U (1) bundle with first Chern class −c1 (Z)/I(Z). Then g = π ∗ h + η ⊗ η is a regular Sasaki-Einstein metric on the simply-connected manifold S, where η is the connection 1-form on S with curvature dη = 2π ∗ ωZ .

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Recall here: Definition 2.2. A Fano manifold is a compact complex manifold Z with positive first Chern class c1 (Z) > 0. The Fano index I(Z) is the largest positive integer such that c1 (Z)/I(Z) is an integral class in the group of holomorphic line bundles Pic(Z) = H 2 (Z, Z) ∩ H 1,1 (Z, R). In particular, K¨ ahler-Einstein manifolds with positive Ricci curvature are Fano. Notice that the principal U (1) bundle in Theorem 2.1 is that 1/I(Z) , where KZ is the canonical line bundle associated to the line bundle KZ of Z. Also notice that by taking a Zm ⊂ U (1) quotient of a simply-connected S in Theorem 2.1, where U (1) acts via the free Reeb action, we also obtain a regular Sasaki-Einstein manifold with π1 (S/Zm ) ∼ = Zm ; this is equivalent to taking the mth power of the principal U (1) bundle, which has associated m/I(Z) . However, the Killing spinors on (S, g) guaranteed by line bundle KZ Theorem 1.14 are invariant under Zm only when m divides the Fano index I(Z). Only in these cases is the quotient Sasaki-Einstein in the stronger sense of admitting a real Killing spinor. Via Theorem 2.1 the classification of regular Sasaki-Einstein manifolds effectively reduces to classifying Fano K¨ahler-Einstein manifolds. This is a rich and deep subject, which is still very much an active area of research. Below we give a brief overview of some key results. 2.2. Homogeneous Sasaki-Einstein manifolds. Definition 2.3. A Sasakian manifold S is said to be homogeneous if there is a transitively acting group G of isometries preserving the Sasakian structure. If S is compact, then G is necessarily a compact Lie group. We then have the following theorem of [15]: Theorem 2.4. Let (S, g  ) be a complete homogeneous Sasakian manifold with Ricg ≥  > −2. Then (S, g  ) is a compact regular homogeneous Sasakian manifold, and there is a homogeneous Sasaki-Einstein metric g on S that is compatible with the underlying contact structure. Moreover, S is the total space of a principal U (1) bundle over a generalized flag manifold K/P , equipped with its K¨ ahler-Einstein metric. Via Theorem 2.1, the converse is also true. Recall here that a generalized flag manifold K/P is a homogeneous space where K is a complex semi-simple Lie group, and P is any complex subgroup of K that contains a Borel subgroup (so that P is a parabolic subgroup of K). It is well-known that K/P is Fano and admits a homogeneous K¨ ahler-Einstein metric [12]. Conversely, any compact homogeneous simply-connected K¨ ahler-Einstein manifold is a generalized flag manifold.

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The metric on K/P is G-invariant, where G is a maximal compact subgroup of K, and one can write K/P = G/H for appropriate subgroup H. In low dimensions Theorem 2.4 leads [14] to the following list, wellknown to supergravity theorists: Corollary 2.5. Let (S, g) be a complete homogeneous Sasaki-Einstein manifold of dimension 2n − 1. Then S is a principal U (1) bundle over (1) CP1 when n = 2, (2) CP2 or CP1 × CP1 when n = 3, (3) CP3 , CP2 × CP1 , CP1 × CP1 × CP1 , SU (3)/T2 , or the real Grassmannian Gr2 (R5 ) of 2-planes in R5 when n = 4. 2.3. Regular Sasaki-Einstein 5-manifolds. As mentioned in the introduction, regular Sasaki-Einstein 5-manifolds are classified [45, 9]. This is thanks to the classification of Fano K¨ ahler-Einstein surfaces due to TianYau [99, 100, 103]. Theorem 2.6. Let (S, g) be a regular Sasaki-Einstein 5-manifold. Then ˜ m , where the universal cover (S, ˜ g) is one of the following: S = S/Z (1) S 5 equipped with its standard round metric. Here Z = CP2 equipped with its standard Fubini-Study metric. (2) The Stiefel manifold V2 (R4 ) ∼ = S 2 × S 3 of 2-frames in R4 . Here Z = 1 1 CP × CP equipped with the symmetric product of round metrics on each CP1 ∼ = S2. (3) The total space Sk of the principal U (1) bundles Sk → Pk , for 3 ≤ k ≤ 8, where Pk = CP2 #kCP2 is the k-point blow-up of CP2 . For each complex structure on these del Pezzo surfaces there is a unique K¨ ahler-Einstein metric, up to automorphism [93, 99, 100, 103], and unique Sasaki-Einstein metric g on Sk ∼ = #k   2 a corresponding 3 S × S . In particular, for 5 ≤ k ≤ 8 by varying the complex structure this gives a complex 2(k − 4)-dimensional family of regular Sasaki-Einstein structures. Cases (1) and (2) are of course the 2 homogeneous spaces listed in (2) of Corollary 2.5, and so the metrics are easily written down explicitly. The Sasaki-Einstein metric in case (2) was first noted by Tanno in [98], although in the physics literature the result is often attributed to Romans [89]. In the latter case the manifold is referred to as T 11 , the T pq being homogeneous Einstein metrics on principal U (1) bundles over CP1 × CP1 with Chern numbers (p, q) ∈ Z ⊕ Z ∼ = H 2 (CP1 × CP1 , Z). This is a generalization of the Kobayashi construction [62], and was further generalized to torus bundles by Wang-Ziller in [109]. The corresponding Ricci-flat K¨ ahler cone over T 11 has 2 the complex structure of the quadric singularity {z1 + z22 + z32 + z42 = 0} ⊂ C4 minus the isolated singular point at the origin. This hypersurface singularity

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is called the “conifold” in the string theory literature, and there are literally hundreds of papers that study different aspects of string theory on this space. The K¨ahler-Einstein metrics on del Pezzo surfaces in (3) are known to exist, but are not known in explicit form. The complex structure moduli simply correspond  to moving  the blown-up points. The fact that Sk is diffeomorphic to #k S 2 × S 3 follows from Smale’s Theorem [95]: Theorem 2.7. A compact simply-connected  5-manifold S with no  2 spin 3 torsion in H2 (S, Z) is diffeomorphic to #k S × S . The homotopy and homology groups of Sk are of course straightforward to compute using their description as principal U (1) bundles over Pk . ˜ g) admits By Theorem 1.14, in each case the simply-connected cover (S, 2 real Killing spinors. Only for m = 3 in case (1) and m = 2 in case (2) do the Zm quotients also admit such Killing spinors [45]. 2.4. Existence of K¨ ahler-Einstein metrics. For a K¨ ahler manifold (Z, h, ω, J) the Einstein equation Rich = κh is of course equivalent to the 2-form equation ρh = κωh , where ρh = −i∂ ∂¯ log det h denotes the Ricci form of the metric h. Since the cohomology class [ρ] ∈ H 1,1 (Z, R) of the Ricci form equals 2πc1 (Z), it follows that on such a K¨ ahler-Einstein manifold 2πc1 (Z) = κ[ω]. Notice that for κ = 0 Z is Calabi-Yau (c1 (Z) = 0) and there is no restriction on the K¨ ahler class, while for κ > 0 or κ < 0 instead Z must be either Fano or anti-Fano (c1 (Z) < 0), respectively, and in either case here the K¨ ahler class is fixed uniquely. Suppose that ω = ωh is a K¨ahler 2-form on Z with κ[ω] = 2πc1 (Z) ∈ H 1,1 (Z, R). By the ∂ ∂¯ lemma, there exists a global real function f ∈ C ∞ (Z) such that ¯ (2.1) ρh − κωh = i∂ ∂f. The function f is often called the discrepancy potential. It is unique up to an additive constant, latter may be conveniently fixed by requiring,  the   f and n−1 for example, Z e − 1 ωh = 0, where dimC Z = n − 1. Notice that f is essentially the same function f appearing in the proof of Proposition 1.10. (More precisely, the function there is the pull-back of f here under the C∗ = R>0 × U (1) quotient C(S) → Z for a regular Sasakian structure with leaf space Z.) On the other hand, if g is a K¨ahler-Einstein metric, with [ωg ] = [ωh ] and ρg = κωg , then the ∂ ∂¯ lemma again gives a real function φ ∈ C ∞ (Z) such that ¯ (2.2) ωg − ωh = i∂ ∂φ. ¯ − κφ), or relating the volume forms as ω n−1 = eF ω n−1 Thus ρh − ρg = i∂ ∂(f g h ∞ with F ∈ C (Z) equivalently ¯ = i∂ ∂(f ¯ − κφ). i∂ ∂F

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This implies F = f − κφ + c with c a constant. Again, this may be fixed by requiring, for example,    ef −κφ − 1 ωhn−1 = 0. (2.3) Z

We have then shown the following: Proposition 2.8. Let (Z, J) be a compact K¨ ahler manifold, of dimension dimC Z = n − 1, with K¨ ahler metrics h, g in the same K¨ ahler class, [ωh ] = [ωg ] ∈ H 1,1 (Z, R), and with κ[ωh ] = 2πc1 (Z). Let f, φ ∈ C ∞ (Z) be the functions defined via (2.1) and (2.2), and with the relative constant of f −κφ fixed by (2.3). Then the metric g is K¨ ahler-Einstein with constant κ if and only if φ satisfies the Monge-Amp`ere equation ωgn−1 = ef −κφ ωhn−1 , or equivalently (2.4)

 det hi¯j +

∂2φ ∂zi ∂ z¯j

det hi¯j

 = ef −κφ ,

where z1 , . . . , zn−1 are local complex coordinates on Z. For κ < 0 this problem was solved independently by Aubin [4] and Yau [110]. Without loss of generality, we may rescale the metric so that κ = −1 and then state: Theorem 2.9. Let (Z, J) be a compact K¨ ahler manifold with c1 (Z) < 0. Then there exists a unique K¨ ahler-Einstein metric with ρg = −ωg . The proof relies on the Maximum Principle. The Calabi-Yau case κ = 0 is substantially harder, and was proven in Yau’s celebrated paper: Theorem 2.10. Let (Z, J) be a compact K¨ ahler manifold with c1 (Z) = 0. Then there exists a unique Ricci-flat K¨ ahler metric in each K¨ ahler class. On the other hand, the problem for κ > 0 is still open. In particular, there are known obstructions to solving the Monge-Amp`ere equation (2.4) in this case. On the other hand, it is known that if there is a solution, it is unique up to automorphism [5]. In the remainder of this section we give a very brief overview of the Fano κ > 0 case, referring the reader to the literature for further details. In [76] Matsushima proved that for a Fano K¨ ahler-Einstein manifold the complex Lie algebra a(Z) of holomorphic vector fields is the complexification of the Lie algebra of Killing vector fields. Since the isometry group of a compact Riemannian manifold is a compact Lie group, in particular this implies that a(Z) is necessarily reductive; that is, a(Z) = Z(a(Z)) ⊕ [a(Z), a(Z)], where Z(a(Z)) denotes the centre. The simplest such obstructed examples

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are in fact the 1-point and 2-point blow-ups of CP2 that are not listed in Theorem 2.6, despite these being Fano surfaces. Matsushima’s result implies that the isometry group of a Fano K¨ ahler-Einstein manifold is a maximal compact subgroup of the automorphism group. Another obstruction, also related to holomorphic vector fields on Z, is the Futaki invariant of [47]. If ζ ∈ a(Z) is a holomorphic vector field then define  (2.5) F(ζ) = ζ(f ) ωhn−1 , Z

where f is the discrepancy potential defined via (2.1). The function F is independent of the choice of K¨ ahler metric h in the K¨ ahler class [ωh ], and defines a Lie algebra homomorphism F : a(Z) → C. For this reason it is also sometimes called the Futaki character. Moreover, Mabuchi [69] proved that the nilpotent radical of a(Z) lies in ker F, so that F is completely determined by its restriction to the maximal reductive subalgebra. Since F is a Lie algebra character, it also vanishes on the derived algebra [a(Z), a(Z)], and therefore the Futaki invariant is determined entirely by its restriction to the centre of a(Z). In practice, F may be computed via localization; see, for example, the formula in [102]. Clearly f is constant for a K¨ ahler-Einstein metric, and thus the Futaki invariant must vanish in this case. Indeed, both the 1-point and 2-point blow-ups of CP2 also have non-zero Futaki invariants, and are thus obstructed this way also. The Futaki invariant will turn out to be closely related to a natural construction in Sasakian geometry, that generalizes to the quasi-regular and irregular cases. Since both obstructions above are related to the Lie algebra of holomorphic vector fields on the Fano Z, there was a conjecture that in the absence of holomorphic vector fields there would be no obstruction to the existence of a K¨ahler-Einstein metric. However, a counterexample was later given by Tian in [101]. In fact it is currently believed that a Fano manifold Z admits a K¨ ahlerEinstein metric if and only if it is stable, in an appropriate geometric invariant theory sense. This idea goes back to Yau [111], and has been developed by Donaldson, Tian, and others. It is clearly beyond the scope of this article to describe this still very active area of research. However, the basic idea is to use KZ−k for k  0 to embed Z into a large complex projective space CPNk via its space of sections (the Kodaira embedding). Then stability of Z is taken in the geometric invariant theory sense, for the automorphisms of these projective spaces, as k → ∞. A stable orbit should contain a zero of a corresponding moment map, and in the present case this amounts to saying that by acting with an appropriate automorphism of CPNk a stable Z can be moved to a balanced embedding, in the sense of Donaldson [40]. For a sequence of balanced embeddings, the pull-back of the Fubini-Study metric on CPNk as k → ∞ should then approach the K¨ ahler-Einstein metric on Z. The precise notion of stability here is called K-stability.

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In practice, even if the above stability conjecture was settled it is difficult to check in practice for a given Fano manifold. More practically, one can sometimes prove existence of solutions to (2.4) in appropriate examples using the continuity method. Thus, for appropriate classes of examples, one can often write down sufficient conditions for a solution, although these conditions are in general not expected to be necessary. This will be the pragmatic approach followed in the next section when we come to discuss the extension to quasi-regular Sasakian manifolds, or rather their associated Fano K¨ ahler-Einstein orbifolds. However, there are two classes of examples in which necessary and sufficient conditions are known. The first is the classification of Fano K¨ ahlerEinstein surfaces already mentioned. One may describe this result by saying that a Fano surface admits a K¨ ahler-Einstein metric if and only if its Futaki invariant is zero. The second class of examples are the toric Fano manifolds. Here a complex (n − 1)-manifold is said to be toric if there is a biholomorphic (C∗ )n−1 action with a dense open orbit. Then a toric Fano manifold admits a K¨ahler-Einstein metric if and only if its Futaki invariant is zero [108]. The K¨ahler-Einstein metric is invariant under the real torus subgroup Tn−1 ⊂ (C∗ )n−1 . We shall see that this result generalizes to the quasi-regular and irregular Sasakian cases, and so postpone further discussion to later in the article. Otherwise, examples are somewhat sporadic (as will be the case also in the next section). As an example, the Fermat hypersurfaces Fd,n = {z0d + · · · + znd = 0} ⊂ CPn are Fano provided d ≤ n, and Nadel [81] has shown that these admit K¨ ahler-Einstein metrics if n/2 ≤ d ≤ n. It is straightforward to compute the homology groups of the corresponding regular SasakiEinstein manifolds Sd,n [15]. In particular, in dimension 7 (n = 4) one finds examples of Sasaki-Einstein 7-manifolds with third Betti numbers b3 (S4,4 ) = 60, b3 (S3,4 ) = 10 (notice S2,n corresponds to a quadric, and is homogeneous). Finally, we stress that in any given dimension there are only finitely many (deformation classes of) Fano manifolds [65]. Thus there are only finitely many K¨ ahler-Einstein structures, and hence finitely many regular SasakiEinstein structures, up to continuous deformations of the complex structure on the K¨ ahler-Einstein manifold. This result is no longer true when one passes to the orbifold category, or quasi-regular case. 3. Quasi-regular Sasaki-Einstein manifolds and hypersurface singularities 3.1. Fano K¨ ahler-Einstein orbifolds. Recall that a quasi-regular Sasakian manifold is a Sasakian manifold whose Reeb foliation has compact leaves, but such that the corresponding U (1) action is only locally free, rather than free. As in the previous section, Theorem 1.5 and Proposition 1.9 imply that the leaf space of a quasi-regular Sasaki-Einstein manifold is a

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compact K¨ ahler-Einstein orbifold (Z, h). The main tool in this section will be the converse result obtained using the inversion Theorem 1.6: Theorem 3.1. Let (Z, h, ωZ ) be a compact simply-connected (π1orb (Z) trivial) K¨ ahler-Einstein orbifold with positive Ricci curvature Rich = 2nh. Let π : S → Z be the principal U (1) orbibundle with first Chern class −c1 (Z)/ 2 (Z, Z). Then (S, g = π ∗ h + η ⊗ η) is a compact simply-connected I(Z) ∈ Horb quasi-regular Sasaki-Einstein orbifold, where η is the connection 1-form on S with curvature dη = 2π ∗ ωZ . Furthermore, if all the local uniformizing groups inject into U (1) then the total space S is a smooth manifold. Here the orbifold Fano index I(Z) is defined in a precisely analogous way to the manifold case: it is the largest positive integer such that c1 (Z)/I(Z) 2 (Z, Z)∩H 1,1 (Z, R). defines an integral class in the orbifold Picard group Horb As in the regular case, the principal U (1) orbibundle appearing here is that 1/I(Z)  associated to the complex line orbibundle KZorb . Here we make the important remark that canonical line orbibundle of Z, KZorb , is not necessarily the same as the canonical line bundle defined in the algebro-geometric sense. The difference between the two lies in the fact that complex codimension one orbifold singularities are not seen by the canonical line bundle, owing to the simple fact that C/Zm ∼ = C as an algebraic variety. More specifically, let the complex codimension one singularities of Z be along divisors Di , and suppose that Di has multiplicity mi in the above sense. In particular, a K¨ ahler-Einstein orbifold metric on Z will have a 2π/mi conical singularity along Di . Then

 1 orb 1− Di . KZ = KZ + mi i

The Di are known as the ramification divisors. A Fano K¨ ahler-Einstein orb orb −1 ifold is then Fano in the sense that KZ is positive, which is not the −1 same condition as KZ being positive. Also, the Fano indices in the two senses will not in general agree. 3.2. The join operation. As already mentioned, the first examples of quasi-regular Sasaki-Einstein manifolds were the quasi-regular 3-Sasakian manifolds constructed in [24]. The first examples of quasi-regular SasakiEinstein manifolds that are not 3-Sasakian were in fact constructed using these examples, together with the following Theorem of [15]: Theorem 3.2. Let S1 , S2 be two simply-connected quasi-regular SasakiEinstein manifolds of dimensions 2n1 − 1, 2n2 − 1, respectively. Then there is a natural operation called the join which produces in general a simplyconnected quasi-regular Sasaki-Einstein orbifold S1  S2 of dimension 2(n1 + n2 ) − 3. The join is a smooth manifold if and only if (3.1)

gcd (ord(Z1 )l2 , ord(Z2 )l1 ) = 1,

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where li = I(Zi )/gcd(I(Z1 ), I(Z2 )) are the relative orbifold Fano indices of the K¨ ahler-Einstein leaf spaces Z1 , Z2 . Recall here that ord(M ) denotes the order of M as an orbifold (see section 1.3). The proof of this result follows from the simple observation that given two K¨ ahler-Einstein orbifolds (Z1 , h1 ), (Z2 , h2 ), the product Z1 × Z2 carries a direct product K¨ ahler-Einstein metric which is the sum of h1 and h2 , after an appropriate constant rescaling of each. The join is then the unique simply-connected Sasaki-Einstein orbifold obtained by applying the inversion Theorem 3.1. The smoothness condition (3.1) is simply a rewriting of the condition that the local uniformizing groups inject into U (1), given that this is true for each of S1 , S2 . In particular, note that if S1 is a regular Sasaki-Einstein manifold (ord(Z1 ) = 1) and I(Z2 ) divides I(Z1 ) then the join is smooth whatever the value of the order of Z2 . The join construction can produce interesting non-trivial examples. For example, the homogeneous Sasaki-Einstein manifold in (2) of Theorem 2.6 is simply S 3  S 3 , with the round metric on each S 3 . More importantly, the join of a quasi-regular 3-Sasakian manifold with a regular Sasaki-Einstein manifold (such as S 3 ) gives rise to a quasi-regular Sasaki-Einstein manifold by the observation at the end of the previous paragraph. However, this particular construction produces new examples of Sasaki-Einstein manifolds only in dimension 9 and higher. 3.3. The continuity method for K¨ ahler-Einstein orbifolds. The Proposition 2.8 holds also for compact K¨ ahler orbifolds, with an identical proof. Thus also in the orbifold category, to find a K¨ ahler-Einstein metric on a Fano orbifold one must similarly solve the Monge-Amp`ere equation (2.4). Of course, since necessary and sufficient algebraic conditions on Z are not even known in the smooth manifold case, for orbifolds the pragmatic approach of Boyer, Galicki, Koll´ ar and their collaborators is to find a sufficient condition in appropriate classes of examples. Also as in the smooth manifold case, one can use the continuity method to great effect. Our discussion in the remainder of this section will closely follow [19]. Suppose, without loss of generality, that we are seeking a solution to (2.4) with κ = 1. Then the continuity method works here by introducing the more general equation   2 det hi¯j + ∂z∂i ∂φz¯j (3.2) = ef −tφ , det hi¯j where now t ∈ [0, 1] is a constant parameter. We wish to solve the equation with t = 1. We know from Yau’s proof of the Calabi conjecture [110] that there is a solution with t = 0. The classic continuity argument works by trying to show that the subset of [0, 1] where solutions exist is both open

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and closed. In fact openness is a straightforward application of the implicit function theorem. On the other hand, closedness is equivalent to the integrals  e−γtφt ω0n−1 Z

being uniformly bounded, for any constant γ ∈ ((n−1)/n, 1). Here ω0 denotes the K¨ ahler form for h0 , the metric given by Yau’s result. Nadel interprets this condition in terms of multiplier ideal sheaves [81]. This was first studied in the case of Fano orbifolds by Demailly-Koll´ ar [38], and indeed it is their results that led to the first examples of quasiregular Sasaki-Einstein 5-manifolds in [17]. The key result is the following: Theorem 3.3. Let Z be a compact Fano orbifold of dimension dimC Z = n−1. Then the continuity method produces a K¨ ahler-Einstein orbifold metric on Z if there is a γ > (n − 1)/n such that for every s ≥ 1 and for every  orb −s  0 holomorphic section τs ∈ H Z, KZ  (3.3) Z

|τs |−2γ/s ω0n−1 < ∞.

For appropriate classes of examples, the condition (3.3) is not too difficult to check. We next introduce such a class. 3.4. Links of weighted homogeneous hypersurface singularities. Let wi ∈ Z>0 , i = 0, . . . , n, be a set of positive integers. We regard these as a vector w ∈ (Z>0 )n+1 . There is an associated weighted C∗ action on Cn+1 given by (3.4)

Cn+1  (z0 , . . . , zn ) → (λw0 z0 , . . . , λwn zn ) ,

where λ ∈ C∗ and the wi are referred to as the weights. Without loss of generality one can assume that gcd(w0 , . . . , wn ) = 1, so that the C∗ action is effective, although this is not necessary. Definition 3.4. A polynomial F ∈ C[z0 , . . . , zn ] is said to be a weighted homogeneous polynomial with weights w and degree d ∈ Z>0 if F (λw0 z0 , . . . , λwn zn ) = λd F (z0 , . . . , zn ). We shall always assume that F is chosen so that the affine algebraic variety (3.5)

XF = {F = 0} ⊂ Cn+1

is smooth everywhere except at the origin in Cn+1 .

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Definition 3.5. The hypersurface given by (3.5) is called a quasihomogeneous hypersurface singularity. The link LF of the singularity is defined to be LF = {F = 0} ∩ S 2n+1 ,  n

2 n+1 is the unit sphere. where S 2n+1 = i=0 |zi | = 1 ⊂ C (3.6)

LF is a smooth (2n − 1)-dimensional manifold, and it is a classic result of Milnor [78] that LF is (n − 2)-connected. Indeed, the homology groups of LF were computed in [78, 79] in terms of the so-called monodromy map. A particularly nice such set of singularities are the so-called BrieskornPham singularities. These take the particular form (3.7)

F=

n 

ziai ,

i=0

where a ∈ (Z>0 )n+1 . Thus the weights of the C∗ action are wi = d/ai where d = lcm{ai }. The corresponding hypersurface singularities are always isolated, as is easy to check. In this case it is convenient to denote the link by LF = L(a). Moreover, to the vector a one associates a graph G(a) with n + 1 vertices labelled by the ai . Two vertices ai , aj are connected if and only if gcd(ai , aj ) > 1. We denote the connected component of G(a) determined by the even integers by Ceven ; all even integer vertices are contained in Ceven , although Ceven may of course contain odd integer vertices also. The following result is due to Brieskorn [30] (although see also [39]): Theorem 3.6. The following are true: (1) The link L(a) is a rational homology sphere if and only if either G(a) contains at least one isolated point, or Ceven has an odd number of vertices and for any distinct ai , aj ∈ Ceven , gcd(ai , aj ) = 2. (2) The link L(a) is an integral homology sphere if and only if either G(a) contains at least two isolated points, or G(a) contains one isolated point and Ceven has an odd number of vertices and ai , aj ∈ Ceven implies gcd(ai , aj ) = 2 for any distinct i, j. A simply-connected integral homology sphere is also a homotopy sphere, by the Hurewicz isomorphism theorem and the Whitehead theorem. Hence by the higher-dimensional versions of the Poincar´e conjecture, a simplyconnected integral homology sphere is in fact homeomorphic to the sphere. In particular, Theorem 3.6 says which a lead to homotopy spheres L(a). Again, classical results going back to Milnor [77] and Smale [94] show that in every dimension greater than 4 the differentiable homotopy spheres form an Abelian group, where the group operation is given by the connected sum. There is a subgroup consisting of those which bound parallelizable manifolds, and these groups are known in every dimension [61]. They are distinguished by the signature τ of a parallelizable manifold whose boundary

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is the homotopy sphere. There is a natural choice in fact (the Milnor fibre in Milnor’s fibration theorem [78]), and Brieskorn computed the signature in terms of a combinatorial formula involving the {ai }. Many of the results quoted in the introduction involving rational homology spheres and exotic spheres are proven this way. Let us give a simple example: Example 3.7. By Theorem 3.6 the link L(6k − 1, 3, 2, 2, 2) is a homotopy 7-sphere. Using Brieskorn’s formula one can compute the signature of the associated Milnor fibre, with the upshot being that all 28 oriented diffeomorphism classes on S 7 are realized by taking k = 1, 2, . . . , 28. Returning to the general case in Definition 3.5, the fact that LF has a natural Sasakian structure was observed as long ago as reference [96]. We begin by noting that Cn+1 (minus the origin) has a K¨ ahler cone metric that is a cone with respect to the weighted Euler vector field (3.8)

r˜∂r˜ =

n 

wi ρ i ∂ ρ i ,

i=0

¯r2 where r˜2 is where zi = ρi exp(iθi ), i = 0, . . . , n. The K¨ahler form is 12 i∂ ∂˜ a homogeneous degree 2 function under (3.8), and a natural choice is r˜2 = n 2/wi . The holomorphic vector field (i 1 + J) r˜∂r˜ of course generates i=0 ρi the weighted C∗ action (3.4), and by construction the hypersurface XF is invariant under this C∗ action. Thus the K¨ ahler metric inherited by XF via its embedding (3.5) is also a K¨ ahler cone with respect to this C∗ action, which in turn gives rise to a Sasakian structure on LF . On the other hand, the quotient of Cn+1 \ {0} by the weighted C∗ action is by definition the weighted projective space P(w) = CPn[w0 ,...,wn ] . There is a corresponding commutative square

(3.9)

2n+1 L ⏐F −→ S ⏐ ⏐ ⏐ ⏐π ⏐   ZF −→ P(w),

where the horizontal arrows are Sasakian and K¨ ahlerian embeddings, respectively, and the vertical arrows are orbifold Riemannian submersions. Here ZF is simply the hypersurface {F = 0}, now regarded as defined in the weighted projective space, so ZF = {F = 0} ⊂ P(w). Thus ZF is a weighted projective variety. We are of course interested in the case in which ZF is Fano: Proposition 3.8. The orbifold ZF is Fano if and only if |w| − d > 0, where |w| = ni=0 wi . This was proven in [22], but a simpler method of proof [53] that bypasses the orbifold subtleties is to use Proposition 1.10. Indeed, XF is Gorenstein

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since the smooth locus XF \{0} is equipped with a nowhere zero holomorphic (n, 0)-form, given explicitly in a coordinate chart in which the denominator is nowhere zero by (3.10)

Ω=

dz1 ∧ · · · ∧ dzn . ∂F/∂z0

One has similar expressions in charts in which ∂F/∂zi = 0, and it is straightforward to check that these glue together into a global holomorphic volume form on XF \ {0}. Since r˜∂r˜zi = wi zi and F has degree d, it follows from (3.10) that (3.11)

Lr˜∂r˜ Ω = (|w| − d) Ω.

As in the proof of Proposition 1.10, positivity of |w| − d is then equivalent to the Ricci form on ZF being positive. Indeed, via a D-homothetic transformation we may define r = r˜a to be a new K¨ahler potential, where a = (|w| − d)/n, so that the new Reeb vector field is n ˜ ξ, ξ= |w| − d and Lξ Ω = inΩ. The resulting K¨ ahler metric on ZF now satisfies [ρZ ] = 2n[ωZ ] ∈ H 1,1 (Z, R). One may then ask when this metric can be deformed to a K¨ahler-Einstein metric, thus giving a quasi-regular Sasaki-Einstein metric on LF . Although in general necessary and sufficient conditions are not known, Theorem 3.3 gives a sufficient condition that is practical to check. 3.5. Quasi-regular Sasaki-Einstein metrics on links. Theorem 3.3 was used by Demailly-Koll´ar in their paper [38] to prove the existence of K¨ ahler-Einstein orbifold metrics on certain orbifold del Pezzo surfaces, realized as weighted hypersurfaces in CP3[w0 ,w1 ,w2 ,w3 ] . More precisely, they produced precisely 3 such examples. The very first examples of quasi-regular Sasaki-Einstein 5-manifolds were constructed in [17] using this result, together with the inversion Theorem 3.1. The differential topology of the corresponding links can be analyzed using the results described in the previous section, together with Smale’s Theorem 2.7, resulting  in 2 non-regular Sasaki-Einstein metrics on S 2 × S 3 , and 1 on #2 S 2 × S 3 . An avalanche of similar results followed [3, 18, 25, 26, 27, 59, 60], classifying all such log del Pezzo surfaces (Fano orbifold surfaces with only isolated orbifold singularities) for which Theorem 3.3 produces a K¨ ahler-Einsteinorbifold metric. This led to quasi-regular Sasaki-Einstein structures on #k S 2 × S 3 for all 1 ≤ k ≤ 9. Compare to Theorem 2.6. In [22] Theorem 3.3 was applied to the Brieskorn-Pham links L(a), giving the following remarkable result: Theorem 3.9. Let L(a) be a Brieskorn-Pham link, with weighted homogeneous polynomial given by (3.7). Denote ci = lcm(a0 , . . . , a ˆi , . . . , an ), bi = gcd(ai , ci ), where as usual a hat denotes omission of the entry. Then L(a)

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admits a quasi-regular Sasaki-Einstein metric if the following conditions hold: n 1 (1) > 1, i=0 ai n 1 n 1 mini { }, 1.

In particular, this determines also the possible torsion groups for simplyconnected Sasaki-Einstein 5-manifolds. Precisely which of the manifolds in the Smale-Barden classification could admit Sasaki-Einstein structures is listed in Corolloary 11.4.14 of the monograph [21], together with those for which existence has been shown. Further results, again using weighted homogeneous hypersurface singularities, have been presented recently in [29]. 4. Explicit constructions 4.1. Cohomogeneity one Sasaki-Einstein 5-manifolds. In the last section we saw that quasi-regular Sasaki-Einstein structures exist in

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abundance, in every odd dimension. It is important to stress that these are existence results, based on sufficient algebro-geometric conditions for solving the Monge-Amp`ere equation (2.4) on the orbifold leaf space of a quasi-regular Sasakian manifold that is transverse Fano. Indeed, the isometry groups of the Sasaki-Einstein manifolds produced via this method in Theorem 3.9 are as small as possible, and this lack of symmetry suggests that it will be difficult to write down solutions in explicit form. On the other hand, given enough symmetry one might hope to find examples of Sasaki-Einstein manifolds for which the metric and Sasakian structure can be written down explicitly in local coordinates. Of course, such examples will be rather special. We have already mentioned in Theorem 2.4 that homogeneous Sasaki-Einstein manifolds are classified. The next simplest case, in terms of symmetries, is that of cohomogeneity one. By definition this means there is a compact Lie group G of isometries preserving the Sasakian structure which acts such that the generic orbit has real codimension 1. In fact the first explicit quasi-regular Sasaki-Einstein 5manifolds constructed were of this form. The construction also gave the very first examples of irregular Sasaki-Einstein manifolds, which had been conjectured by Cheeger-Tian [32] not to exist. The following result was presented in [50]: Theorem 4.1. There exist countably infinitely many Sasaki-Einstein metrics on S 2 ×S 3 , labelled by two positive integers p, q ∈ Z>0 , gcd(p, q) = 1, q < p, given explicitly in local coordinates by g=

 1−y  2 1 q(y) dθ + sin2 θdφ2 + dy 2 + (dψ − cos θdφ)2 6 w(y)q(y) 9 + w(y) [dα + f (y) (dψ − cos θdφ)]2 ,

(4.1) where

w(y) =

2(a − y 2 ) , 1−y

q(y) =

a − 3y 2 + 2y 3 , a − y2

f (y) =

a − 2y + y 2 , 6(a − y 2 )

and the constant a = ap,q is (4.2)

a = ap,q =

1 (p2 − 3q 2 )  2 − 4p − 3q 2 . 2 4p3

The manifolds are cohomogeneity one under the isometric action of a Lie group with Lie algebra su(2)⊕u(1)⊕u(1). The Sasakian structures are quasiregular if and only if 4p2 − 3q 2 = m2 , m ∈ Z; otherwise they are irregular of rank 2. In particular, there are countably infinite numbers of quasi-regular and irregular Sasaki-Einstein structures on S 2 × S 3 . We discovered these manifolds quite by accident, whilst trying to classify a certain class of supergravity solutions [49]. It is not too difficult to check that the metric in (4.1) is indeed Sasaki-Einstein, although a key point is

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that the local coordinate system here is not in fact well-adapted to the Sasakian structure. For example, the Reeb vector field is ξ = 3∂ψ − 12 ∂α . Instead these local coordinates are convenient for analysing when and how this metric extends to a smooth complete metric on a compact manifold. The metric in the first line of (4.1) can in fact be shown to be a smooth complete metric on S 2 ×S 2 , for any value of the constant a ∈ (0, 1), by taking θ ∈ [0, π], y ∈ [y1 , y2 ], and the coordinates φ and ψ to be periodic with period 2π. Here y1 < y2 are the two smallest roots of the cubic appearing in the numerator of the function q(y), the condition that a ∈ (0, 1) guaranteeing in particular that these roots are real. Geometrically, these coordinates naturally describe a 4-manifold which is given by the 1-point compactifications of the fibres of the tangent bundle of S 2 , T S 2 . This results in an S 2 bundle over S 2 that is topologically trivial. There is a natural action of SO(3) × U (1), under which the metric is invariant, in which SO(3) acts in the obvious way on T S 2 and the U (1) acts on the fibre, the latter U (1) being generated by the Killing vector field ∂ψ . The 1-form in square brackets appearing in the second line of (4.1) can then be shown to be proportional to a connection 1-form on the total space of a principal U (1) bundle over S 2 × S 2 , provided a = ap,q is given by (4.2). The integers p and q are simply the Chern numbers of this U (1) bundle, so naturally (p, q) ∈ Z ⊕ Z ∼ = H 2 (S 2 × S 2 , Z). It is important here that w(y) > 0 for all y ∈ [y1 , y2 ]. It is also important to stress that this principal U (1) bundle is not generated by the Reeb vector field, and indeed the metric on S 2 × S 2 in the first line of (4.1) is neither K¨ ahler nor Einstein. Via the Gysin sequence and Smale’s Theorem 2.7, the total space is diffeomorphic to S 2 × S 3 provided gcd(p, q) = 1. In a sense, the Sasaki-Einstein manifolds of Theorem 4.1 interpolate between the two 5-dimensional homogeneous Sasaki-Einstein manifolds given in (1) and (2) of Theorem 2.6. More precisely, setting p = q leads to a SasakiEinstein orbifold S 5 /Z2p , with the round metric on S 5 , while q = 0 instead leads to a Sasaki-Einstein orbifold which is a non-freely acting Zp quotient of the homogeneous Sasaki-Einstein metric on V2 (R4 ). As stated in Theorem 4.1, the resulting Sasaki-Einstein manifolds are cohomogeneity one under the effective isometric action of a compact Lie group G with Lie algebra su(2) ⊕ u(1) ⊕ u(1). In fact we have the following classification result [36]: Theorem 4.2. Let (S, g) be a compact simply-connected Sasaki-Einstein 5-manifold for which the isometry group acts with cohomogeneity one. Then (S, g) is isometric to one of the manifolds in Theorem 4.1. Much is known about the structure of cohomogeneity one manifolds, and also the Einstein equations in this case; a review was presented in a previous article in this journal series [107]. The cohomogeneity one assumption reduces the conditions for having a complete G-invariant Sasaki-Einstein metric to solving a system of ordinary differential equations on an interval, with certain boundary conditions at the endpoints of this interval. Here the

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interval is parametrized by distance t along a geodesic transverse to a generic orbit of G. Denoting the stabilizer group of a point on a generic orbit by H ⊂ G, then the manifold S has a dense open subset that is equivariantly diffeomorphic to (t0 , t1 ) × G/H. At the boundaries t = t0 , t = t1 of the interval the generic orbit collapses to 2 special orbits G/H0 , G/H1 . For this to happen smoothly, H1 /H and H2 /H must both be diffeomorphic to spheres of positive dimension. Choosing an AdH -invariant decomposition g = h + m, 2 (m)H , a G-invariant metric on S is determined by a map from [t0 , t1 ] → S+ where the latter is the space of AdH -invariant symmetric positive bilinear maps on m. This has appropriate boundary conditions at t = t0 and t = t1 that guarantee the metric compactifies to a smooth metric on S. We refer the reader to [36] for a complete discussion in the case of Sasaki-Einstein 5-manifolds. For the cohomogeneity one manifolds in Theorem 4.1 the special orbits are located at y = y1 , y = y2 . 4.2. A higher dimensional generalization. Being Sasaki-Einstein, the Reeb foliation Fξ for the manifolds in Theorem 4.1 is transversely K¨ ahler-Einstein. The K¨ ahler-Einstein metric on a local leaf space is, after an appropriate local change of coordinates, given by  1 1−y  2 dθ + sin2 θdφ2 + dy 2 gT = (4.3) 6 w(y)q(y)

2 w(y)q(y) 1 + dγ + cos θdφ . 4 3 This local K¨ ahler metric is of Calabi form. By definition, the Calabi ansatz [31] takes a K¨ ahler manifold (V, gV , ωV ) of complex dimension m and produces a local K¨ ahler metric in complex dimension m + 1 given by 1 dy 2 + Y (y) (dγ + A)2 , 4Y (y) 1 ωh = (β − y)ωV − dy ∧ (dγ + A) . 2 Here A is a local 1-form on V with dA = 2ωV , Y is an arbitrary function, and β is a constant. If V is compact of course the 1-form A cannot be globally defined on V . However, if (V, ωV ) is a Hodge manifold then by definition ωV is proportional to the curvature 2-form of a Hermitian line bundle over V . In this case the 1-form dγ +A may be interpreted globally as being proportional to the connection 1-form on the total space P of the associated principal U (1) bundle. The K¨ ahler metric (4.4) is then defined on (y1 , y2 ) × P , where the function Y (y) is strictly positive on this interval and y2 < β so that (β − y) is also strictly positive. The K¨ ahler-Einstein metric (4.3) is locally of this form, where one takes (V, gV ) to be the standard Fubini-Study K¨ ahler-Einstein metric on CP1 , normalized to have volume 2π/3, 4Y (y) = w(y)q(y), and β = 1. The metric is also cohomogeneity one, where the isometry group has Lie algebra (4.4)

h = (β − y)gV +

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su(2) ⊕ u(1). In fact this metric was constructed as early as [55], with some global properties being discussed in [85]. One can replace (V, gV ) = (CP1 , gFubini−Study ) by a general Fano K¨ ahler-Einstein manifold of complex dimension m. The metric h in (4.4) is then itself K¨ ahler-Einstein provided Y satisfies an appropriate ordinary differential equation. Remarkably, this equation can be solved explicitly in every dimension [10, 84], leading to a local 1-parameter family of K¨ahler-Einstein metrics. However, in the latter reference it was shown that this local metric extends to a complete metric on a compact Fano manifold only for a particular member of this family, which is then simply the homogeneous Fubini-Study metric on CPm+1 . This family of local metrics was subsequently forgotten. Given the above discussion, it is perhaps then unsurprising that the construction of Sasaki-Einstein manifolds in Theorem 4.1 extends to a construction of infinitely many Sasaki-Einstein manifolds, in every odd dimension 2n − 1 ≥ 5, for every complete Fano K¨ ahler-Einstein manifold (V, gV ) with dimC V = n − 2. The following result was shown in [51], although the more precise statement given here appeared later in [73]: ahler-Einstein manifold Theorem 4.3. Let (V, gV ) be a complete Fano K¨ of complex dimension dimC V = n − 2 with Fano index I = I(V ). Then for every choice of positive integers p, k ∈ Z>0 satisfying Ip/2 < k < pI, gcd(p, k) = 1, there is an associated explicit complete simply-connected SasakiEinstein manifold in dimension 2n − 1. Theorem 4.1 is the special case in which (V, gV ) = (CP1 , gFubini−Study ) and where k = p + q. The proof is almost identical to the proof of Theorem 4.1. One uses the local 1-parameter family of K¨ ahler-Einstein metrics of [10, 84] to write down a local 1-parameter family of Sasaki-Einstein metrics in dimension 2n − 1. This is a local version of the inversion theorem: given a (local) K¨ ahler-Einstein metric h with positive Ricci curvature Rich = 2nh the local metric (4.5)

g = h + (dψ + A)2 ,

is Sasaki-Einstein, where A is a local 1-form with dA = 2ωh . After an appropriate change of local coordinates, one sees that this local metric can be made into a complete metric on a compact manifold for a countably infinite number of members of the family. This manifold is the total space of a principal U (1) bundle over a manifold which is itself an S 2 bundle over V . The latter is obtained from KV−1 by compactifying each fibre, and the integers p, k specify the first Chern class of the principal U (1) bundle. Unlike the case n = 3 in Theorem 4.1, the homology groups of these SasakiEinstein manifolds in general depend on p and k. Determining which of the Sasakian structures are quasi-regular and which are irregular is equivalent to determining whether a certain polynomial of degree n − 1, with integer coefficients depending on p and k, has a certain root which is rational or irrational, respectively [72].

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4.3. Transverse Hamiltonian 2-forms. In [52, 34] it was noted that the last construction may be extended further by replacing the K¨ ahlerEinstein manifold (V, gV ) by a finite set of Fano K¨ ahler-Einstein manifolds ahler metric (Va , ga ), a = 1, . . . , N , and correspondingly extending Calabi’s K¨ ansatz. Moreover, in [37] (see also [70]) an infinite set of explicit cohomogeneity two Sasaki-Einstein metrics were presented on S 2 × S 3 . These have isometry group T3 . In [70] it was realized that there is a single geometric structure that underlies all of these explicit constructions of Sasaki-Einstein manifolds, namely a transverse Hamiltonian 2-form on the K¨ ahler leaf space of the Reeb foliation. Hamiltonian 2-forms were introduced in [2]: Definition 4.4. Let (Z, h, ω, J) be a K¨ ahler manifold. A Hamiltonian 2-form φ is a real (1, 1)-form that solves non-trivially the equation [2]  1 ∇Y φ = d Trω φ ∧ JY − dc Trω φ ∧ Y . 2 Here Y is any vector field, ∇ denotes the Levi-Civita connection of h, and Y = h(Y, ·) is the 1-form dual to Y . In fact Hamiltonian 2-forms on K¨ ahler manifolds are related to another structure that is perhaps rather more well-known, especially to relativists: Proposition 4.5. A (1, 1)-form φ is Hamiltonian if and only if φ + (Trω φ) ω is closed and the symmetric 2-tensor S = J(φ − (Trω φ) ω) is a Killing tensor; that is, Sym ∇S = 0 (in components, ∇(i Sjk) = 0). The proof is an elementary calculation and may be found in [2]. In fact if φ is Hamiltonian then the (1, 1)-form φ − 12 (Trω φ) ω a is conformal Killing 2-form in the sense of [92]. In the relativity literature such a form is also called a conformal Killing-Yano form. Again, this leads to an equivalence between conformal Killing 2-forms of type (1, 1) and Hamiltonian 2-forms. Conformal Killing tensors and forms generalize the notion of conformal Killing vectors. The latter generate symmetries of the metric, and the same is true also of Killing tensors, albeit in a more subtle way. For example, a classic early result was that a Killing form gives rise to a quadratic first integral of the geodesic equation [86]. The key result about Hamiltonian 2-forms is that their existence leads to a very specific form for the K¨ ahler metric h. Moreover, particularly relevant for us is that the K¨ ahler-Einstein condition is then equivalent to solving a simple set of decoupled ordinary differential equations. Below we just sketch how this works, referring the reader to [2] for details. We note that many of the resulting ans¨ atze for K¨ ahler metrics had been arrived at prior to the work of [2], both in the mathematics literature (as pointed out in [2]), and also in the physics literature. The theory of Hamiltonian 2-forms unifies these various approaches.

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One first notes that if φ is a Hamiltonian 2-form, then so is φt = φ − tω for any t ∈ R. One then defines the momentum polynomial of φ to be p(t) =

(−1)m ∗ φm t . m!

Here m is the complex dimension of the K¨ ahler manifold and ∗ is the Hodge operator with respect to the metric h. It is then straightforward to show that {p(t)} are a set of Poisson-commuting Hamiltonian functions for the 1-parameter family of Killing vector fields K(t) = J gradh p(t). For a given point in the K¨ ahler manifold, these Killing vectors will span a vector subspace of the tangent space of the point; the maximum dimension of this subspace, taken over all points, is called the order s of φ. This leads to a local Hamiltonian Ts action on the K¨ ahler manifold, and one may take a (local) K¨ ahler quotient by this torus action. The reduced K¨ ahler space is a direct product of N K¨ ahler manifolds that depends on the moment map level at which one reduces, but only very weakly. The 2s-dimensional fibres turn out to be orthotoric, which is a rather special type of toric K¨ ahler structure. For further details, we refer the reader to reference [2]. However, the above should give some idea of how one arrives at the following structure theorem of [2]: Theorem 4.6. Let (Z, h, ω, J) be a K¨ ahler manifold of complex dimension m with a Hamiltonian 2-form φ of order s. This means that the momentum polynomial p(t) has s non-constant roots y1 , . . . , ys . Denote the remaining distinct constant roots by ζ1 , . . . , ζN , where ζa has multiplicity N ma , s so that p(t) = pnc (t)pc (t) where pnc (t) = i=1 (t − yi ) and pc (t) = a=1 (t − ζa )ma . Then there are functions F1 , . . . , Fs of one variable such that on a dense open subset the K¨ ahler structure may be written ⎡ ⎞2 ⎤ ⎛ N s s  (y )    p (y ) F i i i ⎝ ⎣ h= dy 2 +  pnc (ζa )ga + σj−1 (ˆ yi )θj ⎠ ⎦ . Fi (yi ) i p (yi ) a=1

ω=

N 

i=1

pnc (ζa )ωa +

a=1

s 

j=1

dσi ∧ θi ,

dθi =

N  (−1)i ζas−i ωa . a=1

i=1

Here σi denotes the ith elementary symmetric function of the non-constant roots y1 , . . . , ys , and σj−1 (ˆ yi ) denotes the (j − 1)th elementary symmetric function of the s − 1 roots {yk | k = i}. Moreover, (ga , ωa ) is a positive (or negative) definite K¨ ahler metric on a manifold Va with dimC Va = ma . In fact the Hamiltonian 2-form is simply φ=

N  a=1

ζa pnc (ζa )ωa +

s  i=1

(σi dσ1 − dσi+1 ) ∧ θi ,

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where σs+1 ≡ 0. What is remarkable about this ansatz for a K¨ ahler structure is the following [2]: Proposition 4.7. The K¨ ahler metric in Theorem 4.6 is K¨ ahler-Einstein if for all i = 1, . . . , s the functions Fi satisfy (4.6)

Fi (t) = pc (t)

s 

bj ts−j ,

j=0

where bj are arbitrary constants (independent of i ), and for all a = 1, . . . , N ±(ga , ωa ) is K¨ ahler-Einstein with scalar curvature Scal±ga = ∓ma

s 

bi ζas−i .

i=0

In this case the Ricci form is ρh =

− 12 b0 ω.

Of course, this result follows from direct local calculations. Notice that (4.6) may immediately be integrated to obtain a local K¨ ahler-Einstein metric that is completely explicit, up to the K¨ ahler metrics ga . By taking m = n−1 and b0 = −4n one can lift such local K¨ahler-Einstein metrics of positive Ricci curvature to local Sasaki-Einstein metrics in dimension 2n − 1 using (4.5). One may then ask when this local metric extends to a complete metric on a compact manifold. In fact all known (or at least known to the author) explicit constructions of Sasaki-Einstein manifolds are of this form. The Sasaki-Einstein manifolds in Theorem 4.3 are constructed this way, with s = 1, N = 1. Indeed, this case is precisely the Calabi ansatz (4.4) for the local K¨ahler-Einstein metric, as already mentioned. The generalization in [52, 34] mentioned at the beginning of this section is s = 1 but N ≥ 1. Finally, most interesting is to take s > 1. In particular, for a Sasaki-Einstein 5-manifold this means that necessarily N = 1 and moreover m1 = 0. In other words, the transverse K¨ahler-Einstein metric is orthotoric in the sense of reference [2]. The results described in this section give the explicit local form of such a metric, although it must be stressed that this is not how they were first derived. In fact in [37] the local family of orthotoric K¨ ahler-Einstein metrics was obtained by taking a certain limit of a family of black hole metrics. These black hole solutions themselves possess Killing tensors. On the other hand, in [70] the same local metrics were obtained by taking a limit of the PlebanskiDemianski metrics [87], again a result in general relativity. It is then simply a matter of analyzing when these local metrics extend to complete metrics on a compact manifold. The result is the following [37, 70]: Theorem 4.8. There exist a countably infinite number of explicit SasakiEinstein metric on S 2 × S 3 , labelled naturally by 3 positive integers a, b, c ∈ Z>0 with a ≤ b, c ≤ b, d = a+b−c, gcd(a, b, c, d) = 1, and also such that each of the pair {a, b} is coprime to each of {c, d}. For a = p − q, b = p + q, c = p

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these reduce to the Sasaki-Einstein structures in Theorem 4.1. Otherwise they are cohomogeneity two with isometry group T3 . The proof here is rather different to that of the proof of Theorem 4.1. In fact it is easiest to understand the global structure using the toric methods developed in the next section. For integers {a, b, c} not satisfying some of the coprime conditions one obtains Sasaki-Einstein orbifolds. The conditions under which the Sasakian structures are quasi-regular is not simple to determine explicitly in general, and involves a quartic Diophantine equation. Generically one expects them to be irregular. The next section allows one to characterize the Sasaki-Einstein manifolds in Theorem 4.8: they are all of the simply-connected toric Sasaki-Einstein manifolds with second Betti number b2 (S) = 1. 5. Toric Sasaki-Einstein manifolds 5.1. Toric Sasakian geometry. We begin with the following: Definition 5.1. A Sasakian manifold (S, g) is said to be toric if there is an effective, holomorphic and Hamiltonian action of the torus Tn on the corresponding K¨ ahler cone (C(S), g¯, ω, J) with Reeb vector field ξ ∈ tn = Lie algebra of Tn . Here we have used the same symbol for an element of the Lie algebra tn and the corresponding vector field on C(S) induced by the group action. The abuse of notation should not cause confusion as the meaning should always be clear. The Hamiltonian property means that Tn acts on constant r surfaces in C(S). This in turn implies that there exists a Tn -invariant moment map (5.1)

μ : C(S) → t∗n ,

where μ, ζ = 12 r2 η(ζ),

∀ζ ∈ tn .

Definition 5.1 is taken from [74], and is compatible with the earlier definition of toric contact manifold appearing in [16] (up to a factor of 12 in the moment map definition). The condition on the Reeb vector field, ξ ∈ tn , implies that the image μ(C(S)) ∪ {0} is a strictly convex rational polyhedral cone C ∗ ⊂ t∗n [44, 66]. (Toric symplectic cones with Reeb vector fields not satisfying this condition form a short list and have been classified [66].) By definition this means that C ∗ may be presented as (5.2)

C ∗ = {y ∈ t∗n | y, va  ≥ 0, a = 1, . . . , d} ⊂ t∗n .

Here the rationality condition means that va ∈ ZTn ≡ ker{exp : tn → Tn }. On choosing a basis this means that we may think of va ∈ Zn ⊂ Rn ∼ = tn , and without loss of generality we assume that the {va } are primitive. We also assume that the set {va } is minimal, in the sense that removing any va from the definition in (5.2) would change the polyhedral cone C ∗ . The strictly convex condition means that C ∗ is a cone over a compact convex polytope of

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dimension n − 1. It follows that necessarily the number of bounding hyperplanes is d ≥ n. The polyhedral cone C ∗ is also good, in the sense of [66]. This may be defined as follows. Each face F ⊂ C ∗ may be realized uniquely as the intersection of some number of facets {y, va  = 0} ∩ C ∗ . Denote by va1 , . . . , vaN the corresponding collection of normal vectors in {va }, where N is the codimension of F – thus {a1 , . . . , aN } is a subset of {1, . . . , d}. Then the cone is good if and only if  N   N    n ν A va A | νA ∈ R ∩ Z = ν A va A | νA ∈ Z , A=1

A=1

holds for all faces F. We denote by Int C ∗ the open interior of C ∗ . The Tn action on μ−1 (Int C ∗ ) is free, and moreover the latter is a Lagrangian torus fibration over Int C ∗ . On the other hand, the bounding facets {y, va  = 0} ∩ C ∗ lift to Tn−1 -invariant complex codimension one submanifolds of C(S) that are fixed point sets of the U (1) ∼ = T ⊂ Tn subgroup specified by va ∈ ZTn . The image μ(S) = μ({1} × S ⊂ C(S)) is easily seen from (5.1) to be   1 ∗ μ(S) = y ∈ C | y, ξ = . 2 Here the hyperplane {y ∈ t∗n | y, ξ = 12 } ⊂ t∗n is called the characteristic hyperplane [16]. This intersects the moment cone C ∗ to form a compact n-dimensional polytope Δ(ξ) = μ({r ≤ 1}), bounded by ∂C ∗ and a (n − 1)dimensional compact convex polytope H(ξ) which is the image μ(S) of the Sasakian manifold S in t∗n . Since μ(ξ) = 12 r2 > 0 on C(S) this immediately implies that the Reeb vector field ξ ∈ Int C where C = {ξ ∈ tn | y, ξ ≥ 0, ∀y ∈ C ∗ } ⊂ tn is the dual cone to C ∗ . This is also a convex rational polyhedral cone by Farkas’ Theorem. Recall that in section 1.4 we explained that the space X = C(S)∪{r = 0} can be made into a complex analytic space in a unique way. For a toric Sasakian manifold in fact X is an affine toric variety; that is, X is an affine variety equipped with an effective holomorphic action of the complex torus TnC ∼ = (C∗ )n which has a dense open orbit. The affine toric variety X may be constructed rather explicitly as follows. Given the polyhedral cone C ∗ one defines a linear map A : Rd → Rn via A(ea ) = va , where {ea } denotes the standard orthonormal basis of Rd . The strictly convex condition on C ∗ implies that A is surjective. This induces a corresponding map of tori A˜ : Td → Tn . The kernel ker A˜ is a compact abelian subgroup of Td of rank d−n ˜ ∼ and π0 (ker A) = ZTn /spanZ {va }. Then the affine variety X is simply X = ˜ Spec C[z1 , . . . , zd ]ker A , the ring of invariants. This is a standard construction in toric geometry [46], and goes by the name of Delzant’s Theorem. The

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goodness condition on C ∗ is necessary and sufficient for X\{o} to be a smooth manifold, away from the apex {o} [66]. The fact that X is toric is also clear via this construction: the torus TnC ∼ = TdC / ker A˜C acts holomorphically on X with a dense open orbit. In this algebro-geometric language the cone C is precisely the fan for the affine toric variety X. Let ∂φi , i = 1, . . . , n, be a basis for tn , where φi ∈ [0, 2π) are coordinates on the real torus Tn . We then have the following very explicit description of the space of toric Sasakian metrics [74]: Proposition 5.2. The space of toric K¨ ahler cone metrics on C(S) is a product Int C × H1 (C ∗ ) where ξ ∈ Int C ⊂ tn labels the Reeb vector field and H1 (C ∗ ) denotes the space of homogeneous degree one functions on C ∗ that are smooth up to the boundary (together with the convexity condition below). Explicitly, on the dense open image of TnC we have (5.3)

g¯ =

n 

Gij dy i dy j + Gij dφi dφj ,

i,j=1

where Gij = ∂yi ∂yj G with matrix inverse Gij , and the function (5.4)

G(y) = Gcan (y) + Gξ (y) + ψ(y)

is required to be strictly convex with ψ(y) ∈ H1 (C ∗ ) and 1 y, va  logy, va , Gcan (y) = 2 a=1  d   d   1  1 y, va  log y, va  . Gξ (y) = y, ξ logy, ξ − 2 2 d

a=1

a=1

The coordinates (yi , φi ) are called symplectic toric coordinates. The yi are simply the Hamiltonian functions for ∂φi : yi = μ, ∂φi  =

1 2 2 r η(∂φi ),

ω=

n 

dyi ∧ dφi .

i=1

The function G(y) is called the symplectic potential. Setting G(y) = Gcan (y) gives precisely the K¨ahler metric on C(S) induced via K¨ ahler reduction of d the flat metric on C . That is, C(S) equipped with the metric given by (5.3) and G(y) = Gcan (y) is isomorphic to the K¨ ahler quotient (Cd , ωflat )// ker A˜ d at level zero. The origin of C here projects to the singular point {r = 0} in X. The function Gcan (y) has a certain singular behaviour at the boundary ∂C ∗ of the polyhedral cone. This is required precisely so that the metric compactifies to a smooth metric on C(S). By construction, the K¨ ahler metric

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g¯ in (5.3) is a cone with respect to r∂r = ni=1 2yi ∂yi . On the other hand, the complex structure in these coordinates is

0 −Gij J= , Gij 0 and one easily checks that J(r∂r ) = ni,j=1 2Gij yj ∂φi = ξ, with ξ determined by Gξ (y) in (5.4). Proposition 5.2 extends earlier work of Guillemin [57] and Abreu [1] from the K¨ ahler case to the Sasakian case. The following topological result is from [67]: Proposition 5.3. Let S be a toric Sasakian manifold. Then π1 (S) ∼ = d−n . ZTn /spanZ {va }, π2 (S) ∼ Z = In particular, S is simply-connected if and only if the vectors {va } that define the moment polyhedral cone C ∗ form a Z-basis of ZTn ∼ = Zn . Using the Hurewicz Isomorphism Theorem and Smale’s Theorem 2.7 then gives: Corollary 5.4. Let S be a simply-connected toric Sasakian 5-manifold.  Then S is diffeomorphic to #k S 2 × S 3 where k = d − n. Finally, we note that an affine toric variety is -Gorenstein in the sense of Definition 1.11 if and only if there is a basis for the torus Tn for which va = (, wa ) for each a = 1, . . . , d, and wa ∈ Zn−1 . In particular, for a simplyconnected toric Sasaki-Einstein manifold the affine toric variety X will be Gorenstein, and hence there will exist a basis such that va = (1, wa ). Example 5.5. The Sasaki-Einstein manifolds in Theorem 4.8 are toric, and in fact the proof makes it evident that the corresponding affine toric ∗ varieties are X = Spec C[z1 , z2 , z3 , z4 ]C (a,b,c) , where C∗ (a, b, c) is the 1-dimensional subgroup of (C∗ )4 specified by the lattice vector (a, b, −c, −a− b + c). The fact that the entries in this vector sum to zero is equivalent to X being Gorenstein. 5.2. Sasaki-Einstein metrics. Proposition 5.2 gives a rather explicit description of the space of toric Sasakian metrics on the link of an affine toric singularity. We may then ask which of these are Sasaki-Einstein. In fact a rather more basic question is for which Reeb vector fields ξ ∈ Int C is there a Sasaki-Einstein metric. Notice there was no analogous question in the approach of section 3: there we had a fixed affine variety, namely a weighted homogeneous hypersurface singularity, with a fixed choice of holomorphic Reeb vector field ξ −iJ(ξ), namely that associated to the weighted C∗ action. Without any essential loss of generality, we consider simply-connected Sasakian manifolds. Then we know that the corresponding affine toric variety must be Gorenstein if the cone is to admit a Ricci-flat K¨ ahler cone metric, and hence there is a basis such that va = (1, wa ). We assume we have chosen such a basis.

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The key idea in [74] was that an Einstein metric g on S with Ricci curvature Ricg = 2(n − 1)g is a critical point of the Einstein-Hilbert action  [Scalg + 2(n − 1)(3 − 2n)] dμg , (5.5) I[g] = S

where dμg is the Riemannian volume form associated to the metric g and as earlier Scalg denotes the scalar curvature. We may then restrict this functional to the space of toric Sasakian metrics. The insight in [74] was that this functional in fact depends only on the Reeb vector field ξ of the Sasakian structure. Direct calculation gives: Proposition 5.6. The Einstein-Hilbert action (5.5), restricted to the space of toric Sasakian metrics on the link of an affine toric Gorenstein singularity, induces a function I : Int C → R given by (5.6)

I(ξ) = 8n(n − 1)(2π)n [e1 , ξ − (n − 1)] vol(Δ(ξ)).

Here e1 = (1, 0, 0, . . . , 0) and vol(Δ(ξ)) denotes the Euclidean volume of the polytope Δ(ξ) = μ ({r ≤ 1}). A toric Sasaki-Einstein metric is a critical point of I defined in (5.6). Of course C is itself a cone, and one may first take the derivative of I along the Euler vector field of this cone. Using the fact that vol(Δ(ξ)) is homogeneous degree −n one easily checks that this derivative is zero if and only if e1 , ξ = n. Thus a critical point of I lies on the interior of the intersection of this plane with C. Call the latter compact convex polytope P ⊂ tn . It follows that the Reeb vector field for a Sasaki-Einstein metric is a critical point of I |Int P = 8n(n − 1)(2π)n vol(Δ). This is, up to a constant of also just the Riemannian volume proportionality, n of (S, g). If we write ξ = i=1 ξi ∂φi then it is simple to compute  n ∂vol(Δ)  1 (5.7) = y i dσ, ∂ξi 2ξk ξk H(ξ) k=1  n 2  ∂ vol(Δ) 2(n + 1) (5.8) = y i y j dσ. ∂ξi ∂ξj ξk ξk H(ξ) k=1

Here dσ is the standard measure induced on the (n − 1)-polytope H(ξ) = μ(S) ⊂ C ∗ . Notice that vol(Δ) diverges to +∞ at ∂P . This can be seen rather explicitly from the formula for the volume of the polytope vol(Δ), but more conceptually for ξ ∈ ∂C the vector field ξ in fact vanishes somewhere on C(S). Specifically, the bounding facets of C correspond to the generating rays of C ∗ under the duality map between cones; ξ being in a bounding facet of C implies that the corresponding vector field then vanishes on the inverse image, under the moment map, of the dual generating ray of C ∗ .

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Uniqueness and existence of a critical point of I now follows from a standard convexity argument: I |Int P = 8n(n − 1)(2π)n vol(Δ) is a strictly convex (by (5.8)) positive function on the interior of a compact convex polytope P . By strict convexity, a critical point of I is equivalent to a local minimum, and this in turn is then the unique global minimum on P . Since I diverges to +∞ on the boundary of P , such a critical point must occur in the interior of P . Also notice that I is bounded from below by zero on P , so it must have a global minimum somewhere, and hence a critical point. It follows from these comments that I must have precisely one critical point in the interior of P , and we have thus proven: Theorem 5.7. There exists a unique Reeb vector field ξ ∈ Int C for which the toric Sasakian structure on the link of an affine toric Gorenstein singularity can be Sasaki-Einstein. Having fixed the Reeb vector field, the problem of finding a SasakiEinstein metric now reduces to deforming the transverse K¨ ahler metric to a transverse K¨ ahler-Einstein metric. As in the regular and quasi-regular cases, this is a Monge-Amp`ere problem. To analyze this it is more convenient to introduce complex coordinates. Recall that the complex torus TnC is a dense open subset of C(S). Introducing log complex coordinates zi = xi + iφi on TnC , the K¨ ahler structure is ¯ ω = 2i∂ ∂F,

g¯ =

n 

Fij dxi dxj + Fij dφi dφj .

i,j=1

Here the K¨ ahler potential is F (x) = 14 r2 and Fij = ∂xi ∂xj F . This is related to the symplectic potential G by Legendre transform   n  yi ∂yi G − G (y = ∂x F ). F (x) = i=1

Having fixed the holomorphic structure on C(S) (this being determined uniquely up to equivariant biholomorphism by C ∗ ) and fixing the Reeb vector field to be the unique critical point of I, we may set ψ = 0 in (5.4) to obtain an explicit toric Sasakian metric g0 that is a critical point of I. This is our background metric. We are then in the situation of Proposition 1.4: any other Sasakian metric with the same holomorphic structure on the cone and same Reeb vector field is related to this metric via a smooth basic function ∞ (S). Thus, if g is a Sasaki-Einstein metric with this property then φ ∈ CB ω T − ω0T = i∂B ∂¯B φ, where ω0T and ω T are the transverse K¨ahler forms associated to g0 and g, respectively. The holomorphic volume form on C(S) is [74] (5.9)

Ω = ex1 +iφ1 (dx1 + idφ1 ) ∧ · · · ∧ (dxn + idφn ) ,

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and the critical point condition e1 , ξ = n is equivalent to Lξ Ω = inΩ. Thus using Proposition 1.10 and the transverse ∂ ∂¯ lemma again we may also write (5.10)

ρT0 − 2nω0T = i∂B ∂¯B f,

∞ (S) smooth and basic. Then Proposition 2.8 goes through in with f ∈ CB exactly the same way in the transverse sense, with resulting transverse Monge-Amp`ere equation   2 det g0Ti¯j + ∂z∂i ∂φz¯j (5.11) = ef −2nφ , det g0Ti¯j

where now z1 , . . . , zn−1 are local complex coordinates on the leaf space of the Reeb foliation Fξ . This problem was recently studied in detail in [48]. In fact the MongeAmp`ere problem is almost identical to the case of toric K¨ahler-Einstein manifolds studied in [108]. The moment polytope in the latter case is essentially replaced by the polytope H(ξ) in the Sasakian case. The continuity method is used to prove existence, as in section 3.3, and crucially the work of [43] on extending Yau’s estimates [110] to transverse Monge-Amp`ere equations is appealed to to show that the C 0 estimate for the basic function φ is sufficient to solve the equation. Thus the main step is to prove the C 0 estimate for φ, and this closely follows the proof in the K¨ ahler-Einstein case [108]. The result of [48, 35] is the following: Theorem 5.8. There exists a unique toric Sasaki-Einstein metric on the link of any affine toric Gorenstein singularity. Here uniqueness was proven in [35], and is understood up to automorphisms of the transverse holomorphic structure. Thus the existence and uniqueness problem for toric Sasaki-Einstein manifolds is completely solved. We note that in [35] the authors stated this theorem with the weaker requirement that the affine toric singularity is -Gorenstein. For  > 1 the links of such singularities will not be simply-connected, although the converse is not true. However, more importantly the existence of a Killing spinor implies the Gorenstein condition, as discussed in section 1.6, which is why we have presented Theorem 5.8 this way. If one does not care for the existence of a Killing spinor, one can weaken the Gorenstein condition to Q-Gorenstein (-Gorenstein for some ). Combining Theorem 5.8 with the topological result of Corollory 5.4 leads to: Corollary exist infinitely many toric Sasaki-Einstein struc 5.9. There  tures on #k S 2 × S 3 , for every k ≥ 1. In fact we have now done enough to see that the explicit metrics in Theorem 4.8 are all of the toric Sasaki-Einstein metrics on S 2 × S 3 .

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Example 5.10. We comment on two particularly interesting examples. The Sasaki-Einstein structure on S 2 × S 3 with p = 2, q = 1 in Theorem 4.1 ∗ has corresponding affine variety X = Spec C[z1 , z2 , z3 , z4 ]C (2,2,−1,−3) [71]. Using standard toric geometry methods it is straightforward to see that X \ {o} is the total space of the canonical line bundle over the 1-point blow up of CP2 , minus the zero section. Equivalently, X is obtained from this canonical line bundle by contracting the zero section. The latter is a Fano surface which doesn’t admit a K¨ ahler-Einstein metric, as discussed in section 2.4. It is equivalent to say that the canonical choice of holomorphic Reeb vector field ξ − iJ(ξ) that rotates the C∗ fibre over the del Pezzo surface cannot be the Reeb vector field for a Sasaki-Einstein metric. Indeed, in this example one can easily compute the function I in Proposition 5.6 and show that this choice of ξ is indeed not a critical point. Instead the critical ξ gives an irregular Sasaki-Einstein structure of rank 2 [74]. In the latter case this irregular Sasaki-Einstein structure associated to the 1-point blow-up of CP2 is completely explicit. For the 2-point blow-up there is no known explicit metric, but Theorem 5.8 implies there exists a unique toric Sasaki-Einstein metric on the total space of the principal U (1) bundle associated to the canonical line bundle over the surface. In [74] the critical Reeb vector field was computed explicitly, showing that this is again irregular of rank 2. Finally, although Theorem 5.8 settles the existence and uniqueness of toric Sasaki-Einstein manifolds in general, we point out that prior to this result van Coevering [105] proved the existence of infinite families of distinct toric quasi-regular Sasaki-Einstein structures on #k(S 2 × S 3 ), for each odd k > 1, using a completely different method. He finds certain quasi-regular toric Sasakian submanifolds of 3-Sasakian manifolds obtained via the quotient construction mentioned in section 1.5, and then applies an orbifold generalization of a result of Batyrev-Selivanova [8] to deform the corresponding K¨ ahler orbifold to a K¨ ahler-Einstein orbifold. Thus, although 3-Sasakian geometry plays a role in this construction, the Sasaki-Einstein metrics are not induced from the 3-Sasakian structure. 6. Obstructions 6.1. The transverse Futaki invariant. A toric Sasaki-Einstein metric has a Reeb vector field ξ which is a critical point of the function I in Proposition 5.6. The derivative of I is of course a linear map on a space of holomorphic vector fields, and its vanishing is a necessary (and in the toric case also sufficient) condition for existence of a Sasaki-Einstein metric, or equivalently a transverse K¨ ahler-Einstein metric for the foliation Fξ . Given the discussion in section 2.4 it is then not surprising that the derivative of the function I is essentially a transverse Futaki invariant. This was demonstrated in [75], although our discussion here follows more closely the subsequent treatment in [48].

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Throughout this section we suppose that we have Sasakian structure S 1,1 2 with Reeb foliation Fξ satisfying 0 < cB 1 ∈HB (Fξ ) and 0 = c1 (D)∈H (S, R). Via Proposition 1.10 it is equivalent to say that, after a possible 1,1 D-homothetic transformation, we have 2πcB 1 = n[dη] ∈ HB (Fξ ). Assuming also that S is simply-connected, then by Proposition 1.10 it is also equivalent to say that the corresponding Stein space X = C(S) ∪ {r = 0} is Gorenstein with Lξ Ω = inΩ, where Ω is a nowhere zero holomorphic (n, 0)-form on C(S). Following [48] we begin with: Definition 6.1. A complex vector field ζ on a Sasakian manifold S is said to be Hamiltonian holomorphic if (1) its projection to each leaf space is a holomorphic vector field; and (2) the complex-valued function uζ ≡ 12 η(ζ) satisfies 1 ∂¯B uζ = − iζ dη. 4 Such a function uζ is called a Hamiltonian function. If (x, z1 , . . . , zn−1 ) are coordinates for a local foliation chart Uα then one may write  n−1 n−1   ζ i ∂z i − η ζ i ∂ z i ∂x , ζ = η(ζ)∂x + i=1

i=1

where ζ i are local basic holomorphic functions. It is straightforward to see that ζ + iη(ζ)r∂r is then a holomorphic vector field on C(S). A Hamiltonian holomorphic vector field in the sense of Definition 6.1 is precisely the orthogonal projection to S = {r = 1} of a Hamiltonian holomorphic vector field on the K¨ ahler cone (C(S), g¯, ω) whose Hamiltonian function is basic and homogeneous degree zero under r∂r . As pointed out in [48], the set of all Hamiltonian holomorphic vector fields is a Lie algebra h. Moreover, if the transverse K¨ ahler metric has constant scalar curvature then h is necessarily reductive [82]; this is a transverse generalization of the Matsushima result mentioned in section 2.4. Thus the nilpotent radical of h acts as an obstruction to the existence of a transverse constant scalar curvature K¨ahler metric, and in particular a transverse K¨ ahler-Einstein metric. ¯ Since 2πcB 1 = n[dη], by the transverse ∂ ∂ lemma there exists a discrep∞ ancy potential f ∈ CB (S) such that ρT − ndη = i∂B ∂¯B f. We may then define

 F (ζ) =

(6.1)

S

ζ(f ) dμg .

Here the Riemannian measure is (6.2)

dμg =

1 η ∧ (dη)n−1 . 2n−1 (n − 1)!

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Compare this to the Futaki invariant (2.5): in the regular, or quasi-regular, case (6.1) precisely reduces to (2.5) by integration over the U (1) Reeb fibre, up to an overall proportionality constant. By following Futaki’s original computation [47] it is not difficult to show that F (ζ) is independent of 1,1 the transverse K¨ahler metric in the K¨ ahler class [ω T ] = [ 12 dη] ∈ HB (Fξ ). Thus F : h → C is a linear function on h whose non-vanishing obstructs the existence of a transverse K¨ahler-Einstein metric in the fixed basic K¨ ahler class. This result was extended [48, 28] to obstructions to the existence of Sasakian metrics with harmonic basic kth Chern form, again generalizing the K¨ ahler result to the transverse setting. 6.2. The relation to K¨ ahler cones. The results of section 5 motivated the following set-up in [75]. Fix a complex manifold (C(S) ∼ = R>0 × S, J) where S is compact, with maximal torus Ts ⊂ Aut(C(S), J). Then let KCM(C(S), J) be the space of K¨ahler cone metrics on (C(S), J) which are compatible with the complex structure J and such that Ts acts Hamiltonianly (preserving constant r surfaces) with the Reeb vector field ξ ∈ ts = Lie algebra of Ts . For each metric in KCM(C(S), J) there is then an associated moment map given by μ : C(S) → t∗s ,

1 μ, ζ = r2 η(ζ). 2

The image is a strictly convex rational polyhedral cone [44], and moreover all these cones are isomorphic for any metric in KCM(C(S), J). The toric case is when the rank of the torus is maximal, s = n. For any metric g¯ ∈ KCM(C(S), J) we may consider the volume functional  g) = dμg = vol(S, g). (6.3) Vol : KCM(C(S), J) → R>0 , Vol(¯ S

Here g is the Sasakian metric on S induced from the K¨ ahler cone metric g¯. Alternatively, it is simple to see that   ωn 2 n−1 −r2 /2 (n − 1)! Vol(¯ g) = e dμg¯ = e−r /2 , (6.4) 2 n! C(S) C(S) where ω is the K¨ ahler form for g¯. Of course, the volume of the cone itself is divergent, and the factor of exp(−r2 /2) here acts as a convergence factor. Since 12 r2 is the Hamiltonian function for the Reeb vector field ξ, the second formula in (6.4) takes the form of a Duistermaat-Heckman integral [41, 42]. Corresponding localization formulae were discussed in [75], but we shall not discuss this here. We then have the following from [75]: Proposition 6.2. The functional Vol depends only on the Reeb vector field ξ.

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This perhaps needs some clarification. Via Proposition 1.4, any two K¨ ahler cone metrics g¯, g¯ ∈ KCM(C(S), J) with the same Reeb vector field ξ have corresponding contact forms related by η  = η + dcB φ, ∞ (S) is basic with respect to F . Notice that the K¨ where φ ∈ CB ahler cone ξ metrics will have K¨ahler potentials given by smooth functions r, r , and that these then give different embeddings of S into C(S). Proposition 6.2 is proven by writing r = r exp(tφ/2) and showing that the derivative of Vol ∞ (S). with respect to t at t = 0 is zero, independently of the choice of φ ∈ CB We may next consider the derivatives of Vol [75]:

Proposition 6.3. The first and second derivatives of Vol are given by  (6.5) dVol(χ) = −n η(χ) dμg , S  2 (6.6) d Vol(χ1 , χ2 ) = n(n + 1) η(χ1 )η(χ2 ) dμg . S

More formally, what we mean here by (6.5) is that we have a 1-parameter family {¯ g (t)}−0 is the orbifold Fano index of Z and (S 2n−1 , gstandard ) is the standard round sphere. The condition g¯ ∈ KCM(C(S), Ω) implies that, in the regular or quasiregular case, [ρZ ] = 2n[ω]Z ∈ H 1,1 (Z, R). The above result then follows since c1 (Z) is represented by ρZ /2π. Notice also that we have integrated over the Reeb U (1) fibre. Here the simply-connected condition means that the associated complex line orbibundle has first Chern class −c1 (Z)/I(Z), as in Theorem 3.1, which determines the length of the generic Reeb S 1 fibre. Note that vol(S 2n−1 , gstandard ) = 2π n /(n − 1)!. In the regular or quasi-regular case, the independence of the volume of transverse K¨ ahler transformations follows simply because the volume of the K¨ ahler leaf space depends only on the K¨ ahler class. On the other hand, this is sufficient to prove the more general statement in Proposition 6.2 since the space of quasi-regular Reeb vector fields in KCM(C(S), Ω) is dense in the space of all possible Reeb vector fields. This is simply the statement that an irregular Reeb vector field corresponds to an irrational slope vector in ts , while regular or quasi-regular Reeb vector fields are rational vectors in this Lie algebra. The latter are dense of course. Example 6.7. Our main class of examples will be the weighted homogeneous hypersurface singularities of section 3.4. Thus C(S) is the smooth locus XF \ {o}, with Ω given by (3.10). If g¯ ∈ KCM(C(S), Ω) with Reeb vector field generating the canonical U (1) ⊂ C∗ action given by the corresponding weighted C∗ action, then one can compute

π(|w| − d) n 2d (6.10) vol(S, g) = . w(n − 1)! n The reader should consult section 3.4 for a reminder of the definitions here. One can prove (6.10) either by directly using (6.9), which was done in [11] for the case of well-formed orbifolds (where all orbifold singularities have complex codimension at least 2), or [53] using the methods developed in [75]. We turn now to the related obstruction. Bishop’s theorem [13] implies that for any (2n − 1)-dimensional compact Einstein manifold (S, g) with Ricg = 2(n − 1)g we have (6.11)

vol(S, g) ≤ vol(S 2n−1 , gstandard ).

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In the current set-up the left hand side depends only on the holomorphic structure of the cone C(S) and Reeb vector field ξ. If one picks a Reeb vector field and computes this using, for example, (6.9), and the result violates (6.11), then there cannot be a Ricci-flat K¨ahler cone metric on (C(S), Ω) with ξ as Reeb vector field. Of course, it is not clear a priori that this condition can ever obstruct existence in this way. However, Example 6.7 shows that the condition is not vacuous. Combining (6.10) with (6.11) leads immediately to: Theorem 6.8. The link of a weighted homogeneous hypersurface singularity admits a compatible Sasaki-Einstein structure only if d(|w| − d)n ≤ wnn . It is simple to write down infinitely many examples of weighted homogeneous hypersurface singularities that violate this inequality. For example, take F = z12 +z22 +z32 +z4k . For k odd the link is diffeomorphic to S 5 , while for k even it is diffeomorphic to S 2 × S 3 . The Bishop inequality then obstructs compatible Sasaki-Einstein structures on these links for all k > 20. On the other hand, in [53] we conjectured more generally that for regular Reeb vector fields the Bishop inequality never obstructs. This is equivalent to the following conjecture about smooth Fano manifolds: Conjecture 6.9. Let Z be a smooth Fano manifold of complex dimension n − 1 with Fano index I(Z) ∈ Z>0 . Then   n−1 I(Z) c1 (Z) ≤n c1 (CPn−1 )n−1 = nn , Z

CPn−1

with equality if and only if Z = CPn−1 . This is related to, although slightly different from, a standard conjecture about Fano manifolds. For further details, see [53]. We turn next to another obstruction. To state this, fix a K¨ ahler cone metric g¯ ∈ KCM(C(S), Ω) with Reeb vector field ξ and consider the eigenvalue equation (6.12)

ξ(f ) = iλf.

Here f : C(S) → C is a holomorphic function and we consider λ > 0. The holomorphicity of f implies that f = r λ u where u is a complex-valued homogeneous degree zero function under r∂r , or in other words a complex-valued function on S. Now on a K¨ ahler manifold a holomorphic function is in fact harmonic. That is Δg¯f = 0, where Δg¯ denotes the Laplacian on (C(S), g¯) acting on functions. On the other hand, since g¯ = dr2 +r2 g is a cone we have   Δg¯ = r−2 Δg − r−2n+1 ∂r r2n−1 ∂r ,

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and so (6.12) implies that Δg u = νu, where (6.13)

ν = λ(λ + 2(n − 1)).

In other words, a holomorphic function f on C(S) with definite weight λ, as in (6.12), leads automatically to an eigenfunction of the Laplacian Δg = d∗ d on (S, g) acting on functions. We again appeal to a classical estimate in Riemmanian geometry. Suppose that (S, g) is a compact Einstein manifold with Ricg = 2(n − 1)g. The first non-zero eigenvalue ν1 > 0 of Δg is bounded from below ν1 ≥ 2n − 1. This is Lichnerowicz’s theorem [68]. Moreover, equality holds if and only if (S, g) is isometric to the round sphere (S 2n−1 , gstandard ) [83]. From (6.13) we immediately see that for holomorphic functions f on C(S) of weight λ under ξ, Lichnerowicz’s bound becomes λ ≥ 1. This leads to another potential holomorphic obstruction to the existence of Sasaki-Einstein structures. Again, a priori it is not clear whether or not this will ever serve as an obstruction. In fact for regular Sasakian structures one can prove [53] this condition is always trivial. This follows from the fact that I(Z) ≤ n for any smooth Fano Z of complex dimension n − 1. However, there exist plenty of obstructed quasi-regular examples: Theorem 6.10. The link of a weighted homogeneous hypersurface singularity admits a compatible Sasaki-Einstein structure only if |w| − d ≤ nwmin . Here wmin is the smallest weight. Moreover, this bound can be saturated if and only if (C(S), g¯) is Cn \ {0} with its flat metric. Notice this result is precisely the necessary direction in Theorem 3.10. As another example, consider again the case F = z12 + z22 + z32 + z4k . The coordinate z4 has Reeb weight λ = 6/(k + 2), which obstructs for all k > 4. For k = 4 we have λ = 1, but since in this case the link is diffeomorphic to S 2 × S 3 the Obata result [83] obstructs this marginal case also. In fact these examples are interesting because the compact Lie group SO(3) × U (1) is an automorphism group of the complex cone and acts with cohomogeneity one on the link. The Matsushima result implies this will be the isometry group of any Sasaki-Einstein metric, and then Theorem 4.2 in fact also rules out all k ≥ 3. Indeed, notice that k = 1 corresponds to the round S 5 while k = 2 is the homogeneous Sasaki-Einstein structure on S 2 × S 3 . The reader will find further interesting examples in [53].

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Very recently it has been shown in [33] that for weighted homogeneous hypersurface singularities the Lichnerowicz condition obstructs if the Bishop condition obstructs. More precisely: Theorem 6.11. Let w0 , . . . , wn , d be positive real numbers such that d (|w| − d)n > wnn ,  and d < |w|, where |w| = ni=0 wi , w = ni=0 wi . Then |w| − d > nwmin . In particular, this shows that Conjecture 6.9 is true for smooth Fanos realized as hypersurfaces in weighted projective spaces. As a final comment, we note that more generally the Lichnerowicz obstruction involves holomorphic functions on (C(S), Ω) of small weight with respect to ξ, whereas the Bishop obstruction is a statement about the volume of (S, g), which is determined by the asymptotic growth of holomorphic functions on C(S), analogously to Weyl’s asymptotic formula [11, 53]. 7. Outlook We conclude with some brief comments on open problems in SasakiEinstein geometry. Clearly, one could describe many more. We list the problems in decreasing order of importance (and difficulty). In general, the existence of a K¨ahler-Einstein metric on a Fano manifold is expected to be equivalent to an appropriate notion of stability, in the sense of geometric invariant theory. We discussed this briefly in section 2.4 and described K-stability. The relation between the Futaki invariant and such notions of stability is well-understood. Very recently, in the preprint [90] the Lichnerowicz obstruction of section 6 was related to stability of Fano orbifolds. The stability condition comes from a Kodaira-type embedding into a weighted projective space, as opposed to the Kodaira embedding into projective space used for Fano manifolds. This then leads to a notion of stability under the automorphisms of this weighted projective space. In particular, slope stability leads to some fairly explicit obstructions to the existence of K¨ ahler-Einstein metrics (or more generally constant scalar curvature K¨ ahler metrics) on Fano orbifolds. This includes the Lichnerowicz obstruction as a special case, although the role of the Bishop obstruction is currently rather more mysterious from this point of view. A natural question is how to extend these ideas to Sasaki-Einstein geometry in general. In particular, how should one understand stability for irregular Sasakian structures? Problem 7.1. Develop a theory of stability for Sasakian manifolds that is related to necessary and sufficient conditions for existence of a SasakiEinstein metric. An obvious approach here is to use an idea we have already alluded to in the previous section: one might approximate an irregular Sasakian

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structure using a quasi-regular one, or perhaps more precisely a sequence of quasi-regular Sasakian structures that converge to an irregular Sasakian structure in an appropriate sense. Then one can use the notions of stability for orbifolds developed in [90]. A key difference between the K¨ ahler and Sasakian cases, though, is that one is free to move the Reeb vector field, which in fact changes the K¨ ahler leaf space. Given that the Sasakian description of obstructions to the existence of K¨ahler-Einstein orbifold metrics led to rather simple differentio-geometric descriptions of these obstructions in section 6, one might also anticipate that embedding Sasakian manifolds into spheres might be a beneficial viewpoint. That is, one takes the Sasakian lift of the Kodaira-type embeddings encountered in stability theory. It is just about conceivable that one could classify Sasaki-Einstein manifolds in dimension 5: Problem 7.2. Classify simply-connected Sasaki-Einstein 5-manifolds. We remind the reader that this has been done for regular Sasaki-Einstein 5-manifolds (Theorem 2.6). This includes the homogeneous Sasaki-Einstein 5-manifolds, and moreover cohomogeneity one Sasaki-Einstein manifolds are also classified by Theorem 4.2. The latter two classes are subsets of the toric Sasaki-Einstein manifolds, which are classified by Theorem 5.8. Indeed, recall that the rank of a Sasakian structure is the dimension of the closure of the 1-parameter subgroup of the isometry group generated by the Reeb vector field. If a Sasaki-Einstein 5-manifold has rank 3 then it is toric, and so classified in terms of polytopes by Theorem 5.8. On the other hand, rank 1 are regular and quasi-regular. Is it possible to state necessary and sufficient conditions for the orbifold leaf space of a transversely Fano Sasakian 5-manifold to admit a K¨ ahler-Einstein metric? We remind the reader that those simply-connected spin 5-manifolds that can possibly admit SasakiEinstein structures are listed as a subset of the Smale-Barden classification of such 5-manifolds in [21]; many, but apparently not all, of these can be realized as transversely Fano links of weighted homogeneous hypersurface singularities. The most complete discussion of what is known about existence of quasi-regular Sasaki-Einstein structures in this case appears in [29]. It is rank 2 that is perhaps most problematic. In fact, all known rank 2 Sasaki-Einstein 5-manifolds are toric. This leads to the simpler problem of whether or not there exist rank 2 Sasaki-Einstein 5-manifolds that are not toric. This problem is interesting for the reason that none of the methods for constructing Sasaki-Einstein manifolds described in this paper are capable of producing such an example. Theorem 4.6 gives a fairly explicit local description of K¨ ahler manifolds admiting a Hamiltonian 2-form. Using also Proposition 4.7 one thereby obtains a local classification of Sasaki-Einstein manifolds with a transverse K¨ ahler structure that admits a transverse Hamiltonian 2-form. Via the comments in section 4.3, this implies the existence of a transverse Killing

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Surveys in Differential Geometry XVI

A survey of geometric structure in geometric analysis Shing-Tung Yau∗ The subject of geometric analysis evolves according to our understanding of geometry and analysis. However, one should say that ideas of algebraic geometry and representation theory have been extremely powerful in both global and local geometry. In fact, the spectacular idea of using geometry to understand Diophantine problem has already widened our concept of space. The desire to find suitable geometry to accommodate unified field theory in physics would certainly drastically change the scope of geometry in the near future. In the following lectures, we shall focus on an important branch of geometric analysis: the construction of geometric structures over a given topological space. 1. Part I There are many kinds of geometric structures; most of them can be classified through the theory of groups and their representations. Some of their structures are motivated by physical science. The idea of classifying geometric structures through group theory dated back to the famous Erlangen Program of Felix Klein and the later work of E. Cartan. Most geometric structures are defined by a family of special coordinate charts such that the coordinate transformations or the Jacobian of the coordinate transformations respect some algebraic structure, such as a complex structure, an affine structure, a projective structure or a foliated structure. Special coordinate systems give connections on natural bundles such as the tangent bundles or some bundles construct from tangent bundles. (Projective structure is related to tangent bundle plus the trivial line bundle, for example.) Connections provide ways to covariantly differentiate vector fields along any curve. For any closed loop at a fixed point, parallel transportation ∗ This article is based on lectures given at the University of California at Los Angeles in April 2007.

c 2011 International Press

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along the loop gives rise to a linear transformation of the tangent space at the point to itself. The totality of such transformations forms a group called the holonomy group of the connection. This group reflects the algebraic aspects of the geometric structure. Therefore, a necessary condition for the geometric structure to exist is the existence of a connection with a special holonomy group on some natural bundle. On the other hand, connections give rise to a torsion tensor. In order for the existence of a connection with special holonomy group to become sufficient condition for existence of preferred coordinate systems, we usually require the torsion tensor of the connection to be trivial. In fact, Cartan-K¨ ahler developed an extensive theory of exterior differential systems to provide proofs that, in the real analytic category, existence of a torsion-free connection with special holonomy group is indeed sufficient for the existence of local coordinate systems for most geometric structures. The smooth version of Cartan-K¨ahler theory has not been established in general. The most spectacular work to date was due to Newlander-Nirenberg [49] on the existence of an integrable complex structure, assuming the complex Frobenius condition. In accordance with our previous discussion, Newlander-Nirenberg proved that if the tangent bundle admits a connection with holonomy group U (n) and the torsion form equal to zero, then the manifold admits a complex structure. The paper of Newlander-Nirenberg is the first application of nonlinear partial differential equations to constructing geometric structures. CartanK¨ ahler theory and Newlander-Nirenberg theory are key contributions to the local theory of geometry. The global theory of geometric structures is quite complicated and is far from being completed. Deformation theory of global structure was initiated by Kodaira-Spencer [32]. Calabi-Vesentini [6] and Kuranishi [33] studied deformations of complex structures based on Hodge theory. Calabi, Weil, Borel, Matsushima and others studied deformations of geometric structures on deformation of discrete group which eventually lead to the global rigidity theorems of Mostow [47] and Margulis [45] for locally symmetric spaces. The approaches of using periods of holomorphic forms (Torelli) and geometric invariant theory (Mumford) to study global algebraic structures are very powerful. Geometric invariant theory has modern interpretations in terms of moment maps of group actions on symplectic manifold. Moment maps and symplectic reductions have important consequence on the theory of nonlinear differential equations. The idea is that stability arose from actions of noncompact groups and based on this, I proposed the following point of view. If there is a noncompact group acting behind a system of nonlinear differential equations, the existence question of such system will be related to the question of the stability of some algebraic structure that defines this system of nonlinear equations.

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An important example is the existence of the K¨ ahler-Einstein metric on Fano manifolds where I conjectured [68] to be equivalent to the stability of the algebraic manifolds in the sense of geometric invariant theory. I shall only touch on the part of geometric structures that can be studied by nonlinear differential equations. They are questions that I am fond of. The basic idea is to use nonlinear differential equations to build geometric structures which in turn can be used to solve problems in topology or algebraic geometry. Historically the first global question on geometric structure is the uniformization of conformal structure for domains in the plane. This question dates back to Riemann. It is still an important problem. For instance, we are still trying to understand the structure of moduli space of complex structures over manifolds. For two dimensional domains, the uniformization theorem of conformal structure gives a description of canonical domains which are bounded by circular arcs. Any finitely connected domain must be conformal to such canonical domains. (The moduli space of such canonical domains can be described easily.) On the other hand, we can say that any finitely connected domain admits a conformal metric which is flat and whose boundary has constant geodesic curvature. The question of uniformization is then reduced to proving existence and classifying such conformal metrics. Such differential geometric interpretations of problems in conformal geometry is the approach that we shall follow. For surfaces with higher genus, there are natural conformal metrics that have constant negative curvature. Poincar´e was the first to demonstrate that every metric can be conformally deformed to a unique metric with curvature equal to −1. The construction of the Poincar´e metric has been fundamental in the understanding of the moduli space of Riemann surfaces. The cotangent space of the moduli space are represented by holomorphic quadratic differentials. Using the Poincar´e metric, one can define an inner product among such quadratic differentials and integrate the product over the surface. The resulting metric can be proved to be a K¨ ahler metric called the Weil-Petersson metric. On the Riemann surface, there are simple closed geodesics that will decompose the Riemann surface into a planar domain. The function defined by minus log of the sum of the length of these geodesic defines a convex function along geodesics of the Weil-Petersson geometry. This was observed by Scott Wolpert [66] who used this to re-prove the fact that the universal cover of the moduli space is contractible and is a Stein manifold. However, the moduli space of curves are such important object that their global geometry need to be studied in depth. The recent works of Mumford conjecture due to I. Madsen and M. Weiss [44] is an important example. There are also works on intersection theory of Chern classes of various bundles over the moduli space which has deep algebraic geometric

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meaning. Many of these works such as Witten conjecture, Faber conjecture, etc. are all exciting developments. Holomorphic quadratic differential is very important in classical surface theory. For example, if a map from a Riemann surface into another manifold is (the map is a critical map of the energy), the pulled back metric harmonic hij dxi dxj gives rise to a holomorphic quadratic differential √ h11 − h22 + 2 −1h12 . This well known statement allows one to apply harmonic map to study the geometry of Teichm¨ uller space. Michael Wolf [65] made use of them to give a compactification which is equivalent to the Thurston compactification of the Teichm¨ uller space, which depends on the theory of measured foliation. Another interesting application of holomorphic quadratic differential is to solve the vacuum Einstein equation for spacetime with dimensional two plus one. Given a conformal structure on a Riemann surface and a holomorphic quadratic differential, the Einstein equation gives a path in the cotangent space of the Teichm¨ uller space of the Riemann surface. The Weil-Petersson metric is not complete in general. However, the negative of its Ricci tensor is complete. Liu, Sun and myself [40, 41] proved that it is equivalent to the Teichm¨ uller metric which is obtained by considering extremal quasiconformal maps between Riemann surfaces. It is also equivalent to the canonical K¨ ahler-Einstein metric that I shall discuss later. There has been attempts to find a good representation of Teichm¨ uller space or the moduli space of Riemann surfaces. For genus greater than 23, Harris-Mumford [27] proved that moduli space is of general type. Hence there is no good parametrization of moduli space. Teichm¨ uller space has an embedding into C3g−3 due to Bers [3]. However, it is not explicit and it is not known how smooth the boundary is. If a bounded domain is smooth, the curvature of the canonical K¨ ahler-Einstein metric must be asymptotic to constant negative curvature in a neighborhood of the point where the domain is convex. This was observed by ChengYau [9]. Since the moduli space of Riemann surfaces have a compactification where the divisor at infinity cannot be blown down to a point, the K¨ ahlerEinstein metric cannot be asymptotic to constant negative curvature in any neighborhood. Hence there is no representation of the Teichm¨ uller space as a smooth domain. The question of how to represent a conformal structure on a Riemann surface is quite interesting. Of course one can compute periods of holomorphic differentials over cycles and Torelli theorem asserts that they can determine the conformal structure of a generic surface. However, how to construct the Riemann surface explicitly from the period is not clear. This is especially true if we want to recognize it in R3 . Can we find canonical

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surfaces in three space that represent different conformal structures of the surface? There is another important geometric structure over a two dimensional surface with higher genus. This is the projective structure. They are defined by coordinate neighborhoods whose coordinate transformations are given by projective transformations. There is a map from the universal cover of the surface to RP 2 which preserves the projective structures. If the image is a convex domain, we call the projective structure convex. It turns out that convex projective structures are classified by Riemann surfaces with a cubic holomorphic differentials. Since this classification is a good illustration of how we construct geometric structures, I shall discuss the construction little more detail. Convex projective structure on a manifold has an invariant metric obtained in the following way: The structure is obtained by the quotient of a bounded convex domain Ω in Rn quotiented by a discrete group of projective transformations. A projectively invariant metric on Ω is obtained by solving the following equation      n+2 = − u1 det ∂x∂u i ∂xj u=0

on ∂Ω.

The following metric

 1 ∂2u − dxi ∧ dxj u ∂xi ∂xj

is observed by Loewner and Nirenberg [42] to be invariant under projective transformation. It generalizes the Hilbert model of the Poincar´e disk. Loewner-Nirenberg proved the existence and completeness of the metric for n = 2. The general case was proved by Cheng-Yau [8]. The Ricci curvature of the metric can be proved to be negative [5]. A Legendre transformation will transform  the graph (x, u(x)) to a new convex surface which is an affine sphere . (Affine sphere is a hypersurface where all affine normals converge to a point. Affine normal is a vector transversal to the tangent space invariant under the affine group.) The dis3 crete  group of projective transformation become affine group of R acting on . (This construction was observed by Calabi [5].) The affine metric can be written as ev ds2 where ds2 = eφ | dz |2 is the hyperbolic metric on a Riemann surface. Using the structure equation for affine sphere, C.P. Wang observed [64] that the Pick cubic form in affine geometry is an holomorphic cubic differential Ψdz 3 on the Riemann surface defined by the affine metric so that v + 4 exp(−2v)  Ψ 2 −2 exp(v) − 2K = 0, where K is the Gauss curvature of the conformal metric. (This formulation is essentially due to Tzitzeica [62] in 1908.) Conversely, given a holomorphic

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cubic differential on a Riemann surface, a solution v of the above equation can be used to define an affine sphere in R3 which in turn gives rise to the projective structure. The projective connection is in fact given by

∂v + ∂φ Ψ exp(−v − φ)d¯ z ¯ + ∂φ ¯ Ψ exp(−v − φ)dz ∂v ∂ , ∂∂z¯ }. with respect to the basis { ∂z Hence we have a good classification of convex projective structure over a Riemann surface. In general, there are projective structures which are not convex. Choi has proved that projective structures on surfaces can be uniquely decomposed into several pieces [11, 12]. However, we do not have good understanding of the nonconvex part of the projective structure. The study of the moduli space of convex projective structure on surfaces was due to Hitchin [29], Goldman [18], Labourie [34] and Loftin [43] using different approaches. The above approach relating it to affine spheres was due to Loftin. Compact Riemann surface with higher genus cannot admit affine structures. But open surfaces may admit such a structure. In general, we are interested in affine structures over a compact manifold which may be singular along a codimensional two complex. The coordinate transformations are linear whose Jacobian has determinant equal to one. Motivated by our study of real Monge-Amp`ere equations and K¨ ahler geometry, S.Y. Cheng and I [10] considered in 1979 affine manifolds which may support a metric which we called affine K¨ ahler metric. This is a Riemannian metric which has the property that in each affine chart, there is a convex potential Vα where the metric can be written as

 ∂ 2 uα dxi dxj . ∂xi ∂xj Note that the potentials are well defined up to a linear function. The equation 2

∂ uα det =1 ∂xi ∂xj is well-defined and can be considered as an analogue of the corresponding equation for Calabi-Yau K¨ ahler√metrics. In fact, one simply introduces coordinates yi and define zi = xi + −1yi . Then we can extend uα to be a function on zi and obtain a K¨ ahler metric with zero Ricci curvature. Note that the equation and the affine structures are defined only on the complement of a codimensional two complex. There is a monodromy associated to the equation. The study of existence for the equation with a given monodromy can be considered as an nonlinear analogue of the RiemannHilbert correspondence.

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If the monodromy preserves some lattice structure, we can define a torus bundle over the affine manifold where the total space is a Ricci flat K¨ ahler manifold. Strominger, Zaslow and myself [58] conjectured that for those Calabi-Yau manifolds that admit mirror partner, the total space can be deformed to a complex manifold admitting a (singular) fibration structure, whose fibers are special Lagrangian torus. These are minimal Lagrangian submanifolds and were studied by Harvey and Lawson [28] from different point of view. There is another geometric structure that is of importance in surface theory. This is the line field structure (with singularity) on a surface. An important case is the line field defined by holomorphic quadratic differential and a polynomial vector field. The former case is used by Thurston to form a compactification of the Teichm¨ uller space and the later case is related to the famous Hilbert sixteenth problem which asked the number of limit cycles associated to the vector field. The behavior of the singular points of the line field has practical importance also, e.g., in the study of finger print. The attempts to generalize these structures on Riemann surfaces to higher dimensional manifolds have occupied the activities of geometric analysts in the past thirty years. The fact that there are much more freedom in higher dimensional manifolds mean that there are many different varieties of geometric structures. 2. Part II The concept of geometric structure has been enriched continuously. It has been found that metrics with special holonomy group may not be enough to describe the structure. In order to explain this, I will motivate the idea through the concept of duality in string theory. Let us start with some classical examples. The theory of Lie groups and their discrete subgroups gives rise to Cartan’s theory of locally symmetric and homogeneous spaces. They provide examples with rich properties for geometers and analysts. many important properties of these spaces were obtained when we consider them to be moduli space of other geometric objects. For example, the Siegel upper space can be considered as moduli space of abelian varieties. Occasionally, moduli space of some algebraic manifolds can be locally Hermitian symmetric: Such manifolds include K3-surfaces, Calabi-Yau manifolds obtained by taking branched cover over CP 3 along eight hyperplanes or cubic surfaces. Many hyperK¨ ahler manifolds such as symmetric products of K3 surfaces can be considered as moduli space of semi-stable vector bundles over hyperK¨ahler manifolds. On the other hand, to understand geometric structures, it is important to understand nonlinear transformations between these spaces that are of geometric importance. For example, if H and K are two closed subgroup in a Lie group G, one can construct a natural map from sheaves or cohomology classes of the space G/H to the space G/K by pulling back the objects from

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G/H to G/(H ∩ K). After twisting by some universal object on G/(H ∩ K), one can push the product to objects on G/K: G/(H ∩ K)

LLL LLL LLL L&

rr rrr r r x rr r

G/H

G/K

As was observed by Chern, the classical Kinematic Formulae of Poincar´e, Santalo and Blaschke can be formulated in terms of the above transformation by taking G to be the group of motions on the homogeneous space where incidence relations of submanifolds are considered. This kind of transformations also appeared in many places. A very important one is the case of four dimensional manifold M and we consider the moduli space M of rank two bundles over M whose curvature is selfdual. On the product space M × M, there is a rank two universal bundle V and we can use the second Chern class of V to transform second cohomology of M to M and obtain the Donaldson polynomials. Another important case is the T -duality that has played an important role in number theory and algebraic geometry. Let T n = Rn /Zn be a torus and (T n )∗ = Rn /(Zn )∗ be the dual torus, which can be considered as the moduli space of complex flat line bundles over T n . Then we have the following diagram L 

T n × (T n )∗

Tn

t tt tt t tt y t t

LLL LLL LLL L&

(T n )∗

There is a universal complex line bundle L over T n × (T n )∗ so that L restricted to T n ×{q} is isomorphic to q. We can pull back cohomology classes from T n to T n × (T n )∗ where we multiply the class by exp(c1 (L)). Then we can push the product class to (T n )∗ . Such a transform can be considered as a nonlinear transform between the torus T n and its dual (T n )∗ . It is called T −duality in the recent developments in string theory. Note that when n = 1, this is the duality between circle of radius r to circle of radius 1r . Strominger-Yau-Zaslow [58] found that a certain algebraic manifold M (Calabi-Yau) admits a T 3 fiber structure over S 3 where generic fibers are T 3 . By replacing T 3 by (T 3 )∗ , we obtain another algebraic manifold M ∗ which is also Calabi-Yau.

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By performing a family version of the above T −duality and a Legendre transformation on some affine structures on the base, one obtains a transform that maps one geometric structure over the algebraic manifold M to another geometric structure over M ∗ . (The affine structure on the base space is the one described previously where we have a potential for the metric and the Legendre transform acts on those potentials.) Note M and M ∗ may be topologically distinct. This transformation has many important properties. For example, holomorphic bundles V over M are supposed to be mapped to special Lagrangian cycles C in M ∗ .  In terms of cohomology, the class Ch(V ) T od(M ) in H 0 (M )⊕H 2 (M )⊕ H 4 (M ) ⊕ H 6 (M ) is mapped to cohomology class of [C] in H 3 (M ∗ ). The fact that the algebraic bundles of M are mapped to H 3 (M ∗ ) raise the following question: If M and M ∗ are defined over some number field, will the Frobenius ´ action on the Etale cohomology of H 3 (M ) be mapped to certain action on the K groups defined by algebraic vector bundles? Will the Adams operation play a role? SYZ argued that the above nonlinear transform is the same as the mysterious mirror symmetry that was initiated by Greene-Plesser [19], Candelasde la Ossa-Green-Parkes [7] based on speculations of conformal field theory. Both the analytic and algebraic properties of the mirror transform are spectacular. However, they are not yet well-understood. It would be very useful to understand the above construction to map even dim cohomology of M to odd dim cohomology of M ∗ . On the other hand, it has already produced a powerful method in geometry. For example, it allows algebraic geometers to calculate the number of algebraic curves in a Calabi-Yau manifold. This was a major classical problem in algebraic geometry. It was solved by Candelas et. al., in that they found the right formula. The rigorous mathematical proof came from the works of Liu-Lian-Yau [38] and Givental [17]. In principle, we can extend the above T -duality to a more general situation. For example, T n can be replaced by a K3-surface or other algebraic manifold and (T n )∗ can be replaced by the moduli space of semi-stable holomorphic bundles over that manifold. In this case, L can be replaced by the universal bundle. Gukov-Yau-Zaslow [22] observed that certain manifolds with holonomy group G2 have a fiber structure with fiber given by K3 surfaces and they are dual to algebraic manifolds which are Calabi-Yau. The arguments of SYZ and GYZ are based on brane theory, a quantized version of string theory. The belief that the transformation should work well for fibrations with singularities comes from intuition that arose from physics. Mirror symmetry gives rise to many conjectures in geometry which were proved later by rigorous mathematics. The mathematical proof in turn justifies the intuition of the physicists. Let us now examine how submanifolds can help the construction of geometric structures. It has been an open problem in geometry to construct

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an explicit metric on a K3-surface with holonomy group SU (2). Greene, Shapere, Vafa, and I [20] found an explicit metric (with SU (2) holonomy) on the K3-surface fibered over the two sphere with torus fiber. All the fibers have flat metric. However, our metric is singular along the singular fiber. One can perturb this singular metric to be a smooth one with SU (2) holonomy. The perturbation series is believed to be expressible in terms of areas of holomorphic disks with boundary specified to be a subset of the fiber torus. The motivation comes from the interpretation of our metric as a semi-classical approximation to the quantum theory based on the K3 surface. The holomorphic disks are instanton corrections. There is a similar picture for three-dimensional Calabi-Yau manifolds. In the process of performing the mirror transform, the metric and the complex structure is perturbed by quantities that come from holomorphic cycles or bundles. Hence, it is reasonable to believe that a good geometric structure should include a metric with a certain holonomy group, a space of bundles that have special holonomy group, and a space of cycles such as holomorphic cycles or special Lagrangian cycles. (The Lagrangian that appeared in low energy string theory includes all these quantities and some scalar functions.) Philosophically, we know that certain subspace of functions can determine the space where they are defined. In fact, algebraic geometers use the rings of algebraic functions to determine the algebraic structure of the manifold. Analytically, we can use solutions of differential equations constructed from the metric to determine the geometric structure. Obvious functions are harmonic functions, eigenfunctions, eigenforms or spinors. But there are many naturally defined nonlinear differential operators such as the Monge-Amp`ere operator. Solutions of these nonlinear operators can be directly related to the construction of the metric. The moduli space of self-dual Yang-Mills bundles or Seiberg-Witten equations have been used by Donaldson et. al. to detect the topological structure of the manifold. One expects that more refined properties of geometric structures can be determined by special bundles or special cycles. Intuitions from physics have been very useful. In fact, an ultimate goal of geometry is to find a geometric structure that can describe quantum physics when distance is small and general relativity when distance is large. For such a picture, the classical view of spacetime is expected to be changed drastically. Classical relativity has been verified successfully. The large scale structure of spacetime is therefore in reasonable good shape. However, curvature (or gravity) can drive spacetime to form singularities, which may have to be understood and resolved by quantum physics. The famous conjecture of Penrose says that generic singularity in classical relativity has to be of black hole type.

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Singularities are places where physical laws do not hold. What it means is that classical concept of spacetime is not adequate to describe physics at small scale. For small scale structure of spacetime, quantum field theory has to be brought in and it is likely that all the quantities such as bundles and cycles will contribute. Let me now discuss the approach from the point of view of geometric analysis to construct geometric structures. Two major ways had been developed: one is by gluing structures together and the other one is by calculus of variation or deformation by parabolic equations. Given a smooth manifold, how does one construct geometric structures over such a manifold? Ideally we would like to find necessary and sufficient conditions in terms of algebraic topological data such as homology classes, homotopy groups and characteristic classes of the manifold. This is indeed possible for questions such as the existence of almost complex structure by studying the classifying map of the manifold into the classifying space BU (n). The question is reduced to study the lifting of the map to B(SO(2n)) which classifies the tangent bundle to a map into BU (n). It is a homotopic question and is completely understood when n ≤ 4. BU (n)

u: uu u uu uu  uu / BSO(2n) M

In principle, we can replace U (n) by other Lie subgroups of the orthogonal group in the above discussion. It would be useful to find a necessary and sufficient condition for the existence of G2 -structure on a seven dimensional manifold where the associated three form is closed. The question of existence of geometric structures is very much related to uniqueness. One can of course relax uniqueness to finite dimensionality of the geometric structures. Only in such cases, techniques of elliptic or parabolic theory of differential equations can be useful. Fortunately, most of the geometric structures have this finite dimensionality property. However, it should be pointed out that there can be infinitely many distinct components of complex structures on a fixed compact manifold. It will be useful to classify all the possible Chern classes of such complex structures. Similarly, there may be infinite number of components of symplectic structures on a given compact manifold, all of whose symplectic forms belong to the same cohomology class. The most direct way to construct geometric structures is to perform surgery on manifolds: replacing one handlebody by another handlebody. In the process, one needs to make sure that the new handlebody has compatible geometric structure and the gluing is smooth. The detail of the geometric structure on a manifold with boundary is thus important.

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A beautiful example is Thurston’s approach to constructing hyperbolic metrics on atoroidal irreducible three-manifolds. Thurston found an important generalization of the rigidity theorem of Mostow on hyperbolic manifolds to three-manifolds with geodesic boundary. The hyperbolic structure is determined by its fundamental group and the conformal structure on the boundary. The possibility of gluing two such manifolds is obtained by a fixed point formula on the Teichm¨ uller space. Another example is given by the work of Schoen-Yau [54] and GromovLawson [21] on the classification of manifolds with positive scalar curvature. They prove that surgery on embedded spheres with codimension ≥3 preserves the existence of metrics with positive scalar curvature. (Recently some question was raised on the formula given in [21].) Construction of geometric structures on a manifold by surgery can be powerful, as many tools of algebraic topology can be brought in. However, the gluing procedure usually involves some question of convexity. For example, a ball is convex for most geometric structures, and in order to glue it to another manifold along the boundary, the boundary of the other manifold has to be concave in a suitable manner. However, in conformal geometry, inversion turns the ball inside out. Therefore, one can prove that the connected sum of conformally flat manifolds is still conformally flat. It is much more difficult to glue complex manifolds along a complex submanifold unless the normal bundle of the submanifold is trivial. Even in such cases, it remains to find obstructions to constructing an integrable complex structure on the connected sum of two complex manifolds along the complex submanifold. (If the normal bundle of the complex submanifold is negative, one can perform a contraction and a suitable surgery can be carried out.) The idea of combining methods from geometric analysis and gluing a geometric structure to a given manifold was initiated by the pioneering work of Taubes. He was the first one to construct anti-self dual bundles on four manifolds by gluing the instantons from four spheres to a given four dimensional manifold. This eventually leads to the Donaldson theory, which is the major tool in four manifold theory. In 1992, Taubes [59] was able to perform similar procedure to construct anti-self dual metrics on any four dimensional manifold as long as we glue in enough copies of CP 2 . The twistor space of these manifolds are complex three dimensional manifold fibered over the four manifold with S 2 fibers. Similar technique was later used by Joyce [30, 31] in 1996 to construct seven dimensional manifolds with holonomy group equal to G2 and eight dimensional manifolds with holonomy group equal to Spin(7). The works are all based on singular perturbation method and are very powerful. Unfortunately the perturbation method is not powerful enough to provide the information of the full moduli space of the corresponding structures. And this is the most basic question in order to apply G2 manifolds to M -theory that appeared in string theory.

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Existence and moduli space of affine, projective flat and conformally fiat structure in higher dimension is much more difficult than two dimension, for example, it is not known whether hyperbolic three manifolds admit affine structures. A well-known question whether compact affine manifolds have zero Euler number is not solved. It is known to be true if the connection is complete. Many of the questions are related to developing map from the universal cover of the manifold to Rn , RP n or S n . In general, the map need not be injective. If the map is injective, the manifold with such geometric structure will be equivalent to study of discrete group acting on a domain. Only in one case, we know the developing map is injective. Schoen-Yau [55] proved in 1986, that any conformal map from a complete conformal flat manifold with positive scalar curvature into S n is injective. This property is false without assuming positivity of scalar curvature. Conformally flat manifolds with positive scalar curvature are then quotients of domains in S n by a discrete group of Mobius transformations. The domain is dense in S n with large codimension. When symmetry is imposed, we have much better understanding of the spacetime. In the past twenty years, the most fruitful results have been found for spacetime with supersymmetries. The concept of supersymmetry may not be acceptable to some physicists, but it does provide a beautiful and elegant playing ground for geometers. Many classical questions in geometry were resolved by supersymmetric considerations. A good example is the Seiberg-Witten theory which was motivated by supersymmetric Yang-Mills theory. The invariant created by Seiberg-Witten theory has been very powerful for the study of four manifolds: especially for those four dimensional symplectic four manifolds. In the later case, Taubes proved the deep theorem that creates existence of pseudo-holomorphic curves based on the topological data of SeibergWitten invariants. As a corollary, he proved that there is only one symplectic structure on CP 2 . A.K. Liu [39] was also able to classify all four dimensional symplectic manifolds that support a metric with positive scalar curvature. 3. Part III When the topological method of surgery and gluing fails, we have to find a method that does not depend on the detailed topological information of the manifold. The best example is the proof of the Severi conjecture and the Poincar´e conjecture and the geometrization conjecture. The beauty of the method of nonlinear differential equation is that we can keep on deforming some unknown structure until we can recognize them eventually. The control on this process of deformation depends on careful a priori estimate of the nonlinear equation. However, if the structure is to be changed in a large scale, standard energy estimate usually cannot be used as

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the underlying Sobolov inequality depends on the geometric structure and need not hold in general. Hence maximum principle is used in most cases. A fruitful idea to construct geometric structure is to construct metrics that satisfy the Einstein equation. We demand that the Ricci tensor of the metric be proportional to the metric itself. This can be considered as a generalization of the Poincar´e metric to higher dimensions. This is an elliptic system, if we identify metrics up to diffeomorphisms. The problem of existence of an Einstein metric is really a very difficult but central problem in geometry. One can obtain such metrics by a variational principle: After normalization of the metrics by setting their volume equal to one, we minimize the total scalar curvature in each fixed conformal class; then we vary the conformal class and maximize the (constant) scalar curvature. The first part is called the Yamabe problem and was settled by the works of Trudinger [60], Aubin [2] and Schoen [53]. The most subtle part was the case when the manifold is conformally flat, where Schoen made use of the positive mass conjecture to control the Green’s function of the conformally invariant operator and hence settle this famous analytic problem. The relation of this problem with general relativity is a pleasant surprise and should be considered as an important development in geometric analysis. The second part of maximization among all conformal structure is much more difficult. Schoen and his students, and also M. Anderson have made contributions towards this approach. Let me now discuss the other two major general approaches to constructing Einstein-metrics. The first one is to solve the equation on a space with certain internal symmetries. For such manifolds, the ability to choose a special gauge, such as holomorphic coordinates is very helpful. The space with internal symmetry can be a K¨ ahler manifold or a manifold with special holonomy group. A very important example is given by the Calabi conjecture, where one asked whether the necessary condition for the first Chern class to have definite sign is also sufficient for the existence of K¨ahler-Einstein metric. Algebraic varieties are classified according to the map of the manifold into the complex projective space by powers of the canonical line bundle. If the map is an immersion at generic point, the manifold is called an algebraic manifold of general type. This class of manifolds comprises the majority of algebraic manifolds, and these manifolds can be considered as generalizations of algebraic curves of higher genus. In general, the above canonical map may have a “base point” and hence be singular. However, the minimal model theory of the Italian and Japanese school (Castelnuovo, Fano, Enriques, Severi, Bombieri, Kodaira, Mori, Kawamata, Miyaoka, Inoue) showed that an algebraic manifold of general type can be contracted to a certain minimal model, where the canonical map has no base point. In this case, the first Chern class of the minimal model is non-positive and negative in a Zariski open set.

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Most algebraic manifolds of general type have negative first Chern class. In this case, Aubin and I independently proved the existence and uniqueness of a K¨ahler-Einstein metric. For the general case of minimal models of manifolds of general type, the first Chern class is not negative everywhere. Hence it does not admit a regular K¨ ahler-Einstein metric. However, it admits a canonical K¨ ahler-Einstein metric which may have singularities. This statement was observed by me right after I wrote my paper on the Calabi conjecture, where I also discussed the regularity of degenerate K¨ ahler-Einstein metrics. (Tsuji [61] later reproved this theorem in 1985 using Hamilton’s Ricci flow.) The singularity of this canonical K¨ ahler-Einstein metric that I constructed on manifolds of general type is not so easy to handle. By making some assumption on the divisors, Cheng-Yau and later Tian-Yau contributed to understanding of the structure of these metrics. These metrics give important algebraic geometric informations of the manifolds. In 1976, I [67] observed that the K¨ ahler-Einstein metric can be used to settle important questions in algebraic geometry. An important contribution is the algebraic-geometric characterization of Shimura varieties: quotients of Hermitian symmetric domains by discrete groups. They are characterized by the statement that certain natural bundle, constructed from tensor product of tangent bundles, has nontrivial holomorphic section. The other important assertions are the inequalities between Chern numbers for algebraic manifolds. For an algebraic surface, I proved 3C2 (M ) ≥ C12 (M ), an inequality which was independently proved by Miyaoka by algebraic means. I [67] proved further that equality holds only if M has constant holomorphic sectional curvature. My inequality holds in arbitrary dimension. It is the last assertion that enabled me to prove that there is only one complex structure on the complex projective plane. This statement was a famous conjecture of Severi. The construction of Ricci flat K¨ahler metric has been used extensively in both algebraic geometry and string theory, such as Torelli theorem for K3 surfaces and deformation of complex structure. The construction of K¨ ahler-Einstein metric with positive scalar curvature has been a very active field. In early eighties, I proposed its existence in relation to stability of the manifolds. In the hands of Donaldson, and others, we see that my proposal is close to be realized. It gives new information about the algebraic geometric stability of manifolds. In general, there should be an interesting program to study K¨ ahlerEinstein metrics on the moduli space of either complex structures or stable bundles. It should provide some informations for the moduli space. For example, recently, using this metric, Liu-Sun-Yau [41] proved the Mumford stability of the logarithmic cotangent bundle of the moduli spaces of Riemann surfaces.

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Hence we see that by constructing new geometric structure through nonlinear partial differential equation, one can solve problems in algebraic geometry that are a priori independent of this new geometric structure. A holomorphic coordinate system is a very nice gauge and a K¨ ahler metric is a beautiful metric as it depends only on one function. When we come to the space of Riemannian metrics, we need to understand a large system of nonlinear equations invariant under the group of diffeomorphism. The choice of gauge causes difficulty. The Severi conjecture can be considered as a complex analog of the Poincar´e conjecture. The fact that Einstein metrics were useful in settling the Severi conjecture indicates that these metrics should also be useful for the geometrization conjecture and hence the Poincar´e conjecture. This was what we believed in the late seventies. Many methods motivated by the calculus of variation were proposed. The most promising method was due to Hamilton who proposed to deform any metric along the negative of its Ricci curvature. The development of the Ricci flow has gone through several important stages of development. The first decisive one was Hamilton’s demonstration of the global convergence of the Ricci flow [23] when the initial metric has positive Ricci curvature. This is a fundamental contribution that give confidence on the importance of the equation. To move further, it was immediately clear that one needs to control the singularities of the flow. This was studied extensively by Hamilton. The necessary a priori estimate was based on Hamilton’s spectacular generalization of the works of Li-Yau [37]. Li-Yau introduced a distance function on spacetime to control the precise behavior of the parabolic system near the singularity. The concept appears naturally from the point of view of a priori estimate. For example, if the equation is ∂u = u − V u. ∂t The distance introduced by Li-Yau is given by

1 1 | r˙ |2 d((x, t1 ), (y, t2 )) = inf r 4(t2 − t1 ) 0

1   + (t2 − t1 ) V r(s), (1 − s)t2 + st1 . 0

where V are paths joining (x, t1 ) to (y, t2 ). The kernel of the parabolic equation can then be estimated by this distance function. The potential V is naturally replaced by the scalar curvature in the case of Ricci flow as it appears in the action of gravity. This is what Perelman did later. The idea of Li-Yau-Hamilton come from the careful study of maximum principle. The basic philosophy of LYH is to study the extreme situation. In

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the case of Ricci flow, one looks at the soliton solution and verify some equality holds along the soliton and such equality can be turned to be estimates for general solutions of the parabolic system, via maximum principle. In the nineties, Hamilton [24, 25, 26] was able to classify singularities of the Ricci flow in three dimension and prove the geometrization conjecture if the curvature of the flow is uniformly bounded. These are very deep works both from the point of view of geometry and analysis. Many ideas in geometric analysis were used. This includes the proof of the positive mass conjecture, the injectivity radius estimate and an improved version of the Mostow rigidity theorem. In particular, he introduced the concept of Ricci flow with surgery. In his classification of singularities, Hamilton could not determine the existence or nonexistence of one type of singularity which he called cigar. This type of singularity was proved to be non-existent by Perelman [50] in 2002 in an elegant manner. Perelman [51] then extended the work of Hamilton on flows with surgery. Among many creative ideas, he found a priori estimates for the gradient of the scalar curvature, the concept of reduced volume and a new way to perform surgery with control. The accumulated works of Hamilton-Perelman are spectacular. Today, 5 years after the first preprint of Perelman was available, several groups of mathematicians have put forward their manuscripts explaining their understandings on how Hamilton-Perelman’s ideas can be put together to prove the Poincar´e conjecture; at the same time, other experts are still working diligently on the proof of this century old conjecture. Besides the Poincar´e conjecture, Ricci flow has many other applications: A very important one is the contribution due to Chau, Chen, Ni, Tam and Zhu, towards the proof of the conjecture that every complete noncompact K¨ ahler manifold with positive bisectional curvature is bi-holomorphic to Cn . (I made this conjecture in 1972 as a generalization to higher dimension of the uniformization theorem. Proceeding to the conjecture, there were important works of Greene-Wu to proving Steinness of the complete noncompact K¨ ahler manifold with positive sectional curvature.) More recently, several old problems were solved by using the classical results of Hamilton that were published in 1983, 1986 and 1997. The most outstanding one is the recent result of Brendle-Schoen [4]. They proved that manifolds with pointwise quarter-pinching curvature are diffeomorphic to manifolds with constant positive curvature. This question has puzzled mathematicians for more than half a century. It has been studied by many experts in differential geometry. Back in 1950s, Rauch was the first one who introduce the concept of pinching condition. Berger and Klingenberg proved such a manifold to be homeomorphic to a sphere when it is simply connected. The diffeomorphic type of the manifolds is far more difficult to understand. For example, Gromoll’s thesis achieved a partial result toward settling such a result: he assumed a much stronger pinching condition.

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The result of Brendle-Schoen achieved optimal pinching condition. More remarkably, they only need pointwise pinching condition and do not have to assume simply-connectivity. Both of these conditions are not accessible by the older methods of comparison theorems. This result partially builds on fundamental work by B¨ ohm and Wilking who proved that a manifold with positive curvature operator is diffeomorphic to a spherical space form. Therefore, the program on Ricci flow laid down by Hamilton in 1983 has opened a new era for geometric analysts to build geometric structures. Other obvious problems are to construct geometric structure on other low dimensional manifolds, especially four-dimensional manifolds. Besides the fundamental works based of Donaldson and Seiberg-Witten, we know very little about the geometry of four manifolds. The most fundamental structures on four manifold are complex structures and metrics with antiself-dual curvature. (Most four manifolds are obtained by some simple surgery on complex surfaces. An important operation called log transformation was introduced by Kodaira. It can change the diffeomorphisim type of the four manifold.) The Atiyah-Singer index formula gives very important obstructions for the existence of integrable complex structures on surfaces, as was found by Kodaira. The moduli spaces of holomorphic vector bundles have been a major source for Donaldson to provide invariants for smooth structures. On the other hand, the existence of pseudoholomorphic curves based on SeibergWitten invariant constructed by Taubes is a powerful tool for symplectic topology. It seems natural that one should build geometric structures over a smooth manifold that include all these types of information. The integrability condition derived from Atiyah-Singer formula for almost complex structures in dimC ≥ 3 is not powerful enough to rule out the following conjecture: For dimC ≥ 3, every almost complex manifold admits an integrable complex structure. If this conjecture is true, we need to build geometry over such nonK¨ ahler complex manifolds. This is especially interesting in higher dimension. It is possible to deform an algebraic manifold to another one with different topology by tunneling through nonK¨ ahler structures. A good example is related to the Clemens-Friedman construction that one can collapse rational curves in a Calabi-Yau three manifold to conifold singularity. Then by smoothing the singularity, one obtains nonsingular nonK¨ ahler manifold. Reversing the procedure, one may get another Calabi-Yau manifold. Reid [52] proposed that this procedure may connect all Calabi-Yau manifolds in three dimension. There is perhaps no reason to restrict ourselves only to Calabi-Yau manifolds, but to more general algebraic manifolds on a fixed topological manifold, is there other general construction to deform one algebraic structure from one component of the moduli space of a complex structure to other component through nonK¨ ahler complex structures?

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Non-K¨ ahler complex structures are difficult to handle geometrically. However, there is an interesting concept of Hermitian structure that can be useful. This is the class of balanced structure. An Hermitian metric ω is called balanced iff d(ω n−1 ) = 0. It was first studied by M. Michelsohn [46] and Alessandrini-Bassanelli [1], who observed that twistor space admits a balanced metric and that existence of balanced metric is invariant under birational transformation. Recently, it came up in the theory of Heterotic string, based on a warped product compactification. Strominger [57] suggests that there should be a holomorphic vector bundle that should admit a Hermitian Yang-Mills connection and that there should be Hermitian metric that is conformally balanced. To be more precise, there should be a holomorphic 3-form Ω so that d( Ω ω ω 2 ) = 0, where ω is the Hermitian form. An important link between the bundle and the metric is that connections on both structures give trivial first Chern form √ and the difference between their second Chern forms can be written as ¯ −1∂ ∂ω. This geometric structure constructed for Heterotic string theory is based on construction of parallel spinors and the anomaly equation required by quantization of string theory. On the other hand, general existence theorem for the Strominger system is still not known. An interesting mathematical question is to construct a balanced complex three manifold with a nonvanishing holomorphic 3-form. Then we like to construct a stable holomorphic vector bundle that satisfies all of the above equations of Strominger. Jun Li and I [36] proved the existence of Strominger system by perturbing around the Calabi-Yau metric. The first example on a nonK¨ ahler manifold is due to Fu-Yau [16]. It is obtained by forming a torus fiber bundle over K3 surface (due to DasguptaRajesh-Sethi, Becker-Becker-Dasgupta-Green and Goldstein-Prokushkin). The construction of Strominger system over this manifold can be achieved if we can solve the following complex Monge-Amp`ere equation: Δ(eu −

det ui¯j α −u + μ = 0, f e ) + 4α 2 det gi¯j

where f and μ are given functions on K3 surface S so that f ≥ 0 and S μ = 0. This was achieved by Fu-Yau based on a priori estimates of u, which is more complicated than those used in Calabi conjecture.

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There are nontrivial interpretation of the Fu-Yau example through conformal field theory. The supersymmetric heterotic string gives an SU (3) Hermitian connection on the tangent bundle. But the connection has torsion (which is trace free). If we are interested in G-structure on the tangent bundle with G ⊆ U (n), it can be accomplished by considering the work of Donaldson-Uhlenbeck-Yau [14, 63] on stable holomorphic bundles over K¨ ahler manifold. The work generalizes the work of Narasimhan-Seshadri [48] for algebraic curves. It was extended to nonK¨ ahler manifolds by Li-Yau [35] where the base complex manifold admit a Gauduchon metric ω with ¯ n−1 ) = 0. ∂ ∂(ω If the tangent bundle T is stable and if some irreducible subbundles constructed from tensor product of T admits nontrivial holomorphic section, the structure group can be reduced. The major question is how to control the torsion of this connection by choosing ω suitably. In the other direction, one should mention that Smith-Thomas-Yau [56] succeeded to construct symplectic manifold mirror to the Clemens-Friedman construction. While the Clemens-Friedman [13, 15] construction leads to nonK¨ ahler complex structures over connected sums S 3 × S 3 , the SmithThomas-Yau construction lead to symplectic non-complex structure over connected sums of CP 3 (which may not admit any integrable complex structure). We expect a mirror structure for the Strominger system in symplectic geometry, where we hope to build an almost complex structure compatible to the symplectic form. They should satisfy a good system of equations. We expect that special Lagrangian cycles and pseudoholomorphic curves will play roles in such a new structure which is dual to the above system of equations of Strominger. The inspirations from string theory has given amazingly deep insight into the structure of Calabi-Yau manifolds which are manifolds with holonomy group SU (n). Constructions of geometric structures by coupling metrics with vector bundles and submanifolds should give a new direction in geometry, as they may exhibit supersymmetry. An important idea provided by string theory is that duality exists between supersymmetric manifolds. Duality allows us to compute difficult geometric information by perturbation methods on the dual objects. General relativity and string theory have inspired a great deal of geometric ideas and it has been very fruitful. Nature also tells us everything vibrates and there should be intrinsic frequency associated to our geometric structure. In the classical geometry, we have an elliptic operator associated to deformation of the structure. For space of Einstein metrics, it is called

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the Lichnerowicz operator. It will be interesting to study the spectrum of this operator. Quantum gravity may provide a deeper concept. A successful construction of quantum geometry will change our scope of geometric structures. Pure logical thinking cannot yield us any knowledge of the empirical world. All knowledge of reality starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality. —Einstein (Herbert Spencer lecture at Oxford in 1933)

References [1] L. Alessandrini and G. Bassanelli, Small deformations of a class of compact nonKahler manifolds. Proc. Am. Math. Soc. 109(1990), no. 4, 1059–1062. [2] T. Aubin, The scalar curvature. Differential geometry and relativity (Cahen and Flato, eds.), Reider, 1976. [3] L. Bers, Simultaneous uniformization. Bull. Amer. Math. Soc. 66(1960), 94–97. [4] S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22(2009), 287–307. [5] E. Calabi, Complete affine hyperspheres I. Instituto Nazionale di Alta Matematica Symposia Mathematica, 10(1972), 19–38. [6] E. Calabi and E. Vesentini, On compact, locally symmetric K¨ ahler manifolds. Ann. of Math. (2) 71(1960), 472–507. [7] P. Candelas, de la Ossa, X. C., P. S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Phys. B 359(1991), no. 1, 21–74. [8] S. Y. Cheng and S.-T. Yau, On the regularity of the Monge-Amp`ere equation det(∂ 2 u/∂xi ∂sxj ) = F (x, u). Comm. Pure Appl. Math. 30(1977), no. 1, 41–68. [9] S. Y. Cheng and S.-T. Yau, On the existence of a complete K¨ ahler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Comm. Pure Appl. Math. 33(1980), no. 4, 507–544. [10] S. Y. Cheng and S.-Y. Yau, The real Monge-Amp`ere equation and affine flat structures, in Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, China (1982), Gordon and Breach Science Publishing Company, New York , pp. 339–370. [11] S. Choi, Convex Decompositions of Real Projective Surfaces. I: π-annuli and Convexity, J. Diff. Geom. 40(1994), no. 1, 165–208. [12] S. Choi, Convex Decompositions of Real Projective Surfaces. II: π-annuli and Convexity, J. Diff. Geom. 40(1994), no. 2, 239–283. [13] C. H. Clemens, Double solids. Adv. in Math. 47(1983), no. 2, 107–230. [14] 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. [15] R. Friedman, Simultaneous resolution of threefold double points. Math. Ann. 274(1986), no. 4, 671–689. [16] J. X. Fu and S.-T. Yau, The theory of superstring with flux on non-Kahler manifolds and the complex Monge-Ampere equation. J. Differential Geom. 78(2008), 369–428. arXiv:hep-th/0604063. [17] A. B. Givental, Equivariant Gromov-Witten invariants. Internat. Math. Res. Notices 1996, no. 13, 613–663.

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