Regular and Irregular Holonomic D-Modules (London Mathematical Society Lecture Note Series) 1316613453, 9781316613450

Table of contents :
Contents
Introduction
1 A review on sheaves and D-modules
1.1 Sheaves
1.2 D-modules
2 Indsheaves
2.1 Ind-objects
2.2 Indsheaves
2.3 Ring action
2.4 Sheaves on the subanalytic site
2.5 Some classical sheaves on the subanalytic site
3 Tempered solutions of D-modules
3.1 Tempered de Rham and Sol functors
3.2 Localization along a hypersurface
4 Regular holonomic D-modules
4.1 Regular normal form for holonomic modules
4.2 Real blow up
4.3 Regular Riemann–Hilbert correspondence
4.4 Integral transforms with regular kernels
4.5 Irregular D-modules: an example
5 Indsheaves on bordered spaces
5.1 Bordered spaces
5.2 Operations
6 Enhanced indsheaves
6.1 Tamarkin’s construction
6.2 Convolution products
6.3 Enhanced indsheaves
6.4 Operations on enhanced indsheaves
6.5 Stable objects
6.6 Constructible enhanced indsheaves
6.7 Enhanced indsheaves with ring action
7 Holonomic D-modules
7.1 Exponential D-modules
7.2 Enhanced tempered holomorphic functions
7.3 Enhanced de Rham and Sol functors
7.4 Ordinary linear differential equations and Stokes phenomena
7.5 Normal form
7.6 Enhanced de Rham functor on the real blow up
7.7 De Rham functor: constructibility and duality
7.8 Enhanced Riemann–Hilbert correspondence
8 Integral transforms
8.1 Integral transforms with irregular kernels
8.2 Enhanced Fourier–Sato transform
8.3 Laplace transform
References
Notations
Index

Citation preview

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid Mathematics Institute University of Warwick, United Kingdom The titles below are available from booksellers or from Cambridge University Press at www.cambridge.org/mathematics 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370

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London Mathematical Society Lecture Note Series: 433

Regular and Irregular Holonomic D-Modules M A S A K I K A S H I WA R A Research Institute for Mathematical Sciences, Kyoto University PIERRE SCHAPIRA Pierre and Marie Curie University, Paris

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781316613450 © Masaki Kashiwara and Pierre Schapira 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication Data Names: Kashiwara, Masaki, 1947– | Schapira, Pierre, 1943– Title: Regular and irregular holonomic D-modules / Masaki Kashiwara, RIMS, Kyoto University, Japan, Pierre Schapira, University Pierre et Marie Curie, Paris. Description: Cambridge : Cambridge University Press, 2016. | Series: London Mathematical Society lecture note series ; 433 | Includes bibliographical references and index. Identifiers: LCCN 2016011164 | ISBN 9781316613450 (pbk. : alk. paper) Subjects: LCSH: D-modules. | Modules (Algebra) | Sheaf theory. | Geometry, Algebraic. Classification: LCC QA247.3 .K385 2016 | DDC 514/.23–dc23 LC record available at https://lccn.loc.gov/2016011164 ISBN 978-1-316-61345-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication, and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents

Introduction

page 1

1 1.1 1.2

A review on sheaves and D-modules Sheaves D-modules

2 2.1 2.2 2.3 2.4 2.5

Indsheaves Ind-objects Indsheaves Ring action Sheaves on the subanalytic site Some classical sheaves on the subanalytic site

18 18 20 23 26 29

3 3.1 3.2

Tempered solutions of D-modules Tempered de Rham and Sol functors Localization along a hypersurface

39 39 42

4 4.1 4.2 4.3 4.4 4.5

Regular holonomic D-modules Regular normal form for holonomic modules Real blow up Regular Riemann–Hilbert correspondence Integral transforms with regular kernels Irregular D-modules: an example

44 44 48 51 55 56

5 5.1 5.2

Indsheaves on bordered spaces Bordered spaces Operations

58 58 60

6 6.1 6.2

Enhanced indsheaves Tamarkin’s construction Convolution products

63 63 64 v

7 7 9

vi

Contents

6.3 6.4 6.5 6.6 6.7

Enhanced indsheaves Operations on enhanced indsheaves Stable objects Constructible enhanced indsheaves Enhanced indsheaves with ring action

67 69 72 74 77

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Holonomic D-modules Exponential D-modules Enhanced tempered holomorphic functions Enhanced de Rham and Sol functors Ordinary linear differential equations and Stokes phenomena Normal form Enhanced de Rham functor on the real blow up De Rham functor: constructibility and duality Enhanced Riemann–Hilbert correspondence

79 79 80 81 82 86 88 89 94

8 8.1 8.2 8.3

Integral transforms Integral transforms with irregular kernels Enhanced Fourier–Sato transform Laplace transform

97 97 98 100

References Notations Index

104 107 110

Introduction

This book develops the contents of a series of lectures given at the Institut des ´ Hautes Etudes Scientifiques in February and March 2015 (see [KS 15]), based on [Ka 84], [DK 13] and [KS 14]. They are addressed to readers familiar with the language of sheaves and D-modules, in the derived sense. As announced in the title, the subject of this book is holonomic D-modules. The theory of D-modules appeared in the 1970s with the thesis of [Ka 70] and Bernstein’s paper [Be 71]. However, already in the 1960s, Mikio Sato had the main ideas of the theory in mind and gave talks at Tokyo University on these topics. Unfortunately, Sato did not write anything and it seems that his ideas were not understood at this time. (See [An 07, Sc 07].) A left coherent DX -module on a complex manifold X is locally represented by (the cokernel of ) a matrix of differential operators acting on the right. Hence, D-module theory is essentially the algebraic study of systems of linear partial differential equations. It seems that algebraic geometers were frightened by the non-commutative nature of the sheaf of rings DX , and it may be the reason why one had to wait untill the 1970s until the theory appeared. But once one realizes that the ring DX has a natural filtration (by the order of the operators) and that the associated graded ring is commutative, it is not too difficult to apply the tools of algebraic geometry to this non-commutative setting. In particular, one can define the characteristic variety char(M ) of a coherent DX -module, a closed C× -conic complex analytic subset of the cotangent bundle T ∗ X and a fundamental result of the theory is that this variety is coisotropic (or involutive). Partial results in this direction (involutivity at generic points) were first obtained by Guillemin, Quillen, and Sternberg [GQS 70]. The general case was obtained later by Sato, Kawai, and Kashiwara [SKK 73], using tools of microlocal analysis such as microdifferential operators of infinite order. Then Gabber proposed a purely algebraic proof of this result in [Ga 81],

1

2

Introduction

and there is also now another totally different proof based on the involutivity of the microsupport of sheaves on a real manifold (see [KS90]). Once one knows that the characteristic variety of a coherent DX -module is co-isotropic, it is natural to study with a special attention those modules whose characteristic variety is as small as possible, that is, Lagrangian, and these are the holonomic D-modules. They were first called “maximally overdetermined systems” in [SKK 73], and they are the natural generalization in higher dimension of the classical theory of ordinary differential equations. An ordinary differential equation may also be regarded as a connection with poles, and among them, there are the connections with regular singularities or, equivalently, the Fuchsian differential operators. In this framework, the Riemann–Hilbert problem is, roughly speaking, to construct a Fuchsian operator on a Riemann surface when the monodromy of its holomorphic solutions is prescribed. A natural question is to generalize the theory of Fuchsian equations to higher dimensions. A first important step is the book of Deligne [De 70], in which he solves the Riemann–Hilbert problem for regular connections with singularities on hypersurfaces. A second important step is the constructibility theorem of [Ka 75], which asserts that the functor “holomorphic solutions” sends the derived category of holonomic DX -modules to that of constructible sheaves on X. More precisely, denote by Dbhol (DX ) the bounded derived category of left DX -modules with holonomic cohomology and by DbC-c (CX ) the bounded derived category of sheaves of C-vector spaces with constructible cohomologies. Then it is proved in [Ka 75] that the (contravariant) functor SolX ( • ) = RH om D ( • , OX ), when restricted to Dbhol (DX ), takes its values in DbC-c (CX ). It is also noticed in this paper that if an object of DbC-c (CX ) is in the image of the abelian category Modhol (DX ) of holonomic DX -modules, then it satisfies the properties which are now called the perversity conditions. → DbC-c (CX ) is not It is well known that the functor SolX : Dbhol (DX )op − 1 faithful. For example, if X = A (C), the complex line with coordinate t, P = t2 ∂t − 1 and Q = t2 ∂t + t, then the two holonomic DX -modules DX /DX P and DX /DX Q have the same sheaves of solutions. Hence, a natural question is to look for a full triangulated category of Dbhol (DX ) on which SolX is fully faithful and induces an equivalence with DbC-c (CX ). A precise formulation was formulated in 1977 by the same author (see [Ra 78, p. 287]), and a detailed sketch of proof of the theorem establishing this correspondence (in the regular case) appeared in [Ka 80] where the functor Thom of tempered cohomology was introduced; a detailed proof appears in [Ka 84]. Many tools used in the

Introduction

3

proof of this result were first elaborated in [KK 81]. Note that a quite different proof to this correspondence was obtained by Mebkhout in [Me 84]. The functor Thom is thus an essential tool in the original proof of the regular Riemann–Hilbert correspondence. Its functorial properties as well as w

the construction of the Whitney tensor product ⊗, a kind of “dual functor” of Thom, are systematically studied in [KS 96]. These two functors are in fact better understood in the language of indsheaves of [KS 01]. They correspond to the indsheaves OXt and OXw of tempered holomorphic functions and Whitney holomorphic functions. For example, OXt is constructed as the Dolbeault complex with tempered distributions as coefficients. Of course, the presheaf of tempered distributions (on a real analytic manifold) is not a sheaf for the usual topology, but it becomes a sheaf for a suitable Grothendieck topology, called the subanalytic topology, and one can naturally embed the category of subanalytic sheaves in that of indsheaves. Already, in early 2000, it became clear that the indsheaf OXt is an essential tool for the study of irregular holonomic modules. A toy model was studied in [KS 03], where the indsheaf of tempered holomorphic solutions of the ordinary differential operator t2 ∂t + 1 is calculated. However, on X = A1 (C), the two holonomic DX -modules DX exp(1/t) and DX exp(2/t) have the same tempered holomorphic solutions, which shows that OXt is not precise enough to treat irregular holonomic D-modules. This difficulty is overcome in [DK 13] by adding an extra variable in order to capture the growth at singular points. This is done first by adapting to indsheaves a construction of Tamarkin [Ta 08], leading to the notion of “enhanced indsheaves”, then by defining the “enhanced indsheaf of tempered holomorphic functions”. Using fundamental results of Mochizuki [Mo 09, Mo 11] (see also Sabbah [Sa 00] for preliminary results and see Kedlaya [Ke 10, Ke 11] for the analytic case), this leads to the solution of the Riemann–Hilbert correspondence for (not necessarily regular) holonomic D-modules. First, we shall recall the main results of the theory of indsheaves and subanalytic sheaves and we shall explain with some detail the operations on D-modules and their tempered holomorphic solutions. As an application, we obtain the Riemann–Hilbert correspondence for regular holonomic D-modules as well as the fact that the de Rham functor commutes with integral transforms. Second, we do the same for the sheaf of enhanced tempered solutions of (no longer necessarily regular) holonomic D-modules. For that purpose, we first recall the main results of the theory of indsheaves on bordered spaces and its enhanced version. Let us describe with some details the contents of this book.

4

Introduction

Chapter 1 is a review on the theory of sheaves and D-modules. Sheaf theory is now so classical that it does not seem necessary to recall it, and our aim is essentially to establish the notation and to recall the main formulas of constant use. Reference for this subject is made to [KS90]. On the other hand, D-module theory is not so well known. Our presentation of the subject here may be considered as an invitation to the reading of [Ka 03]. In Chapter 2, extracted from [KS 96, KS 01], we briefly describe the category of indsheaves on a locally compact space and the six operations on indsheaves. A method for constructing indsheaves on a subanalytic space is the use of the subanalytic Grothendieck topology, a topology for which the open sets are the open relatively compact subanalytic subsets and the coverings are the finite coverings. On a real analytic manifold M, this allows us to construct the indsheaves of Whitney functions, tempered C∞ -functions and tempered distributions. On a complex manifold X, by taking the Dolbeault complexes with such coefficients, we obtain the indsheaf (in the derived sense) OXw of Whitney holomorphic functions and the indsheaf OXt of tempered holomorphic functions. Then, in Chapter 3, also extracted from [KS 96, KS 01], we study the tempered de Rham and Sol (“Sol” for solutions) functors; that is, we study these functors with values in the sheaf of tempered holomorphic functions. We prove two main results which will be the main tools to treat the regular Riemann–Hilbert correspondence later. The first one is Theorem 3.1.1, which calculates the inverse image of the tempered de Rham complex. It is a reformulation of a theorem of [Ka 84], a vast generalization of the famous Grothendieck theorem on the de Rham cohomology of algebraic varieties. The second result, Theorem 3.1.5, is a tempered version of the Grauert direct image theorem. In Chapter 4, we give a proof of the main theorem of [Ka 80, Ka 84] on the Riemann–Hilbert correspondence for regular holonomic D-modules (see Corollary 4.3.4). Our proof is based on Lemma 4.1.9, which essentially claims that to prove that regular holonomic D-modules have a certain property, it is enough to check that this property is stable by projective direct images and is satisfied by modules of “regular normal forms”, that is, modules associated with equations of the type zi ∂zi − λi or ∂zj . The Riemann–Hilbert correspondence as formulated in [Ka80, Ka84] is not enough to treat integral transform, and we have to prove a “tempered” version of it (Theorem 4.3.2). We then collect all results on the tempered solutions of D-modules in a single formula which, roughly speaking, asserts that the tempered de Rham functor commutes with integral transforms whose kernel is regular holonomic (Theorem 4.4.2). We end this chapter with a detailed study of the irregular

Introduction

5

holonomic D-module DX · exp(1/z) on A1 (C), following [KS 03]. This case shows that the solution functor with values in the indsheaf OXt gives much information on the holonomic D-modules, but not enough: it is not fully faithful. As seen in the next chapters, in order to treat irregular case, we need the enhanced version of the setting discussed in this chapter. Chapter 5, extracted from [DK 13], treats indsheaves on bordered spaces. A  of good topological spaces with M ⊂ M  an bordered space is a pair (M, M)  is the quotient open embedding. The derived category of indsheaves on (M, M)  by that of indsheaves on M  \ M. Indeed, of the category of indsheaves on M contrary to the case of usual sheaves, this quotient is not equivalent to the derived category of indsheaves on M. The main way of treating the irregular Riemann–Hilbert correspondence is to replace the indsheaf OXt with an enhanced version, the object OXE . Roughly speaking, this object (which is no longer an indsheaf) is obtained as the image t , in a suitable of the complex of solutions of the operator ∂t − 1 acting on OX×C category, namely that of enhanced indsheaves. Chapter 6, also extracted from [DK 13], defines and studies the triangulated category Eb (IkM ) of enhanced indsheaves on M, adapting to indsheaves a construction of Tamarkin [Ta 08]. Denoting by R∞ the bordered space (R, R) in which R is the two-point compactification of R, the category Eb (IkM ) is the quotient of the category of indsheaves on M × R∞ by the subcategory of indsheaves which are isomorphic to the inverse image of indsheaves on M. Chapter 7, mainly extracted from [DK 13], treats the irregular Riemann– Hilbert correspondence. Similarly as in the regular case, an essential tool is Lemma 7.5.5, which asserts that to prove that holonomic D-modules have a certain property, it is enough to check that this property is stable by projective direct images and is satisfied by modules of “normal forms”, that is, D-modules of the type DX · exp ϕ where ϕ is a meromorphic function. This lemma follows directly from the fundamental results of Mochizuki [Mo 09, Mo 11] (in the algebraic setting) and later Kedlaya [Ke 10, Ke 11] in the analytic case, after preliminary results by Sabbah [Sa 00]. The proof of the irregular Riemann– Hilbert correspondence is rather intricate and uses enhanced constructible sheaves and a duality result between the enhanced solution functor and the enhanced de Rham functor. However, as formulated in [DK 13], this theorem is not enough to treat irregular integral transform and we have to prove an “enhanced” version of it (Theorem 7.8.1, extracted from [KS 14]). In Chapter 8, extracted from [KS 14], we apply the preceding results. The main formula (8.1.4) asserts, roughly speaking, that the enhanced de Rham

6

Introduction

functor commutes with integral transforms with irregular kernels. In a previous paper [KS 97] we had already proved (without the machinery of enhanced indsheaves) that given a complex vector space V, the Laplace transform induces an isomorphism of the Fourier–Sato transform of the conic sheaf associated with OVt with the similar sheaf on V∗ (up to a shift). We obtain here a similar result in a non-conic setting, replacing OVt with its enhanced version OVE . For that purpose, we extend first the Tamarkin non-conic FourierSato transform to the enhanced setting. Comments. As already mentioned, most of the results discussed here are already known. We sometimes do not give proofs or give only a sketch of the proof. However, Theorems 2.5.13 and 6.6.4 and Corollaries 2.5.15 and 7.7.2, proving the R-constructibility of tempered and Whitney holomorphic solutions of (irregular) holonomic D-modules, are new.

1 A review on sheaves and D-modules

As already mentioned in the Introduction, we assume the reader is familiar with the language of sheaves and D-modules, in the derived sense. Hence, the aim of this chapter is mainly to establish some notation.

1.1 Sheaves We refer to [KS90] for all notions of sheaf theory used here. For simplicity, we denote by k a field, although most of the results would remain true when k is a commutative ring of finite global dimension. A topological space is good if it is Hausdorff, locally compact, countable at infinity and has finite flabby dimension. Let M be such a space. For a subset A ⊂ M, we denote by A its closure and Int(A) its interior. One denotes by Mod(kM ) the abelian category of sheaves of k-modules on M and by Db (kM ) its bounded derived category. Note that Mod(kM ) has a finite homological dimension. For a locally closed subset A of M, one denotes by kA the constant sheaf on A with stalk k extended by 0 on X \ A. For F ∈ Db (kM ), one sets FA := F ⊗kA . One denotes by Supp(F) the support of F. We shall make use of the dualizing complex on M, denoted by ωM , and the duality functors DM := RH om ( • , kM ),

DM := RH om ( • , ωM ).

(1.1.1)

Recall that, when M is a real manifold, ωM is isomorphic to the orientation sheaf shifted by the dimension.

7

8

A review on sheaves and D-modules

We have the two internal operations of internal hom and tensor product: RH om ( • , • ) : Db (kM )op × Db (kM ) − → Db (kM ), •

⊗

•

: Db (kM ) × Db (kM ) − → Db (kM ).

Hence, Db (kM ) has a structure of commutative tensor category with kM as unit object, and RH om is the inner hom of this tensor category. Now let f : M − → N be a morphism of good topological spaces. One has the functors → Db (kM ) inverse image, f −1 : Db (kN ) − f ! : Db (kN ) − → Db (kM ) extraordinary inverse image, R f ∗ : Db (kM ) − → Db (kN ) direct image, R f ! : Db (kM ) − → Db (kN ) proper direct image. We get the pairs of adjoint functors (f −1 , R f ∗ ) and (R f ! , f ! ). The operations associated with the functors ⊗, RH om , f −1 , f ! , R f ∗ , and R f ! are called Grothendieck’s six operations. For two topological spaces M and N, one defines the functor of external tensor product •

 • : Db (kM ) × Db (kN ) − → Db (kM×N )

−1 by setting F  G := q−1 1 F ⊗ q2 G, where q1 and q2 are the projections from M × N to M and N, respectively. → Denote by pt the topological space with a single element and by aM : M − pt the unique morphism. One has the isomorphism

kM a−1 M kpt ,

! ωM aM kpt .

There are many important formulas relying on the six operations. In particular we have the formulas below in which F, F1 , F2 ∈ Db (kM ), G, G1 , G2 ∈ Db (kN ):   RH om (F ⊗ F1 , F2 ) RH om F, RH om (F1 , F2 ) , R f ∗ RH om (f −1 G, F) RH om (G, R f ∗ F), R f ! (F ⊗ f −1 G) (R f ! F) ⊗ G !

f RH om (G1 , G2 ) RH om (f

(projection formula), −1

G1 , f ! G2 ),

1.2 D-modules

9

and for a Cartesian square of good topological spaces, M g

 M

f

/ N

 f

g

(1.1.2)

 /N

we have the base change formulas for sheaves g−1 R f ! Rf! g−1

and

g ! R f ∗ Rf∗ g ! .

In this book, we shall also encounter R-constructible sheaves. References are made to [KS90, ch. 8]. Let M be a real analytic manifold. On M there is the family of subanalytic sets due to Hironaka and Gabrielov (see [BM 88, VD 98] for an exposition). This family is stable by all usual operations (finite intersection and locally finite union, complement, closure, interior) and contains the family of semi-analytic sets (those locally defined by analytic inequalities). If f:M− → N is a morphism of real analytic manifolds, then the inverse image of a subanalytic set is subanalytic. If Z is subanalytic in M and f is proper on the closure of Z, then f (Z) is subanalytic in N. A sheaf F is R-constructible if there exists a subanalytic stratification M =  j∈J Mj such that for each j ∈ J, the sheaf F|Mj is locally constant of finite rank. One defines the category DbR-c (kM ) as the full subcategory of Db (kM ) consisting of objects F such that H i (F) is R-constructible for all i ∈ Z and one proves that this category is triangulated. The category DbR-c (kM ) is stable by the usual internal operations (tensor product, internal hom), and the duality functors in (1.1.1) induce antiequivalences on this category. If f : M − → N is a morphism of real analytic manifolds, then f −1 and f ! send R-constructible objects to R-constructible objects. If F ∈ DbR-c (kM ) and f is proper on Supp(F), then R f ! F ∈ DbR-c (kN ).

1.2 D-modules References for D-module theory are made to [Ka 03]. See also [Ka 70, Bj 93, HTT 08]. Here, we shall briefly recall some basic constructions in the theory of D-modules that we shall use. Note that many classical functors that shall appear in this chapter will be extended to indsheaves in Chapter 3 and the subsequent chapters. In this section, the base field is the complex number field C.

10

A review on sheaves and D-modules

1.2.1 Basic constructions Let (X, OX ) be a complex manifold. We denote as usual by • dX the complex dimension of X, • X the invertible sheaf of differential forms of top degree, ⊗−1 −1 • X/Y the invertible OX -module X ⊗f −1 OY f (Y ) for a morphism f:X− → Y of complex manifolds, the sheaf of holomorphic vector fields,  X • the sheaf of algebras of finite-order differential operators, the subring of D • X Hom (OX , OX ) generated by OX and X . Denote by Mod(DX ) the abelian category of left DX -modules and by op Mod(DX ) that of right DX -modules. There is an equivalence op r : Mod(DX ) −∼ → Mod(DX ),

M → M r := X ⊗OX M . (1.2.1)

By this equivalence, it is enough to study left DX -modules.

1.2.2 Filtrations and characteristic variety The ring DX is endowed with the filtration by the order. Denoting by F DX this filtered ring, Fm DX is the sheaf of differential operators of order ≤ m. One can also define this filtration by F−1 DX = {0}, Note that 

Fm DX = {P ∈ DX ; [P, OX ] ∈ Fm−1 DX }.

F 0 D X = OX , F 1 D X = OX ⊕  X , (1.2.2) Fm DX · Fl DX ⊂ Fm+l DX , [Fm DX , Fl DX ] ⊂ Fm+l−1 DX .

We denote by gr DX the associated graded ring:  gr DX = Fi DX / Fi−1 DX , i

→ gr DX the “principal symbol map” and by σm : Fm DX − → by σ : F DX − gr m DX the map “symbol of order m.” The ring gr DX is a commutative graded ring. Moreover, gr 0 DX OX and gr 1 DX X . Denote by SOX (X ) the symmetric OX -algebra associated with the locally free OX -module X . By the universal property of symmetric algebras, the → gr DX may be extended to a morphism of symmetric algebras morphism X − SOX (X ) − → gr DX , and one easily proves that the morphism (1.2.3) is an isomorphism.

(1.2.3)

1.2 D-modules

11

Denote by π : T ∗ X − → X the projection. There is a natural monomorphism X → π∗ OT ∗ X . Indeed, a vector field on X is a section of the tangent bundle TX, hence defines a linear function on T ∗ X. By the universal property of symmetric algebras, we get a monomorphism SOX (X ) → π∗ OT ∗ X and thus an embedding of CX algebras: gr DX → π∗ OT ∗ X . The ring gr DX is coherent, and one easily deduces that the ring DX is (right and left) coherent. One denotes by Modcoh (DX ) the thick abelian subcategory of Mod(DX ) consisting of coherent modules. A good filtration on a coherent DX -module M is a filtration which is locally the image of a finite free filtration. Hence, a filtration F M on M is good if and only if ⎧ ⎨locally on X, Fj M = 0 for j  0, (1.2.4) F M is OX -coherent, ⎩ j locally on X, (Fk DX ) · (Fj M ) = Fk+j M for j  0 and all k ≥ 0. One proves that for a coherent DX -module M endowed with a good filtration F M , the induced filtration on a coherent submodule N of M , defined by Fj N = N ∩ Fj M , is good. Definition 1.2.1. The characteristic variety of M , denoted char(M ), is the closed subset of T ∗ X characterized as follows: for any open subset U of X such that M |U is endowed with a good filtration, char(M )|T ∗ U is the support of gr M |U . Note that the support of gr M does not depend on the choice of a good filtration on M . Theorem 1.2.2. (i) char(M ) is a closed C× -conic analytic subset of T ∗ X. (ii) char(M ) is co-isotropic (one also says involutive ) for the symplectic structure of T ∗ X, and in particular, codim(char(M )) ≤ dX . →M − → M  − → 0 is an exact sequence of coherent DX (iii) If 0 − → M − modules, then char(M ) = char(M  ) ∪ char(M  ). The involutivity property is a central theorem of the theory and is due to [SKK 73]. A purely algebraic proof was obtained later in [Ga 81].

12

A review on sheaves and D-modules

If char(M ) is Lagrangian, M is called holonomic. It follows immediately from Theorem 1.2.2 that the full subcategory Modhol (DX ) of Modcoh (DX ) consisting of holonomic D-modules is a thick abelian subcategory. A DX -module M is quasi-good if, for any relatively compact open subset U ⊂ X, M |U is a sum of coherent (OX |U )-submodules. A DX -module M is good if it is quasi-good and coherent. The subcategories of Mod(DX ) consisting of quasi-good (resp. good) DX -modules are abelian and thick.

1.2.3 Internal operations Let us denote by Db (DX ) the bounded derived category of Mod(DX ). One has the triangulated categories  b b j • Dcoh (DX ) =  M ∈ D (DX ) ; H (M ) is coherent for all j ∈ Z , b b j • Dhol (DX ) = M ∈ D (DX ) ; H (M ) is holonomic for all j ∈ Z , b b j • Dq-good (DX ) = M ∈ D (DX ) ; H (M ) is quasi-good for all j ∈ Z ,  b b j • Dgood (DX ) = M ∈ D (DX ) ; H (M ) is good for all j ∈ Z . One may also consider the unbounded derived categories D(DX ), D− (DX ) and D+ (DX ) and the full triangulated subcategories consisting of coherent, holonomic, quasi-good and good modules. We have the functors → D+ (CX ), RH om DX ( • , • ) : Db (DX )op × Db (DX ) − •

L

⊗DX

•

: Db (DX ) × Db (DX ) − → D− (CX ). op

We also have the functors D

•

⊗

•

⊗

D

•

: D− (DX ) × D− (DX ) − → D− (DX ),

•

: D− (DX ) × D− (DX ) − → D− (DX ). op

op

constructed as follows. The (DX , DX ⊗DX )-bimodule structure on DX ⊗OX DX gives M ⊗OX N (DX ⊗OX DX ) ⊗DX ⊗DX (M ⊗ N )

(1.2.5)

the structure of a DX -module for M and N two left DX -modules. One denotes D

by M ⊗ N the OX -module M ⊗OX N endowed with the structure of a left DX -module given by (1.2.5). There is a similar construction when N is a right DX -module.

1.2 D-modules

13

One defines the duality functor for D-modules by setting DX M = RH om DX (M , DX ⊗OX ⊗−1 )[dX ] ∈ Db (DX ) X op

DX N = RH om D op (N , X ⊗OX DX )[dX ] ∈ Db (DX ) X

for M ∈ Db (DX ), op

for N ∈ Db (DX ).

1.2.4 De Rham and Spencer complexes Recall first the classical de Rham complex •

d

d

d

→ 0X −→ 1X −→ · · · −→ dXX − → 0, DRX (OX ) := 0 −

(1.2.6)

where d is the differential. → 1X ⊗O Let M be a left DX -module. One defines the differential d : M − M as follows. In a local coordinate system (x1 , . . . , xdX ) on X, the differential d is given by

dxi ⊗ ∂i m M − → 1X ⊗O M , m → i

and one checks easily that this does not depend on the choice of the local coordinate system. One then defines the de Rham complex of M , denoted by • DRX (M ), as the complex •

DRX (M ) :=

d

0− → 0X ⊗O M −→ · · · − → dXX ⊗O M − → 0,

(1.2.7)

where dXX ⊗O M is in degree 0 and the differential d is characterized by d(ω ⊗ m) = dω ⊗ m + (−)p ω ∧ dm, •

p

ω ∈ X , m ∈ M .

op

Note that DRX (DX ) ∈ Cb (Mod(DX )), the category of bounded complexes of right DX -modules, and •

•

DRX (M ) DRX (DX ) ⊗D M .

(1.2.8)

The right DX -module structure on X induces a natural right DX -linear → X . Moreover, one checks easily that the composition morphism X ⊗O DX − → dXX ⊗O DX − → X XdX −1 ⊗O DX − op

is zero. Hence, we get a morphism in Cb (Mod(DX )) •

DRX (DX ) − → X ,

(1.2.9)

and one checks, using the classical tool of Koszul complexes, that this morphism is a quasi-isomorphism.

14

A review on sheaves and D-modules •

Notation 1.2.3. We denote by DRX (M ) the image of DRX (M ) in the op derived category Db (DX ). •

op

Since DRX (DX ) is a flat resolution of the DX -module X , we deduce that for a left coherent DX -module M , L

DRX (M ) X ⊗D M

in Db (CX ).

Let us apply the contravariant functor Hom D op ( • , DX ) [dX ] to the complex X • DRX (DX ). One sets •

•

SpX (OX ) := Hom D op (DRX (DX ), DX ) [dX ] X

(1.2.10)

•

and calls SpX (OX ) the Spencer complex of OX . More explicitly, we have •

SpX (OX ) := 0 − → DX ⊗O

dX 

d

X −→ · · · − → DX ⊗O X − → DX − → 0, (1.2.11)

where DX is in degree 0. Note that the morphism DX − → OX extends to a morphism of complexes •

→ OX , SpX (OX ) −

(1.2.12) •

and this last morphism is a quasi-isomorphism. Hence SpX (OX ) is a resolution of OX by locally free DX -modules. Let M be a left DX -module. One sets •

D

•

SpX (M ) := SpX (OX ) ⊗ M .

(1.2.13)

•

Therefore, SpX (M ) is the complex 0− → DX ⊗O

dx 

d

d

X ⊗OX M −→ · · · −→ DX ⊗O X ⊗OX M d

−→ DX ⊗OX M − →0 in which the differential d is given by the formula   d P ⊗ (v1 ∧ · · · ∧ vl ) ⊗ u

= (−1)k−1 Pvk ⊗ (v1 ∧ · · · ∧ vk ∧ · · · ∧ vl ) ⊗ u 1≤k≤l

−

(−1)k−1 P ⊗ (v1 ∧ · · · ∧ vk ∧ · · · ∧ vl ) ⊗ vk u

1≤k≤l

+

1≤i