Fourier-Mukai and Nahm transforms in geometry and mathematical physics

Progress in Mathematics Volume 276

Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein

Claudio Bartocci Ugo Bruzzo Daniel Hernández Ruipérez

Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics

Birkh¨auser Boston • Basel • Berlin

Claudio Bartocci Dipartimento di Matematica Università di Genova Genova, Italy [email protected]

Ugo Bruzzo Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare Trieste, Italy [email protected]

Daniel Hernández Ruipérez Departamento de Matemáticas and Instituto Universitario de Fisica Fundamental y Matemáticas Universidad de Salamanca Salamanca, Spain [email protected]

ISBN 978-0-8176-3246-5 e-ISBN 978-0-8176–4663-9 DOI 10.1007/b11801 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926479 Mathematics Subject Classification (2000): 14-02, 14D21, 14D20, 14E05, 14F05, 14J28, 14J32, 14J81, 14K05, 18E30, 19K56, 53C07, 58J20 © Birkhäuser Boston, a part of Springer Science +Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Cover Design by Joseph Sherman Printed on acid-free paper Birkhäuser Boston is part of Springer Science+Business Media (www.springer.com)

Contents Preface

xi

Acknowledgments

xv

1 Integral functors

1

1.1

Notation and preliminary results . . . . . . . . . . . . . . . . . . .

2

1.2

First properties of integral functors . . . . . . . . . . . . . . . . . .

5

1.2.1

Base change formulas . . . . . . . . . . . . . . . . . . . . .

8

1.2.2

Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Fully faithful integral functors . . . . . . . . . . . . . . . . . . . . .

15

1.3.1

Preliminary results . . . . . . . . . . . . . . . . . . . . . . .

15

1.3.2

1.3

1.4

1.5

Strongly simple objects . . . . . . . . . . . . . . . . . . . .

19

The equivariant case . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.4.1

Equivariant and linearized derived categories . . . . . . . .

24

1.4.2

Equivariant integral functors . . . . . . . . . . . . . . . . .

29

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . .

30

2 Fourier-Mukai functors 2.1

2.2

31

Spanning classes and equivalences . . . . . . . . . . . . . . . . . . .

32

2.1.1

Ample sequences . . . . . . . . . . . . . . . . . . . . . . . .

35

2.1.2

Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Orlov’s representability theorem

. . . . . . . . . . . . . . . . . . .

44

2.2.1

Resolution of the diagonal . . . . . . . . . . . . . . . . . . .

44

2.2.2

Uniqueness of the kernel . . . . . . . . . . . . . . . . . . . .

51

2.2.3

Existence of the kernel . . . . . . . . . . . . . . . . . . . . .

54

vi

Contents 2.3

2.4

Fourier-Mukai functors

. . . . . . . . . . . . . . . . . . . . . . . .

60

2.3.1

Some geometric applications of Fourier-Mukai functors . . .

61

2.3.2

Characterization of Fourier-Mukai functors . . . . . . . . .

71

2.3.3

Fourier-Mukai functors between moduli spaces . . . . . . .

76

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . .

78

3 Fourier-Mukai on Abelian varieties

81

3.1

Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.2

The transform

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.3

Homogeneous bundles . . . . . . . . . . . . . . . . . . . . . . . . .

90

3.4

Fourier-Mukai transform and the geometry of Abelian varieties . .

91

3.4.1

Line bundles and homomorphisms of Abelian varieties . . .

91

3.4.2

Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . .

94

3.4.3

Picard sheaves . . . . . . . . . . . . . . . . . . . . . . . . .

95

Some applications of the Abelian Fourier-Mukai transform . . . . .

97

3.5.1

Moduli of semistable sheaves on elliptic curves . . . . . . .

97

3.5.2

Preservation of stability for Abelian surfaces . . . . . . . . 102

3.5.3

Symplectic morphisms of moduli spaces . . . . . . . . . . . 104

3.5.4

Embeddings of moduli spaces . . . . . . . . . . . . . . . . . 106

3.5

3.6

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . . 108

4 Fourier-Mukai on K3 surfaces

111

4.1

K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2

Moduli spaces of sheaves and integral functors

4.3

Examples of transforms . . . . . . . . . . . . . . . . . . . . . . . . 122

. . . . . . . . . . . 116

4.3.1

Reflexive K3 surfaces . . . . . . . . . . . . . . . . . . . . . . 124

4.3.2

Duality for reflexive K3 surfaces . . . . . . . . . . . . . . . 125

4.3.3

Homogeneous bundles . . . . . . . . . . . . . . . . . . . . . 131

4.3.4

Other Fourier-Mukai transforms on K3 surfaces . . . . . . . 133

4.4

Preservation of stability . . . . . . . . . . . . . . . . . . . . . . . . 139

4.5

Hilbert schemes of points on reflexive K3 surfaces . . . . . . . . . . 142

4.6

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . . 145

5 Nahm transforms 5.1

147

Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Contents

5.2

5.3

5.4

5.5

vii

5.1.1

Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.1.2

Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.1.3

The Hitchin-Kobayashi correspondence

5.1.4

Dirac operators and index bundles . . . . . . . . . . . . . . 155

The Nahm transform for instantons . . . . . . . . . . . . . . . . . . 158 5.2.1

Definition of the Nahm transform . . . . . . . . . . . . . . . 158

5.2.2

The topology of the transformed bundle . . . . . . . . . . . 161

5.2.3

Line bundles on complex tori . . . . . . . . . . . . . . . . . 161

5.2.4

Nahm transform on flat 4-tori . . . . . . . . . . . . . . . . . 164

Compatibility between Nahm and Fourier-Mukai . . . . . . . . . . 165 5.3.1

Relative differential operators . . . . . . . . . . . . . . . . . 165

5.3.2

Relative Dolbeault complex . . . . . . . . . . . . . . . . . . 166

5.3.3

Relative Dirac operators . . . . . . . . . . . . . . . . . . . . 170

5.3.4

K¨ahler Nahm transforms

6.2

. . . . . . . . . . . . . . . . . . . 171

Nahm transforms on hyperk¨ ahler manifolds . . . . . . . . . . . . . 173 5.4.1

Hyperk¨ ahler manifolds . . . . . . . . . . . . . . . . . . . . . 173

5.4.2

A generalized Atiyah-Ward correspondence . . . . . . . . . 174

5.4.3

Fourier-Mukai transform of quaternionic instantons . . . . . 178

5.4.4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . . 181

6 Relative Fourier-Mukai functors 6.1

. . . . . . . . . . . 153

183

Relative integral functors . . . . . . . . . . . . . . . . . . . . . . . 184 6.1.1

Base change formulas . . . . . . . . . . . . . . . . . . . . . 185

6.1.2

Fourier-Mukai transforms on Abelian schemes . . . . . . . . 188

Weierstraß fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.2.1

Todd classes

. . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.2.2

Torsion-free rank one sheaves on elliptic curves . . . . . . . 192

6.2.3

Relative integral functors for Weierstraß fibrations . . . . . 193

6.2.4

The compactified relative Jacobian . . . . . . . . . . . . . . 197

6.2.5

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.2.6

Topological invariants . . . . . . . . . . . . . . . . . . . . . 201

6.3

Relatively minimal elliptic surfaces . . . . . . . . . . . . . . . . . . 204

6.4

Relative moduli spaces for Weierstraß elliptic fibrations . . . . . . 208

viii

Contents Semistable sheaves on integral genus one curves . . . . . . . 208

6.4.2

Characterization of relative moduli spaces

. . . . . . . . . 213

6.5

Spectral covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.6

Absolutely stable sheaves on Weierstraß fibrations . . . . . . . . . 220

6.7 7

6.4.1

6.6.1

Preservation of absolute stability for elliptic surfaces . . . . 221

6.6.2

Characterization of moduli spaces on elliptic surfaces . . . . 225

6.6.3

Elliptic Calabi-Yau threefolds . . . . . . . . . . . . . . . . . 228

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . . 231

Fourier-Mukai partners and birational geometry

233

7.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.2

Integral functors for quotient varieties . . . . . . . . . . . . . . . . 238

7.3

Fourier-Mukai partners of algebraic curves . . . . . . . . . . . . . . 242

7.4

Fourier-Mukai partners of algebraic surfaces

7.5

7.6

7.7

. . . . . . . . . . . . 242

7.4.1

Surfaces of Kodaira dimension 2 . . . . . . . . . . . . . . . 245

7.4.2

Surfaces of Kodaira dimension −∞ that are not elliptic . . 245

7.4.3

Relatively minimal elliptic surfaces . . . . . . . . . . . . . . 248

7.4.4

K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.4.5

Abelian surfaces . . . . . . . . . . . . . . . . . . . . . . . . 253

7.4.6

Enriques surfaces . . . . . . . . . . . . . . . . . . . . . . . . 254

7.4.7

Nonminimal projective surfaces . . . . . . . . . . . . . . . . 256

Derived categories and birational geometry . . . . . . . . . . . . . 257 7.5.1

A removable singularity theorem . . . . . . . . . . . . . . . 258

7.5.2

Perverse sheaves . . . . . . . . . . . . . . . . . . . . . . . . 264

7.5.3

Flops and derived equivalences . . . . . . . . . . . . . . . . 272

McKay correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.6.1

An equivariant removable singularity theorem . . . . . . . . 276

7.6.2

The derived McKay correspondence . . . . . . . . . . . . . 277

Notes and further reading . . . . . . . . . . . . . . . . . . . . . . . 279

A Derived and triangulated categories

281

A.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.2 Additive and Abelian categories . . . . . . . . . . . . . . . . . . . . 283 A.3 Categories of complexes . . . . . . . . . . . . . . . . . . . . . . . . 287

Contents

ix

A.3.1 Double complexes . . . . . . . . . . . . . . . . . . . . . . . 292 A.4 Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 A.4.1 The derived category of an Abelian category . . . . . . . . 295 A.4.2 Other derived categories . . . . . . . . . . . . . . . . . . . . 300 A.4.3 Triangles and triangulated categories . . . . . . . . . . . . . 303 A.4.4 Differential graded categories . . . . . . . . . . . . . . . . . 307 A.4.5 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . 312 A.4.6 Some remarkable formulas in derived categories . . . . . . . 328 A.4.7 Support and homological dimension . . . . . . . . . . . . . 335 B Lattices

339

B.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 B.2 The discriminant group . . . . . . . . . . . . . . . . . . . . . . . . 341 B.3 Primitive embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 342 C Miscellaneous results

347

C.1 Relative duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 C.2 Pure sheaves and Simpson stability . . . . . . . . . . . . . . . . . . 351 C.3 Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 D Stability conditions for derived categories

359

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 D.2 Bridgeland’s stability conditions

. . . . . . . . . . . . . . . . . . . 362

D.2.1 Definition and Bridgeland’s theorem . . . . . . . . . . . . . 363 D.2.2 An example: stability conditions on curves . . . . . . . . . . 369 D.2.3 Bridgeland’s deformation lemma . . . . . . . . . . . . . . . 371 D.3 Stability conditions on K3 surfaces . . . . . . . . . . . . . . . . . . 373 D.3.1 Bridgeland’s theorem . . . . . . . . . . . . . . . . . . . . . . 374 D.3.2 Construction of stability conditions . . . . . . . . . . . . . . 375 D.3.3 The covering map property . . . . . . . . . . . . . . . . . . 380 D.3.4 Wall and chamber structure . . . . . . . . . . . . . . . . . . 382 D.3.5 Sketch of the proof of Theorem D.19 . . . . . . . . . . . . . 383 D.4 Moduli stacks and invariants of semistable objects on K3 surfaces . 385 D.4.1 Moduli stack of semistable objects . . . . . . . . . . . . . . 385 D.4.2 Sketch of the proof of Theorem D.35 . . . . . . . . . . . . . 386

x

Contents D.4.3 Counting invariants and Joyce’s conjecture for K3 surfaces

391

D.4.4 Some ideas from the proof of Theorem D.45 . . . . . . . . . 392 References

397

Subject index

419

Preface A fundamental question in geometry is to find invariants for a given class of geometric objects. In the case of algebraic varieties, natural invariants are the Chow groups and the algebraic K-theory. In looking for finer invariants one could think of the category of coherent sheaves; however, it is known that two projective varieties are isomorphic if and only if the respective categories of coherent sheaves are equivalent [120]. A straightforward extension of this idea is to look at the derived category of coherent sheaves. That notion was used by Grothendieck and Verdier as an appropriate framework for their theory of duality; moreover, in the late 1970s Be˘ılinson gave a simple characterization of the derived categories of projective spaces. Derived categories of coherent sheaves appeared again in the fundamental paper by Shigeru Mukai [224], where the integral functor now called “Fourier-Mukai transform” was introduced. A radical change of perspective in connection with derived categories took place with a result of Bondal and Orlov, according to which the derived category of a projective variety, whose canonical or anticanonical bundle is ample, fully determines the variety. Subsequent work by Orlov showed that any equivalence of derived categories of coherent sheaves on two projective varieties X and Y is an integral functor — i.e., a kind of “correspondence” induced by an object in the derived category of the product X × Y . This prompts us to consider the more general question: to what extent does the derived category of coherent sheaves determine the underlying algebraic variety? And also, what is the relationship between the group of automorphisms of the derived category of a projective variety and the group of isomorphisms of the variety? In the second half of the 1990s, physicists working in string theory got interested in triangulated categories. Actually, the quantum theories corresponding to some string models admit solitonic states (branes), which can be geometrically characterized as coherent sheaves supported on subvarieties of the compactification space of the string (in most cases a Calabi-Yau threefold). Thus, derived categories of coherent sheaves naturally come into play, and integral functors can be exploited to describe some mirror dualities.

xii

Preface

This book is about the study of integral functors. Given two projective varieties X and Y , an integral functor between the derived categories D(X) and D(Y ) is a functor of the type •

L

• ∗ • • ΦK X→Y (E ) = RπY ∗ (πX E ⊗ K ) ,

where K• is an object in D(X × Y ) (a complex of coherent sheaves on X × Y , called the kernel of the integral functor) and πX , πY are the projections onto the two factors of X × Y (for precise definitions, the reader is referred to Chapter 1). When an integral functor is an equivalence of categories, we shall call it a Fourier-Mukai functor, and if in addition the kernel K• is concentrated (i.e., it reduces to a single coherent sheaf on X × Y ), the integral functor will be called a Fourier-Mukai transform. The prototype of this kind of transform was defined by Mukai in 1981: X is an Abelian variety, Y its dual variety, and the kernel is the Poincar´e bundle on the product X × Y . Besides developing the basic theory of integral functors, we shall put special emphasis on some of their applications to geometry and mathematical physics. As a matter of fact, one proves that whenever two projective varieties X and Y have equivalent derived categories (i.e., they are Fourier-Mukai partners), then Y is a coarse moduli space of coherent sheaves on X, and vice versa. Thus, we shall be mostly concerned with applications to moduli spaces of sheaves. A first example is Mukai’s original transform, which establishes the equivalence between ˆ which is the derived categories of an Abelian variety X and of its dual variety X, indeed the moduli space of flat line bundles on X. Other examples are provided by the construction of Fourier-Mukai transforms for K3 surfaces, and by the relative Fourier-Mukai transforms. The latter play an important role in the so-called homological mirror symmetry in string theory and in some constructions appearing in the theory of algebraically completely integrable systems. We now give a cursory presentation of the contents of the book. Chapter 1 is devoted to the foundations of the theory of integral functors. A key notion is that of strongly simple object, which provides a characterization of fully faithful integral functors (Theorem 1.27). This result — due to Bondal and Orlov [48] — will be a cornerstone on which we shall build many of the most fundamental theorems proved in this book. In Chapter 2 we further develop the theory of these functors, considering the case when they are equivalences of derived categories. Orlov’s representability theorem [242] — whose proof involves the notion of spanning classes for a derived category of coherent sheaves — shows that every equivalence of derived categories is an integral functor. This explains the pervasiveness of Fourier-Mukai functors in the study of the geometry of algebraic varieties. While the two initial chapters are framed in a rather general setting, in Chapters 3 and 4 we study in some detail the cases of Abelian varieties and K3 surfaces. In particular, in Chapter 3 we review Mukai’s original construction and

Preface

xiii

present some applications. The first nontrivial instance of Fourier-Mukai transform was obtained on K3 surfaces by the current authors [24]. In this case, X is a K3 surface whose N´eron-Severi group satisfies certain restrictions, Y is 2-dimensional compact component of the moduli space of stable bundles over X (which one proves to be a K3 surface as well) and the kernel is the relevant universal sheaf. A remarkable feature of the Fourier-Mukai transform in the case of both Abelian and K3 surfaces is that, under suitable assumptions, it preserves the stability of the sheaves it operates on. Consequently, it supplies a helpful tool to investigate the structure of moduli spaces of stables sheaves, as we show, e.g., in Section 4.5. Chapter 5 is a digression in the realm of complex differential geometry. On a compact K¨ahler manifold, Hermitian-Yang-Mills bundles and stable bundles are related by the celebrated Hitchin-Kobayashi correspondence. Regarding an Abelian surface as a flat 4-dimensional real torus T and applying this correspondence, Mukai’s original transform translates into Nahm’s transform, introduced in the early 1980s by the physicist Werner Nahm to study periodic instantons on R4 [230]. This transform is based on an index-theoretic construction: thinking of the dual torus Tb as a space parameterizing a family of twisted Dirac operators, one can associate an index bundle to any instanton on T . After extending Nahm’s transform to a more general setting, we show how it relates with a Fourier-Mukai transform. We examine in some detail the case when the base manifold carries a hyperk¨ahler structure. Besides being interesting on its own, this perspective sheds new light on some results obtained in the algebro-geometric framework: for instance, in Section 5.4, we provide a different proof of the preservation of stability for bundles on Abelian and K3 surfaces. In Chapter 6 we develop the machinery of relative Fourier-Mukai functors for algebraic B-schemes. These are a particular kind of integral functors, whose kernel is an element of the derived category of the fibered product X ×B Y (here X and Y are schemes over a base scheme B). We offer a quite comprehensive treatment of elliptic fibrations X → B, dealing separately with the case when the fibration admits a Weierstraß model (allowing the base scheme to be of arbitrary dimension) and the case of relatively minimal elliptic surfaces. In the first situation, we study the moduli spaces of relatively semistable sheaves and discuss the notion of spectral cover. If the total space X of the Weierstraß fibration has dimension 2 or 3, the Fourier-Mukai transform establishes a correspondence between relatively semistables sheaves on X and spectral data on the compactified relative Jacobian associated with the fibration. When X is an elliptically fibered Calabi-Yau threefold, this construction — originally obtained by Friedman, Morgan and Witten [113, 114] — is relevant to string theory. The study of the Fourier-Mukai functor on elliptic surfaces is part of a much wider research program, which we pursue in Chapter 7. Two projective varieties are said to be Fourier-Mukai partners if there is an exact equivalence of trian-

xiv

Preface

gulated categories between their bounded derived categories; in view of Orlov’s representability theorem, this amounts to the existence of a Fourier-Mukai functor between them. Section 7.4 is devoted to the classification of Fourier-Mukai partners of algebraic surfaces, a result that for minimal surfaces was first obtained by Bridgeland and Maciocia. The problem of the birational invariance of the derived category of a projective variety is dealt with in Section 7.5, while BridgelandKing-Reid’s interpretation of the McKay correspondence is presented in Section 7.6. The results presented in Chapter 7 are a clear indication of the significance of Fourier-Mukai transforms in algebraic geometry, as it has been recently pointed out by Kawamata and others. We have made an effort to be self-contained, devoting three appendices to present some preliminary material (respectively, derived categories, integral lattices and a miscellany of results in algebraic geometry), and proving most of the fundamental theorems from scratch. Generically, prerequisites for reading this book reduce to a basic knowledge of algebraic geometry at the level of Hartshorne [141] and, for Chapter 5, to some rudiments of differential geometry (manifolds, bundles, connections, and on a somehow more advanced level, some theory of elliptic operators on spaces of sections of a vector bundle). In addition, we use quite extensively some basic categorial language, and occasionally, we employ spectral sequences. On the other hand, we did not aim at giving an exhaustive treatment of the subject, which appears to be widely ramified and steadily growing. Among the most conspicuous omissions, we do not deal with the case of Fourier-Mukai functors on singular varieties and we leave out the important topic of autoequivalences of derived categories. Furthermore, we mention only cursorily the recent developments related to differential graded categories and derived algebraic geometry. We have tried to put a remedy to major and minor omissions adding a “notes and further reading” section at the end of each chapter. In this connection one should also cite D. Huybrecht’s book on Fourier-Mukai transforms [153]. Should one try to compare the two books in terms of their contents, one would notice that our book is more abundant in technical details, and somehow aims at a reasonably selfcontained treatment of the arguments it touches. On the other hand, the choice of topics in the two books is somehow different; in this sense, we believe that the two books nicely complement each other. A final appendix serves as an introduction to one of the most interesting recent developments related to integral functors, namely, the notion of stability condition for derived categories. Claudio Bartocci Ugo Bruzzo Daniel Hern´andez Ruip´erez April 2008

Acknowledgments The project of writing this book saw the light in 2002 at a conference in Cookeville, Tennessee, organized by our friend Rafal Ablamowicz. It is very apt that he is the first person whom we thank. The project started with a fourth author, Marcos Jardim, who at some stage preferred to withdraw. We owe many thanks to Marcos: without him this book would not have been written. Appendix A, an introduction to derived categories that originates from a set of notes for a course at the School on Algebraic Geometry and Physics in Salamanca in September 2003, has been written by Fernando Sancho. Appendix D, an introduction to stability conditions for derived categories, has been written by Emanuele Macr`ı. We are deeply thankful to Emanuele and Fernando for their contributions to this work. Very special thanks are due to David Ploog, who was patient enough to read this book twice during the final stage of its redaction, pointed out several mistakes and inconsistencies, and proposed several improvements. Without David’s help this book would definitely have been worse. We also thank Benoit Charbonneau, Stefano Guerra, Daniel Hern´ andez Serrano, Adrian Langer, Cristina L´opez Mart´ın, Tony Pantev, Dar´ıo S´ anchez G´ omez, Justin Sawon, Edoardo Sernesi, Carlos Tejero Prieto and the anonymous referee for pointing out mistakes and suggesting improvements. We have included many results obtained jointly with our collaborators Bj¨orn Andreas, Antony Maciocia, Jos´e M. Mu˜ noz Porras and Fabio Pioli: we are grateful to all of them. We also thank Bj¨ orn for clarifying some physics-related issues. Many thanks are due to Birkh¨ auser, especially in the person of Ann Kostant, for the enthusiasm with which this project was considered, and for patiently waiting until it was over. Writing this book has required many exchanges of visits among the authors (very likely, the most pleasant aspect of the enterprise). These have been made possible by funding provided by the University of Genova, the International School for Advanced Studies in Trieste and the University of Salamanca, by the Italian

xvi

Acknowledgments

National Research Projects (P.R.I.N.) “Geometric methods in the theory of nonlinear waves” and “Geometry on algebraic varieties,” and by the Spanish Research Projects “Geometrical integral transforms and applications” (BFM2003-00097), “Applications of integral functors to geometry and physics” (GCyL-SA114/04) and “Coherent sheaves, derived categories and integral functors in birational geometry and string theory” (MTM2006-04779). A strong initial impulse to the writing of the book was given by a “Research in Pairs” stay of the first and second author at Mathematische F¨ orschunginstitut Oberwolfach, which we deeply thank, especially in the person of its director, Professor Martin Greuel. The editing of the book was finalized when the second author was spending the 2008 spring term at the Department of Mathematics of the University of Pennsylvania in Philadelphia. Warm thanks are due to the scientists and staff of the department for their collaboration and the nice atmosphere they create and to the university for providing support.

Chapter 1

Integral functors Introduction The first instance of an integral functor is to be found in Mukai’s 1981 paper on the duality between the derived categories of an Abelian variety and of its dual variety [224]. Integral functors have also been called “Fourier-Mukai functors” or “Fourier-Mukai transforms.” However, we shall give these terms specific meanings that we shall introduce in Chapter 2. The core idea in the definition of an integral functor is very simple: if we have two varieties X and Y , we may take some “object” on X, pull it back to the product X × Y , twist it by some object (“kernel”) in X × Y and then push it down to Y (i.e., we integrate on X). This is what happens with the Fourier transform of functions: one takes a function f (x) on Rn , pulls it back to Rn × Rn , multiplies it by the kernel ei x·y and then integrates over the first copy of Rn , thus obtaining a function fˆ(y) on the second copy. This naive resemblance between an integral functor and the Fourier transform is the reason for the “Fourier” appearing in the original name of these functors (the christening was done by Mukai in [224]). But the analogy goes further, since for integral functors one can talk (under suitable conditions) about an inverse functor, convolutions (composing two such functors amounts to convolute the kernels in a suitable way), a Parseval theorem (which states that, in the appropriate sense, an integral functor is an “isometry”), etc. Technically, if we denote by D− (X) the derived categories of complexes of coherent sheaves on X that are bounded on the right (here X and Y are proper algebraic varieties over a field k), an integral functor is a functor •

− − ΦK X→Y : D (X) → D (Y )

C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_1, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

1

2

Chapter 1. Integral functors

of the form •

L

• ∗ • • ΦK X→Y (E ) = RπY ∗ (πX E ⊗ K )

where K• is an object in D− (X × Y ). The morphisms πX , πY are projections onto the factors of the Cartesian product. Our aim in this chapter is to develop a fairly general theory of such functors, without making detailed assumptions on the varieties X and Y and on the kernel K• . In Chapter 2 we shall consider the case of integral functors that are exact equivalences of categories. More specific situations (e.g., X and Y Abelian varieties, or K3 surfaces, or hyperk¨ahler varieties, in each case with a kernel of appropriate type, etc.) will be studied in the following chapters. Let us briefly describe the structure of this chapter. After fixing in Section 1.1 some notation, conventions and definitions, in the next section we introduce the integral functors and study their first properties (base change and adjoint functors). In Section 1.3 we prove a criterion for the testing whether an integral functor is fully faithful. Finally we introduce equivariant integral functors. This will be used in Chapter 7 to study the McKay correspondence.

1.1

Notation and preliminary results

By algebraic variety X we mean a separated scheme of finite type over an algebraically closed field k. We do not assume a priori that X is irreducible or reduced. The symbol D(X) will denote the derived category of complexes of quasi-coherent OX -modules with coherent cohomology sheaves. (Details about the construction of the category D(X) are given in Appendix A; here we only set the basic notation.) We denote by Db (X), D+ (X) and D− (X) the corresponding derived categories of bounded complexes, bounded on the left and bounded on the right complexes, respectively; their objects will be denoted by such symbols as E • , and the set of morphisms between two objects in one of these derived categories by HomD(X) (E • , F • ). It is a remarkable fact that Db (X) is equivalent to the bounded derived category of the Abelian category of coherent sheaves on X. If f : X → Y is a proper morphism to another projective variety, we shall denote by Rf∗ : D+ (X) → D+ (Y ), Lf ∗ : D− (Y ) → D− (X) the associated derived functors (properness is necessary because otherwise the direct image of a coherent sheaf may fail to be coherent). Since every sheaf has only a finite number of nonvanishing higher direct images and this number is bounded by the maximum of the dimensions of the fibers, the derived direct image can be extended to a functor Rf∗ : D(X) → D(Y ) and induces functors Rf∗ : Db (X) → Db (Y ), Rf∗ : D− (X) → D− (Y ). If the morphism f has finite Tor-dimension, that is, if there is a number that bounds the number of nonvanishing higher inverse images for every sheaf on Y , then Lf ∗ : D− (Y ) → D− (X) extends to a functor Lf ∗ : D(Y ) → D(X) and induces Lf ∗ : Db (Y ) → Db (X).

1.1. Notation and preliminary results

3

If the morphism f is flat, the inverse image functor is exact, hence it does not need to be derived, so that one actually has Lf ∗ = f ∗ . Analogously, if f is an affine morphism (for instance, a closed immersion), the direct image functor does not need to be derived, and we have Rf∗ = f∗ . The bifunctor induced by the left derived functor of the tensor product will be L

denoted by ⊗ . One can also derive the homomorphism sheaf bifunctor, obtaining the functor RHom•OX (E • , F • ) (for either E • bounded on the right or F • bounded on the left, see Appendix A for details). For any object M• in D(X) we have a “dual object” in the derived category D(X), M•∨ = RHom•OX (M• , OX ). Note that when M• is a complex concentrated in degree zero, the cohomology sheaves of M•∨ are the local Ext sheaves, Hi (M•∨ ) = ExtiOX (M0 , OX×Y ). So we shall use the symbol ∨ for the dual in derived category, while for the dual of a sheaf we shall use the notation ∗ (but for the dual of a line bundle L we shall sometimes also write L−1 ). Assume now that the variety X is smooth. Then any complex M• in Db (X) is isomorphic in the derived category to a bounded complex E • of locally free sheaves, i.e., to a perfect complex (cf. Definition A.42). In this situation the dual of an object of Db (X) is still an object of Db (X), and, remarkably, all objects in the bounded derived category are “reflexive,” that is, M• ' (M•∨ )∨ (see Propositions A.75 and A.87). If X is also proper, the Chern characters of an object M• of Db (X) are defined by X (−1)i chj (E i ) ∈ Aj (X) ⊗ Q , chj (M• ) = i

where Aj (X) is the degree-j summand of the Chow ring (when k = C, the group Aj (X) ⊗ Q is the algebraic part of the rational cohomology group H 2j (X, Q)). This definition is well posed since it is independent of the choice of the bounded complex E • of locally free sheaves. By definition the rank of M• is the integer number rk(M• ) = ch0 (M• ). For a complex E • in Db (X) we define its Mukai vector as p v(E • ) = ch(E • ) · td(X) ,

(1.1)

where td(X) ∈ A• (X) ⊗ Q is the Todd class of X. There is a natural involution on the Chow ring ∗ : A• (X) → A• (X) which acts on the degree-j summand Aj (X) as the multiplication by (−1)j . Given v ∈ A• (X), the element v ∗ is called the Mukai dual of v. If E • is an object of Db (X), one has ch(E • )∗ = ch(E •∨ ). Due to the identity p p td(X) = ( td(X))∗ · exp( 12 c1 (X))

4

Chapter 1. Integral functors

it turns out that v(E •∨ ) = v(E • )∗ · exp( 12 c1 (X)) . In particular, for a locally free sheaf E one has v(E ∗ ) = v(E)∗ ·

(1.2) exp( 12 c1 (X)).

We define on the rational Chow group A• (X) ⊗ Q a symmetric bilinear form h·, ·i, called Mukai pairing, by setting Z v ∗ · w · exp( 12 c1 (X)) . (1.3) hv, wi = − X

2

We shall write v instead of hv, vi. When k = C, the involution ∗ and the Mukai pairing naturally extend to the even rational cohomology ⊕j H 2j (X, Q). When X is a surface we can write a Mukai vector v in the form v = (v0 , v1 , v2 ) with vj ∈ Aj (X) ⊗ Q, and the Mukai pairing takes the form hv, wi = v1 · w1 − v0 · (w2 − 12 w1 · c1 (X))− w0 · (v2 − 12 v1 · c1 (X)) − 14 v0 · w0 · c1 (X)2 .

(1.4)

When the first Chern class of the surface is trivial (X is K3 or Abelian), the Mukai pairing takes the form Z hv, wi = − v ∗ · w = v1 · w1 − v0 · w2 − w0 · v2 . X

We shall make use of these expressions mainly in Chapter 4. As a first application of the notion of Mukai vector, we use it to express the Euler characteristic of two objects E • and F • of the bounded derived category Db (X) of a smooth proper variety X. This is defined as X (−1)i dim HomiD(X) (E • , F • ) (1.5) χ(E • , F • ) = i

where HomiD(X) (E • , F • ) = HomD(X) (E • , F • [i]), cf. eq. (A.12). When E sheaves, one has HomiD(X) (E, F) ' ExtiX (E, F), see Proposition A.68.

and F are The Euler

characteristic χ(F) of a sheaf coincides with χ(OX , F). mula

Note that the Grothendieck-Hirzebruch-Riemann-Roch theorem gives the forZ ch(E •∨ ) · ch(F • ) · td(X) = −hv(E), v(F)i . (1.6) χ(E • , F • ) = X

When X is a surface, the Euler characteristic of E • and F • can be computed using eq. (1.6) by means of the explicit expression (1.4) of the Mukai pairing. In particular, if rk(E • ) = rk(F • ) = 0 (when E • and F • are sheaves this means that they are torsion sheaves), we obtain χ(E • , F • ) = −c1 (E • ) · c1 (F • ) .

(1.7)

1.2. First properties of integral functors

5

The symbol Ox will denote as usual the skyscraper sheaf of length 1 at the closed point x ∈ X: this is the structure sheaf of x as a closed subscheme of X. These sheaves satisfy the following properties: HomiD(X) (Ox1 , Ox2 ) = 0 for all i ∈ Z if x1 6= x2 ; ( ∧i Tx X for 0 ≤ i ≤ dim X i HomD(X) (Ox , Ox ) ' 0 otherwise

(1.8) (1.9)

where Tx X is the tangent space to X at x. This is proved by resolving Ox locally by the Koszul complex.

1.2

First properties of integral functors

Let X, Y be proper algebraic varieties over k; the projections of the Cartesian product X × Y onto the factors X, Y are denoted by πX , πY respectively. Let K• be an object in the derived category D− (X × Y ). We define the functor •

− − ΦK X→Y : D (X) → D (Y )

by letting •

L

• ∗ • • ΦK X→Y (E ) = RπY ∗ (πX E ⊗ K ) . •

The complex K• will be called the kernel of the functor, and ΦK X→Y will be called the associated integral functor. Remark 1.1. One may also define integral functors D(Qco(X)) → D(Qco(Y )) (with a kernel in D(Qco(X×Y ))) because the derived tensor product is well defined also for unbounded complexes of quasi-coherent sheaves (see Section A.4.5). 4 Example 1.2. Let δ : X ,→ X × X denote the diagonal immersion, and write ∆ for its image. Let K• be the complex in Db (X × X) concentrated in degree zero with • K0 = O∆ = δ∗ OX . It is easy to check that ΦK X→X is isomorphic to the identity ∗L functor. More generally, if L is a line bundle on X, the functor ΦδX→ X acts as the twist by L. Let us also note that, given a proper morphism f : X → Y , by taking as K• the structure sheaf of the graph Γf ⊂ X × Y , one has isomorphisms of • K• ∗ 4 functors ΦK X→Y ' Rf∗ and ΦY→X ' Lf . A useful feature of integral functors is that the composition of two of them is still an integral functor, whose kernel may be expressed as a kind of “convolution product” of the two original kernels. This may be seen as an analogue of the composition law for correspondences [209]. Let us describe this property. If Z is

6

Chapter 1. Integral functors

another proper variety, we consider the diagram X × Y ×NZ NNN pp p p NπNXZ p πY,Z p NNN p p p N& xp  X ×Y Y ×Z X ×Z πXY

Given kernels K• in D− (X × Y ) and L• in D− (Y × Z), we define the convolution of L• and K• as the kernel in D− (X × Z) L

∗ L• ∗ K• = RπXZ∗ (πXY K• ⊗ πY∗ Z L• ) .

Proposition 1.3. There is a natural isomorphism of functors •

•

•

•

K K ∗L . ΦL Y→Z ◦ ΦX→Y ' ΦX→Z

Proof. Given E • in D− (X), one has: •

L

•

L

K • ∗ ∗ • • • ΦL Y→Z (ΦX→Y (E )) = RπZ∗ (πY [RπY ∗ (πX E ⊗ K )] ⊗ L ) L

L

∗ ∗ • (πX E ⊗ K• )) ⊗ L• ) ' RπZ∗ (RπY,Z∗ (πXY L

L

∗ ∗ • (πX E ⊗ K• ) ⊗ πY∗ Z L• )) ' RπZ∗ (RπY,Z∗ (πXY L

L

∗ ∗ • ∗ πX E ⊗ πXY K• ⊗ πY∗ Z L• ) ' RπZ∗ RπXZ,∗ (πXZ L

∗ • E ⊗ (L• ∗ K• )) . ' RπZ,∗ (πX

The isomorphism in the second line is cohomology flat base change (Proposition A.85), those in the third and fifth lines are the projection formula in derived category (Proposition A.83), while in the fourth line the isomorphism is due to  the obvious identities πX ◦ πXY = πX ◦ πXZ and πZ ◦ πXZ = πZ ◦ πY Z . A natural issue to be addressed is when a kernel defines an integral functor which maps Db (X) to Db (Y ). We say that a kernel K• in D− (X × Y ) is of finite Tor-dimension as a complex of OX -modules if it is isomorphic in the derived category to a bounded complex of sheaves that are flat over X. Proposition 1.4. Let K• an object of D− (X × Y ) of finite Tor-dimension as a • complex of OX -modules. The integral functor ΦK X→Y has the following property: there exist integer numbers z and n ≥ 0 such that for every coherent sheaf F on • / [z, z + n]. X, the cohomology sheaves Hi (ΦK X→Y (F)) vanish for i ∈ Proof. Take a bounded complex E • ≡ E z → E z+1 → · · · → E z+m of flat sheaves over X isomorphic to K• in the derived category. If r is the dimension of X, • then for every coherent sheaf F on X the cohomology sheaves Hi (ΦK X→Y (F)) = ∗ F ⊗ E • )) vanish unless z ≤ i ≤ z + n with n = m + r.  Hi (RπY ∗ (πX

1.2. First properties of integral functors

7

Corollary 1.5. If K• is of finite Tor-dimension as a complex of OX -modules, the • b b functor ΦK X→Y maps D (X) to D (Y ). Note that the condition of finite Tor-dimensionality is always fulfilled if X and Y are smooth and K• is an object of Db (X × Y ), because in this case K• is isomorphic in the derived category to a bounded complex of locally free sheaves and the projection πX is a flat morphism. Another important feature of integral functors is that they are exact as functors of triangulated categories (see Definition A.46 and Proposition A.62), since they are compositions of exact functors: derived tensor product, inverse image under a flat morphism, and direct image in derived category. In particular, for any exact sequence 0 → F → E → G → 0 of coherent sheaves in X we obtain an exact sequence · · · → Φi−1 (G) → Φi (F) → Φi (E) → Φi (G) → Φi+1 (F) → . . .

(1.10)

•

where we have written Φi (•) = Hi (ΦK X→Y (•)) for simplicity. In the next chapters we shall be often interested in studying cases where an integral functor applied to a complex or a sheaf yields a concentrated complex. •

• in D− (X) satisfies Definition 1.6. Given an integral functor ΦK X→Y , a complex F the WITi condition (or is WITi ) if there is a coherent sheaf G on Y such that • • ΦK X→Y (F ) ' G[−i] in D(Y ), where G[−i] is the associated complex concentrated in degree i. We say that F • satisfies the ITi condition if in addition G is locally free. 4

The acronym “IT” stands for “index theorem” and “W” stands for “weak.” The reason for this terminology (which is not entirely appropriate in the most general case of nonconcentrated complexes) will be made clear in Chapter 5, where a link between the integral functors and index theory is established. Proposition 1.7. Assume that the kernel is a locally free sheaf Q on the product. A coherent sheaf F on X is ITi if and only if H j (X, F ⊗ Qy ) = 0 for all y ∈ Y and for all j 6= i, where Qy denotes the restriction of Q to X × {y}. Furthermore, F is WIT0 if and only if it is IT0 . Proof. Both statements follow from the cohomology base change theorem [141, ∗ III.12.11] taking into account that πX F ⊗ Q is flat over Y .  If X and Y are smooth proper varieties, and K• is a kernel in Db (X × Y ), by • b the Grothendieck-Riemann-Roch theorem the integral functor ΦK X→Y : D (X) →

8

Chapter 1. Integral functors

Db (Y ) gives rise to the commutative diagram •

Db (X)

ΦK X→Y

v

 A• (X) ⊗ Q

/ Db (Y ) (1.11)

v

f

K•

 / A• (Y ) ⊗ Q

•

where f K is the homomorphism of Q-vector spaces defined by •

∗ f K (α) = πY ∗ (πX α · v(K• ))

(1.12)

and v(K• ) is the Mukai vector (1.1) of K• . Notice that √ √ ∗ v(K• ) = πX td X · ch(K• ) · πY∗ td Y . •

The map f K depends functorially on the kernel, i.e., fK

•

∗L•

•

•

= fL ◦ fK .

(1.13)

When the base field k is the field C of the complex numbers, we can extend the diagram (1.11) to a diagram •

Db (X)

ΦK X→Y

v

 H • (X, Q)

/ Db (Y ) (1.14)

v

f

K•

 / H • (Y, Q)

where the homomorphism in the bottom line is defined by a formula like (1.12) and maps the even cohomology ring H 2• (X, Q) to H 2• (Y, Q). The analogue of Equation (1.13) holds true.

1.2.1

Base change formulas •

− − We wish to generalize the notion of integral functor ΦK X→Y : D (X) → D (Y ) − − to “relative” integral functors D (T × X) → D (T × Y ) where T is a (proper) variety of “parameters.” Although in further chapters we shall extend this notion to nontrivial families, we stick for the time being to this simple situation because it is general enough for our present purposes.

For any variety T we shall write XT = T ×X and denote by πT the projection XT = T × X → T , unless confusion can arise. Given a kernel K• ∈ D− (X × Y )

1.2. First properties of integral functors

9 K•

∗ we define KT• = πX×Y K• and consider the functor ΦT = ΦXTT→YT : D− (XT ) → − D (YT ) given by L

∗ E • ⊗ KT• ) , ΦT (E • ) = RπYT ∗ (πX T

where πXT : (X × Y )T = T × X × Y → XT = T × X πYT : (X × Y )T = T × X × Y → YT = T × Y πX×Y : (X × Y )T = T × X × Y → X × Y are projections. The functor ΦT can be regarded as an integral functor with kernel i∗ KT• in D− (XT × YT ), where i : XT ×T YT ,→ XT × YT is the closed immersion of the fiber product. The notions of WITi and ITi introduced in Definition 1.6 apply to this new situation. What makes relative integral functors interesting is their compatibility with base changes. Let f : S → T be a morphism and denote by fZ the induced morphism S × Z → T × Z for any Z. Proposition 1.8. For every object E • in D− (T × X) there is a functorial isomorphism ∗ • E ) LfY∗ ΦT (E • ) ' ΦS (LfX in the derived category of YS . Proof. One has L

L

∗ ∗ ∗ LfY∗ ΦT (E • ) = LfY∗ RπYT ∗ (πX E • ⊗ KT• ) ' RπYS ∗ LfX×Y (πX E • ⊗ KT• ) T T

by base change in the derived category (Proposition A.85). Then L

L

∗ ∗ ∗ ∗ • (πX E • ⊗ KT• ) ' πX (LfX E ) ⊗ KS• , LfX×Y T S

whence the statement follows.



Note that we do not need to assume that the base change morphism f : S → T is flat. If the original kernel K• is of finite Tor-dimension as a complex of OX modules, then KT• is of finite Tor-dimension as a complex of OXT -modules so that Proposition 1.4 implies that ΦT is bounded and can be extended to a functor ΦT : D(T × X) → D(T × Y ) for every T and maps Db (T × X) → Db (T × Y ). Base change compatibility means that if we think of an object E • ∈ D(T ×X) as a family of objects Ljt∗ E • ∈ D(X), then the relative integral functor ΦT (E • ) is the family of integral functors Φt (Ljt∗ E • ), that is, Ljt∗ ΦT (E • ) ' Φt (Ljt∗ E • ) .

(1.15)

10

Chapter 1. Integral functors

We shall often be interested in transforming a complex that reduces to a single sheaf E on T ×X. We can derive from the base change (1.15) the relationship between the sheaves ΦiT (E)t = jt∗ ΦiT (E) and the sheaves Φit (Et ), where Et = jt∗ E. Note that ΦiT (E) = 0 for every sheaf E when i > n = dim X + m, where Hm (K• ) is the highest nonzero cohomology sheaf of the kernel. Corollary 1.9. Let E be a sheaf over XT = T × X. The formation of ΦnT (E) is compatible with base change for sheaves, that is, one has ΦnT (E)t ' Φnt (Et ) for every point t ∈ T . Moreover, if E is flat over T , the following results hold true. 1. For every point t in T there is a convergent spectral sequence q q−p q−p T E2−p,q (t) = TorO (Et ) . p (ΦT (E), Ot ) =⇒ E∞ (t) = Φt

2. Assume that E is WITi and write Eb = ΦiT (E). Then for every t ∈ T there are isomorphisms of sheaves over Xt i−j T b TorO j (E, Ot ) ' Φt (Et ) ,

j ≤ i.

3. The restriction Et to the fiber Xt is WITi for every t ∈ T if and only if E is WITi and Eb = ΦiT (E) is flat over T . In that case the formation of Eb is b t ' Ebt for every point t ∈ T . compatible with base change, that is, (E) Proof. Let us write Lp ft∗ (F) = H−p (Lft∗ (F)) for any sheaf F. For every point ¯ −p,q (t) = t ∈ B there exist two spectral sequences E2−p,q (t) = Lp jt∗ (ΦqT (E)), E 2 q−p ¯ q−p (t) = (t) = Hq−p (Ljt∗ ΦT (E) and E Φq (Lp j∗ E) converging respectively to E∞ ∞ n (t) and Φnt (Et ) ' Hq−p (Φt (Ljt∗ E)). One deduces isomorphisms ΦnT (E)t ' E∞ ¯ n (t), so that Φn (E)t ' Φn (Et ) by Proposition 1.8. For the rest of the proof, E ∞ t T ¯ −p,q (t) degenerates and one has E is flat over T . Then the spectral sequence E 2 statements 1 and 2. Regarding 3, the only point worthy of a comment is that if Et is WITi for every t, then E is WITi and Eb = Φi (E) is flat over T . Let q0 be the maximum of the q’s with Φq (E) 6= 0. Then E20,q0 (t) = Φq0 (E) ⊗OT Ot 6= 0 for some point t. Since E2−2,q0 +1 (t) = 0, every nonzero cycle in the term E20,q0 (t) survives q0 (t) 6= 0 and then q0 = i. The same argument proves that at infinity, whence E∞ p,i E2 (t) = 0 for every point t and every p > 0, so that Eb is flat over T . Since E2−2,i (t) = 0, any nonzero cycle E20,i−1 (t) survives at infinity as before; this imi−1 (t) 6= 0 which is absurd. Then Φi−1 (E) ⊗OT Ot = E20,i−1 (t) = 0 for every plies E∞ point t so that Φi−1 (E) = 0. Proceeding as above one proves that Φj (E) = 0 for every j < i. The last part follows now from Equation (1.15). 

1.2. First properties of integral functors

11

The following formula will be used to prove that in many cases the kernel K• is uniquely determined, up to isomorphism in the derived category, by the integral • functor ΦK X→Y . •

• Proposition 1.10. Let Φ = ΦK X→Y an integral functor with kernel K . Then

K• ' ΦX (O∆ ) • . where ΦX is the integral functor from D− (X × X) to D− (X × Y ) with kernel KX

Proof. Let δ : X ,→ X × X the diagonal immersion and δY : X × Y ,→ X × X × Y the induced immersion. Then, L

L

∗ ∗ δ (OX ) ⊗ K• X ) ' RπYX ∗ (δY ∗ πX OX×X ⊗ K• X ) ΦX (O∆ ) ' RπYX ∗ (πX X ∗

' RπYX ∗ (δY ∗ (δY∗ K• X )) ' K• where the second isomorphism is base change and the third is the projection formula.  The following result can be found in [61, Lemma 4.3]. Let us denote by jt the immersion X ,→ T × X as the fiber Xt = πT−1 (t)). Proposition 1.11. Let F • be an object in Db (YT ). If the restriction Ljt∗ F • is a concentrated complex Ft for every (closed) point t ∈ T , then F • is concentrated as well, and is a coherent sheaf F on YT , flat on T , and such that jt∗ F ' Ft for every t ∈ T . Proof. For every t ∈ T there is a spectral sequence E2−p,q = Lp jt∗ Hq (F • ) convergq−p = Hq−p (Ljt∗ F • ). Let q0 be the maximum of the indexes q such that ing to E∞ q • H (F ) 6= 0. Then for some t ∈ T one has E20,q0 6= 0 and E20,q0 survives to infinity so that Hq0 (Ljt∗ F • ) 6= 0. Then q0 = 0. Now any nonzero element in E2−1,0 survives −1 = 0, one has E2−1,0 = 0 which implies that F = H0 (F • ) to infinity and since E∞ is flat over T . Thus E2−p,0 = 0 for every p ≥ 1. Finally, if q1 < 0 is the largest strictly negative index q with Hq (F • ) 6= 0, then E20,q1 6= 0 for some t and E20,q1 survives to infinity so that Hq1 (Ljt∗ F • ) 6= 0; this is absurd, hence no such q1 exists  and F • ' F in the derived category. We can easily derive a simple consequence of this fact. Corollary 1.12. Let K• ∈ Db (X × Y ) be of finite Tor-dimension over X and let • b b ΦK X→Y : D (X) → D (Y ) be the corresponding integral functor. Assume that for

12

Chapter 1. Integral functors •

every (closed) point x ∈ X one has ΦK X→Y (Ox ) ' Oy for some (closed) point y ∈ Y . Then there is a morphism f : X → Y and a line bundle L on X such that •

• • ΦK X→Y (E ) ' Rf∗ (E ⊗ L)

for every object E • in Db (X). •

• Proof. Since Ljx∗ K• ' ΦK X→Y (Ox ) ' Oy , by Proposition 1.11 the kernel K reduces to a single sheaf K on X × Y flat over X. Moreover K is a flat family of skyscraper sheaves on Y of length 1. Since Y is a fine moduli space for its own points and the diagonal ∆ ,→ Y × Y is a universal family, there is a morphism f : X → Y ∗ L for some line bundle L on X. Now the proof such that K ' (f × 1)∗ (O∆ ) ⊗ πX is completed by a simple computation. 

1.2.2

Adjoints

The issue of the existence of right and left adjoint functors of an integral functor can be naturally addressed within the framework of the Grothendieck-Serre theory of duality [139, 291], which we describe in Section C.1. We shall use indeed Grothendieck-Serre duality to prove that under quite general conditions, integral functors have left and right adjoints. Proposition 1.13. Let X, Y be proper algebraic varieties, of dimensions m and n, and let K• be a kernel in Db (X × Y ). 1. If K• is of finite Tor-dimension over Y and X is smooth, the functor ∗ K•∨ ⊗πX ωX [m]

ΦY→X

: Db (Y ) → Db (X)

•

is right adoint to the functor ΦK X→Y . 2. If K• is of finite Tor-dimension over X and Y is smooth, the functor ∗ K•∨ ⊗πY ωY [n]

ΦY→X

: Db (Y ) → Db (X)

•

is left adjoint to the functor ΦK X→Y . Proof. In both cases, K• is of finite Tor-dimension over X ×Y , and the same is true L

for the derived dual K•∨ . As a consequence, (−) ⊗ K•∨ is both left and right adjoint L

to (−) ⊗ K• as functors on Db (X × Y ) by Corollary A.88. We only prove the first •

L

• ∗ part, since the second is analogous. Since ΦK X→Y = RπY ∗ ◦ ((−) ⊗ K ) ◦ πX , a right K• adjoint to ΦX→Y is the composition in the reverse order of the right adjoints to the

L

∗ K•∨ ⊗πX ωX [m]

∗ ωX [m]) ◦ πY∗ ] = ΦY→X factors, namely, RπX∗ ◦ ((−) ⊗ K• ) ◦ [((−) ⊗ πX

.



1.2. First properties of integral functors

13

Remark 1.14. If X and Y are smooth and proper with the same dimension and the canonical bundles ωX and ωY are trivial, the left and right adjoint coincide. 4 Let X be a smooth proper variety. Since any object on Db (X) has finite Tor-dimension, applying Equation (C.13) we have that the functor SX : Db (X) → Db (X) F • 7→ F • ⊗ ωX [n] provides a natural isomorphism HomD(X) (F • , G • ) ' HomD(X) (G • , SX (F • ))∗

(1.16)

where F • and G • are complexes in Db (X). Proposition 1.15. If X and Y are smooth and proper, for every complex F • in • •∨ • ∨ •∨ ' ΦK )). Db (X) there is an isomorphism (ΦK X→Y (F )) Y→X (SX (F Proof. This follows from the chain of isomorphisms L

•

• ∨ ∗ = RHom•OY (RπY ∗ (πX F • ⊗ K• ), OY ) (ΦK X→Y (F ))

L

∗ ∗ F • ⊗ K• , πX ωX [m]) ' RπY ∗ RHom•OX×Y (πX L

∗ ∗ F • ⊗ K• )∨ ⊗ πX ωX [m]) ' RπY ∗ ((πX L

∗ (F • ∨ ⊗ ωX [m]) ⊗ K•∨ ) ' RπY ∗ (πX •∨

•∨

•∨ •∨ ' ΦK ⊗ ωX [m]) = ΦK )) Y→X (F Y→X (SX (F

where the first isomorphism is duality for πY .



The isomorphism (1.16) fits into a commutative diagram / HomD(X) (G • , SX (F • ))∗ HomD(X) (F • , G • ) UUUU UUUU UUUU UUUU SX *  HomD(X) (SX (F • ), SX (G • )) .

(1.17)

An easy consequence of Serre duality is the following formula involving the Euler characteristic of two bounded complexes (see Eq. (1.5)) χ(E • , F • ) = χ(F • , E • ⊗ ωX ) , where E • and F • are in Db (X).

(1.18)

14

Chapter 1. Integral functors

The functor SX is the model for an abstract definition of Serre functor for triangulated categories [45]. We remind that a functor F : A → B is an equivalence of categories when it has a quasi-inverse functor, that is a functor G : B → A such that G ◦ F ' IdA and F ◦ G ' IdB . In this case G is both a left and a right adjoint to F . Definition 1.16. Let A be a k-linear triangulated category whose sets of homomorphisms are finite-dimensional vector spaces. An exact autoequivalence SA : A → A is a Serre functor if for all a, b in A there is a bifunctorial isomorphism HomA (a, b) ' HomA (b, SA (a))∗ . 4 A Serre functor automatically satisfies a compatibility condition expressed by a diagram analogous to (1.17). A Serre functor, if it exists, is unique up to functorial isomorphisms. Serre functors provide a handy tool for describing adjoints of exact functors. Lemma 1.17. Let F : A → B be an exact functor between triangulated categories with Serre functors SA and SB . Then F has a left adjoint G if and only if it has −1 . a right adjoint H and one has H = SA ◦ G ◦ SB Proof. We prove the statement in one direction, the other being analogous. If G is a left adjoint to F , then −1 −1 −1 (b)) ' HomA (G ◦ SB (b), a)∗ ' HomB (SB (b), F (a))∗ HomA (a, SA ◦ G ◦ SB

' HomB (b, SB ◦ F (a))∗ ' HomB (F (a), b) −1 is a right adjoint to F . so that H = SA ◦ G ◦ SB



In particular, when F is an equivalence of triangulated categories, since its left and right adjoints coincide, we get the following result. Corollary 1.18. Let F : A → B be an exact equivalence of triangulated categories with Serre functors SA , SB . Then SB ◦ F ' F ◦ SA . Remark 1.19. If X is a proper Gorenstein variety, one can define the functor SX : Db (X) → Db (X) by letting SX (F • ) = F • ⊗ ωX [n], where n = dim X as in the smooth case. However, this is a Serre functor for Db (X) in the sense of Definition 1.16 only if X is smooth. Indeed, if SX is a Serre functor, for every point x ∈ X, every i ∈ Z and every coherent sheaf M on X, one has isomorphisms n−i ∗ ∗ ExtiX (Ox , M) ' Extn−i X (M, Ox ⊗ ωX ) ' ExtX (M, Ox ) .

Then ExtiX (Ox , M) = 0 for all i > n and any M, so that Ox has finite Tordimension. Hence x cannot be a singular point. 4

1.3. Fully faithful integral functors

15

We have proved explicitly that integral functors admit right and left adjoints. One should note however that every exact functor Db (X) → Db (Y ) between the bounded derived categories of smooth projective varieties has a right adjoint (and hence, since both categories have Serre functors, a left adjoint as well). This follows from the fact that these categories are saturated. Let us recall this result. Let A be a triangulated category. A cohomological functor K from A to the category of k-vector spaces Vect(k) is said to be of finite type if for all a ∈ A, the P sum i dimk K(a[i]) is finite (the notion of cohomological functor is defined in Appendix A). Moreover A is said to be saturated if every cohomological functor of finite type is representable (the notions of cohomological and representable functors are given in Appendix A, see Definitions A.47 and A.1). A remarkable result by Bondal and Van den Bergh [52] states that for any smooth projective variety X, the bounded derived category Db (X) is saturated. Proposition 1.20. Let X and Y be smooth projective varieties. Every exact functor F : Db (X) → Db (Y ) has a right and a left adjoint. Proof. For every complex G • in Db (Y ), the functor K : Db (X) → Vect(k) given by K(E • ) = HomDb (Y ) (F (E • ), G • ) is a cohomological functor of finite type, so that it is representable by an object M• in Db (X), unique up to isomorphisms. By Yoneda’s lemma, by setting H(G • ) = M• one defines a functor H : Db (Y ) → Db (X) which is right adjoint to F . Lemma 1.17 implies the existence of a left adjoint as well. 

1.3

Fully faithful integral functors

Our next step is to establish a criterion for an integral functor to be fully faithful. This will be expressed by Theorem 1.27.

1.3.1

Preliminary results

We recall that a functor F : A → B is fully faithful if the map F : HomA (a1 , a2 ) → HomB (F (a1 ), F (a2 )) is bijective for any pair of objects a1 , a2 in A. If the functor F : A → B has a left adoint G, then F is fully faithful if and only if the natural adjunction map γ : G◦F → IdA is an isomorphism. Analogously, if F has a right adjoint H, it is fully faithful if and only if the natural adjunction map η : IdA → H ◦ F is an isomorphism. Then, given an object a in A, the induced morphism γ(a) : G(F (a)) → a is zero if and only if F (a) = 0. Similarly, η(a) : a → H(F (a)) is zero if and only if F (a) = 0.

16

Chapter 1. Integral functors

Remark 1.21. One can state a handy criterion for checking if a functor F : A → B is fully faithful. If F has a left adoint G, there is a commutative diagram HomA (a1 , a2 )

F

/ HomB (F (a1 ), F (a2 ))

G◦F

 HomA ((G ◦ F )(a1 ), (G ◦ F )(a2 ))

γ(a2 )



'

/ HomA ((G ◦ F )(a1 ), a2 ) .

We can derive two consequences of this. The first is that if the map G ◦ F : HomA (a1 , a2 ) → HomA ((G ◦ F )(a1 ), (G ◦ F )(a2 ) is injective, then the map HomA (a1 , a2 ) → HomB (F (a1 ), F (a2 )) is injective as well. The second is that if G ◦ F is fully faithful, so that both the first vertical arrow and the bottom arrow are isomorphisms, then F is also fully faithful. 4 To prove Theorem 1.27 we need a preliminary lemma which characterizes objects of the derived category supported on a closed subvariety. Lemma 1.22. [48, Prop. 1.5] Let j : Y ,→ X be a codimension d closed immersion of irreducible smooth algebraic varieties and K• an object of Db (X). Assume that 1. Ljx∗ K• = 0 for every (closed) point x ∈ X − Y , where jx : {x} ,→ X denotes the embedding of x into X; 2. Li jx∗ K• = 0 when either i < 0 or i > d for every (closed) point x ∈ Y . Then there is a sheaf on X whose topological support is Y which is isomorphic to K• in Db (X). Proof. Recall that Li jx∗ K• denotes the cohomology sheaf H−i (jx∗ L• ) where L• is a bounded complex of flat sheaves quasi-isomorphic to K• . Let us write Hq = Hq (K• ). For every point x ∈ X there is a spectral sequence −p+q = Lp−q jx∗ K• . E2−p,q = Lp jx∗ Hq =⇒ E∞

If q0 is the maximum of the q’s with Hq 6= 0, then a nonzero element in jx∗ Hq0 q survives up to infinity in the spectral sequence. Since E∞ = L−q jx∗ K• = 0 for every x ∈ X and q > 0, one has q0 ≤ 0. We now show that the topological support of all the sheaves Hq is contained in Y . Assume indeed that this is not true and consider the maximum q1 of the q’s such that jx∗ Hq 6= 0 for a certain point x ∈ X − Y ;

1.3. Fully faithful integral functors

17

then a nonzero element in jx∗ Hq1 survives up to infinity in the spectral sequence, which is absurd since Ljx∗ K• = 0. Let q2 ≤ q0 be the minimum of the q’s with Hq 6= 0. We know that Hq2 is topologically supported on a closed subset of Y . Take a component Z ⊆ Y of the support. Let d be the codimension of Y in X. If c ≥ d is the codimension of Z, then, for every x ∈ Z outside a closed subset of Z of codimension greater or equal to c + 2 in X, one has Lp jx∗ Hq2 +1 = 0 for every p ≥ c + 2 and Lc jx∗ Hq2 6= 0. This follows from the fact that the m-singularity set of a coherent sheaf is a closed subscheme of dimension smaller than or equal to m; this is proved in [184, Thm. 5.8] in the complex case, while for a proof in the general case, the reader may refer, e.g., to [144, Prop. 1.13]. For such a point x ∈ Z any nonzero element of Lc jx∗ Hq2 survives in the spectral sequence up to infinity and gives Lc−q2 jx∗ K• 6= 0. Thus c − q2 ≥ 0 which implies q2 ≥ c − d ≥ 0 and then q2 = q0 = 0. So K• ' H0 in Db (X). We already know that the topological support of H0 is contained in Y ; actually it is the whole of Y : if this were not true, since Y is irreducible, the support would have a component Z ⊂ Y of codimension c > d and one could find, by reasoning as above, a point x ∈ Z such that Lc jx∗ K• 6= 0. Therefore c ≤ d, but this is absurd.  We recall the notion of Kodaira-Spencer map for families of sheaves. Let f : Z → S be a morphism of algebraic varieties and F a coherent sheaf on Z, flat over S, and let D = Spec k[ε]/ε2 be the double point scheme. The tangent space Ts S at a closed point s ∈ S is the space of the morphisms v : D → S that map the closed point s0 of D to s. A tangent vector v ∈ Ts S induces a morphism vD = (v × Id) : D ×S Z → Z, which defines an infinitesimal deformation ∗ F of Fs . The set of such deformations is identified with the space Ext1Zs (Fs , Fs ) vD (cf. Proposition A.70 for a similar statement in derived category), and the KodairaSpencer map at the point s is the resulting morphism KSs (F) : Ts S → Ext1Zs (Fs , Fs ) . Assume now that A is a full subcategoy of the category of coherent sheaves on an algebraic variety X and that M is a functor associating to any variety T the set M(T ) of coherent sheaves E on T × X flat over T such that Et is an object of A for every (closed) point t ∈ T . As usual, two such families E and E 0 are considered equivalent if E ' E 0 ⊗ π1∗ L, where L is a line bundle on T . We also assume that M has a fine moduli space, that is, it is represented by an algebraic variety M . Then, there is a universal sheaf Q on M × X, flat over M inducing an equivalence Hom(T, M ) ' M(T ) φ 7→ [(φ × Id)∗ (Q)] ,

(1.19)

18

Chapter 1. Integral functors

where square brackets denote equivalence clases. Now, Equation (1.19) gives an equivalence between Ts M ' Hom(D, M )s and the infinitesimal deformations of Qs , that is, an isomorphism Ts M ' Ext1X (Qs , Qs ), which coincides by its very definition with the Kodaira-Spencer map KSs (Q) for the universal family Q at the point s. In other words, the Kodaira-Spencer map for the universal family Q at any point s ∈ M is an isomorphism: KSs (Q) : Ts M ' Ext1X (Qs , Qs ) .

(1.20)

The following result is essentially contained in [61]. Lemma 1.23. Assume that the base field k has characteristic zero. Let S and X be algebraic varieties, with X projective, and let F be a coherent sheaf on S × X, flat over S and schematically supported on a closed subscheme j : Z ,→ S × X. Suppose that for every closed point s ∈ S the support Z ∩ ({s} × X) of the sheaf Fs is topologically a single point x ∈ X and that HomX (Fs , Ox ) ' k . Then F is a line bundle on its support, i.e., F ' j∗ L for a line bundle L on Z. Moreover, if for all pairs of distinct closed points s1 , s2 ∈ S the sheaves Fs1 , Fs2 are not isomorphic, then the Kodaira-Spencer map of the family F is injective at some point s ∈ S. Proof. Let us first prove that for every s ∈ S, Fs is the structure sheaf of a zerodimensional closed subscheme of X, namely, that there is a surjective morphism f : OX → Fs ; this will imply that Fs ' OZs . Let g : Fs → Ox and f : OX → Fs be nonzero morphisms. If f is not surjective, coker f is a nonzero sheaf supported topologically at x and then there is a nonzero map coker f → Ox . This induces a nonzero morphism h : Fs → Ox such that h◦f = 0. Now since HomX (Fs , Ox ) ' k, the morphism g is a multiple of h and then g ◦ f = 0, i.e., f takes values in ker g. Since dim H 0 (X, ker g) = dim H 0 (X, Fs ) − 1, there exist morphisms f : OX → Fs not coming from H 0 (X, ker g) and these are surjective. The remaining issues are local, so we may assume that S is affine. Let s ∈ S be a closed point. By hypothesis there exists a surjective morphism σs : OX → Fs . The natural morphism H 0 (S×X, F) → H 0 (X, Fs ) is surjective and therefore there exists a global section σ of F mapping to σs . For a suitable open neighborhood U of s, σU : OU → FU is still surjective, that is, FU is the structure sheaf of a closed subscheme of U × X. Then one must have FU ' OZU , where ZU = Z ∩ (U × X), which proves that F is a line bundle on its support. Moreover, FU induces a map α : U → HilbP (X) to the Hilbert scheme of zero-dimensional subschemes of X whose Hilbert polynomial P is equal to that of the sheaf Fs . The morphism α is characterized by the condition F ' (1 × α)∗ Q, where Q is a universal sheaf on

1.3. Fully faithful integral functors

19

HilbP (X) × X. By hypothesis the map α is injective on closed points, and the tangent map Ts α : Ts U → Tα(s) HilbP (X) is injective at least at a point s ∈ U (one here uses essentially the fact that k has characteristic zero). Moreover, the Kodaira-Spencer map at s for the family FU is the composition of the tangent map Ts α with the Kodaira-Spencer map for the universal family Q. Since the latter is an isomorphism because of the universality of Q (cf. Eq. (1.20)), KSs (F) is injective.  Let X, Y be proper varieties and F a coherent sheaf on X × Y , flat over X. The flatness of F implies Fx ' ΦF X→Y (Ox ) for every closed point x ∈ X. Lemma 1.24. [61] The integral functor Φ = ΦF X→Y induces a morphism Φ : Ext1X (Ox , Ox ) → Ext1Y (Fx , Fx )

(1.21)

which coincides with the Kodaira-Spencer morphism for the family F. Proof. Note that in view of (1.8) one has the identification Ext1X (Ox , Ox ) = Tx X. Given a tangent vector v ∈ Tx X, the corresponding deformation of the sheaf ∗ (O∆ ) of the structure sheaf of the diagonal ∆ ⊂ X × Ox is the pullback vX X. The morphism (1.21) sends v to the deformation of Φ(Ox ) induced by it, ∗ (O∆ )). By base change (Proposition 1.8), one namely, to the deformation ΦD (vX ∗ ∗ ∗ (F), where the last equality is due to has ΦD (vX (O∆ )) ' vD (ΦX (O∆ )) ' vD Proposition 1.10. But this is the image of v by the Kodaira-Spencer map of the family F. 

1.3.2

Strongly simple objects

In this section the base field k has characteristic zero. Let K• be a kernel in Db (X × Y ). By taking its restrictions to the fibers of X × Y over X we obtain a family Ljx∗ K• of objects in Db (Y ) parameterized by points in X. The notion of strong simplicity, originally stated for sheaves in [202], expresses a kind of orthonormality condition for the elements of this family. Definition 1.25. A kernel K• in Db (X × Y ) is strongly simple over X if it satisfies the following two conditions: 1. HomiD(Y ) (Ljx∗1 K• , Ljx∗2 K• ) = 0 unless x1 = x2 and 0 ≤ i ≤ dim X; 2. Hom0D(Y ) (Ljx∗ K• , Ljx∗ K• ) = k. 4

20

Chapter 1. Integral functors •

Since Ljx∗ K• ' ΦK X→Y (Ox ), one can check the conditions in this definition on • K• the groups HomiD(Y ) (ΦK X→Y (Ox1 ), ΦX→Y (Ox2 )). Remark 1.26. For every pair of points x1 , x2 in X there is an isomorphism HomiD(Y ) (Ljx∗1 K•∨ , Ljx∗2 K•∨ ) ' HomiD(Y ) (Ljx∗2 K• , Ljx∗1 K• ) . Therefore, a kernel K• in Db (X × Y ) is strongly simple over X if and only if its 4 dual K•∨ is so. The following crucial result was originally proved by Bondal and Orlov [48]. Theorem 1.27. Let X and Y be smooth projective algebraic varieties, and let K• • • be a kernel in Db (X × Y ). The functor ΦK is X→Y is fully faithful if and only if K strongly simple over X. •

Proof. If the functor ΦK X→Y is fully faithful, one has •

•

K HomiD(Y ) (Ljx∗1 K• , Ljx∗2 K• ) ' HomiD(Y ) (ΦK X→Y (Ox1 ), ΦX→Y (Ox2 ))

' HomiD(X) (Ox1 , Ox2 ) and then K• is strongly simple over X. Let us assume in turn that K• is strongly simple over X. We need to prove • that the functor ΦK X→Y is fully faithful. We know from Proposition 1.3 that ∗ K•∨ ⊗πY ωY [n]

ΦY→X

•

•

M ◦ ΦK X→Y ' ΦX→X ,

L

∗ ∗ where M• = Rπ13∗ (π12 K• ⊗ π23 (K•∨ ⊗ πY∗ ωY [n])) and πij denotes the projection of X × Y × X onto the (i, j)-th factor.

The strategy of the proof is as follows: first we prove that M• is topologically supported on the image ∆ of the diagonal morphism δ : X ,→ X × X. Next, we show that M• is the push-forward of a line bundle N supported on a closed subscheme Z of X × X. We then prove that Z coincides with the diagonal ∆. M• So the functor ΦX→ X is the twist by N , as we saw in Example 1.2; this is an • equivalence of categories and then ΦK X→Y is fully faithful by Remark 1.21. (a) M• ' M for a sheaf M topologically supported on the diagonal. Let us fix a point x1 ∈ X. For every point x2 ∈ X we have •

•

i M ∗ M (Li jx∗2 ΦX→ X (Ox1 )) ' HomD(X) (ΦX→X (Ox1 ), Ox2 ) •

•

K ' HomiD(Y ) (ΦK X→Y (Ox1 ), ΦX→Y (Ox2 )) ,

which is zero unless x1 = x2 and 0 ≤ i ≤ m by the strong simplicity of K• . By M• applying Lemma 1.22 to the immersion {x1 } ,→ X one sees that ΦX→ X (Ox1 ) '

1.3. Fully faithful integral functors

21

Ljx∗1 M• is a sheaf topologically supported at x1 . By Proposition 1.11, M• ' M where M is a sheaf on X × X flat over X for the first projection p1 : X × X → X. Moreover, since jx∗ M is topologically supported on x, M is topologically supported on the diagonal. (b) Let us denote by δ¯ : Z ,→ X × X the schematic support of M, so that M = δ¯∗ N for a coherent sheaf N on Z. We show that N is a line bundle. For every closed point x ∈ X the sheaf jx∗ M ' ΦM X→X (Ox ) is topologically supported on x, and moreover one has •

•

0 K K HomX (ΦM X→X (Ox ), Ox ) ' HomD(X) (ΦX→Y (Ox ), ΦX→Y (Ox ) ' k

(1.22)

because K• is strongly simple. So by Lemma 1.23, N is a line bundle. (c) The scheme Z coincides with the diagonal ∆. A first step is to show that the sheaf p1∗ (M) is a line bundle on X. The diagonal embedding δ factors through a closed immersion η : X ,→ Z which topologically is a homeomorphism. The morphism p¯1 = p1 ◦ δ¯ : Z → X being finite, the condition that M is flat over X through p1 implies that p1∗ (M) ' p¯1∗ (N ) is a locally free sheaf. Let r be its rank. To see that r = 1, it is enough to prove that ΦM X→X (Ox ) ' Ox for at least one closed point x ∈ X. Let us consider the exact sequence 0 → Px → ΦM X→X (Ox ) → Ox → 0 where the last morphism is the adjunction and Px is the kernel. We want to prove that for some point x the sheaf Px is zero. Since Px is supported at x, it suffices to see that HomX (Px , Ox ) = 0. Taking homomorphisms in Ox , and in view of Equation (1.22), we have an exact sequence 0 → HomX (Px , Ox ) → Hom1D(X) (Ox , Ox ) → Hom1D(X) (ΦM X→X (Ox ), Ox ) , so that we have to show that Hom1D(X) (Ox , Ox ) → Hom1D(X) (ΦM X→X (Ox ), Ox ) is injective. By Remark 1.21, we need only to prove that the morphism 1 1 M M ΦM X→X : HomD(X) (Ox , Ox ) → HomD(X) (ΦX→X (Ox ), ΦX→X (Ox ))

is injective. Lemma 1.24 tells us that this morphism is the Kodaira-Spencer map for the family M, which is injective at some point x by Lemma 1.23. Finally, we show that η : X → Z is an isomorphism, which is equivalent to prove that the finite morphism p¯1 : Z → X is an isomorphism. This follows from the fact that the direct image of the line bundle N is also a line bundle.  This characterization of fully faithful integral functors allows us to prove rather easily that the product of two such functors is again fully faithful. Let X, f• be kernels in Db (X ×Y ) ˜ Y˜ be smooth projective varieties, and let K• , K Y and X,

22

Chapter 1. Integral functors L

˜ × Y˜ ), respectively. These define a kernel K•  K f• in Db (X × X ˜ ×Y × Y˜ ), and Db (X L

• ∗ f• is the “box product” π ∗ f• where K•  K ˜ Y˜ K . X×Y K ⊗ πX×

˜•

•

K Lemma 1.28. If the integral functors ΦK ˜ Y ˜ are fully faithful, then the X→Y and ΦX→ •

L

˜•

K functor ΦK ˜ Y ×Y ˜ is fully faithful. X×X→

Proof. The K¨ unneth formula (see Theorem A.89) L

L

HomhD(Y ×Y˜ ) (E •  F • , G •  M• ) ' M

HomiD(Y ) (E • , G • ) ⊗ HomjD(Y˜ ) (F • , M• )

i+j=h

˜ • is strongly simple over X, ˜ then implies that if K• is strongly simple over X and K L ˜ • is strongly simple over X × X. ˜  K•  K The definition of strongly simple object in Db (X × Y ) basically implies that such an object is simple (thus justifying the teminology), since the restriction to every fiber {x} × Y is simple. A more formal proof of this fact may be obtained as follows. Proposition 1.29. A strongly simple object K• in Db (X × Y ) is simple. •

Proof. By Theorem 1.27 the functor Φ = ΦK X→Y is fully faithful. The explicit expression for the right adjoint H of Φ (cf. Proposition 1.13) shows that it changes base, that is, the base-changed functor HX is a right adoint to the functor ΦX = • KX b b ΦX×X→ X×Y : D (X × X) → D (X × Y ), so that the latter is fully faithful. Since K• ' ΦX (O∆ ) by Proposition 1.10, we have HomDb (X×Y ) (K• , K• ) ' HomDb (X×X) (O∆ , O∆ ) ' k .  A different proof is given in [249, Lemma 1.12]. When the kernel is a coherent sheaf the notion of strong simplicity in Definition 1.25 looks more familiar. Assume that Q is a coherent sheaf on X × Y , flat over X. Then, for any point x ∈ X, the derived category restriction Ljx∗ Q is merely the sheaf Qx = jx∗ Q on {x} × Y and the two conditions in Definition 1.25 are equivalent to the following: 1. HomiD(Y ) (Qx1 , Qx2 ) = 0 for every i ∈ Z whenever x1 6= x2 ;

1.3. Fully faithful integral functors

23

2. Qx is simple for every x ∈ X, i.e., all its automorphisms are constant multiples of the identity, HomOX (Q, Q) ' k. Condition 1 is equivalent to condition 1 in Definition 1.25 because HomiD(Y ) (Qx1 , Qx2 ) ' ExtiOX (Qx1 , Qx2 ) and since Y is smooth, we have HomiD(Y ) (Qx1 , Qx2 ) = 0 for any x1 , x2 unless 0 ≤ i ≤ n = dim Y . Definition 1.30. If Q is a sheaf on X × Y , flat over X, which is strongly simple as a complex, we call it a strongly simple sheaf over X. 4 Remark 1.31. By Remark 1.26, the dual of a locally free strongly simple sheaf is strongly simple. 4 Example 1.32. If we take X = Y , the structure sheaf O∆ of the diagonal ∆ ⊂ X × X is strongly simple over both factors. 4 In the literature one usually finds a particular case of Theorem 1.27; this is what we shall mostly use in the sequel. Theorem 1.33. A coherent sheaf Q on X × Y which is flat over X is strongly b b simple over X if and only if the integral functor ΦQ X→Y : D (X) → D (Y ) is fully faithful. The following result is known as “Parseval formula” because it is similar to the formula for the ordinary Fourier transform for functions on a torus bearing • the same name. We recall that if Φ = ΦK X→Y is an integral functor and E is WITi i for Φ, we denote by Eb = Φ (E) the unique nonzero Fourier-Mukai sheaf, and that b one has Φ(E) ' E[−i]. •

Proposition 1.34. Let Φ = ΦK X→Y a fully faithful integral functor. 1. One has HomhD(X) (F • , E • ) ' HomhD(Y ) (Φ(F • ), Φ(E • )) for any pair of objects F • and E • of Db (X). 2. Let F a WITi sheaf and G a WITj sheaf on X. Then b G) b . ExthX (F, G) ' Exth+i−j (F, Y for every h ≥ 0. In particular, there is an isomorphism b F) b ExthX (F, F) ' ExthY (F, for every h ≥ 0.

24

Chapter 1. Integral functors

Proof. Since HomhD(X) (F • , E • ) = HomD(X) (F • , E • [h]), the first statement holds true because Φ is fully faithful. As for the second, we use the first formula together b b with Φ(F) ' F[−i], Φ(E) ' E[−j] and Yoneda’s formula (cf. Proposition A.68). 

1.4

The equivariant case

If an algebraic (typically, finite) group G acts on an algebraic variety X, one may define the equivariant derived category of coherent sheaves on X (cf. [39]). This is defined in terms of coherent sheaves carrying a linearized action of G, compatible with the action on X. In this section we provide the basics of this theory, which we shall then use in Chapter 7, notably in connection with the proof of the McKay correspondence.

1.4.1

Equivariant and linearized derived categories

Let G be an algebraic group. We denote by µ : G × G → G and e : Spec(k) ,→ G the product and the unity of G, respectively. If X is an algebraic variety, a left action of G on X is given by a morphism σ : G × X → X such that • the following diagram is commutative G×G×X

/ G×X

µ×IdX

σ

IdG ×σ

 G×X

σ

e×Id

 /X σ

• the composition Spec(k) × X ' X −−−−X →G×X − → X is the identity. If g ∈ G is a closed point, we denote also by g the composition X ' {g}×X ,→ µ G×X − → X; it acts on closed points as g(x) = g · x = µ(g, x). A G-linearization of a sheaf E of OX -modules is an isomorphism of sheaves on G × X ∼ π∗ E , ψ : σ∗ E → 2 (where π2 : G × X → X is the projection) satisfying the cocycle condition (µ × ∗ (ψ) ◦ (σ × IdG )∗ (ψ). IdX )∗ (ψ) = π23 We shall be particularly interested in the situation where G is a finite group. Then a linearization ψ of E, or better, the inverse isomorphism λ = ψ −1 , is charac∼ g ∗ E fulfilling the conditions λE = Id terized by a family of isomorphisms λEg : E → 1

1.4. The equivariant case

25

and λEgh = h∗ (λEg ) ◦ λEh . By a G-linearized sheaf we understand a pair (E, λE ) where E is a G-equivariant sheaf and λE is (the inverse of) a G-linearization for E. Example 1.35. If X is a smooth variety acted on by a finite group G, the canonical line bundle ωX is canonically linearized. The isomorphism g : X → X induces an 4 isomorphism of sheaves g ∗ ωX ' ωX , and we take λωX as the inverse. Example 1.36. If a finite group G acts trivially on an algebraic variety X (i.e., the action morphism σ : G × X → X is the projection π2 ), a G-linearization of E is a representation G → AutOX (E), that is, is an action of G on E. We can then 4 denote by E G the subsheaf of G-invariant sections of E. In the remainder of this section we assume that G is finite (see [39] for the general case). If (E, λE ) and (F, λF ) are two linearized OX -modules, the local Hom sheaf HomOX (E, F) has a natural linearization defined by letting HomOX (E,F )

λg

∗ E −1 (φ) = λF g ◦ g (φ) ◦ (λg )

on every invariant open subset U ⊆ X. By taking global sections one obtains a right action of G on HomX (E, F), HomOX (E,F ) given by φg = λg (φ). One can then introduce the category ModG (X) of G-linearized OX -modules by defining HomModG (X) ((E, λE ), (F, λF )) as the invariant part HomG X (E, F) of HomX (E, F). A simple computation shows that if G φ ∈ HomX (E, F), the linearizations λEg induce linearizations on both ker(φ) and coker(φ); using this fact one easily checks that ModG (X) is an Abelian category. One can also define the full Abelian subcategories QcohG (X) and CohG (X) of G-linearized quasi-coherent and coherent sheaves. The linearizations of E and F also endow the tensor product E ⊗ F with a natural linearization, so that the tensor product gives a functor QcohG (X) × QcohG (X) → QcohG (X). Let ϕ : G → H be a morphism of finite groups. If Y is an algebraic variety acted on by H, a morphism f : X → Y is ϕ-equivariant if it is compatible with the actions of G and H, that is, if the following diagram is commutative: G×X

ϕ×f

σ

 X

f

/ H ×Y 

σ

/Y.

When no confusion can arise, we shall simply say that f is equivariant. ∗ If F is an H-linearized sheaf on Y , the inverse images f ∗ (λF h ): f F → ∗ ∗ ∗ f (h F) ' h (f F) give an H-linearization for f F. By using ϕ we get a Glinearization for f ∗ . Therefore, the inverse image yields a right exact functor ∗

∗

26

Chapter 1. Integral functors

f ∗ : ModH (Y ) → ModG (X) which maps QcohH (Y ) and CohH (Y ) to QcohG (X) and CohG (X), respectively. If we now take a G-linearized sheaf E on X, the direct images f∗ (λEg ) : f∗ E → f∗ (g E) ' g ∗ (f∗ E) give a G-linearization for f∗ E. In order to induce an Hlinearization we impose that ϕ is surjective. The kernel ker(ϕ) acts trivially on Y , and one can consider the sheaf of ker(ϕ)-invariant sections (f∗ E)ker(ϕ) (cf. Example 1.36). The G-linearization of f∗ (E) induces then an H-linearization of (f∗ E)ker(ϕ) . Then the direct image defines a left exact functor ∗

f ker(ϕ)

∗ −−→ ModH (Y ) ModG (X) −−

ker(ϕ)

E 7→ f∗

(E) = (f∗ E)ker(ϕ) ,

which maps QcohG (X) to QcohH (Y ). We call it the equivariant direct image. If ker(ϕ) in addition f is proper, the equivariant direct image f∗ maps CohG (X) to H Coh (Y ). Example 1.37. Let G be a finite group acting freely on a projective smooth variety X. Then there is a geometric quotient Y = X/G and it is also smooth and projective. The quotient morphism φ : X → Y is equivariant (with respect to the natural projection f : G → {1}) and one easily checks that the functors π ∗ : Coh(Y ) → CohG (X) ,

π∗G : CohG (X) → Coh(Y )

are quasi-inverse of each other and hence, equivalences of categories (cf. [229]). 4 By a classical result of Grothendieck [133, Prop. 5.1.2], the category QcohG (X) has enough injectives. One can then take G-linearized injective resolutions to right derive any left-exact functor. Examples are the global G-invariant homomorphisms ker(ϕ) for an equivariant map F 7→ HomG X (E, F) and the equivariant direct image f∗ f (when ϕ is surjective). The derived functors of the G-invariant homomorphisms are usually denoted ExtG,i X (E, F), and they are naturally isomorphic to the Ginvariant parts of the ordinary ExtiX (E, F) for its natural G-action. If we assume that X is projective, since G is finite there exists a G-equivariant ample line bundle, and this implies that every G-invariant sheaf has a left resolution by locally free G-equivariant sheaves. Thus, we can left derive any right exact functor on QcohG (X). In the case of the functor HomG we obtain the Ginvariant Ext’s as defined before. We can also left-derive the tensor product ⊗ and the inverse image f ∗ under an equivariant map. The G-linearized categories, being Abelian, have associated derived categories. In particular, we can consider the full subcategory DG (X) of the derived category D(QcohG (X)) consisting of complexes with coherent cohomology sheaves, and the corresponding subcategories DG,b (X), DG,+ (X) and DG,− (X) of

1.4. The equivariant case

27

bounded, bounded below, and bounded above complexes, respectively. The natural functor D(CohG (X)) → D(QcohG (X)) induces an equivalence between the bounded derived categories Db (CohG (X)) and DG,b (X). By proceeding as in the case of usual derived categories of coherent sheaves, L

we can introduce the G-equivariant derived functors E • ⊗ F • and RHomOX (E • , F • ) for complexes E • and F • in DG,b (X). Much in the same way as in Corollary A.88 one shows that if E • is an object in DG,b (X) of finite homological dimension, then L

the functor (−) ⊗ E •∨ : DG,b (X) → DG,b (X) is both left and right adjoint to the L

functor (−) ⊗ E • : DG,b (X) → DG,b (X); namely, there are functorial isomorphisms L

L

HomDG,b (X) (F • , G • ⊗ E • ) ' HomDG,b (X) (F • ⊗ E •∨ , G • ) L

L

(1.23)

HomDG,b (X) (F • , G • ⊗ E •∨ ) ' HomDG,b (X) (F • ⊗ E • , G • ) for F • and G • in DG,b (X). Let us fix in the rest of this section a surjective morphism of groups ϕ : G → H, two algebraic varieties X, Y acted on by G and H respectively, and a proper equivariant morphism f : X → Y . We then have the derived functors ker(ϕ)

Rf∗

Lf ∗ : DH,b (Y ) → DG,b (X) .

: DG,b (X) → DH,b (Y ) ,

Both functors are compatible with composition of equivariant morphisms. Let ψ : H → K be another group morphism and f¯: Y → Z a ψ-equivariant morphism of algebraic varieties (where K acts on Z); then, for every object F • in DK,− (Z) there is a functorial isomorphism Lf ∗ (Lf¯∗ F • ) ' L(f¯ ◦ f )∗ (F • ) in DG,− (X). For the direct image, we have to assume that ψ is surjective. Then, for every object E • ∈ DG (X), there is a functorial isomorphism ker(ψ)

Rf¯∗

ker(ϕ)

(Rf∗

ker(ψ◦ϕ) (E • )) ' R(f¯ ◦ f )∗ (E • ) .

(1.24)

in DK (Z). ker(ϕ)

As in the ordinary case, Rf∗ functorial isomorphism

is a right adjoint to Lf ∗ , that is, there is a ker(ϕ)

HomDG,b (X) (Lf ∗ F • , E • ) ' HomDH,b (Y ) (F • , Rf∗

E •) .

(1.25)

One also has an equivariant projection formula in DH,b (Y ) ker(ϕ)

Rf∗

L

ker(ϕ)

(E • ) ⊗ F • ' Rf∗

for E • in DG,b (X) and F • in DH,b (Y ).

L

(F • ⊗ Lf ∗ F • ) ,

(1.26)

28

Chapter 1. Integral functors

Example 1.38. Let G be a finite group acting freely on a projective smooth variety X and φ : X → Y quotient morphism. A direct consequence of Example 1.37 is that the functors Lπ ∗ : Db (Coh(Y )) → Db (CohG (X)) ,

Rπ∗G : Db (CohG (X)) → Db (Coh(Y )) ,

are quasi-inverse of each other and hence, equivalences of categories. Moreover, they induce equivalences of categories Lπ ∗ : Db (Y ) ' DG,b (X) ,

Rπ∗G : DG,b (X) ' Db (Y ) ,

which are also inverse of each other. An analogous statement is true for the derived categories of bounded below, bounded above and unbounded complexes. 4 The last standard property we would like to mention is the equivariant flat base change. If ψ : K → H is another group morphism, Z is an algebraic variety acted on by K and φ : Z → Y is a ψ-equivariant morphism, we can consider diagrams K ×H G

¯ ψ

/G

Z ×Y X

ϕ

ϕ ¯

φX

fZ

/X f

    φ ψ /Y /H Z K of group morphisms and of morphisms of algebraic varieties, respectively. Each morphism in the second diagram is equivariant with respect to the corresponding morphism in the first diagram. Note that ϕ¯ is still surjective and ker ϕ¯ ' ker ϕ. Then an equivariant analogue to Proposition A.85 holds true, namely, if either f or φ is flat, there is a functorial isomorphism in DK,b (Z) ker(ϕ)

φ∗ Rf∗

ker(ϕ)

E • ' RfZ∗

(φ∗X E • )

(1.27)

for any complex E • in DG,b (X). A more delicate issue concerns Grothendieck duality for a proper equivariant morphism f : X → Y . Luckily enough, the general duality formalism developed in [234] also applies to this case, once one checks that the equivariant direct image functor is compatible with small coproducts; this can be seen as in the case of the ordinary direct image. Then, as a consequence of the results in [234], the equivker(ϕ) ariant derived direct image functor Rf∗ : DG,b (X) → DH,b (Y ) has a right ker(ϕ),! H,b G,b : D (Y ) → D (X), that is, there is a functorial isomorphism adjoint f ker(ϕ)

HomDH,b (Y ) (RfX∗

F • , G • ) ' HomDG,b (X) (F • , f ker(ϕ),! G • ) .

(1.28)

The complex f ker(ϕ),! OY is called the G-linearized dualizing complex of f , and whenever f is smooth of relative dimension n, one has f !,ker(ϕ) OY ' ωX/Y [−n] in DG,b (X), where we endow the ordinary relative dualizing sheaf ωX/Y with its natural linearization, which is defined as in Example 1.35.

1.4. The equivariant case

1.4.2

29

Equivariant integral functors

Since we have defined direct and inverse images and tensor product for linearized complexes, one can define integral functors in the equivariant setting. Let G, H be finite groups acting on two proper algebraic varieties X and Y , respectively. Then G × H acts naturally on the product X × Y , and the projections πX : X × Y → X, πY : X × Y → Y are equivariant with respect to the surjective group morphisms φG : G × H → G and φH : G × H → H, respectively. Note that ker φG ' H and ker φH ' G. We can now consider “linearized kernels,” that is, objects K• in the linearized derived category DG×H,− (X × Y ). The equivariant integral functor with kernel • ,G×H : DG,− (X) → DH,− (Y ) given by K• is the functor ΦK X→Y •

L

,G×H ∗ • ΦK (E • ) = RπYG∗ (πX E ⊗ K• ) , X→Y

(1.29)

where all the functors involved are taken in the linearized sense. Most of the results about integral functors previously described apply to equivariant integral functors, due to the properties described in Section 1.4.1. We report here a few properties that will be relevant to the proof of the derived McKay correspondence, which we shall discuss in Chapter 7. When K• is of finite Tor-dimension as a complex of •OX -modules, one can ,G×H is bounded and proceed as in the proof of Proposition 1.4 to prove that ΦK X→Y K• ,G×H G H : D (X) → D (Y ) which maps DG,b (X) can be extended to a functor ΦX→Y H,b to D (Y ). As for ordinary integral functors, the composition of two linearized integral functors is obtained by convoluting in the linearized sense the corresponding kernels; if Z is another algebraic variety acted on by a finite group K, given two kernels K• in DG×H,− (X × Y ) and L• in DH×K,− (Y × Z) corresponding to equivariant integral functors Φ and Ψ, the composition Ψ ◦ Φ : DG,− (X) → DK,− (Z) has kernel L H ∗ (πXY K• ⊗ πY∗ Z L• ) L• ∗ K• = RπXZ∗ in DG×K,− (X × Z), where the morphisms πXY , πXZ∗ and πY Z are the projections of the product X × Y × Z onto the fiber products X × Y , X × Z and Y × Z, and all functors are taken in the linearized sense. The proof is analogous to that of Proposition 1.3, and uses Equation (1.24), together with the linearized flat base change formula (1.27) and the linearized projection formula (1.26). Adjoints to equivariant integral functors can be computed as for ordinary integral functors (cf. Proposition 1.13), using the properties of Grothendieck duality in the linearized setting, and in particular Equation 1.28. For future reference let us describe the adjunction property we shall need. We consider a linearized kernel K• in DG×H,b (X × Y ) of finite Tor-dimension as a complex of OX -modules, and the corresponding linearized integral functor. Using the adjunction properties

30

Chapter 1. Integral functors

given by Equations (1.23), (1.25) and (1.28), and proceeding as in the proof of Proposition 1.13, we obtain the following description of the adjoint of a linearized integral functor. Proposition 1.39. Assume that K• is of finite Tor-dimension as a complex of OX -modules and that Y is smooth. The kernel K•∨ ⊗ πY∗ ωY [n] is an object of DG×H,b (X × Y ) of finite Tor-dimension, and the corresponding linearized integral functor K•∨ ⊗π ∗ ω [n],H×G ΦY→X Y Y : DH,b (Y ) → DG,b (X) , •

,G×H . where n = dim Y , is a left adjoint to ΦK X→Y

1.5

Notes and further reading

The first systematic investigation of integral functors is contained in Mukai’s seminal paper [224], where X is an Abelian variety, Y its dual variety and the kernel in Db (X × Y ) is provided by the normalized Poincar´e bundle (cf. Chapter 3). Subsequently, some authors started studying integral functors in more generality, notably Maciocia [202] and Bondal-Orlov [48, 49]. In particular, in [202] the notion of strong simplicity was first introduced, although it had already been used implicitly in [48, 50]. The theory has been somehow settled by Bridgeland’s paper [61]. Serre functors have been studied to some extent by Bondal and Kapranov [45]. The characterization of fully faithful integral functors in terms of the kernel (cf. Theorem 1.27) has been generalized to Gorenstein varieties [144] and CohenMacaulay varieties [143]. Equivariant integral functors and autoequivalences of equivariant derived categories have been studied by Ploog in [250].

Chapter 2

Fourier-Mukai functors Introduction According to a fundamental theorem due to D. Orlov, any equivalence between derived categories of coherent sheaves of two smooth projective varieties is an integral functor. This “representability” result — which lies at the heart of the current chapter — opens the way to the investigation of the geometric consequences of the equivalence between the derived categories of two varieties. Section 2.1 is quite technical; it presents and develops the notions of spanning class, ample sequence and convolution (of a complex of objects in the derived category). These will be mainly used in the next section and may at first be given little attention, concentrating on definitions and main results, leaving proofs and details to a second reading. Section 2.2 pivots around Orlov’s theorem. An important ingredient of the proof that we provide for this result is a construction of the resolution of the diagonal of a projective variety which generalizes Be˘ılinson’s resolution of the diagonal of projective space. This generalization is originally due to Kapustin, Kuznetsov and Orlov and has been further formalized by Kawamata. In Section 2.3 we specialize our attention to those integral functors which establish an equivalence of categories. We call such functors Fourier-Mukai functors, reserving the term Fourier-Mukai transforms to the cases where the associated kernel is a concentrated complex (i.e., it is a sheaf). One of the main objectives of the section is to state and prove (basically following [61]) a criterion for testing whether a fully faithful integral functor is a Fourier-Mukai functor. The existence of an equivalence between the derived categories of two varieties of course severely constrains their geometry, and indeed one proves that whenever two smooth projective varieties have equivalent derived categories, and one of them has an ample canonical bundle, then they are isomorphic. The section also provides other geoC. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_2, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

31

32

Chapter 2. Fourier-Mukai functors

metric applications, some of them concerning moduli spaces of sheaves.

2.1

Spanning classes and equivalences

We introduce the notion (due to Bridgeland [61] but already implicit in [48]) of spanning class for a triangulated category. Definition 2.1. Let A be a triangulated category. A subclass Σ ⊂ Ob(A) is a spanning class if the following two properties are satisfied: 1. if HomiA (σ, a) = 0 for all σ ∈ Σ and all i ∈ Z, then a = 0; 2. if HomiA (b, σ) = 0 for all σ ∈ Σ and all i ∈ Z, then b = 0. 4 Example 2.2. The skyscraper sheaves Ox form a spanning class for the derived category Db (X) of a smooth projective variety, as we shall see in Proposition 2.52. Moreover, if L is an ample sheaf, then {Li }i∈Z is also a spanning class for Db (X) by virtue of Proposition 2.9. In particular {OPn (i)}i∈Z is a spanning class for the bounded derived category of the projective n-space. (Actually, for any m ∈ Z the 4 collection {Li }i<m is a spanning class as well.) Definition 2.3. A triangulated category A is decomposable if there exist two triangulated nontrivial full subcategories A1 , A2 such that 1. For every object a in A there exist objects ai in Ai and a triangle 0

→ a1 [1] ; a1 → a → a2 − 2. For every pair of objects a1 , a2 in A1 , A2 one has HomiA (a1 , a2 ) = HomiA (a2 , a1 ) = 0 ,

for any i ∈ Z.

We then write A ' A1 ⊕ A2 . A triangulated category A is indecomposable if it is not decomposable. 4 Lemma 2.4. Let F : A → B be an exact fully faithful functor with a right adjoint H and a left adjoint G. Assume that B is indecomposable and that A is nontrivial. Then F is an equivalence if and only if the condition H(c) = 0 for an object c in B implies G(c) = 0.

2.1. Spanning classes and equivalences

33

Proof. If F is an equivalence, then G ' H, and there is nothing to prove. For the converse, define full subcategories B1 , B2 consisting of the objects b in B such that F H(b) ' b and H(b) = 0 respectively. If b1 , b2 are objects in B1 , B2 , one has HomiB (b1 , b2 ) ' HomiB (F H(b1 ), b2 ) ' HomiA (H(b1 ), H(b2 )) = 0 and HomiB (b2 , b1 ) ' HomiB (b2 , F H(b1 )) ' HomiA (G(b2 ), H(b1 )) = 0 because G(b2 ) = 0 by hypothesis. Moreover, for any object b in B the counit morphism (b) : F H(b) → b can be embedded into a triangle (b)

δ

→ F H(b)[1] . F H(b) −−→ b → c − Applying H we get H(c) = 0; in fact, H((b)) : HF H(b) → H(b) is an isomorphism since F is fully faithful. Then F H(b) is an object of B1 and c is in B2 . Moreover the H(δ)

composition of 0 = H(c) −−−→ HF H(b)[1] ' Hb[1] is the morphism corresponding δ

→ F H(b)[1], and then δ = 0. Since B is indecomposable, either by adjunction to c − B1 or B2 is trivial. If B1 is trivial, then B2 = B so that H = 0; then for every object a of A we have HomA (a, a) ' HomB (F (a), F (a)) ' HomA (a, H(F (a))) = 0 and this implies that A is trivial. But this is impossible, and therefore B2 is ∼ b for any object b in B. Thus trivial, which means that c = 0 and (b) : F H(b) →  F H ' IdB , and F is an equivalence. As shown in Corollary 1.18, exact equivalences of categories intertwine Serre functors. Whenever this intertwining property holds true for all objects in a spanning class, under some additional assumptions one has a converse statement. Proposition 2.5. Let F : A → B be an exact fully faithful functor of triangulated categories with Serre functors SA , SB . Assume that B is indecomposable, A is not trivial and that F has a right adjoint H. Then F is an equivalence if and only if SB F (σ) = F SA (σ) for all σ in some spanning class Σ ⊂ Ob(A). −1 ◦ H ◦ SB . For any Proof. By Lemma 1.17, F has a left adjoint given by G = SA object b in B, any σ ∈ Σ and any i ∈ Z, we have

HomiA (σ, G(b)) ' HomiA (G(b), SA (σ))∗ ' HomiA (b, F SA (σ))∗ ' HomiA (b, SB F (σ))∗ ' HomiA (F (σ), b) ' HomiA (σ), H(b)) . Then G(b) = 0 if and only if H(b) = 0, so that F is an equivalence by Lemma 2.4.  Spanning classes may be used to test whether a functor is fully faithful.

34

Chapter 2. Fourier-Mukai functors

Theorem 2.6. [61] Let F : A → B be an exact functor of triangulated categories, admitting a left and a right adjoint. The functor F is fully faithful if and only if the morphism F : HomiA (σ, τ ) → HomiB (F (σ), F (τ )) is an isomorphism for every σ, τ in some spanning class Σ for A. Proof. Let H, G be a right and a left adjoint to F and consider the corresponding units and counits η : IdA → H ◦ F

 : F ◦ H → IdB

ξ : IdB → F ◦ G

δ : G ◦ F → IdA

We have a commutative diagram η(τ )∗

/ HomA (σ, H ◦ F (τ )) HomiA (σ, τ ) SSSS S SSSFS β ' δ(σ)∗ SSSS SS)   ' HomiA (G ◦ F (σ), τ ) α / HomiB (F (σ), F (τ ))

(2.1)

where α = ξ(F (σ))∗ ◦ F and β = (F (τ ))∗ ◦ F are the isomorphisms given by adjunction. Since F is an isomorphism for all σ, τ ∈ Σ, all morphisms in diagram 2.1 are isomorphisms for all σ, τ ∈ Σ. Taking this into account we proceed in three steps: (1) We prove that δ(σ) : G ◦ F (σ) → σ is an isomorphism for every σ ∈ Σ. δ(σ)

Indeed, δ(σ) fits into the triangle G ◦ F (σ) −−−→ σ → ρ → G ◦ F (σ)[1]. For every τ ∈ A we have an exact sequence δ(σ)∗

−1 · · · → Hom−1 A (σ, τ ) −−−→ HomA (G ◦ F (σ), τ ) → δ(σ)∗

HomA (ρ, τ ) → HomA (σ, τ ) −−−→ HomA (G ◦ F (σ), τ ) → . . . If τ ∈ Σ, then all morphisms δ(σ)∗ in this exact sequence are isomorphisms, so that HomiA (ρ, τ ) = 0 for every i ∈ Z and ρ = 0. Thus δ(σ) is an isomorphism. (2) Next we show that η(τ ) : τ → H ◦ F (τ ) is an isomorphism for every η(τ )

τ ∈ Ob(A). We can embed η(τ ) into the triangle τ −−−→ H ◦ F (τ ) → ρ → τ [1] and get for every σ ∈ Σ the exact sequence η(τ )∗

· · · → HomA (σ, τ ) −−−→ HomA (σ, H ◦ F (τ )) → HomA (σ, ρ) → η(τ )∗

Hom1A (σ, τ ) −−−→ Hom1A (σ, H ◦ F (τ )) → . . .

2.1. Spanning classes and equivalences

35

As σ ∈ Σ, by (1) δ(σ) is an isomorphism, hence δ(σ)∗ is an isomorphism. By diagram (2.1) F is an isomorphism, so that η(τ )∗ is an isomorphism as well for every σ ∈ Σ, τ ∈ Ob(A). By the above exact sequence one has HomiA (σ, ρ) = 0 for every i ∈ Z, σ ∈ Σ and thus ρ = 0 and η(τ ) is an isomorphism. (3) Finally, we prove that F is fully faithful. Since η(τ ) is an isomorphism for every τ ∈ Ob(A), η(τ )∗ is an isomorphism for every σ, τ ∈ Ob(A), and then F is an isomorphism by diagram (2.1). 

2.1.1

Ample sequences

Let A be a k-linear Abelian category. Definition 2.7. A sequence {Pi }i∈Z of objects in A is said to be ample if for every object C of A there is an integer i0 = i0 (C) such that the following conditions hold for i < i0 : 1. the natural morphism HomA (Pi , C) ⊗ Pi → C is surjective; 2. HomjDb (A) (Pi , C) = 0 for j 6= 0; 3. HomA (C, Pi ) = 0. 4 Condition 1 is equivalent to the existence of an exact sequence α

ρ

Pj⊕s − → Pi⊕k − → A → 0. The most important example is provided by the sequence {Li }i∈Z in the category of quasi-coherent sheaves on a projective variety where L is an ample line bundle (here Pi = L−i ). The following results are used in the proof of Orlov’s representability theorem 2.15. Lemma 2.8. [242] Let {Pi }i∈Z be an ample sequence in A. An object A• of Db (A) is isomorphic to an object of A (i.e., it is isomorphic in Db (A) to a complex concentrated in degree zero) if and only if HomjDb (A) (Pi , A• ) = 0 for j 6= 0 and i  0. Proof. The “only if” part is clear by the definition of ample sequence. Now assume that HomjDb (A) (Pi , A• ) = 0 for j 6= 0 and i  0. Embed A into an Abelian category with enough injectives, so that HomjDb (A) (Pi , C) = Extj (Pi , C), where the Exts are computed in the larger category. Since A• has bounded cohomology, we can

36

Chapter 2. Fourier-Mukai functors

find i  0 such that Extj (Pi , H q (A• )) = 0 for every q. Then there is a convergent p+q = Homp+q (Pi , A• ). If spectral sequence E2p,q = Extp (Pi , H q (A• )) with E∞ D b (A) H q (A• ) 6= 0, we have E20,q = Hom(Pi , H q (A• )) 6= 0 for i  0 by the first condition in Definition 2.7, so that any nonzero element in E20,q survives to infinity yielding q = HomqDb (A) (Pi , A• ). Thus q = 0 and A• ' H 0 (A• ) in a nonzero element in E∞  Db (A). Proposition 2.9. Let {Pi }i∈Z be an ample sequence in A. Assume that Db (A) has a Serre functor. Then the class Σ = {Pi }i∈Z ⊂ Ob(Db (A)) is a spanning class for Db (A). Proof. Let A• be an object of Db (A) and assume that HomjDb (A) (Pi , A• ) = 0 for every i and j. By Lemma 2.8, A• is isomorphic to an object A in A. By the first condition in Definition 2.7, one has A = 0. Now take a complex A• in Db (A) such that HomjDb (A) (A• , Pi ) = 0 for every i and j. If S is the Serre functor of the category Db (A), then HomjDb (A) (Pi , S(A• )) ' HomDb (A) (Pi , S(A• [j])) ' HomDb (A) (A• [j], Pi )∗ ' Hom−j (A• , Pi )∗ = 0 for every i and j. Then by the D b (A) •  previous argument we have S(A [j]) = 0 for every j, so that A• = 0. We shall also denote by Σ the full subcategory of Db (A) whose objects are {Pi }i∈Z . Proposition 2.10. [242, Prop. 2.16] Let {Pi }i∈Z be an ample sequence of objects in ∼ A. Let F : Db (A) → Db (A) be an exact equivalence. Every isomorphism h : IdΣ → ∼ b F|Σ can be extended to an isomorphism IdDb (A) → F on D (A). ∼ F (P ) depending functoriProof. We have for every i an isomorphism hPi : Pi → i ally on Pi . Our task is to extend these isomorphisms to functorial isomorphisms hA• : A• → F (A• ) for every object A• in Db (A). We divide this rather long proof into four steps. (1) A• ' ⊕nk=1 Pik . In this case F (A• ) ' ⊕nk=1 Pik and we simply set hA• to be the direct sum of the given hPik . (2) A• is isomorphic to an object A of A. We first note that F (A) is an object of A since HomjDb (A) (Pi , F (A)) ' HomjDb (A) (F (Pi ), F (A)) ' HomjDb (A) (Pi , A) due to Pi ' F (Pi ), and we can apply Lemma 2.8. By definition of ample sequence there is an exact sequence α

ρ

Pj⊕s − → Pi⊕k − →A→0

2.1. Spanning classes and equivalences

37

for i, j  0. We then have a diagram Pj⊕s

/ P ⊕k

α

s ' hPj

 F (Pj⊕s )

ρ

/A

F (ρ)

/ F (A) .

i

/0

' hk Pi

F (α)

 / F (Pi )⊕k

Since F (ρ) ◦ hkPi ◦ α = F (ρ) ◦ F (α) ◦ hsPj = F (ρ ◦ α) ◦ hsPj = 0, there is an iso∼ F (A) which completes the diagram. Moreover h is unique morphism hA : A → A as Hom(A, F (A)) ,→ Hom(F (Pi )⊕k , F (A)). This also implies that hA is indepenρ dent of the choice of the surjection Pi⊕k − → A → 0. One can easily check that hA depends functorially on A. (3) A• ' A[n] for an object A of A. Since F (A[n]) ' F (A)[n] we simply set hA[n] = hA [n]. (4) A• is any object in Db (A). We proceed by induction on the length N = `(A ), which is the number of nonzero cohomology objects of A• . The case N = 1 corresponds to the objects of A and has been considered in the previous steps. Let us then take N > 1. •

Let q be the maximum of the integers such that H q (A• ) 6= 0. Then we can find an index i and a surjective morphism Pi⊕k → ker dq inducing a morphism φ : Pi⊕k [−q] → A• ≤q ' A• in the derived category. We can also choose the index i so that: (a) the induced morphism H q (φ) : Pi⊕k → H q (A• ) is surjective; (b) HomjDb (A) (Pi , H p (A• )) = 0 for all j 6= 0 and for all p; (c) HomjDb (A) (H q (A• ), Pi ) = 0 for all j 6= 0 and for all q. φ

We can now embed φ into an exact triangle B• → Pi⊕k [−q] − → A• → B • [1]. Since • `(B ) = N − 1, induction provides a commutative diagram B• [−1] ' hB• [−1]



F (B • )[−1]

/ P ⊕k [−q]

φ

i

/ A•

ψ

' hP ⊕k [−q]



/ B• ' hB •

i

/ F (P ⊕k )[−q]F (φ) / F (A• ) i

F (ψ)

 / F (B • ) .

∼ F (A• ) making the above diagram Then there exists an isomorphism hA• : A• → into a morphism of triangles. Moreover, this morphism is the only one satisfying the condition F (ψ) ◦ hA• = hB• ◦ ψ, as HomDb (A) (A• , F (Pi⊕k )[−q])

'

HomDb (A) (A• , Pi⊕k [−q])

'

HomqDb (A) (A• , Pi⊕k ) = 0

38

Chapter 2. Fourier-Mukai functors

Again, one easily proves that the morphism hA• does not depend on the choice of the morphism φ : Pi⊕k [−q] → A• ≤q ' A• . Finally, we prove that this construction is functorial. Let us take a morphism $ : A• → C • with `(C • ) ≤ N . We must prove that the diagram $

A• ' hA•

 F (A• )

/ C• ' hC•

F ($)

 / F (C • )

(2.2)

is commutative. Let q be as above the maximum of the integers such that H q (A• ) 6= 0 and p the maximum of the integers such that H p (C • ) 6= 0. We consider separately the cases p < q and p ≥ q. The case p < q. We take as above a morphism φ : Pi⊕k → A• and the corresponding exact triangle φ

ψ

→ A• − → B• → Pi⊗k [−q + 1] . Pi⊗k [−q] − We can assume HomDb (A) (Pi , H j (C • )) = 0 for all j, so that HomDb (A) (Pi , C • ) = 0. Then $ factors through a morphism ρ : B• → C • . Since `(B • ) = `(A• ) − 1, the diagram ρ

B• ' hB•

 F (B • )

/ C• ' hC•

F (ρ)

 / F (C • )

commutes by induction. Then (2.2) is commutative as well. The case p ≥ q. We proceed by induction on p − q, the case p − q = −1 being included in the preceding case. We can find an index i and a surjective morphism Pi⊕k → ker dp . This gives a morphism φ : Pi⊕k [−p] → C • ≤p ' C • in the derived category. Moreover, the induced morphism H p (φ) : Pi⊕k → H p (C • ) is surjective. We can also assume that HomDb (A) (H j (A• ), Pi⊕k ) = 0 for all j, so that HomDb (A) (A• , Pi⊕k [−p]) = 0. We have an exact triangle φ

β

Pi⊕k [−p] − → C• − → M• → Pi⊕k [−p + 1] and the maximum of the integers p˜ with H p˜(M• ) 6= 0 is smaller than p. Let us

2.1. Spanning classes and equivalences

39

consider the diagram 9 C LLL tt LL β $ ttt LL LL tt t L% tt β◦$ / M• A• •

' hC•

 F (C • ) ' hM• JJ v: JJF (β) F ($) vv JJ vv JJ vv J%   vv F (β◦$) / F (M• ) . F (A• ) ' hA•

By induction, the rectangular-shaped subdiagram is commutative, and by the construction of the morphisms hC • the lozenge-shaped subdiagram on the right is commutative as well. Then F (β) ◦ (hC • ◦ $ − F ($) ◦ hA• ) = 0 and since HomDb (A) (H p (A• ), Pi⊕k ) = 0 we obtain that hC • ◦ $ − F ($) ◦ hA• = 0, that is, the diagram (2.2) is commutative.  Proposition 2.11. [176, Lemma 6.5] Let F, F¯ : Db (A) → B, be exact functors, where B is a k-linear triangulated category. Assume that: 1. Db (A) has a Serre functor; ¯ and right adjoint functors H, H ¯ 2. F, F¯ have left adjoint functors G and G (note that by Lemma 1.17, if B has a Serre functor, then right adjoints if and only if left adjoints exist); 3. F is fully faithful; 4. there is an ample sequence {Pi }i∈Z of objects of A and an isomorphism of ∼ F¯ , where Σ is the full subcategory of Db (A) whose functors fΣ : F|Σ → |Σ objects are {Pi }i∈Z . ∼ F¯ extending f . Then there is an isomorphism of functors f : F → Σ Proof. Since the restrictions of F and F¯ to Σ are isomorphic and {Pi }i∈Z is a spanning class by Proposition 2.9, Theorem 2.6 implies that F¯ is fully faithful. ∼ HF Since F and F¯ are fully faithful, there are functor isomorphisms IdDb (A) → ∼ ¯ F¯ → IdDb (A) . and G ¯ is left adjoint to H F¯ . Since both composed functors are isomorphic Now, GF to the identity when rectricted to Σ, again by Theorem 2.6 they are fully faithful. Whenever a fully faithful functor admits a left adjoint which is fully faithful, then it is an equivalence of categories. Thus, H F¯ is an exact equivalence. By Proposition

40

Chapter 2. Fourier-Mukai functors

∼ H F¯ can be extended to a functor isomorphism 2.10, the isomorphism IdΣ → |Σ ∼ IdDb (A) → H F¯ . Composing from the left with F and using adjunction, we have a morphism f : F → F¯ extending fΣ . Let us check that f is an isomorphism. For any object A• in Db (A), let fA• → F¯ (A• ) → c → F (A• )[1] F (A• ) −−

be an exact triangle. Since H(fA• ) is an isomorphism we have H(c) = 0 and then ¯ HomDb (A) (Pi , H(c)) ' HomB (F¯ (Pi ), c) ' HomB (F (Pi ), c) ' HomDb (A) (Pi , H(c)) = 0 ¯ for every i, so that H(c) = 0. Thus HomB (F¯ (A• ), c) = 0, and F (A• ) ' F¯ (A• ) ⊕ c from the above exact triangle. Since HomB (F (A• ), c) ' HomDb (A) (A• , H(c)) = 0  we deduce that c = 0 and then fA• is an isomorphism.

2.1.2

Convolutions

Given an object in Abelian category, we are used to associate with it a complex of objects in the derived category; we also know how to associate an object of the derived category with a double complex. Sometimes, dealing with (bounded) complexes of objects in the derived category d−m

d−m+1

d−1

(A• )−m −−−→ (A• )−m+1 −−−−→ · · · → (A• )−1 −−→ (A• )0 , one wishes to construct objects a of the derived category which somehow represent them; one also requires that when the objects of the complex we start with are in the original Abelian category, d−m

d−m+1

d−1

A• ≡ A−m −−−→ A−m+1 −−−−→ · · · → A−1 −−→ A0 the new object a is just the image A• of the complex in the derived category. This is possible under very mild requirements, and the process is called convolution. Let us consider at first a complex d−1

A• ≡ A−1 −−→ A0 of objects of A. If Cone(d−1 ) is the cone of d−1 , the natural morphism A0 → Cone(d−1 ) induces an isomorphism in the derived category ∼ Cone(d ) . A• → −1

2.1. Spanning classes and equivalences

41

Then Cone(d−1 ) represents the complex A• in the derived category Db (A). This is definitely a trivial observation, but this “cone construction” may be straightforwardly extended to complexes of objects in the derived category: given a complex d−1

(A• )−1 −−→ (A• )0 of objects of Db (A), we can take a cone Cone(d−1 ) of d−1 and we have a natural morphism (A• )0 → Cone(d−1 ) and an exact triangle d−1

(A• )−1 −−→ (A• )0 → Cone(d−1 ) → (A• )−1 [1] . We can iterate this process to define the right convolution of a bounded complex of objects of the derived category. Actually, we do not need to work with a derived category, since any triangulated category will do. Let then B be a triangulated category and d−(m−1)

d−m

d−1

a−m −−−→ a−(m−1) −−−−−→ a−(m−2) → · · · → a−1 −−→ a0

(2.3)

a complex of objects of B (that is, the composition of any two consecutive morphisms vanishes). Assume also that one has HomB (a−p [r], a−q ) = 0 ,

for every p > q and r > 0.

(2.4)

Following Orlov and Kawamata [242, 176] we can define the right convolution of the complex 2.3 as the pair formed by the object a of B and the morphism d0 : a0 → a constructed by induction on the length m as follows: •

If m = 0, then a = a0 and d0 is the identity.

•

If m ≥ 1, we let a−(m−1) = Cone(d−m ), so that there is an exact triangle g−(m−1)

d−m

a−m −−−→ a−(m−1) −−−−−→ a−(m−1) → a−m [1] . After taking homomorphisms we have an exact sequence HomB (a−m [1], a−(m−2) ) → HomB (a−(m−1) , a−(m−2) ) → HomB (a−(m−1) , a−(m−2) ) → HomB (a−m , a−(m−2) ) . Since dm−2 ◦ dm−1 = 0 there is a morphism dm−1 : a−(m−1) → a−(m−2) such that dm−1 ◦ gm−1 = dm−1 ; due to condition (2.4) one also has HomB (a−m [1], a−(m−2) ) = 0 and then the morphism is unique. Hence, we obtain a new complex d−(m−1)

d−1

a−(m−1) −−−−−→ a−(m−2) → · · · → a−1 −−→ a0

(2.5)

which also fulfils condition (2.4). •

We iterate the previous steps from this new complex.

Note that a0 remains unchanged during the process. Summing up, we have

42

Chapter 2. Fourier-Mukai functors

Lemma 2.12. Let d−(m−1)

d−m

d−1

a−m −−−→ a−(m−1) −−−−−→ a−(m−2) → · · · → a−1 −−→ a0 be a complex of objects of B that fulfil the condition (2.4). There exists a right convolution d0 : a0 → a in B, which is uniquely determined up to isomorphism. When one works with the derived category B = Db (A) of an Abelian category and the objects a−p of Db (A) are just objects A−p of the Abelian category A, then any complex d−(m−1)

d−m

d−1

A• ≡ A−m −−−→ A−(m−1) −−−−−→ A−(m−2) → · · · → A−1 −−→ A0 fulfils the condition (2.4) and the right convolution a of A• is the complex A• itself, together with the obvious morphism A0 → a = A• . Lemma 2.13. Let d−m

a−m 

f−m

b−m

de−m

/ a−(m−1) 

d−(m−1)

/ ...

f−(m−1)

/ b−(m−1)

de−(m−1)

/ ...

d−1

/ a−1 

/ a0

f−1

/ b−1

de−1



f0

/ b0

be a morphism between complexes of objects of B fulfilling condition (2.4) and let d0 : a0 → a, de0 : b0 → b be right convolutions. If one has HomB (a−p [r], b−q ) = 0

for every p > q and r > 0 ,

(2.6)

then for any morphism h : b → b0 there exists a morphism f : a → b0 in B such that the diagram d a0 0 / a ? ?? ??f (2.7) f0 ?? ?   de0 / b h / b0 b0 is commutative. Moreover, if HomB (a−p [r], b0 ) = 0

for every p > 0 and r > 0 ,

(2.8)

then the morphism f is the only satisfying that property. Proof. The morphism f is constructed inductively. If m = 0, then f = hf0 . If m ≥ 1, we have a commutative diagram a−m f−m



b−m

/ a−(m−1) 

g−(m−1)

/ a−(m−1)

f−(m−1)

/ b−(m−1)

g e−(m−1)

/ b−(m−1)

/ a−m [1] 

f−m [1]

/ b−m [1]

2.1. Spanning classes and equivalences

43

where a−(m−1) and b−(m−1) are cones of the corresponding differentials as before. By the axioms of the triangulated categories, there exists a morphism (not uniquely determined) f −(m−1) : a−(m−1) → b−(m−1) completing the diagram. If we consider e −(m−1) : b−(m−1) → b−(m−2) the morphisms d−(m−1) : a−(m−1) → a−(m−2) and d constructed above, we have e −(m−1) ge−(m−1) f−(m−1) = de−(m−1) f−(m−1) e −(m−1) f−(m−1) g−(m−1) = d d e −(m−1) g−(m−1) = f−(m−2) d−(m−1) = f−(m−2) d e −(m−1) f−(m−1) = f−(m−2) d e −(m−1) . Thus, we have a morphism of comand then d plexes d−(m−1)

/ a−(m−2)

a−(m−1) 

f−(m−1)

/ ...

f−(m−2)

 e −(m−1) d / −(m−2) −(m−1)

b

d−(m−2)

de−(m−2)

b

/ ...

/ a−1 

d−1

/ a0

f−1

/ b−1

de−1



f0

/ b0 .

One easily checks that this morphism of complexes fulfils condition (2.6), and then we obtain the morphism f : a → b0 by induction. If the condition (2.8) is satisfied, f is uniquely determined by the commutativity of diagram (2.7) because  f0 : a0 → b0 does not change during the process. Since right convolutions are constructed out of exact triangles and compositions of morphisms, they are compatible with exact functors. Remark 2.14. Let d−m

d−(m−1)

d−1

a−m −−−→ a−(m−1) −−−−−→ a−(m−2) → · · · → a−1 −−→ a0 be a complex of objects of B fulfilling condition (2.4) and let d0 : a0 → a be the right convolution. If F : B → C is an exact functor to another triangulated category and the complex F (d−m )

F (d−1 )

F (a−m ) −−−−−→ F (a−(m−1) ) → · · · → F (a−1 ) −−−−−→ F (a0 ) of objects of C also fulfils condition (2.4), then its right convolution is F (d0 ) : F (a0 ) → F (a). This happens for instance if F is fully faithful, because in this case HomΣ (F (a−p )[r], F (a−q )) ' HomB (a−p [r], a−q ) = 0 . 4

44

2.2

Chapter 2. Fourier-Mukai functors

Orlov’s representability theorem

This section is devoted to the proof of the following fundamental result by Orlov [242, Thm. 2.2] and some related issues. Theorem 2.15. Let X and Y be smooth projective varieties. Any fully faithful exact functor Ψ : Db (X) → Db (Y ) is an integral functor. This is indeed a deep result since in general functors between triangulated categories are quite difficult to describe. It should be pointed out that much stronger results than Orlov’s theorem hold true in the more flexible setting of dg-categories, which provide an enhancement of the usual derived categories. In this framework, roughly speaking, all functors are integral functors, as Theorem A.57 and, more in particular, Equation A.10 show (see Section A.4.4 and the “Notes and further reading” at the end of the current chapter). Remark 2.16. In our proof of Theorem 2.15 we shall use the fact that any exact functor Db (X) → Db (Y ) has a right adjoint (and hence, since the categories Db (X) and Db (Y ) have Serre functors, a left adjoint as well). This has been proved by Bondal and Van den Bergh [52]. The original result by Orlov was weaker in that this property was assumed in the hypotheses of the theorem. 4

2.2.1

Resolution of the diagonal

One of the first important results about derived categories is Be˘ılinson’s construction of a resolution of the structure sheaf of the diagonal of the product Pn × Pn of two copies of the projective space [33, 34], and its implications for the computation of the derived category Db (Pn ). We present here Be˘ılinson’s resolution as a particular case of the Koszul sequence associated to certain sections of a locally free sheaf. Let E be a locally free sheaf of rank n on an algebraic variety X and e : OX → E a global section. The zero locus of e is the closed subvariety Z of X defined by the exact sequence e∗ E ∗ −→ OX → OZ → 0 . We have a Koszul complex 0→

n ^

i

e E ∗ −→

n−1 ^

i

i

e∗

e e . . . −→ E ∗ −→ OX → OZ → 0 E ∗ −→

(2.9)

where ie is the inner product with e, i.e., for every affine open subset U and Vp ∗ E , we have sections e1 , . . . , ep on U of E and Ωp on U of (ie Ωp )(e1 , . . . , ep ) = Ωp (e, e1 , . . . , ep ) . As a consequence of the theory of Koszul complexes we get the following result.

2.2. Orlov’s representability theorem

45

Corollary 2.17. Assume that every point x of Z has an affine neighborhood in X where E ∗ has a local basis (ω1 , . . . , ωn ) such that (ω1 (e), . . . , ωn (e)) is a regular sequence in OX (U ). Then the Koszul complex (2.9) is exact, thus providing a finite resolution of the structure sheaf OZ of the zero locus of e by locally free sheaves. Now take X = Pn × Pn and write Pn as the projective spectrum of the symmetric algebra Sym(V ), where V is a k-vector space of dimension n + 1. Let us write for simplicity O(m) = OPn (m) and Ω = ΩPn . One has the Euler exact sequence α → V ⊗k O → O(1) → 0 (2.10) 0 → Ω(1) − and taking duals α∗

0 → O(−1) → V ∗ ⊗k O −−→ Ω∗ (−1) → 0 . This gives Γ(Pn , O(1)) ' V and Γ(Pn , Ω∗ (−1)) ' V ∗ so that Γ(X, π1∗ O(1) ⊗ π2∗ Ω∗ (−1)) ' V ⊗k V ∗ ' End(V ) , where π1 , π2 are the projections onto the two factors. It follows that the identity on V defines a global section e of E = π1∗ O(1) ⊗ π2∗ Ω∗ (−1) to which we can apply the precedent discussion on Koszul complexes. If we take a basis (x0 , . . . , xn ) of V , in the open subset Ui where the ho(i) mogenous coordinate xj do not vanish, we have affine coordinates yh = xh /xi . (i) (i) On Ui the morphism α is given by dyh ⊗ x∗i 7→ xh − yh xi (h 6= i) and on (j) Ui × Uj a local basis for E ∗ is {π1∗ (x∗i ) ⊗ π2∗ (dyh ⊗ xj )}. Since the identity e on P ∗ (j) V as a section of E is e = ` π1 (x` ) ⊗ π2∗ ((δ`h − yh ∂y(j) ⊗ x∗j ), computing the h morphism e∗ : E ∗ → OX on Ui × Uj we see that the ideal of the zero set Z of (i) (i) (j) e is generated by π1∗ (yh ) ⊗ 1 − π1∗ (yj ) ⊗ π2∗ (yh ) (h 6= j), and then Z is the n n diagonal ∆ of X = P × P . Moreover, on Ui × Ui we get that those elements are (i) (i) π1∗ (yh ) ⊗ 1 − 1 ⊗ π2∗ (yh ), and they form a regular sequence. Corollary 2.17 implies the existence of a resolution of the diagonal, usually called Be˘ılinson resolution of the diagonal of the projective space. Proposition 2.18. There is an exact sequence 0 → π1∗ O(−n) ⊗ π2∗ Ωn (n) → π1∗ O(−(n − 1)) ⊗ π2∗ Ωn−1 (n − 1) → . . . δ∗

→ π1∗ O(−1) ⊗ π2∗ Ω(1) → OX −→ O∆ → 0 . This provides a resolution of the structure sheaf of the diagonal ∆ ,→ X = Pn × Pn by locally free sheaves.

46

Chapter 2. Fourier-Mukai functors Let j > 0 be a positive integer and consider the exact sequence

0 → π1∗ O(−n) ⊗ π2∗ Ωn (n + j) → π1∗ O(−(n − 1)) ⊗ π2∗ Ωn−1 (n − 1 + j) → . . . δ∗

→ π1∗ O(−1) ⊗ π2∗ Ω(1 + j) → π1∗ O ⊗ π2∗ O(j) −→ O∆ ⊗ π2∗ O(j) → 0 , obtained as the tensor product of the Be˘ılinson resolution by π2∗ O(j). Since all the sheaves Ωp (p + j) (0 ≤ p ≤ n) are acyclic, after taking direct images by π1 we have an exact sequence 0 → O(−n) ⊗k H 0 (Pn , Ωn (n + j)) → O(−(n − 1)) ⊗k H 0 (Pn , Ωn−1 (n − 1 + j)) → · · · → O(−1) ⊗k H 0 (Pn , Ω(1 + j)) → O ⊗k H 0 (Pn , O(j)) → O(j) → 0 , so that there is an exact sequence 0 → O(−j) → V0j ⊗k O → V1j ⊗k O(1) → . . . j ⊗k O(n − 1) → Vnj ⊗k O(n) → 0 → Vn−1

(2.11)

where Vpj = H 0 (Pn , Ωp (p + j))∗ . We shall use this resolution of O(−j) later on. Be˘ılinson’s resolution has been generalized by Kapustin, Kuznetsov and Orlov in [171] and further formalized and studied by Kawamata [176, Theorem 3.2]. We shall consider here Kawamata’s formulation in a way which is sufficient to our purposes. In order to state and prove this generalization we need some preliminaries. Let A = ⊕n∈N An be a graded k-algebra, with A0 = k. Let us define recursively vector spaces Bn , with n ∈ N, as the kernels Bn = ker(Bn−1 ⊗ A1 → Bm−2 ⊗ A2 ) for n ≥ 2, and Bn = An for n = 0, 1. There are natural homomorphisms Bn ⊗ A[−n] → Bn−1 ⊗ A[−n + 1] where A[n] is the shifted module A[n]m = An+m . Definition 2.19. The graded algebra A is Koszul if the sequence · · · → Bn ⊗ A[−n] → Bn−1 ⊗ A[−n + 1] → · · · → B1 ⊗ A[−1] → A → k → 0 (2.12) is exact.

4

2.2. Orlov’s representability theorem

47

A line bundle L on a projective variety X is said to be Koszul if its associated homogeneous coordinate ring M H 0 (X, nL) A= n∈N

is Koszul. Theorem 2 in [33] implies that Lk is Koszul for k big enough if L is ample. By composing the morphisms Bn → Bn−1 ⊗ A1 (tensored by OX ) and A1 ⊗ OX → L, one has morphisms ψn : Bn ⊗ OX → Bn−1 ⊗ L. So we can define sheaves Rn on X (with n ∈ N) as Rn = ker ψn . We have an exact sequence 0 → Rn → Bn ⊗ OX → Bn−1 ⊗ L → · · · → B1 ⊗ Ln−1 → Ln → 0 .

(2.13)

Lemma 2.20. [176] There is an exact sequence 0 → A0 ⊗ Rn → A1 ⊗ Rn−1 → · · · → An−1 ⊗ R1 → An ⊗ R0 → Ln → 0 . (2.14) Proof. For any given n ≥ 0 we introduce a double complex of sheaves ( Ap ⊗ Bn−p−q ⊗ Lq for p, q, n − p − q ≥ 0 p,q G(n) = 0 otherwise. The differentials δ1 , δ2 are induced by the morphisms in the sequences (2.12) and (2.13). If one studies the associated spectral sequences 0E(n) , 00E(n) one sees that the second spectral sequence degenerates at the first step, and ( n L for p = 0, q = n 00 p,q E(n)1 = 0 otherwise. Therefore the cohomology of the total complex T • is H n (T • ) = Ln in degree n and 0 otherwise. The first spectral sequence at first step is ( Ap ⊗ Rn−p for q = 0 0 p,q E(n)1 = 0 otherwise and degenerates at the second step. This implies that δ

δ

1 1 Ap+1 ⊗ Rn−p−1 ) = im(Ap−1 ⊗ Rn−p+1 −→ Ap ⊗ Rn−p ) ker(Ap ⊗ Rn−p −→

for p < n, and δ

1 An ⊗ R0 ) Ln ' (An ⊗ R0 )/ im(An−1 ⊗ R1 −→

so that the exactness of (2.14) follows.



48

Chapter 2. Fourier-Mukai functors

Proposition 2.21. [176] Let X be a projective variety, ∆ ,→ X × X the diagonal of the product, and let L be a Koszul line bundle on X. Define sheaves Rm for every integer m ≥ 0 as before. Then there is an exact sequence dm−1

d

→ π1∗ L−(m−1) ⊗ π2∗ Rm−1 −−−→ . . . · · · → π1∗ L−m ⊗ π2∗ Rm −−m d

d

δ

2 1 −→ π1∗ L−1 ⊗ π2∗ R1 −→ π1∗ OX ⊗ π2∗ OX − → O∆ → 0

of coherent sheaves on X × X. Moreover, we can choose L so that H i (X, Ls ) = 0 for ever i > 0 and s ≥ 1. Proof. Let us define the complex F • Fm

=

π1∗ Lm+1 ⊗ π2∗ R−m−1

0

=

O∆

m

=

0

F F

for m ≤ −1

for m > 0 .

Let m0 be a (nonpositive) integer such that Hm0 (F • ) 6= 0 and let k be an integer which is big enough to ensure that Rp π2∗ (Hq (F • ) ⊗ π1∗ Lk ) = 0 p

k+q

)=0

0

m0

•

H (X, L

R π2∗ (H

(F ) ⊗

for π1∗ Lk )

for

p > 0, q ≥ m0 − dim X

p > 0, q ≥ m0 − dim X 6= 0.

We may associate two spectral sequences to these data. The first has second term 0 p,q E2

= Rp π2∗ (Hq (F • ) ⊗ p∗1 Lk )

and converges to R• π2∗ (F • ⊗ π1∗ Lk ). One has 0E2p,q = 0 for p > 0 and q ≥ m0 − dim X while 0E20,m0 6= 0, so that Rm0 π2∗ (F • ⊗ π1∗ Lk ) 6= 0. The second sequence has first term 00 p,q E1

= Rq π2∗ (F p ⊗ π1∗ Lk )

and converges to R• π2∗ (F • ⊗ π1∗ Lk ). One has  k  L 00 p,q E1 = H q (X, Lk+p+1 ) ⊗ R−p−1   0

for p = 0 and q = 0 for p ≤ −1 otherwise.

As a consequence, 00E1p,0 = Ak+p+1 ⊗ R−p−1 for p ≤ −1 and 00E1p,q = 0 for p + 1 ≥ m0 −dim X and q > 0. However Lemma 2.20 implies that 00E2p,q = 0 for p+q = m0 , a contradiction. 

2.2. Orlov’s representability theorem

49

By reasons that will be evident later on, we need to consider the truncated complex d

C • (m) ≡ π1∗ L−m ⊗ π2∗ Rm −−m → ... d

d

2 1 −→ π1∗ L−1 ⊗ π2∗ R1 −→ π1∗ OX ⊗ π2∗ OX .

(2.15)

Let Tm = ker dm (m ≥ 1) and let αm : Tm [m] → C • (m) be the morphism induced by the immersion Tm ,→ π1∗ L−m ⊗ π2∗ Rm . Then the cone Cone(αm ) is isomorphic to the complex d

d

d

2 1 → . . . −→ π1∗ L−1 ⊗ π2∗ R1 −→ π1∗ OX ⊗ π2∗ OX Tm ,→ π1∗ L−m ⊗ π2∗ Rm −−m

with Tm at the −(m + 1)-th place, and is then quasi-isomorphic to O∆ . Assume now that X is smooth and choose m ≥ 2 dim X. Then in the exact triangle α Tm [m] −−m → C • (m) → Cone(αm ) ' O∆ → Tm [m + 1] the last morphism vanishes, since HomD(X) (O∆ , Tm [m+1]) ' Extm+1 (O∆ , Tm ) = 0 because m+1 > 2 dim X and X is smooth. As a consequence, C • (m) is a biproduct of O∆ and Tm [m] in the derived category, C • (m) ' O∆ ⊕ Tm [m] . Moreover if we call d0 : OX×X → c(m) the convolution of C • (m) in Db (X × X) (which is C • (m) itself, see Section 2.1.2), we can write the above formula in the following form, which we shall shortly use. c(m) ' O∆ ⊕ Tm [m] .

(2.16)

Let us consider now the complex d0

→ ... π1∗ Lsk ⊗ C • (m) ' π1∗ Lsk−m ⊗ π2∗ Rm −−m d0

d0

2 1 −→ π1∗ Lsk−1 ⊗ π2∗ R1 −→ π1∗ Lsk ⊗ π2∗ OX ,

where d0p = 1 ⊗ dp . By Remark 2.14 we have d0

0 ∗ sk ⊗ c(m) π1∗ Lsk ⊗ π2∗ OX −→π 1L

'(π1∗ Lsk ⊗ O∆ ) ⊕ ((π1∗ Lsk ⊗ Tm )[m]) , where d00 = 1 ⊗ d0 , is a convolution of π1∗ Lsk ⊗ C • (m) .

(2.17)

50

Chapter 2. Fourier-Mukai functors

Now fix a value of m (to be specified later) and let s > m. Then all sheaves Lsk−p (0 ≤ p ≤ m) are acyclic, so that after applying Rπ2∗ to the complex π1∗ Lsk ⊗ C • (m) we obtain the complex π2∗ (d0 )

m → ... Rπ2∗ (π1∗ Lsk ⊗ C • (m) ) ' Γ(X, Lsk−m ) ⊗k Rm −−−−−

π2∗ (d0 )

π2∗ (d0 )

2 1 −−−−− → Γ(X, Lsk−1 ) ⊗k R1 −−−−− → Γ(X, Lsk ) ⊗k OX .

(2.18)

Again by Remark 2.14, Rπ2∗ (d0 )

Γ(X, Lsk ) ⊗k OX −−−−−−0→ Rπ2∗ (π1∗ Lsk ⊗ c(m) ) ' Lsk ⊕ Rπ2∗ (π1∗ Lsk ⊗ Tm [m]) is a convolution of Rπ2∗ (π1∗ Lsk ⊗ C • (m) ). Recall that the isomorphism in the formula holds when X is smooth and m > 2 dim X − 1. We finish this section with a lemma that generalizes the argument used in the proof of (2.16). Lemma 2.22. Let X be a smooth variety. 1. If E • , F • are objects of Db (X) such that Hi (E • ) = 0 for i > z and Hq (F • ) = 0 for q ≤ z + 2 dim X for an integer z, then HomD(X) (F • , E • ) = 0. 2. Let E • be an object of Db (X). If there exist integers z and s > 2 dim X such that Hi (E • ) = 0 for z < i ≤ z + s, then for every p ∈ [z, z + s] one has an isomorphism E • ' E • ≤p ⊕ E • ≥p in Db (X).

Proof. 1. We proceed by induction on the sum n = `(E • ) + `(F • ) of the lengths of E • and F • (recall that the length is the number of nonzero cohomology sheaves of E • and F • ). The first case is n = 2, since we can assume that E • and F • are nonzero. Then E • ' E[−q] for q ≤ z and F • ' F[−m] for m > z + 2 dim X, where E and F are sheaves. It follows that HomD(X) (F • , E • ) ' HomD(X) (F, E[m − q]) ' Extm−q (F, E) and this is zero because m − q > 2 dim X and X is smooth. X Take n > 2. Then either E • or F • have at least two nonzero cohomology sheaves. If `(F • ) ≥ 2, let q0 be the first of the indexes q such that Hq (F • ) 6= 0. Then there is a exact triangle α

F • ≤q0 − → F • → Cone(α) → F • ≤q0 [1] .

2.2. Orlov’s representability theorem

51

Now Hq (F • ≤q0 ) = 0 for q 6= q0 and Hq (Cone(α)) = 0 for q ≤ q0 , so that they are in the same hypotheses as F • . Moreover `(E • )+`(F • ≤q0 ) = `(E • )+1 < n and `(E • )+ `(Cone(α) = n−1. Then HomD(X) (Cone(α), E • ) = 0 and HomD(X) (F • ≤q0 , E • ) = 0 by induction so that HomD(X) (F • , E • ) = 0. If `(E • ) ≥ 2, we take i0 as the first of the integers i such that Hi (E • ) 6= 0 and proceed as above using the exact triangle α

→ E • → Cone(α) → E • ≤i0 [1] . E • ≤i0 − 2. There is a exact triangle α

→ E • → Cone(α) ' E • ≥p → E • ≤p [1] , E • ≤p − where the isomorphism Cone(α) ' E • ≥p is a consequence of Hp (E • ) = 0. Moreover Hi (E • ≤p [1]) ' Hi+1 (E • ≤p ) = 0 for i > z and Hq (E • ≥p ) = 0 for q ≤ z + s ≤ z + 2 dim X. Thus HomD(X) (E • ≥p , E • ≤p [1]) = 0 by the first part, and then E • decomposes as a biproduct in the derived category as claimed. 

2.2.2

Uniqueness of the kernel

As a first step in the proof of Orlov’s theorem, and for later use, we want to prove that an integral functor completely determines its kernel. More precisely, let K• be a kernel in D− (X × Y ); we denote simply by Φ the associated integral functor • ΦK X→Y . By using Proposition 1.10 and the base change compatibility of the integral functors (Proposition 1.8), we shall prove that K• is completely determined by Φ. To do so we slightly change notation from Section 1.2. Let us write πij for the projection of X × X × Y onto its (i, j)-th factor. Then by Proposition 1.10 the kernel K• can be recovered as π ∗ K•

•

KX 23 (O∆ ) K• ' ΦX×X→ X×Y (O∆ ) = Φ π ∗ K•

•

23 where we have written ΦKX = ΦX×X→ X×Y for simplicity.

In the rest of this section we assume that X is smooth and that K• is of finite Tor-dimension as a complex of OX -modules (cf. Section 1.2), so that Φ • maps Db (X) into Db (Y ). Then ΦKX maps Db (X × X) into Db (X × Y ). Lemma 2.23. There exist integer numbers p, m0 such that for every m > m0 , one has a natural isomorphism •

K• ' (ΦKX (c(m) ))≥p .

•

•

•

Proof. By (2.16), we have ΦKX (c(m) ) ' ΦKX (O∆ )⊕ΦKX (TM [m]) for m ≥ 2 dim X. • Now, Proposition 1.4 for ΦKX implies the existence of integer numbers z and

52

Chapter 2. Fourier-Mukai functors •

n ≥ 0 depending only on Φ such that the cohomology sheaves Hi (ΦKX (O∆ )) • vanish for i ∈ / [z, z + n] and the cohomology sheaves Hi (ΦKX (Tm [m])) vanish for i∈ / [z −m, z +n−m]. Now take p < z and m0 = max{2 dim X, z +n−p}. Then the • • natural morphism ΦKX (O∆ ) → (ΦKX (O∆ ))≥p is an isomorphism in the derived • • • category and ΦKX (c(m) ) → ΦKX (O∆ ) induces an isomorphism ΦKX (c(m) )≥p ' •  (ΦKX (O∆ ))≥p for m ≥ m0 as desired. •

Let us prove that ΦKX (c(m) ) depends only on Φ. Since c(m) is a convolution • of C (m) , we consider the complex •

ΦKX (dm )

•

ΦKX (π1∗ L−m ⊗ π2∗ Rm ) −−−−−−→ · · · → •

•

ΦKX (d1 )

•

ΦKX (π1∗ L−1 ⊗ π2∗ R1 ) −−−−−−→ ΦKX (π1∗ OX ⊗ π2∗ OX ) •

of objects of Db (X × Y ), obtained by applying ΦKX to the complex C • (m) . By base change (Proposition 1.8) one has isomorphisms •

ΦKX (π1∗ L−i ⊗ π2∗ Ri ) ' π1∗ L−i ⊗ π2∗ Φ(Ri ) so that one also has a complex Φ(dm )

π1∗ L−m ⊗ π2∗ Φ(Rm ) −−−−→ · · · → Φ(d1 )

π1∗ L−1 ⊗ π2∗ Φ(R1 ) −−−→ π1∗ OX ⊗ π2∗ Φ(OX ) .

(2.19)

•

The objects of the complex depend only on Φ and not on ΦKX ; we are going to show that under appropriate conditions, even the morphisms depend only on Φ. For every pair of indices p > q and integer r ≥ 0, we have HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ Φ(Rp )[r], π1∗ L−q ⊗ π2∗ Φ(Rq )) ' HomDb (X×Y ) (π2∗ Φ(Rp )[r], π1∗ Lp−q ⊗ π2∗ Φ(Rq )) ' HomDb (Y ) (Φ(Rp )[r], π2∗ (π1∗ Lp−q ) ⊗ Φ(Rq )) p−q

' HomDb (Y ) (Φ(Rp )[r], Γ(X, L

) ⊗k Φ(Rq ))

p−q

' HomDb (Y ) (Φ(Rp )[r], Φ(Γ(X, L

(2.20)

) ⊗k Rq ))

where we have used adjunction between inverse and direct images and the projection formula. Analogously one has HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ Rp [r], π1∗ L−(q) ⊗ π2∗ Rq ) ' HomDb (X×Y ) (π2∗ Rp [r], π1∗ Lp−q ⊗ π2∗ Rq ) ' HomDb (Y ) (Rp [r], π2∗ (π1∗ Lp−q ) ⊗ Rq ) ' HomDb (Y ) (Rp [r], Γ(X, Lp−q ) ⊗k Rq ) .

(2.21)

2.2. Orlov’s representability theorem

53

Then the natural morphism Φ

→ HomDb (X) (Rp [r], Γ(X, Lp−q ) ⊗k Rq ) − HomDb (Y ) (Φ(Rp )[r], Φ(Γ(X, Lp−q ) ⊗k Rq ))

(2.22)

induces a morphism Φ

HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ Rp [r], π1∗ L−q ⊗ π2∗ Rq ) − → HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ Φ(Rp )[r], π1∗ L−q ⊗ π2∗ Φ(Rq )) .

(2.23)

•

Taking in particular q = p − 1 and r = 0, we see that ΦKX (dp ) = Φ(dp ). We have thus proved the following result. Lemma 2.24. There is an isomorphism π1∗ L−m ⊗ π2∗ Φ(Rm )

K• X

Φ



/

Φ(dm )

/ π1∗ L−1 ⊗ π2∗ Φ(R1 )

...

'

(π1∗ L−m ⊗

•

K Φ X (dm ) π2∗ Rm )

/

/

...

K• X

Φ



/

Φ(d1 )

π1∗ OX ⊗ π2∗ Φ(OX )

' •

K Φ X (d1 )

(π1∗ L−1 ⊗ π2∗ R1 )

/

K• X

Φ



'

(π1∗ OX ⊗ π2∗ OX ) •

of complexes of objects of Db (X × Y ). As a consequence, the image under ΦKX of • the complex C • (m) given by (2.15) depends only on Φ and not on ΦKX . Assume that Φ is fully faithful. Then (2.23) implies that the complex π∗

Φ(dm )

Φ(d1 )

1 L−1 ⊗ π2∗ Φ(R1 ) −−−→ π1∗ OX ⊗ π2∗ Φ(OX ) π1∗ L−m ⊗ π2∗ Φ(Rm ) −−−−→ . . . −→

d

0 (1⊗Φ)(c(m) ). Moreover, has a convolution, which we denote π1∗ OX ⊗π2∗ Φ(OX ) −→ by Remark 2.14 a convolution of the complex (2.19) exists and is given by •

ΦKX (d0 )

•

•

ΦKX (π1∗ OX ⊗ π2∗ OX ) −−−−−−→ ΦKX (c(m) ) . Now by Lemma 2.12 there is commutative diagram π1∗ OX ⊗ π2∗ Φ(OX )

•



d0

α≥p

K• X

Φ

 (d0 ) • / ΦKX (c(m) )

/ ((1 ⊗ Φ)(c(m) ))≥p γ≥p '

γ '

'

ΦKX (π1∗ OX ⊗ π2∗ OX )

/ (1 ⊗ Φ)(c(m) )

α≥p

 • / (ΦKX (c(m) ))≥p ' K•

for a certain isomorphism γ (not uniquely determined). Here the last isomorphism in the bottom row is due to Lemma 2.23 and the morphisms α≥p are the natural

54

Chapter 2. Fourier-Mukai functors

epimorphisms to the truncations. Since HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ Φ(Rp )[r], K• ) ' HomDb (X×Y ) (π2∗ Φ(Rp [r]), π1∗ Lp ⊗ K• ) ' HomDb (Y ) (Φ(Rp )[r], Rπ2,∗ (π1∗ Lp ⊗ K• )) = HomDb (Y ) (Φ(Rp )[r], Φ(Lp )) ' HomDb (X) (Rp [r], Lp ) = 0 we deduce from Lemma 2.12 that the composed diagonal morphism α≥p ◦ γ≥p = α≥p ◦ γ : (1 ⊗ Φ)(c(m) ) → K• is the unique morphism making the diagram commutative. Then, the isomorphism γ≥p is uniquely characterized by the commutativity of the diagram π1∗ OX ⊗ π2∗ Φ(OX )

•



α≥p ◦d0

/ ((1 ⊗ Φ)(c(m) ))≥p γ≥p '

'

ΦKX (π1∗ OX ⊗ π2∗ OX )

K• X

α≥p ◦Φ

 (d0 ) • / (ΦKX (c(m) ))≥p ' K• .

(2.24)

•

This eventually implies the desired uniqueness result. Let f K = γ≥ p ◦ α≥p ◦ d0 . Theorem 2.25. Let X, Y be projective varieties, K• be a kernel in D− (X × Y ) and • let Φ = ΦK X→Y be the corresponding integral functor. Assume that X is smooth, K• is of finite Tor-dimension as a complex of OX -modules and that Φ is fully faithful. Then the kernel K• is uniquely determined by the functor Φ. Moreover ˜ • is another kernel in D− (X × Y ), of finite Tor-dimension as a complex of if K ˜• • K ˜ • in OX -modules, such that Φ = ΦX→ Y , there is a unique isomorphism η : K ' K Db (X × Y ) making the diagram K•

f / K• π1∗ OX ⊗ π2∗ Φ(OX ) NNN NNN NNN η ' ˜• NNN  fK ' ˜• K

commutative. Proof. Since ((1 ⊗ Φ)(c(m) ))≥p ' K• one has that K• is uniquely determined by Φ. The second part follows straightforwardly from the above discussion. 

2.2.3

Existence of the kernel

In this section we conclude the proof of Orlov’s Theorem 2.15 by constructing the kernel that realizes the given fully faithful functor as an integral functor. Let X,

2.2. Orlov’s representability theorem

55

Y be smooth projective varieties and F : Db (X) → Db (Y ) an exact fully faithful functor. As we shall see in Proposition 2.31, the functor F is bounded, so that there exist integer numbers z and n ≥ 0 such that for every coherent sheaf F on X, the / [z, z + n]. Then F can be extended cohomology sheaves Hi (F (F)) vanish for i ∈ to a functor D(X) → D(Y ). We can consider the complex F (π2∗ (d0 ))

F (Rπ2∗ (π1∗ Lsk ⊗ C • (m) )) ≡ Γ(X, Lsk−m ) ⊗k F (Rm ) −−−−−−m−→ . . . F (π2∗ (d0 ))

F (π2∗ (d0 ))

−−−−−−2−→ Γ(X, Lsk−1 ) ⊗k F (R1 ) −−−−−−1−→ Γ(X, Lsk ) ⊗k F (OX )

(2.25)

of objects in Db (Y ). By Remark 2.14, the morphism F (π2∗ (d0 ))

Γ(X, Lsk ) ⊗k F (OX ) −−−−−−0−→ F (Rπ2∗ (Lsk ⊗ c(m) )) is a convolution of (2.25). Moreover for m ≥ 2 dim X, (2.16) implies that F (Rπ2∗ (Lsk ⊗ c(m) )) ' F (Lsk ) ⊕ F (Rπ2∗ (π1∗ Lsk ⊗ Tm ))[m] .

(2.26)

Let us fix p < z and choose m > max{z + n − p, 2 dim X}. i Lemma 2.26. Take s > m such that Rπ2,∗ (π1∗ Lsk ⊗ Tm ) = 0 for i > 0 and k ≥ 1. i sk Then H (F (Rπ2∗ (L ⊗ c(m) ))) = 0 unless i ∈ [z − m, z + n − m] or i ∈ [z, z + n]. Moreover F (Rπ2∗ (Lsk ⊗ c(m) )) → F (Lsk ) induces an isomorphism

βkF : F (Rπ2∗ (Lsk ⊗ c(m) ))≥p ' F (Lsk ) in Db (Y ). Proof. One has Hi (F (Rπ2∗ (Lsk ⊗ c(m) ))) ' Hi (F (π2∗ (π1∗ Lsk ⊗ Tm ))) for i ∈ [z − m, z + n − m] , Hi (F (Rπ2∗ (Lsk ⊗ c(m) ))) ' Hi (F (Lsk ))

for i ∈ [z, z + n] ,

Hi (F (Rπ2∗ (Lsk ⊗ c(m) ))) = 0 for the remaining values of i. The result follows.



Let us take m > n + 2(1 + dim X + dim Y ) so that we can apply Lemma 2.26. We consider the truncated complex C • (m) given by (2.15). Since F is fully faithful, we have an isomorphism F

→ HomDb (X) (Rp [r], Γ(X, Lp−q ) ⊗k Rq ) − HomDb (Y ) (F (Rp )[r], F (Γ(X, Lp−q ) ⊗k Rq )) .

56

Chapter 2. Fourier-Mukai functors

Then (2.20) remains true if we replace Φ by F , and one obtains, as in Section 2.2.2, an isomorphism F : HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ Rp [r], π1∗ L−q ⊗ π2∗ Rq ) ' HomDb (X×Y ) (π1∗ L−p ⊗ π2∗ F (Rp )[r], π1∗ L−q ⊗ π2∗ F (Rq )) .

(2.27)

Taking in particular q = p − 1 and r = 0, we see that dp induces a morphism F (dp ) : π1∗ L−p ⊗ π2∗ F (Rp ) → π1∗ L−(p−1) ⊗ π2∗ F (Rp−1 ) and we have a complex F (dm )

(1 ⊗ F )(C • (m) ) ≡ π1∗ L−m ⊗ π2∗ F (Rm ) −−−−→ · · · → F (d1 )

π1∗ L−1 ⊗ π2∗ F (R1 ) −−−−→ π1∗ OX ⊗ π2∗ F (OX )

(2.28)

of objects of Db (X ×Y ). Furthermore, (2.27) implies that there exists a convolution F (d0 )

π1∗ OX ⊗ π2∗ F (OX ) −−−−→ Fc(m) of (1 ⊗ F ))(C • (m) ). Analogously for any s and k, one has a complex F (dm )

π1∗ Lsk ⊗ (1 ⊗ F )(C • (m) ) ≡ π1∗ Lsk−m ⊗ π2∗ F (Rm ) −−−−→ · · · → F (d1 )

π1∗ Lsk−1 ⊗ π2∗ F (R1 ) −−−−→ π1∗ Lsk ⊗ π2∗ F (OX )

(2.29)

1⊗F (d0 )

of objects of Db (X × Y ), and π1∗ Lsk ⊗ π2∗ F (OX ) −−−−−→ π1∗ Lsk ⊗ Fc(m) is a convolution of π1∗ Lsk ⊗ (1 ⊗ F )(C • (m) ). Lemma 2.27.

1. There is a (not uniquely determined) isomorphism

Fc ∼ F (Rπ (π ∗ Lsk ⊗ Fc )) (Lsk ) = Rπ2∗ (π1∗ Lsk ⊗ Fc(m) ) → ηk : ΦX→(m) 2∗ 1 Y (m)

in the derived category Db (Y ) which makes the diagram π2∗ (π1∗ Lsk ⊗ π2∗ F (OX ))

Rπ2∗ (1⊗F (d0 ))

X→Y

ηk '

'

 Γ(X, Lsk ) ⊗k F (OX )

/ Rπ2∗ (π1∗ Lsk ⊗ Fc(m) )= ΦFc(m) (Lsk )

F (π2∗ (d00 ))

 / F (Rπ2∗ (π1∗ Lsk ⊗ Fc(m) )) (2.30)

commutative. 2. Hi (Fc(m) ) = 0 unless i ∈ [z − m, z + n − m] or i ∈ [z, z + n]. Moreover Fc(m) can be expressed as a biproduct Fc(m) ' (Fc(m) )≥p ⊕ (Fc(m) )≤p .

2.2. Orlov’s representability theorem

57

Proof. 1. By applying Rπ2∗ to the complex π1∗ Lsk ⊗ (1 ⊗ F )(C • (m) ) one obtains the complex F (Rπ2∗ (Lsk ⊗ C • (m) )) described in (2.25). Then, by Remark 2.14, ∼ F (Rπ (π ∗ Lsk ⊗ Fc )) we have an isomorphism ηk : Rπ2∗ (π1∗ Lsk ⊗ Fc(m) ) → 2∗ 1 (m) between the convolutions which makes the diagram (2.30) commutative. 2. Assume that there is an integer i not in the prescribed rank such that Hi (Fc(m) )) 6= 0. Then for k  0 one has 0 6= π2∗ (π1∗ Lsk ⊗ Hi (Fc(m) )) ' π2∗ (Hi (π1∗ Lsk ⊗Fc(m) )) and Rj π2∗ (π1∗ Lsk ⊗Hi−1 (Fc(m) )) ' Rj π2∗ (Hi−1 (π1∗ Lsk ⊗ Fc(m) )) = 0 for every j > 0. There is a spectral sequence E2p,q = Rp π2∗ (Hq (π1∗ Lsk ⊗ p+q = Hp+q (Rπ2∗ (π1∗ Lsk ⊗ Fc(m) )). Since E −2,i+1 = Fc(m) )) converging to E∞ 2,i−1 E2 = 0, every nonzero element in E20,i is a cycle that survives to infinity. Then 0,i E2 6= 0 implies that Hi (Rπ2∗ (π1∗ Lsk ⊗ Fc(m) )) 6= 0. If we also take the preceding part 1 into account, this contradicts Lemma 2.26. Now, since m − n − 2 > 2(dim X + dim Y ) and X × Y is smooth, Lemma  2.22 gives the decomposition Fc(m) ' (Fc(m) )≥p ⊕ (Fc(m) )≤p . •

• ' (Fc(m) )≥p as we saw in Lemma 2.23. It Note that if F = ΦK X→Y then K is therefore convenient in our general situation to define the following objects of Db (X × Y ) (2.31) K• ' (Fc(m) )≥p , Q• ' (Fc(m) )≤p

(see Corollary A.35). Then we have an isomorphism of functors •

Fc

•

Q ' ΦK ΦX→(m) Y X→Y ⊕ ΦX→Y . Fc

•

sk (Lsk ) → ΦK ) induces for every Lemma 2.28. The natural morphism ΦX→(m) Y X→Y (L k an isomorphism • Fc sk (Lsk ))≥p ' ΦK ). βkΦ : (ΦX→(m) Y X→Y (L

/ [z − m, z + n − m]. Then Proof. By Lemma 2.27, one has Hq (Q• ) = 0 for q ∈ ∗ sk L ⊗ Q• )) = 0 for q ∈ / [z − m, z + n − m + dim X]. Since m > Rp πY ∗ (Hq (πX • sk )) = 0 for i ≥ p and the result follows.  z + n − p + dim X, one has Hi (ΦQ X→Y (L As a consequence, (2.30) can be completed to a commutative diagram π2∗ (π1∗ Lsk ⊗ π2∗ F (OX )) '

 Γ(X, Lsk ) ⊗k F (OX )

F (π2∗ (d00 ))

Φ

βk / ΦFc(m) (Lsk ) / ΦK• (Lsk ) X→Y X→Y RRR RRR fk RRR ηk ' γk ' RRR RRR   F ) β / F (Rπ2∗ (π1∗ Lsk ⊗ Fc(m) )) k / F (Lsk ) (2.32)

Rπ2∗ (1⊗F (d0 ))

58

Chapter 2. Fourier-Mukai functors

where βkF is the isomorphism given by Lemma 2.26 and Fc

(Lsk ) → F (Lsk ) . fk = γk ◦ βkΦ = βkF ◦ ηk : ΦX→(m) Y Even if ηk is not unique, the morphism γk is, in view of the following lemma. •

sk ) ' F (Lsk ) in the derived category Lemma 2.29. The isomorphism γk : ΦK X→Y (L b D (Y ) is uniquely determined by the commutativity of the diagram (2.32). Thus the morphisms γk are functorial, in the sense that, for every morphism α : Lsk → Ls` , there is a commutative diagram in Db (Y ) •

K•

sk

ΦX→Y (L )

ΦK X→Y (α)

γk '

 F (Lsk )

/ ΦK• (Ls` ) X→Y γ` '

F (α)

 / F (Ls` ) .

Proof. Since HomDb (Y ) (π2∗ (π1∗ Lsk−p ⊗ π2∗ F (Rp ))[r], F (Lsk )) ' HomDb (Y ) (Γ(Y, Lsk−p ) ⊗k F (Rp ))[r], F (Lsk )) ' HomDb (X) (Γ(Y, Lsk−p ) ⊗k Rp [r], Lsk ) = 0 as F is fully faithful, Lemma 2.12 implies that fk is the unique morphism which makes the diagram (2.32) commutative. Under this condition the morphism γk is unique as well. Functoriality follows straightforwardly.  Since X and Y are smooth, the categories Db (X) and Db (Y ) have Serre functors. Moreover {Lsk }k∈Z is an ample sequence in Db (X), and then we can apply Proposition 2.11 to obtain an existence result. The uniqueness of the kernel is given by Theorem 2.25. Proposition 2.30. Let X, Y be smooth projective varieties and let F : Db (X) → Db (Y ) be an exact fully faithful functor. If F is bounded, then it is an integral functor and its kernel is uniquely determined in Db (X × Y ) up to isomorphism. Orlov’s representability theorem 2.15 follows from Proposition 2.30 due to the following result: Proposition 2.31. If X, Y are projective varieties and X is smooth, every exact functor F : Db (X) → Db (Y ) is bounded.

2.2. Orlov’s representability theorem

59

Proof. As we have already noticed, the functor F admits a left adjoint G : Db (Y ) → Db (X), cf. Remark 2.16. Take an embedding Y ,→ PN induced by a very ample line bundle L. Pulling back to Y the right resolution (2.11), we have a right resolution 0 → L−j → V0j ⊗k OX → V1j ⊗k L → · · · → VNj −1 ⊗k LN −1 → VNj ⊗k LN → 0

(2.33)

for every integer j > 0. Then, for every j > 0 there is a spectral sequence E2p,q = p+q = Hp+q (G(L−j )). Since each Hp (Vqj ⊗k G(Lq )) = Vqj ⊗k G(Lq )) converging to E∞ q G(L ) is bounded, there exist integers p0 ≤ p1 such that Hp (G(Lq )) = 0 for every / [p0 , p1 + N ] q ∈ [0, N ] if p ∈ / [p0 , p1 ]. This implies that Hk (G(L−j )) = 0 for k ∈ and for every j > 0. A similar spectral sequence argument implies that for every sheaf F on X one has HomiDb (Y ) (L−j , F (F)) = HomiDb (X) (G(L−j ), F) = 0

(2.34)

for i ∈ / [−p1 − N, −p0 + dim X] and for every j > 0, since X is smooth and then HomiDb (X) (G, F) = 0 for any sheaf G on X and any i > dim X. Again, for every value of j there is a spectral sequence with E2p,q = ExtpY (L−j , Hq (F (F))) p+q = Homp+q (L−j , F (F)). Since F (F) is bounded, for j  0 converging to E∞ D b (Y ) p,q one has E2 = 0 for all values of q; then the spectral sequence degenerates yielding an isomorphism HomY (L−j , Hq (F (F))) ' HomqDb (Y ) (L−j , F (F)). If Hq (F (F)) 6= 0, we can find j  0 such that HomY (L−j , Hq (F (F))) 6= 0. Thus, (2.34) implies  that Hq (F (F)) = 0 unless q ∈ [−p1 − N, −p0 + dim X]. Remark 2.32. Given two kernels K• , G • in Db (X × Y ), any morphism f : K• → G • • G• in the derived category induces a morphism of functors ΦfX→Y : ΦK X→Y → ΦX→Y which in turn induces another morphism f¯: K• → G • . This morphism may fail to • G• coincide with f . Moreover, the groups HomDb (X×Y ) (K• , G • ) and Hom(ΦK X→Y , ΦX→Y ) may not be isomorphic. In other words, the functor that maps the kernel K• to • the integral functor ΦK X→Y is not fully faithful in general. Take for instance for X = Y an elliptic curve, K• = O∆ and G • = O∆ [2]. The Serre functor of X × X consists in the shift by 2, so that HomDb (X×X) (O∆ , O∆ [2]) ' HomDb (X×X) (O∆ , O∆ )∗ ' k . •

•

G On the other hand, ΦK X→Y is the identity functor IdD b (X) and ΦX→Y ' IdD b (X) [2]. If g : IdDb (X) → IdDb (X) [2] is a functor morphism, then for every sheaf F on X, the induced morphism g(F) : F → F[2] is zero since HomDb (X) (F, F[2]) = Ext2X (F, F) = 0 because X is a curve. One easily proves that the morphism g(F • ) is also zero for every bounded complex F • ; thus g = 0, that is, •

•

G Hom(ΦK X→Y , ΦX→Y ) ' Hom(IdD b (X) , IdD b (X) [2]) = 0 .

60

Chapter 2. Fourier-Mukai functors

The fact that the functor mapping kernels to integral functors may fail to be fully faithful can be regarded as an intrinsic limitation of the triangulated structure of the derived category. Actually, if we pass to the setting of dg-categories, the corresponding functor (suitably defined) turns out to be an equivalence. See Theorem A.57 and the comments in “Notes and further reading.” 4

2.3

Fourier-Mukai functors

The following definition introduces the objects that will be our main concern in this book. •

b b Definition 2.33. An integral functor ΦK X→Y : D (X) → D (Y ) is called a FourierMukai functor if it is an exact equivalence of derived categories. If in addition the kernel is a concentrated complex, the functor will be said to be a Fourier-Mukai transform. 4

We give now some basic properties of the Fourier-Mukai functors. Later on we shall describe some geometric applications of Fourier-Mukai functors and shall establish a criterion for testing whether an integral functor is a Fourier-Mukai functor. The composition of two Fourier-Mukai functors is a Fourier-Mukai functor as well (and, as we know, the kernel of the composition is the convolution of the two kernels, cf. Proposition 1.3). However, the composition of two Fourier-Mukai transforms may fail to be a Fourier-Mukai transform, because its kernel may not be a concentrated complex, as we shall see in Example 2.59. Fourier-Mukai functors behave well with respect to the WIT condition. Let • b b Φ = ΦK X→Y : D (X) → D (Y ) be a Fourier-Mukai functor; the functor i-th cohoi i mology sheaf Φ (•) = H (Φ(•)) will be called the i-th Fourier-Mukai functor. Given b : Db (Y ) → Db (X) of Φ, we have an isomorphism a quasi-inverse Φ • b )) ' E • Φ(Φ(E

in the derived category. When E • is a sheaf E in degree zero, the above isomorphism means that there is a convergent spectral sequence ( p,q p q b (Φ (E)) =⇒ E if p + q = 0 (2.35) E2 = Φ 0 otherwise. Proposition 2.34. If Φ is a Fourier-Mukai functor and E is a WITi sheaf, then the b b ' b −i (E) unique nonzero Fourier-Mukai sheaf Eb is a WIT−i sheaf. Moreover Eb = Φ E.

2.3. Fourier-Mukai functors

61

We shall also need the following result. Proposition 2.35. Let Φ : Db (X) → Db (Y ) be a Fourier-Mukai functor and assume that X is smooth of dimension n. For every (closed) point x ∈ X the following inequality holds true: X dim Hom1D(Y ) (Φi (Ox ), Φi (Ox )) ≤ n . i

Proof. There is a spectral sequence E2p,q =

L

i

HompD(Y ) (Φi (Ox ), Φi+q (Ox )) con-

p+q verging to E∞ = Homp+q D(Y ) (Φ(Ox ), Φ(Ox )). The exact sequence of the lower 1 terms yields 0 → E21,0 → E∞ . By the Parseval formula (Proposition 1.34), one 1  has HomD(Y ) (Φ(Ox ), Φ(Ox )) ' Hom1D(X) (Ox , Ox ) ' kn .

2.3.1

Some geometric applications of Fourier-Mukai functors

The existence of a Fourier-Mukai functor between the derived categories of two smooth algebraic varieties (or the equivalent conditions that the two algebraic varieties have equivalent derived categories, cf. Theorem 2.15) imposes strong constraints on their geometry. A first manifestation of this fact is Theorem 2.38 which states that the two (smooth projective) varieties X and Y have the same dimension, and that their canonical bundles satisfy some stringent conditions. Corollary 2.40 will establish that the rational Chow rings of X and Y are isomorphic (as Q-vector spaces). According to Theorem 2.49, under a condition on the Kodaira dimension of X, the varieties X and Y are birational. If the hypotheses are strengthened by assuming that X has an ample canonical bundle, then X and Y are isomorphic (Theorem 2.51). The following definition is commonly adopted for varieties with equivalent bounded derived categories. Definition 2.36. Two projective varieties X and Y are Fourier-Mukai partners if ∼ Db (Y ). 4 there is an exact equivalence of triangulated categories F : D b (X) → Note that we do not impose that any of the varieties is smooth. However, if one is smooth, the other is smooth as well. Lemma 2.37. Let X be a smooth projective variety. 1. Every Fourier-Mukai partner of X is smooth. 2. A projective variety Y is a Fourier-Mukai partner of X if and only if there • ∼ b b is Fourier-Mukai functor ΦK X→Y : D (X) → D (Y ).

62

Chapter 2. Fourier-Mukai functors

Proof. Assume that X is a smooth variety and that there is an equivalence of categories F : Db (Y ) → Db (X). For every (closed) point y ∈ X and every sheaf F on Y , one has Homi (Oy , F) ' Homi (F (Oy ), F (F)). Since X is smooth and F (Oy ) and F (F) are bounded, there is only a finite number of indexes i with Homi (Oy , F) 6= 0. Then Oy is of finite homological dimension, and hence Y is smooth at y by Serre’s Theorem [266], [215, 19.2]. This proves the first part. The second follows from Orlov’s representability theorem 2.15.  We briefly recall the notion of determinant bundle for an object M• of Db (X) where X is a smooth projective variety. This is defined by O i (det(E i ))(−1) det(M• ) = i

where E • is any bounded complex of locally free sheaves isomorphic to M• in the derived category and det(E i ) is the highest exterior power of E i . A direct computation shows that •

det(M• ⊗ L) = det(M• ) ⊗ Lrk(M

)

(2.36)

for every line bundle L. Theorem 2.38. Let X, Y be smooth projective varieties that are Fourier-Mukai • b b partners, so that there is a Fourier-Mukai functor ΦK X→Y : D (X) → D (Y ). •

1. The right and left adjoints to ΦK X→Y are functorially isomorphic ∗ K•∨ ⊗πX ωX [m]

ΦY→X

∗ K•∨ ⊗πY ωY [n]

' ΦY→X

•

(here m = dim X and n = dim Y ) and they are both quasi-inverses to ΦK X→Y . 2. X and Y have the same dimension, m = n. k 3. ωX and ωY have the same order, that is, ωX is trivial if and only if ωYk is trivial. Thus, ωX is trivial if and only if ωY is trivial and in this case the • K•∨ [n] functor ΦY→X is a quasi-inverse to ΦK X→Y . r 4. ωX ' OX and ωYr ' OY with r = rk(K• ).

Proof. 1. Since a quasi-inverse is both a right and a left adjoint, the uniqueness of adjoints together with Proposition 1.13 yields the statement. 2. Applying the above functorial isomorphism to the skyscraper sheaf Oy we obtain Ljy∗ K•∨ ⊗ ωX [m] ' Ljy∗ K•∨ [n]. Since the functors we have applied are equivalences of categories, both objects are nonzero in Db (X). Then there is an

2.3. Fourier-Mukai functors

63

integer q0 which is the minimum of the q’s with Hq (Ljy∗ K•∨ ) 6= 0 and one gets q0 + m = q0 + n so that m = n. k is trivial; the other case is proved analogously. 3. Assume for instance that ωX • • k k K• −1 −1 is a , where (ΦK By Corollary 1.18, one has SY ' ΦK X→Y ◦ SX ◦ (ΦX→Y ) X→Y ) K• k k • • quasi-inverse to ΦX→Y . Since ωX is trivial, SX (F ) ' F [kn] and then SYk ' • K• −1 ' [kn]. Therefore ωYk ' OY . ΦK X→Y ◦ [kn] ◦ (ΦX→Y )

4. Taking determinant bundles in the expression Ljy∗ K•∨ ⊗ ωX ' Ljy∗ K•∨ , we r have det(Ljy∗ K•∨ ) ' det(Ljy∗ K•∨ ) ⊗ ωXy with ry = rk(Ljy∗ K•∨ ) by Equation (2.36), ry and therefore ωX ' OX . Now the functoriality of the Chern classes gives ry =  rk(K•∨ ) = r. The proof of the second formula is analogous. As we recalled at the beginning of this chapter, when K• is a sheaf Q concentrated in degree zero the dual complex may be different from the concentrated complex given by the dual sheaf Q∗ in degree zero. This point deserves a comment. Example 2.39. Take K• = O∆ , the structure sheaf of the diagonal ∆ ⊂ X × X. ∆ The integral functor ΦO X→X is isomorphic to the identity functor as we have seen ∗ = 0 because O∆ is a torsion sheaf, so that in Example 1.2. We have that O∆ ∗ O∆ ⊗π1∗ ωX [n] ∆ ΦX→X = 0 and this cannot be a quasi-inverse to ΦO X→X . The identity functor is of course a quasi-inverse to itself, and according to Theorem 2.38 it must coincide ∨ O∆ ⊗πi∗ ωX [n] with ΦX→ for i = 1, 2. Let us check that this is indeed the case. Since X X is smooth, the diagonal is a regular embedding and then a standard local computation using the Koszul complex yields the formulas ( 0 for q 6= n i ExtOX×X (O∆ , OX×Y ) ' (2.37) ∗ δ∗ (ωX ) for q = n. ∨ ∨ Thus, O∆ ' δ∗ (ωX )[−n] in the derived category, and therefore O∆ ⊗ πi∗ ωX [n] ' ∨ O∆ ⊗πi∗ ωX [n] ∆ O∆ so that ΦX→X ' ΦO X→X is the identity.

Looking at things the other way round, one should say that Theorem 2.38 ∨ ⊗πi∗ ωX [n] ' together with the uniqueness of the kernel (Theorem 2.25) proves O∆ O∆ and therefore yields the formula 2.37 without resorting to the Koszul complex. 4 Theorem 2.38 allows us to prove an important property of the map f K defined in Equation (1.12). •

•

b Corollary 2.40. Let X, Y be smooth projective varieties and let ΦK X→Y : D (X) → b K• • • D (Y ) be a Fourier-Mukai functor. The induced map f : A (X)⊗Q → A (Y )⊗Q is an isomorphism of Q-vector spaces. Moreover, if k = C, the induced f -map in • cohomology f K : H • (X, Q) → H • (Y, Q) is also an isomorphism of Q-vector spaces

64

Chapter 2. Fourier-Mukai functors

which induces an isomorphism of vector spaces between the even cohomology rings. K•∨ ⊗π ∗ ω [m]

•

is a quasi-inverse to ΦK Proof. By Theorem 2.38, ΦY→X X X X→Y , so that the • •∨ ∗ convolution K ∗ (K ⊗ πX ωX [m]) is isomorphic to O∆ in the derived category because of the uniqueness of the kernel (Theorem 2.25). The functoriality of the • •∨ ∗ map f (cf. Eq. (1.13)) yields f K ◦ f K ⊗πX ωX [m] = f O∆ . Since v(O∆ ) = δ∗ (1) by Grothendieck-Riemann-Roch for the diagonal immersion δ, we have f O∆ = Id (here v is the Mukai vector defined in Eq. (1.1)). One analogously proves that •∨ ∗ • f K ⊗πX ωX [m] ◦ f K = Id. The cohomology statement is proved in a similar way.  Part 4 of Theorem 2.38 implies that whenever the kernel K• is not of rank r of the canonical bundle of X has to be trivial, with zero, a certain power ωX r 6= 0. This is a strong geometric constraint: if X is a curve, it has to be elliptic (and then ωX ' OX ); if X is a surface, it has to be Abelian, K3 (in which cases 2 12 ' OX ) or bielliptic (for which ωX ' OX ) ωX ' OX ), Enriques (for which ωX (cf. [141, Thm. 6.3]). In dimension 3 the most important example is provided by Calabi-Yau varieties (for which, by definition, ωX ' OX ). This is the reason why the Fourier-Mukai transform has been mostly studied for this kind of variety. r (with nonzero expoHowever, this by no means implies that if all powers ωX K• b nent) are nontrivial, then Fourier-Mukai functors ΦX→Y : D (X) → Db (Y ) cannot exist. Rather they do exist, but the kernel K• must be of rank zero, as in the case of the structure sheaf of the diagonal.

We shall deal with the case of rank zero kernels when in Chapter 6 we shall study integral transforms for families, or relative integral transforms. Given two families of varieties X → S and Y → S, we shall define an integral functor • b b • in the derived category ΦK X→Y : D (X) → D (Y ) by means of a relative kernel K b D (X ×S Y ) of the fiber product. That transform will be defined as the ordinary integral functor with kernel i∗ K• , where i : X ×S Y ,→ X × Y is the natural • immersion. Even when the integral functor ΦK X→Y is an equivalence of categories we cannot use Theorem 2.38 to get information about ωX , because as a complex in Db (X × Y ) we have rk(i∗ K• ) = 0. Here one needs a relative version of Theorem 2.38 where the relative canonical sheaves ωX/S and ωY /S replace the absolute canonical sheaves. •

Let X, Y be smooth projective varieties and ΦK X→Y a Fourier-Mukai functor. ∗ K•∨ ⊗πX ωX K•∨ ⊗π ∗ ω By Theorem 2.38, a quasi-inverse is given by ΦY→X = ΦY→X Y Y . Let W and W ∨ be the supports of K• and K•∨ , respectively (see Definition A.90). Proposition 2.41. One has W = W ∨ and the two projections πX |W : W → X, πY |W : W → Y are surjective.

2.3. Fourier-Mukai functors

65

Proof. There is a convergent spectral sequence with E2p,q = ExtpOX×Y (Hp (K• ), O) p+q = Hp+q (K•∨ ). This proves that W ∨ ⊆ W . Reversing the roles of K• and E∞ • •∨ K• and K we get that W ⊆ W ∨ . Now, since ΦK Y→X is an equivalence, ΦY→X (Ox ) = Ljx∗ K• 6= 0 for every x ∈ X; this proves that the morphism πX |W : W → X is surjective. The surjectivity of πY |W : W → Y is proved analogously by using the •∨ fact that ΦK  Y→X is also a Fourier-Mukai functor by Theorem 2.38. •

b b The existence of a Fourier-Mukai functor ΦK X→Y : D (X) → D (Y ) (which is equivalent to the existence of an exact equivalence by Theorem 2.15) has interesting effects on the geometry of X and Y . Our next aim is to prove some strong results in that direction due to Orlov and Kawamata

We begin by recalling some standard definitions. If X is a smooth projective variety, and L is a line bundle on X, then for n  0 the dimension of the global sections Γ(X, Ln ) of Ln is a polynomial in n of a certain degree d ≤ dim X; the degree of the null polynomial is −∞ by decree. The degree of such polynomial is called the Kodaira dimension of L and it is denoted by κ(X, L). So one knows that κ(X, L) ≤ dim X. In particular, the Kodaira dimension of X is defined as κ(X) = κ(X, ωX ). One can also define the Kodaira dimension in terms of the projective rational maps defined by Ls (s ≥ 0), assuming that they exist. In that case (which corresponds to κ(X, L) > −∞), κ(X, L) is the maximum of the dimensions of the images of those maps, and it is also the transcendence degree of the graded ring R(X, L) = ⊕s≥0 Γ(X, Ls ) minus 1. ∗ ) = dim X), Lemma 2.42. (Kodaira’s Lemma) If κ(X) = dim X (resp. κ(X, ωX there exist an ample divisor H and an integer s0 such that for any integer s ≥ s0 s ∗ s there is an effective divisor Ds such that ωX ' OX (H) ⊗ OX (Ds ) (resp. (ωX ) ' OX (H) ⊗ OX (Ds )).

Proof. Assume that κ(X) = dim X. Let H ,→ X be a smooth ample divisor, which exists by Bertini’s theorem [141, 8.18], and consider the exact sequence s s (−H) → ωX → ωX s|H → 0 . 0 → ωX s ) is a polynomial in s of degree κ(X) = dim X, and χ(H, ωX s|H ) is Since χ(X, ωX a polynomial in s of degree κ(H, ωX |H ) ≤ dim H = dim X − 1, we see that for s s (−H) has a section. Thus, ωX (−H) ' OX (Ds ) for some s  0 the line bundle ωX  effective divisor Ds . The other case is analogous.

Recall that a line bundle L on a projective variety is numerically effective or nef if for any morphism φ : C → X where C is a projective curve, one has deg φ∗ L ≥ 0. We can consider only closed immersions C ,→ X, because we can always replace φ : C → X by its image.

66

Chapter 2. Fourier-Mukai functors

Lemma 2.43. Let f : Y → X be a projective morphism. 1. If L is a nef line bundle on X, then f ∗ L is nef on Y . 2. If f is surjective, then a line bundle L on X is nef if and only if f ∗ L is nef on Y . Proof. The first claim is obvious. For the second, let C be a projective curve, φ : C → X a morphism and N a very ample line bundle for the projective morphism fC : C ×Y X → C. For n  0, N n has a section which defines a divisor H ,→ C ×Y X. The curve C˜ = H r (r = dim Y − dim X) intersects every fiber in a finite number of points, so that the projection π : C˜ → C is a finite morphism. Moreover the composition φ ◦ π : C˜ → X factors as f ◦ ρ, where ρ : C˜ → Y is the induced morphism. Since f ∗ L is nef, deg ρ∗ f ∗ L ≥ 0, then deg φ∗ L ≥ 0 as well.  We can define the numerical Kodaira dimension of a line bundle L on a projective variety as the maximum ν(X, L) of the integer numbers m such that there is a proper morphism ϕ : T → X from a variety T of dimension m with the property ϕ∗ (c1 (L))m · T 6= 0. The intersection numbers ϕ∗ (c1 (L))m · T can be defined in terms of the Snapper polynomial. To this end, let us recall that for any line bundle N on a m-dimensional projective variety T , the Euler characteristic χ(T, N n ) of Ln is a polynomial in n of a certain degree d ≤ m, called the Snapper polynomial [119, Ex. 18.3.6], and that χ(T, N n ) =

1 c1 (N )m · T nm + terms of lower degree. m!

It is clear that we can define the numerical Kodaira dimension of L by considering only closed immersions ϕ : T ,→ X. Moreover, the numerical Kodaira dimension of any power of a line bundle L equals that of L, namely, ν(X, L) = ν(X, Ls ) for any s 6= 0. When L is nef, the numerical Kodaira dimension is the maximum of the integers m such that c1 (L)m is not numerically trivial. In this case, the numerical Kodaira dimension is bounded by the Kodaira dimension, ν(X, L) ≤ κ(X, L). If X is a projective Gorenstein variety, the numerical Kodaira dimension of X is defined as ν(X) = ν(X, ωX ). Lemma 2.44. Let f : Y → X be a projective morphism and L a line bundle on X. Then ν(Y, f ∗ L) ≤ ν(X, L). Moreover, if f is surjective, one has ν(Y, f ∗ L) = ν(X, L). Proof. The first claim is obvious. The proof of the second is similar to that of Lemma 2.43. Let ϕ : T → X be a proper morphism such that ϕ∗ (Lm ) · T 6= 0 with

2.3. Fourier-Mukai functors

67

m = dim T , and let N be a very relatively ample line bundle for the projective morphism fT : T ×Y X → T . For n  0, N n has a section which defines a divisor H ,→ T ×Y X. Then T˜ = H r (r = dim Y − dim X) intersects every fiber in a finite number of points, so that the projection π : T˜ → T is a finite morphism. Moreover the composition ϕ ◦ π : T˜ → X factors as f ◦ ρ, where ρ : T˜ → Y is the induced morphism. It follows that ρ∗ (f ∗ Lm ) · T˜ ' π ∗ (ϕ∗ Lm ) · T˜ 6= 0, and then  ν(X, L) ≤ ν(Y, f ∗ L), so that ν(Y, f ∗ L) = ν(X, L) as claimed. Finally, we need a technical result whose proof we include although it is standard. Lemma 2.45. Let Z be a normal variety and F a rank r coherent sheaf on Z. If L1 and L2 are line bundles on Z such that F ⊗ L1 ' F ⊗ L2 , then Lr1 ' Lr2 . Proof. Modding the torsion out we can assume that F is torsion-free. Since Z is normal there is a codimension two closed subset Z 0 such that F is locally free of rank r on U = Z − Z 0 . By the theorem on generic smoothness [141, III.10.7], U can be assumed to be smooth. Taking determinants, we get det(F|U ) ⊗ Lr1|U ' det(F|U ) ⊗ Lr2|U and thus Lr1|U ' Lr2|U . Since Z is normal and Z 0 has codimension 2, this isomorphism can be extended to an isomorphism Lr1 ' Lr2 (cf. [140, Theorem 3.8]).  •

Let X and Y be smooth projective varieties and ΦK X→Y a Fourier-Mukai functor. For every irreducible component Z of the support W of K• , we denote by Ze → Z its normalization and by p˜X : Ze → X, p˜Y : Z˜ → Y the induced maps. r ' p˜∗Y ωYr for some r > 0. In particular p˜∗X KX and Lemma 2.46. One has p˜∗X ωX ∗ p˜Y KY are Q-linearly equivalent. Moreover, we can chose an irreducible component ZX (K• ) of W such that pX = πX |ZX (K• ) : ZX (K• ) → X and then also p˜X : ZeX (K• ) → X, are dominant.

Proof. By Theorem 2.38, one has dim Y = dim X and if we denote by n this ∗ • K•∨ ⊗πX ωX [n] K•∨ ⊗π ∗ ω [n] dimension, a quasi-inverse to ΦK ' ΦY→X Y Y . The X→Y is given by ΦY→X ∗ ωX ' K•∨ ⊗ πY∗ ωY , uniqueness of the kernel (Theorem 2.25) implies that K•∨ ⊗ πX i •∨ ∗ i •∨ ∗ so that H (K ) ⊗ πX ωX ' H (K ) ⊗ πY ωY for every i. If ρ : Ze → X × Y is the composition of Ze → Z and the immersion Z ,→ X × Y , we see that ρ∗ (Hi (K•∨ ) ⊗ r ' p˜∗Y ωYr where r is the p∗X ωX ) ' ρ∗ (Hi (K•∨ ) ⊗ p∗Y ωY ). By Lemma 2.45, p˜∗X ωX rank of Hi (K•∨ )|Z , which is not zero. By Proposition 2.41, πX |W : W → X is surjective. Then we can choose an  irreducible component ZX (K• ) of W which dominates X.

68

Chapter 2. Fourier-Mukai functors

The following Proposition 2.48 and Theorem 2.49 express a result due to Kawamata, usually known as “D-equivalence implies K-equivalence.” We now give the precise definition of K-equivalent algebraic varieties. Definition 2.47. Two smooth projective algebraic varieties X and Y are K-equivalent if there are a normal variety Ze and projective birational morphisms p˜X : Ze → X, p˜Y : Ze → Y such that p˜∗ KX and p˜∗ KY are Q-linearly equivalent, that is, X

Y

rp˜∗X KX and rp˜∗Y KY are linearly equivalent for some r 6= 0.

4

Notice that if F • is a simple object of Db (Y ), i.e., HomD(Y ) (F • , F • ) = k, the support of F • is connected. This can be seen as follows: assume that Supp F • = ` Y1 Y2 ; if we represent F • as a finite complex E • of locally free sheaves of finite rank, the natural map E • → j1∗ j1∗ E • ⊕ j2∗ j2∗ E • is a quasi-isomorphism. Then F • ' j1∗ j1∗ E • ⊕ j2∗ j2∗ E • in the derived category, so that F • is not simple. Proposition 2.48. Let ZX (K• ) be an irreducible component of W = Supp K• such that pX : ZX (K• ) → X is dominant. −1 1. There is a nonempty open subset U of X such that p−1 X (x) = πX (x) ∩ W ' • Supp ΦK X→Y (Ox ) for every point x ∈ U .

2. If dim ZX (K• ) = dim X, then p˜X : ZeX (K• ) → X and p˜Y : ZeX (K• ) → Y are birational and p˜∗X KX and p˜∗Y KY are Q-linearly equivalent. That is, X and Y are K-equivalent. Proof. If Z1 , . . . , Zs are the irreducible components of W other than Z = ZX (K• ), • and T = ∪i (Z ∩ Zi ), we take U = X − pX (T ). Since ΦK X→Y (Ox ) is simple, its • −1 support Supp ΦK X→Y (Ox ) ' πX (x) ∩ W is connected as we have just seen. Then −1 K• (x) ∩ Z = p−1 Supp ΦX→Y (Ox ) has to be contained in πX X (x) if x ∈ U ; this proves the first part. We now prove the second part. We have to prove that both pX and pY are birational. Consider first the projection pX : Z → X. Since dim Z = dim X, pX is generically finite. By Zariski’s main theorem [141, 11.4], to prove that it is birational, we need only to check that it is generically injective. By the first part, −1 −1 x ∈ X, and then πX (x) = {(x, y1 ), . . . , (x, ys )} p−1 X (x) = πX (x) ∩ W for generic • is finite. Thus, Ljx∗ K• = ΦK X→Y (Ox ) is supported at the points y1 , . . . , ys , and • K• HomDb (Y ) (ΦK X→Y (Ox ), ΦX→Y (Ox )) is of dimension s. By the Parseval formula (cf. Proposition 1.34), one has •

•

K HomDb (Y ) (ΦK X→Y (Ox ), ΦX→Y (Ox )) ' HomD b (X) (Ox , Ox ) = k ,

so that s = 1. This proves that pX is generically injective. Our last step is to show that pY : Z → Y is also birational. If we prove that it is dominant, taking into account that Z is also an irreducible component of W ∨ =

2.3. Fourier-Mukai functors

69

Supp K•∨ (cf. Proposition 2.41), we deduce as above that pY is birational. Since pX is birational, there is an open subset U 0 ⊆ U such that πX induces an isomorphism −1 0 0 0 between p−1 X (U ) = πX (U ) ∩ W and U . If we assume that pY is not dominant, then dim pY (Z) < dim Y and there exist distinct points x1 , x2 of U 0 such that −1 −1 (x1 )∩W ) = πY (πX (x2 )∩W ). Thus, the point y is the support of both y = πY (πX • K• K• K• ΦX→Y (Ox1 ) and ΦX→Y (Ox2 ), so that HomiDb (Y ) (ΦK X→Y (Ox1 ), ΦX→Y (Ox2 )) 6= 0 for some integer i. By the Parseval formula, this implies that HomiDb (X) (Ox1 , Ox2 ) 6= 0, which is absurd. r = p˜∗Y ωYr , we eventually obtain that p˜∗X KX and p˜∗Y KY Finally, since p˜∗X ωX are Q-linearly equivalent. 

Theorem 2.49. [175, 176] Let X, Y be smooth Fourier-Mukai partners. ∗ 1. The line bundle ωX (resp. ωX ) is nef if and only if ωY (resp. ωY∗ ) is nef.

2. X and Y have the same numerical Kodaira dimension, ν(X) = ν(Y ). ∗ ) = dim X), 3. If the Kodaira dimension κ(X) is equal to dim X (or if κ(X, ωX then X and Y are K-equivalent. •

• Proof. By Lemma 2.37, there is a Fourier-Mukai functor F = ΦK X→Y . Let ZX (K ) be • an irreducible component of the support W of K which dominates X (cf. Lemma 2.46; we use the same notation as in this lemma).

1. If ωY is nef so is ωYr , then p˜∗Y ωYr is nef by Lemma 2.43 and since p˜X is r r ' p˜∗Y ωYr , the same lemma implies that ωX , and hence ωX , surjective and p˜∗X ωX ∗ is nef. The case when ωY is nef is proved analogously. 2. By Lemma 2.44, ν(X, ωX ) = ν(ZX (K• ), p∗X ωX ) = ν(ZX (K• ), p∗Y ωY ) ≤ ν(Y, ωY ). Reversing the roles of X and Y one proves the converse statement. 3. Assume first that κ(X) = dim X. By Kodaira’s Lemma 2.42, one can take m m ' p˜∗Y ωYm and ωX ' OX (H) ⊗ m > 0 such that the two isomorphisms p˜∗X ωX OX (D), where H is an ample divisor and D is effective, hold true. Let us see that p˜Y : ZeX (K• ) → Y is quasi-finite (i.e., it has finite fibers) • outside p˜−1 X (D). First note that since pX and pY are the restrictions to ZX (K ) of the projections of X × Y onto its factors, no curve can be contracted by both of them; since normalization is a finite morphism, the same happens for p˜X and p˜Y . Assume now that there is a curve C contained in a fiber p˜−1 Y (y) and not entirely −1 (D); then we have p ˜ K · C = 0 and contained in p˜−1 Y X Y

m˜ p−1 p−1 ˜−1 ˜−1 ˜−1 Y KY · C = m˜ X KX · C = p X H ·C +p X D·C ≥p X H ·C. p−1 Since C cannot be contracted by p˜X and H is ample, we get m˜ Y KY · C > 0, • • e which is a contradiction. Thus dim ZX (K ) = dim ZX (K ) ≤ n = dim Y so that

70

Chapter 2. Fourier-Mukai functors

∗ ) = dim ZX (K• ) = n, and we conclude by Proposition 2.48. The case κ(X, ωX dim X is analogous. 

Remark 2.50. The variety ZeX (K• ) in Lemma 2.46 may be assumed to be smooth possibly by replacing it with a resolution of its singularities. 4 A consequence of Kawamata’s Theorem 2.49 is a celebrated “reconstruction theorem” due to Bondal and Orlov [49]. Theorem 2.51. Let X, Y be smooth Fourier-Mukai partners. If either ωX or ωY is ample or anti-ample, there is an isomorphism X ' Y . Proof. Assume that ωX is ample or anti-ample. If C ,→ ZeX (K• ) is a curve conm ˜∗Y ωYm |C ' OC , which is impossible because ωX is tracted by p˜Y , then p˜∗X ωX |C ' p ample or anti-ample and C is not contracted by p˜X as we have seen in the proof of Theorem 2.49. Hence p˜Y is an isomorphism and we have a birational morphism m ' ωYm with m > 0 when ωX is f : Y → X of smooth varieties such that f ∗ ωX ample. Let us now prove that f is actually an isomorphism. In the exact sequence of differentials df f ∗ ΩX −→ ΩY → ΩY /X → 0 the morphism df is injective (because it is injective at the generic point and f ∗ ΩX is locally free); then ΩY /X is a torsion sheaf supported by a closed subscheme Y 0 6= Y which coincides with the zeroes of the determinant of df . This determinant −1 −1 m . Then (det(df ))m is a section of (ωY ⊗ f ∗ ωX ) ' OY is a section of ωY ⊗ f ∗ ωX 0 0 vanishing on Y . Thus Y is empty, so that ΩY /X = 0 and f is smooth of relative dimension zero, and being also birational, it is an isomorphism.  We have chosen to give a detailed account of this proof of this important theorem because the techniques introduced and its underlying ideas will be useful elsewhere in this book, in particular when we shall study the Fourier-Mukai partners of a variety. However, the original proof by Bondal and Orlov in [49] is more direct and does not make use of Orlov’s representability theorem 2.15. It also enlightens how the derived category of a variety encodes information about its points and the line bundles on it. For this reason we offer here a brief sketch of that proof. One starts by defining the point objects and invertible objects in a triangulated category. One shows that when the latter is the derived category Db (X) of coherent sheaves of a smooth algebraic variety with ample canonical or anticanonical sheaf, the point objects are the complexes of the form Ox [i] with i ∈ Z. On the other hand, when point objects have this form, one shows that the invertible objects are the objects of the form L[i] with i ∈ Z, where L is a line bundle. Assume now that X is a smooth projective algebraic variety with ample or anti-ample canonical bundle, and Y is a Fourier-Mukai partner of X, i.e., that there

2.3. Fourier-Mukai functors

71

is an exact equivalence of triangulated categories F : Db (X) → Db (Y ). One proves that F maps point objects to point objects, and this in turn implies that the point objects in Y are again exactly the shifted skyscraper sheaves. Along the same lines, one proves that F maps invertible objects to invertible objects. The result about point objects implies now that line bundles are mapped into shifted line bundles. By suitably redefining the functor F , one can obtain that skyscraper sheaves are mapped to skyscrapers, thus providing a set-theoretic identification of X with Y , and that line bundles are mapped to line bundles. The latter property implies that X and Y are homeomorphic. After redefining F again, we can assume that i ) ' ωYi F (OX ) ' OY , and since F commutes with the Serre functor, one has F (ωX for all i. Since ωX is ample or anti-ample, the topology of X has a basis formed j i , ωX ), i, j ∈ Z; by definition, Uα is the by open sets Uα labeled by α ∈ HomX (ωX set of points in X where α does not vanish. The properties of F that one has so far proved imply that the open sets Vα ⊂ Y defined in the same way in terms of ωY are also a basis of the topology of Y . A theorem by Illusie [158] implies that ωY is ample or anti-ample, respectively. Moreover, the equivalence F induces an isomorphism between the graded i ) and ⊕i≥0 HomY (OY , ωYi ). When ωX and canonical algebras ⊕i≥0 HomX (OX , ωX ωY are ample, this gives rise to an algebraic isomorphism i ∼ Proj(⊕ Hom (O , ω i )) ' Y . X ' Proj(⊕i≥0 HomX (OX , ωX )) → i≥0 Y Y Y

When ωX and ωY are anti-ample, one proceeds in a similar way with the algebras −i ) and ⊕i≥0 HomY (OY , ωY−i ). It is worth saying that this proof ⊕i≥0 HomX (OX , ωX does not makes use of Orlov’s representability theorem 2.15.

2.3.2

Characterization of Fourier-Mukai functors

We have already seen that a spanning class may be used to test if an exact fully faithful functor is an equivalence of categories. But one can also find a suitable spanning class for the derived category Db (X) and use it to state conditions for an integral functor to be a Fourier-Mukai functor. The results in the first part of this section are valid in arbitrary characteristic, while starting from Proposition 2.56 we need to assume that the characteristic is zero. The following two propositions are taken from [61]. Proposition 2.52. On a smooth proper variety X the skyscraper sheaves Ox form a spanning class for the derived category Db (X). Proof. For every M• ∈ Db (X) and every point x ∈ X there is a spectral sequence p+q • • = Homp+q E2p,q = ExtpOX (H−q (M• ), Ox ) =⇒ E∞ D(X) (M , Ox ). If M 6= 0, then

72

Chapter 2. Fourier-Mukai functors

there exists an integer q such that Hq (M• ) 6= 0. Let q0 be the maximum of such q’s and let x be a point in the support of Hq0 (M• ). Then we have a nonzero element e ∈ E20,−q0 = HomD(X) (Hq0 (M• ), Ox ), and that element survives to give 0 −q0 • a nonzero element of E∞ = Hom−q D(X) (M , Ox ). This implies that the skyscraper sheaves Ox form a spanning class on the right for the derived category Db (X), that is, they satisfy Condition 2 in Definition 2.1. Note that this does not require X to be smooth, while this is necessary to prove Condition 1. Take then M• 6= 0. Since X is smooth, ωX is a line bundle, so that N • = M• ⊗ ωX 6= 0 and we can q0 • ¯ such that Ext−¯ apply the above argument to N • and find q¯0 and x ¯) = OX (N , Ox −¯ q0 n+¯ q −¯ q 0 0 HomD(X) (N • , Ox¯ ) 6= 0. By Serre duality HomD(X) (Ox , N • ) ' ExtOX (N • , Ox¯ )∗ 6= 0, thus finishing the proof.  We also need to ascertain when the derived category Db (X) is indecomposable. Proposition 2.53. Let X be a smooth proper variety. Then Db (X) is indecomposable if and only if X is connected. ` Proof. If X is not connected, write X = X1 X2 , and then Db (X) ' Db (X1 ) ⊕ Db (X2 ). Assume now that X is connected and that there exist full nontrivial subcategories A1 , A2 with Db (X) ' A1 ⊕ A2 . For any integral closed subvariety Y ,→ X, the sheaf OY is indecomposable, so that it is isomorphic to an object either in A1 or A2 . Moreover, for every point y ∈ Y , the sheaf Oy is isomorphic to an object in the same Aj as OY , because otherwise HomD(X) (OY , Oy ) = 0 and this is not true. Let Xj be the union of all integral subvarieties Y such that OY is ` isomorphic to an object of A2 . Then X1 , X2 are closed subsets and X = X1 X2 , because if y ∈ X1 ∩ X2 , then Oy is isomorphic both to an object of A1 and an object of A2 , and this is absurd. Since X is connected, one of the Xj ’s, say X2 , is empty. Then for every object K• in Db (X2 ) one has HomiD(X) (K• , Ox ) = 0 ,

for any i ∈ Z, x ∈ X

and therefore K• ' 0 because the skyscraper sheaves form a spanning class by Proposition 2.52.  Let X be a smooth projective variety. Definition 2.54. A sheaf F on X is special if F ⊗ ωX ' F. An object F • of Db (X) 4 is special if F • ⊗ ωX ' F • in Db (X). Then, when the canonical bundle ωX is trivial, every object of Db (X) is special. An object F • of Db (X) is special if and only if its cohomology sheaves Hi (F • ) are special sheaves, as the following proposition shows.

2.3. Fourier-Mukai functors

73

Proposition 2.55. Let F • be a complex in Db (X) such that all its cohomology sheaves Hi (F • ) are special sheaves. If f i : Hi (F • ) → Hi (F • ) ⊗ ωX are isomorphisms of sheaves, there is an isomorphism f : F • → F • ⊗ ωX in the derived category such that Hi (f ) = f i for every i. Proof. We proceed by induction on the number of nonzero cohomology sheaves. If Hn (F • ) is the highest nonzero cohomology sheaf, we can assume that F m = 0 for m > n. Assume that F • has only a nonzero cohomology sheaf. Then F • ' Hm (F • )[m] in Db (X) and we set f = f m [m] : F • → F • ⊗ ωX . In the general case, we can assume by induction that there is an isomorphism f˜: F • ≤(n−1) → F • ≤(n−1) ⊗ ωX in the derived category inducing in cohomology the morphisms f i for i ≤ n − 1. Consider the exact triangle in K(Qco(X)) β

i

F • ≤(n−1) → − F • → Cone(i) − → F • ≤(n−1) [1] . Note that β is homotopic to zero, so β = 0 in K(Qco(X)) and then also in the derived category. Since Cone(i) ' Hm (F • )[m] in the derived category, we have a commutative diagram in Db (X) whose arrows are exact triangles F • ≤(n−1)

i

/ F•

' f m [m]

' f0



F • ≤(n−1)

/ Cone(i)

i

/ F•

 / Cone(i)

β=0

/ F • ≤(n−1) [1] ' f 0 [1]

 / F • ≤(n−1) [1] .

β=0

Then, there is an isomorphism f : F • → F • ⊗ ωX in Db (X) which completes the diagram.  From this point on, we assume that the base field k has characteristic zero. Proposition 2.56. [202, 61] Let X and Y be smooth projective varieties of the same dimension n, and let K• be a kernel in Db (X × Y ). The following conditions are equivalent: •

1. ΦK X→Y is a Fourier-Mukai functor; •

∗ • 2. ΦK is a special object of Db (Y ) for all x ∈ X. X→Y is fully faithful and Ljx K

In particular, if Q is a sheaf on X × Y strongly simple over X, then ΦQ X→Y is a Fourier-Mukai transform if and only if Qx is special for all x ∈ X. Proof. We can assume that Y is connected so that Db (Y ) is indecomposable by Proposition 2.53. To prove that 2 implies 1, in view of Propositions 2.52 and 2.5 we

74

Chapter 2. Fourier-Mukai functors •

•

K need to show that ΦK X→Y SX (Ox ) ' SY ΦX→Y (Ox ) for every closed point x. Indeed, ∗ • by the speciality of the complexes Ljx K , we have •

•

∗ • ∗ • K ΦK X→Y SX (Ox ) ' Ljx K [n] ' Ljx K ωY [n] ' SY ΦX→Y (Ox ) .

The fact that 1 implies 2 is proved similarly.



The results of the previous proposition will be mostly used in the following form. Proposition 2.57. Assume that X and Y are smooth projective varieties of the same dimension with trivial canonical bundles and that K• is an object in Db (X × Y ) • strongly simple over X. Then the functor ΦK X→Y is a Fourier-Mukai functor and •∨ • K [n] the functor ΦY→X is a quasi-inverse to ΦK X→Y . In particular, if Q is a locally free Q∗ [n] sheaf on X × Y strongly simple over X, the functor ΦY→X is a quasi-inverse for ΦQ X→Y . Corollary 2.58. Let X, Y be as in Proposition 2.57 and let K• be an object in Db (X × Y ) which is strongly simple over X. Then K• is strongly simple over Y • K•∨ [n] . as well and ΦK Y→X is a Fourier-Mukai functor with inverse ΦX→Y K•∨ [n]

Proof. By Proposition 2.57, ΦY→X is an exact equivalence, and then K•∨ is strongly simple over Y by Theorem 1.27. By Remark 1.26, K• is strongly simple over Y , so that the statement follows again from Proposition 2.57.  We are now in a position to show that the composition of two Fourier-Mukai transforms may fail to be a Fourier-Mukai transform. Example 2.59. Let X be a K3 surface. We shall give an introduction to the geometry of K3 surfaces in Chapter 4; what we shall need here is that ωX ' OX and b b ∆ H 1 (X, OX ) = 0. Let us consider the integral functor Φ = ΦIX→ X : D (X) → D (X), where I∆ is the ideal sheaf of the diagonal in X × X. One easily checks that I∆ is strongly simple, so that Φ is a Fourier-Mukai functor by Corollary 2.58. Again a straightforward computation shows that Φ(OX ) ' OX [−2]. Moreover, if L is a line bundle on X which has no cohomology in every degree, one has Φ(L) ' L[−1]. Comparing the two results, we see that for k big enough, the kernel of iterated 4 composition Φk is not a shifted sheaf. Let X and Y be smooth projective varieties and K• a kernel in Db (X × Y ). ˜ Y˜ of smooth projective varieties and another kernel We consider another pair X, L ˜ × Y˜ ). We then have a kernel K•  K ˜ • in Db (X × X ˜ × Y × Y˜ ). ˜ • in Db (X K Lemma 1.28 about the product of integral functors can be strengthened in the case of Fourier-Mukai functors.

2.3. Fourier-Mukai functors

75 •

˜•

K Corollary 2.60. If the integral functors ΦK ˜ Y ˜ are Fourier-Mukai funcX→Y and ΦX→ •

L

˜•

K tors, then the functor ΦK ˜ Y ×Y ˜ is a Fourier-Mukai functor as well. X×X→

Proof. In view of Lemma 1.28 and Proposition 2.56 we need only to show that for L ˜ • ) is a special object. ˜ the restriction Lj ∗ (K•  K evey closed point x, x ˜) ∈ X × X (x,˜ x) L

∗ • Lj(x,˜ x) (K

From one side, one has the isomorphism the other, as ωY ×Y˜ ' ωY  ωY˜ , we have L

L

˜ • ) ' Lj ∗ K•  Lj ∗ K ˜ • . From K x x ˜ L

∗ • ∗ • ∗ ˜• ˜• Lj(x,˜ ˜ K ⊗ ωY˜ ) . x) (K  K ) ⊗ ωY ×Y˜ ' (Ljx K ⊗ ωY )  (Ljx

 In a number of important examples that will be thoroughly investigated in the next chapters, the hypotheses of Proposition 2.56 are met when Y is a connected component of the moduli spaces of simple sheaves on X, X is a connected component of the moduli spaces of simple sheaves on Y , and Q is the corresponding bi-universal family (provided it exists). In the case of surfaces, there are particular results that will be very useful. We describe here some of them. Let Y be a smooth projective surface and X a fine moduli space of special stable sheaves on Y with fixed Mukai vector v (cf. Eq. (1.1)). Let Q be a universal sheaf on X × Y for the corresponding moduli problem, so that Q is flat over X and Qx is a stable special sheaf on Y with Mukai vector v. Given closed points x and z in X, one has χ(Qx , Qz ) = −v 2 by Equation (1.7). The following result can be found in [70]. Proposition 2.61. Assume that X is a projective surface. 1. X is smooth if and only if v 2 = 0. b b 2. In this case, the integral functor ΦQ X→Y : D (X) → D (Y ) is a Fourier-Mukai functor.

Proof. Let x be a closed point of X. Since the Hom1X (Qx , Qx ) is the tangent space at x to the moduli space X of the sheaves {Qx } on Y , one has that X is smooth at x if and only if dim Hom1X (Qx , Qx ) = 2. To compute this dimension, we note that the sheaf Qx is stable and special, so that HomY (Qx , Qz ) ' k by the stability and Hom2X (Qx , Qx ) ' k by Serre duality. It follows that −v 2 = χ(Qx , Qx ) = 2 − dim Hom1X (Qx , Qx ) which proves the first claim. Assume now that v 2 = 0. If x and z are different closed points of X, one has that HomY (Qx , Qz ) = 0 because Qx and Qz are nonisomorphic stable sheaves with the same Chern characters. Since the sheaves QX are special, Serre duality

76

Chapter 2. Fourier-Mukai functors

gives Hom2X (Qx , Qz ) = 0. Finally, from χ(Qx , Qz ) = −v 2 = 0 we deduce that also Hom1X (Qx , Qz ) = 0, so that Q is strongly simple over X. The claim follows now from Proposition 2.56.  Remark 2.62. Proposition 2.61 holds also true for pure stable sheaves in the sense of Simpson as described in Section C.2, because pure stable sheaves have the properties of torsion-free stable sheaves we have used in its proof. 4

2.3.3

Fourier-Mukai functors between moduli spaces

We would like to show that in many cases, integral transforms define in a natural way algebraic morphisms between moduli spaces. Let Φ : Db (X) → Db (Y ) be an integral functor, where X and Y are smooth projective varieties. Let MX,P be the functor associating to any variety T the set of equivalence classes of all coherent sheaves E on T × X, flat over T and whose restrictions Et = jt∗ E to the fibers Xt ' X of πT : T × X → T have Hilbert polynomial P . Here, two sheaves E and E 0 are considered to be equivalent if E ' E 0 ⊗ πT∗ L for a line bundle L on T . Furthermore, let MX be a subfunctor of MX,P parameterizing WITi sheaves for a certain index i. Corollary 1.9 implies that if E is in MX (T ), the sheaves Eb = ΦiT (E) are flat over T , so that for a fixed i the b t ' Ebt have the same Hilbert polynomial Pˆ . Moreover Φi (E ⊗ π ∗ L) ' fibers (E) T T i ΦT (E) ⊗ πT∗ L. Thus, ΦiT maps MX (T ) to MY,Pˆ (T ). The polynomial Pˆ can be computed by the Grothendieck-Riemann-Roch formula in terms of P , the Chern classes of the kernel K• of Φ and the Todd class of X. By compatibility of integral functors with base change (Proposition 1.8), Φi induces a morphism of functors ΦiM : MX → MY,Pˆ . Proposition 2.63. Assume that MX has a coarse moduli scheme MX . 1. If there is a subfunctor MY ⊂ MY,Pˆ containing the image of ΦiM that is also coarsely representable by a moduli scheme MY , then the integral functor Φ gives rise to an algebraic morphism of schemes ΦiM : MX → MY . 2. If Φ is a Fourier-Mukai functor, then the image functor MY = Φi (MX ) is coarsely representable by a moduli scheme MY , and Φ induces a scheme isomorphism ∼M . ΦiM : MX → Y Moreover MX is a fine moduli scheme (that is, it represents the moduli functor MX ) if and only if MY is a fine moduli scheme.

2.3. Fourier-Mukai functors

77

Proof. 1. Since MY is coarsely represented by MY , there exists a morphism of functors MY → Hom(−, MY ), where the latter is the functor of points of MY . The composition with Φi is a morphism of functors MX → Hom(−, MY ) which, by the definition of coarse moduli, factors in a unique way through a morphism of functors Hom(−, MX ) → Hom(−, MY ). This corresponds to a scheme morphism Φi : MX → MY . Part 2 is straightforward, due to the uniqueness of the coarse moduli of a functor.  An important example is given by the moduli functor of skyscraper sheaves Ox on X. Assume that the skyscraper sheaves are all WITi ; this happens for instance if the kernel of Φ is a concentrated complex, in which case they are WIT0 . Then we have: Corollary 2.64. If Φ is a Fourier-Mukai functor, then X is a fine moduli space for the moduli functor MY of the sheaves Φi (Ox ) over Y . To illustrate another example, let X and Y be polarized smooth projective ss varieties (see Section C.2), and let Mss X,P , MY,Pˆ be the corresponding moduli functors of (Gieseker) semistable sheaves. Assume that all semistable sheaves F i in Mss X,P are WITi and that their images Φ (F) are semistable. We have: Corollary 2.65.

ss ss → MY, . 1. Φi induces a morphism of schemes ΦiM : MX,P Pˆ

ss 2. If Φ is a Fourier-Mukai functor and Φi (Mss X,P ) = MY,Pˆ , the induced mor∼ phism is an isomorphism of schemes Φi : M ss → M ss . M

X,P

Y,Pˆ

Corollary 2.65 implies that Φi transforms S-equivalent semistable sheaves on X to S-equivalent sheaves on Y (for the notion of S-equivalence, see Section C.2). However, Φi may transform non-S-equivalent semistable sheaves on X to S-equivalent sheaves on Y , even if Φ is a Fourier-Mukai functor. Thus, in general ss ss → MY, induced by a a Fourier-Mukai functor may fail the morphism ΦiM : MX,P Pˆ to be injective or surjective. There is however a partial result. Corollary 2.66. If Φ is a Fourier-Mukai functor and there is a stable sheaf F s such that Φi (F) is stable, the functor Φi induces a surjective birational in MX,P morphism ˜X → M ˜Y , ΦiM : M ˜ X and M ˜ Y are the irreducible components of M ss and M ss which where M X,P Y,Pˆ contain [F] and [Φi (F)], respectively. Proof. If G is a semistable sheaf on X and ΦiM ([G]) = [Φi (F)], then Φi (G) is Sequivalent to F, and thus Φi (G) ' Φi (F) because Φi (F) is stable. Hence, G ' F by

78

Chapter 2. Fourier-Mukai functors

the invertibility of Φ, and the fiber of ΦiM over the point [Φi (F)] is a single point. ss ss and MY, are projective (see Theorem C.6), by Zariski’s main theorem Since MX,P Pˆ [136, 4.4.3] there exist open neighborhoods V of [Φi (F)] and U = (ΦiM )−1 (V ) of ∼ V . Moreover, Φi (M ˜) = M ˜ Y because Φi (M ˜ ) is irreducible [F] such that ΦiM : U → M M i i ˜X → M ˜ Y is birational.  and contains [Φ (F)]. Then ΦM : M

2.4

Notes and further reading

Historical remarks. As we already mentioned, the first instance of a Fourier-Mukai transform is contained in Mukai’s 1981 paper [224]. A Fourier-Mukai transform on K3 surfaces was first constructed by the authors of this book in 1994 [24] (see also [26]). Later a similar construction was done by Mukai [228]. We shall study Fourier-Mukai transforms on K3 surfaces in Chapter 4. Other examples of FourierMukai transforms will be described in Chapter 6 as relative integral functors. The talk by Bondal and Orlov at the 2002 International Congress of Mathematicians [50] is a nice review of work done by them and others on equivalences between derived categories of coherent sheaves. Spherical objects. Fourier-Mukai functors can be constructed by using the socalled spherical objects, first introduced by Bondal and Polishchuk [51]. A complex E • in Db (X) is spherical if: (i) HomiDb (X) (E • , E • ) is equal to k for i = 0, dim X and ∼ E • . For instance, the structure to zero otherwise; (ii) E • is special, i.e., E • ⊗ ωX → sheaf of a Calabi-Yau variety is a spherical object. We exploited this property in Example 2.59, which indeed generalizes to any Calabi-Yau variety. For any object E • of Db (X), one defines the twist functor TE • as the integral L

functor whose kernel is the cone of the evaluation morphism E •∨  E • → O∆ . Whenever E • is spherical, the twist functor TE • is a Fourier-Mukai functor, as proved by Seidel and Thomas [265]. Results for singular varieties. Kawamata proved a generalization of Orlov’s representability theorem 2.15 to stacks associated with normal varieties with quotient singularities [176]. A characterization of Fourier-Mukai functors on CohenMacaulay varieties was given in [144, 143]. An alternative setting: differential graded categories. Though they are a powerful tool in algebraic geometry as well as in algebraic analysis, representation theory and several other branches of mathematics, derived categories suffer from a number of drawbacks. In particular, the underlying triangulated structure appears too poor to allow for an entirely satisfactory description of functors between these categories and of natural algebraic or homotopical operations. Bondal and Kapranov [46] proposed the idea of using differential graded categories as an “enhancement” of derived categories in order to provide a more flexible and rich environment.

2.4. Notes and further reading

79

Differential graded categories — whose first appearance in the literature dates back to the 1960s [179] — can be thought of as “differential graded algebras with many objects,” pretty much in the same vein as additive categories can be thought of as “rings with many objects.” The basics of differential graded categories are briefly presented in Section A.4.4. For a more detailed overview the reader is referred to Keller’s beautiful exposition [178]. Here we shall limit ourselves to focus attention on a few issues that appear to be more relevant to our purposes. The category dgcatk of small differential graded k-categories (Definition A.50) admits a structure of model category, whose weak equivalences are the quasiequivalences (Theorem A.51). One denotes by Ho(dgcatk ) the localization of dgcatk with respect to quasi-equivalences. L

As proved in [105, 284], the monoidal category (Ho(dgcatk ), ⊗ ) admits an internal Hom-functor RHom (see Theorem A.56). Within this framework, To¨en [284] has recently worked out a version of derived Morita theory, where the morphisms between dg-categories of modules over two dg-categories E, F are described as the dg-category of (E-F)-bimodules. As an application of his theory, To¨en proved some results that can be viewed as a strengthening of Orlov’s representability theorem 2.15. Let us consider the Abelian category Qco(X) of quasi-coherent sheaves on an algebraic variety X. We shall denote by Ddg (X) the dg-derived category Ddg (Cdg (Qco(X))) (see Definition A.54); one has that the homotopy category H 0 (Ddg (X)) is equivalent to D(X). As Theorem A.57 shows, given two algebraic varieties X, Y , there is natural isomorphism in Ho(dgcatk ) Ddg (X × Y ) ' RHomc (Ddg (X), Ddg (Y )) , where RHomc denotes the full subcategory of RHom consisting of coproduct preserving quasi-functors. In particular, when X and Y are smooth and projective, it turns out (Equation A.10) that parf dg (X ×k Y ) ' RHom(parf dg (X), parf dg (Y )) , where parf dg (X) is the full sub-dg-category of Ddg (X) whose objects are the perfect complexes. We can rephrase this equivalence by saying that, in the dg environment, all functors are integral functors. One should compare these results with the representability theorem [47, 6.8] proved by Bondal, Larsen, and Lunts. Actually, the dg-category parf dg (X) can be viewed as a standard enhancement of the derived category D(X) in the sense of Definition [47, 5.1].

Chapter 3

Fourier-Mukai on Abelian varieties Introduction Mukai’s 1981 paper [224] contains, in one way or another, in a more or less explicit form, many of the ideas that have been introduced and developed in subsequent years in connection with Fourier-Mukai transforms. Thus these ideas are often at the core of the theory of integral functors that we have quite systematically developed in the first chapter of this book. As a result, we can prove most of the results presented in this chapter quite straightforwardly. So, while the treatment of this chapter is fairly close in spirit to the original paper by Mukai [224], the details of the arguments are often rather different. In this chapter we review very briefly the basic definitions concerning Abelian varieties, the dual Abelian variety, and the Poincar´e bundle P (later in the chapter we shall also discuss the notion of polarization of an Abelian variety). We then introduce the integral functor associated with the kernel P and show immediately that it is a Fourier-Mukai transform. We study some properties of this transform and apply it to study some classes of sheaves on Abelian varieties (unipotent and homogeneous bundles, Picard sheaves). The chapter ends with the description of a property of Fourier-Mukai transforms that we encounter here for the first time: under suitable conditions, it preserves the stability of the sheaves it acts on. We shall meet this property again in other situations, e.g., for Fourier-Mukai transforms on K3 surfaces and for relative Fourier-Mukai transforms on elliptic fibrations. In Chapter 6, devoted to relative integral functors, we shall discuss a FourierC. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_3, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

81

82

Chapter 3. Fourier-Mukai on Abelian varieties

Mukai transform on Abelian schemes (an example again due to Mukai [226]) which is a quite straightforward generalization of the transform studied in this chapter. A good deal of information about Fourier-Mukai transforms on Abelian varieties may be found in the book by Polishchuk [251], where several topics not considered here are covered, such as Theta functions, a proof of the Torelli theorem by means of the Fourier-Mukai transform, and others.

3.1

Abelian varieties

An Abelian variety X over a field k is a proper integral (i.e., irreducible and reduced) algebraic group, i.e., a proper integral variety for which there are morphisms • mX : X × X → X (the group law), • ιX : X → X (the inverse morphism), • e : Spec k → X (the identity point), such that usual relations are fulfilled: 1. (associativity) mX ◦ (mX × IdX ) = mX ◦ (IdX × mX ); 2. (existence of the unit) mX ◦ (IdX × e) = m ◦ (e × IdX ) = IdX ; 3. (property of the inverse) denoting by (IdX × ιX ), (ιX × IdX ) : G → G × G the morphisms (IdX × ιX )(x) = (x, ιX (x)) and (ιX × IdX )(x) = (ιX (x), x), one has mX ◦ (IdX × ιX ) = mX ◦ (IdX × ιX ) = e ◦ p, where p : G → Spec k is the natural projection onto a point. A homomorphism φ : X → Y between two Abelian varieties X, Y is a morphism which is also a group homomorphism. If φ is surjective and its kernel is finite, it is called an isogeny. In this case one defines the exponent e(φ) as the smallest positive integer such that e(φ) x = 0 for all x ∈ ker(φ). As a matter of fact, the condition that X is a proper variety implies that this group structure is commutative (which motivates the name of these varieties) [229, §4, Cor. 2]. So we shall use an additive notation and will write mX (x, x0 ) = x + x0 and ιX (x) = −x. The translation morphism τ is defined as τx (x0 ) = x + x0 . The identity element will be denoted 0. We shall also use the notation µX (x, x0 ) = x − x0 , and will denote by π1 , π2 the projections onto the factors of X × X. Since the translation morphism τx is an isomorphism for every x ∈ X, and smooth points in X exist, any Abelian variety is nonsingular.

3.1. Abelian varieties

83

We wish to show that any Abelian variety is projective. Given a line bundle L on X, we consider the line bundle on X × X Q = m∗X L ⊗ π1∗ L−1 ⊗ π2∗ L−1 . It is a general fact that there exists a closed subvariety K(L) ⊂ X which is the largest subscheme of X such that the restriction of Q to X × K(L) is trivial. K(L) is a subgroup of X (this follows from the so-called theorem of the square [229, §6, Cor. 4]), and its points x are characterized by the condition τx∗ L ' L. One has the following result. Lemma 3.1. If L is effective, then it is ample if and only if K(L) is finite. Proof. Here we just sketch the proof; for details cf. [229, Ch. 2]. Let D be an effective divisor such that L = OX (D) and define GD = {x ∈ X | τx∗ (D) = D}. If K(L) is finite, GD ⊂ K(L) is finite as well. From this one proves that the linear system |2D| has no base points and defines a finite morphism into a projective space. By general theory this implies that L is ample [134, 2.6.1 or 4.4.2]. On the other hand, if L is ample, one proves that K(L) is finite. Indeed, if this is not true, let Y be the connected component of K(L) containing 0. Then Y is a nontrivial Abelian variety. One proves quite easily that L0 = L|Y ⊗(−1)∗Y (L|Y ) is ample. On the other hand since Y ⊂ K(L), pulling back Q|Y ×Y to Y by the morphism y 7→ (y, −y), one shows that the line bundle L0 is trivial, which is a contradiction.  By using this we may show that any Abelian variety X is projective, as for instance is proved in [251, Thm. 8.12]. One can see that there exists an effective divisor D ,→ X whose complement U in X is affine and contains the origin. Let Y be the connected component of the origin in K(O(D)). Then the restriction of O(D) to Y is trivial, so that D is not contained in Y , and it does not intersect it. This implies that Y is contained in the affine variety U , which in turn implies that Y , being proper, reduces to a point, so that K(O(D)) is finite. By the previous lemma, X is projective. ˆ and the Poincar´e bundle P on X × X ˆ may be The dual Abelian variety X introduced as the solution to the problem of representing the Picard functor. This is the functor which to any variety T associates the group of equivalence classes of line bundles on X × T , where two line bundles are identified when they are isomorphic up to tensoring by the pullback of a line bundle on T . It is a general fact that when X is projective, this functor is representable [3] by an algebraic group Pic(X). The connected component of the latter containing the origin is denoted by Pic0 (X) and represents line bundles that are topologically equivalent to the trivial line bundle (i.e., line bundles whose first Chern class vanishes). In

84

Chapter 3. Fourier-Mukai on Abelian varieties

the case at hand, where X is an Abelian variety, Pic0 (X) is an Abelian variety as ˆ and called the dual Abelian variety to X. The fact well; it is usually denoted X ˆ represents the Picard functor means that there is a universal line bundle P that X ˆ called the Poincar´e bundle. Universality means that given a variety T on X × X, and a line bundle L on X × T , whose restrictions to the fibers of pT : X × T → T ˆ such have vanishing first Chern class, there exists a unique morphism φ : T → X ∗ ∗ that L ' (IdX × f ) P ⊗ pT N , where N is a line bundle on T . Thus, if a point ˆ corresponds to a line bundle L on X, one has ξ∈X Pξ = P|X×{ξ} ' L . Analogously, we shall denote Px = P|{x}×Xˆ if x ∈ X. The Poincar´e bundle can be normalized by the condition that P0 = P|{0}×Xˆ ˆ If we consider P as a relative line bundle with is the trivial line bundle on X. ˆ ˆ → X, we obtain a morphism X → Pic0 (X) respect to the projection X × X ˆ ˆ which is actually an isomorphim canX : X → X. Let us consider the line bundle m∗X L ⊗ π1∗ L−1 on X × X; by the universal ˆ property of the Poincar´e bundle, there is a unique scheme morphism φL : X → X such that (1 × φL )∗ P ' m∗X L ⊗ π1∗ L−1 ⊗ π2∗ N for certain line bundle N on X, which is actually isomorphic to L−1 as a consequence of the normalization of the Poincar´e bundle. Thus, (1 × φL )∗ P ' m∗X L ⊗ π1∗ L−1 ⊗ π2∗ L−1 .

(3.1)

We then have φL (x) = τx∗ L ⊗ L−1 . One can also check that φL is actually a group morphism (this is again a consequence of the theorem of the square [229, §6, Cor. 4]), i.e., it is a homomorphism of Abelian varietes. The kernel ker φL is easily shown to coincide with the scheme K(L). The latter is a subgroup of X, hence it acts freely on X, and φL factors through the quotient X → X/K(L).

3.2

The transform

ˆ be Let X be an Abelian variety over k, whose dimension we denote by g, and let X its dual Abelian variety. In the rest of the chapter we assume that k has characteristic zero. Mukai’s seminal idea [224] was to use the normalized Poincar´e bundle P to define an integral functor between the derived categories of coherent sheaves ˆ which turns out to be an equivalence of triangulated categories. on X and X,

3.2. The transform

85

b b ˆ Theorem 3.2. The integral functor ΦP ˆ : D (X) → D (X) is a Fourier-Mukai X→X P ∗ [g] P transform. The functor ΦX→ ˆ. ˆ X is quasi-inverse to ΦX→X

ˆ (cf. Definition 1.30) and Proof. The Poincar´e bundle is strongly simple over X the canonical bundle of an Abelian variety is trivial, so that by Corollary 2.58, which can be applied because k has characteristic zero, ΦP ˆ is a Fourier-Mukai X→X functor.  It is therefore natural to give the following definitions: Definition 3.3. The Abelian Fourier-Mukai transform is the functor b b ˆ S = ΦP ˆ : D (X) → D (X) . X→X

The dual Abelian Fourier-Mukai transform is the functor b = ΦPˆ ∗ : Db (X) ˆ → Db (X) . S X→X 4 b the functor S e : Db (X) e= ˆ → Db (X) given by S Mukai considers instead of S e . The relation between S and S is given by the isomorphism of functors ΦX→ ˆ X P

e ◦ S ' ι∗ˆ ◦ [−g] . S X This is easily proved as a consequence of the isomorphism (id × ιXˆ )∗ P ' P ∗ . Recall (Definition 1.6) that a coherent sheaf F on X is WITi if its Abelian Fourier-Mukai transform reduces to a single coherent sheaf Fb located in degree i, b that is S(F) ' F[−i], and that it is ITi if in addition Fb is locally free. By base change theory, a coherent sheaf F is ITi if and only if H j (X, F ⊗ Pξ ) = 0 for j 6= i ˆ and every point ξ ∈ X. The notions of WIT and IT sheaf also apply to the dual Fourier-Mukai transb A consequence of the invertibility of the Fourier-Mukai transform (see form S. Proposition 2.34) is b Corollary 3.4. If F is WITi , then Fb is WITg−i . Moreover Fb ' F. Moreover, the spectral sequence (2.35) assumes the form ( E if p + q = g p,q p q ˆ E2 = S (S (E)) =⇒ 0 otherwise . Corollary 3.5. For every sheaf E on X, the sheaf S0 (E) is WITg , while the sheaf Sg (E) is WIT0 (and hence IT0 by Proposition 1.7).

86

Chapter 3. Fourier-Mukai on Abelian varieties

Example 3.6. Since Ox is a skyscraper sheaf, one has H j (X, Ox ⊗ Pξ ) = 0 for cx ' Px . By Corollary 3.4, Px ˆ Then Ox is IT0 and O j 6= 0 and every point ξ ∈ X. b is WITg for every point x ∈ X and Px ' Ox ; this is an example of a sheaf which is WIT but not IT. cξ ' P ∗ ' P−ξ ˆ one has that Oξ is IT0 with O Reversing the roles of X and X, ξ ˆ Then Pξ is WITg with P b∗ ' where −ξ is the opposite of ξ for the group law in X. ξ

O−ξ , in other words S(Pξ ) ' O−ξ [−g]. In particular, S(OX ) = S(Pˆ0∗ ) ' Oˆ0 [−g] ˆ where ˆ0 denotes the origin of the Abelian variety X. 4 Theorem 3.2, whose proof is very simple thanks to the machinery developed in Chapter 1, encodes some nontrivial information about the cohomology of the Poincar´e bundle. Indeed, the following result [229, §13, Cor. 1] can be easily deduced from it. Proposition 3.7. One has ( ˆ P) = H (X × X, i

0

if i 6= g

k

if i = g .

Proof. We have seen that S(OX ) ' Oˆ0 [−g], or, in other terms, ( 0 if i 6= g i R πX∗ ˆ P = Oˆ0 if i = g . ˆ P) ' H 0 (X, ˆ Ri π ˆ P), By using the Leray spectral sequence we obtain H i (X × X, X∗ whence the claim follows.  As a matter of fact, Theorem 3.2 and Proposition 3.7 are essentially equivalent, since Mukai uses the latter to prove the first. Example 3.8 (Unipotent bundles). A locally free sheaf U on X is said to be unipotent if there is a filtration 0 = U0 ⊂ U1 ⊂ · · · ⊂ Un−1 ⊂ Un = U such that the quotients Ui /Ui−1 are isomorphic to OX for every i ≥ 1. Applying the Abelian Fourier-Mukai transform to the sequences 0 → Ui−1 → Ui → OX → 0 (cf. Eq. (1.10)) we can prove by induction on i that Ui is WITg for every i and that its Fourier-Mukai transform Ubi is a skyscraper sheaf supported at the origin ˆ ˆ0 ∈ X.

3.2. The transform

87

ˆ supported at the Conversely, if C is a skyscraper sheaf of length n on X origin, we have a filtration 0 = C0 ⊂ · · · ⊂ Cn−1 ⊂ Cn = C ∼ O for every i. The exact sequence of Fourier-Mukai transforms with Ci /Ci−1 → ˆ 0 b is a unipotent bundle of rank n. Thus the gives that C is IT0 and that U = C Fourier-Mukai transform establishes a one-to-one correspondence between unipotent bundles of rank n a skyscraper sheaves of length n supported at the origin ˆ0. As a consequence of Proposition 2.63, the moduli scheme of unipotent bundles of a given degree is isomorphic to the moduli scheme of skyscraper sheaves of length n supported at the origin ˆ 0. 4 Mukai derived from the Parseval formula (Proposition 1.34) many interesting consequences [224]. We list here some of them: Proposition 3.9. If F is a WITi sheaf on X, then b (Oξ , F) H j (X, F ⊗ Pξ ) ' Extj+g−i ˆ X and ˆ Fb ⊗ P−x ) ExtjX (Ox , F) ' H j−i (X, ˆ and j ≥ 0. for every x ∈ X, ξ ∈ X Proof. One has H j (X, F ⊗ Pξ ) ' ExtjX (Pξ∗ , F) ' ExtjX (P−ξ , F) as Pξ is locally j+g−i b free. By Example 3.6 and the Parseval formula, ExtjX (P−ξ , F) ' ExtX (Oξ , F). The second formula is similar.  Corollary 3.10. If F is an ITi sheaf on X, then the Euler characteristic of F and the rank of Fb are related by b . χ(X, F) = (−1)i rk(F) P P j+g−i j j j b Proof. One has χ(X, F) = (Oˆ0 , F). j (−1) h (X, F) = j (−1) dim ExtX ˆ b ' H i−j (X, Fb∗ ⊗ Oˆ ). (Oˆ0 , F) Since Fb is locally free, Serre duality gives Extj+g−i 0 ˆ X This vanishes unless j = i, and in that case we have dim H 0 (X, Fb∗ ⊗ O0ˆ ) = b rk(F).  The topological invariants of the Abelian Fourier-Mukai transform can be computed by means of the Grothendieck-Riemann-Roch theorem in terms of the first Chern class of the Poincar´e bundle. In this case the map f : A• (X) ⊗ Q → ˆ ⊗ Q introduced in Section 1.2 (which according to Corollary 2.40 is an A• (X) isomorphism) takes the form ∗ f (α) = πX∗ ˆ (ch(P) · πX (α)) .

88

Chapter 3. Fourier-Mukai on Abelian varieties

More explicitly, the Chern character of the Fourier-Mukai transform of a complex F • in Db (X) is given by chk (S(F • )) =

  1 πX∗ c1 (P)k · chg−k (F • ) = f (chg−k (F • )) . ˆ k!

(3.2)

unneth type (1, 1); in particular, by embedding End(A1 (X)) Note that c1 (P) has K¨ 2 ˆ it corresponds to the identity of A1 (X). Since f as a submodule of A (X × X) g−k ˆ ⊗ Q, one can say that the Fourier-Mukai (X) ⊗ Q with Ak (X) identifies A transform “flips” the Chern character of the complex it acts on. The Fourier-Mukai transform exchanges tensor products by line bundles of degree zero with translations by the group law. This parallels the property of the Fourier transforms which interchanges products with sums. To prove this, let us denote by τx : X → X the translation τx (y) = x+y. Similarly we have translations ˆ →X ˆ and τ(x,ξ) : X × X ˆ → X × X. ˆ τξ : X The following property of the Poincar´e bundle is well known: Lemma 3.11. One has ∗ ∗ P ' P ⊗ πX τ(x, ˆ ˆ Px 0) ∗ ∗ τ(0,ξ) P ' P ⊗ πX Pξ

ˆ Moreover, for all x ∈ X, ξ ∈ X. ∗ ∗ P ⊗ π23 P, (mX × 1)∗ P ' π13

ˆ onto the ij-th components. where πij are the projections of X × X × X ∗ ˆ the universal property of P Proof. Since (τ(x, P)ξ ' Pξ for every point ξ in X, ˆ 0) ˆ The latter is shown implies that τ ∗ P ' P ⊗ π ∗ N for some line bundle N on X. (x,ˆ 0)

ˆ X

to coincide with Px by the normalization property P0 ' OXˆ . This proves the first formula, the second is similar. For the third we use those formulas together with the cube theorem [229] (this states that if X, Y are proper integral varieties, Z any variety, (x0 , y0 , z0 ) ∈ X × Y × Z, and L is a line bundle on X × Y × Z such that all restrictions of L on {x0 } × Y × Z, X × {y0 } × Z and X × Y × {z0 } are trivial, then L is trivial).  Proposition 3.12. [224] There are isomorphisms S ◦ τx∗ ' (⊗P−x ) ◦ S ,

S ◦ (⊗Pξ ) ' τξ∗ ◦ S

ˆ Similarly, there are isomorphisms of functors Db (X) → Db (X). b, b ◦ τ ∗ ' (⊗Pξ ) ◦ S S ξ ˆ → Db (X). of functors Db (X)

b ◦ (⊗P−x ) ' τ ∗ ◦ S b S x

3.2. The transform

89

∼ τ ∗ π ∗ M• so that ∗ ∗ τx M• → Proof. Let M• be an object of Db (X). Then πX (x,ˆ 0) X ∗ ∗ ∗ π ∗ M• ⊗ P) ' RπX,∗ (π ∗ M• ⊗ τ(−x, P) S(τx∗ M• ) = RπX,∗ ˆ (τ(x,ˆ ˆ τ(x,ˆ ˆ 0) X 0) X 0) ∗ ∗ ∗ • ∗ (π ∗ M• ⊗ P ⊗ πX ' RπX,∗ ˆ τ(x,ˆ ˆ (πX M ⊗ P ⊗ πX ˆ P−x ) ' RπX,∗ ˆ P−x ) 0) X ∗ • • ' RπX,∗ ˆ (πX M ⊗ P) ⊗ P−x = S(M ) ⊗ P−x

where the third equality follows from Lemma 3.11, the fourth from πXˆ ◦τ(x,ˆ0) = πXˆ and the projection formula and the fifth again by the projection formula. Reversing b = (⊗Pξ ) ◦ S. b Then b ◦ τ ∗ ' (⊗P ∗ ) ◦ S ˆ we now find S the roles of X and X ξ −ξ ∗ ∗ b ◦ τ ◦ S[g] by Theorem 3.2, and therefore S ◦ (⊗Pξ ) ' S ◦ S b ◦ τ ◦ S[g] ' (⊗Pξ ) ' S ξ ξ  [−g] ◦ τξ∗ ◦ S[g] ' τξ∗ ◦ S. The same reasoning yields the last formula. The Pontrjagin product of two coherent sheaves E and F on X is the sheaf E ? F = mX∗ (π1∗ E ⊗ π2∗ F). This product has a derived functor R

?

Db (X) × Db (X) −→ Db (X) L

R

(E • , F • ) 7→ E • ? F • = RmX∗ (π1∗ E • ⊗ π2∗ F • ) Proposition 3.13. [224] The Abelian Fourier-Mukai transform intertwines the Pontrjagin and the tensor product, that is: R

L

L

S(E • ? F • ) ' S(E • ) ⊗ S(F • )

R

S(E • ⊗ F • ) ' S(E • ) ? S(F • )[g] .

Proof. We have only to prove the first isomorphism. We use the isomorphism ∗ ∗ ˆ onto (mX × 1)∗ P ' π13 P ⊗ π23 P where πij are the projections of X × X × X the ij-th components (Lemma 3.11). The formula is obtained by successive base changes and using the identities πXˆ ◦ (mX × 1) = πXˆ ◦ π13 , π1 ◦ π12 = πX ◦ π13 and π2 ◦ π12 = πX ◦ π23 . L

R

∗ ∗ • ∗ • S(E • ? F • ) ' RπX∗ ˆ (πX (RmX∗ (π1 E ⊗ π2 F ) ⊗ P)) L

∗ ∗ • ∗ • ' RπX∗ ˆ (R(mX × 1)∗ (π12 (π1 E ⊗ π2 F )) ⊗ P) L

∗ ∗ • ∗ ∗ • ∗ ' RπX∗ ˆ R(mX × 1)∗ (π13 πX E ⊗ π23 πX F ⊗ (mX × 1) P) L

∗ ∗ • ∗ ∗ • ∗ ∗ ' RπX∗ ˆ Rπ13∗ (π13 πX E ⊗ π23 πX F ⊗ π13 P ⊗ π23 P) L

∗ • ∗ ∗ • ' RπX∗ ˆ (πX E ⊗ P ⊗ Rπ13∗ π23 (πX F ⊗ P)) L

L

∗ • ∗ ∗ • • • ' RπX∗ ˆ (πX E ⊗ P ⊗ πX ˆ (πX F ⊗ P)) ' S(E ) ⊗ S(F ) . ˆ RπX∗



90

3.3

Chapter 3. Fourier-Mukai on Abelian varieties

Homogeneous bundles

A sheaf M on an Abelian variety is called homogeneous if it is invariant under ∼ M for every point x ∈ X. We can give the same deftranslations, that is, τx∗ M → b inition for objects in D (X), saying that an object M• in Db (X) is homogeneous when τx∗ M• ' M• in Db (X) for every x. Since translations are isomorphisms they commute with homology, so that if M• is homogeneous, then all the cohomology sheaves Hi (M• ) are homogeneous. Conversely, if the cohomology sheaves Hi (M• ) are homogeneous, induction on the number of nonzero cohomology sheaves together with the exact triangle M• ≤n−1 → M• ≤n → Hn (M• )[−n] → M• ≤n−1 [1] (where Hn (M• ) is the last cohomology sheaf) shows that the complex M• is homogeneous. Proposition 3.12 allows one to characterize homogeneous sheaves and more generically homogeneous objects of Db (X). We need a preliminary result, which we prove using the notion of determinant bundle already considered in Chapter 1. ˆ then Lemma 3.14. Let M• an object of Db (X). If M• ⊗ Pξ ' M• for every ξ ∈ X • i • rk(M ) = 0 and all the cohomology sheaves H (M ) are skyscraper sheaves. In ˆ particular a sheaf M on X verifies the condition M ⊗ Pξ ' M for every ξ ∈ X if and only it is a skyscraper sheaf. ˆ if M is a skyscraper sheaf. For the Proof. Clearly M ⊗ Pξ ' M for every ξ ∈ X remaining statements, taking determinants in M• ⊗ Pξ ' M• we have det(M• ) ' rk(M• ) rk(M• ) ˆ (see Eq. (2.36)) so that Pξ is trivial for every ξ ∈ X. det(M• ) ⊗ Pξ • rk(M ) ˆ →X ˆ given Since P ' Prk(M• )ξ we have that the image of the morphism X ξ

by ξ 7→ rk(M• )ξ, reduces to the origin. This is absurd unless rk(M• ) = 0. When M• reduces to a sheaf M, then M has to be a skyscraper. Since M• ⊗ Pξ ' M• implies Hi (M• ) ⊗ Pξ ' Hi (M• ), we then have that the sheaves Hi (M• ) are skyscrapers as well.  Proposition 3.15. The Abelian Fourier-Mukai transform of a skyscraper sheaf M ˆ is a homogeneous locally free sheaf on X. Conversely, a homogeneous sheaf F on X on X is WITg and locally free and its Fourier-Mukai transform Fb is a skyscraper. c is a locally free sheaf on X. ˆ is IT0 so that M Proof. A skyscraper sheaf M on X ∗ c Since M⊗P−x ' M by Lemma 3.14, then τx M ' M by Proposition 3.12, and this c is homogeneous and WITg . For the converse, if F is homogeneous, proves that M then S(F) ⊗ Px ' S(F) again by the same proposition, and therefore all the cohomology sheaves Si (F) = Hi (S(F)) are skyscrapers by Lemma 3.14. Then

3.4. Fourier-Mukai transform and the geometry of Abelian varieties

91

b p (Sq (F)) (which b p (Sq (F)) = 0 for p > 0 and the spectral sequence E p,q = S S 2 p+q p+q converges to E∞ = F for p + q = g and E∞ = 0 otherwise) degenerates,  proving that Sq (F) = 0 for q 6= g. Corollary 3.16. The Abelian Fourier-Mukai transform induces an equivalence between the category of homogeneous sheaves on X and the category of skyscraper ˆ sheaves on X. As an application of the above results, Mukai ([224]) gave the following characterization of the homogeneous sheaves. Corollary 3.17. A sheaf F on X is homogeneous if and only if it is isomorphic to Ln i=1 Ui ⊗ Pi , where the Ui ’s are unipotent bundles and the Pi ’s line bundles of degree zero. Proof. If F is homogeneous, then it is WITg and Fb is a skyscraper, Fb = M01 ⊕· · ·⊕ ˆ We can write M0 = τ ∗ Mi M0i where each M0i is supported at a point ξi ∈ X. i ξi where Mi is supported at the origin, so that the invertibility of the FourierL b L b ∗ Mukai transform gives F = i S(τ ξ i Mi ) ' i S(Mi ) ⊗ Pξi by Proposition 3.12. b Moreover Ui = S(Mi ) is a unipotent bundle by Example 3.8. The converse is similar. 

3.4

Fourier-Mukai transform and the geometry of Abelian varieties

We can give an alternative approach to some general properties of Abelian varieties by means of the Abelian Fourier-Mukai transform.

3.4.1

Line bundles and homomorphisms of Abelian varieties

ˆ Let us recall that a line bundle L on X defines a flat homomorphism φL : X → X, whose kernel is K(L). Proposition 3.18. One has ˆ 1. φL is constant (and then equal to the zero map) if and only if L ∈ X. 2. If L and N are line bundles on X, then φL⊗N = φL + φN . 3. φL = φN if and only if L and N are algebraically equivalent, that is, L ⊗ ˆ or c1 (L) = c1 (N ). N −1 ∈ X

92

Chapter 3. Fourier-Mukai on Abelian varieties

Proof. 1. φL is constant if and only if L is a homogeneous line bundle. By Corollary ˆ that is, L ∈ X. ˆ 3.17, this amounts to saying that L ' Pξ for some ξ ∈ X, 2. One has (1 × φL⊗N )∗ P ' m∗X (L ⊗ N ) ⊗ π1∗ L−1 ⊗ π1∗ N −1 ⊗ π2∗ L−1 ⊗ π2∗ N −1 ∗ ∗ ' (1 × φL )∗ P ⊗ (1 × φN )∗ P ' (1 × (φL , φN ))∗ (π13 P ⊗ π23 P)

ˆ ×X ˆ is the morphism given by (x, x0 ) 7→ where 1 × (φL , φN ) : X × X → X × X 0 ∗ ∗ P ⊗π23 P obtained (x, φL (x), φN (x )). Now we apply the formula (1×mXˆ )∗ P ' π13 ˆ Thus by applying Lemma 3.11 to X. (1 × φL⊗N )∗ P ' (1 × (φL , φN ))∗ ((1 × mXˆ )∗ P) ' (1 × (φL + φN ))∗ P . By the universal property of P we get φL⊗N = φL + φN . 3. It follows directly from 1. and 2.



We also recall that due to the fact that the tangent bundle to an Abelian variety is trivial, the Grothendieck-Riemann-Roch formula for a line bundle takes the well-known form 1 χ(X, L) = c1 (L)g . g! It follows that χ(X, L−1 ) = (−1)g χ(X, L). A line bundle L on X is said to be nondegenerate when the morphism φL is finite. Note that in this case φL is a separable isogeny, and in particular is ´etale, cf. [229]. Proposition 3.19. Let L be a line bundle on X. R

1. If L is nondegenerate, then j∗ (L|K(L) )[−g] ' L−1 ? L, where j : K(L) ,→ X is the immersion. 2. If L is nondegenerate, there exists an integer i = i(L) such that H ` (X, L) = 0 for ` 6= i. Moreover i(L−1 ) = g − i(L). 3. deg φL = χ(X, L)2 . Moreover, χ(X, L) = 0 when L is degenerate. 4. If L is nondegenerate, then it is ITi with i = i(L) and Lb is locally free of rank (−1)i χ(X, L). Moreover φ∗L Lb ' H i (X, L) ⊗k L−1 . 5. If L is ample, then it is effective. Proof. 1. Since L is nondegenerate the morphism φL is flat, and one has j∗ OK(L) ' φ∗L Oˆ0 ' Lφ∗L Oˆ0 . Example 3.6, together with base change and Equation (3.1), now

3.4. Fourier-Mukai transform and the geometry of Abelian varieties

93

gives j∗ OK(L) [−g] ' φ∗L S(OX ) ' Rπ2∗ ((1 × φL )∗ P) ' Rπ2∗ (m∗X L ⊗ π1∗ L−1 ) ⊗ L−1 ' Rπ2∗ ((π1 , mX )∗ (m∗X L ⊗ π1∗ L−1 )) ⊗ L−1 ' RmX,∗ (π1∗ L−1 ⊗ π2∗ L) ⊗ L−1 whence the statement follows. 2. By the first part, and by the definition of the Pontrjagin product, we have that Ri m∗ (π1∗ L−1 ⊗π2∗ L) = 0 for i 6= g and that Rg m∗ (π1∗ L−1 ⊗π2∗ L) ' j∗ (L|K(L) ). Taking cohomology, we have H p (K(L), L|K(L) ) ' H p+g (X × X, π1∗ L−1 ⊗ π2∗ L) M H j (X, L−1 ) ⊗k H i (X, L) . '

(3.3)

j+i=p+g

If L is nondegenerate, the first member vanishes for p > 0 because K(L) is finite and we thus deduce the result. 3. If L is nondegenerate, so that K(L) is finite, the previous formula gives deg φL = length(K(L)) = hi (X, L)hg−i (X, L−1 ) = (−1)i χ(X, L)(−1)g−i χ(X, L−1 ) = χ(X, L)2 , where i is the index of L. If L is degenerate, deg φL = 0 because φL is not finite and we have to prove that χ(X, L) = 0. Since K(L) is not finite, we can find for any integer n an algebraic subgroup G ,→ K(L) of order n. Then 1 × φL factors through X × X → ˆ so that m∗ L ⊗ π ∗ L−1 = (1 × φL )∗ (P ⊗ π ∗ L) is the inverse X × X/G → X × X 2 1 X image of a line bundle on X ×X/G. Thus χ(X ×X, m∗X L⊗π2∗ L−1 ) is divisible by n and since n is arbitrary large, this implies that χ(X ×X, m∗X L⊗π2∗ L−1 ) = 0. Now, (µX , π2 ) : X × X → X × X is an isomorphism, and then χ(X × X, π1∗ L ⊗ π2∗ L−1 ) = χ(X × X, (µX , π2 )∗ (m∗X L ⊗ π2∗ L−1 )) = 0. Moreover χ(X × X, π1∗ L ⊗ π2∗ L−1 ) = χ(X, L)χ(X, L−1 ) = (−1)g χ(X, L)2 , and then χ(X, L) = 0. 4. Proceeding as in the first part we have φ∗L S(L) ' Rπ2∗ (π1∗ L ⊗ (1 × φL )∗ P) ' Rπ2∗ (m∗X L ⊗ π2∗ L−1 ) ' Rπ2,∗ ((mX , π2 )∗ (π1∗ L ⊗ π2∗ L−1 )) ' Rπ2,∗ (π1∗ L ⊗ π2∗ L−1 ) ' RΓ(X, L) ⊗k L−1 ' H i (X, L) ⊗k L−1 [−i]

94

Chapter 3. Fourier-Mukai on Abelian varieties

where the last equality is due to the second part. Then φ∗L (Hj (S(L))) = 0 for j 6= i and φ∗L (Hi (S(L))) is locally free. Since φL is flat and surjective, it is is faithfully flat, so that Hj (S(L)) = 0 for j 6= i and Hi (S(L)) is locally free; thus L is ITi . Moreover, φ∗L Lb ' H i (X, L) ⊗k L−1 . 5. Let n be an integer such that Ln is very ample. By Lemma 3.1, Ln is nondegenerate. Since φLn = [n] ◦ φL , the line bundle L is nondegenerate as well. By the corollary at page 159 of [229] one has i(L) = i(Ln ) = 0, so that h0 (L) = χ(L) =

1 c1 (L)g > 0 . g! 

3.4.2

Polarizations

A class of algebraic equivalence of ample line bundles H = [L] (or of ample divisors [D]) is called a polarization. By Proposition 3.18, two line bundles are algebraically ˆ Then the equivalent if and only if they define the same morphism φL : X → X. morphism is actually associated to the class, and we denote it by φ[L] . A polarization [L] is said to be principal if φ[L] is an isomorphism of Abelian ∼ X. ˆ We have seen that this is equivalent to either deg φL = 1 or varieties, φ[L] : X → to χ(X, L) = 1. An important example of a principally polarized Abelian variety is the Jacobian J(C) of a smooth projective curve C, the scheme which parameterizes the lines bundle on C having degree zero. In this case the principal polarization is given by the equivalence class of the so-called Θ divisor on J(C). Given a polarization H = [L] on X, we may use the Fourier-Mukai transform ˆ with a polarization. to endow the dual variety X ˆ −1 on Corollary 3.20. If L is an ample line bundle on X, the line bundle det(L) ˆ X is ample as well. Proof. We have φ∗L Lˆ∗ ' H 0 (X, L) ⊗k L. Now, the locally free sheaf H 0 (X, L) ⊗k L is ample (for the definition and main properties of ample locally free sheaves see [138]). Since a locally free sheaf is ample if and only if its pullback under a finite surjective morphism is ample, the locally free sheaf Lˆ∗ is ample. Then the line ˆ −1 is ample as well.  bundle det(L) ˆ = [det(L) ˆ −1 ] is a polarization for X. ˆ If one identifies X with X ˆˆ by So H ˆˆ on X. canX , we may iterate the construction, obtaining a polarization H ˆˆ Corollary 3.21. H = (−1)∗X (H).

3.4. Fourier-Mukai transform and the geometry of Abelian varieties

95

ˆ Denoting Φ ˆ = ΦPˆ ∗ , by Equation (3.2) we have Proof. Let M = det−1 (L). X→X ˆˆ ˆ = (−1)∗X (c1 (L)) = (−1)∗X (H) . H = c1 (Φ(M))  When we are given a principally polarized Abelian variety (X, [L]) we shall ˆ ˆ by the isomorphism φ[L] . Then X × X is identified with X × X identify X with X and under this identification we have P ' m∗X L ⊗ π1∗ L−1 ⊗ π2∗ L−1 .

(3.4)

Thus the Abelian Fourier-Mukai transform S and the dual Abelian Fourier-Mukai b are autoequivalences of D(X). transform S

3.4.3

Picard sheaves

In this section, C is a smooth projective curve of genus g ≥ 1. For any integer d we denote by Jd the Picard scheme parameterizing degree d line bundles on C and by p : C × Jd → C, q : C × Jd → Jd the projections. The universal Poincar´e line bundle on C × Jd is denoted by Pd . It is determined up to twisting by pullbacks of line bundles on Jd ; however if we fix a point x0 ∈ C, we can normalize Pd by imposing that Pd|{x0 }×Jd ' OJd . The Jacobian of C is the Abelian variety J(C) = J0 . In this case, the normalized Poincar´e bundle P0 is just the restriction of the normalized Poincar´e bundle on J(C)×J(C) to C ×J(C), where we identify J(C) with its dual via the principal polarization given by the Θ divisor, and C is embedded into J(C) via the Abel map. All the Picard schemes are isomorphic in a noncanonical way. The choice of a point x0 ∈ C gives isomorphisms ∼ J(C) λ d : Jd → (3.5) [L] 7→ [L ⊗ OC (−dx0 )] . These isomorphisms are induced by the line bundle Pd ⊗ p∗ OC (−dx0 ). The normalization of the Poincar´e sheaves give Pd ⊗ p∗ OC (−dx0 ) ' (1 × λd )∗ P0 .

(3.6)

Pd b b We consider the integral functors Φd = ΦC→ Jd : D (C) → D (Jd ).

Definition 3.22. The Picard sheaves are the cohomology sheaves Ed,n = Φ0d (OC (nx0 )) , of the transformed complex Φd (OC (nx0 )).

Fd,n = Φ1d (OC (nx0 )) , 4

96

Chapter 3. Fourier-Mukai on Abelian varieties

Picard sheaves were introduced by Schwarzenberger [264] and have been the subject of study by many authors in connection with the geometry of Abelian varieties (cf. [224, 180, 181]). By base change in the derived category we have isomorphisms (1×λ )∗ P0 ⊗p∗ OC (dx0 )

Φd (OC (nx0 )) ' ΦC→Jd d

(OC (nx0 )) ' λ∗d Φ0 (OC ((n + d)x0 )) (3.7)

and then Ed,n ' λ∗d (E0,d+n ) ,

Fd,n ' λ∗d (F0,d+n ) .

(3.8)

The sheaf p∗ ωC ⊗ Pd∗ induces an isomorphism ∼J θd : Jd → 2g−2−d

(3.9)

[L] 7→ [p∗ ωC ⊗ L−1 ] which gives p∗ ωC ⊗ Pd∗ ' (1 × θd )∗ P2g−2−d

(3.10)

due to the normalization. Again by base change in the derived category we have isomorphisms P ∗ ⊗p∗ ωC

d θd∗ (Φ2g−2−d (OC (nx0 ))) ' ΦC→ Jd

P∗

d (OC (nx0 )) ' ΦC→ Jd (ωC ⊗ OC (nx0 )) . (3.11)

Proposition 3.23. There is an isomorphism θd∗ Φ2g−2−d (OC (nx0 ))∨ ' λ∗d Φ0 (OC ((d − n)x0 ))[1] where as usual

∨

denotes the dual in the derived category.

Proof. By Proposition 1.15 P∗

−1 ∨ d ' Φd (SC (ωC ⊗ OC (−nx0 ))) ' Φd (OC (−nx0 )[1]) . ΦC→ Jd (ωC ⊗ OC (nx0 ))

One then applies (3.7).



For any d > 0, the Abel morphism Symd C → Jd in degree d (where Symd C is the symmetric product of d copies of C) may be identified with the projective morphism associated with a coherent OJd -module Md (cf. [3]). Moreover Md is univocally characterized by the property HomT (f ∗ Md , N ) ' HomC×T ((1 × f )∗ Pd∗ , qT∗ N ) for every morphism f : T → Jd and every quasi-coherent sheaf N on T , where qT : C × T → T is the projection. The following result is easily proved [264].

3.5. Some applications of the Abelian Fourier-Mukai transform

97

Proposition 3.24. There is an isomorphism ˜ 1 (ωC ) ' θ∗ F0,2g−2−d . Md ' Φ d d ∗

d ˜ d = ΦP where Φ C→Jd .

Proof. Relative duality and base change give ˜ d (ωC ))[1], N ) ' HomDb (J ) (Φ ˜ d (ωC )[1], f∗ (N )) HomDb (T ) (f ∗ (Φ d ' HomDb (C×Jd ) (p∗ ωC ⊗ Pd∗ [1], q ∗ f∗ (N ) ⊗ p∗ ωC [1]) ' HomDb (C×Jd ) (Pd∗ , q ∗ f∗ (N )) ' HomDb (C×Jd ) (Pd∗ , (1 × f )∗ qT∗ (N )) ' HomDb (C×T ) ((1 × f )∗ Pd∗ , qT∗ (N )) . There is a spectral sequence whose second term is ˜ d (ωC ))[1]), N ) E2p,q = HompC×T (H−q (f ∗ (Φ p,q ˜ d (ωC ))[1], N ). Since Φ ˜ 1 (ωC ) is the highest = Homp+q (f ∗ (Φ converging to E∞ d D b (T ) ˜ d (ωC ) one has E p,q = 0 for q < 0, so that cohomology sheaf of Φ 2

HomC×T ((1 × f )

∗

Pd∗ , qT∗ (N ))

˜ 1 (ωC )), N ) , ' HomT (fT∗ (Φ d

˜ 1 (ωC ). The second isomorphism follows from (3.11).  which proves that Md ' Φ d

3.5

Some applications of the Abelian Fourier-Mukai transform

We discuss in this section two applications of the Abelian Fourier-transform: a proof of Atiyah’s characterization of semistable bundles on elliptic curves, and the preservation of stability in the case of Abelian surfaces.

3.5.1

Moduli of semistable sheaves on elliptic curves

The structure of µ-stable and µ-semistable vector bundles on an elliptic curve was determined by Atiyah [15] in a paper which now is considered to be a classic. His results were used by Tu [285] to study the moduli spaces of µ-semistable sheaves on an elliptic curve. It turns out that all those results can be obtained in a very simple way as an application of the Abelian Fourier-Mukai transform on an elliptic curve. This approach was introduced in [251] and [142] but our treatment is somehow different.

98

Chapter 3. Fourier-Mukai on Abelian varieties

Let X be an elliptic curve, understood as an Abelian variety of dimension 1. In other words, X is a smooth curve of genus 1 with a fixed point x0 which we take as the zero of the group law on X. The line bundle L = OX (x0 ) induces a ˆ by the isomorphism principal polarization on X so that we can identify X with X φ[L] . Under this identification the Poincar´e bundle takes the form P ' OX×X (∆ι ) ⊗ π1∗ OX (−x0 ) ⊗ π2∗ OX (−x0 ) ,

(3.12)

where ∆ι is the graph of the isomorphism ι : X → X which maps a point x to the opposite point ι(x) = −x (cf. (3.4)). We can then consider both the Abelian b as Fourier-Mukai transform S and the dual Abelian Fourier-Mukai transform S autoequivalences of D(X). Since X has dimension 1, an object E • of Db (X) has only two topological invariants, namely, the rank n = ch0 (E • ) and the degree d = ch1 (E • ) (which is naturally identified with an integer number), so that the Chern character of E • can be written as ch(E • ) = (n, d). Applying Equation (3.2) or computing directly by Grothendieck-RiemannRoch from Equation (3.12), we have: Proposition 3.25. If the Chern character of E • is (n, d), then the Chern character of the Abelian Fourier-Mukai transform S(E • ) is (d, −n). When the object E • reduces to a single sheaf E, one has: b = −1/µ(E). Corollary 3.26. If E is WITi for some i = 0, 1 and d 6= 0, then µ(E) Corollary 3.27. If E is WIT0 , then d(E) ≥ 0, and d(E) = 0 if and only if E = 0. If E is WIT1 , then d(E) ≤ 0. The key point for the study of µ-semistable sheaves on an elliptic curve is the following result [251, Lemma 14.5], whose proof is based on the HarderNarasimhan filtration. Proposition 3.28. Any indecomposable torsion-free sheaf on an elliptic curve X is semistable. Proof. Let E be a torsion-free sheaf on X and let 0 ⊂ E1 ⊂ · · · ⊂ En = E be its Harder-Narasimhan filtration (cf. [155, 1.3]). Then the quotient sheaves Gi = Ei /Ei−1 are µ-semistable with µ(Gi ) > µ(Gi+1 ). It follows that HomX (Gi , Gi+1 ) = 0 and then Serre duality implies Ext1X (Gi+1 , Gi ) = 0. Thus the Harder-Narasimhan filtration splits. If E is indecomposable, then it is semistable.  Corollary 3.29. Let E be a semistable sheaf of rank n and degree d on an elliptic curve X.

3.5. Some applications of the Abelian Fourier-Mukai transform

99

b and both transforms 1. If d < 0, then E is IT1 with respect to both S and S, are semistable. b and both transforms 2. If d > 0, then E is IT0 with respect to both S and S, are semistable. 3. If d 6= 0, then E is locally free. 4. If d = 0 and E is stable, then E is a line bundle. Thus, any semistable sheaf of degree 0 is WIT1 and Eb is a skyscraper sheaf. Moreover a torsion-free sheaf of degree 0 is semistable if and only if it is a homogeneous bundle. 5. If d = 0, E is S-equivalent to a direct sum of degree 0 line bundles: X M i L⊕n , ni = n . E∼ i i

i

ˆ one has H 0 (X, E ⊗ Pξ ) ' HomX (P ∗ , E) = 0 Proof. 1. For every point ξ ∈ X ξ since E is semistable of negative degree. By Proposition 1.7, E is IT1 . To prove the semistability part, we can assume that E is indecomposable; then Eb is indecomposable as well, so that it is semistable by Proposition 3.28. An analogous argument b proves the statement for S. 2. By Serre duality, one has isomorphisms H 1 (X, E ⊗ Pξ )∗ ' HomX (E ⊗ Pξ , OX ) ' HomX (E, Pξ∗ ) . As above, the latter group is zero since E is semistable of positive degree, and then b is E is IT0 . The semistability of Eb is proved as in the first part. The proof for S similar. 3. Notice that if d 6= 0, by parts 1 or 2 E is ITi with respect to S and b Then b E is semistable of nonzero degree, so that it is IT1−i with respect to S. 1−i i b E ' S (S (E)) is locally free. 4. Since one has H 0 (X, E ⊗Pξ ) ' HomX (Pξ∗ , E), if E is stable of degree 0, then H (X, E ⊗ Pξ ) = 0 unless E ' Pξ∗ . It follows that if E is not a line bundle, it is IT1 ; the transform Eb is locally free of rank 0 by Proposition 3.25 so that Eb = 0. By the invertibility of S, one has E = 0. This proves the first part. Now if E is semistable of degree 0, it has a Jordan-Holder filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = E whose quotients Gi = Ei /Ei−1 are stable of degree 0. Then the sheaves Gi are line bundles ˆ Since the sheaves Pξ are WIT1 of degree 0, that is, Gi ' Pξi for a point ξi ∈ X. i bξ ' Oι(ξ ) (cf. Example 3.6), we deduce that E is WIT1 and Eb is a skyscraper and P i i sheaf. By Proposition 3.15, E is a homogenous bundle. A similar argument proves b and that S b 1 (E) is a skyscraper sheaf. that E is also WIT1 with respect to S 0

5. The argument used in part 4 proves the statement.



100

Chapter 3. Fourier-Mukai on Abelian varieties

Let us denote by Cohss n,d (X) the full subcategory of the category Coh(X) of coherent sheaves on X whose objects are semistable sheaves of rank n and degree d. Unlike the category Cohss µ (X) of semistable sheaves with slope µ, the category (X) is not additive. However, this category will be useful to our present Cohss n,d purpose, which is to prove Atiyah’s results on the characterization of semistable sheaves on X. We denote by Skyn (X) the category of skyscraper sheaves on X of length n. Corollary 3.29 implies directly the following result. Proposition 3.30. The Abelian Fourier-Mukai transform S induces equivalences of categories ss Cohss n,d (X) ' Cohd,−n (X) ,

Cohss n,0 (X)

if d > 0

' Skyn (X) .

Moreover, the twist by L = OX (x0 ), which is a Fourier-Mukai transform b b ∗L Ψ = ΦδX→ X : D (X) ' D (X), also induces an equivalence of categories ss Cohss n,d (X) ' Cohn,d+n (X) .

By appropriately composing the integral functors S and Ψ we obtain the following result. Proposition 3.31. For every pair (n, d) of integers (n > 0), there is a Fourier˜ : Db (X) ' Db (X) which induces an equivalence of categories Mukai functor Φ ss Cohss n,d (X) ' Cohn ¯ ,0 (X) ,

˜ induces then an equivalence where n ¯ = gcd(n, d). The integral functor Φ = S ◦ Φ of categories Cohss ¯ (X) . n,d (X) ' Skyn Proof. If d = 0, we just take Φ = S (cf. Proposition 3.30). Assume now that d > 0. If n ≤ d, we reproduce the method which computes the greatest common divisor n ¯ of (n, d); there is a sequence of Euclidean divisions d = a0 n + d1 n = a1 d1 + d2 .. .

with with

d1 < n d2 < d1

¯ with n ¯ < ds−1 ds−2 = as−1 ds−1 + n ds−1 = as n ¯. ˜ = Ψbs ◦S◦Ψbs−1 ◦S · · ·◦Ψb1 ◦S◦Ψa0 If we denote bj = (−1)j−1 aj , the composition Φ is the required integral functor. If n > d, we just start from the second step by ˜ = Ψbs ◦ S ◦ Ψbs−1 ◦ S · · · ◦ Ψb1 ◦ S. Finally, if d < 0, by taking d1 = d, that is, Φ applying S we reduce to the case (−d, n) and we can then proceed as above. 

3.5. Some applications of the Abelian Fourier-Mukai transform

101

Remark 3.32. Via Proposition 3.31, the multiplicative properties of the Poincar´e bundle can be used to deduce Atiyah’s multiplicative structure of Cohss n,d (X), see [142]. 4 Corollary 3.33. Let E be a torsion-free sheaf of rank n and degree d on X. The following conditions are equivalent: 1. E is stable; 2. E is simple; 3. E is semistable and gcd(n, d) = 1. Thus, the integral functors of Proposition 3.31 map stable sheaves to stable sheaves. Proof. If E is stable, obviously it is simple. If E is simple, it is indecomposable, so that it is semistable by Proposition 3.28. By Proposition 1.34, Φ(E) is simple as well. Since Φ(E) is a skyscraper sheaf of length n ¯ = gcd(n, d), one has n ¯ = 1. If E is semistable and gcd(n, d) = 1, then E is evidently stable.



We can also derive the structure of the coarse moduli space Mss (n, d) of semistable sheaves of rank n and degree d on X. Let us denote by Symn X the n-th symmetric product of X. Corollary 3.34. For every pair (n, d) of integers (n > 0), there is a Fourier-Mukai ∼ Db (X) which induces an isomorphism of moduli spaces ˜ : Db (X) → functor Φ Mss (n, d) ' Mss (¯ n, 0) , ˜ induces an isomorphism where n ¯ = gcd(n, d). The integral functor Φ = S ◦ Φ Mss (n, d) ' Symn¯ X . Proof. The first part follows directly from Propositions 3.31 and 2.63. To prove the second part we need to show that the symmetric product Symn¯ X is a coarse moduli space for the moduli functor of skyscraper sheaves of length n ¯ on X. Indeed, once this result is established, we may use Propositions 3.31 and 2.63 to get the claim. We note that the moduli functor of skyscraper sheaves of length n ¯ on X of Simpson semistable sheaves with concoincides with the moduli functor Mss X,¯ n stant Hilbert polynomial P (m) = n ¯ (cf. Section C.2). Then, this functor has a ss coarse moduli space MX,¯ n whose closed points are S-equivalence classes. Since the only simple skyscraper sheaves are the structure sheaves of the closed points, P ¯ = i ni ), so any skyscraper sheaf is S-equivalent to a direct sum ⊕i Oxnii (with n

102

Chapter 3. Fourier-Mukai on Abelian varieties

ss that closed points of MX,¯ n are in a one-to-one correspondence with closed points n ¯ of Sym X. Though it is a standard result, for completeness we prove that this correspondence is induced by an algebraic isomorphism.

First, we recall that the Hilbert-Chow morphism Hilbn¯ (X) → Symn¯ X, which maps a zero-cycle of length n to its support, is an isomorphism, since X is a smooth curve. Secondly, if T is a scheme and F is a sheaf on X × T , flat over T and such ¯ on Xt for every t ∈ t, then the modified that Ft is a skyscraper sheaf of length n ¯ support Supp0 (F) (see Definition C.9) is a subscheme of X × T flat of degree n over T , that is, a T -valued point of the Hilbert scheme Hilbn¯ (X). We then have a functor morphism Mss X,¯ n → Hom(

•

, Hilbn¯ (X)) ' Hom(

•

, Symn¯ X)

n ¯ ss and thus, an algebraic morphism MX,¯ n → Sym X. We know that this morphism n ¯ is one-to-one on closed points; since Sym X is smooth, the morphism is an isomorphism by Zariski’s main theorem [141, 11.4]. 

We shall consider similar properties for relatively semistable sheaves on an elliptic fibration in Section 6.4.

3.5.2

Preservation of stability for Abelian surfaces

We analyze in this section a first instance of an important feature of the FourierMukai transforms for Abelian surfaces, i.e., that in suitable circumstances they preserve the stability of sheaves. We follow mainly [203] (cf. also [108]). The main result is the following. Let X be an Abelian surface with a fixed ˆ with the dual polarization polarization H. We consider the dual Abelian surface X ˆ we have defined in Section 3.4.2. H Theorem 3.35. Let E be a µ-stable locally free sheaf on X with µ(E) = 0 which is not a flat line bundle (i.e., it is not a line bundle with vanishing first Chern class). Then E is IT1 , and its Fourier-Mukai transform Eˆ is µ-stable with respect to the ˆ dual polarization H. The key result in this context is the following lemma. Lemma 3.36. If E is an IT0 sheaf on X then deg(E) ≥ 0, with equality if and only if E is a skyscraper. Proof. Let T be the torsion subsheaf of E, and F = E/T . Then µ(F) ≤ µ(E) with equality if and only if T is a skyscraper (or it is zero).

3.5. Some applications of the Abelian Fourier-Mukai transform

103

Let us assume that F is nonzero and µ(F) ≤ 0. We have S2 (T ) = 0 so that applying the functor S to the exact sequence 0→T →E →F →0 and using that E is IT0 , we obtain that F is IT0 as well. Thus, Fˆ = S0 (F) is ˆ ≥ 0. locally free, WIT2 (cf. Corollary 3.4) and satisfies µ(F) ˆ = F ⊗ O0 6= 0, there is a nonzero morphism Fˆ → O ˆ , hence Since H 2 (X, F) X ˆ F is not µ-stable. Let 0 → K → Fˆ → G → 0 ˆ ≥ 0. By be a destabilizing sequence, where K is µ-stable, so that µ(K) ≥ µ(F) b Fˆ ∗ ) ' F ∨ = RHomO (F, OX ), whence by taking Proposition 1.15, one has S( X cohomology and taking into account that F is torsion-free, we obtain b 2 (Fˆ ∗ ) ' Ext2 (F, OX ) = 0 S OX ˆ Fˆ ⊗ Px ) ' H 2 (X, ˆ Fˆ ∗ ⊗ P−x ) = 0 for all x ∈ X. Then H 0 (X, ˆ K⊗ so that H 0 (X, 2 ˆ Px ) = 0 for all x, so that K is not a flat line bundle. Moreover H (X, K ⊗ Px ) = 0 as K is µ-stable, so that K is IT1 . On the other hand we have S0 (G) ' S1 (K), where S1 (K) is WIT1 by Corollary 3.4, while S0 (G) is WIT2 by Corollary 3.5. This is a contradiction, so that either F = 0 or µ(F) > 0. In the first case, E has nonnegative degree and has degree zero if and only if it is a skyscraper. In the second case, we have deg(E) > 0.  Corollary 3.37. If E is a WIT2 sheaf on X then deg(E) ≤ 0, with equality if and only if E is a homogeneous bundle. Proof. This follows from the previous lemma and Proposition 3.15.



Proposition 3.38. If E is a µ-semistable WIT1 sheaf on X with µ(E) = 0, then Eˆ is µ-semistable. Proof. Let us assume that Eˆ is not µ-semistable, and let 0 → F → Eˆ → G → 0 be a destabilizing sequence; possibly by modding its torsion out, we may assume ˆ to this sequence we obtain that G is torsion-free. Applying the functor S ˆ 1 (F) → E → S ˆ 1G → S ˆ 2 (F) → 0 ˆ 0 (G) → S 0→S

(3.13)

ˆ 0 (F) = S ˆ 2 (G) = 0. By Corollary 3.5, the sheaf S ˆ 0 (G) is WIT2 and together with S 0 ˆ (G)) ≤ 0 with equality if and only if hence by Corollary 3.37 it satisfies deg(S ˆ 2 (F)) ≥ 0 with equality if ˆ 0 (G) is a homogeneous bundle, and analogously deg(S S ˆ 2 (F) is a skyscraper. and only if S

104

Chapter 3. Fourier-Mukai on Abelian varieties

ˆ 0 (G) and C = S ˆ 2 (F)/S ˆ 1 (G), so that the sequence ˆ 1 (F)/S Now let K = S ˆ 0 (G)) ≥ 0 ˆ 1 (F)) − deg(S 0 → K → E → C → 0 is exact. We have deg(K) = deg(S 2 0 ˆ (G) is a homogeneous bundle, ˆ (F) is a skyscraper, S with equality if and only if S and deg(F) = deg(G) = 0. Note that deg(K) = 0 since otherwise, as E is µ-semistable, one has rk(K) = rk(E), so that deg(C) ≥ 0 and deg(K) ≤ 0 which is a contradiction unless deg(K) = 0. Therefore deg(F) = 0 so Eˆ is µ-semistable.  We prove now Theorem 3.35. Since H 0 (X, E ⊗ Pξ ) = H 2 (X, E ⊗ Pξ ) = 0 ˆ as E is µ-stable, locally free of zero degree, E is IT1 . Let again for all ξ ∈ X 0 → F → E → G → 0 be a destabilizing sequence, this time with G µ-stable. We may repeat the proof of Proposition 3.38 but this time if K is a proper subsheaf of E we have deg(K) < 0, so that necessarily either K = 0 or rk(K) = rk(E). ˆ 2 (G) = 0, so that H 2 (X, G ⊗ Pξ ) = 0 for all ξ ∈ X, ˆ while on the Note that S other hand H 0 (X, G ∗∗ ⊗ Pξ ) = 0 since G ∗∗ is µ-stable of nonpositive degree, so that H 0 (X, G ⊗ Pξ ) = 0 and G is IT1 . Hence we have an exact sequence ˆ 2 (F) → 0 . ˆ 1 (F) → E → Gˆ → S 0→S

(3.14)

ˆ 2 (F) is a skyscraper sheaf; but the sequence ˆ 1 (F) = 0 and S If K = 0, then S (3.14) implies that E is not locally free, whence we may exclude this case. Thus ˆ 1 (F) and since Gˆ is locally free, the morphism E → Gˆ rk(K) = rk(E). But K ' S ˆ 2 (F) which is absurd because the first sheaf is WIT1 while vanishes, and Gˆ ' S  the second is IT0 .

3.5.3

Symplectic morphisms of moduli spaces

The preservation of stability expressed by Theorem 3.35 provides morphisms between different components of the moduli space of stable sheaves on an Abelian surface. These morphisms happen to be symplectic with respect to the so-called Mukai’s symplectic structure on the moduli space [225]. Let X be a (complex) Abelian or K3 surface equipped with a polarization H, and let M = MH (v) be the moduli space of H-stable sheaves on X with Mukai vector v = (r, `, s). If the greatest common divisor of the numbers r, ` · H and s is 1, then M is a smooth quasi-projective variety. If [F] is a point in M , the tangent space T[F ] (M ) may be identified with the vector space Ext1 (F, F). The Yoneda product (see Eq. (A.13)) provides a skew-symmetric map Ext1 (F, F) ⊗ Ext1 (F, F) → Ext2 (F, F). Moreover one has a trace morphism Ext2 (F, F) → Ext2 (OX , OX ) dual to the natural  − Hom(F, F). This defines a holomorphic 2-form ς morphism C = Hom(OX , OX ) → on M which turns out to be closed and nondegenerate, thus defining a holomorphic symplectic 2-form.

3.5. Some applications of the Abelian Fourier-Mukai transform

105

Actually, the previous isomorphisms come from Serre duality, which involves the choice of an isomomorphism H 2 (X, ωX ) ' C, which in this case is provided by integration on X of a form of type (2, 2) representing the cohomology class. The morphism Ext2 (F, F) → C is the composition trace

λ

Ext2 (F, F) −−−→ H 2 (X, OX ) ' H 0,2 (X, C) − → H 2,2 (X, C) ' C

(3.15)

where λ is the wedging by a symplectic form ω on X. This may also be written as the composition ∼ Hom(F, F)∗ → ∼C Ext2 (F, F) → (3.16) where the first isomorphism is Serre duality and the second is the dual of the isomorphism  introduced before. Remark 3.39. Let us note for future use that if X is a symplectic variety of dimension n > 2, we can anyway define symplectic structures on the moduli spaces of stable sheaves on X as above, by taking for λ in Equation (3.15) the ¯ m−1 , where ω is the symplectic form of X, and m = n/2 [184]. wedging by ω m ∧ ω 4 Let X be an Abelian surface with a polarization H, and let MH (v) be the µ (v) be the moduli space of H-stable sheaves on X with Mukai vector v. Let MH subset of MH (v) formed by locally free µ-stable sheaves on X. It is known that µ (v) is open in MH (v) in the Zariski topology [155]. We shall use the same MH ˆ equipped notation for the moduli space of sheaves on the dual Abelian variety X ˆ defined in Section 3.4.2. If ` · H = 0, by Theorem 3.35 with the polarization H µ (v) → MHˆ (ˆ v )µ , induced by the Abelian Fourier-Mukai we have a morphism ς : MH transform S. Here, as consequence of Equation (3.2), we have vˆ = −(s, `, r) if ˆ Z)). v = (r, `, s) (having identified H 2 (X, Z) with H 2 (X, Proposition 3.40. The morphism ς is symplectic. Proof. One has a commutative diagram Ext1 (F, F) ⊗ Ext1 (F, F) S×S



Ext1 (S(F), S(F)) ⊗ Ext1 (S(F), S(F))

/ Ext2 (F , F) 

S

/ Ext2 (S(F), S(F))

/ Hom(F, F)∗ ' C O S∗

/ Hom(S(F), S(F))∗ ' C .

The first square commutes by (A.14), and the second by the compatibility of the integral functors with Serre duality.  We shall observe in Chapter 4 that a similar result holds in the case of the Fourier-Mukai transform on K3 surfaces.

106

3.5.4

Chapter 3. Fourier-Mukai on Abelian varieties

Embeddings of moduli spaces

The integral functor Φ0 : Db (C) → Db (J(C)) defined in Section 3.4.3, where C is a smooth projective curve of genus g > 1, and J(C) its Jacobian variety, has the property of preserving the stability of bundles it acts on, according to a result of Li. We may use this result to construct an embedding of the moduli space of stable bundles on C as a subvariety of a component of the moduli space of stable bundles on J(C), which is isotropic, and in some cases Lagrangian, with respect to the natural symplectic structure of the moduli space of stable bundles on J(C). Here we basically follow [76]. We start by recalling Li’s result [196, Thm. 4.11]. For the proof we refer the reader to Li’s paper. Proposition 3.41. Let C be a smooth projective curve of genus g > 1. If E is a stable bundle on C of rank r and degree d such that d > 2rg, then E is WIT0 with respect to the integral functor Φ0 , and the transformed sheaf Eˆ = Φ0 (E) is locally free and µ-stable with respect to the polarization given by the Θ divisor on J(C). If E has rank r and degree d, the Chern character of Eˆ is readily computed, obtaining ˆ = (d + r(1 − g), −rΘ, 0, . . . , 0) ch(E) (3.17) where Θ is the cohomology class associated with the Θ divisor in J(C). Therefore by Proposition 3.41 we obtain a map µ (r, d) , j : MC (r, d) → MJ(C)

(3.18)

where MC (r, d) is the moduli space of stable sheaves on C with rank r and degree µ (r, d) is the subset of the moduli space of Θ-stable sheaves on J(C) d, and MJ(C) with Chern character as in (3.17) that is formed by µ-stable locally free sheaves. Proposition 3.42. Let d > 2rg. The morphism (3.18) is an embedding (i.e., both j and its tangent map are injective). Proof. Let α : C → J(C) be the embedding given by the Abel map. This induces a functor α ˜ : Db (C) → Db (J(C)). Then we have an isomorphism of functors Φ0 ' P ˜. ΦJ(C)→J(C) ◦ α On the other hand, if E is a stable bundle on C of rank r and degree d, the direct image α∗ (E) is a stable (in Simpson’s sense) pure sheaf of dimension 1 on J(C), with Chern character ch(α∗ (E)) = (0, . . . , 0, −rΘ, d + r(1 − g) (see Section C.2). Since ΦP J(C)→J(C) is an equivalence of categories, the injectivity of j follows from the fact that α∗ (E) ' α∗ (F) implies E ' F.

3.5. Some applications of the Abelian Fourier-Mukai transform

107

As far as the injectivity of the differential of j is concerned, we note that the latter may be regarded as a map Ext1C (E, E) → Ext1J(C) (α∗ (E), α∗ (E)). If an extension 0 → E → F → E → 0 maps to zero, then α∗ (F) ' α∗ (E) ⊕ α∗ (E), and this implies F ' E ⊕ E.  Proposition 3.43. Assume that d > 2rg, where g is the genus of C. If g is even, µ 0 (r, d) of MJ(C) (r, d), and the map j embeds MC (r, d) into the smooth locus MJ(C) the subvarieties MC (r, d) are isotropic with respect to any of the symplectic forms defined in Remark 3.39. In particular, when g = 2, the subvarieties MC (r, d) are µ (r, d). Lagrangian with respect to the Mukai form of MJ(C) µ (r, d) (cf. Section 3.5.3) Proof. The symplectic form on the moduli space MJ(C) vanishes on the image of MC (r, d) because the Yoneda map

Ext1 (F, F) ⊗ Ext1 (F, F) → Ext2 (F, F) vanishes when F is of the form F = α∗ (E) for a vector bundle E on C. This is shown by a direct computation; details may be found in [247]. In the case g = 2, the moduli space is smooth by the results in [225]; moreover, µ (r, d) = 2(r2 + 1) = 2 dim MC (r, d) . dim MJ(C)

 Remark 3.44. If we consider the moduli space MC (r, ξ) of stable bundles on C of rank r and fixed determinant isomorphic to ξ, then the result is trivial: the variety 4 MC (r, ξ) is Fano, so that it carries no holomorphic forms. Let us now briefly elaborate on the case g = 2. One can characterize situations µ where the moduli space MJ(C) (r, d) is compact. This happens for instance in the following case. Proposition 3.45. Assume g = 2, d > 4r and that ρ = d − r is a prime number. Then every Gieseker-semistable sheaf on J(C) with Chern character (d−r, −rΘ, 0) is µ-stable. Moreover, if d > r2 + r, every such sheaf is locally free (this always happens when r = 1, 2, 3). Proof. Since d − r is prime, every sheaf in MJ(C) (r, d) is properly stable. Let [F] ∈ MJ(C) (r, d) and assume that the subsheaf G destabilizes F. Let ch(G) = (σ, ξ, s). Standard computations show that if F is not µ-stable, then 2r ξ·Θ =− σ ρ

and

s < 0.

108

Chapter 3. Fourier-Mukai on Abelian varieties

Setting n = ξ · Θ we have |n| = 2rσ/ρ, with σ < ρ and ρ > 3r. This is impossible whenever ρ is prime. The statement about local freeness follows from the Bogomolov inequality: from the exact sequence 0 → F → F ∗∗ → Q → 0 we have ch(F ∗∗ ) = (d − r, −rΘ, λ) where λ is the total length of the torsion sheaf Q. Since F ∗∗ is µ-stable it satisfies the Bogomolov inequality [155], which in this case reads r2 ≥ λ(d − r). Together with the condition d > r2 + r this forces λ = 0, i.e., F is locally free.  Remark 3.46. In the case g = 2, the complex Lagrangian embedding j : MC (r, d) → µ (r, d) provides examples of special Lagrangian submanifolds. We refer the MJ(C) reader to [76] for this aspect. 4 Remark 3.47. In the case g = 4, one can use the fact that MC (r, d) embeds µ µ (r, d) to show that at a smooth point [E] of MJ(C) (r, d) isotropically into MJ(C) 2 one has h (End0 (E)) > 0, that is, the sufficient condition for the smoothness of the µ (r, d) is not satisfied. Let us set A(r) = dim h2 (End0 (E)) (as we shall space MJ(C) see this depends only on the rank). The isotropicity condition yields the inequality dim MJ(C) (r, d) ≥ 6r2 + 2. On the other hand, if [E] is a smooth point of the moduli space corresponding to a vector bundle E, we have dim MJ(C) (r, d)

=

h1 (End(E))

=

1 + 3 + 21 A(r) ≥ 6r2 + 2,

whence the inequality A(r) ≥ 12r2 − 4 > 0 4

follows.

3.6

Notes and further reading

A fairly comprehensive treatment of the Abelian Fourier-Mukai transform is given by A. Polishchuk in the book [251], which also contains an extensive bibliography. It is also worth mentioning that the classical Torelli theorem about the characterization of a projective curve in terms of its Jacobian has been proved by using the Abelian Fourier-Mukai transform in [36]. A generalization of that paper to the characterization of curves from their Prym varieties is contained in [233].

3.6. Notes and further reading

109

The Abelian Fourier-Mukai transform was used by Beauville [32] to study the Chow ring of an Abelian variety. The group of autoequivalences of the derived category of an Abelian variety X over a field k has been characterized by Orlov in [243] as an extension of the ˆ k is ˆ k , where (X × X) group of isometric isomorphisms of X × X by Z ⊕ (X × X) ˆ the group of rational points of X × X. The preservation of Gieseker stability under the Abelian Fourier-Mukai transform has been studied by Maciocia in the case of Abelian surfaces [203]. The results are largely negative: Gieseker stability is not well-behaved under the action of the transform. The situation improves if one considers a notion of twisted Gieseker stability, as shown by Yoshioka [294, 295].

Chapter 4

Fourier-Mukai on K3 surfaces Introduction In looking for examples of Fourier-Mukai transforms on varieties other than the Abelian ones, it is natural to consider K3 surfaces, especially in view of Theorem 2.38 and the subsequent discussion. A forerunner of a Fourier-Mukai functor for K3 surfaces (which in our notation is a morphism of the type f Q : H • (X, Z) → H • (Y, Z), cf. Eq. (1.12)) was introduced by Mukai in [227]. When trying to define a Fourier-Mukai functor in the proper sense, one realizes that it is necessary to limit the class of K3 surfaces one considers; essentially one needs to require that the Picard lattice contains some preferred sublattice. A first example was given in [24] where a class of K3 surfaces called (strongly) reflexive was introduced. Another example by Mukai appeared later [228]. In this chapter we start by giving a general introduction to K3 surfaces, considering at first the general K¨ ahlerian case and then specializing to the algebraic case. We also provide the basic elements of the theory of moduli spaces of stable sheaves on K3 surfaces. The core part of the chapter is the construction of the Fourier-Mukai transform for reflexive K3 surfaces. The transformation is constructed by realizing a reflexive K3 surface X as a fine moduli space of stable bundles on X itself, and using the corresponding universal bundle as integral kernel. Applications are then given to the study of a class of bundles, called homogeneous, which extend the notion of homogeneous bundles on Abelian varieties, and to the Hilbert schemes of points of a (reflexive) K3 surface. Section 4.3.4 also contains Mukai’s example, which is very similar in nature. Section 4.4 shows that the Fourier-Mukai transform for reflexive K3 surC. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_4, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

111

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Chapter 4. Fourier-Mukai on K3 surfaces

faces shares another important feature with the Abelian Fourier-Mukai transform, namely, under suitable conditions it preserves the stability of the sheaves it acts on. The base field is always C and all characteristic classes take values in the cohomology ring. For simplicity, in this chapter by “(semi)stable” we mean “Gieseker(semi)stable.”

4.1

K3 surfaces

Here we wish to collect some basic results on K3 surfaces in the most general setting, i.e., we do not assume they are algebraic (which is not always the case, of course) or K¨ahlerian (which on the other hand is always the case by a wellknown theorem of Y.-T. Siu [270], cf. our Theorem 4.7). We denote by TX the (holomorphic) tangent bundle of X and ΩpX the sheaf of germs of holomorphic ∗ . p-forms, i.e., the sheaf of sections in the locally free sheaf Λp TX Definition 4.1. A K3 surface is a compact connected smooth complex surface X 4 such that q = dim H 1 (X, OX ) = 0 and the canonical bundle ωX is trivial. It follows from the definition that pg = dim H 0 (X, ωX ) = 1 and c1 (X) = −c1 (ωX ) = 0. By using Noether’s formula 1 − q + pg =

1 (c1 (X)2 + c2 (X)) 12

we get c2 (X) = 24 (as usual, we have identified H 4 (X, Z) with Z by integrating over the fundamental class of X). As a consequence, one obtains that the P topological Euler characteristic χtop (X) = i (−1)i bi is equal to 24. Since b1 = dim H 1 (X, R) is either 2q or 2q − 1 according to whether it is even or odd [22, Theorem IV.2.7], one necessarily has b1 = 0. Proposition 4.2. One has H 1 (X, Z) = H1 (X, Z) = H 3 (X, Z) = H3 (X, Z) = 0, and H 2 (X, Z) ' H2 (X, Z) is a free Z-module of rank 22. Proof. The only nontrivial fact to prove is that H1 (X, Z) = 0. Since we already know that H1 (X, R) = 0, then H1 (X, Z) may only be a torsion module. But the existence of a torsion element of order k > 1 implies the existence of a compact complex surface Y which is a k-fold covering of X and would violate Noether’s formula. By the universal coefficient theorem, the torsion submodule of H 2 (X, Z)  is isomorphic to the torsion submodule of H1 (X, Z).

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113

The intersection form H 2 (X, Z) × H 2 (X, Z) → Z (γ, γ 0 ) 7→ γ · γ 0 defines a lattice structure on H 2 (X, Z). Poincar´e duality is precisely the statement that this lattice is unimodular. The index τ (X) = b+ − b− of the intersection is readily computed through the Hirzebruch formula: τ (X) =

1 2 (c − 2c2 ) = −16 . 3 1

So, we get b+ = 3 and b− = 19. Since the second Stiefel-Whitney class of X vanishes (indeed, w2 (X) = c1 (X) mod 2 = 0), Wu’s formula γ · γ = w2 · γ mod 2

for all γ ∈ H 2 (X, Z)

tells us that the intersection form is even. The classification theorem B.2 for indefinite unimodular even lattices yields the following result. Proposition 4.3. The Z-module H 2 (X, Z) of a K3 surface X endowed with the intersection form is a lattice isomorphic to Σ = U ⊕ U ⊕ U ⊕ E8 h−1i ⊕ E8 h−1i, where U is the rank 2 hyperbolic lattice, and E8 is the rank 8 lattice whose intersection form is the Cartan matrix associated to the exceptional Lie algebra e8 (cf. Eq. (B.1)). Example 4.4. Let us consider a smooth quartic surface X in P3 . By Bott’s homotopic version of Lefschetz’s hyperplane theorem [56], the fundamental group of X is trivial. Hence, 0 = b1 (X) = 2q(X). Moreover, a direct computation using the adjunction formula [131, p. 146] gives c1 (X) = 0. But, since q(X) = 0, the first Chern class classifies holomorphic line bundles on X. Thus, the canonical bundle is trivial, and X is a K3 surface. 4 Example 4.5. An important class of K3 surfaces is provided by the Kummer surfaces. Let us consider the involution ι : (z, z 0 ) 7→ (−z, −z 0 ) on a complex 2-torus T = C2 /Λ. The quotient X 0 = T /ι has 16 conical singularities corresponding to the fixed points of the involution. Blowing up these singularities one gets a smooth 4 surface X, having b1 (X) = 0 and trivial canonical bundle (see [22, V.16]). Example 4.6. Let C be a smooth sextic in P2 ; one has g(C) = 10. To the sextic C we associate a section τ ∈ H 0 (P2 , OP2 (6H)) such that (τ ) = C; we fix an isomorphism OP2 (3H) ⊗ OP2 (3H) ' OP2 (6H). In the total space of the fibration p : OP2 (3H) → P2 we consider the locus X = {(x, λ)|λ ⊗ λ = τ (x)}. So, X is a smooth double cover of P2 branched along C. By the Hurwitz formula the

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canonical bundle of X is trivial. Moreover, we have p∗ OX ' OP2 ⊕ OP2 (−3H) [22, I.17.2]. Since p is a finite morphism, the sheaves OX and p∗ OX have isomorphic 4 cohomologies, so that H 1 (X, OX ) = 0, and X is a K3 surface. According to a general result, sometimes called the Kodaira conjecture, a compact complex surface admits a K¨ ahler metric if and only if b1 (X) is even [77, 192] (see also [22, IV.3.1]). Since for a K3 surface b1 = 0, one has at once the following result. Theorem 4.7. Every K3 surface X admits a K¨ ahler metric. This result was first proved by Siu in 1983 [270]. It implies that the decomposition of the cohomology space H 2 (X, C) of a K3 surface X H 2 (X, C) = H 2,0 (X) ⊕ H 1,1 (X) ⊕ H 0,2 (X)

(4.1)

(which can be defined on any compact complex surface for the Fr¨ohlicher spectral sequence degenerates at E1 -level [22, IV.2.8]) coincides with the usual Hodge decomposition for any choice of the K¨ ahler metric. The Hodge numbers hp,q = p,q dimC H (X) are readily calculated. One has h0,2 = h2,0 = 1, and h1,1 = b2 − h2,0 − h0,2 = 20. The C-linear extension of the intersection form on H 2 (X, C) coincides with the cup product of cohomology classes of differential forms. Its restriction to H 1,1 (X) ∩ H 2 (X, R) has signature (1, h1,1 − 1) = (1, 19). ∗ ) the Picard For any algebraic surface, one denotes by Pic(X) = H 1 (X, OX group of isomorphism classes of line bundles over X. By the Lefschetz theorem on (1, 1) classes, the image of Pic(X) in H 2 (X, Z) coincides with the intersection H 1,1 (X)∩j ∗ (H 2 (X, Z)), where j ∗ : H 2 (X, Z) → H 2 (X, C) is the natural injection. This sublattice of H 2 (X, Z) is called the N´eron-Severi group of the surface X. Its rank is called the Picard number of X. If X is a K3 surface one has q = 0, hence the map c1 : Pic(X) → H 2 (X, Z) is injective, and one can identify Pic(X) with the N´eron-Severi group. We shall often use the common terminology Picard lattice.

A useful characterization of Pic(X) is provided by the following criterion, which is an easy consequence of the orthogonality of the Hodge decomposition (4.1). Proposition 4.8. A class in H 2 (X, Z) is in Pic(X) if and only if it is orthogonal to H 2,0 (X). Definition 4.9. The orthogonal complement to Pic(X) in H 2 (X, Z) is called the transcendental lattice, which will be denoted by T(X). 4 A class d ∈ Pic(X) is said to be nodal if d2 = −2. The Riemann-Roch theorem implies that d or −d is the class of a unique curve C which is irreducible [22, Prop. VIII.3.7]. The curve C is said to be nodal and is smooth and rational.

4.1. K3 surfaces

115

The importance of nodal curves in the theory of K3 surfaces stems from the fact that the set of effective classes on a K3 surface is the semigroup generated by the nodal classes and the integral points in the closure of the positive cone [22, Prop. VIII.3.8]. The lattice H 2 (X, Z) endowed with its natural weight-two Hodge structure given by the decomposition 4.1 contains essentially all the information about the geometric structure of the K3 surface X. The choice of an isometry φ : H 2 (X, Z) → Σ is called a marking of the K3 surface X. The line φ(H 2,0 (X, C)) ⊂ Σ ⊗ C determines a point (called the period of X) in the period domain ∆ ⊂ P(Σ ⊗ C), which is defined by the equations α · α = 0,

α·α ¯>0

(this assignment is called the period map, and is surjective, cf. [22]). Here α is the image under φ of a generator of H 2,0 (X, C)). According to the idea underlying the Torelli theorem, two K3 surfaces X and Y are isomorphic if and only if they can be given markings such that the corresponding points in period domain coincide. Actually, this statement can be strengthened to a more precise and deeper result, the ˇ ˇ (global) Torelli theorem, which was first proved by Pjatecki˘ı-Sapiro and Safareviˇ c [248] in the projective case and by Burns and Rapoport [81] in the K¨ahler case (see [91] or [22] for a detailed account). Let X and Y be K3 surfaces. A group isomorphism φ : H 2 (X, Z) → H 2 (Y, Z) is a Hodge isometry if it preserves both the intersection forms and the natural Hodge structures (the second requirement is equivalent to saying that the Clinear extension of φ to H 2 (X, C) preserves the Hodge decomposition). A Hodge isometry is said to be effective if it maps the K¨ ahler cone of X to the K¨ahler cone of Y . By means of the notion of Hodge isometry, one can state a Torelli theorem for K3 surfaces, both in strong and weak form [22, Thm. 11.1, Cor. 11.2]. Theorem 4.10. (Torelli theorem) Let X and Y be K3 surfaces, and let φ : H 2 (Y, Z) → H 2 (X, Z) be an effective Hodge isometry. Then, there is a unique isomorphism f : X → Y such that f ∗ = φ. A weaker form of this result, which follows from the previous one, will be useful in the sequel. Corollary 4.11. (Weak Torelli theorem) Let X and Y be K3 surfaces whose lattices H 2 (X, Z) and H 2 (Y, Z) are Hodge isometric. Then X and Y are isomorphic. The Torelli theorem is a consequence of the fact that there is a well-behaved theory of deformations for K3 surfaces, both local and global. In particular, it turns out all K3 surfaces are deformation equivalent to each other, so that they are all diffeomorphic as differentiable 4-manifolds. This shows that every K3 is

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simply connected, since this is the case for the particular example of the quartic surface in P3 (Example 4.4). By using the Torelli theorem one is able to construct a universal family of (K¨ahlerian) marked K3 surfaces. Its base space (the moduli space of marked K¨ahlerian K3 surfaces) is a smooth, non-Hausdorff analytic space of dimension 20 [22]. Algebraic marked K3 surfaces form a subfamily of dimension 19.

4.2

Moduli spaces of sheaves and integral functors

In this section we introduce moduli spaces of sheaves on algebraic K3 surfaces; when such moduli space are fine, so that they carry a universal sheaf, the latter may be used as the kernel of a Fourier-Mukai transform. We need to fix some preliminary notation and definitions. The Todd class of a K3 surface X is
1 1 td(X) = 1, c21 (X), (c21 + c2 (X)) = (1, 0, 2) ∈ H 2• (X, Z) , 2 12 p so that td(X) = (1, 0, 1). Hence, p the Mukai vector of a coherent sheaf E on X is the element v(E) = ch(E) td(X) = (rk E, c1 (E), χ(E) − rk(E)), which still belongs to the even part of the integer cohomology of X. We recall that the Mukai pairing (Definition 1.3) is a symmetric bilinear form h·, ·i on the even part of the cohomology of X that for a K3 surface takes the form Z hv, wi = − v∗ · w , X ∗

0

1

2

where v = (v , −v , v ) is the Mukai dual of v. Definition 4.12. Let e 0,2 (X) = H 0,2 (X) , H

e 2,0 (X) = H 2,0 (X) H

e 1,1 (X) = H 0 (X) ⊕ H 1,1 (X) ⊕ H 4 (X) . H e p,q (X)} is denoted by Then the resulting weight-two structure {H 2• (X, Z), H • e 4 H (X, Z). e • (X, Z) respects the Hodge structures. The natural inclusion H 2 (X, Z) ,→ H • e The space H (X, Z) endowed with the Mukai pairing h·, ·i is an even lattice isomorphic to Σ ⊕ U (so it has signature (4, 20)). The restriction of the Mukai pairing to H 2 (X, Z) coincides with the intersection product. We shall also use the notation ˜ 1,1 (X) ∩ H 2 (X, Z) ' Pic(X) ⊕ U . ˜ 1,1 (X, Z) = H H

4.2. Moduli spaces of sheaves and integral functors

117

Since the canonical sheaf of X is trivial, Serre duality for Ext groups together with Equation (1.6) yields the formula hv(E), v(E)i = dim Ext1 (E, E) − 2 dim HomOX (E, E)

(4.2)

for any coherent sheaf E on X. From this we see that dim Ext1 (E, E) is always an even integer, for the Mukai pairing is even. If the sheaf E is simple, then HomOX (E, E) ' C, so that v 2 (E) ≥ −2. The space Ext1 (E, E) is canonically isomorphic to the Zariski tangent space to the infinitesimal deformations of E. We fix a polarization H on X and we consider the (coarse) moduli space MH (v)ss of S-equivalence classes of sheaves which are semistable with respect to H and have Mukai vector equal to v. By Maruyama’s general results [211, 212], we know that MH (v)ss is a (possibly empty) projective variety. In most applications and examples we will be more interested in studying an open subscheme of MH (v)ss , namely the (coarse) moduli space MH (v) parameterizing stable sheaves. The following fundamental fact follows from results due to Mukai [225]. e 1,1 (X, Z). The moduli space MH (v) of Theorem 4.13. Let v be an element in H stable sheaves whose Mukai vector is v is a smooth scheme of dimension v 2 + 2 (possibly empty). The canonical bundle of MH (v) is trivial. Corollary 4.14. If v = (r, `, s) with v 2 = 0 and r ≥ 2, and E is a µ-stable sheaf on X (with respect to some polarization) with v(E) = v, then E is locally free. Proof. If E is not locally free, its double dual E ∗∗ has Mukai vector v 0 = (r, `, s+λ), where λ is the length of the support of the quotient E ∗∗ /E. Since E ∗∗ is µ-stable, the moduli space MH (v) is nonempty of dimension (v 0 )2 + 2 = −2rλ + 2. Since this cannot be negative, one must have λ = 0, i.e., E is locally free.  Mukai’s results provide a quite exhaustive description of the moduli spaces of dimension 0 or 2. We need to state a technical result [227, Prop. 2.14]. Lemma 4.15. Let E be a torsion-free coherent sheaf on X. The following inequality holds: dim Ext1 (E ∗∗ , E ∗∗ ) + 2 length(E ∗∗ /E) ≤ dim Ext1 (E, E) . Recall that on a smooth surface the dual of a coherent sheaf is always locally free. e 1,1 (X, Z) such that v 2 = −2. If there Theorem 4.16. Let v be an element in H exists at least one stable sheaf E on X whose Mukai vector is equal to v, then the ss (v) is a single reduced point and E is locally free. moduli space MH

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Chapter 4. Fourier-Mukai on K3 surfaces

Proof. Assume that F is a semistable sheaf with v(F) = v. By Equation (1.6) we have χ(E, F) = −hv(F), v(E)i = −v 2 = 2. It follows (using Serre duality) that dim Hom(E, F) + dim Hom(F, E) > 0. Since E stable and F are semistable of the same Hilbert polynomial, we conclude that E ' F. Now Lemma 4.15 implies that  length(E ∗∗ /E) = 0, so that E is locally free. Remark 4.17. Kuleshov shows in [191] that for every Mukai vector v = (r, `, s), with r > 1 and v 2 = −2, there exists a simple µ-semistable bundle E such that v(E) = v. 4 We now address the issue which has the greatest interest to us, namely the case of the two-dimensional components of the moduli space. The results proved by Mukai in the papers [225, 227] suggest the existence of a notion of “duality” for K3 surfaces similar, under many respects, to that holding for Abelian varieties. This idea has been further developed in [24, 26, 228]. ss (v) are not irreducible, and the In general, the moduli spaces MH (v) and MH first is strictly contained into the latter. However in dimension 2 the existence of a compact component implies that every semistable sheaf is actually stable and that the moduli space is irreducible. The following result is a consequence of [227, Prop. 4.4] and of Theorem 4.13 (a concise proof can be found in [155, p. 144]).

e 1,1 (X, Z) such that v 2 = 0. Suppose there exists a comTheorem 4.18. Let v ∈ H ponent M of the moduli space MH (v) which is compact and irreducible. Then ss (v), and this surface is smooth, irreducible and compact with M = MH (v) = MH trivial canonical bundle. Therefore, it is either an Abelian surface or a K3 surface. A universal family for the moduli of stable sheaves on a projective variety may not exist (the definition of universal family is given below). In order to circumvent this obstacle, one introduces the notion of quasi-universal family (see [227]). Let S be a scheme and let M be a connected component of the (coarse) moduli space of stable sheaves on S (with respect to a fixed polarization). We denote by y a point in M and by Ey the corresponding stable sheaf on S. Definition 4.19. A coherent sheaf Q on S × M is a quasi-universal family if the following conditions are satisfied: 1. Q is flat over M ; 2. for all y ∈ M there exists an positive integer ν such that Qy ' Ey⊕ν ; 3. for every scheme T and for every sheaf Q0 over S × T flat over T with 0 Q0t ' Etν for some stable sheaf Et ∈ M and for all t ∈ T , ν 0 being a positive integer independent of t, there exists a unique morphism u : T → M and two locally free sheaves F and F 0 on T such such that u∗ Q ⊗ F ' Q0 ⊗ F 0 . 4

4.2. Moduli spaces of sheaves and integral functors

119

The number ν is called the similitude of the family. A quasi-universal family with ν = 1 is nothing but a universal family in the usual sense. Building on work by Maruyama [212], Mukai proved a general result on the existence of a quasi-universal family for the moduli functor of simple sheaves [227, Thms. A.5, A.6], which we state only in the case of K3 surfaces, and for the moduli of stable sheaves. Theorem 4.20. Let M be a connected component of the moduli space MH (v) of stable sheaves over a K3 surface X. Then there exists a quasi-universal family Q on X ×M . Moreover, if the greatest common divisor of the integers rk(E), c1 (E)·D and χ(E) (where E ∈ M and D runs over all divisors in X) is 1, then there exists a universal family on X × M and M is a fine moduli space. We may now use a universal family Q as a kernel to define an integral functor b b ΦQ M→X : D (M ) → D (X) .

Proposition 4.21. Assume that dim M = 2. The integral functor is fully faithful if and only if the family Q is universal. Moreover, in this case both functors b b ΦQ M→X : D (M ) → D (X)

and

b b ΦQ X→M : D (X) → D (M )

are Fourier-Mukai transforms. Proof. If Q is universal, then for every y ∈ M the sheaf Qy is simple. Moreover, if y1 , y2 are distinct points in M , we have Hom(Qy1 , Qy2 ) = Ext2 (Qy1 , Qy2 ) = 0, while Ext1 (Qy1 , Qy2 ) = 0 since v 2 = 0. So Q is strongly simple over M , and the functor ΦQ M→X is fully faithful by Theorem 1.33. Conversely, if ΦQ M→X is fully faithful, then Q is strongly simple over M , so that all sheaves Qy are simple and the similitude ν must be 1. Q The functors ΦQ M→X and ΦX→M are exact equivalences by Corollary 2.58.



Remark 4.22. If Q is a quasi-universal it may be convenient to regard the moduli space as a stack (or gerbe). In this picture, it becomes a fine moduli space. See, e.g., [82, 83, 37]. For a “gerby” Fourier-Mukai transform see [97]. See also Section 6.7. 4 ˆ ◦ Φ ' [−2] and Φ ◦ Φ ˆ ' [−2], If Q is universal and locally free, we have Φ Q Q∗ ˆ where Φ = ΦX→M and Φ = ΦM→X . In this case one has the results expressed by the following corollaries. Corollary 4.23. If F is WITi with respect to Φ, then Fb is WIT2−i with respect to b ˆ Moreover F b ' F. Φ.

120

Chapter 4. Fourier-Mukai on K3 surfaces The spectral sequence (2.35) takes the form ( p,q p q ˆ (Φ (E)) =⇒ E if p + q = 2 E2 = Φ 0 otherwise .

(4.3)

Corollary 4.24. For every sheaf F on X, the sheaf Φ0 (F) is WIT2 with respect to ˆ (and hence IT0 by Proposition Φ, while the sheaf Φ2 (F) is WIT0 with respect to Φ 1.7). We can use the integral functor ΦQ X→M , when the family Q is universal, as a handy tool to study the two-dimensional components of the moduli space of stable sheaves on X. When v 2 = 0, Theorem 4.18 implies that the existence of a compact and irreducible component of MH (v) is equivalent to MH (v) itself being compact and irreducible. The following result was originally proved by Mukai [227]. Our proof relies on techniques developed in Chapter 1. Theorem 4.25. Assume v 2 = 0. If MH (v) is compact and there exists a universal family Q on X × MH (v), then MH (v) is a K3 surface. Proof. Let us write M for MH (v). By Proposition 4.21 the functor ΦQ X→M is an equivalence of triangulated categories. Hence, by Corollary 2.40 the induced map in cohomology f = f Q : H • (X, Q) → H • (M, Q) ∗ α · v(Q)) α 7→ πM ∗ (πX

is an isomorphism of Q-vector spaces. By Theorem 4.18, M is either an Abelian or a K3 surface, but the first case cannot occur since X is K3 and the cohomologies of the two kinds of surfaces are different.  ∨

Note that in view of Equation (1.13), the inverse map f −1 is f −1 = f Q (we need to take the dual in the derived category because Q may fail to be locally free). ˆ the moduli space MH (v). In the hypotheses of Theorem 4.25 we denote by X The functor Φ = ΦQ ˆ is a Fourier-Mukai transform by Proposition 4.21 and X→X Theorem 4.25. We now prove that the map f , defined in the proof of Theorem 4.25, induces e • (X, ˆ Z). e • (X, Z) and H an isometry between the lattices H Lemma 4.26. [227, Lemma 4.7] The Mukai vector v(F) is integral for any sheaf ˆ As a consequence, the Mukai vector v(E • ) is integral for any object F on X × X. • b ˆ E ∈ D (X × X).

4.2. Moduli spaces of sheaves and integral functors

121

Proof. This amounts to proving that ch(F) is integral. Let us denote by ψ i,j the ˆ have trivial canonical (i, j) K¨ unneth component of ψ = ch(F). As X and X bundles, (ψ 2,0 )2 and (ψ 0,2 )2 are even, so that ch2 (F) is integral. Now one has X ∗ ˆ 2i,4 = (ψ · πX (−1)i chi (Rj πX∗ ˆ, ˆ F) ⊗ $ ˆ td(X)) j

ˆ This implies that ch4 (F) and ψ 2,4 are where $ ˆ is the fundamental class of X. ˆ one shows that ψ 4,2 is integral. The integral. Interchanging the roles of X and X second statement is straightforward since X is smooth.  Proposition 4.27. [227, Thm. 1.5][242, Prop. 3.5] ∼H e • (X, ˆ Z). e • (X, Z) → 1. The map f yields a Hodge isometry f : H ˆ 2. f (v ∗ ) = $, ˆ where $ ˆ is the fundamental class of X. ∼ H 2 (X, ˆ Z). 3. f induces an isometry (v ⊥ /v, h , i) → Proof. 1. By Lemma 4.26, the Mukai vector v(Q) is integral and then the map f is defined over Z. Using the projection formula, one has Z Z ∗ ∗ ∗ w∗ f (u) = − πX hw, f (u)i = − ˆ (w )v(Q)πX (u) ˆ ˆ X×X ZX Z ∗ ∗ =− πXˆ (w)v(Q)∗ πX (u∗ ) = − f −1 (w)u = hf −1 (w), ui , ˆ X×X

X

as v(Q)∗ = v(Q∨ ). This shows that f is a Hodge isometry. ˆ we have ΦQˆ ∨ (Oy ) = Lj ∗ (Q∨ ) ' (Qy )∨ , where Qy = 2. For a fixed y ∈ X, y X→X ∨ Q|X×{y} . This in turn implies that ΦQ ˆ ((Qy ) ) ' Oy . By the commutativity of X→X ˆ the diagram (1.11) one has f (v ∗ ) = v(Oy ) = $. 3. Since the Mukai dual v 7→ v ∗ is an isometry, v ⊥ /Zv is isometric to (v ) /Zv ∗ . By the previous point, there are isometries ∗ ⊥

∼ H 2 (X, ∼$ ˆ Z) . (v ∗ )⊥ /Zv ∗ → ˆ→ ˆ ⊥ /Z$  Corollary 4.28. [227, p. 347] The map f yields a Hodge isometry between the ∼ T(X). ˆ ˆ i.e., f|T(X) : T(X) → transcendental lattices of X and X, ∼ ˜ 1,1 (X, Z) → Proof. Since v(Q) is integral, the map f provides Hodge isometries H ∼ T(X). b Z) and T(X) → ˆ ˜ 1,1 (X,  H

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Chapter 4. Fourier-Mukai on K3 surfaces

Orlov has proved that the existence of such an isometry between the trascendental lattices of two K3 surfaces X and Y is equivalent to the fact that X and Y have equivalent derived categories. We shall prove this result in Chapter 7 (Theorem 7.24).

4.3

Examples of transforms

In this section we consider the case where the “partner” of the K3 surface is a 2-dimensional moduli space of µ-stable locally free sheaves and the integral kernel is provided by a universal family on the product. Other Fourier-Mukai transforms for K3 surfaces will be introduced later on. We start with some technical results. Let X be an algebraic K3 surface with a polarization H. We have the following existence result (cf. [227, Thms. 5.1, 5.2]). Recall that a Mukai vector v is primitive if it is not an integer multiple of any other Mukai vector. Lemma 4.29. If v = (r, `, s) is a primitive Mukai vector with v 2 = 0 and r ≥ 1, there exists a µ-semistable simple sheaf E on X with v(E) = v. Moreover, for every divisor class D of X, E can be chosen so that D · c1 (F) D·` ≥ rk(F) rk(E) for every torsion-free rank 1 quotient sheaf F of E such that µ(E) = µ(F). We reproduce from [26] the following lemma. Lemma 4.30. Let E be a simple µ-semistable sheaf with v(E) = (2, `, s), ` · H = 0, v(E)2 = 0 and s odd. Then E is locally free. Proof. Note that v(E) is primitive. If E is not locally free, [227, Prop. 3.9] implies that E ∗∗ is rigid, i.e., Ext1 (E ∗∗ , E ∗∗ ) = 0, and that there is an exact sequence 0 → E → E ∗∗ → Ox → 0

(4.4)

for a point x ∈ X. It follows that ch2 (E ∗∗ ) = ch2 (E)+1, so that v(E ∗∗ ) = (2, `, s+1) and v(E ∗∗ )2 = −4. By [227, Prop. 3.2] E ∗∗ is not simple, hence is not stable. We then have a destabilizing sequence 0 → OX (D1 ) → E ∗∗ → IZ (D2 ) → 0

(4.5)

where IZ is the ideal sheaf of a zero-dimensional subscheme, and D1 , D2 are divisors of degree zero such that ` = D1 + D2 and D12 ≥ s − 1.

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123

The divisors D1 , D2 are not numerically equivalent, since we would have ` ≡ 2D1 and then D12 = s, which is absurd because s is odd; thus by the Hodge index theorem [22, 141], 0 > (D1 − D2 )2 = D12 + D22 − 2D1 · D2 = `2 − 4D1 · D2 = 4s − 4D1 · D2 , so that D1 · D2 > s. Since c2 (E ∗∗ ) = s + 1, the exact sequence (4.5) yields s + 1 = D1 · D2 + length(Z) > s + length(Z) so that length(Z) = 0 and D1 · D2 = s + 1. Then, (D1 − D2 )2 = −4 and we have dim Ext1 (O( D2 ), OX (D1 )) = −χ(X, OX (D1 − D2 )) = 0 , ∼ O (D ) ⊕ O (D ). Then, the exact sequence (4.4) implies which implies E ∗∗ → X 1 X 2 ∼ O (D ) ⊕ I (D ), which is absurd since ∼ that either E → Ix (D1 ) ⊕ OX (D2 ) or E → X 1 x 2 E is simple.  e 1,1 (X, Z), such that v 2 = 0 Let us now take a Mukai vector v = (r, `, s) in H ˆ = MH (v) of stable and r > 0. Moreover we assume that the moduli space X sheaves with Mukai vector v is compact and irreducible and that there is a universal ˆ So X ˆ is an algebraic K3. We show that X ˆ carries a natural family Q on X × X. polarization. e 1,1 (X, Z), the Hodge isometry The composition of the injection Pic(X) ,→ H ∼ • • 1,1 e (X, ˆ Z) and the projection of H (X, Z) onto its direct summand e (X, Z)→ H f: H ˆ defines a morphism µ : Pic(X) → Pic(X). ˆ This may be explicitly computed Pic(X) as 2,2 α) µ(α) = πX∗ ˆ (γ

(4.6)

unneth component of γ = ch(Q). where γ i,j is the (i, j) K¨ ˆ correspond to locally free sheaves, Proposition 4.31. Assuming that all points in X ˆ = −µ(H) ∈ Pic(X) ˆ is ample. the class H ˆ is ample by comparing it with the first Chern class of the Proof. We prove that H determinant bundle Lm = det(Φ(OX (mH)))−1 ⊗ det(Φ(OX (−mH))) , where Φ = ΦQ ˆ , which is known to be ample for m  0 by a theorem of X→X Donaldson [101, §5]. Indeed a simple computation using Grothendieck-Riemannˆ  Roch shows that c1 (Lm ) = mH. ˆ with a One can also give a transcendental proof of this fact by identifying H ˆ positive multiple of the class of the Weil-Petersson metric on X [24, Prop. 6].

124

4.3.1

Chapter 4. Fourier-Mukai on K3 surfaces

Reflexive K3 surfaces

The case of the so-called reflexive K3 surfaces provides an example of an explicit construction of a Fourier-Mukai transform for K3 surfaces. Definition 4.32. A K3 surface is reflexive if it carries a polarization H and a divisor 4 ` such that H 2 = 2, H · ` = 0, `2 = −12, and ` + 2H is not effective. Reflexive K3 surfaces are particular instances of the K3 surfaces mentioned in Example 4.6. Our first step is to prove that for any reflexive K3 surface X there is a ˆ of the moduli space of stable sheaves on X which is a K3 surface component X made of locally free µ-stable sheaves. Let us fix the Mukai vector v = (2, `, −3). Proposition 4.33. There exists a µ-semistable simple locally free sheaf E on X with v(E) = v such that ` · c1 (F) ≥ −6 for every torsion-free rank 1 quotient sheaf F of E of degree 0. Proof. By Lemma 4.29, taking D = `, there is µ-semistable simple sheaf E on X fulfilling the remaining conditions. Moreover, E is locally free by Lemma 4.30.  Proposition 4.34. There exists a stable locally free sheaf E on X with v(E) = v (so that the moduli space MH (X, v) is not empty). Moreover, every element in MH (X, v) is locally free. Proof. Let E be the sheaf provided by Proposition 4.33. If it is not stable, there exists an exact sequence 0 → OX (D1 ) → E → IZ (D2 ) → 0 , where IZ is the ideal of a zero-dimensional subscheme, D1 , D2 are divisors of degree 0, and D12 ≥ −5, so that D12 ≥ −4 since D12 is even. From ` = D1 + D2 , −1 = c2 (E) = D1 · D2 + length(Z) and ` · D2 ≥ −6, we obtain −4 ≤ D12 ≤ −5 + length(Z), so that length(Z) ≥ 1. Moreover D1 , D2 are not numerically equivalent, since we would have ` ≡ 2D1 and then D12 = −3, which is absurd; thus by the Hodge index theorem, 0 > (D1 − D2 )2 = −12 − 4D1 · D2 , so that D1 · D2 > −3 and length(Z) < 2. Then length(Z) = 1 and D12 = D22 = −4. Since ` + 2H = (D1 + H) + (D2 + H) and D1 + H, and D2 + H are linearly equivalent to nodal curves of degree 2, this contradicts the fact that ` + 2H is not effective. So the moduli space MH (X, v) is not empty. Moreover every element in  MH (X, v) is locally free by Lemma 4.30.

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125

Since the greatest common divisor mentioned in Theorem 4.20 is 1, we can apply Proposition 4.21 and Theorem 4.25 to get the following result. Proposition 4.35. If X is a reflexive K3 surface with polarization H, and v = (2, `, −3), then there is a locally free rank 2 universal sheaf Q on X × MH (v) → MH (v) making MH (v) a fine moduli scheme parameterizing locally free stable ˆ = MH (v) is a projective K3 surface and the sheaves with vector v. Moreover, X Q integral functor ΦX→Xˆ is a Fourier-Mukai transform. Note that since ˆ Z) ˆ + terms in H 4 (X, −f (H) = H one has ˆ 2 = f (H)2 = H 2 = 2 H

(4.7)

because f is an isometry.

4.3.2

Duality for reflexive K3 surfaces

We have chosen to call “reflexive” the K3 surfaces previously introduced because ˆ = MH (2, `, −3) is a reflexive they are self-dual in the sense that the moduli space X ˆ stable K3 surface as well. One can realize X as a moduli space of sheaves on X ˆ ˆ with respect to the natural polarization H; the universal family on X ×X, suitably normalized, will turn out to be isomorphic to Q∗ . To this end we need to strengthen a little bit our assumptions about the K3 surface X. ˆ the sheaf Qy = Q|X×{y} is a µ-stable locally free sheaf Since for every y ∈ X of degree zero, one has h0 (X, Qy ) = 0, h2 (X, Qy ) = 0 and h1 (X, Qy ) = 1. So the ˆ structure sheaf OX is IT1 and the sheaf R1 πX∗ ˆ Q is a line bundle on X. We can therefore normalize the universal bundle Q by setting R1 πX∗ ˆ Q = OX ˆ .

(4.8)

We shall henceforth assume that this normalization has been fixed. Definition 4.36. A K3 surface is strongly reflexive if it carries a polarization H and a divisor ` such that H 2 = 2, H · ` = 0, `2 = −12, and there are in X no nodal curves of degree 1 or 2. 4 Strong reflexivity is a generic condition. Indeed, the ample divisor H defines a double cover of P2 branched over a sextic; the image of a nodal curve of degree 1 is a line tritangent to the sextic, while the image of a nodal curve of degree 2 is a conic, tangent to the sextic at six points. Neither situation can arise in the general case. A coarse moduli space parameterizing strongly reflexive K3 surfaces, which

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Chapter 4. Fourier-Mukai on K3 surfaces

is an irreducible quasi-projective scheme of dimension 18, is constructed in [25]. On a generic Kummer surface one can choose divisors H and ` making it strongly reflexive. Strongly reflexive K3 surfaces are reflexive. Lemma 4.37. If X is a strongly reflexive K3 surface the class E = ` + 2H is not effective. Proof. Since E 2 = −4, if E is effective it is not irreducible and E = D + F for some nodal curve D. Then D · H = 3 and F · H = 1, so that F is also irreducible. It follows that F 2 ≥ −2. If F 2 ≥ 0, then D · F ≤ −1, so that D = F which is absurd. Thus, F 2 = −2 and F is a nodal curve of degree 1, a situation we are excluding.  ˆ = MH (v) On strongly reflexive K3 surfaces the sheaves in the moduli space X is Proposition 4.35 are µ-stable. Proposition 4.38. Any stable bundle E on X with v(E) = (2, `, −3) is µ-stable. Proof. If E is not µ-stable, it can be destabilized by a sequence 0 → OX (D1 ) → E → IZ (D2 ) → 0 , where IZ is the ideal of a zero-dimensional subscheme, D1 , D2 are divisors of degree 0, and D1 · D2 = −1 − length(Z). Since E is stable, D12 < −5

and

D22 > −5 + 2 length(Z) .

Moreover, since D2 6= 0, we have χ(X, OX (D2 )) ≤ 0, so that D22 ≤ −4. It follows that length(Z) = 0, and we have an exact sequence 0 → OX (D1 ) → E → OX (D2 ) → 0 , with D1 , D2 of degree 0 and D12 = −6, D22 = −4. Then, D2 + H is a nodal curve of degree 2, which is a contradiction.  Proposition 4.35 now takes the form Proposition 4.39. If X is a reflexive K3 surface with polarization H, and v = (2, `, −3), then there is a locally free rank 2 universal sheaf Q on X × MH (v) → MH (v) making MH (v) a fine moduli scheme parameterizing locally free µ-stable ˆ = MH (v) is a projective K3 surface and the sheaves with vector v. Moreover, X Q integral functor ΦX→Xˆ is a Fourier-Mukai transform.

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127

We start now an analysis which will culminate in the proof that the K3 ˆ Henceforth we surface X is itself a fine moduli space of µ-stable bundles on X. assume that X is strongly reflexive. ˆ Since H 2 (X, E(H)) = 0 and χ(E(H)) = 1, Let E = Qy for a point y ∈ X. the sheaf E(H) has at least one section. Lemma 4.40. For every section of E(H) there is an exact sequence 0 → OX → E(H) → Ix (` + 2H) → 0 . Moreover, dim H 0 (X, E(H)) = 1, so that the point x depends only on the sheaf E, and dim Ext1 (Ix (` + 2H), OX ) = 1. Proof. Given a section of E(H), we have an exact sequence 0 → OX → E(H) → K → 0. By taking double duals, we obtain the exact sequence 0 → OX (D) → E(H) → IZ ⊗ OX (` + 2H − D) → 0 ,

(4.9)

where Z is a zero-dimensional subscheme and D is an effective divisor of degree 1, so that D2 ≥ −2. Moreover, H − 2D 6≡ 0, so that Hodge index theorem implies that 4D2 − 2 < 0 and we have two cases, D2 = 0 and D2 = −2. If D2 = 0, then (H − 2D)2 = −2, so that either H − 2D or 2D − H is effective, which is absurd. Thus, D2 = −2, and D is a nodal curve of degree 1, a situation we are excluding. Hence D = 0 and length(Z) = 1. As far as the second statement is concerned, since H 0 (X, O(` + 2H)) = 0, one has H 0 (X, Ix (` + 2H)) = 0 so that from the sequence (4.9) we obtain dim H 0 (X, E(H)) = 1. Moreover, by Riemann-Roch we have H i (X, O(` + 2H)) = 0 also for i = 1, 2, so that Serre duality gives dim Ext1 (Ix (` + 2H), OX ) =  dim H 1 (X, Ix (` + 2H)) = 1. ˆ ˆ Corollary 4.41. The line bundle OX (H) is IT0 and O\ ˆ (−` − H), where X (H) ' OX 0,2 ˆ unneth part of the Chern character ch Q. −` = γ is the (0,2) K¨ Proof. By Lemma 4.40, OX (H) is IT0 and its Fourier-Mukai transform is a line bundle N . The sheaf N is identified by computing its first Chern class by Grothendieck-Riemann-Roch.  ˆ → X, Lemma 4.40 implies that there is a one-to-one set-theoretic map Ψ : X given by Ψ(y) = x, where x is the point determined by E = Qy . Proposition 4.42. The map Ψ is an isomorphism of schemes.

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Chapter 4. Fourier-Mukai on K3 surfaces

∗ ∗ \ Proof. The natural morphism πX ˆ N → Q⊗πX OX (H) where N = OX (H) provides a section τ ∗ ∗ −1 0 → OX×Xˆ → Q ⊗ πX OX (H) ⊗ πX → K → 0. ˆN

ˆ be the closed subscheme of zeroes of τ , and let p = π◦j : Z → X, Let j : Z ,→ X×X ˆ be the proper morphisms induced by the projections πX and pˆ = π ˆ ◦ j: Z → X πXˆ . ∼ X ˆ is an isomorphism of It is enough to show that the morphism pˆ: Z → ˆ → X. Now, one schemes and the map Ψ is the composite morphism p ◦ pˆ−1 : X τy ˆ easily sees that for every (closed) point y ∈ X, τ induces a section 0 → OX → E(H) → Ky → 0 of E(H). By Lemma 4.40, Ky ' Ip(y) (` + 2H) for a well-defined ˆ consists of a single point and point p(y) ∈ X. Then, every closed fiber of x ˆ: Z → X ˆ is x ˆ is a proper finite epimorphism of degree 1 by Zariski’s main theorem. Since X −1 smooth, pˆ is an isomorphism. Moreover one has Ψ = p ◦ pˆ . As a consequence of ˆ the fiber Ψ−1 (Ψ(y)) is a single point. Lemma 4.40, for every (closed) point y ∈ X ˆ is the scheme-theoretic image of Ψ, X ˆ → Ψ(X) ˆ is a finite epimorphism If Ψ(X) ˆ = 2 and Ψ(X) ˆ = X. The smoothness of of degree 1 as above, so that dim Ψ(X) X yields the result.  Corollary 4.43. Let E be a sheaf which fits into an exact sequence 0 → OX → E(H) → Ix (` + 2H) → 0 , where Ix is the ideal sheaf of a point x ∈ X. Then E is µ-stable and locally free ˆ and Ψ(y) = x. with v(E) = v = (2, `, −3) so that it defines a point y ∈ X Proof. According to Proposition 4.39, it is enough to prove that E(H) is stable, which is easily checked.  Lemma 4.40 suggests that the universal bundle Q can be obtained as an ˆ Let IΨ be the ideal extension of suitable torsion-free rank 1 sheaves on X × X. ˆ ,→ X × X ˆ of Ψ. sheaf of the graph ΓΨ : X 1 ∗ Lemma 4.44. The direct image πX∗ ˆ )] is a line ˆ [Ext (IΨ ⊗ πX OX (` + 2H), OX×X ˆ bundle L on X.

Proof. Write E = ` + 2H and OΨ = (ΓΨ )∗ OXˆ . Then, ∗ ∗ Ext1 (IΨ ⊗ πX OX (E), OX×Xˆ ) ' OΨ ⊗ πX OX (−E) . ∗ By Lemma 1, Ri πX∗ ˆ πX OX (−E) = 0 for i ≥ 0, hence, from the exact sequence ∗ ∗ ∗ 0 → IΨ ⊗ πX OX (−E) → πX OX (−E) → OΨ ⊗ πX OX (−E) → 0 ,

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129

∼ 1 ∗ ∗ we obtain πX∗ ˆ (OΨ ⊗ πX OX (−E)) → R πX∗ ˆ (IΨ ⊗ πX OX (−E)). But for every ˆ one has H 1 (X, IΨ ⊗ π ∗ OX (−E) ⊗ Oy = H 1 (X, Ix (−E)), where x = Ψ(y), y ∈ X, X and one concludes by Lemma 4.29 and by Grauert’s cohomology base change theorem.  ∗ ∗ −1 OX (` + 2H), πX )) has a section, It follows that the sheaf Ext1 (IΨ ⊗ πX ˆ (L so that there is an extension ∗ −1 ∗ ) → P → IΨ ⊗ πX OX (` + 2H) → 0 . 0 → πX ˆ (L

(4.10)

∗ ˆ Moreover, Lemma 4.40 implies that P ⊗πX OX (−H) is a universal sheaf on X × X; ∼ ∗ ∗ ˆ The sheaves L and G thus P → Q ⊗ πXˆ G ⊗ πX OX (H) for a line bundle G on X. are readily determined; by applying πX∗ ˆ to the sequence above, one obtains

∼ ˆ ˆ L−1 = O\ ˆ (−` − H) ⊗ G . X (H) ⊗ G → OX Now, by restricting the exact sequence (4.10) to a fiber π −1 (x), we obtain c1 (G) = ˆ Then we have −H. ∗ ∼ ˆ ˆ πX∗ ˆ (OΨ ⊗ πX OX (−E)) → OX ˆ (` + 2H)

(4.11)

and ˆ Proposition 4.45. The sequence of coherent sheaves on X × X ∗ ∗ ∗ ∗ ˆ ˆ ˆ 0 → πX ˆ (−`−2H) → Q⊗πX ˆ (−H)⊗πX OX (H) → IΨ ⊗πX OX (`+2H) → 0 ˆ OX ˆ OX

is exact. This implies that there is an exact sequence ˆ → Iy (`ˆ + 2H) ˆ →0 0 → OXˆ → F(H) ˆ = 1 and dim Ext1 (Iy (`ˆ + 2H), ˆ O ˆ ) = 1. so that dim H 0 (X, Q∗x (H)) X Proposition 4.45 allows us to compute the Chern character γ = ch Q of Q. In particular, we obtain that the (2, 2) K¨ unneth part of γ is ˆ + H ∪ `ˆ − ι , γ 2,2 = (` + 2H) ∪ H ˆ Z) is the element corresponding to the isometry Ψ∗ : where ι ∈ H 2 (X, Z) ⊗ H 2 (X, 2 2 ˆ ˆ = −π ˆ (γ 2,2 H), one has H (X, Z) → H (X, Z). Taking into account that H X∗ ˆ Moreover, Equation (4.11) gives Ψ∗ (−` − 2H) = `ˆ+ 2H. ˆ + 2`. ˆ From Ψ∗ (H) = 5H this we get the symmetric relations ˆ Ψ∗ (H) = 2`ˆ + 5H ˆ = Ψ∗ (2` + 5H) H

ˆ Ψ∗ (`) = −5`ˆ − 12H `ˆ = Ψ∗ (−5` − 12H)

(4.12)

ˆ A first We need to show that the sheaves Q∗x are µ-stable with respect to H. step is the following.

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Chapter 4. Fourier-Mukai on K3 surfaces

ˆ is reˆ equipped with the divisors H ˆ and `, Proposition 4.46. The K3 surface X, flexive. ˆ = 0. Moreover, since Proof. By Equation (4.12) we have `ˆ2 = −12 and `ˆ · H ∗ ˆ ˆ = −Ψ (` + 2H), this divisor has negative degree with respect to the ` + 2H  polarization Ψ∗ H, so that it is not effective. ˆ the most In order to prove that X is a moduli space of stable sheaves on X, natural thing to do would be to use Corollary 4.43. However we cannot do that ˆ is strongly reflexive. because we do not know a priori if the reflexive K3 surface X This problem is circumvented as follows. ˆ v ) of stable sheaves on X Proposition 4.35 implies that the moduli space MHˆ (ˆ ˆ (with respect to H) with Mukai vector vˆ is nonempty and connected and consists of locally free sheaves. v ) is µ-stable. Otherwise, proceeding as in the Actually any sheaf F in MHˆ (ˆ proof of Proposition 4.38, one could see that it can be destabilized by a sequence 0 → OXˆ (D1 ) → F → OXˆ (D2 ) → 0 , ˆ and D2 = −6, D2 = −4, D1 · D2 = −1. with D1 , D2 of degree 0 with respect to H 1 2 ˆ is a nodal curve and (D2 + H) ˆ · Ψ∗ H = 0, which is absurd because Then D2 + H Ψ∗ H is ample. Thus F is µ-stable. We may now proceed as in the proof Lemma 4.40; the sheaf F fits into an exact sequence ˆ → Iy (`ˆ + 2H) ˆ →0 0 → OXˆ → F(H) ˆ unless F is given by an extension for a well-defined point y ∈ X ˆ → IZ (`ˆ + 2H ˆ − D) → 0 0 → OXˆ (D) → F(H) ˆ = 1 and Z is a zero-dimensional closed where D is a nodal curve with D · H ˆ ˆ IZ (`ˆ + 2H ˆ − D)) = 0 for i ≥ 0 so subscheme of X. In the latter case, H i (X, 2 ˆ ˆ that length(Z) = 0 and −4 = (` + 2H − D) . Then, D · `ˆ = −3 and D · Ψ∗ H = ˆ = −1, which is absurd since Ψ∗ H is ample. Then, one has ˆ + 2`) D · (5H ˆ → Iy (`ˆ + 2H) ˆ → 0, 0 → OXˆ → F(H) ˆ ˆ and F ' Q∗ = ΦQ∗ ˆ (Ox ), with x = Ψ(y), since dim Ext1 (Iy (`+ for a point y ∈ X, x X→X ˆ O ˆ ) = 1. 2H), X

Q This implies that the Fourier-Mukai transform ΦX→ ˆ X , maps F to Ox . By v ) → X. applying Proposition 2.63 this induces an immersion of schemes MHˆ (ˆ Since the two schemes are irreducible and have the same dimension, they are isomorphic. As a consequence, we eventually obtain the sought-for result.

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131

ˆ (polarized by Theorem 4.47. X is a fine moduli space of µ-stable bundles on X ˆ −3), and the relevant universal sheaf is Q∗ . ˆ with invariants (2, `, H) The main result we have so far obtained in this section is the following: if X is a strongly reflexive K3 surface, we have realized X as a moduli space of locally free µ-stable sheaves on X itself. In this sense, one could say that strongly reflexive K3 surfaces are “self-dual.” We finish this section with the computation of the topological invariants of • b the Fourier-Mukai transform ΦQ ˆ (F ) of an object in D (X) in terms of those of X→X • F . The formula is obtained by the Riemann-Roch theorem, taking into account that we can compute the Chern character γ of Q from Proposition 4.45. Proposition 4.48. Let u = (ρ, c1 , σ) = (rk(F • ), c1 (F • ), rk F • + ch2 (F • )) be the • ˆ = v(ΦQ ρ, cˆ1 , σ ˆ ), one has Mukai vector of F • , and d = c1 · H. If u ˆ (F )) = (ˆ X→X ρˆ = −3ρ + 2σ + ` · c1 , b + (ρ + d − s)`ˆ − Ψ∗ (c1 ), cˆ1 = (` · c1 + 2d)H σ ˆ = 2ρ − 3σ − ` · c1 . • • Then χ(ΦQ ˆ (F )) = −χ(F ). X→X

We also recover that u ˆ2 = u2 , something we already know since the map f is an isometry. ˆ = (−1)i+1 χ(F) and c1 (F) · Corollary 4.49. If F is a WITi sheaf on X, then χ(F) ˆ · H. ˆ In particular, the Euler characteristic and the degree of H = (−1)i+1 c1 (F) WIT1 sheaves are preserved.

4.3.3

Homogeneous bundles

In Section 3.3, we have considered homegeneous bundles on Abelian varieties, namely, bundles that are invariant under translations. These can also be characterized as the coherent sheaves whose Abelian Fourier-Mukai transform is a skyscraper sheaf. By means of the Fourier-Mukai we have introduced in the previous sections we can generalize this notion to the case of strongly reflexive K3 surfaces. So we consider a strongly reflexive K3 surface X and the universal bundle Q given by Proposition 4.54. Lemma 4.50. If E is a µ-stable locally free sheaf of degree zero, and v(E ∗ ) 6= (2, `, −3), then E is IT1 . In particular, every zero-degree line bundle on X is IT1 . If E is a µ-stable coherent nonlocally free sheaf of degree zero, then E is IT1 unless ˆ and a there is an exact sequence 0 → E → Q∗ξ → OZ → 0, for a point ξ ∈ X zero-dimensional closed subscheme Z ,→ X.

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Chapter 4. Fourier-Mukai on K3 surfaces

ˆ we have H 2 (X, F ⊗ Qξ )∗ ' Hom(Qξ , F ∗ ). Since F and Proof. For every ξ ∈ X Qξ are µ-stable, if there exists a nonzero morphism Qξ → F ∗ , then it is an isomorphism, which is incompatible with the condition in the statement. The same argument also shows that H 0 (X, F ⊗ Qξ ) ' Hom(F ∗ , Qξ ) = 0, thus concluding the proof of the first claim. For the second claim, if E is a µ-stable nonlocally free sheaf of degree zero, for ∼ Hom(E, Q∗ ). Since E and Q∗ are µ-stable, ˆ we have H 2 (X, E ⊗ Qξ )∗ → every ξ ∈ X ξ ξ for any nonzero homomorphism f ∈ Hom(E, Q∗ξ ) there is an exact sequence f

0 → E → Q∗ξ → K → 0 ∼ O for a zerowith rk(K) = 0. Moreover K has degree zero, and then K → Z dimensional closed subscheme Z, a situation we are excluding. Then H 2 (X, E ⊗ ∼ Hom(Q∗ , E). If ˆ On the other hand, H 0 (X, E ⊗ Qξ )∗ → Qξ )∗ = 0 for every ξ ∈ X. ξ f ∈ Hom(Q∗ξ , E) is nonzero, we have as above f

0 → Q∗ξ → E → K0 → 0 with rk(K0 ) = 0. After dualizing we get again a contradiction. Then H 0 (X, E ⊗ ˆ thus concluding the proof.  Qξ )∗ = 0 for every ξ ∈ X, Definition 4.51. A coherent torsion-free sheaf E on X is homogeneous if there is a filtration by coherent sheaves 0 = E0 ⊂ E1 ⊂ · · · ⊂ Es = E such that every sheaf (Ei+1 /Ei )∗ is µ-stable with Mukai vector v, namely, it is a sheaf defining points of ˆ X. 4 Note that homogeneous sheaves are µ-semistable (since the quotients of their filtration are µ-stable). ˆ looking at X as a modAn analogous definition applies for sheaves on X, ˆ uli space of stable sheaves on X with universal bundle Q∗ . These homogeneous sheaves play, in a sense, the same role as homogeneous sheaves on Abelian surfaces described in Section 3.3. In that case, sheaves that admit a filtration by line bundles of zero degree are just homogeneous sheaves, that is, sheaves invariant under translations by points of the Abelian surface. The relevance of this definition is shown by the following result. Proposition 4.52. If T is a coherent sheaf on X with zero-dimensional support, it is IT0 , and its Fourier-Mukai transform Tˆ is homogeneous. Conversely, if E is a ˆ then it is WIT2 , and its Fourier-Mukai transform Eˆ has homogeneous sheaf on X, zero-dimensional support. Proof. The second statement is the dual of the first one. We prove the first statement by induction on the length m of the support of T . For m = 1, this reduces

4.3. Examples of transforms

133

to the statement that ΦQ ˆ (Ox ) ' Qx , which we already know. Otherwise, if p is X→X a point of the support, there is an exact sequence 0 → T 0 → T → Op → 0 , where T 0 has a zero-dimensional support of length m−1. By the inductive hypothesis, T 0 is IT0 and Tˆ0 is quasi-homogeneous. From the previous exact sequence, we see that T is IT0 , and that there is an exact sequence cp → 0 , 0 → Tˆ0 → Tˆ → O which implies that Tˆ is quasi-homogeneous as well.



Corollary 4.53. Let E be a µ-stable sheaf of degree zero. If E is locally free, it is either WIT2 or IT1 according to whether v(E ∗ ) = (2, `, −3) or not. If E is neither locally free nor IT1 , there is an exact sequence 0 → E → Q∗ξ → OZ → 0 , ˆ and a zero-dimensional closed subscheme Z ,→ X, so that for a point ξ ∈ X 0 1 Φ (E) = 0, Φ (E) is quasi-homogeneous and Φ2 (E) ' Oξ , where Φ = ΦQ ˆ. X→X

4.3.4

Other Fourier-Mukai transforms on K3 surfaces

Mukai’s construction Strongly reflexive K3 surfaces have been the first example of a class of K3 surfaces supporting a Fourier-Mukai transform. Another example was provided by Mukai [228]. The description we give here of that example is quite different from the original treatment by Mukai since we can take advantage of the results proved in Chapter 1 and in Section 4.1. Let X be an algebraic K3 surface, and assume there exist coprime positive integers r, s and a polarization H in X such that H 2 = 2rs. Let us consider the ˆ = MH (r, H, s) of stable sheaves on (X, H), with Mukai vector moduli space X ˆ = MH (r, H, s) is nonempty v = (r, H, s). Note that v 2 = 0. The moduli space X as a consequence of the following result. Proposition 4.54. If (X, H) is a polarized K3 surface, and v = v(r, H, s) is a ˆ = MH (v) is nonempty. primitive Mukai vector with v 2 = 0, the moduli space X ˆ is a K3 surface, and there is a universal family Q on X × X. ˆ Moreover, X Proof. The first claim is [227, Thm. 5.4]. To prove the second, note that the greatest common divisor of the numbers r, H 2 = 2rs and s is 1 since r and s are ˆ is compact. Again because r and coprime. As a consequence the moduli space X ˆ exists. By s are coprime, Theorem 4.20 applies, and a universal family on X × X ˆ Theorem 4.25, X is a K3 surface. 

134

Chapter 4. Fourier-Mukai on K3 surfaces

Q Remark 4.55. Note that by Proposition 4.21, both integral functors ΦX→ ˆ X and Q 4 ΦX→Xˆ are Fourier-Mukai transforms.

Let M be the lattice having generators e, f with intersection numbers e2 = −2r,

e · f = s + 1,

f2 = 0 .

By the surjectivity of the period map [22, Chap. 8], there exists a K3 surface X with Picard lattice isomorphic to M (cf. also [93]). Lemma 4.56. The class h = e + rf ∈ Pic(X) is ample. Proof. The Grothendieck-Riemann-Roch formula shows that, possibly after replacing the pair {e, f } with the pair {−e, −f }, the class f may be assumed to be effective. Moreover, the generic element in the linear system |f | is an irreducible elliptic curve. Let C be the class of an irreducible curve, and let C = me + nf ; then, m ≥ 0 as C · f = ms. Notice that h · C = ns. Since C 2 ≥ −2, we have ms(n − mr) ≥ −1.

(4.13)

If m > 0, we have n > 0 by (4.13), while if m = 0, C = nf implies again n > 0. Therefore h · C > 0, and since also h2 = 2rs > 0, h is ample by Nakai’s criterion.  Notice that in this K3 surface X there are divisors of degree 2s (for instance, D = 2f ). The previous results apply to the K3 surface X (note in particular that ˆ = Mh (r, h, s) of stable sheaves h2 = 2rs), so we may consider the moduli space X Q on X. Then for every m ∈ Z, the complex ΦX→Xˆ (OX (mh − D)) has rank zero. As a consequence, the determinant bundle −1 Lm = (det ΦQ ⊗ (det ΦQ ˆ (OX (mh − D)) ˆ (OX (−mh − D)) X→X X→X

is independent of the choice of the universal family. Moreover, for m big enough, Lm is ample [101]. ˆ in X ˆ by letting Let us define a divisor class h ˆ = −µ(h) + 2sφ , h where µ is the morphism defined in Equation (4.6) and φ is the K¨ unneth (0,2) part of the Chern class c1 (Q). A direct computation by the Grothendieck-RiemannRoch theorem yields ˆ, Lm = 2m h ˆ is ample. so that h

4.3. Examples of transforms

135

Now, following Mukai [228], and in analogy with the result in Proposition ˆ Let G the sheaf on X associated with the 4.45, we construct a morphism X → X. presheaf U Ext1X×U (I∆ ⊗ π1∗ OX (e + f ), π1∗ OX (f )) where I∆ is the ideal sheaf of the diagonal ∆ ⊂ X × X. The morphisms π1 , π2 are here the projections onto the factors of X × X. Since Hom(Ix (e), OX ) = Ext2 (Ix (e), OX ) = 0 for every x ∈ X, the sheaf G is locally free, and G ⊗ k(x) ' Ext1 (Ix (e), OX ). Then there is a coherent sheaf E on X × X fitting into an exact sequence 0 → π1∗ OX (f ) ⊗ π2∗ G ∗ → E → I∆ ⊗ π1∗ OX (e + f ) → 0 . For every x ∈ X, let Ex = E|X×{x} . Note that Ex fits into the exact sequence 0 → OX (f )⊕(r−1) → Ex → Ix (e + f ) → 0 .

(4.14)

One has v(Ex ) = (r, h, s). Moreover, Proposition 4.57. For every x ∈ X, the sheaf Ex is locally free and h-stable. Proof. Starting for the exact sequence (4.14) one proves that Ex ' (Ex )∗∗ , so that Ex is locally free. To prove the second claim, let F be a proper torsion-free quotient ⊕(r−1) → Ex (−f ) → F, of Ex (−f ) and let F0 be the image of the composition OX where the first arrow is the morphism in the sequence (4.14). We have a diagram 0

0

0

0

 / K1

 / K2

 / K3

/0

0

 / O⊕(r−1)

 / Ex (−f )

 / Ix (e)

/0

0

 / F0

 /F

 / F 00

/0

 0

 0

 0

X

The sheaf K3 either has rank 1, or is zero. In the second case, rk(F0 ) = rk(F) − 1, and c1 (F) = c1 (F0 ) + e, and µ(F) ≥

h·e h·e > = µ(Ex (−f )) , rk(F) r

136

Chapter 4. Fourier-Mukai on K3 surfaces ⊕(r−1)

so that Ex is stable (note that deg(F0 ) ≥ 0 since F0 is a quotient of OX

).

If rk(K3 ) = 1, then rk(F0 ) = rk(F). Now, let us notice that det(F0 ) is ⊕(r−1) effective, since by taking the ρ-th exterior power of the morphism OX → F0 in the first column of the previous diagram, a nonzero morphism  one obtains  r−1 ⊕N O → det(F0 ) (here ρ = rk(F0 ), and N = ). ρ Let c1 (F0 ) = me + nf . Due to Lemma 4.59, one has µ(F) ≥ µ(F0 ) =

m(rs − 1) h · (me + nf ) ≥ ≥ ms . rk(F) rk(F)

Now, if m ≥ 1 then µ(F) > s − 1 = µ(Ex (−f )) so that Ex is stable. So let m = 0. As we have already noticed, that the generic member in the linear system |f | is an irreducible elliptic curve, hence h0 (OX (nf )) = ⊕(ρ+1) n + 1. In view of Lemma 4.60 there is a morphism OX → F0 which is surjective out of a finite sets of points. The kernel of this morphism is isomorphic to (det(F0 ))−1 , so that we obtain an exact sequence ⊕(ρ+1)

0 → (det(F0 ))−1 → OX

→ F00 → 0 .

The associated cohomology long exact sequence contains the segment ⊕(ρ+1)

0 → H 1 (X, F00 ) → H 2 (X, (det(F0 ))−1 ) → H 2 (X, OX

) → H 2 (X, F00 ) → 0 .

We may assume that F is stable, and then F0∗ is stable as well. Since deg(F0∗ ) ≤ 0 we have H 0 (X, F0∗ ) ' H 2 (X, F0 ) = 0. As F0 /F00 is supported on points, we also have H 2 (X, F00 ) = 0. As a consequence, n + 1 = h0 (det(F0 )) ≥ ρ + 1, so that µ(F) ≥ µ(F0 ) =

h · nf ≥ f · h = s + 1 > µ(Ex (−f )) . rk(F) 

So we have a “classification morphism” Ψ, mapping x ∈ X to the sheaf Ex ˆ = Mh (r, h, s). in X Corollary 4.58. The classification morphism Ψ is an isomorphism. Proof. It is enough to prove that Ex ' Ey implies x = y. Indeed if this is the ˆ is connected, Ψ is an isomorphism. case, Ψ is an open embedding, and since X ∼ E maps the subsheaf To prove this claim, let us note that an isomorphism Ex → y ⊕(r−1) ⊕(r−1) OX ⊂ Ex to the subsheaf OX ⊂ Ey , since Hom(OX , OX (e)) = 0. Then  Ix ' Iy , and x = y.

4.3. Examples of transforms

137

We prove now the two lemmas that have been used to prove Proposition 4.57. Lemma 4.59. If a divisor class D = me + nf is effective, then n ≥ 0 and h · D ≥ (rs − 1)m. Proof. The proof is a simple computation. See [228], Lemma 3.3.



Lemma 4.60. Let X be a smooth projective surface X such that H 1 (X, OX ) = 0. Let G be a sheaf on X whose torsion is supported on points, and assume that there is a morphism V ⊗ OX → G (where V is a finite-dimensional vector space) whose cokernel is supported on points. Then there is a morphism V 0 ⊗ OX → G (where V 0 is a vector space of dimension rk(G) + 1) whose cokernel is supported at a finite set of points. Proof. We prove this result first for rk(G) = 1 and then extend it by induction. We may assume that dim V ≥ 2. Composing the morphism V ⊗ OX → G with the natural morphism G → G ∗∗ , we obtain a morphism V ⊗ OX → G ∗∗ whose cokernel is supported on points, since the torsion of G is supported on points. This means that there exist s1 , s2 ∈ V whose corresponding divisors, when they are regarded as sections of the line bundle G ∗∗ , intersect at finite number of points. If V 0 = hs1 , s2 i, the cokernel of the morphism V 0 ⊗ OX → G is supported on points. To trigger the induction mechanism we need to show that if rk(G) ≥ 2, a s → G → G 0 → 0 such generic element of s ∈ V induces an exact sequence 0 → OX − that the torsion of G 0 is supported on points. To prove this, let U ⊂ X be the open subset where G is locally free and the morphism V ⊗ OX → G is surjective (under our hypotheses, the complement of U is a finite set of points). By [119, Example 12.1.11], for a generic s ∈ V the zero locus Z of s|U is empty or is a finite number of points. Then G 0 is locally free on U − Z. Now we can draw a commutative diagram 0

/ OX

0

/ OX

s

/ V ⊗ OX

/ W ⊗ OX

/0

 /G

 / G0

/0

where W and V are vector spaces of dimension rk(G) and rk(G) + 1, respectively, and the cokernel Q0 of the rightmost vertical arrow is supported on points by the induction hypothesis. The morphism V ⊗ OX → G exists because a morphism OX → G 0 can be lifted to a morphism OX → G since H 1 (X, OX ) = 0. The cokernel  of the morphism V ⊗ OX → G is isomorphic to Q0 . In this way, we have constructed another example of a family of K3 surfaces that are “self-dual,” i.e., each of them is isomorphic to a component of the moduli

138

Chapter 4. Fourier-Mukai on K3 surfaces

space of stable bundles on it, having suitable topological invariants. Moreover, these components of the moduli space are fine, and the corresponding universal sheaves define Fourier-Mukai transforms (cf. Remark 4.55). A Fourier-Mukai transform by extension of the kernel It is not difficult to construct new Fourier-Mukai transforms out of given ones, for instance by taking extensions of the kernels. We give here an example based on strongly reflexive K3 surfaces (taken from [73] with some changes). ˆ Q) = 1, The normalization Rˆ π∗ Q ' O ˆ [−1] implies that dim H 1 (X × X, X

as the Leray spectral sequence shows immediately. So there is a unique nontrivial extension (4.15) 0 → Q → U → OX×Xˆ → 0 . Proposition 4.61. The restrictions of the sheaf U to the varieties X × {ξ}, with ˆ and {p} × X, ˆ with p ∈ X, are all stable. ξ ∈ X, Proof. We consider only the second type of restriction, since the proof is the same in the two cases. The sheaf Uξ = UX×{ξ} is µ-semistable with vanishing degree. Let F be a destabilizing proper subsheaf of Uξ , which we may assume to be stable. Then χ(F)/ rk(F) ≥ χ(Uξ )/rk(Uξ ) = 13 . Let f : F → OX be the composite morphism. One necessarily has f 6= 0, otherwise there would be a nonzero morphism F → Qξ and then χ(F)/ rk(F) ≤ − 12 , which contradicts the previous inequality. But then then F ' OX , which implies in turn Uξ ' OX ⊕ Qξ . From this we get ˆ 0 (Uξ ) = 0 , Φ

ˆ 1 (Uξ ) ' OX , Φ

ˆ 2 (Uξ ) ' Oξ . Φ

But then the spectral sequence (2.35) (or, to be more precise, the spectral sequence ˆ degenerates at the second step, yielding a associated to the composition Φ ◦ Φ) contradiction.  By the general theory this implies that the kernel U gives rise to a FourierU∗ ˆ Mukai transform Ψ = ΦU ˆ . Let Ψ = ΦX→ ˆ X be the inverse transform. The transX→X form Ψ has some nice features. ˆ satisfy the Proposition 4.62. The Fourier-Mukai transform Ψ and its inverse Ψ following properties. 1. Ψ(OX ) ' OXˆ [−2]. ˆ ˆ ξ ) ' Q∗ [−1] for all ξ ∈ X. 2. Ψ(I ξ

4.4. Preservation of stability

139

Proof. The first claim is proved by inspection of the long exact sequence that one obtains by applying the functor Rˆ π∗ to the sequence (4.8). To prove the second ˆ ξ ) ' U ∗ . Then ˆ ) ' O by the previous result, and that Ψ(O claim note that Ψ(O ˆ X ξ X ˆ to the exact sequence 0 → Iξ → O ˆ → Oξ → 0, we obtain by applying Ψ X

0 → OX →

Uξ∗

ˆ1

→ Ψ (Iξ ) → 0

which proves the claim.



ˆ The second property implies that the ideal sheaf IZ of a zero-cycle Z in X ˆ is IT1 with respect to Ψ and its transform is the dual of the quasi-homogeneous bundle on X corresponding to Z.

4.4

Preservation of stability

In this section we study the behavior of µ-(semi)stable sheaves on reflexive K3 surfaces under the Fourier-Mukai transform. Our treatment is inspired by techniques developed by Maciocia for Abelian surfaces [203] already described in Section 3.5.2. Another approach which involves transcendental techniques will be developed in Chapter 5. Lemma 4.63. Let E be a coherent sheaf on X. 1. If E is IT0 , then deg(E) ≥ 0, and deg(E) = 0 if and only if E has zerodimensional support. 2. If E is WIT2 , then deg(E) ≤ 0, and deg(E) = 0 if and only if E is a quasihomogeneous sheaf. b = (−1)i+1 deg(E) for a WITi sheaf by Corollary 4.49. Proof. We recall that deg(E) 1. Let us consider the exact sequence 0 → T → E → F → 0,

(4.16)

where T is the torsion subsheaf of E. Then deg(T ) ≥ 0 so that deg(F) ≤ deg(E) with equality if and only if T is supported in dimension zero. Suppose that deg(E) ≤ 0; then deg(F) ≤ 0. Since T has rank zero, H 2 (X, T ⊗ Qξ ) = 0 for every ˆ so that Φ2 (T ) = 0 and the exact sequence (1.10) applied to (4.16) proves ξ ∈ X, b = − deg(F) ≥ 0. that F is IT0 . Then Fb is WIT2 and locally free of degree deg(F) This implies that ∼ H 2 (X, ˆ Fb ⊗ Q∗ )∗ 6= 0 b Qp ) → Hom(F, p for every point p ∈ X, so that there exists a nonzero morphism Fb → Qp .

140

Chapter 4. Fourier-Mukai on K3 surfaces

We prove that Fb is µ-stable. Suppose indeed that there is a destabilizing sequence 0 → L → Fb → M → 0 , (4.17) b ≥ 0. It is enough to see where L is a stable locally free sheaf with µ(L) ≥ µ(F) 2 b 0 (M) ' Φ b 1 (L), which is b that Φ (L) = 0 because in this case L is WIT1 and Φ 0 1 2 b b b absurd since Φ (M) is WIT2 and Φ (L) is WIT1 . If Φ (L) 6= 0, there exists a ˆ L ⊗ Q∗ ) 6= 0, so that there is a nonzero morphism point p ∈ X such that H 2 (X, p

f : L → Qp . This implies µ(L) ≤ µ(Qp ) = 0, and then µ(L) = 0 and f is an b 2 (L) ' Op , and applying Φ b to the exact sequence (4.17) we isomorphism. Then Φ 0 1 b b b 2 (M) → 0. Thus Φ b 1 (M) ' Op , get Φ (M) = 0 and 0 → Φ (M) → Op → F → Φ because F is torsion-free. We now consider the spectral sequence given by Equation b q (M), E 2 ' M, and E p+q = 0 for p + q 6= 2. Since (4.3); one has E2p,q = Φp Φ ∞ ∞ 2,0 E2 = 0, any nonzero element in E20,1 ' Φ0 (Op ) ' Qp is a cycle which survives 1 6= 0, which is absurd. to infinity; thus E∞ Since Fb and Qp are both µ-stable, the existence of a nonzero morphism b b = 0. Then deg(F) = 0, so that deg(T ) = 0; as a F → Qp implies that µ(F) consequence, T has zero-dimensional support, and the same is true for the sheaf E. b ≥ 0 with equality if and only 2. If E is WIT2 , then Eb is IT0 , so that deg(E) b ˆ if E has zero-dimensional support, by 1 on X. Hence, by Proposition 4.52 we have deg(E) ≤ 0 with equality if and only if E is quasi-homogeneous.



Proposition 4.64. Let E be a coherent sheaf on X. If E is µ-semistable and WIT1 with deg(E) = 0, then its Fourier-Mukai transform Eb is µ-semistable. b = 0, if Eb is not µ-semistable, there is an exact sequence Proof. Since deg(E) 0 → F → Eb → G → 0 ,

(4.18)

where G is torsion-free and deg(F) ≥ 0 ≥ deg(G). In general, we do not get strict inequalities because we cannot assume that Eb is torsion-free; the equalities deg(F) = 0 = deg(G) hold only if Eb has torsion and F has zero-dimensional b 0 (F) = Φ b 2 (G) = 0 support. The long exact sequence 1.10 applied to (4.18) gives Φ and exact sequences b 0 (G) → Φ b 1 (F) → K → 0 , 0→Φ

b 1 (G) → Φ b 2 (F) → 0 . 0→K→E →Φ

In particular, F is not IT0 and, by Proposition 4.52, its support is not zerodimensional, whence deg(F) > 0. Thus Eb is torsion-free. b 2 (F) is IT0 by Corollary 4.24, so that Lemma b 0 (G) is WIT2 and Φ Now, Φ 0 b 2 (F)) ≥ 0. Then deg(Φ b 1 (F)) = deg(F) + b (G)) ≤ 0 and deg(Φ 4.63 implies deg(Φ 2 1 0 b (F)) − deg(Φ b (G)) > 0, thus contradictb (F)) > 0, so that deg(K) = deg(Φ deg(Φ ing the semistability of E.



4.4. Preservation of stability

141

Lemma 4.65. If F is locally free and T has zero-dimensional support, every exact sequence 0 → F → K → T → 0 splits. Proof. By local duality the sheaves ExtiOX (T , F) vanish for i 6= 2. Then Ext1 (T , F) = 0 and the exact sequence splits.  Corollary 4.53 characterizes the µ-stable IT1 sheaves of degree zero on X. The next proposition and its corollaries study the preservation of stability of such sheaves. Proposition 4.66. Let E be a coherent µ-stable IT1 locally free sheaf of degree zero on X. The Fourier-Mukai transform Eb is µ-stable. Proof. Since Eb is locally free, if it is not µ-stable it can be destabilized by a sequence 0 → F → Eb → G → 0 , b and G is torsion-free. where F is µ-stable and locally free, rk(F) < rk(E) We have two cases: b 2 (G) = 0 and two exact sequences (a) F is IT1 . We have Φ b 0 (G) → Fb → K → 0 , 0→Φ

b 1 (G) → 0 . 0→K→E →Φ

(4.19)

b 0 (G)) ≤ 0, so that b 0 (G) is WIT2 and deg(Φ Again, Φ b − deg(Φ b 0 (G)) ≥ 0 . deg(K) = deg(F) Since E is µ-stable, the only possibility is deg(K) = 0 and rk(K) = rk(E), so that b 1 (G)) = 0, Φ b 1 (G) has zero-dimensional support, hence b 1 (G)) = 0. Since deg(Φ rk(Φ b 0 (G)) = it is IT0 by Proposition 4.52. On the other hand, deg(K) = 0 implies deg(Φ 0 b 0, and then Φ (G) is quasi-homogeneous, by Lemma 4.63 again. The first sequence in (4.19) shows that K is WIT1 and induces the exact sequence b 0 (G)) → 0 . b → Φ2 (Φ 0→F →K b 0 (G)) has zero-dimensional support, and since F is By Proposition 4.52, Φ2 (Φ b is torsion-free, Lemma 4.65 forces Φ b 0 (G) = 0, and then G is locally free and K 1 b WIT1 . But then Φ (G) is WIT1 as well, and, since we have proved that it is IT0 , b we have G = 0, which is absurd as rk(F) < rk(E). (b) F is not IT1 . Then, by Corollary 4.53 and Proposition 4.52, F is WIT2 ∼ κ(p) for a closed point p ∈ X. The long exact sequence (1.10) shows that and Fb → G is WIT1 and yields the exact sequence b 1 (G) → κ(p) → 0 . 0→E →Φ b 1 (G) is WIT1 and This sequence splits by Lemma 4.65 , which is absurd since Φ E ⊕ κ(p) is not. 

142

Chapter 4. Fourier-Mukai on K3 surfaces

Corollary 4.67. Let E be a IT1 torsion-free sheaf of degree zero on X. If E is strictly µ-semistable, then its Fourier-Mukai transform Eb is strictly µ-semistable. Proof. Eb is µ-semistable by Proposition 4.64. Since E is not µ-stable, Proposition ˆ implies that if Eb is µ-stable it cannot be IT1 . Then by Corollary 4.53 4.66 for X b  E is WIT2 , which is absurd. Corollary 4.68. If E is a nonlocally free IT1 µ-stable torsion-free sheaf of degree zero on X, its transform Eb is not µ-stable. Proof. The sheaf Eb is locally free and WIT1 . If it is µ-stable, by Corollary 4.53 it  is IT1 . But this contradicts the fact that E is not locally free. The moduli space MH (v) of H-stable bundles on a K3 (or Abelian) surface X with Mukai vector v carries a naturally defined symplectic holomorphic structure, defined as follows. If [F] is a point in MH (v), then tangent space T[F ] (MH (v)) may be identified with the vector space Ext1 (F, F). The cup product provides a skew-symmetric map Ext1 (F, F) ⊗ Ext1 (F, F) → Ext2 (F, F). Moreover one has a trace morphism Ext2 (F, F) → Ext2 (OX , OX ) dual to the natural morphism Hom(OX , OX ) → Hom(F, F). Since Ext2 (OX , OX ) ' C, this defines a holomorphic 2-form on MH (v) which turns out to be closed and nondegenerate, thus defining a holomorphic symplectic 2-form. Remark 4.69. In Section 3.5.3, we introduced symplectic structures of moduli spaces of sheaves on surfaces in terms of the Yoneda pairing on the Ext groups. Since the Fourier-Mukai transform is well behaved with respect to these groups, it should induce symplectomorphisms between the moduli spaces, as we indeed were able to prove in the case of Abelian surfaces. The same happens in the case of (strongly reflexive) K3 surfaces. Let X be an H-polarized strongly reflexive K3 µ surface, fix a Mukai vector v, and let MH (v) be the subset of MH (v) formed by locally free µ-stable sheaves on X. We also assume that v = (r, c, s) is such that c µ (v) is nonempty. All sheaves parameterized by has degree zero, c · H = 0, and MH µ points in MH (v) are IT1 with respect to the Fourier-Mukai transform ΦQ ˆ . By X→X µ (v) → Proposition 4.66, the Fourier-Mukai transform defines a morphism ς : MH µ ˆ Z) is given by Proposition 4.48. Again, v ), where the Mukai vector vˆ ∈ H • (X, MH ˆ (ˆ one proves that ς is symplectic. 4

4.5

Hilbert schemes of points on reflexive K3 surfaces

As a geometric application of the Fourier-Mukai transform on strongly reflexive K3 surfaces, we show that the Hilbert scheme Hilbn (X) of such a surface X (parameterizing zero-dimensional subschemes of X of length n) may be realized as a

4.5. Hilbert schemes of points on reflexive K3 surfaces

143

moduli space of stable bundles on X. For the definition of the Hilbert scheme see [135], or [155] for a more modern and readable account. Since X is a projective smooth surface, Hilbn (X) is smooth and projective as well. We shall prove the following result. Theorem 4.70. For any n ≥ 1, the Hilbert scheme Hilbn (X) is isomorphic to the ˆ 1 − 3n) of H-semistable ˆ ˆ with sheaves on X moduli space Mn = MHˆ (1 + 2n, −n`, ˆ Mukai vector (1 + 2n, −n`, 1 − 3n). As a matter of fact, one sees that all points of Mn correspond to stable locally free sheaves. The result established by Theorem 4.70 can be compared with several results about the birationality of the Hilbert scheme of points of a polarized K3 surface (X, H) with a moduli space of H-stable sheaves of X. For instance, in [297] 2 2 a birational map MH (2, 0, −1 − n2 H 2 ) → Hilb2n H +3 (X) is constructed. Now we prove Theorem 4.70. We shall follow [73]. Let Z be a zero-cycle in X. The standard tricks show that the ideal sheaf IZ is IT1 , so that by applying the Fourier-Mukai transform to the sequence 0 → IZ → OX → OZ → 0 we get bZ → IbZ → O b → 0. 0→O X

(4.20)

Lemma 4.71. The FM transform IbZ is stable. Proof. Since ch(IbZ ) = (1, 0, −n), by the formulas in Proposition 4.48 we obtain rk IbZ = 1 + 2n,

ch2 IbZ = −5n ,

and moreover, denoting by P (IbZ ) = χ(IbZ )/ rk IbZ , we have P (IbZ ) =

1 2−n >− . 1 + 2n 2

Let A be a destabilizing subsheaf of IbZ , which we may assume to be stable with a torsion-free quotient. Then we have P (A) ≥ P (IbZ ) > − 12 . Let f denote the composite A → IbZ → OXˆ . There are two cases: bZ . Let gk : A → Qp be the composition (i) f = 0. Then there is a map A → O k of this map with the canonical projection onto Qpk . Since P (A) > − 12 = P (Qpk ) and both sheaves are stable, we obtain gk = 0 for all k, which is absurd. (ii) f 6= 0. We divide this into two further cases: rk A = 1 and rk A > 1. If rk A = 1 we have A∗ ' OXˆ ; hence the sequence (4.20) splits, which c contradicts the inversion theorem IbZ ' IZ .

144

Chapter 4. Fourier-Mukai on K3 surfaces If rk A > 1, we consider the exact sequences h

→B→0 0 → K1 → A −

and

0 → B → OXˆ → K2 → 0,

where rk K2 = 0, 1. If rk K2 = 1 then B = 0, i.e., f = 0 which is absurd, so that rk K2 = 0, and B has rank one. We have an exact commuting diagram 0O

0O

0O

0

/ K3 O

/ K4 O

/ K2 O

/0

0

/O bZ O

/ Ib Z O

/ Oˆ OX

/0

g

0

(4.21)

h0

/ K1 O

/A O

/B O

0

0

0

/0

with µ(K1 ) = 0, 0 < rk K1 < 2n and f = h0 ◦ h. bZ is µ-stable, but this is a contradiction. For n > 1, we may If n = 1, then O bZ . Then K3 assume that K1 is µ-semistable so that it is a direct summand of O is locally free and rk K1 ≥ 2. Moreover, µ(B) ≤ 0 because B injects into OXˆ , and µ(B) ≥ 0 because µ(K1 ) ≤ 0. Then µ(B) = µ(K1 ) = 0. Since K3 is locally free, the support of K2 is not zero-dimensional. So µ(B) = 0 implies K2 = 0 and K3 ' K4 . Finally, we consider the middle column in (4.21)). The sheaf A has rank greater than 2, and is stable, so that it is IT1 . But IbZ is WIT1 while K4 is WIT2 .  Then A ' IbZ , but this is a contradiction. Note that IbZ is never µ-stable because (4.20) destabilizes it. ˆ 1 − 3n) of stable sheaves on Let Mn be the moduli space MHˆ (1 + 2n, −n`, n ˆ X. The previous construction yields a map Hilb (X) → Mn which is algebraic because Fourier-Mukai transform yields a natural isomorphism of moduli functors and so gives rise to an isomorphism of (coarse or fine) moduli schemes. This map is injective due to the inversion theorem of the Fourier-Mukai transform. We shall now show that this map is surjective as well. Lemma 4.72. Any element F ∈ Mn is WIT1 . Proof. Since P (F) > − 12 and P (Qp ) = − 12 , there is no map F → Qp . This means ˆ F ⊗ Q∗ ) = 0. We consider now nonzero morphisms Qp → F. Any that H 2 (X, p

4.6. Notes and further reading

145

such map is injective; otherwise it would factorize through a rank 1 torsion-free sheaf B with µ(B) > 0 (because Qp is µ-stable) and µ(B) ≤ 0 (because F is µ-semistable), which is impossible. Then Qp is a locally free element of a JordanHolder filtration of F. Since any such filtration has only a finite number of terms, and the associated grading gr(F)∗∗ is unique, there is only a finite number of p’s giving rise to nontrivial morphisms, i.e., Hom(Qp , F) ' H 0 (X, F ⊗ Q∗p ) does not vanish only for a finite set of points p. This suffices to prove that F is WIT1 due to [227, Prop. 2.26].  Lemma 4.73. The Fourier-Mukai transform Fb of F is torsion-free. b so one has an exact sequence Proof. Let T be the torsion subsheaf of F, 0 → T → Fb → G → 0 .

(4.22)

Since T is supported at most by a divisor, and Fb is WIT1 , the sheaf T is WIT1 as well. Moreover deg(T ) ≥ 0. If deg T = 0, then T is IT0 , i.e., T = 0. Hence, we assume deg(T ) > 0. The rank 1 sheaf G is torsion-free and, by embedding it into its double dual, we see that it is IT1 . Then, applying the inverse ˆ to (4.22) we get 0 → Tb → F → Gb → 0. Since F is Fourier-Mukai transform Φ µ-semistable, we see that deg T = deg Tb ≤ 0, which is a contradiction.  Now the Chern character of Fb is (1, 0, −n), so that Fb is the ideal sheaf of a zero-dimensional subscheme of X of length n. We have therefore shown that the Fourier-Mukai transform surjects as a map Hilbn X → Mn . Altogether, this establishes Theorem 4.70. This theorem and its proof have some immediate consequences which we can state in the following proposition, where n is any positive integer. Proposition 4.74. Let X be a K3 surface, and let H, ` be divisors that make it strongly reflexive. The moduli space Mn of H-stable sheaves on X with Chern character (1 + 2n, −n`, −5n) is connected and projective. All its points correspond to locally free sheaves, and it contains no µ-stable sheaves. Proof. Only the last claim has not yet been demonstrated. To prove it one uses the fact that any µ-semistable sheaf of the given Chern character admits a surjection to OX and so fits into a sequence of the form (4.20). 

4.6

Notes and further reading

More on preservation of stability. A more general theorem about preservation on µ-stability on K3 surfaces has been given in [154]. This result requires the

146

Chapter 4. Fourier-Mukai on K3 surfaces

introduction of some categories naturaly attached to the choice of an ample divisor on X; this goes beyond the scope of this chapter, see however Remark D.28. The preservation of twisted Gieseker stability under the Fourier-Mukai transform has been shown by Yoshioka also in the case of K3 surfaces [294, 295, 296]. See also Section D.3.2. Hilbert schemes. Our result about the isomorphism of some moduli spaces of stable sheaves on a K3 surface X, and Hilbert schemes of points of X, is a particular case of general results about the birationality of such spaces proved by Zuo [297], Qin [255], G¨ottsche and Huybrechts [129] and O’Grady [238]. Fourier-Mukai partner K3 surfaces. In the examples of Fourier-Mukai functors constructed in this chapter, the two “partner” K3 surfaces turn out to be isomorphic. This is not necessarily the case. This is discussed in some detail in Chapter 7, especially Section 7.4.4. K3 fibrations. The material presented in this chapter is relevant to study of relative integral functors on varieties that are fibered in K3 surfaces. There is not much mathematical literature about this subject. In [71] Bridgeland and Maciocia show that a relative moduli space for a variety fibered in K3 surfaces or Abelian surfaces is smooth if and only if it is fine. In [277] Thomas studies some moduli spaces of stable sheaves on K3 fibrations. Calabi-Yau varieties fibered in K3 surfaces have been on the other hand quite extensively studied in the physical literature as compactification spaces for heterotic string theories; among the main contributions one can cite [183, 11, 151, 125, 21, 174].

Chapter 5

Nahm transforms Introduction The original Nahm transform, i.e., a mechanism that starting from an instanton on a 4-dimensional flat torus produces an instanton on the dual torus, was introduced by Nahm in 1983 [230]. This construction was formalized by Schenk [263] and Braam and van Baal [57] in later years. Their descriptions show that the Nahm transform is essentially an index-theoretic construction: given a vector bundle E on flat torus X, equipped with an anti-self-dual connection ∇, one considers the ˆ as a space parameterizing a family of Dirac operators twisted by ∇. dual torus X ˆ Taking the index of this family yields, under suitable conditions, the instanton ∇ ˆ on X. Braam-van Baal and Schenk already hinted at a connection between the Nahm and the Fourier-Mukai transforms. A first description of their relation was given by Donaldson and Kronheimer [102]. From an abstract point of view, the bridge between the two constructions is provided by a relation between index bundles and higher direct images, very much in the spirit of Illusie’s definition of the “analytical index” of a relative elliptic complex [158, Appendix II]. The fact that the Nahm transforms maps instantons to intantons then corresponds, via the Hitchin-Kobayashi correspondence, to the fact that sometimes the Fourier-Mukai transform preserves the condition of stability. The purpose of this chapter is to embed Nahm’s construction into a more general class of transforms, which we call K¨ ahler Nahm transforms. This will allow us to compare in a precise manner the Nahm and Fourier-Mukai transforms. After that we further develop the theory, introducing a special case of such transforms when the manifolds involved have a hyperk¨ ahler structure. We consider a generC. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_5, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

147

148

Chapter 5. Nahm transforms

alization of the notion of instanton (the quaternionic instantons) and prove that the “hyperk¨ahler Fourier-Mukai transform” preserves the quaternionic instanton condition. The Nahm transform has been widely used to study instantons admitting symmetries — e.g., instantons that are periodic in one or more directions. We shall not cover here these applications, just restricting ourselves to provide some relevant bibliography in the “Notes and further reading” section. In the first section we provide the reader with some notions that will be needed in the chapter — basically, the concept of instanton, a cursory view of the Hitchin-Kobayashi correspondence, and a review of Dirac operators and index bundles.

5.1 5.1.1

Basic notions Connections

In this section we shall consider complex vector bundles E on differentiable manifolds. We shall at first use the same symbol E also for the associated sheaf of C ∞ sections, even though later on in this chapter we shall need to resort to a more precise notation. Let X be a differentiable manifold. We shall denote by ΩkX the sheaf of differential k-forms on X. If E is a smooth complex vector bundle on X, a connection ∇ on E is a C-linear sheaf morphism ∇ : E → Ω1X ⊗ E satisfying the Leibniz rule ∇(f σ) = f ∇(σ) + df ⊗ σ for every section σ of E and every function f on X. (The tensor product of C ∞ vector bundles is taken over the sheaf of complex-valued smooth functions.) For every k ≥ 1, the connection ∇ yields in a natural way C-linear morphisms ⊗ E satisfying the Leibniz rule ∇ : ΩkX ⊗ E → Ωk+1 X ∇(ω ⊗ σ) = dω ⊗ σ + (−1)k ω ∧ ∇(σ) . The morphism F∇ : E → Ω2X ⊗ E , ∞ CX -linear

F∇ = ∇ ◦ ∇ ,

called the curvature of ∇, is and therefore may be regarded as a global section of the sheaf Ω2X ⊗ End(E). If the curvature F∇ vanishes, the connection is said to be flat.

5.1. Basic notions

149

We may also consider connections on principal bundles. We recall that a principal bundle over a differentiable manifold X with structure group a Lie group G is a manifold P carrying a free action of G such that the quotient P/G is isomorphic to X, and the projection π : P → X is locally trivial. For every u ∈ P , the vertical tangent space Vu P is defined as ker(π∗ : Tu P → Tπ(u) X). Definition 5.1. A connection Γ on a principal G-bundle P is a smooth G-invariant distribution {Hu P ⊂ Tu P }u∈P such that for all u ∈ P one has Tu P = Vu P ⊕Hu P . 4 To the connection Γ we may associate a g-valued 1-form ωΓ on P , where g is the Lie algebra of G. The 1-form ωΓ is defined as the annihilator of the distribution Γ. The curvature of Γ is the g-valued differential 2-form on P defined by RΓ = dωΓ + 12 [ωΓ , ωΓ ] . The 2-form RΓ is horizontal (i.e., RΓ (α, β) = 0 if α ∈ Vu P and β ∈ Tu P ) and G-equivariant, i.e., Rg∗ RΓ = Adg−1 RΓ for all g ∈ G; here Rg is the right action of the group, Rg (u) = ug). If H is a closed subgroup of G, a principal H-bundle Q with an H-equivariant fiber-preserving inclusion j : Q → P is said to be a reduction of the structure group G to H. If a connection Γ on P induces a connection on some Q for some closed proper subgroup H, we say that Γ is reducible. If Γ is not reducible, we say it is irreducible. In the vector bundle case a connection ∇ on a vector bundle E is said to be reducible if there is a direct sum decomposition E = E 0 ⊕ E 00 such that ∇(E 0 ) ⊂ Ω1X ⊗ E 0 . If F is a space over which the group G acts via a representation ρ, and P is a principal G-bundle, we may form the associated bundle E(P, F ) = P ×G F as the quotient of P × F under the equivalence relation (u, v) ∼ (ug, ρ(g −1 )v). If F is a linear space and ρ is a linear representation, E = E(P, F ) is a vector bundle. In this case, a connection on P induces a connection on E in the former sense. See [185] for details. Note that in particular if G = Gl(n, C) the datum of a principal Gl(r, C)-bundle is equivalent to that of a rank r complex vector bundle E. Example 5.2. An important example of associated bundle is the adjoint bundle Ad P . This is the bundle with standard fiber g associated to P via the adjoint representation of G on g. We shall denote by Ωk (Ad P ) the space of differential k-forms on X with values in Ad P . Note that the curvature RΓ of a connection on P may be regarded as an element in Ω2 (Ad P ), and analogously, the difference of two connections in an element in Ω1 (Ad P ). The adjoint bundle Ad P enters the Atiyah sequence 0 → Ad P → T P/G → T X → 0

150

Chapter 5. Nahm transforms

and one may regard a connection on P as a splitting of this sequence, cf. [14]. 4 A vertical automorphism of a principal bundle P is a G-equivariant vertical diffeomorphism φ : P → P (i.e., it verifies π ◦ φ = π and φ(ug) = φ(u)g). The group of smooth (vertical) automorphisms of P , which we denote by G, is usually called the gauge group. It acts naturally on a connection Γ and on the curvature RΓ by pullback. An analogous situation prevails in the vector bundle case; so, if φ ∈ G, we shall denote by φ∗ (∇) and φ∗ (F∇ ) the transformed connection and curvature, respectively. If U ⊂ X is an open subset over which E trivializes, after fixing a trivialization on U the curvature F∇ may be regarded as a matrix-valued 2-form, while the restriction of φ to U is described by a smooth map g : U → Gl(r, C), where r is the rank of E. The transformed curvature may be written on U as φ∗ (F∇ ) = Adg−1 (F∇ ) = g −1 F∇ g . Let as assume now that X is oriented and is equipped with a Riemannian metric γ. This allows one to introduce the Hodge duality operator ∗ as the map ∗ : ΩkX → Ωn−k (where n = dim X) defined on a global k-form α by the condition X α ∧ ∗β = (α, β) vol(γ) for all k-forms β (in the right-hand side ( , ) is the scalar product given by the Riemannian metric, and vol(γ) is the Riemannian volume form). Note that ∗2 = (−1)k(n−k) .

5.1.2

Instantons

We can now define the notion of instanton. Definition 5.3. Let X be an orientable 4-manifold, equipped with a Riemannian metric, and let E be a vector bundle on X. An instanton on E is a connection ∇ whose curvature F∇ is anti-self-dual with respect to Hodge duality, ∗F∇ = −F∇ . (This makes sense since F∇ is a 2-form with values in End(E).) Analogously, an instanton on a principal G-bundle P on X is a connection Γ on P whose curvature RΓ is anti-self-dual, that is, ∗RΓ = −RΓ (here one regards RΓ as an element in Ω2 (Ad P ), so that it makes sense to apply the Hodge duality operator to it). 4 Remark 5.4. We might as well define instantons as connections with self-dual curvature. Since a reversal of the orientation swaps self-dual with anti-self-dual 2-forms, as far as there is no distinguished orientation the two notions are interchangeable. When X is a complex manifold, so that a preferred orientation does exist, and E is a vector bundle, it is preferable to choose the anti-self-dual condition since in that case instantons relate to stable bundles via the so-called Hitchin-Kobayashi correspondence, see Section 5.1.3. 4

5.1. Basic notions

151

We briefly recall the construction of moduli spaces of instantons, i.e., of a space which parameterizes gauge equivalence classes of instantons. For technical reasons we assume that the structure group G is compact and semisimple. The first step is to consider the space of all connections on the bundle P . This is an affine infinite-dimensional space A, modeled on the vector space Ω1 (Ad P ). By introducing suitable Sobolev norms it can be given the structure of Hilbert manifold, cf. [109, 102]. To avoid pathological behaviors, we restrict A to its subspace A] whose elements are irrreducible connections. The next step is to take quotient by the natural action of the gauge group G. Provided that one takes completions with respect to suitable norms, the quotient B = A] /G has a structure of smooth, infinite-dimensional Hilbert manifold; it is called the orbit space. (For details on the analytical aspects of this construction the reader may refer to [109] and [102].) Its points represent gauge equivalence classes of irreducible connections on P . The space of gauge equivalence classes of irreducible anti-self-dual connections M is a subset of B. It is a remarkable fact that M is a finite-dimensional, smooth differentiable manifold. If we want to make a precise statement we need to make assumptions of some sort on the base manifold X. One possibility is to assume that the (oriented and compact) Riemannian 4-manifold (X, γ) is antiself-dual, i.e., its Weyl curvature 2-form is anti-self-dual and the scalar curvature of (X, γ) is nonnegative (this is the case discussed in [17]); the Weyl curvature 2-form is an invariant of the conformal structure of (X, γ) which is constructed from the Riemannian curvature, see, e.g., [40]. Under all these assumptions, one can prove that M is smooth of dimension dim M = p1 (Ad P ) − (1 − b1 + b+ ) dim G where p1 (Ad P ) is the first Pontrjagin class of the adjoint bundle Ad P , b1 = dim H 1 (X, R) is the first Betti number of X, and b+ is the number of positive eigenvalues of the intersection form (the quadratic form defined on H 2 (X, R) by the cup product). Equivalently, b+ is the dimension of the space of harmonic selfdual 2-forms on X. The moduli space M has a natural metric, called the Weil-Petersson metric. Since the structure group G is semisimple, the Killing-Cartan form κ on the Lie algebra g of G is nondegenerate. One starts by equipping the affine space A with a metric by noticing that the tangent space T∇ A at a point ∇ ∈ A may be identified with the space Ω1 (Ad P ). The Weil-Petersson metric Φ∇ is defined as Z κ(α1 , α2 ) vol(γ) Φ∇ (α1 , α2 ) = X

where κ(α1 , α2 ) is obtained by first applying the Killing-Cartan form to α1 and α2 , thus getting a 2-form on X, and then making it into a function using the metric

152

Chapter 5. Nahm transforms

γ on X. Moreover, vol(γ) is the volume form on X given by the metric γ. The metric Φ∇ is gauge-invariant, hence descends to the orbit space B, and induces a metric on M by restriction. The product manifold X × B carries a universal G-bundle Q equipped with a universal connection ∇ : if b ∈ B is a point corresponding to the gauge equivalence class of a connection ∇ on a principal G-bundle P on X, then Q|X×{b} ' P and the connection ∇ |X×{b} is gauge-equivalent to ∇. A very neat and concise treatment of the main properties of the universal pair (Q, ∇ ) is given in [19]. Let us here briefly summarize the construction of the pair (Q, ∇ ). We consider the action of the gauge group G on the product P × A] . This action has no fixed points so that one can define a principal bundle Q0 Q=

P × A] . G

The action of the structure group G of P commutes with the action of the gauge group, so that G acts on the bundle Q. Since we are considering only irreducible connections, this action is free and defines a principal bundle with total space Q and base manifold Q/G ' X × B. The connection ∇ is defined as follows. Consider on the space X × A] the metric given by the metric γ on X and the Weil-Petersson metric on A] . This is invariant under the natural action of G × G and hence descends to a G-invariant metric on Q. The connection is obtained by considering the distribution in T Q which is orthogonal to the fibers of Q. The universality of (Q, ∇ ) means the following. Let R be a principal Gbundle R on X × Y , where Y is any compact manifold, with the property that ˜ be a connection on R. Then there exist RX×{y} ' P for all y ∈ Y , and let ∇ a map f : Y → B and a principal bundle map f˜: R → (IdX × f )∗ Q such that ˜ ∇) = ∇. f˜∗ (∇ The curvature F of the universal connection ∇ may be described explicitly. It is convenient to split it into its K¨ unneth components with respect to the product X × B, namely, F = F2,0 + F1,1 + F0,2 . One has: 1. if b ∈ B, then F2,0 |X×{b} = F∇ , where ∇ is a connection whose gauge equivalence class is b; 2. if (v, α) ∈ T(x,b) (X × B) then F1,1 (v, α) = α(v), after identifying α with an element in Ω1 (Ad P ) such that ∇∗ (α) = 0, where ∇ is a connection whose gauge equivalence class is b (one should note that Tb B ' Ω1 (Ad P )/ ker ∇∗ ); 3. if α, β ∈ Tb B, then F0,2 (α.β) = G∇ (λα (β)), where G∇ is the Green function of the trace (or “rough”) Laplacian ∇∗ ◦ ∇ : Ω0 (Ad P ) → Ω0 (Ad P ) (i.e.,

5.1. Basic notions

153

G∇ = (∇∗ ◦ ∇)−1 ), while the map λα : Ω0 (Ad P ) → Ω1 (Ad P ) is defined by λα (s) = [α, s]. Let us eventually notice that if we choose a representation ρ : G → Aut(V ) of G, then we may define on X × B a universal bundle QV with standard fiber V and a universal connection. The bundles Q, QV and the corresponding connections may be restricted to X × M ⊂ X × B, obtaining universal bundles with connections for the instanton moduli spaces.

5.1.3

The Hitchin-Kobayashi correspondence

In some cases, as we saw in Chapters 3 and 4, the Fourier-Mukai transform preserves the stability of the sheaves it acts on. A parallel property of the Nahm transform is that sometimes it maps instantons to instantons. As we shall see in this chapter, the two transforms may be related, and the above-mentioned preservation properties are intertwined by the correspondence between instantons, or more general, Hermitian-Yang-Mills bundles, and stable bundles — the so-called Hitchin-Kobayashi correspondence, which we proceed now to discuss briefly. Comprehensive references on this subject are the monographs [184, 200]. Let X be an n-dimensional compact K¨ ahler manifold, with K¨ahler form ω. One can give a notion of µ-stability for coherent sheaves F on X exactly as in the projective case by defining the degree of F as Z γ1 (F) ∧ ω n−1 deg(F) = X

where γ1 (F) is here any closed 2-form on X whose cohomology class is c1 (F) (one should notice that c1 (F) can be introduced for every coherent sheaf F even when X is not projective by defining it as the first Chern class of the determinant bundle det(F)). On the other hand, let E be a holomorphic vector bundle on X equipped with a Hermitian fiber metric h. The latter, together with the complex structure of E, singles out a unique connection ∇ on E, which has the property of being compatible with both the metric h, meaning that dh(s, t) = h(∇(s), t) + h(s, ∇(t)) for all sections s, t of E, and with the complex structure of E, which in turn means that ∇0,1 = ∂¯E , where ∂¯E is the Dolbeault (Cauchy-Riemann) operator of the bundle E. This connection is called the Chern connection of the Hermitian bundle (E, h) [184].

154

Chapter 5. Nahm transforms

Let Λ be the adjoint of the map given by wedging by the K¨ahler form, i.e., (Λ(α), β) = (α, ω ∧ β) for all forms α, β on X. Definition 5.5. A Hermitian vector bundle (E, h) is said to satisfy the HermitianYang-Mills condition if there exists a complex constant c such that Λ(F∇ ) = c IdE where F∇ is the curvature of the Chern connection ∇.

4

Remark 5.6. The constant c is fixed by the topology of the bundle, and one has indeed Z 1 2nπ µ(E) with vol(X) = ωn . c= n! vol(X) n! X 4 Remark 5.7. The notion of Hermitian-Yang-Mills bundle generalizes that of instanton: indeed, if n = 2 one may see that Hermitian-Yang-Mills bundles of zero degree are exactly the instantons. 4 Definition 5.8. A coherent sheaf F is said to be polystable if it is a direct sum of µ-stable sheaves having the same slope. 4 Proposition 5.9. The sheaf of holomorphic sections of a Hermitian-Yang-Mills bundle is polystable. The proof of this result is not difficult, and may be found, e.g., in Kobayashi [184]. A much deeper result is the converse. Theorem 5.10. A µ-stable bundle E on a compact K¨ ahler manifold admits a Hermitian metric h (unique up to homotheties) such that (E, h) satisfies the HermitianYang-Mills condition. The proof given by Donaldson first for projective surfaces, and then for projective varieties of any dimension [99, 100], considers the space of all Hermitian structures on E and defines a parabolic flow on it by introducing a suitable functional. The proof that the flow admits a limit, which is the sought-for HermitianYang-Mills metric, relies on the Mehta-Ramanathan theorem about the restriction of semistable sheaves to divisors in certain linear systems, and therefore confines the validity of the proof to the projective case. This techniques is nicely illustrated in Kobayashi [184]. A proof which works on general compact K¨ahler manifolds was later given by Uhlenbeck and Yau [286]. Example 5.11. As an application of the Hitchin-Kobayashi correspondence we may show how Proposition 4.74 yields a statement on some moduli spaces of instantons. Let X be a K3 surface, and let H, ` be divisors that make it strongly reflexive (see

5.1. Basic notions

155

Chapter 4). We saw in Proposition 4.74 that the moduli space Mn of H-stable sheaves on X with Chern character (1 + 2n, −n`, −5n) is connected and projective and that it contains no µ-stable sheaves. Identifying µ-stable bundles of zero degree with irreducible instantons, this means that on X there are no irreducible U (2n + 1)-instantons, with fixed determinant OX (−n`) and second Chern character −5n, and the moduli space of all instantons with this type is isomorphic to the n-th symmetric product S n X. This last fact follows from the structure of the so-called Uhlenbeck compactification of the instanton moduli space, see, e.g., [102]. 4

5.1.4

Dirac operators and index bundles

Finally, we review another basic construction needed to define the Nahm transform, namely, the Dirac operator. A suitable reference on this topic, among many others, is Lawson and Michelsohn’s book [194]. If X is an orientable Riemannian manifold, with Riemannian metric γ, we may consider the principal bundle SO(X) of oriented orthonormal frames (bases of tangent spaces at the points of X), whose structure group is the special orthogonal group SO(n), where n = dim X. A spin structure on X is a principal Spin(n)-bundle Spin(X) on X together with a bundle homomorphism Spin(X) → SO(X) which on the fibers reduces to the spin covering homomorphism Spin(n) → SO(n). The second Stieffel-Whitney class w2 (X) ∈ H 2 (X, Z2 ) is an obstruction to the existence of a spin structure on X. When w2 (X) = 0 the isomorphism classes of spin structures on X are classified by the group H 1 (X, Z2 ). Let Cl(X, γ) be the Clifford algebra bundle of (X, γ), that is, the bundle of Clifford algebras on X whose fiber at the point x is the Clifford algebra associated with the vector space Tx X equipped with the quadratic form given by the scalar product γ(x). A Clifford bundle S on X is a bundle of Cl(X, γ)-modules equipped with a Hermitian metric and a connection ∇ satisfying suitable and natural compatibility conditions, see [194]. The Dirac operator is the first-order differential operator D : Γ(S) → Γ(S) defined by the composition of the maps γ −1 Cl ∇ Γ(S) −→ Γ(Ω1X ⊗ S) −−→ Γ(T X ⊗ S) −−→ Γ(S) .

Here Cl : Γ(T X ⊗S) → Γ(S) is the multiplication morphism given by the Cl(X, γ)module structure of S and the immersion T X ,→ Cl(X, γ). An important example of Clifford bundle S is provided by the spinor bundle, i.e., the bundle associated with the principal bundle Spin(X) via the spin representation of the complexified Clifford algebra Cl(Rn , γ0 ) ⊗ C, where γ0 is the standard scalar product in Rn . The bundle S has complex rank 2[n/2] , and as a consequence of the Z2 -gradation of the Clifford algebra it inherits a Z2 -gradation as well, S = S+ ⊕ S− (using a terminology coming from physics, we may call S±

156

Chapter 5. Nahm transforms

the bundles of spinors of positive or negative chirality). The Dirac operator has odd degree with respect to this gradation. By abuse of notation, usually one writes D for the operator Γ(S+ ) → Γ(S− ); the adjoint D∗ then coincides with the term of the full Dirac operator mapping Γ(S− ) to Γ(S+ ). In order to apply standard techniques in analysis one needs to complete the space Γ(S) to a Hilbert space. Assuming that X is compact and using the Riemannian metric on X and the Hermitian fiber metric in S, we may consider the L2 norm on the space Γ(S), and then complete the latter in this norm, obtaining the Hilbert space L2 (S). More generally, for every integer p ≥ 0 we may consider the Hilbert space L2p (S) — called the Sobolev space of sections of S of weight p — formed by those sections of S whose p-th covariant derivative has finite L2 norm. The Dirac operator extends to an operator L21 (S) → L2 (S), or more generally, to an operator L2p (S) → L2p−1 (S) for all integers p ≥ 1. Elliptic regularity implies that the kernel of any of these operators coincides with the kernel of D : Γ(S) → Γ(S). Moreover, due to the fact that D is a Fredholm operator, this kernel is finitedimensional. The same is true for the cokernel of this operator, so that it makes sense to introduce the index of the Dirac operator as the integer number ind(D) = dim ker(D) − dim coker(D) . This number is actually a topological invariant of the manifold X, and is computed by the celebrated Atiyah-Singer index theorem (we assume henceforth the the dimension of X is even): Z ˆ A(X) (5.1) ind(D) = X

ˆ where A(X) is a characteristic class that may be expressed in terms of the Ponˆ trjagin classes of X. A simple way of writing a formula for A(X) (in de Rham cohomology) is    i ˆ R A(X) = p 2π where R is the curvature of a connection on the tangent bundle to X, and p is the polynomial which expresses the formal Taylor expansion of the function 1√ z f (z) = 2 1 √ sinh 2 z around z = 0 up to order n/2. A nontrivial consequence of the index formula (5.1) ˆ is that the right-hand side (called the A-genus of X) is an integer. There exists a twisted version of Atiyah-Singer index theorem: assume that a vector bundle E is given, with a connection ∇ on it, and define the twisted Dirac operator D∇ D∇ (s ⊗ ψ)

:

Γ(E ⊗ S+ ) → Γ(E ⊗ S− ),

=

∇(s) · ψ + s ⊗ D(ψ)

5.1. Basic notions

157

where again the product · is obtained by first applying the inverse Riemannian metric to ∇(s) and then performing the Clifford product. The formula (5.1) should now be replaced by Z ˆ ch(E) A(X) . ind(D∇ ) = X

A far reaching generalization of these formulas is the Atiyah-Singer index theorem for families. In this case we deal with a family of Dirac operators, which we may regard as a continuous map D : T → H, where T is a topological space (playing the role of parameter space) and H is the (separable) Hilbert space of bounded operators L21 (E ⊗ S+ ) → L2 (E ⊗ S− ). Let us denote Dt = D(t), i.e., Dt is the Dirac operator corresponding to the parameter t ∈ T . By assigning to each point t ∈ T the virtual vector space ker(Dt ) − coker(Dt ) one constructs a class ind(D) in the topological K-theory group K(T ) of the parameter space T . The Atiyah-Singer theorem for families is a formula which computes the Chern character of the virtual bundle ind(D). We shall give here that formula only in a specific but very important case. Let W be a vector bundle on the product X × T , equipped with a connection ∇ . Assume that the restriction W|X×{t} is isomorphic to E for every t ∈ T . Then the connections ∇t = ∇ |X×{t} yield a family of connections on E, and we then twist the Dirac operator with that family. In that case one has Z ˆ ch(ind(D)) = ch(W) A(X) ; (5.2) X

R here the integral X may be regarded as the integration along the fibers of the projection X → Y → Y . One can define a generalization of the spin structures, and introduce a related Atiyah-Singer index theorem, by using complex geometry. Let X be an ndimensional complex manifold with a Hermitian metric h, and define the complex vector bundle S = ⊕nk=0 Ω0,k X with the obvious Z2 -gradation. This becomes a Clifford module by letting, for every vector field v on X and every section η of S √ v · η = 2(h(p) ∧ η − iq η) where p and q are the (1,0) and (0,1) parts of v, respectively. One can prove that if the pair (X, h) is a K¨ ahler manifold, then the Dirac operator of this Clifford module may be identified with the operator √ 2(∂¯ + ∂¯∗ ) where ∂¯ is the Dolbeault (Cauchy-Riemann) operator. If we twist this operator with a complex holomorphic vector bundle E we obtain a twisted Dirac operator

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Chapter 5. Nahm transforms

DE , and one can easily show that the index of this operator coincides with the holomorphic Euler characteristic of E, namely, ind(DE ) = χ(E) =

n X (−1)i dim H i (X, E) . i=0

The Atiyah-Singer index theorem (5.2) in this case takes the form Z χ(E) = ch(E) td(X) , X

where td(X) is the Todd class of X, i.e., it reproduces the Hirzebruch-RiemannRoch theorem for a holomorphic vector bundle (we are of course assuming that X is compact). In the same way, if X is a compact K¨ahler manifold, Y is a complex manifold, and E is a holomorphic complex vector bundle on X × Y , the Atiyah-Singer index theorem for families reproduces the Grothendieck-RiemannRoch theorem for the Chern character of the higher direct images of E: Z n X i i (−1) ch(R π∗ E) = ch(E) td(X) i=0

X

where π : X × Y → Y is the projection.

5.2 5.2.1

The Nahm transform for instantons Definition of the Nahm transform

Let (X, γ) be a smooth oriented Riemannian spin 4-manifold such that the corresponding scalar curvature Rγ is nonnegative at every point of X. For simplicity, we assume that X is compact. We denote as before by S± the spinor bundles of positive and negative chirality. Moreover, let Y be a smooth manifold parameterizing a family of anti-self-dual connections on a fixed complex vector bundle W → X. We assume that the bundle W is equipped with a Hermitian metric, and that the connections we are considering on it are compatible with this metric (so that they are unitary connections). Associating to any connection in this family its gauge equivalence class, we obtain a map f : Y → M, where M is a moduli space of instantons on W . We assume that the map f is smooth. We also assume that Y has a Riemannian metric compatible via the map f with the Weil-Petersson metric on M. So for every t ∈ Y one has an anti-self-dual connection ∇t on the bundle W . The Nahm transform from X to Y is a mechanism that transforms Hermitian vector bundles with unitary anti-self-dual connections on X into Hermitian

5.2. The Nahm transform for instantons

159

vector bundles with unitary connections on Y . If Y parameterizes a family of flat connections over X, we will say that the transform is flat; otherwise, we will say that the Nahm transform is nonflat. Let us now describe the transform in detail. Let E be a Hermitian complex vector bundle on X with an anti-self-dual connection ∇ compatible with the Hermitian metric (i.e., we have a unitary anti-self-dual connection). On the tensor bundle E ⊗ W → X, we have a twisted family of anti-self-dual connections ˜ t = ∇ ⊗ IdW + IdE ⊗ ∇t . We further assume an irreducibility condition. ∇ ˜ t } is 1-irreducible if for all t ∈ Y the condition Definition 5.12. The family {∇ ˜ 4 ∇t s = 0 implies s = 0. In other terms, for all values of t, the tensor bundle E ⊗W has no covariantly ˜ t . We consider the family D of constant sections with respect to the connection ∇ twisted Dirac operators Dt : L2p (E ⊗ W ⊗ S+ ) → L2p−1 (E ⊗ W ⊗ S− ), and denote as usual by let Dt∗ the adjoint Dirac operator. The Dirac Laplacian ˜l ∗ ∇ ˜ t via the Weitzenb¨ock formula: Dt∗ Dt is related to the trace Laplacian ∇ t Dt∗ Dt

˜ ∗∇ ˜+ 1 ˜ = ∇ t t − Ft + 4 Rγ ˜ ∗∇ ˜ t + 1 Rγ = ∇ t

4

(5.3) (5.4)

˜ t vanishes. Applying (5.3) to since the self-dual part F˜t+ of the curvature F˜t of ∇ 2 a section s ∈ Lp (E ⊗ W ⊗ S+ ) and integrating by parts we obtain Z 2 2 1 ||Dt s|| = ||∇t s|| + 4 Rγ hs, si ≥ 0 (5.5) X

with equality if and only if s = 0, since Rγ ≥ 0. Therefore, we conclude that ker Dt = {0} for all t ∈ Y . This means that ˆ = −ind(D) is a well-defined complex vector bundle over Y ; the fiber E ˆt is given E ˆ ± denote the trivial Hilbert bundle over Y with fibers given by coker(Dt ). Let H ˆ as a subbundle of H ˆ − ; we by the spaces L2p−1 (E ⊗ W ⊗ S± ). One can think of E have an exact sequence of bundles ι

ˆ− → H ˆ+ → 0 ˆ→ − H 0→E which may be split using the metric on the bundle H − , thus defining a projection ˆ This provides E ˆ with a natural Hermitian metric, and we can also ˆ − → E. P: H ˆ ˆ via the projection formula define a unitary connection ∇ on E ˆ = P ◦ dH − ◦ ι ∇

(5.6)

160

Chapter 5. Nahm transforms

ˆ − , i.e., the exterior differwhere dH − denotes the trivial covariant derivative on H ential. It is easily checked that this construction behaves well with respect to gauge transformations of W . This means that if we apply an automorphism of the bundle W , thus getting a new family ∇0t of connections on W , and a new family of Dirac operators D0 , there is a natural isomorphism coker(D) ' coker(D0 ), so that the ˆ descends to a bundle on the quotient Y /G, where G denotes the index bundle E group of gauge transformations of W (we are assuming here that this quotient ˆ is defined on this bundle. For this is well behaved). Moreover, a connection ∇ reason, we may assume that Y parameterizes a family of gauge equivalence classes of anti-self-dual connections on the fixed vector bundle W sucht that the family of connections twisted by ∇ is 1-irreducible. ˆ A) ˆ is called the Nahm transform of (E, A). Definition 5.13. The pair (E,

4

The Nahm transform is well behaved also with respect to the gauge transformations of E. Lemma 5.14. If ∇ and ∇0 are two gauge-equivalent connections on the vector bunˆ and ∇ ˆ 0 are gauge equivalent connections on the transformed dle E → X, then ∇ ˆ bundle E → Y . Proof. Let h be a a bundle automorphism h : E → E which makes ∇ and ∇0 gauge equivalent and let g = h ⊗ IdW be the induced automorphism of E ⊗ W . ∗ g, for all t ∈ Y . Thus if {Ψi } is a basis for ker Dt∗ , then Then Dt∗0 = g −1 DA t 0 −1 {Ψi = g Ψi } is a basis for ker Dt∗0 . So g can also be regarded as an automorphism ˆ It is then easy to see that of the transformed bundle E.

ˆ 0 = P 0 ◦ dH − ◦ ι0 = g −1 P 0 g ◦ dH − ◦ g −1 ι0 g = g −1 ∇g ˆ ∇ since dH − ◦ g −1 = 0, for g does not depend on t.



The construction performed in this section provides the following result. Theorem 5.15. Let Y be a connected component of the moduli space of irreducible instanton connections on a smooth oriented Riemannian spin 4-manifold X with nonnegative scalar curvature, and let E be a Hermitian complex vector bundle on X. Let W be a Hermitian complex vector bundle on X carrying a 1-irreducible family of anti-self-dual connections. The corresponding Nahm transform yields a well-defined map from M(E), the moduli space of gauge equivalence classes of ˆ gauge equivalence unitary anti-self-dual connections on E, into the space BY (E) ˆ classes of (unitary) connections on the Nahm transform E.

5.2. The Nahm transform for instantons

5.2.2

161

The topology of the transformed bundle

The Atiyah-Singer index theorem for families allows us to compute the Chern character of the transformed bundle. Let us assume that there exists a complex vector bundle W on X × Y with a connection ∇ , such that for all t ∈ Y one has W|X×{t} ' W and ∇ |X×{t} = ∇t . Then the formula (5.2) yields Z ˆ ˆ=− ch(E) ch(W) A(X) ch E X

ˆ is the bundle of cokernels. This formula where the minus sign is needed because E ˆ depends only on the topology shows that the topology of the transformed bundle E of the original bundle E. Example 5.16. Let us now briefly analyze the Nahm transform for the simplest possible compact spin 4-manifold with nonnegative scalar curvature, the round 4-dimensional sphere S 4 . So let X = S 4 , and let Y be the moduli space of SU (2) instantons over S 4 with charge one; as a Riemannian manifold, Y is a hyperbolic 5-ball B5 [109]. Let E → S 4 be a complex vector bundle of rank n ≥ 2, equipped ˆ → B 5 of with an instanton ∇ of charge k ≥ 1. Nahm transform gives a bundle E 5 rank 2k + r, by the index formula (5.2). Since B is contractible, this is the only nontrivial topological invariant of the transformed bundle. 4 Remark 5.17. (Differential properties of the transformed connection.) Given that the original connection ∇ satisfies a nonlinear first-order differential equation (the anti-self-duality condition), it is reasonable to expect that the transformed connection will also satisfy some kind of strict differential or algebraic condition. However, since it does not seem possible to write a formula for the curvature of ˆ which depends explicitly on the curvature of the the transformed connection ∇ original connection ∇, it is in general very difficult to characterize any particular ˆ properties of ∇. For instance, when the parameter space Y is 4-dimensional, one would like to know whether F∇ ˆ is anti-self-dual. This seems to be a very hard question in general. However, when M is a hyperk¨ ahler manifold, complex analytic methods can be used to show that this is indeed the case. This will be shown later on in a more general setting. The results we shall discuss will include as a special case the original Nahm transform on flat tori, which is known to map instantons on a 4-torus to instantons on the dual torus [57, 102, 263]. This case will be discussed in Section 5.2.4, 4

5.2.3

Line bundles on complex tori

As a preparation for studying the Nahm transform on tori, we analyze in this section a very handy description of U (1) line bundles on complex tori in terms of

162

Chapter 5. Nahm transforms

their automorphy factors. A standard reference about this theory is [42]. Let V be a g-dimensional complex vector space, and Ξ a nondegenerate lattice in it. Then the quotient T = V /Ξ has a natural structure of g-dimensional complex manifold, and is said to be a complex torus of dimension g. If some conditions are satisfied (the Riemann bilinear conditions), T is actually algebraic, and then it is an Abelian variety, whose origin is the image of the origin of V in T . However in this section we consider the general case where T may not be algebraic. Since any generator of the lattice Ξ corresponds to a loop in T , we have a natural identification of Ξ with the fundamental group π1 (T ), and hence with the homology group H1 (T, Z). As a result, we also have identifications H k (T, Z) ' Λk Ξ∗ . Let H(T ) be the space of Hermitian forms H : V × V → C that satisfy the condition Im(H(Ξ, Ξ)) ⊂ Z. Under the inclusion H 2 (T, Z) ⊂ Hom(V × V, C) we have an identification of H(T ) with the image of the morphism c1 : Pic(T ) → H 2 (T, Z) , i.e., H(T ) coincides with the N´eron-Severi group NS(T ). Definition 5.18. A semicharacter associated with an element H ∈ H(T ) is a map χ : Ξ → U (1) such that χ(λ + µ) = χ(λ) χ(µ) eiH(λ,µ) . An element H ∈ H(T ) and an associated semicharacter χ define a automorphy factor a: V × Ξ

→

U (1)

a(v, λ)

=

χ(λ) eπH(v,λ)+ 2 H(λ,λ) .

π

4 If H = 0, then an associated semicharacter is just a character of the lattice Ξ. In the same way as functions on T are just periodic functions on the universal cover V , sections of a line bundle L on T are functions on V that satisfy a “generalized periodicity condition” given by an automorphy factor. We have indeed: Proposition 5.19. The holomorphic functions s on V that satisfy the condition s(v + λ) = a(v, λ) s(v) for all v ∈ V and λ ∈ Ξ, where a is an automorphy factor associated with an element H ∈ H(T ), are in a one-to-one correspondence with sections of a line bundle L on T such that c1 (L) = H.

5.2. The Nahm transform for instantons

163

For a proof see [42]. In this framework the dual torus T ∗ may be built as follows. Let Ω be the conjugate dual space of V , and let Ξ∗ = {` ∈ Ω | `(Ξ) ⊂ Z} be the lattice dual to Ξ. Then we set T ∗ = Ω/Ξ∗ . One has a natural isomorphism T ∗ ' HomZ (Ξ, U (1)), and an exact sequence 0 → T ∗ → Pic(T ) → NS(T ) → 0 which shows that the dual torus T ∗ parameterizes flat U (1) bundles on T (and indeed when T is algebraic, T ∗ is the complex torus underlying the dual Abelian variety Tˆ). If ξ ∈ T ∗ , the line bundle on T parameterized by ξ may be described by the automorphy factor aξ (v, λ) = eπ w(λ)

(5.7)

where w is a representative of ξ in Ω. On the basis of Proposition 5.19 we can give a very explicit description of the Poincar´e bundle P on T × T ∗ . We choose the element H ∈ H(T × T ∗ ) given by H(v, w, α, β) = β(v) + α(w)

(5.8)

where v, w ∈ V , α, β ∈ Ω, and an associated semicharacter χ(λ, µ) = eiπ µ(λ) .

(5.9)

Note that H 2 (T × T ∗ , Z) contains the K¨ unneth component H 1 (T, Z) ⊗ H 1 (T ∗ , Z) 1 which can be identified with EndZ (H (T, Z)), and under this identification H corresponds to the identity endomorphism. The Poincar´e bundle is by definition the line bundle P corresponding to the Hermitian form (5.8) and semicharacter (5.9). Thus, its automorphy factor is aP (v, w, λ, µ) = eiπ µ(λ) eπ(µ(v)+w(λ)) . Let us check that that P|T ×{ξ} is isomorphic to the flat line bundle on T corresponding to ξ ∈ T ∗ . Indeed P|T ×{ξ} admits an automorphy factor given by aP (v, w, λ, 0) = eπ w(λ) . By comparing with Definition 5.18 and Equation 5.7 we see that P|T ×{ξ} is isomorphic to the line bundle parameterized by ξ.

164

Chapter 5. Nahm transforms

5.2.4

Nahm transform on flat 4-tori

Let us describe here the special case of the general construction of the Nahm transform described in Section 5.2.1 that one obtains when the manifold X, Y are a flat 4-tours and its dual. This is the original transform defined by Nahm [230] and later formalized by Braam and van Baal [57], Schenk [263] and Donaldson and Kronheimer [102]. Let us equip X with a compatible complex structure (in other terms, since X is a quotient V /Ξ, where V is 4-dimensional vector space and Ξ ∈ V and nondegenerate lattice, we equip V with a complex structure), and identify Y with the dual complex torus. So Y parameterizes flat U (1) line bundles on X, and we have the Poincar´e bundle P on X × Y . For every ξ ∈ Y the line bundle Pξ is trivial as a C ∞ bundle, so that we may regard the collection {Px i} a trivial bundle W with a varying complex structure and Hermitian metric. Let E be a holomorphic Hermitian bundle on X with a compatible antiself-dual connection ∇. Donaldson and Kronheimer considered the following irreducibility condition. Definition 5.20. The pair (E, ∇) is said to be without flat factor if there is no ∇-compatible splitting E = E 0 ⊕ L where L is a flat line bundle. 4 As a matter of fact this is just 1-irreducibility in disguise: Proposition 5.21. (E, ∇) is without flat factors if and only if the family of connec˜ t = ∇ ⊗ Id + Id ⊗ ∇ξ } (where ∇ξ is the connection on Pξ ) is 1-irreducible. tions {∇ ˜ t } is not 1-irreducible. Then for some ξ ∈ Y there is a Proof. Assume that {∇ ˜ ξ 6= 0. After tensoring by P−ξ , this splits off a flat section s of E ⊗ Pξ such that ∇ parallel summand of E (i.e., E = E 0 ⊕ P−ξ ) and contradicts the fact the (E, ∇) is without flat factors. Conversely, if E has a parallel splitting E = E 0 ⊕ L with L flat, then L ' Pξ for some ξ ∈ Y . Let s be a nonzero covariantly constant section of L, i.e., ∇ξ (s) = ˜ ξ (s) = 0.  0. Then s, regarded as a section of E, satisfies ∇ So, if the pair (E, ∇) is without flat factors, we may apply the general theory ˆ on Y equipped of Section 5.2.1 and obtain a holomorphic Hermitian bundle E ˆ ˆ with a compatible connection ∇. One can prove that ∇ is anti-self-dual. Since this is particular case of the general result proved in Section 5.4.3, we shall not repeat it here. Moreover, in this case the Nahm transform is invertible, its inverse being exactly the same transform when we identify the dual torus to Y as X, and the corresponding Poincar´e bundle on Y × X as P ∗ . The proof that this actually provides an inverse to the Nahm transform from X to Y may be given in terms of a direct computation, as in [57] (which follow closely [230]). When X is algebraic,

5.3. Compatibility between Nahm and Fourier-Mukai

165

we can use the identification of the Nahm transform with the Abelian FourierMukai transform which follows from Section 5.3, and then use the fact that, as shown in Chapter 3, the Abelian Fourier-Mukai transform, being an equivalence of categories, is invertible (in Chapter 3 we consider Abelian varieties, but the proof also works for complex tori). This is very much in the spirit of the proof given by Donaldson and Kronheimer [102].

5.3

Compatibility between Nahm and Fourier-Mukai

Given a submersive morphism of complex manifolds f : Z → Y , and a complex vector bundle E on Z, there is a relationship between the higher direct images of E (the sheaf of holomorphic sections of E) and the index of the relative Dolbeault complex twisted by E. In this section we analyze this correspondence. This will be used to study the relationship between the Nahm transform and the Fourier-Mukai functors.

5.3.1

Relative differential operators

Let f : Z → Y be a submersion of differentiable manifolds, and let E → Z and F → Z be complex vector bundles. Let us denote by E ∞ , F ∞ the locally free CZ∞ -modules of sections of E and F . Definition 5.22. A relative differential operator of order k is a morphism D : f∗ E ∞ → f∗ F ∞ of CY∞ -modules that factors through the direct image of the sheaf J k (E ∞ /Z) of sections of the k-order (relative) jet bundle J k (E/Z) → Z, f∗ E ∞

/ f∗ F ∞ 9 r r rr r r r rrr D

 J k (E ∞ /Z)

4 Definition 5.23. Assume that E and F have Hermitian fiber metrics. The adjoint D∗ : f∗ F ∞ → f∗ E ∞ of a relative differential operator D : f∗ E ∞ → f∗ F ∞ is defined by letting (u, Dv) = (D∗ u, v) for each pair of sections u, v, of f∗ E ∞ and f∗ F ∞ respectively, on an open subset V ⊂ Y . 4 If y ∈ Y is a point in the parameter space Y , we denote Zy = f −1 (y) and by Ey the restricted fiber bundle E ×Y {y} → Zy . One easily checks that

166

Chapter 5. Nahm transforms

J k (E ∞ /Z)y = Γ(Zy , J k (Ey ∞ /Zy )). A relative differential operator D : f∗ E ∞ → f∗ F ∞ induces differential operators (in the usual sense) Dy : Γ(Zy , Ey ∞ ) → Γ(Zy , Fy ∞ ) and may be therefore regarded as a family of differential operators parameterized by points of Y . Definition 5.24. A relative differential operator D : f∗ E ∞ → f∗ F ∞ is elliptic if at 4 any point y ∈ Y the differential operator Dy is elliptic. We recall that a differential operator D : Γ(U ) → Γ(V ), where U , V are vector bundles on some differentiable manifold X, is said to be elliptic if for all ξ ∈ and ξ ∈ Tx∗ X the associated symbol map σx,ξ (D) : Ux → Vx is a linear isomorphism. For details on this notion, the reader may consult, e.g., [54]. If f : Z → Y is proper, then, given an elliptic relative differential operator, the Atiyah-Singer index theory for families provides an element ind(D) ∈ K(Y ) called the index of D. If either one of ker Dy or coker Dy has constant rank, then ker D and coker D are locally free CY∞ -modules of finite rank and one has that ind(D) = [ker D] − [coker D] , where [ ] denotes a class within K(Y ) (cf. [18], or [158, Appendix II]).

5.3.2

Relative Dolbeault complex

Let Z be a complex manifold and E a holomorphic vector bundle on Z. The sheaf of holomorphic sections of E will be denoted by E. In other words, E ∞ = E ⊗OZ CZ∞ . On the other hand, a C ∞ vector bundle E → Z whose sheaf of sections is E ∞ has a compatible holomorphic structure if there exists a locally free OZ -submodule ∼ E⊗ ∞ ∞ E ,→ E ∞ of finite rank that induces an isomorphism E ∞ → OZ CZ of CZ modules. We recall the following standard result in the theory of complex vector bundles (cf. [184]). Proposition 5.25. A C ∞ Hermitian vector bundle E → Z admits a compatible holomorphic structure if and only if there exists a Hermitian connection on E → Z whose curvature is of type (1, 1). If Ω1Z denotes the sheaf of C ∞ 1-forms on Z and ∇ : E ∞ → Ω1Z ⊗CZ∞ E ∞ is the aforementioned connection, the sheaf of holomorphic sections of E is given by ∞ ∞ is the (0, 1) component of ∇. E = ker ∂¯E , where ∂¯E = ∇0,1 : E ∞ → Ω0,1 Z ⊗CZ E

5.3. Compatibility between Nahm and Fourier-Mukai

167

This is actually part of a more general statement, namely, the exactness of the Dolbeault sequence of sheaves of OZ -modules: ∂¯

∂¯

∂¯

E ∞ E ∞ E 0 → E → E∞ → Ω0,1 → Ω0,2 → ... Z ⊗E Z ⊗E

Let f : Z → Y be a holomorphic morphism of complex manifolds. Let us ∗ the inverse images in the categories of Abelian sheaves, denote by f −1 , fh∗ , f∞ holomorphic sheaves and C ∞ sheaves, respectively. That is, for every OY -module E, fh∗ E is the OZ -module fh∗ E = f −1 E ⊗f −1 (OY ) OZ , and for every CY∞ -module ∗ ∞ ∗ ∞ E is the CZ∞ -module f∞ E = f −1 E ∞ ⊗f −1 CY∞ CZ∞ . In particular, if E ∞ = E ∞ , f∞ ∞ ∗ ∞ ∼ ∗ E ⊗OY CY , then f∞ E → fh E ⊗OZ CZ∞ . Definition 5.26. The sheaf of relatively holomorphic functions on Z is the sheaf fh∗ CY∞ . Analogously, we say that a C ∞ vector bundle E → Z has a relative holomorphic structure if there exists a locally free finite-rank fh∗ CY∞ -submodule E r ,→ E ∞ ∼ E r ⊗ ∗ ∞ C∞. that induces an isomorphism of CZ∞ -modules E ∞ → 4 fh CY Z We shall henceforth assume that the map f : Z → Y is submersive. Then we have an exact sequence of OZ -modules f∗

∗ Ω1Y → Ω1Z → Ω1Z/Y → 0 . 0 → f∞

The pullback of 1-forms via a holomorphic map preserves the Hodge decomposition, so that we can define a Hodge decomposition for the sheaf of relative 1-forms, 1,0 Ω1Z/Y = Ω0,1 Z/Y ⊕Ω Z/Y and the corresponding Hodge decomposition for the exterior V m 1 m powers ΩZ/Y = ΩZ/Y , Ωm Z/Y =

M

Ωp,q Z/Y .

p+q=m

If E → Z is a C ∞ vector bundle, and E ∞ is the CZ∞ -module of its sections, a relative connection for E is a morphism ∇E/Y : E ∞ → Ω1Z/Y ⊗CZ∞ E ∞ of f −1 CY∞ -modules satisfying an obvious Leibniz condition. The (0,1) component of the relative connection (with respect to the relative Hodge decomposition) is a morphism of fh∗ CY∞ -modules ∂¯E/Y : E ∞ → Ω1Z/Y ⊗CZ∞ E ∞ which defines a sequence of sheaf morphisms ∂¯E/Y

∂¯E/Y

∂¯E/Y

∞ ∞ → Ω0,2 → ... 0 → ker ∂¯E/Y → E ∞ → Ω0,1 Z/Y ⊗ E Z/Y ⊗ E

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Chapter 5. Nahm transforms

The kernel ker ∂¯E/Y is the sheaf of sections of E that are holomorphic along the fibers of Z → Y . If the (0,2) component of the curvature R = ∇E/Y ◦ ∇E/Y vanishes, the sequence (5.3.2) is a complex, called then relative Dolbeault complex. 2 = 0 is the integrability condition of the equation Moreover the equation ∂¯E/Y ¯ ∂E/Y (s) = 0, so that the relative Dolbeault complex is actually exact, and provides a resolution of ker ∂¯E/Y by fine sheaves. As a consequence, the higher direct images • ∞ ∞ Ri f∗ ker ∂¯E/Y are the cohomology sheaves of the complex f∗ (Ω0, Z/Y ⊗CZ E ). A Hermitian relative connection is a relative connection compatible with a given Hermitian metric in the usual sense. Proposition 5.25 has now a relative analogue. Summing up, we have proved the following results. Proposition 5.27. A Hermitian C ∞ vector bundle E → Z admits a compatible relative holomorphic structure if and only if there exists a Hermitian relative connection on E whose curvature is of type (1, 1). Proposition 5.28. Let E → Z be a vector bundle endowed with a Hermitian metric and a relative holomorphic structure. The higher direct images Ri f∗ E r are the • ∞ ∞ cohomology sheaves of the relative Dolbeault complex f∗ (Ω0, Z/Y ⊗CZ E ). Definition 5.29. Let E → Z be a C ∞ vector bundle endowed with a relative holomorphic structure. 1. E satisfies the i-th weak index theorem condition (i.e., it is WITi ) if Rj f∗ E r = 0 for every j 6= i; 2. E satisfies the i-th index theorem condition (i.e., it is ITi ) if H j (Zy , Ey ) = 0 for every j 6= i and for all points y ∈ Y , where Ey is the locally free OZy module Ey = E ⊗ OZy . Moreover, we say that E satisfies the even (odd resp.) WIT condition if Rj f∗ E r = 0 for all odd (resp. even) j, or that it satisfies the even (resp. odd) IT condition if 4 H j (Zy , Ey ) = 0 for all y ∈ Y and all odd (resp. even) j. We need the following technical result: Lemma 5.30. Let Y be a complex manifold. Then CY∞ is a faithfully flat sheaf of OY -modules. Proof. The proof is easy but quite dull. Since a flat morphism of local rings is faithfully flat, one has only to prove that the local ring Cy∞ is flat over (OY )y for every point y ∈ Y . A result by Malgrange [207] asserts that Cy∞ is flat over the subring of germs of (complex-valued) real analytic functions. That subring is isomorphic with the subring of convergent series C{z1 , . . . , zn , z¯1 , . . . , z¯n }. The

5.3. Compatibility between Nahm and Fourier-Mukai

169

problem then reduces to proving that C{z1 , . . . , zn , z¯1 , . . . , z¯n } is flat over (OY )y = C{z1 , . . . , zn }. If C{t1 , . . . , tn } is the ring of convergent series in a neighborhood of the origin in Rn , we have to prove that C{t1 , . . . , tn , tn+1 } is flat over C{t1 , . . . , tn }. Now, there is a chain of ring morphisms C{t1 , . . . , tn } ,→ C{t1 , . . . , tn }[tn+1 ] ,→ C{t1 , . . . , tn }[tn+1 ](tn+1 ) ,→ C{t1 , . . . , tn , tn+1 } . The first morphism is obviously flat, the second is the localization by the ideal generated by tn+1 , so it is flat as well. Let us notice that if A is a local Noetherian ring, and Aˆ is the completion of A in the topology of the maximal ideals, then Aˆ is flat over A, hence is faithfully flat. Now, the third morphism is flat because it is an immersion of local Noetherian rings that induces an iso∼ C[[t , . . . , t , t morphism C[[t1 , . . . , tn , tn+1 ]] → 1 n n+1 ]] between their completions in the topology of maximal ideals.  Theorem 5.31. Let f : Z → Y be a proper morphism of complex manifolds. A holomorphic bundle E → Z has a relative holomorphic structure given by E r = fh∗ CY∞ ⊗OZ E, and one has: 1. The holomorphic higher direct images Ri f∗ E are coherent sheaves of OY modules, and the natural map Ri f∗ E → Ri f∗ E r induces an isomorphism of CY∞ -modules ∼ Ri f E r = Ri f (f ∗ C ∞ ⊗ E) . C∞ ⊗ Ri f E → Y

OY

∗

∗

∗

h Y

OZ

2. E is WITi if and only if its holomorphic higher direct images vanish for j 6= i, for every j 6= i . Rj f∗ E = 0, Let us assume that f : Z → Y is flat as a morphism of complex manifolds. 3. For any i, E is ITi if and only if it is both WITi and the only nonvanishing holomorphic higher direct image Ri f∗ E is a locally free OY -module. 4. In particular, E is IT0 if and only if it is WIT0 . Proof. The coherence of the holomorphic higher direct images for a proper morphism is Grauert’s semicontinuity theorem [130]. Since CY∞ is a flat OY -module (Lemma 5.30), the formula in (1) is a direct adaptation of the algebraic projection formula [229]. Property (2) is a consequence of (1) and of the faithful flatness of CY∞ as an OY -module (Lemma 5.30). Part (3) and (4) follow now straightforwardly from Grauert’s cohomology base change theorem.  Remark 5.32. Points 1, 2, 3 in the previous theorem hold true if WITi is replaced 4 by even or odd WIT, and ITi by even or odd IT.

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Chapter 5. Nahm transforms

5.3.3

Relative Dirac operators

Let us assume that f : Z → Y is a proper holomorphic submersive morphism, ahler structure, i.e., it has a Hermitian and that the bundle Ω1Z/Y has a relative K¨ metric such that the corresponding 2-form in Ω2Z/Y is closed under the relative exterior differential. We define locally free CZ∞ -modules M 0,k M 0,k ΩZ/Y , Σ+ = ΩZ/Y , Σ− = k odd

Σ = Σ+ ⊕ Σ− .

k even

The Hermitian metric on Ω1Z/Y induces Hermitian metrics on the bundles Σ± . Let now E → Z be a relatively holomorphic vector bundle with a Hermitian structure. By considering the relative Dolbeault operator as a morphism ∂¯E/Y : f∗ (E ∞ ⊗CZ∞ Σ) → f∗ (E ∞ ⊗CZ∞ Σ) , we define a relative Dirac operator ∗ D = ∂¯E/Y + ∂¯E/Y : f∗ (E ∞ ⊗CZ∞ Σ+ ) → f∗ (E ∞ ⊗CZ∞ Σ− ) .

Theorem 5.33. Let E → Z be a relatively holomorphic vector bundle with a Hermitian structure, and let D be the corresponding relative Dirac operator. If E satisfies the odd IT condition, then ker D = 0. As a consequence, −ind(D) is a vector bundle and −ind(D) = coker D = ker D∗ . Moreover, there is a natural isomorphism of CY∞ -modules M ∼ −ind(D) = coker D → Ri f∗ E r . odd i

Proof. Since the higher direct images of E r are computed by the direct image of the relative Dolbeault complex (cf. Proposition 5.28), a direct calculation shows L that ker D = even i Ri f∗ E r = 0, thus proving the first claim. Analogously, one has M ∗ ∼ R i f∗ E r . → ker D∗ = ker ∂¯E/Y ∩ ker ∂¯E/Y odd i

 Corollary 5.34. If, in addition to the hypotheses of Theorem 5.33, f : Z → Y is flat, and E is holomorphic, then the bundle −ind(D) admits a natural holomorphic L Ri f E is a locally free OY -module of finite rank, and structure. Indeed, L odd i i ∗ L the natural map odd i R f∗ E → odd i Ri f∗ E r induces an isomorphism of CY∞ modules M M ∼ ∼ − ind(D) . Ri f∗ E ⊗OY CY∞ → R i f∗ E r → (5.10) odd i

odd i

5.3. Compatibility between Nahm and Fourier-Mukai

171

An analogous result is obtained by replacing “odd” with “even” and inverting the roles of ker D and coker D. Proof. For every y ∈ Y , let ψyi : Ri f∗ E ⊗OY Oy → H i (Zy , Ey ) be the natural map. Since H i (Zy , Ey ) = 0 for even i, Grauert’s semicontinuity theorem implies that Ri f∗ E = 0 for even i, and then the morphisms ψyi are surjective for even i. Cohomology base change implies that every ψyi is an isomorphism for odd i, and the for odd i, the sheaves Ri f∗ E are locally free. The isomorphism (5.10) now follows. 

5.3.4

K¨ahler Nahm transforms

By means of the techniques developed in the previous section, one can define a high-dimensional version of the Nahm transform introduced in Section 5.2. As a consequence, we also establish the compatibility between the Fourier-Mukai and Nahm transforms for K¨ ahler surfaces. Let X, Y be compact K¨ ahler manifolds. Fix on Z = X ×Y a Hermitian vector bundle Q equipped with a connection ∇Q whose curvature is of type (1, 1). Thus ∇Q induces a holomorphic structure on Q, and let Q → Z denote its locally free sheaf of holomorphic sections. We shall denote as usual by πX , πY the projections of X × Y onto its factors. Moreover, let (E, ∇) → X be a Hermitian vector bundle with a connection whose curvature is of type (1, 1) on X. Again, ∇ induces a holomorphic structure on E. We shall again denote by E the corresponding sheaf of holomorphic sections. ∗ E ⊗ Q satisfies Let us assume further that the Hermitian vector bundle πX the odd IT condition with respect to the projection πY . Then, as we saw in Theorem 5.33, minus the index of the relative Dirac operator D associated with these ˆ Moreover, we data is a holomorphic vector bundle on Y , which we denote E. ˆ is isomorphic with the sheaf know that the sheaf Eˆ of holomorphic sections of E L i ∗ ˆY ∗ (πX E ⊗ Q). In this context, we have: odd i R π

Proposition 5.35. Let X, Y be compact K¨ ahler manifolds and let ΦQ X→Y be the Fourier-Mukai functor with kernel Q. Then the sheaf Eˆ of holomorphic sections of ˆ is isomorphic to ΦQ (E). E X→Y The geometric data we have fixed at the outset also induce a Hermitian ˆ on the holomorphic bundle E; ˆ clearly metric and a compatible Chern connection ∇ ˆ ˆ ˆ the curvature of ∇ is of type (1,1). The pair (E, ∇) will be called the K¨ ahler Nahm transform of (E, ∇). A notation for this transform which is in line with Q (E, ∇). It can be our notation for the integral functors of Chapter 1 is KNX→Y thought as a map from the space of gauge equivalence classes of connections on ˆ →Y. E → X whose curvature are of type (1,1) to the same space on E

172

Chapter 5. Nahm transforms

We examine now this construction more closely. We claim that the induced ˆ can also be obtained via a projection formula. Chern connection ∇ Indeed, for every y ∈ Y let Qy = Q|X×{y} and let ∇Qy be the connection induced on it. Moreover let us define bundles S± , S by M 0,k M 0,k S− = ΩX , S+ = ΩX , S = S+ ⊕ S− . k odd

k even

We have of course S+ ' (Σ+ )|X×{y} for all y ∈ Y , etc., where the Σ’s are the bundles defined in Section 5.3.3. Note that the K¨ ahler metric of X induces Hermitian metrics on the bundles S± and S, as well as compatible connections on them. By coupling the connections ∇, ∇Qy and the induced connections on S± , we obtain connections on the bundles E ⊗ S± ⊗ Qy and a family of (twisted) Dirac operators Dy : Γ(E ⊗ Qy ⊗ S+ ) → Γ(E ⊗ Qy ⊗ S− )

(5.11)

which are no more than the specializations to the fibers of πY of the relative Dirac ˜ ± on operator introduced in Section 5.3.1, twisted by the coupled connections ∇ ∗ E ⊗ Q ⊗ Σ± . Here Γ denotes the spaces of global C ∞ sections. The the bundle πX spaces Γ(E ⊗ Qy ⊗ S± ) have natural inner products given by the K¨ahler and the various Hermitian metrics, so that they may be completed to Hilbert spaces Hy± . Since the holomophic bundle E satisfies the odd IT condition, we have for every y ∈ Y an exact sequence (cf. Eq. (5.6)) Dy∗

ˆy → Hy− −−→ Hy+ → 0 . 0→E

(5.12)

The spaces Hy± may be regarded as the fibers of vector bundles H ± on Y of infinite rank and the exact sequence (5.12) is then an exact sequence of vector bundles ∗

D ˆ → H− − 0→E −→ H + → 0 .

The inner products in the spaces Hy− induce Hermitian inner products in the ˆ This also defines a ˆ has a Hermitian fiber metric h. ˆy so that the bundle E fibers E − ˆ projector Π : H → E. Let us now come to the connection. On the bundles H ± one can define connections ∇± according to the following covariant derivative rule: e± ∇± α (s) = ∇αX (s) where s is regarded as a section of H ± in the left-hand side, and as a section in ∗ E ⊗ Q ⊗ Σ± ) on the right. α is a vector (the Hilbert space completion of) Γ(πX X ˆ is now field on Y , and α is its natural lift to Z = X × Y [43]. The connection ∇ defined as ˆ = (Π × Id) ◦ ∇− . ∇

5.4. Nahm transforms on hyperk¨ ahler manifolds

173

The operators Dy∗ vary holomorphically with y. Then standard arguments (cf. [102, Theorem 3.2.8]) show that this connection is compatible both with the Hermitian ˆ and therefore coincides with the Chern metric and the holomorphic structure of E, ˆ ˆ h). connection of the holomorphic Hermitian bundle (E, Now assume that X and Y fit into the framework of Section 5.2, i.e., dim X = 2, Y is a connected component of the moduli space of instantons on X (with respect to the K¨ahler metric on X) and ∇ is anti-self-dual. Let (Q, ∇Q ) → X × Y be the universal bundle with connection, as described in Section 5.2.2. We have (see [162, Theorem 3]): Proposition 5.36. If (Q, ∇Q ) → X × Y is the universal bundle with connection, as described in Section 5.2.2, then its curvature is of type (1, 1). Proof. The proof follows from some easy computations, by taking into account the explicit form of the curvature of the universal connection (see Section 5.1.2) and the complex structure of the moduli space Y .  As discussed in Section 5.2, if X has nonnegative scalar curvature, then ∗ E ⊗ Q satisfies the odd IT condition with respect to the projection πY . It is πX Q (E, ∇) of (E, ∇) in fact then easy to see that the K¨ ahler Nahm transform KNX→Y ˆ ∇), ˆ the Nahm transform of (E, ∇) as discussed in Section 5.2. coincides with (E,

5.4

Nahm transforms on hyperk¨ahler manifolds

When the base manifold X has a hyperk¨ ahler structure, we may address the problem of whether the Nahm transform maps instantons to instantons. As a matter of fact the presence of the hyperk¨ ahler structure will allow us to consider a generalization of the notion of instanton (that of quaternionic instanton) which makes sense on a hyperk¨ ahler manifold of any dimension. We shall be able to see that the Nahm transform preserves the property of being a quaternionic instanton, thus generalizing the property of preservation of the instanton condition on 4-tori as proved in [230, 57, 263, 102].

5.4.1

Hyperk¨ahler manifolds

Let us start by recalling the basic notions concerning hyperk¨ahler manifolds. General references about hyperk¨ ahler manifolds are [262, 40, 147] We recall that a 4k-dimensional Riemannian manifold (M, g) is said to be a quaternionic K¨ ahler manifold (resp. a hyperk¨ ahler manifold ) if its holonomy group is contained in the group Sp(k) Sp(1) = (Sp(k) × Sp(1))/Z2 (resp. Sp(k)). A

174

Chapter 5. Nahm transforms

hyperk¨ahler manifold may be alternatively defined as Riemannian manifold (X, γ) equipped with three complex structures I, J, K that fulfil the following properties: 1. they satisfy the algebra of the quaternions, i.e., IJ = K,

KI = J,

JK = I,

I 2 = J 2 = K 2 = −Id ;

2. each of the complex structures I, J, K makes the Riemannian manifold (X, γ) into a K¨ ahler manifold. Let us denote by ωI , ωJ , ωK the three corresponding K¨ahler forms. One easily checks that the 2-form ΩI = ωJ + iωK is holomorphic with respect to the complex structure I, and is therefore a holomorphic symplectic form. As a consequence, the canonical bundle of X equipped with the holomorphic structure I is trivial. Analogous statements are obtained by cyclically permuting the complex structures I, J, K. One should also notice that if a, b, c are real numbers such that a2 + b2 + c2 = 1, then aI + bJ + cK is a complex structure which again makes (X, γ) into a K¨ahler manifold. Therefore any hyperk¨ ahler manifold comes with a family of ahler K¨ahlerian structures parameterized by a 2-sphere S 2 . We call this the hyperk¨ family of complex structures of X.

5.4.2

A generalized Atiyah-Ward correspondence

In order to study the Nahm transform for hyperk¨ahler manifolds, we shall need a generalization of the classical Atiyah-Ward correspondence, which relates instantons on S 4 to holomorphic vector bundles on the complex projective 3-space P3 , satisfying suitable properties [20, 16]. This will rely on the treatment of [27], which in turn mainly (but not only) reproduces results in [262, 40, 208, 236]. We shall start by introducing the twistor space Z of a (connected) quaternionic K¨ahler manifold X, following [147]. Let P be holonomy bundle of X, i.e., the principal bundle obtaining by reducing the structure group of the bundle of linear frames of X to the holonomy group G [185, 40]. The space Z can be defined as the sphere bundle of the associated bundle W = P ×G sp(1), where G acts on the Lie algebra sp(1) by its adjoint action. (To be more precise, note that G is a subgroup of Sp(k) Sp(1). The Lie algebra of the latter group is isomorphic to sp(k)⊕sp(1), hence one can restrict the adjoint action of G to sp(1); one can check that sp(1) is preserved by such action.) Any orthonormal basis of local sections of W pointwise defines a triple {Ii } of complex structures in the tangent spaces Tx X, which satisfy the quaternionic algebra Ii Ij = −δij Id + ijk Ik .

5.4. Nahm transforms on hyperk¨ ahler manifolds

175

In terms of this trivialization, one makes the identifications u = (x, z) and Tu Z = Tx X ⊕ Tz P1 , and the tangent space Tu Z has the complex structure   1 − z z¯ z + z¯ z − z¯ I1 + I2 + i I3 , I 0 , (5.13) 1 + z z¯ 1 + z z¯ 1 + z z¯ where I0 is the complex structure of P1 . The resulting almost complex structure of Z is integrable (if k = 1, one needs the additional assumption that M is half conformally flat, i.e., the self-dual part of the Weyl 2-form must vanish, cf., e.g., [40]). According to (5.13), the points u ∈ Zx = p−1 (x) parameterize complex structures in Tx X: the complex structure labeled by u is the one induced by the projection p∗ : Tu Z → Tp(u) X. If k = 1 all complex structures of Tx M are recovered by varying u in Zx , while for k > 1 this is not anymore true. On the twistor space there is a naturally defined antihomolorphic involution τ : Z → Z which preserves the fibers of p. Its differential acts as α, ι∗ (β)), τ∗ (α, β) = (¯ where ι is the antipodal map ι : P1 → P1 . This implies the identities τ ◦ ∂ = ∂¯ ◦ τ,

τ ◦ ∂¯ = ∂ ◦ τ.

(5.14)

P3 The endomorphism i=1 Ii ⊗ Ii of Λ2 Tx∗ X has real eigenvalues 3 and −1, and correspondingly Λ2 Tx∗ X splits into the eigenspaces [208] Λ2 Tx∗ X = (e1 )x ⊕ (e2 )x . The space (e1 )x ' sp(n) admits the further identification \ ∗ (e1 )x ' Λ1,1 u Tx X

(5.15)

u∈Zx ∗ where Λ1,1 u Tx X is the space of 2-forms of type (1,1) with respect to the complex structure in Tx X parameterized by the point u ∈ Zx .

Definition 5.37. A rank n quaternionic instanton on quaternionic K¨ahler manifolds X is a pair (E, ∇), with E a rank n complex smooth vector bundle on X, and ∇ a connection on E such that its curvature at x ∈ X takes values in (e1 )x ⊗ End(Ex ). 4 By using the Hitchin-Kobayashi correspondence (Section 5.1.3) and a result by Verbitsky [290], one can easily prove the following characterization of quaternionic instantons on hyperk¨ ahler manifolds as “hyperstable bundles.”

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Chapter 5. Nahm transforms

Proposition 5.38. A complex vector bundle on a hyperk¨ ahler manifold X is an irreducible quaternionic instanton if and only it is a holomorphic stable bundle of degree zero with respect to any K¨ ahler structure in the hyperk¨ ahler family of X. Remark 5.39. Contrary to what happens in the case of complex dimension two, in the higher dimensional case a line bundle on a hyperk¨ahler manifold is not necessarily a quaternionic instanton. Indeed, its first Chern class may fail to be orthogonal to all the three basic K¨ ahler forms of X. 4 Let us now study the generalized Atiyah-Ward correspondence for quaternionic instantons. We start by considering the Hermitian case, namely, we assume that E is a complex vector bundle on a quaternionic K¨ahler manifold, and that E is equipped with a Hermitian metric h and a quaternionic instanton ∇ which is compatible with the Hermitian metric h. Because of (5.15), the pullback connection p∗ ∇ induces a holomorphic structure on the pullback bundle W = p∗ E (so that (p∗ ∇)0,1 = ∂¯W ). Then W is holomorphically trivial along the fibers of p. We define a bundle morphism σ : W → W ∗ by letting σ(s1 )(s2 ) = (p∗ h)(s1 , τ 0 (s2 )) , where the antiholomorphic bundle automorphism τ 0 : W → W is the lift of the involution τ . Then σ is an antilinear antiholomorphic bundle isomorphism covering τ , and induces a positive definite Hermitian form on the spaces H 0 (Zx , Wx ). The map σ is a real positive form according to Atiyah’s terminology [16]. Thus, with any quaternionic instanton on X we may associate a holomorphic vector bundle on Z, holomorphically trivial along the fibers of p, carrying a positive real form. Let us now describe the inverse correspondence. Let W be such a bundle on Z. By the triviality requirement, there is a bundle E on X such that W = p∗ E. Since two points e1 , e2 ∈ Ex may be regarded as elements in H 0 (Zx , Wx ), the real positive form σ induces a Hermitian structure h on E. ˜ be the connection on W uniquely determined by the compatibility Let ∇ with the holomorphic structure of W and the Hermitian structure induced on W ˜ is the Chern connection of the by σ (which obviously coincides with p∗ h); i.e., ∇ ˜ descends to a connection on E if Hermitian bundle (W, p∗ h). The connection ∇ and only if the curvature F∇ ˜ is horizontal with respect to p. Let us briefly show that this is indeed the case. Let Ix be the ideal of the fiber Zx . As in [16] one can produce an isomorphism ∼ H 0 (Z , F ⊗ O /I ) , H 0 (Zx , F ⊗ OZ /Ix2 ) → x Z x

(5.16)

where F ⊗ OZ /Ix may be identified with the sheaf of holomorphic sections of Fx . Due to the isomorphism (5.16), we may choose in a neighborhood U of Zx a basis {si } of holomorphic sections of F which restricted to Zx yields a unitary

5.4. Nahm transforms on hyperk¨ ahler manifolds

177

basis of holomorphic sections of Fx . If ωij is the matrix-valued connection 1-form ˜ we have of ∇, 0,1 ∈ Ix2 Ω0,1 (U ), ωij

ωij + ω ¯ ji ∈ Ix2 Ω1 (U ),

(si , sj ) − δij ∈ Ix2 .

By computing the local curvature forms we see that the curvature is horizontal. The induced connection ∇ on E is compatible with the Hermitian metric h, and we need only to show that at each point x ∈ M the curvature F∇ of ∇ takes values in (e1 )x ⊗ EndEx . Now, the two-form (F∇ ˜ )u is of type (1,1) at any point u ∈ Zx , so that the two-form (F∇ )x at x is of type (1,1) with respect to the complex structure on Tx M parameterized by u. Since this happens for all u ∈ Zx , Equation (5.15) implies that (R∇ )x lies in (e1 )x . We have therefore proved the following result. Theorem 5.40. There is a one-to-one correspondence between the following objects: 1. gauge equivalence classes of rank r Hermitian quaternionic instantons on a quaternionic K¨ ahler manifold X; 2. isomorphism classes of rank r holomorphic vector bundles on the twistor space Z of X, holomorphically trivial along the fibers of Z, carrying a positive real form. Remark 5.41. In [236] a similar result was proved, but instead of the condition of holomorphic triviality along the fibers of p, it is assumed there that the holomorphic Hermitian bundle (F, k) restricted to the fibers is flat. The equivalence of the two constructions is easily established. 4 For the sake of completeness we also treat the non-Hermitian case. Clearly, if (E, ∇) is a quaternionic instanton on X, we may construct on the twistor space Z a holomorphic bundle W , which is holomorphically trivial along the fibers of the projection p. We show how to recover the quaternionic instanton (E, ∇) from such data on Z (in particular now we have no real form at our disposal). We define a differential operator D : Γ(W ) → Γ(W ⊗ Ω1,0 Z ) by letting D(s) = τ 0 ◦ ∂¯W ◦ τ 0 (s). Due to the identities (5.14), this operator satifies the Leibniz rule D(f s) = ∂f ⊗ s + f D(s), ˜ = D + ∂¯W is a connection on W , compatible with its complex structure so that ∇ 2 ˜ = 0, the curvature F∇ (in particular, since D2 = ∂¯W ˜ of ∇ is of type (1,1)). By the ˜ same argument as in the previous case, ∇ descends to a connection ∇ on E, such that (E, ∇) is a quaternionic instanton. One has therefore:

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Chapter 5. Nahm transforms

Theorem 5.42. There is one-to-one correspondence between the following objects: 1. gauge equivalence classes of quaternionic instantons of rank r on a quaternionic K¨ ahler manifold X; 2. isomorphism classes of holomorphic vector bundles of rank r on the twistor space Z of X which are holomorphically trivial along the fibers of Z.

5.4.3

Fourier-Mukai transform of quaternionic instantons

We introduce now a version of the Fourier-Mukai transform which is apt to study quaternionic instantons in that it uses in an essential way the twistor space. We restrict here to hyperk¨ ahler manifolds. There are several reasons for doing so: on hyperk¨ahler manifolds we can make full use of techniques from complex geometry; the product of two hyperk¨ ahler manifolds has a natural hyperk¨ahler structure; ahler manifold X is isomorphic to X × P1 as and the twistor space ZX of a hyperk¨ a smooth manifold, a condition which we shall need in the sequel. This Fourier-Mukai transform will map quaternionic instantons on a compact hyperk¨ahler manifold X to quaternionic instantons on a second hyperk¨ahler manifold Y . The product X × Y carries a natural hyperk¨ahler structure, namely, the only one compatible with the isomorphism of complex manifolds ZX×Y ' ZX ×P1 ZY . One has a commutative diagram ZX o

t1

t2

q

p1

 Xo

ZX×Y

π1

 X ×Y

/ ZY p2

π2

 /Y

where the horizontal arrows are holomorphic morphisms while the vertical ones are just smooth. Moreover we denote by ρ1 : ZX → P1 ,

ρ2 : ZY → P1 ,

$ : ZX×Y → P1

the holomorphic projections of the twistor spaces onto the projective line. Let now E be the sheaf of holomorphic sections of a Hermitian quaternionic instanton (E, ∇) on X, and let (P, ∇ ) be a Hermitian quaternionic instanton on X ×Y . Let E˜ and P˜ denote the sheaves of holomorphic sections of the bundles p∗1 E and q ∗ P , respectively. After setting W = π ∗ E ⊗ P , and endowing the bundle W with the product connection, we denote by Wz the sheaf of holomorphic sections

5.4. Nahm transforms on hyperk¨ ahler manifolds

179

of W determined by the the complex structure Iz on X, where z ∈ P1 . If one sets ˜ = t∗ E˜ ⊗ P, ˜ there is an identification of holomorphic vector bundles W| ˜ $−1 (z) = W 1

Wz . We assume now that (E, ∇) satisfies one of the even or odd IT conditions (cf. Definition 5.29; when this happens, we say that (E, ∇) “is IT”). One should notice here that this does not depend on the choice of a complex structure in the hyperk¨ahler family of X; in fact, the cohomology groups H i (Xz × {y}, Wz |Xz ×{y} ) do not depend on the choice of z (here Xz is X with the complex structure z, i.e., Xz = ρ−1 (z)). This is proved in [290]. According to the results of the previous sections, the smooth vector bundle underlying the holomorphic bundle ⊕i Ri π2∗ (Wz ) may be identified with the bundle ±ind(Dz ), where {Dz }z∈P1 is a family of Dirac operators in the sense of Atiyah-Singer’s index theorem; the plus (minus) sign holding when the even (odd) IT condition is satisfied. All the smooth Hermitian bundles ±ind(Dz ) are isomorphic, since all the operators Dz∗ Dz have isomorphic (co)kernels, and will be denoted by ±ind(D). Furthermore, this bundle is endowed with a Hermitian connection, which is compatible with all complex structures on X. We notice that, if u ∈ ZY , then t−1 2 (u) = Xz , with z = ρ2 (u); therefore, we ˜ −1 = W|X ×{p (u)} . We then have: make the identifications W| z 2 t (u) 2

˜ is IT with respect to t2 : ZX×Y → ZY . Lemma 5.43. If (E, ∇) is IT, then W ˜ t −1 (u) ) = H i (Xz , W|X ×{u} ). Proof. It suffices to observe that H i (t2 −1 (u), W z 2



We consider the following commutative diagram: Xz × Yz

jz

π2

 Yz

t2 iz

/ P1 x< x xx xxρ2 x  x

/ ZX×Y

$

/ ZY

where iz and jz are the natural holomorphic embeddings. The square on the left is Cartesian; indeed, Xz × Yz = $−1 (z) and Yz = so that ZX×Y ×ZY Yz = t2 −1 (Yz ) = Xz × Yz . Therefore, although iz is not ˜ → Ri π2∗ Wz (notice that j ∗ W ˜ ' Wz ). flat, we have natural maps µi : i∗z Ri t2∗ W z ˜ is ±ind(D), ˜ We remark that the smooth bundle underlying ⊕i Ri t2∗ (W) ˜ where D is a family of Dirac operators that, when restricted to Xz , coincides ρ−1 2 (z),

with Dz . Thus we have proved: Lemma 5.44. The natural morphism ⊕i µi is an isomorphism of holomorphic Her˜ ' ⊕i Ri π2∗ Wz . mitian vector bundles i∗z ⊕i Ri t2∗ W

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Chapter 5. Nahm transforms

˜ For any z ∈ P1 , one has an isomorphism of Let us denote G˜ = ⊕i Ri t2∗ W. ∗ ˜∞ smooth bundles (iz ◦ p2 ) G ' p2 ∗ (±ind(D)). By applying the Riemann-Roch formula, it is easy to check that the restriction of G˜∞ to each P1 -fiber is trivial (indeed, c1 (G˜∞ |{y}×P1 ) = 0 for any y ∈ Y ). In this way, we see that G˜∞ is the pullback of a smooth bundle on Y , which can be identified with ±ind(D). The pullback of the Hermitian connection on ±ind(D) is a Hermitian connection on G˜∞ , which is compatible with the holomorphic structure of G˜ by virtue of Lemma 5.44. This implies that G˜ is holomorphically trivial along the fibers of the twistor projection p2 . In particular, the smooth vector bundle ±ind(D) carries the structure of a quaternionic instanton on Y . Summing up, we have proved: Theorem 5.45. Let (E, ∇) be a quaternionic instanton on X which satisfies one of the IT conditions. Then ⊕i Ri π2∗ (π1∗ E ⊗ P) is the sheaf of holomorphic sections of a quaternionic instanton on Y .

5.4.4

Examples

The only compact hyperk¨ ahler surfaces are the complex 2-tori and the K3 surfaces (see, e.g., [40]). The case of complex 2-tori was considered in Section 5.2.4. Another example is provided by the strongly reflexive K3 surfaces of Chapter 4. Using ˆ which is the Hitchin-Kobayashi correspondence, the “dual” reflexive surface X, a moduli space of µ-stable locally free sheaves, may be identified with a moduli ˆ space of instantons (note indeed that the locally free sheaves parameterized by X have zero degree). According to results given in [200], the two moduli spaces can be identified as complex manifolds. Moreover, the universal instanton bundle Q ˆ admits a holomorphic structure, since the curvature of the on the product X × X universal connection ∇ is of Hodge type (1,1), and its sheaf of holomorphic section may be identified with the universal sheaf that we have constructed in Chapter 4. Now, every choice of complex structure in the hyperk¨ahler family of X induces a ˆ and in this way we obtain a hyperk¨ahler structure on the complex structure in X, ˆ product X × X. Arguments given in [162] imply that the pair (Q, ∇) is a Hermitian quaternionic instanton. Therefore we have all the ingredients for building up a hyperk¨ahler Fourier-Mukai transform on X. Our construction however shows that this coincides with the Fourier-Mukai transform built in Chapter 4, and under this correspondence, Theorem 5.45 coincides with Proposition 4.66. (Note that in this case, in the statement of Theorem 5.45 we need only to say “instanton” instead of “quaternionic instanton” since dim X = 2). We may consider the hyperk¨ ahler Fourier-Mukai transform for hyperk¨ahler tori of higher dimension (i.e., complex tori of even dimension). However we can remark here that, contrary to what happens in complex dimension 2, in higher

5.5. Notes and further reading

181

dimensions the Fourier-Mukai transform is not well behaved on stable bundles. ahler tori. Consider for instance a stable rank-two vector Let T1 , T2 be hyperk¨ bundle E on T1 with vanishing first Chern class, and a flat line bundle L on T2 . unneth formula [134, 6.7.8.1], one shows that Since E is IT1 and L is WIT2 by a K¨ the bundle F = E  L on T1 × T2 is WIT3 , and by considering its restrictions to subsets of the type T1 × {x2 } and {x1 } × T2 , one proves that it is stable. However, c has rank 1 c2 (E)c1 (L)2 = 0, i.e., it is a torsion the Fourier-Mukai transform W 2 sheaf.

5.5

Notes and further reading

Physics literature. There is an extensive physical literature relating Nahm transform and fundamental problems in physics, like quark confinement in QCD and string dualities. For the reader interested in these issues, we recommend for instance [121] (among other papers by Pierre van Baal) for the relevance of Nahm transform in QCD on the lattice and [88, 92, 172, 292] for the relations between Nahm transform and string theory. In [13] a version of Nahm transform for instantons over noncommutative 4-tori was introduced. Nahm transform in K-Theory. We would like to notice that the construction presented above is essentially topological, in the sense that its main ingredient is simply index theory. All the geometric structures used in Section 5.2 (spin structure, positivity of scalar curvature, hyperk¨ ahler metric, etc.) were needed either because a particular differential operator was used (i.e., the Dirac operator), or because we selected those objects (i.e., anti-self-dual connection over hyperk¨ahler manifolds) that yielded very particular transforms (anti-self-dual connections). One can conceive, for instance, a similar construction either based on a different pseudodifferential elliptic operator, other than the Dirac operator, or allowing for classes in K(T ), rather than actual vector bundles over the parameter space. Translation invariant instantons. There exists an ample literature on Nahm transforms of translation invariant instantons. Roughly speaking, these Nahm transforms yield a one-to-one correspondence between instantons on R4 invariant under the action of a lattice Λ and Λ∗ -invariant instantons on the dual R4 . Let us cite some cases: • The “trivial” case Λ = {0} is closely related to the celebrated ADHM construction of instantons, as described by Donaldson and Kronheimer [102]; in this case, Λ∗ = R4 , and an instanton on R4 corresponds to some algebraic data (ADHM data). This has been worked out by Corrigan and Goddard [90]. • Λ = R corresponds to monopoles; here, Λ∗ = R3 , and the transformed

182

Chapter 5. Nahm transforms object is, for SU (2) monopoles, an analytic solution of Nahm’s equations defined over the open interval (−1, 1) and with simple poles at the endpoints. This case was extensively studied by several authors, including Hitchin [146], Donaldson [98], Hurtubize and Murray [152], and Nakajima [231].

abo in his thesis [272] • The case Λ = R3 was treated by Sz´ • For Λ = Z4 we have the Nahm transform of Schenk [263], Braam and van Baal [57], and Donaldson and Kronheimer [102], defining a correspondence between instantons over two dual 4-dimensional tori, as discussed in Section 5.2.4. • Λ = Z corresponds to the so-called calorons, studied by Nahm [230], van Baal [287] and others (see [237] and the references therein and [86]); the transformed object is the solution of Nahm-type equations on a circle. • The case Λ = Z2 (doubly periodic instantons) has been analyzed in great detail in [41, 163, 165, 164]; here, Λ∗ = Z2 × R2 , and the Nahm transform gives a correspondence between doubly-periodic instantons and certain tame solutions of Hitchin’s equations on a 2-torus. • Λ = R × Z gives rise to the periodic monopoles considered by Cherkis and Kapustin [88]; in this case, Λ∗ = Z × R, and the Nahm dual data are given by certain solutions to Hitchin’s equations on a cylinder. • More recently, the case Λ = Z3 (spatially periodic instantons) has been studied by Charbonneau [85]; the transformed object is a singular monopole on a 3-torus. Previous work in that case had been done by van Baal [287, 288]. Instantons on ALE spaces. Asymptotically locally Euclidean spaces (ALE spaces) are hyperk¨ahler manifolds that are obtained as resolutions of singularities of quotients C2 /Γ, where Γ is a discrete subgroup of SU (2), acting on C2 in the standard way [189]. Instantons on ALE spaces have been first studied by Kronheimer and Nakajima [190] and have since then received a lot of attention. A Nahm transform for instantons on ALE spaces has been constructed by Bartocci and Jardim [30]. Other examples. The Nahm transform has been applied to study vortexes (in particular, some holomorphic triples over elliptic curves) by Garc´ıa-Prada, Hern´andez Ruip´erez, Pioli and Tejero [122]. A related construction is the Nahm transform for Higgs bundles as defined in [165] (Bonsdorff and Bartocci-Biswas studied FourierMukai tranforms for Higgs bundles in [23] [53]). Another significant application to the integer quantum Hall effect is due to Tejero [276].

Chapter 6

Relative Fourier-Mukai functors Introduction In this chapter we offer a quite comprehensive study of the relative Fourier-Mukai functors. We consider (proper) morphisms of algebraic schemes X → B, Y → B, and use an element in the derived category of the fibered product X ×B Y as a kernel to define an integral functor from the derived category of X to the derived category of Y . This generalizes what we have already seen in Chapter 1 when the morphisms X → B, Y → B are projections onto a factor of a product. We start by giving some general properties, in particular base change formulas, and a first example: Mukai’s relative transform for Abelian schemes [226]. We then move over to the case of elliptic fibrations (where by “elliptic fibration” we mean a proper flat morphism whose fibers are Gorenstein curves of arithmetic genus 1). Our treatment here may be divided in two parts, where we are able to achieve different degrees of generality in different directions. Indeed, we consider at first Weierstraß fibrations, leaving the dimension of the base scheme arbitrary. Given a Weierstraß fibration X → B, we construct a Poincar´e sheaf on the fibered product X ×B X and use this as a kernel to define a relative FourierMukai transform Db (X) → Db (X). We use this to provide a direct construction of the Altman-Kleiman compactified relative Jacobian of a Weierstraß fibration (proving that it is actually isomorphic to the original fibration). We also study this Fourier-Mukai transform in some detail, in particular computing the topological invariants of the transforms for elliptic surfaces and elliptic Calabi-Yau threefolds. The second approach we study in connection with elliptic fibrations is due to Bridgeland, and does not require the fibration to be of the Weierstraß type. While this works only in the case of elliptic surfaces, it has some advantages; for C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_6, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

183

184

Chapter 6. Relative Fourier-Mukai functors

instance, it may be used to study Fourier-Mukai partners of elliptic surfaces, as we shall do in Chapter 7. Then, we prove a generalization of Atiyah’s characterization of semistable sheaves on elliptic curves to the case of Weierstraß (possibly singular) elliptic curves. Moreover we prove the preservation of (semi)stability under the FourierMukai transform for such curves. This partly relies on our paper [29], where a result of this type was proved. This allows us to characterize the category of semistable sheaves on Weierstraß elliptic curves. These results are then generalized to Weierstraß fibrations, in particular describing all moduli spaces of relatively semistable sheaves, assuming that the base scheme B is normal. Some of the results given along this line in Section 6.4 are new. Partial results were contained in [29, 145], and in papers by Friedman, Morgan and Witten, by Donagi, by Bridgeland and by Yoshioka [114, 94, 60, 293]. The next topic we consider is the spectral cover construction. In particular we study how stable bundles on an elliptic fibration (in dimension 2 and 3) may be built out of spectral data. Spectral covers are particularly interesting for relatively semistable torsion-free sheaves of degree 0. In this case the spectral cover is a finite cover of the base, of degree equal to the rank of the sheaf. The FourierMukai transform of such relatively semistable sheaf turns out to be a rank one torsion-free sheaf on the spectral cover. Due to the invertibility of the transform, the sheaf may be recovered from its spectral data (the pair formed by the spectral cover together with a rank one torsion-free sheaf on it). These results may be used to study sheaves on the total space of a Weierstraß fibration which are (semi)stable in an “absolute” sense. The analysis is limited here to elliptic surfaces and elliptic Calabi-Yau threefolds since we need to know how the Fourier-Mukai transform acts on the topological invariants of the sheaves. We prove some instances of the preservation of this kind of (semi)stability under the Fourier-Mukai transform. This analysis is also useful to provide examples of absolutely µ-stable sheaves on elliptic Calabi-Yau threefolds, which are obtained out of spectral data. This is relevant to the construction of compatifications of the heterotic string. In this chapter we assume that the ground field k has characteristic zero; this allows us to apply Proposition 1.27 about fully faithful integral functors. However, most of Section 6.1 is true in arbitrary characteristic.

6.1

Relative integral functors

Let p : X → B, q : Y → B be proper morphisms of algebraic varieties. We denote ˜Y the projections of the fiber product X ×B Y onto its factors and by by π ˜X , π

6.1. Relative integral functors

185

˜Y the projection of X ×B Y onto the base scheme B. We have ρ=p◦π ˜X = q ◦ π a Cartesian diagram . X ×B HY HH π˜ vv H v HHY vv HH vv v H# zv ρ X II Y vv II p q vv II II vv II  vvv $ zv B

(6.1)

π ˜X

Given a “relative kernel” K• in the derived category D− (X ×B Y ), the relative integral functor with kernel K• is the functor Φ : D− (X) → D− (Y ) given by L

∗ • πY ∗ (L˜ πX E ⊗ K• ) . Φ(E • ) = R˜

This can be regarded as an integral functor with kernel j∗ K• in the derived category D− (X × Y ), where j : X ×B Y ,→ X × Y is the closed immersion of the fiber product. We can then apply all results about integral functors described in Chapter 1 to relative integral functors. In particular, WITi and ITi notions introduced in Definition 1.6 apply to this new situation. Assume now that K• is of finite Tor-dimension as a complex of OX -modules. As j∗ K• may fail to have this property, we cannot apply Proposition 1.4. Nevertheless, we can modify the proof of that proposition to show that Φ is bounded and can be extended to a functor Φ : D(X) → D(Y ) which maps Db (X) to Db (Y ). As in the absolute case, the composition of two relative integral functors is obtained by convoluting the corresponding kernels. So, given two kernels K• in D− (X ×B Y ) and L• in D− (Y ×B Z) corresponding to relative integral functors Φ and Ψ, the composition Ψ ◦ Φ has kernel in D− (X ×B Z) L

∗ πXZ∗ (L˜ πXY K• ⊗ L˜ πY∗ Z L• ) L• ∗B K• = R˜

where the morphisms π ˜XY , π ˜XZ and π ˜Y Z are the projections of the fiber product X ×B Y ×B Z onto the fiber products X ×B Y , X ×B Z and Y ×B Z.

6.1.1

Base change formulas

As we saw in Section 1.2.1, what makes relative integral functors interesting is their compatibility with base change. Let f : S → B be a morphism. For any morphism g : Z → B (a scheme over B), we denote by gS : ZS = Z ×B S → S and ∗ K• gives rise to a fZ : ZS → Z the induced morphisms. The kernel KS• = LfX× BY

186

Chapter 6. Relative Fourier-Mukai functors

relative integral functor ΦS : D− (XS ) → D− (YS ) L

∗ πYS ∗ (L˜ πX E • ⊗ KS• ) . ΦS (E • ) = R˜ S

If the original kernel K• is of finite Tor-dimension as a complex of OX -modules, then KS• is of finite Tor-dimension as a complex of OXS -modules, so that ΦS is bounded, and for every f : S → B it can be extended to a functor ΦS : D(XS ) → D(YS ), mapping Db (XS ) to Db (YS ). In the rest of this section we assume indeed that K• is of finite Tor-dimension as a complex of OX -modules. The proof of the following base change compatibility result is analogous to that of Proposition 1.8. Note that because of the flatness condition, base change in the derived category (Proposition A.85) can be applied. It is worth observing that, if the morphism p : X → B is flat, there is no need to assume that the base change morphism is flat (this fact is often neglected). Proposition 6.1. Assume either that f : S → B or p : X → B is flat. For every object E • in Db (X) there is a functorial isomorphism ∗ • E ) LfY∗ Φ(E • ) ' ΦS (LfX

in the derived category of YS . Let us assume that p : X → B is flat. If E • ∈ Db (X), by denoting by jt the immersions of both fibers Xt = p−1 (t) and Yt = q −1 (t) over a closed point t ∈ B into X ×B Y , one has (6.2) Ljt∗ Φ(E • ) ' Φt (Ljt∗ E • ) . Whenever the morphism q : Y → B is flat, from the base change formula (Proposition A.85) we also have (6.3) jt∗ Φt (G • ) ' Φ(jt∗ G • ) for every G • ∈ D(Xt ). A straightforward consequence of Proposition 1.11 and Equation (6.2) is the following result. Corollary 6.2. Assume that p : X → B is flat, and let E • be an object in Db (X). Then the derived restriction Ljt∗ E • to the fiber Xt is WITi for every t if and only if E • is WITi and Φi (E) is flat over B. When we transform a complex that reduces to a single sheaf E on X, we have Φi (E) = 0 for i > n = dim p + m0 , where dim p is the maximum of the dimensions of the fibers of p and m0 is the biggest index m such that Hm (K• ) 6= 0. We have a result analogous to Corollary 1.9.

6.1. Relative integral functors

187

Corollary 6.3. Let p : X → B be a flat morphism and E be a sheaf on X. The functor Φn is compatible with base change for sheaves, that is, one has Φn (E)t ' Φnt (Et ) , for every (closed) point t ∈ B, where Et = jt∗ E. Moreover, if E is flat over B one has: 1. for every (closed) point t in B there is a convergent spectral sequence i−j i i−j B E2−j,i (t) = TorO j (Φ (E), Ot ) =⇒ E∞ (t) = Φt (Et ) .

2. Assume that E is WITi and write Eb = Φi (E). Then for every t ∈ B there are isomorphisms of sheaves over Xt i−j B b TorO j (E, Ot ) ' Φt (Et ) ,

j ≤ i.

3. The restriction Et to the fiber Xt is WITi for every (closed) point t ∈ B if and only if E is WITi and Eb = Φi (E) is flat over B. In that case the functor b t ' Ebt for every Φi is compatible with base change for sheaves, that is, (E) point t ∈ B. Proposition 6.4. Let p : X → B be a flat morphism and E be a sheaf on X flat over B. The set U of points in B such that the restriction Et of E to the fiber Xt is WITi has a natural structure of open subscheme of B. Proof. Given a point t ∈ B, we consider the flat base change B{t} = Spec OB,t ,→ B where OB,t is the local ring of B at the point t. By Proposition 6.1, the restriction Φj (E)B{t} of Φj (E) to the fiber product Y{t} = B{t} ×B Y is isomorphic to ΦjB{t} (EB{t} ). Then Corollary 6.3 applied to p : X{t} = B{t} ×B X → B{t} implies that Et is WITi for a closed point t ∈ B, if and only if Φj (E)B{t} = 0 for j 6= i and Φi (E)B{t} is flat over B{t} . By the generic flatness criterion [214, 22.B], the set of the points t ∈ B such that the two last conditions are fulfilled is open.  We are now going to apply Proposition 6.4 to the particular situation of a rela• b tive integral functor induced by an ordinary integral functor Φ = ΦK X→Y : D (X) → b • D (Y ), where X and Y are smooth connected proper varieties and K is an object of Db (X × Y ) of finite Tor-dimension as a complex of OX -modules. Following

188

Chapter 6. Relative Fourier-Mukai functors

Section 1.2.1, we consider the base change diagram X × X ×NY NNN p NNπ˜N13 p NNN p p p N& xpp ρ X × XN X ×Y NNN p π1 pppp NNπN1 NNN pp NN'  xppppp X π ˜ 12 pppp

,

where π ˜ij denote the projection onto the (i, j)-factor, and the relative integral • ∗ =π ˜23 K• . functor ΦX from D− (X × X) to D− (X × Y ) with kernel KX Proposition 6.5. The set U of points x in X such that the skyscraper sheaf Ox is WITi with respect to Φ has a natural structure of open subscheme of X. Proof. We apply Proposition 6.4 to the integral functor ΦX : Db (X×X) → Db (X×  Y ) taking as E the structure sheaf O∆ of the diagonal in X × X. By Lemma 2.46, there is an irreducible component Z of the support of K• such that pX = πY |Z : Z → X is dominant. Proposition 6.6. Assume that there is a (closed) point x ∈ X such that Φ(Ox ) ' Oy [i] for some (closed) point y ∈ Y and some integer i. If Ze is the normalization of Z, the induced morphisms p˜X : Ze → X and p˜Y : Ze → Y are birational. Thus, X and Y are K-equivalent (cf. Definition 2.47). −1 Proof. Since pX is dominant, dim p−1 X (x) ≥ dim Z − dim X. Since pX (x) = {y}, one has dim Z = dim X. Since X and Y have the same dimension (cf. Theorem 2.38), we can apply Proposition 2.48 to conclude. 

6.1.2

Fourier-Mukai transforms on Abelian schemes

An instance of a relative integral functor is provided by the Fourier-Mukai transform on Abelian schemes [226]; this has indeed been the first example of a such a transform. An Abelian scheme p : X → B over a scheme B is a proper flat morphism such that there exist morphisms of B-schemes mX : X × X → X,

ιX : X → X,

e: B → X

so that the relations described at the beginning of Section 3.1 are satisfied. In analogy with the absolute case, one proves the existence of an Abelian scheme ˆ = Pic0 (X/B) → B which is a fine moduli space for line bundles whose pˆ: X restrictions to the fibers of p have vanishing first Chern class. Universality implies

6.2. Weierstraß fibrations

189

ˆ which we normalize by the existence of a Poincar´e line bundle P on X ×B X, ˆ of the projection X ×B X ˆ→ imposing that its restriction to the section e(B)×B X ˆt ˆ is trivial. For every closed point t ∈ B the restriction of P to the fiber Xt × X X ˆt. ˆ coincides with the normalized Poincar´e bundle on Xt × X of X ×B X ˆ The line bundle P defines a relative integral functor Φ : D− (X) → D− (X) by letting ∗ • πX∗ πX E ⊗ P) . Φ(E • ) = R˜ ˆ (˜

ˆ For the same reason, Since P is a line bundle, the functor Φ maps Db (X) to Db (X). and using formula (C.12), one proves as in Proposition 1.13 that the relative ∗ ˆ → Db (X) defined by the kernel P ∗ ⊗ π ˜X ωX/B [g] integral functor from Ψ : Db (X) is a right adjoint to the functor Φ (here g is the relative dimension of the Abelian scheme p : X → B). Proposition 6.7. The relative integral functor Φ is a Fourier-Mukai transform. Proof. We know that the composition Ψ ◦ Φ : Db (X) → Db (X) is the relative ∗ πX ωX/B [g])∗B P, integral functor with kernel given by the convolution M• = (P ∗ ⊗˜ b ˆ b ˆ and that the composition Φ ◦ Ψ : D (X) → D (X) is the relative integral functor ∗ ˜X ωX/B [g]). We first prove that Φ is fully faithful. By with kernel N • = P ∗B (P ∗ ⊗ π Remark 1.21 it suffices to show that Ψ ◦ Φ is fully faithful. Actually, we shall prove that it is an equivalence of categories. In view of the base change property given in ˆt) Proposition 6.1, for any point t ∈ B the composition Ψt ◦ Φt : Db (Xt ) → Db (X is the integral functor with kernel Ljt∗ M• . By Theorem 3.2 the functor Φt is a Fourier-Mukai transform, the composition Ψt ◦ Φt is isomorphic to the identity functor, and its kernel is isomorphic to O∆t , where ∆t is the diagonal in Xt × Xt . Proposition 1.11 then implies that M• is isomorphic in the derived category to a sheaf M, flat over B, such that jt∗ M ' O∆t for every point t ∈ B. Moreover, M is topologically supported on the image of the diagonal immersion δ : X ,→ X ×B X. Let us denote by L = δ ∗ M the restriction of M to the diagonal; we have an epimorphism M → δ∗ L → 0. The condition jt∗ M ' O∆t implies that jt∗ L ' OXt for every t ∈ B and that M → δ∗ L induces an isomorphism jt∗ M ' jt∗ δ∗ L for every point t ∈ B. Since M and δ∗ L are flat over B, the morphism M → δ∗ L is an isomorphism as well. As a consequence, the composed functor Ψ ◦ Φ coincides with the operation of tensoring by L, so it is an equivalence of categories. A similar argument proves that Φ◦Ψ is an equivalence of categories, and that Ψ is fully faithful as well. This implies that Φ is an equivalence of categories. 

6.2

Weierstraß fibrations

Elliptic fibrations yield examples of relative Fourier-Mukai transforms that are of great interest in view of their geometric and physical applications [28, 60, 29, 71,

190

Chapter 6. Relative Fourier-Mukai functors

82, 145, 6]. We shall adopt the following definition of elliptic fibration (not the most general). Definition 6.8. Let B be an integral and projective scheme. An elliptic fibration over B is a proper flat morphism of schemes p : X → B whose fibers are Gorenstein curves of arithmetic genus 1. 4 If X is smooth, the generic fiber of p is a smooth elliptic curve, but singular fibers are allowed. The simplest nontrivial examples are provided by elliptic surfaces. Definition 6.9. A relatively minimal elliptic surface is an elliptic fibration p : X → B such that B is a smooth projective curve, X is smooth, and there are no (−1)4 curves (i.e., rational curves C with C 2 = −1) contained in the fibers. Relatively minimal elliptic surfaces were classified by Kodaira [186], who described all types of singular fibers which may occur (the so-called Kodaira curves). Elliptic fibrations whose base is a smooth surface have been studied by Miranda [219], who showed that the configuration of singular fibers can be more complicated than in the case of elliptic surfaces. We say that a sheaf over an elliptic fibration p : X → B is relatively torsionfree if it is flat over B and its restriction to every fiber is torsion-free. In an analogous way one defines the notion of relative µS -(semi)stability (cf. Definitions C.3 and C.4). If E • in Db (X) is a complex of finite Tor-dimension its relative degree is the intersection number (6.4) d(E • ) = c1 (E • ) · f , where f ∈ Am (X) is the class of the generic fiber of p (here m = dim B). If F is a sheaf on X flat over B, its relative degree is the degree of the restriction Ft to any fiber Xt of p. The pair (rk(E • ), d(E • )) (cf. Section 1.1 for the definition of rank in the derived category) is called the relative Chern character of E • . If rk(E • ) 6= 0, the rational number µ(E • ) = d(E • )/ rk(E • ) is the relative slope.

6.2.1

Todd classes

We now focus on a particular, though very important, kind of elliptic fibration. Definition 6.10. A Weierstraß fibration is an elliptic fibration p : X → B such that the fibers of p are geometrically integral and there exists a section σ : B ,→ X of p whose image Θ = σ(B) does not contain any singular point of the fibers. 4 We notice that the singular fibers can have at most one singular point, either a cusp or a simple node.

6.2. Weierstraß fibrations

191

By cohomology base change one shows that the sheaf p∗ ωX/B is a line bundle and ωX/B ' p∗ (p∗ ωX/B ). Adopting standard notation, we set ω = R1 p∗ OX ' (p∗ ωX/B )∗ , where the isomorphism is given by the Grothendieck-Serre duality for p (cf. Eq. (C.12)). Then (6.5) ωX/B ' p∗ ω ∗ . ¯ = c1 (p∗ ωX/B ) = −c1 (ω), the adjunction formula for Θ ,→ X gives If K ¯. Θ2 = −Θ · p∗ K

(6.6)

By [220, Lemma II.4.3], a Weierstraß fibration p admits a Weierstraß form, which one can construct in the following way. Let us consider the projective bundle p¯: P = P(E ∗ ) = Proj(S • (E)) → B, where ∼ O ⊕ ω ⊗2 ⊕ ω ⊗3 . E = p∗ OX (3Θ) → B The divisor 3Θ is relatively very ample and induces a closed immersion of Bschemes j : X ,→ P such that j ∗ OP (1) = OX (3Θ). The normal sheaf to the local complete intersection j, is ∼ p∗ ω −⊗6 ⊗ O (9Θ) . NX/P → X To prove this one takes the Euler exact sequence 0 → ΩP/B → p¯∗ E(−1) → OP → 0 . and apply relative duality (Proposition C.1) to the sheaf ^ ∼ p¯∗ ω ⊗5 (−3) . ΩP/B → ωP/B = The morphism p : X → B is a local complete intersection (in the sense of Fulton [119, 6.6]) and has a virtual relative tangent bundle TX/B = [j ∗ TP/B ] − [NX/P ] in the K-group K • (X). Though TX/B is not a genuine sheaf, it still has Chern classes; in particular, it has a Todd class which can be readily computed [145]. Proposition 6.11. The Todd class of the virtual tangent bundle TX/B is ¯ + 1 (12Θ · p∗ K ¯ + 13p∗ K ¯ 2) p∗ K 12 1 ¯ 2 + terms of higher degree. − Θ · p∗ K 2

td(TX/B ) = 1 −

1 2

If B is smooth, its Todd class is given by the formula 1 1 1 td(B) = 1 + c1 (B) + (c1 (B)2 + c2 (B)) + c1 (B)c2 (B) + . . . . 2 12 24

192

Chapter 6. Relative Fourier-Mukai functors

Thus, we obtain an expression for the Todd class of X: 1 ¯ td(X) = 1 + p∗ (c1 (B) − K) 2 1 ¯ + p∗ (c1 (B)2 + c2 (B))) ¯ + 13p∗ K ¯ 2 − 3p∗ (c1 (B) · K) + (12Θ · p∗ K 12 1 ¯ · (c1 (B)2 + c2 (B))) + 12Θ · p∗ (K ¯ · c1 (B)) + [p∗ (c1 (B)c2 (B)) − p∗ (K 24 ¯ 2 ) − 6Θ · p∗ (K ¯ 2 · c1 (B))] + p∗ (c1 (B) · K + terms of higher degree. (6.7)

6.2.2

Torsion-free rank one sheaves on elliptic curves

Let Xt be a fiber of a Weierstraß fibration p : X → B. So, Xt can be any geometrically integral Gorenstein curve of arithmetic genus 1 (as we already noted, Xt has at most one singular point). We denote x0 = σ(t) ∈ Xt ; this is a smooth point in Xt . For a torsion-free sheaf L of rank one on Xt , the Riemann-Roch theorem yields χ(L) = h0 (L) − h1 (L) . When Xt is smooth, or when is singular and L is of finite Tor-dimension, we have deg L = χ(L). We then adopt this formula as the definition of the degree of a torsion-free sheaf L of rank one on any fiber Xt . Moreover, since the dualizing sheaf of Xt is trivial, the Grothendieck-Serre duality (C.10) implies that H 1 (Xt , L)∗ ' HomXt (L, O) = H 0 (Xt , L∗ ) ,

(6.8)

where we write O = OXt for simplicity. Lemma 6.12. Let L be a torsion-free sheaf of rank one and degree zero on Xt . Then, H 0 (Xt , L) = H 1 (Xt , L) = 0 unless L ' O. Proof. If H 0 (Xt , L) 6= 0 there is an exact sequence 0 → O → L → K → 0. It follows that K has rank zero and length `(K) = χ(K) = χ(L) − χ(O) = 0. Therefore, K = 0 and L is trivial. Moreover, if h0 (Xt , L) = 0, then h1 (Xt , L) = 0 by Riemann-Roch. 

6.2. Weierstraß fibrations

193

Lemma 6.13. Let L be a rank one torsion-free sheaf on Xt . There is a point x ∈ Xt and an isomorphism L ' mx ⊗ O((d + 1)x0 ) , where d = deg L and mx is the ideal sheaf of x in Xt . Proof. Assume d = −1. In this case, h0 (L) = 0 and h1 (L) = 1. By Equation (6.8), there is a nonzero morphism L → O which is injective because L is torsion-free. Thus, there is an exact sequence 0 → L → O → K → 0. The sheaf K is a quotient of O of rank zero and length `(K) = χ(K) = χ(O) − χ(L) = 1, so that it is the skyscraper sheaf of a point, K ' Ox . Thus, L ' mx . In the general case, L⊗O(−(d+1)x0 ) has degree −1, so that L⊗O(−(d+1)x0 ) ' mx by the previous argument. 

6.2.3

Relative integral functors for Weierstraß fibrations

Let us consider the commutative diagram /X . X ×B X HH HH ρ HH p π ˜1 HH H$   p /B X π ˜2

(6.9)

Definition 6.14. Let I∆ be the ideal sheaf of the diagonal immersion δ : X ,→ X ×B X. The relative Poincar´e sheaf for the elliptic fibration p is the sheaf ˜1∗ OX (Θ) ⊗ π ˜2∗ OX (Θ) ⊗ ρ∗ ω −1 P = I∆ ⊗ π 4 We shall show in Section 6.2.4 that P is a universal sheaf for a moduli prob˜2 are torsion-free sheaves lem. The restrictions of P to the fibers of either π ˜1 or π of rank one. Moreover, we have twisted the ideal sheaf of the diagonal so as to ensure that P satisfies the normalization condition P|Θ×B X ' P|X×B Θ ' OX .

(6.10)

. Proposition 6.15. The relative Poincar´e sheaf P has the following properties:

194

Chapter 6. Relative Fourier-Mukai functors

1. it is flat over both factors of X ×B X; 2. its dual sheaf coincides with its dual in the derived category D(X ×B X), i.e., P ∗ ' P ∨ ; 3. P ∗ is flat over both factors; 4. it is reflexive, i.e., P ' P ∗∗ . Proof. 1. It follows from the definition of P. 2. One has to check that ExtiOX×

BX

(P, OX×B X ) = 0 for i ≥ 1. This is a local

issue, so by Definition 6.14 it is enough to show that ExtiOX× X (I∆ , OX×B X ) = 0 B for i ≥ 1. Let us consider the exact sequence 0 → I∆ → OX×B X → δ∗ OX → 0 where δ : X ,→ X ×B X is the diagonal immersion. By dualizing we obtain an exact sequence ∗ 0 → OX×B X → I∆ → Ext1OX×

BX

(δ∗ OX , OX×B X ) → 0

(6.11)

and isomorphisms ExtiOX×

BX

(I∆ , OX×B X ) ' Exti+1 OX×

BX

(δ∗ OX , OX×B X )

for i ≥ 1.

These sheaves are the cohomology sheaves of the derived homomorphism complex RHomOX×B X (δ∗ OX , OX×B X ), which, by Equation (C.7), is isomorphic to the direct image under δ of the dualizing complex δ ! OX×B X . Since π1 ◦ δ = IdX , Equation (C.6) implies that OX ' δ ∗ p∗2 ωX/B [1]⊗δ ! OX×B X , and thus δ ! OX×B X ' −1 [−1]. This proves that ExthOX× X (δ∗ OX , OX×B X ) = 0 for h 6= 1. ωX/B B

∗ is flat over each factor. The previous computation also 3. One has to prove that I∆ 1 −1 , so that the sequence (6.11) takes the yields ExtOX× X (δ∗ OX , OX×B X ) ' δ∗ ωX/B B form −1 ∗ → δ∗ ωX/B → 0. (6.12) 0 → OX×B X → I∆ ∗ has the required property. So, I∆ ∗∗ . By Equation (6.12) there is an exact sequence 4. We need to prove that I∆ ' I∆ ∗∗ 0 → I∆ → OX×B X → Ext1OX×

BX

−1 (δ∗ ωX/B , OX×B X ) → 0 .

By applying relative duality we get −1 −1 RHomOX×B X (δ∗ ωX/B , OX×B X ) ' δ∗ RHomOX (ωX/B , δ ! OX×B X ) −1 −1 ' δ∗ RHomOX (ωX/B , ωX/B [−1]) ' δ∗ OX [−1] .

Hence, Ext1OX×

BX

−1 ∗∗ (δ∗ ωX/B , OX×B X ) ' δ∗ OX and I∆ ' I∆ .



6.2. Weierstraß fibrations

195

We consider the relative integral functor Φ : D− (X) → D− (X) defined by Φ(E • ) = R˜ π2∗ (˜ π1∗ E • ⊗ P) . Since p is flat, the morphism π ˜1 is flat as well, so that we do not need to derive π ˜1∗ . Moreover, P being flat over both factors of X ×B X (Proposition 6.15), the functor Φ can be extended to a functor Φ : D(X) → D(X), which induces a functor between the bounded derived categories Φ : Db (X) → Db (X). We can also regard Φ as an “absolute” integral functor: • ∗ • ∗P Φ(E • ) = ΦjX→ X (E ) = Rπ2∗ (π1 E ⊗ j∗ P) .

Here, j : X ×B X ,→ X ×X is the natural immersion and π1 , π2 are the projections of X × X onto its factors. ∗P b b W can prove that the functor Φ = ΦjX→ X : D (X) → D (X) is an equivalence of categories. To this end we fix some notation. As in Chapter 1 we denote by jx : X ,→ X × X the immersion of the fiber π1−1 (x) = {x} × X, whilst ˜x : Xt ,→ ˜1−1 (x) ' Xt . Since P and then j∗ P X ×B X will be the immersion of the fiber π are flat over both factors, one has that

L˜ ∗x P ' ˜∗x P ' Px ,

Ljx∗ j∗ P ' jx∗ j∗ P ' (j∗ P)x

and also (j∗ P)x ' jt∗ Px , where jt : Xt ,→ X is the natural immersion. Assume now that X is smooth. By Proposition C.1, the base variety B is Cohen-Macaulay and ωX ' p∗ ωB ⊗ ωX/B ' p∗ (ωB ⊗ ω −1 )

(6.13)

where the second isomorphism is induced by Equation (6.5). Lemma 6.16. If X is smooth, the sheaf j∗ P is strongly simple over both factors of X × X. Proof. Since by Definition 6.14 P is invariant under the permutation of factors, it is enough to check that j∗ P is strongly simple over the first factor. We already know that it is flat over X, so we only need to compute the groups HomiD(X) ((j∗ P)x , (j∗ P)y ) for x, y in X (cf. Definition 1.30). Let us write t = p(x), s = p(y). One has HomiD(X) ((j∗ P)x , (j∗ P)y ) ' HomiD(X) (jt∗ Px , js∗ Py ) ' HomiD(Xt ) (Ljt∗ jt∗ Px , Py ) .

196

Chapter 6. Relative Fourier-Mukai functors

Thus, HomiD(X) ((j∗ P)x , (j∗ P)y ) = 0 for any i if s 6= t. Assume now s = t. We have two possible cases: either x 6= y or x = y. In the first case, either x or y is a smooth point of Xt . If x is smooth in Xt , then Px is a line bundle, which implies Ljt∗ jt∗ Px ' jt∗ jt∗ Px ' Px . So, by Lemma 6.12 HomiD(X) ((j∗ P)x , (j∗ P)y ) ' HomiD(Xt ) (Px , Py ) ' H i (Xt , Px∗ ⊗ Py ) = 0 for every i. Assume on the other hand that x is not smooth in Xt . Since X is smooth, it has a Serre functor, and we have HomiD(X) (jt∗ Px , jt∗ Py )∗ ' Homn−i D(X) (jt∗ Py , jt∗ Px ⊗ ωX ) . Now, jt∗ Px ⊗ ωX ' jt∗ (Px ⊗ jt∗ ωX ) ' jt∗ Px because jt∗ ωX ' OXt by (6.13). Thus HomiD(X) (jt∗ Px , jt∗ Py )∗ ' Homn−i D(X) (jt∗ Py , jt∗ Px ) = 0 for every i because y is a smooth point of Xt and we can apply the previous argument. Finally, if x = y, adjunction between inverse and direct images of sheaves implies HomX (jt∗ Px , jt∗ Px ) ' HomXt (jt∗ jt∗ Px , Px ) ' HomXt (Px , Px ) ' k , where the last isomorphism follows from the stability of Px .



Lemma 6.17. (j∗ P)∨ ⊗ π1∗ ωX [m + 1] ' j∗ (P ∗ ⊗ ρ∗ ω −1 )[1], where m = dim B. Proof. Since ωX is a line bundle, one has (j∗ P)∨ ⊗ π1∗ ωX [m + 1] ' RHomOX×X (j∗ P, π1∗ ωX [m + 1]) . Now there are isomorphisms RHomOX×X (j∗ P, π1∗ ωX [m + 1]) ' RHomOX×X (j∗ P, π2! OX ) ' j∗ RHomOX×B X (P, j ! π2! OX ) ˜2! OX ) ' j∗ RHomOX×B X (P, π ˜1∗ ωX/B [1]) ' j∗ RHomOX×B X (P, π ˜1∗ ωX/B [1]) ' j∗ (P ∨ ⊗ ρ∗ ω −1 )[1] , ' j∗ (P ∨ ⊗ π where the first isomorphism is relative duality for π2 together with Equation (C.3), the second is relative duality for j, the third is due to Equation (C.5) and the fourth is relative duality for π ˜2 together with Equation (C.3). In this way we conclude, ∗ ∨  since P ' P by Proposition 6.15.

6.2. Weierstraß fibrations

197

Theorem 6.18. If X is smooth, the relative integral functor Φ is a Fourier-Mukai transform. The quasi-inverse of Φ is the relative Fourier-Mukai functor with relative kernel Q[1], where Q is the sheaf P ∗ ⊗ ρ∗ ω −1 on X ×B X. Proof. The sheaf j∗ P is strongly simple by Lemma 6.16, so that Φ is fully faithful by Theorem 1.27. Moreover, for every point x ∈ X, one has (j∗ P)x ⊗ ωX ' jt∗ (Px ) ⊗ ωX ' jt∗ (Px ⊗ jt∗ ωX ) ' jt∗ (Px ) ' (j∗ P)x because jt∗ ωX is trivial by (6.13). By Proposition 2.56, Φ is an equivalence of categories and, by Lemma 6.17, its quasi-inverse is the functor (j P)∨ ⊗π1∗ ωX [m+1]

∗ ΦX→ X

∗Q ' ΦjX→ X ,

where m = dim B.



Pt : Db (Xt ) → Db (Xt ) is an equivCorollary 6.19. The integral functor Φt = ΦX t→Xt alence for every closed point t ∈ S. Its quasi-inverse is the Fourier-Mukai functor with kernel Pt∗ [1]. Q[1]

Proof. Let H = ΦX→X be the quasi-inverse of Φ. Then, the unit morphism Id → H ◦ Φ is an isomorphism and one has an isomorphism jt∗ G • → (H ◦ Φ)(jt∗ G • ) for every object G • of Db (Xt ). Since (H ◦ Φ)(jt∗ G • ) ' jt∗ (Ht ◦ Φt )(G • ) by Equation (6.3) and jt is a closed immersion, the unit morphism G • → (Ht ◦ Φt )(G • ) is an isomorphism; this proves that Φt is fully faithful. Since H is also a left adjoint P ∗ [1] to Φ, a similar argument proves that Ht = ΦXtt→Xt is actually a quasi-inverse of  Φt . b : Db (X) → Db (X) the relative Fourier-Mukai transform We shall denote by Φ P∗ b with kernel Q. Thus, Φt = ΦXtt→Xt for any closed point t ∈ B. Theorem 6.18 implies that if a sheaf F on X is WITi with respect to Φ b and Φ b 1−i (F) b ' F. The (i = 0, 1), then Fb = Φi (F) is WIT1−i with respect to Φ b holds true as well. analogous statement intertwining Φ and Φ

6.2.4

The compactified relative Jacobian

Let p : X → B a Weierstraß fibration. Altman and Kleiman proved in [3] that there is an algebraic variety pˆ: J¯0 (X/B) → B, the so-called Altman-Kleiman compactification of the relative Jacobian, whose points parameterize torsion-free, rank one and degree zero sheaves on the fibers of X → B. It is a straightforward consequence of Lemma 6.13 that the natural morphism of B-schemes $ X −→ J¯0 (X/B)

x 7→ mx ⊗ OXt (σ(t))

t = p(x)

(6.14)

198

Chapter 6. Relative Fourier-Mukai functors

is an isomorphism (recall that mx is the ideal sheaf of the point x in the fiber Xt ) We shall provide a direct proof of the fact that X is a compactification of the relative Jacobian without resorting to Altman-Kleiman’s theory. The Poincar´e sheaf P will turn out to be a universal object. Let f : T → B be a scheme morphism. We denote by pT : XT = X ×B T → T and fX : XT → X the projections. Theorem 6.20. Let L be a sheaf on XT , flat over X, whose restriction Lt to any fiber Xt is a torsion-free sheaf of rank one and degree zero. There exists a unique morphism of B-schemes ψL : T → X such that (1 × ψL )∗ P ' L ⊗ p∗T M for a line bundle M on T . Here, 1×ψL is the induced morphism X×B T → X×B X. ∗ Proof. By Equation (C.3) the relative dualizing sheaf for pS is fX ωX/B . Moreover, ∗ ∗ −1 ∗ fX ωX/B ' pT ωT where ωT ' f ω (see (6.5)). Cohomology base change implies ∗ OX (−Θ)) is a line bundle, and by relative duality that N = R1 pT ∗ (L ⊗ fX ∗ N −1 ' pT ∗ HomOXT (L ⊗ fX OX (−Θ), p∗T ωT−1 ) .

Let us consider the natural morphism ∗ p∗T N −1 → HomOXT (L ⊗ fX OX (−Θ), p∗T ωT−1 ) .

Its restriction to every fiber Xt is nonzero, so that it induces a section of ∗ HomOXT (L ⊗ fX OX (−Θ), p∗T (ωT−1 ⊗ N )) , ∗ that is, a morphism L ⊗ fX OX (−Θ) → p∗T (ωT−1 ⊗ N ), whose restriction to every fiber Xt is nonzero. There is an exact sequence g

∗ L ⊗ fX OX (−Θ) ⊗ p∗T (ωT ⊗ N −1 ) − → OXT → OY → 0

for some closed subscheme η : Y ,→ XT = X ×B T .The restriction of g to every fiber Xt is an injective morphism gt : Lt ⊗ OXt (σ(t)) → OXt . Then, OY is flat over T , i.e., the projection pT ◦ η : Y → T is flat. Moreover, Lt ⊗ OXt (−σ(t)) has degree −1, and by Lemma 6.13 Lt ⊗ OXt (−σ(t)) ' mx for a certain point x ∈ Xt . Therefore, we have an exact sequence gt

0 → mx −→ OXt → OYt → 0 where Yt = Xt ∩ Y . So, OYt is the skyscraper sheaf Ox . Since it is flat, pT ◦ η is an isomorphism; thus, Y is the graph of a morphism ψL = fX ◦η ◦(pT ◦η)−1 : T → X.

6.2. Weierstraß fibrations

199

Moreover, the flatness of OY and the injectivity of all the restrictions gt imply ∗ OX (−Θ) ⊗ p∗T (ωT ⊗ N −1 ) ' IY is the ideal that g is injective and then L ⊗ fX ∗ sheaf of Y . Now, since (1 × ψL ) I∆ ' IY , one has ∗ (1 × ψL )∗ P ' L ⊗ p∗T (ψL OX (Θ) ⊗ N −1 ) .

 The Poincar´e sheaf P is the unique universal sheaf on X ×B X for the above moduli problem verifying the normalization condition P|Θ×B X ' OX imposed in (6.10). ∼ J¯ (X/B) the involution defined by taking the We denote by ι : J¯0 (X/B) → 0 ∼ J¯ (X/B) the morphism ι defines an involution dual. Via the identification $ : X → 0

of X, that we denote by the same symbol. There is a functorial description of this isomorphism: by the universality property, the dual P ∗ of the relative Poincar´e ˆ2∗ M for a sheaf defines a morphism ι = ψP ∗ : X → X such that (1 × ι)∗ P ' P ∗ ⊗ π line bundle on X. The normalization condition implies that M is trivial, so that (1 × ι)∗ P ' P ∗ .

(6.15)

Remark 6.21. Whenever a fiber Xt is smooth, the fiber J¯0 (X/B)t of the compactibt (which is isomorphic to fication of the relative Jacobian is the dual elliptic curve X bt Xt ). Moreover, the restriction Pt of the relative Poincar´e bundle sheaf to Xt × X coincides with the Poincar´e line bundle defined in Chapter 3. 4

6.2.5

Examples

Assume that X is smooth, so that we can apply Theorem 6.18. We compute the action of the Fourier-Mukai transform on relatively torsion-free sheaves. Lemma 6.22. Let L be a torsion-free rank one and degree 0 sheaf on a fiber Xt of the Weierstraß fibration p : X → B. Then the direct image Lt = jt∗ L is WIT1 for Φ, and Φ1 (Lt ) ' Ox∗ , where x∗ = ι(x) is the point corresponding to L∗ by the isomorphism $. b and Proof. By the invertibility of Φ it is enough to prove that Ox∗ is WIT0 for Φ 0 ∗ b b ∗ that Φ (Ox ) ' Lt . We know that Φ(Oι(x) ) ' Ljι(x)∗ j∗ Q ' j∗ Qι(x) because the b is flat over the first factor. Since Qι(x) ' P ∗ ' kernel Q = P ∗ ⊗ ρ∗ ω −1 of Φ ι(x)

b ι(x) ) ' Lt . Px ' L, one has Φ(O



Let f : T → B a scheme morphism. Let L be a sheaf on XT flat over X and whose restrictions Lt to any fiber Xt are torsion-free sheaves of rank one and degree zero. Let ψL : T → X be the morphism induced by the universal property

200

Chapter 6. Relative Fourier-Mukai functors

(Theorem 6.20), so that (1 × ψL )∗ P ' L ⊗ p∗T M for a certain line bundle M on T . The morphism ψL∗ : T → X corresponding to the dual sheaf by the universal property is ψL∗ = ι ◦ ψL . Thus, (1 × ψL∗ )∗ P ∗ ' L ⊗ p∗T M .

(6.16)

We denote by Γ : T ,→ XT = X ×B T the graph of ψL∗ , which is a section of ¯ : XT → the projection pT : XT → T . By base change, Γ induces an immersion Γ ˜2T . One also has XT ×T XT ' (X ×B X)T which is a section of the projection π ¯ 1 × ψL∗ = fX×B X ◦ Γ where fX×B X : (X ×B X)T → X ×B X is the natural projection. Proposition 6.23. L is WIT1 for ΦT , and Φ1T (L) ' OΓ(T ) ⊗ p∗T (ωT ⊗ M−1 ). bT Proof. By the invertibility of ΦT it is enough to prove that OΓ(T ) is WIT0 for Φ −1 0 ∗ b and that ΦT (OΓ(T ) ) ' L ⊗ pT (ωT ⊗ M). Since ∗ ∗ ∗ ¯ ∗ (Γ ¯∗f ∗ π ˜1T OΓ(T ) ⊗ PT∗ ' Oπ˜ −1 (Γ(T )) ⊗ fX× P∗ ' Γ X×B X P ) BX 1T

¯ ∗ (L ⊗ p∗ M) ¯ ∗ ((1 × ψL∗ )∗ P ∗ ) ' Γ 'Γ T by (6.16), we have b T (OΓ(T ) ) ' R˜ ¯ ∗ (L ⊗ p∗ M)) ⊗ p∗ ω −1 simeqL ⊗ p∗ (M ⊗ ω −1 ) . Φ π2T ∗ (Γ T T T T T  Example 6.24. Take T = B, f = Id and L = OX . The associated morphism ψOX : B → X is nothing but the section σ and Γ coincides with σ as well. Moreover (1 × σ)∗ P = P|X×B Θ ' OX by the normalization condition (6.10). Then Φ0 (OX ) = 0 ,

Φ1 (OX ) = OΘ ⊗ p∗ ω . 4

Example 6.25. Take T = X and f = p : X → B. The morphism associated with L = P ∗ is ι, and the one associated with P is the identity. The section Γ is the diagonal δ : X ,→ X ×B X in the first case, and the composition δ˜ = (1×ι)◦δ : X ,→ X ×B X in the second case. Hence, Φ0X (P) = 0 , Φ0X (P ∗ )

= 0,

Φ1X (P) = δ˜∗ (OX ) ⊗ ρ∗ ω , Φ1X (P ∗ ) = δ∗ (OX ) ⊗ ρ∗ ω . 4

6.2. Weierstraß fibrations

6.2.6

201

Topological invariants

In this section we assume that X is smooth. So, every object in Db (X) has a well defined Chern character. We can compute the Chern character of the complex ˜2 since p : X → B is a local Φ(E • ) by using the Riemann-Roch theorem for π complete intersection morphism, as we have seen in Section 6.2.1. It should be clear that also π ˜2 is a local complete intersection morphism, whose virtual relative tangent bundle is π ˜1∗ TX/B , where TX/B is the virtual relative tangent bundle to p described in Section 6.2.1. By the singular Riemann-Roch theorem [119, Cor. 18.3.1] the Chern character of Φ(E • ) is ˜2∗ (˜ π1∗ (ch E • ) · ch(P) · π ˜1∗ td(TX/B )) . ch(Φ(E • )) = π

(6.17)

The Todd class of TX/B is given by Proposition 6.11, while the Chern character of P can be computed from Definition 6.14. One has 1 ∗ 2 1 ∗ 2 ch(P) = ch(I∆ ) · (1 + π ˜ Θ + . . . ) · (1 + π ˜ Θ + ...) ˜1∗ Θ + π ˜2∗ Θ + π 2 1 2 2 ˜ + 1 ρ∗ K ˜2 − ...). · (1 − ρ∗ K 2

(6.18)

Therefore we need to compute the Chern character ch(I∆ ). Lemma 6.26. The Chern character of the ideal sheaf I∆ of the diagonal immersion δ : X ,→ X ×B X is ¯ + ∆˜ ¯ + 5 ∆ρ∗ K ¯2 ch(I∆ ) =1 − ∆ − 12 ∆ρ∗ K π2∗ (Θp∗ K) 6 ¯ 2 ) + 23 ∆ρ∗ K ¯ 3 + terms of higher degree + 12 ∆˜ π2∗ (Θ p∗ K 24 where ∆ = δ∗ (1) is the class of the diagonal. Proof. Note first that ch(I∆ ) = 1 − ch(δ∗ OX ). The singular Riemann-Roch theorem gives π1∗ td(TX/B ) = δ∗ (ch(OX )) = δ∗ (1) = ∆ . ch(δ∗ OX )˜ By using the expression for td(TX/B ) given by Proposition 6.11, one has ¯ − ∆π ¯ − 5 ∆ ρ∗ K ¯2 ch(δ∗ OX ) =∆ + 12 ∆ ρ∗ K ˜2∗ (Θ · p∗ K) 6 ¯ 2 ) − 23 ∆ ρ∗ K ¯ 3 + terms of higher degree. − 12 ∆ π ˜2∗ (Θ p∗ K 24 

202

Chapter 6. Relative Fourier-Mukai functors

Computation for elliptic surfaces In this case the scheme B is a smooth projective curve and X a smooth projective surface. Let us denote by e the degree of the line bundle p∗ ωX/B ; recall that ¯ = c1 (p∗ ωX/B ). We have Θ · p∗ K ¯ = e = −Θ2 and c1 (ωX/B ) = p∗ K ¯ ≡ e f. The K Todd class of the virtual relative tangent bundle of p (Proposition 6.11) is given by the formula ¯ + ew, (6.19) td(TX/B ) = 1 − 12 p∗ K where w is the fundamental class of X. The Todd class of X is td(X) = 1 + 12 (c1 − e)f + e w ,

(6.20)

where c1 = c1 (B). Finally, by Lemma 6.26, the Chern Character of I∆ is 1 ¯ + e δ∗ (w) . ch(I∆ ) = 1 − ∆ − δ∗ (p∗ K) 2 If E • is an object of Db (X), the Chern character of Φ(E • ) is given by ¯ + e δ∗ (w))e w) ch(Φ(E • )) =π2∗ [π1∗ (ch E • ) · (1 − δ∗ (1) − 12 δ∗ (p∗ K) ¯ + ew)] · (1 + Θ − 1 w) · (1 + e f) . · (1 − 1 p∗ K 2

2

Thus, the topological invariants of Φ(E • ) are ch0 (Φ(E • )) = d ¯ + (d − n)Θ + (c − ch1 (Φ(E • )) = −c1 (E • ) + d p∗ K •

ch2 (Φ(E )) = (−c − de +

1 2

1 2

ed + s) f

(6.21)

ne)w ,

where n = ch0 (E • ), d = c1 (E • ) · f is the relative degree, c = c1 (E • ) · Θ and ch2 (E • ) = s w. Similar calculations for the inverse relative Fourier-Mukai transform yield the formulas b • )) = d ch0 (Φ(E b • )) = (c1 (E • )) − np∗ K ¯ − (d + n)Θ + (s + ne − c − ch1 (Φ(E b • )) = −(c + de + ch2 (Φ(E

1 2

1 2

ed)f

(6.22)

ne)w .

Computation for elliptic Calabi-Yau threefolds We now consider the case of a Weiertraß elliptic fibration p : X → B, where B is a smooth projective surface and X is a projective Calabi-Yau threefold. In this case, the existence of a section of p imposes constraints on the base surface B: it has

6.2. Weierstraß fibrations

203

to be del Pezzo surface (a surface whose anticanonical divisor −KB is ample), or a Hirzebruch surface (a rational ruled surface), or an Enriques surface (a minimal surface B for which pa (B) = χ(OB ) − 1 = 0, pg = h0 (B, KB ) = 0 and 2KB = 0) or a blowup of a Hirzebruch surface (see for instance [95] or [223]). Since ωX ' OX , we have ω ' ωB . By Proposition 6.11, one has td(TX/B ) = 1 − 12 c1 +

2 1 12 (13 c1

+ 12 Θ c1 ) − 12 Θ c21

(6.23)

with c1 = p∗ c1 (B) = −p∗ (KB ). The Todd class of X admits the following expression: 1 ∗ p (c2 + 11 c21 + 12 Θ c1 ) (6.24) td(X) = 1 + 12 with c2 = p∗ (c2 (B)). Finally, the Chern character of the ideal sheaf I∆ (Lemma 6.26) takes the form ch(I∆ ) = 1 − ∆ − 12 ∆ · π ˜2∗ c1 + ∆ · π ˜2∗ (Θ c1 ) + 56 ∆ · π ˜2∗ (c21 ) + 12 ∆ · π ˜2∗ (Θ c21 ) . We shall consider for simplicity objects E • in D(X) with Chern characters have the form ch0 (E • ) = nE • ch1 (E • ) = xE • Θ + p∗ SE •

(6.25)

ch2 (E • ) = Θp∗ ηE • + aE • f ch3 (E • ) = sE •

where ηE • , SE • ∈ A1 (B) ⊗Z Q, sE • ∈ A3 (X) ⊗Z Q ' Q and f ∈ A2 (X) ⊗Z Q is the class of a fiber of p. These assumptions are met in the majority of applications. Now Equation (6.17) and the corresponding formula for the inverse elliptic relative b • ), namely: b give the Chern character of Φ(E • ) and Φ(E Fourier-Mukai transform Φ ch0 (Φ(E • )) = xE • ch1 (Φ(E • )) = −nE • Θ + p∗ ηE • − 12 xE • c1 ch2 (Φ(E • )) = ( 12 nE • c1 − p∗ SE • )Θ + (sE • − 12 p∗ ηE • c1 Θ +

2 1 • 12 xE c1 Θ)f

(6.26)

ch3 (Φ(E • )) = − 16 nE • Θc21 − aE • + 12 Θc1 p∗ SE • and b • )) = xE • ch0 (Φ(E b • )) = −nE • Θ + p∗ ηE • + 1 xE • c1 ch1 (Φ(E 2 b • )) = (− 1 nE • c1 − p∗ SE • )Θ + (sE • + 1 p∗ ηE • c1 Θ + ch2 (Φ(E 2 2 • 2 ∗ 1 1 b • • • ch3 (Φ(E )) = − nE Θc − aE − Θc1 p SE + xE • Θc2 . 6

1

2

1

2 1 • 12 xE c1 Θ)f

(6.27)

204

6.3

Chapter 6. Relative Fourier-Mukai functors

Relatively minimal elliptic surfaces

In this section, following [28] and [60], we describe a Fourier-Mukai transform for smooth elliptic surfaces which need not admit a Weierstraß model. In Chapter 7, we shall use this transform to study the Fourier-Mukai partners of an elliptic surface. Unfortunately this transform does not easily extend to higher dimensions, with the exception of a few cases in dimension three (cf. [71, 82]). Since elliptic surfaces (Definition 6.9) allow for nonintegral and even reducible fibers, a compactified relative Altman-Kleiman Jacobian as described in Section 6.2.4 may fail to exist. To circumvent this problem, following a suggestion by Morrison [222], one considers a moduli space of pure sheaves supported on the fibers in the sense of Simpson (Definition C.2). Let p : X → B be our relatively minimal elliptic surface. The dualizing sheaf of X can be computed as in the following formula: X (6.28) ωX ' p∗ L ⊗ OX ( (mi − 1)fi ) , where mi fi are the multiple fibers of p and L is a line bundle on B [22, V.12.3]. As a consequence of Equation (6.28), if E • is an object of Db (X) whose cohomology sheaves are all supported on a fiber Xt , then E • and E • ⊗ ωX have the same Chern character. Hence, Equations (1.18) and (1.6) imply that for any object F • of Db (X), the equality χ(E • , F • ) = χ(F • , E • ) (6.29) holds true. Let us fix some notation. For any object E • of Db (X) we write its Chern character in the form ch(E • ) = (r, c, s) ∈ Z ⊕ A1 (X) ⊕ Q , where r is the rank, c is the first Chern class and ch2 (E • ) = s w with w the fundamental class of X. We denote by λX/B the highest common divisor of the relative degrees d(E • ) = c1 (E • ) · f of the objects E • of Db (X) (cf. Eq. (6.4)). Equivalently, λX/B is the smallest positive number d such that there is a divisor D with d = D · f. Since the divisor D + βf is effective for β  0 and has the same intersection with the fiber as D, we can also say that λX/B is the smallest positive relative degree d = D · f of an effective divisor D in X (a d-multisection). Let us fix integer numbers r > 0 and d such that d is coprime to rλX/B . We also fix a polarization H in X having relative degree h = H · f such that rh is coprime to d. To prove that such polarizations actually exist, take an arbitrary polarization H 0 in X. By the very definition of λX/B , the fiber degree h0 = H 0 · f is a multiple of λX/B ; since d is coprime to rλX/B , by adding if necessary a suitable

6.3. Relatively minimal elliptic surfaces

205

multiple of a λX/B -multisection to H 0 , we obtain a new polarization H satisfying our requirements. By Theorem C.6, there exists a coarse relative moduli scheme q : M (X/B, r, d) → B which parameterizes S-equivalence classes of sheaves on the fibers of p that are relatively semistable with respect to H and have relative polarized rank r (Definition C.5) and degree d. This means that the Hilbert polynomial of the sheaves Et is P (s) = rhs + d. The coprimality condition ensures that all sheaves in M (X/B, r, d) are relatively stable and that M (X/B, r, d) is a fine moduli space (cf. Proposition C.7). To be more precise: q is a projective morphism; the (closed) points of M (X/B, r, d) are in a one-to-one correspondence with pure sheaves of polarized rank r and degree d on the fibers of p that are µS -stable with respect to the polarization induced by H; and there is a universal relative sheaf P on X ×B M (X/B, r, d) → M (X/B, r, d), flat over M (X/B, r, d), such that for every point y ∈ M (X/B, r, d) the restriction Py of P to the fiber Xq(t) × {y} is the stable sheaf corresponding to y. Definition 6.27. The compactified relative Jacobian of type (r, d) is the union JX/B (r, d) of the connected components of M (X/B, r, d) that contain the direct image it∗ (E) of a stable locally free sheaf E of rank r and degree d on a generic 4 fiber Xt of p. In Remark 6.33, we shall compare this compactified Jacobian with the AltmanKleiman compactified relative Jacobian J¯0 (X/B) previously introduced. We also denote by P the restriction to X × JX/B (r, d) of the universal sheaf on X ×B M (X/B, r, d). Again by Theorem C.6 and the coprimality condition, there also exists a projective variety M (X, r, d) which is a fine moduli scheme for pure dimensional sheaves on X with Chern character v = (0, rf, dw) (where w is the fundamental class of X) and stable with respect to H (cf. Proposition C.7). Let i : X ×B JX/B (r, d) ,→ X × JX/B (r, d) be the natural immersion. The direct e = i∗ P is flat over JX/B (r, d) and for every point y ∈ JX/B (r, d) its fiber image P ey ' jt∗ (Py ) (where t = q(y)) is pure-dimensional and stable with respect to H. P e corresponds to a morphism Moreover, it has Chern character (0, rf, dw). Then P  : JX/B (r, d) → M (X, r, d) from JX/B (r, d) to the “absolute” moduli scheme M (X, r, d). ey = it∗ (Py ) for are special (Definition For any point y ∈ Y , the sheaves P 2.54), namely, they have the property that Py ' Py ⊗ ωX . This can be seen as follows: if Py is supported on a smooth fiber Xt of p, then it∗ (Py ) ⊗ ωX ' it∗ (Py ⊗ i∗t ωX ) ' it∗ (Py )

206

Chapter 6. Relative Fourier-Mukai functors

because i∗t ωX ' OXt by Equation 6.28. For a general y, there is always a morphism ey ⊗ ωX because y 7→ dim HomX (P ey , P ey ⊗ ωX ) is an upper-semicontinuous ey → P P e ey ⊗ ωX are stable with the same function [141, III.12.8] [136, 7.7.5]. Since Py and P c1 , the above morphism has to be an isomorphism. The following result is proved in [60]. Proposition 6.28. q : JX/B (r, d) → B is an elliptic fibration and JX/B (r, d) is a smooth surface. Moreover,  : JX/B (r, d) → M (X, r, d) is an isomorphism. Proof. Let us write for simplicity Y = JX/B (r, d). Let U ⊆ B be the largest open set such that p : XU = p−1 (U ) → U is smooth. For every point t ∈ U the fiber q −1 (t) is a moduli space of stable sheaves of rank r and degree d on the elliptic curve Xt , and then it is isomorphic to Xt by [15]. Now, q is dominant, so that it is surjective and flat, B being a smooth curve [141, III.9.7], and hence there is a component of Y that dominates B. Any other connected component of Y must contain sheaves supported on a smooth fiber, and this is impossible because the fiber q −1 (t) is connected for t ∈ U as we have seen. Then Y is connected and elliptically fibered over B, which proves the first statement. As for the second statement, note that the support of every sheaf E in M (X, r, d) is contained in a fiber, because its Chern character is (0, rf, d w) and the support of a stable sheaf is connected. Then  is one-to-one on closed points. It follows that M (X, r, d) is a surface. Since v 2 = 0, M (X, r, d) is smooth by Proposition 2.61 (see also Remark 2.62). Zariski’s main theorem [141, 11.4] implies that  is an isomorphism, and then Y is also smooth.  The relative universal sheaf P defines a relative integral functor − − P Φ = ΦY→ X : D (Y ) → D (X) . e

As in many other situations, Φ is defined over the whole of the derived category of Y , and maps Db (Y ) to Db (X). By applying again Proposition 2.61 and Remark 2.62 we obtain the following result. P Proposition 6.29. The relative integral functor Φ = ΦY→ X is an equivalence of categories. e

Corollary 6.30. The elliptic surface q : JX/B (r, d) → B is relatively minimal. Proof. Let us write Y = JX/B (1, d) as above. If the claim is not true, there is a rational curve C with C 2 = −1 contained in a fiber Yt = q −1 (t). Then KY ·C = −1, so that χ(OC , OY ) = χ(C, ωY |C ) = 0, whereas χ(OY , OC ) = χ(C, OC ) = 1. It follows that χ(Φ(OC ), Φ(OY )) = 0 and χ(Φ(OY ), Φ(OC )) = 1 because Φ is fully faithful. Since p : X → B is relatively minimal, this contradicts (6.29) because

6.3. Relatively minimal elliptic surfaces

207

Φ(OC ) and Φ(OC )⊗ωY have the same Chern character, all the cohomology sheaves  of Φ(OC ) being supported on the fiber Xt . Remark 6.31. The relative moduli scheme JX/B (r, d) depends on the polarization H used to define relative stability. Recall that the relative degree h of H has to sat0 (r, d) is the moduli isfy the coprimality condition gcd(d, rh) = 1. However, if JX/B defined as above with respect to another polarization H 0 (with d coprime to rh0 ), 0 (r, d), we have seen that there is an isomorphism of schemes JXU /U (r, d) ' JX U /U −1 where U ⊆ B is the open set where p : XU = p (U ) → U is smooth. Since 0 (r, d) → B are relatively minimal, in (Corollary 6.30) JX/B (r, d) → B and JX/B view of [220, II.1.2] this isomorphism extends to an isomorphism of elliptic surfaces 0 (r, d). Indeed, JX/B (r, d) is independent of the polarization H JX/B (r, d) ' JX/B (as long as d is coprime to rh). 4 The elliptic surface JX/B (d) = JX/B (1, d) is the relative Picard scheme defined by Friedman [110]. Proposition 6.32. Let r, d integers with r > 0 and d coprime to rλX/B . There are isomorphisms JX/B (r, d) ' JX/B (d) ' JX/B (d + λX/B ) of elliptic surfaces over B. Thus, if d¯ denotes the residue class of d modulo λX/B , ¯ of elliptic surfaces over B. there is an isomorphism JX/B (r, d) ' JX/B (d) Proof. Let U ⊆ B be the smooth locus of p. The restriction of P 0 to XU ×U JXU /U (r, d) is locally free and its determinant is a line bundle parameterizing line bundles of degree d on the fibers of p : XU = p−1 (U ) → U . This gives an isomorphism JXU /U (r, d) → JXU /U (d) which extends to an isomorphism of elliptic surfaces JX/B (r, d) ' JX/B (d) by a similar argument to the one used in Remark 6.31. Let us now onsider a λX/B -multisection Θ. After twisting by OX (Θ) one  obtains an isomorphism JX/B (d) ' JX/B (d + λX/B ). Remark 6.33. We have seen that in order to ensure that Friedman’s relative Picard scheme JX/B (d) = JX/B (1, d) is projective and a fine moduli space, we have to impose that d is coprime to λX/B and to the relative degree h of the chosen polarization. In particular, d = 0 forces λX = 1 and h = 1. The first condition is equivalent to the fact that X → B has a section, thus preventing X → B from having multiple fibers; the second imposes that there is a polarization H that intersects every fiber at one point. Since a polarization must meet all the irreducible components of a fiber, this implies that all fibers are irreducible. Thus, the elliptic surface X → B turns out to be, in this case, a Weierstraß surface. Friedman’s relative Picard scheme JX/B (0) is then isomorphic to the AltmanKleiman compactified relative Jacobian J¯0 (X/B) as defined in Section 6.2.4, and

208

Chapter 6. Relative Fourier-Mukai functors

by Equation (6.14) we have isomorphisms X ' J¯0 (X/B) ' JX/B (0) . Thus, in the case of Weierstraß surfaces, the Fourier-Mukai functors defined in this section are a generalization of those considered in Section 6.2.3 (which have, however, the advantage of existing in arbitrary dimension). 4 If we drop the coprimality condition on the polarization H, strictly µS semistable sheaves on the fibers with polarized rank 1 and degree d may exist, and the moduli space q : M (X/B, 1, d) → B can be considered as a “compactified relative Jacobian,” in the sense that it is a compactification of the moduli space of relatively µS -stable sheaves of relative polarized rank 1 and degree d.

6.4

Relative moduli spaces for Weierstraß elliptic fibrations

Let p : X → B be a Weierstraß elliptic fibration. We use the relative elliptic Fourier-Mukai transform defined in Section 6.2.3 to study relative moduli spaces of sheaves on X. In this section we write “(semi)stability” to mean “µS -(semi)stability” (cf. Definition C.4). We start by computing the effect of the Fourier-Mukai transform on the relative Chern character. Proposition 6.34. Let E • a complex in Db (X) of relative Chern character (n, d) (cf. Eq. (6.4)). The relative Chern character of the Fourier-Mukai transform Φ(E • ) is (d, −n). Proof. To compute the relative invariants of Φ(E • ), we take for S a point t ∈ B with smooth fiber Xt and apply the Riemann-Roch theorem with respect to the  projection of Xt × Xt onto the second factor. Let F be a rank n sheaf on X flat over B. b = −1/µ(F). Corollary 6.35. If F is WITi and d 6= 0, then µ(F) Corollary 6.36. If F is WIT0 , then d(F) ≥ 0, and d(F) = 0 if and only if F = 0. If F is WIT1 , d(F) ≤ 0.

6.4.1

Semistable sheaves on integral genus one curves

Our aim is to characterize the moduli spaces of µ-semistable sheaves on geometrically integral curves of arithmetic genus one, that is, on fibers of a Weierstraß

6.4. Relative moduli spaces for Weierstraß elliptic fibrations

209

elliptic fibration p : X → B. We need to assume that X is smooth so that for every closed point t ∈ B, the integral functors Φt : Db (Xt ) → Db (Xt ) defined for every fiber are equivalences of categories (cf. Corollary 6.19). When Xt is smooth, it is an elliptic curve, with origin at the point σ(t). The moduli spaces of µ-semistable sheaves on it are then characterized, as have seen in Section 3.5.1, by using the Abelian Fourier-Mukai transform S, whose kernel we denote now by Pab to avoid confusion with the kernel Pt of the elliptic FourierMukai transform Φt . Comparing the expressions of Pab and Φt given respectively by Equation (3.12) and Definition 6.14, we see that Pab ' (1 × ι)∗ Pt , where ι : X → X is the involution which maps a point x to the opposite point for the group law. Thus, b ' ι∗ ◦ Φt , S ' ι∗ ◦ Φ bt . S bt Most arguments developed in Section 3.5.1 can be carried through, by using Φ b and Φt instead of the Abelian Fourier-Mukai transforms S and S. Although the proofs given there depend on the smoothness of the curve, we can modify them so that most of the results hold true in the singular case as well. It should be noticed that when Xt is singular, there exist indecomposable sheaves on Xt (even locally free) which are not semistable (cf. [78]). Thus Proposition 3.28 does not generalize to the singular fibers, and we need to slightly change the strategy of Section 3.5.1. Let us recall that any torsion-free rank-one sheaf on a fiber Xt is of the form L ' Px for a point x ∈ Xt , and that it is WIT1 , with transform Φ1t (Px ) = Ox∗ , where x∗ = ι(x) (see Lemma 6.22). To prove that µ-semistable sheaves on a fiber Xt of positive degree are WIT0 we need a preliminary result (cf. also [60]). Lemma 6.37. A coherent sheaf E on Xt is WIT0 if and only if HomXt (E, Px ) = 0 for every x ∈ Xt . Proof. Since Px is WIT1 and Φ1t (Px ) = Ox∗ , the Parseval formula (Proposition 1.34) implies that HomXt (E, Px ) ' HomDb (Xt ) (Φt (E), Ox∗ [−1]) . If E is not WIT0 , there is a point x ∈ Xt together with a nonzero morphism Φ1t (E) → Ox . This gives rise to a nonzero morphism Φt (E) → Ox [−1] in the derived category, so that HomXt (E, Px∗ ) 6= 0. The converse is straightforward. 

210

Chapter 6. Relative Fourier-Mukai functors

We are now ready to generalize Corollary 3.29 to the case of Weierstraß curves. As previously discussed, this extends results by Friedman, Morgan and Witten [114], since we consider non-locally free (torsion-free) sheaves. Proposition 6.38. Let E be a torsion-free µ-semistable sheaf of rank n and degree d on Xt . b t and in both cases Eb 1. If d > 0, then E is WIT0 with respect to both Φt and Φ is µ-semistable . b t and in both cases Eb 2. If d < 0, then E is WIT1 with respect to both Φt and Φ is µ-semistable. 3. If d = 0 and E is µ-stable, then E is of rank one. Thus, any µ-semistable sheaf of degree 0 is WIT1 and Eb is a skyscraper sheaf. 4. A µ-semistable sheaf E of degree d = 0 is S-equivalent to a direct sum of degree zero, rank one torsion-free sheaves: E∼

m M

i L⊕n , i

i=1

m X

ni = n .

i=1

Moreover Eb is supported at the points {x∗1 , . . . , x∗m } corresponding to the sheaves L∗i under the identification X ' J¯0 (X/B) of (6.14) and E ' E1 ⊕ · · · ⊕ Em , where Ei are µ-semistable subsheaves of degree 0 of E, such that i for every i. Ei ∼ L⊕n i Proof. 1. The fact that E is WIT0 follows from Lemma 6.37, since E is µ-semistable of positive degree. An analogous argument proves that E is WIT0 as well with ˆ t . The proof of the semistability of Eb in both cases is postponed until respect to Φ the end of the proof of part 2. b i (Φjt (E)) (cf. (2.35)) gives rise to an exact 2. The spectral sequence E i,j = Φ 2

t

sequence 0 → E21,0 → E → E20,1 → E22,0 = 0 . b 1 (Φ0 (E)) is WIT0 , so that it has positive degree Moreover, the sheaf E21,0 = Φ t t by Corollary 6.36. E is µ-semistable of negative degree, E21,0 must be zero. Thus, b 1 (Φ0 (E))) = 0, and therefore E is WIT1 . Φ0t (E) ' Φ0t (Φ t t Let us check that Eb is torsion-free and µ-semistable. If the torsion subsheaf T of Eb is nonzero, it is WIT0 (cf. Lemma 6.22), and Tb is a subsheaf of E having degree zero by Corollary 6.36. This contradicts the semistability of E. Thus Eb is torsion-free of degree n = rk(E) > 0 by Proposition 6.34. If Eb is not µ-semistable, there is a destabilizing sequence 0 → F → Eb → G → 0 ,

6.4. Relative moduli spaces for Weierstraß elliptic fibrations

211

b The sheaf G is WIT0 ; moreover, in view with F µ-semistable and µ(F) > µ(E). of point 1, F is WIT0 because it is µ-semistable of positive degree. Thus, we have an exact sequence 0 → Fb → E → Gb → 0 , b ≤ µ(E) because of the semistability of E. By Corollary 6.35 this so that µ(F) b The proof for Φ b is similar. contradicts the inequality µ(F) > µ(E). We now complete the proof of part 1. Then E has positive degree d > 0 and b t . If Eb = Φ0 (E) we have already seen that it is WIT0 with respect to both Φt and Φ t is not µ-semistable, there is a destabilizing sequence 0 → F → Eb → G → 0 , b The sheaf F is WIT1 , so that d(F) ≤ 0 by with F µ-semistable and µ(F) > µ(E). Corollary 6.36 and one has exact sequences 0 → Φ0t (G) → Fb → K → 0 ,

0 → K → E → Φ1t (G) → 0 .

If d(F) = 0, Fb has rank zero, so that K is a torsion sheaf. Then K = 0 and b since the Φ0 (G) is WIT1 and Fb is WIT0 , one has Fb = 0 and then Φ0t (G) ' F; t F = 0 which is absurd. Thus, d(F) < 0 and Fb is µ-semistable by part 2. It follows b Moreover µ(K) ≥ µ(E) by the semistability of E, and then that µ(K) ≤ µ(F). b µ(E) ≤ µ(F). Again by Corollary 6.35 this in contradiction with the inequality b The proof for Φ b is similar. µ(F) > µ(E). 3. Since E is µ-stable of degree 0, we have HomXt (E, Pξ ) = 0 unless E ' Pξ∗ . Lemma 6.37 implies that if E is not of rank one, it is IT0 ; by Proposition 3.25 the transform Eb = Φ0 (E) is then a locally free sheaf of rank 0, so that Eb = 0, and E = 0 by the invertibility of Φ. This proves the first statement. If E is µ-semistable of degree 0, it has a Jordan-H¨ older filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = E, with quotients Gi = Ei /Ei−1 being µ-stable of degree 0. The sheaves Gi are torsion-free rank one sheaves of degree 0, that is, Gi ' Pξi for a point ξi ∈ X. Since the sheaves bξ ' Oι(ξ ) , we deduce that E is WIT1 and Eb is a skyscraper Pξi are WIT1 and P i i sheaf. 4. The argument used in part 3 proves the statement about S-equivalence and that Eb is supported at the points {x∗1 , . . . , x∗m }. Then Eb ' ⊕m i=1 Fi where Fi ˆ 0 (Fi ). is a skyscraper sheaf of length ni supported at x∗i . One then takes Ei = Φ  Corollary 6.39. A sheaf E on a fiber Xt of zero degree and rank n ≥ 1 is torsion-free and µ-semistable if and only if it is WIT1 . Proof. If E is WIT1 all its subsheaves are WIT1 as well; then E has neither subsheaves supported on dimension zero, nor torsion-free subsheaves of positive de-

212

Chapter 6. Relative Fourier-Mukai functors

gree, so that it is torsion-free and µ-semistable. The converse is part of Proposition 6.38.  The result about categories of µ-semistable sheaves for smooth elliptic curves proved in Section 3.5.1 can now be extended straightforwardly to the case of general integral curves of genus one. Let us denote by Cohss n,d (Xt ) the full subcategory of the category Coh(Xt ) of coherent sheaves on Xt whose objects are µ-semistable sheaves on Xt of rank n and degree d. We also denote by Skyn (Xt ) the category of skyscraper sheaves on Xt of length n. By Proposition 6.38, one has the following result. Proposition 6.40. The Fourier-Mukai transform Φt induces equivalences of categories ss Cohss n,d (Xt ) ' Cohd,−n (Xt ) ,

if d > 0

Cohss n,0 (Xt ) ' Skyn (Xt ) . ∼ Db (X ), with L = O (σ(t)); δ∗ Lt : Db (Xt ) → Consider the functor Ψt = ΦX t t Xt t→Xt so, Ψt (E • ) ' E • ⊗ Lt . By composing in a suitable way the integral functors Φt and Ψt and proceeding as in the proof of Proposition 3.31, we obtain the following results. Proposition 6.41. For every pair (n, d) of integers (n > 0), there is an integral ∼ Db (X ) which induces an equivalence of categories ˜ t : Db (Xt ) → functor Φ t ss Cohss n,d (Xt ) ' Cohn ¯ ,0 (Xt ) ,

˜ t induces an equivalence of catewhere n ¯ = gcd(n, d). The integral functor Φt ◦ Φ gories Cohss ¯ (Xt ) . n,d (Xt ) ' Skyn Corollary 6.42. Let E be a torsion-free sheaf on Xt of rank n and degree d. The following conditions are equivalent: 1. E is µ-stable; 2. E is simple; 3. E is µ-semistable and gcd(n, d) = 1. Thus, the integral functors of Proposition 6.41 map µ-stable sheaves to µ-stable sheaves.

6.4. Relative moduli spaces for Weierstraß elliptic fibrations

6.4.2

213

Characterization of relative moduli spaces

Let p : X → B a Weierstraß elliptic fibration whose total space X is smooth. In this section we prove that the relative integral functor Φ : Db (X) → Db (X), which we know to be a relative Fourier-Mukai transform by Theorem 6.18, preserves the relative (semi)stability of sheaves. We shall also use this property to compute the relative moduli spaces of µ-semistable sheaves on the fibers of p. Most of the material has been taken from [145, 29]; some results are also contained in [113, 114, 110]. We rely on the results about µ-semistable sheaves on integral genus one curves described in Section 6.4.1. Recall that if E is a sheaf on X flat over B, the restriction Et of E to the fiber Xt is WITi for every closed point t ∈ B if and only if E is WITi and Eb = Φi (E) is flat over B (cf. Corollary 6.3). By using this fact together with Proposition 6.38 and Corollary 6.42, we obtain directly the following result. Proposition 6.43. Let E be a relatively µ-(semi)stable sheaf on X. b and in both cases Eb 1. If d > 0, then E is WIT0 with respect to both Φ and Φ, is relatively µ-(semi)stable. b and in both cases Eb 2. If d < 0, then E is WIT1 with respect to both Φ, and Φ, is relatively µ-(semi)stable. 3. If d = 0 and E is relatively µ-stable, then E is of rank one. Thus, any relatively µ-semistable sheaf of degree 0 is WIT1 and Eb is a flat family of skyscraper sheaves of length N = rk(E). Let us denote by Mss (X/B, n, d) the coarse relative moduli space of rank n and degree d µ-semistable sheaves on the fibers of p (thus, a section of this space as a fibration on B corresponds to a relatively µ-semistable sheaf on X). ¯ ) is the coarse moduli space of skyscraper sheaves of In particular, Mss (X/B, 0, n length n ¯ on the fibers of p, or in other words, the moduli space of µ-semistable sheaves having constant Hilbert polynomial P (m) = n ¯. Proposition 6.44. ˜ : Db (X) → Db (X) which induces 1. There is relative Fourier-Mukai functor Φ an isomorphism of B-schemes Mss (X/B, n, d) ' Mss (X/B, n ¯ , 0) , where n ¯ = gcd(n, d).

214

Chapter 6. Relative Fourier-Mukai functors

2. The relative Fourier-Mukai transform Φ induces an isomorphism Mss (X/B, m, 0) ' Mss (X/B, 0, m) . ˜ induces an isomorphism of BThus, the relative integral functor Φ ◦ Φ schemes ¯) . Mss (X/B, n, d) ' Mss (X/B, 0, n Proof. This follows from Proposition 6.41 and Corollary 6.3.



Next we extend Corollary 3.34 to the relative setting. Since any skyscraper sheaf of length n ¯ on a fiber Xt of p is S-equivalent to a direct sum ⊕i Oxnii (with P ¯ ) are in n ¯ = i ni ), the closed points of the relative moduli space Mss (X/B, 0, n a one-to-one correspondence with the closed points of the relative n ¯ -symmetric ¯ . We shall prove that this correspondence is actually induced by an product SymnB algebraic isomorphism. Let us consider the relative Hilbert scheme Hilbn¯ (X/B) → B of B-flat subschemes of X of relative dimension 0 and length n ¯ . There is a Hilbert-Chow mor¯ X mapping a zero-cycle of length n ¯ to the n ¯ points phism Hilbn¯ (X/B) → SymnB defined by the zero-cycle. Contrary to what happens for a smooth curve, this morphism is not an isomorphism, but is only birational. However, it induces an ¯ Xsm , where Xsm → B is the smooth locus isomorphism Hilbn¯ (Xsm /B) ' SymnB of p : X → B. Let T → B be a scheme morphism. If E is a sheaf on X ×B T → T defining ¯ ), then the modified support Supp0 (E) (see a T -valued point of Mss (Xsm /B, 0, n Definition C.9) is a subscheme of X × T which is flat over T and has degree n ¯ , that is, a T -valued point of the relative Hilbert scheme Hilbn¯ (X/B). Therefore, we have ¯ ) → HomB ( •, Hilbn¯ (Xsm /B)), inducing an a functor morphism Mss (Xsm /B, 0, n algebraic morphism between the moduli spaces ¯ ¯ ) → Hilbn¯ (Xsm /B) ' SymnB Xsm . ζ : Mss (Xsm /B, 0, n

(6.30)

Lemma 6.45. Assume that the base scheme B is normal. The relative moduli space ¯ ) of skyscraper sheaves of length n ¯ is normal as well. Mss (X/B, 0, n Proof. If B is a point, X is an elliptic curve and Mss (X, 0, n ¯ ) is smooth (cf. Corollary 3.34). Assume that dim B ≥ 1. We first note that since the fibers of the mor¯ ) → B are smooth and B is normal, Mss (Xsm /B, 0, n ¯) phism Mss (Xsm /B, 0, n ¯ ) not in Mss (Xsm /B, 0, n ¯ ), is normal as well. If ξ is a point of Mss (X/B, 0, n and t = p(ξ), the fiber Xt has to be singular and ξ belongs to the image of the ¯ − 1) ,→ Mss (Xt , 0, n ¯ ) given by F 7→ F ⊕ Ox0 , closed immersion f : Mss (Xt , 0, n where x0 is the unique singular point of the fiber Xt . This proves that the di¯ − 1) ,→ Mss (Xt , 0, n ¯ ) is smaller than n ¯ − 1. Hence, mension of Mss (Xt , 0, n

6.4. Relative moduli spaces for Weierstraß elliptic fibrations

215

¯ ) − Mss (Xsm /B, 0, n ¯ ) is smaller that n ¯−1+ the dimension of Mss (X/B, 0, n 0 0 dim B , where B ,→ B is the closed integral subscheme of the points t ∈ B whose fiber Xt is a singular curve. Since dim B 0 < dim B, we conclude that the ¯ ) − Mss (Xsm /B, 0, n ¯ ) is greater than one. Thus, codimension of Mss (X/B, 0, n ss ¯ ) is regular in codimension one. It remains only to prove that the M (X/B, 0, n ¯ ) is greater or equal to 2 at every point ξ of Mss (X/B, 0, n ¯) depth of Mss (X/B, 0, n ss ¯ ). Since t = p(ξ) belongs to B 0 , it is not the generic point of −M (Xsm /B, 0, n ¯ ) has depth ≥ 1 at ξ. Since ξ B, and we need only to show that that Mss (Xt , 0, n lies in the image of the closed immersion f , the result is proved by induction on n ¯.  Proposition 6.46. If B is normal, the morphism ζ of Equation (6.30) is an isomorphism and extends to an isomorphism of B-schemes ∼ Symn¯ X . ¯) → ζ : Mss (X/B, 0, n B Qn¯ Proof. Let T → B be a scheme morphism and ψ : T → B X a morphism of Bschemes, i.e., a family of morphisms ψi : T → X. Denoting by Γi ,→ X ×B T the graph of ψi , the sheaf E = ⊕i OΓi is flat over T and restricts to a skyscraper sheaf of Qn¯ ¯ ) given length n ¯ on every fiber. Thus, there is a morphism B X → Mss (X/B, 0, n ¯ by ψ 7→ E, which is equivariant under the natural action of the symmetric group S b n ¯ ss and, therefore, induces a morphism of B-schemes η : SymB X → M (X/B, 0, n ¯ ). ¯ ) is By Proposition 6.40, η is one-to-one on closed points. Since Mss (X/B, 0, n normal by Lemma 6.45, η is an isomorphism by Zariski’s main theorem [141, ¯ Xsm is the inverse 11.4]. Moreover, we can see that the restriction of η to SymnB of ζ.  Propositions 6.44 and 6.46 enable us to describe the structure of the relative moduli spaces of semistable-sheaves on the fibers of p : X → B (cf. [145, Theorem 2.1] for the case of degree 0 and [29] for the case of nonzero degree). Corollary 6.47. Assume that the base variety B is normal. 1. The relative Fourier-Mukai transform Φ induces an isomorphism of B-schemes ∼ Symn¯ X ¯ , 0) → ζ : Mss (X/B, n B for every positive integer n ¯. 2. For every pair (n, d) of integers (n > 0), there is an isomorphism of Bschemes ¯ X, Mss (X/B, n, d) ' SymnB where n ¯ = gcd(n, d).

216

Chapter 6. Relative Fourier-Mukai functors

We now study the isomorphism ζ in some more detail. Let E be a µ-semistable ¯ , 0). torsion-free sheaf on a fiber Xt , which represents a closed point of Mss (X/B, n Lr By Proposition 6.38, E is S-equivalent to a direct sum E ∼ i=0 (Li ⊕ .n.i. ⊕ Li ) of P torsion-free, rank one sheaves of degree zero (¯ n = i ni ). The isomorphism ζ can be explicitly described (on close points) as the assignment ζ

¯ ¯ , 0) − → SymnB X Mss (X/B, n

(6.31)

E 7→ n0 x∗0 + · · · + nr x∗r ,

where x∗i is the point of Xt that corresponds to L∗i under the identification ∼ J¯ (X/B) of (6.14). $: X → 0 ∼ nΘ), one defines an isomorphism J (X/B) → By sending L to L∗ ⊗ O (¯ X

n ¯

J0 (X/B). Corollary 6.48. There is a commutative diagram of B-schemes ¯ , 0) Mss (Xsm /B, n

∼

/ Symn¯ (Xsm ) , B φn ¯

det

 J0 (X/B) o

∼

 Jn¯ (X/B)

where det is the determinant morphism and φn¯ is the Abel morphism of degree n ¯. Let Ln¯ be a universal line bundle over q : X ×B Jn¯ (X/B) → Jn¯ (X/B). The ¯ and defines Picard sheaf Pn¯ = R1 q∗ (L−1 n ⊗ ωX/B ) is a locally free sheaf of rank n a projective bundle P(Pn¯∗ ) = Proj S • (Pn¯ ). We have a commutative diagram ¯ , 0) Mss (XU /U, _ n

∼

 ∼ Mss (Xsm /B, n ¯ , 0) det

 J0 (X/B) o

/ Symn¯ XU U _  / Symn¯ (Xsm )   B Abel

∼

∼

 Jn¯ (X/B)

/ P(Pn¯∗ |U ) _

 / P(Pn¯∗ ) k kkk kkk k k kkk k u kk dense

where U ,→ B is the open subset supporting the smooth fibers of p : X → B and XU = p−1 (U ). The immersions of the symmetric products into the projective bundles follow from the structure of the Abel morphism (cf. [3] and Proposition en¯ the locally free sheaf on J0 (X/B) induced by Pn¯ via 3.24). Let us denote by P ∼ J (X/B). the isomorphism Jn¯ (X/B) → 0
en¯ ' (det)∗ OMss (X /B,¯n,0) (Θn¯ ,0 ) . Corollary 6.49. P sm

6.5. Spectral covers

217

∼ (p O (nH))∗ , we obtain the following en ) → By using the isomorphism σ ∗ (P ∗ X theorem, whose proof is given in [114]. ¯ , OX ) (resp. Mss (XU /U, n ¯ , OX )) be the subCorollary 6.50. Let Mss (Xsm /B, n ss ¯ , 0) (resp. Mss (XU /U, n ¯ , 0)) with trivial scheme of the sheaves in M (Xsm /B, n ¯ , OX ) ,→ determinant. There is a dense immersion of B-schemes Mss (Xsm /B, n nH)). Moreover, this morphism induces an isomorP(Vn¯ ), where Vn¯ = p∗ (OX (¯ ¯ , OX ) ' P(Vn¯ |U ). phism of U -schemes Mss (XU /U, n Part 1 of Corollary 6.47 generalizes [110, Theorem 3.8] and can be considered as a global version of the results obtained in Section 4 of [114] about the relative moduli space of locally free sheaves on X → B whose restrictions to the fibers have rank n and trivial determinant. We can get such results also from Corollary 6.48 by making use of the standard properties of the Abel morphism.

6.5

Spectral covers

Let p : X → B be a Weierstraß fibration, with X smooth and B normal as in ∼ Symn X Section 6.4.2. The isomorphism ζ : Mss (Xsm /B, n, 0) → sm provided by B Proposition 6.46 may be considered from a different perspective. Let Ft be a µsemistable torsion-free sheaf of rank n and degree 0 on a fiber Xt which defines a closed point of Mss (Xsm /B, n, 0), that is, Ft is S-equivalent to a direct sum of line bundles of degree 0. Then ζ(Ft ) is a cycle of length n defined by the modified support of the only non-vanishing Fourier-Mukai transform Φ1t (Ft ) of Ft (this modified support is by definition the closed subscheme defined by 0-th Fitting ideal of Φ1t (Ft ), cf. Definition C.9). Recall that, by Proposition 6.38, Φ1t (Ft ) is supported on a finite number of points. When Ft moves in a flat family F, the modified support of Φ1t (Ft ) defines a n-covering of the parameter space of the family. We shall study this kind of coverings, which we shall call spectral covers, and see how F can be reconstructed out of its “spectral data.” Before giving the precise definition of spectral cover, we describe some properties relating the WIT1 condition with relative semistability. The following result is a direct consequence of Proposition 6.5 and Corollary 6.39. Proposition 6.51. Let F be a sheaf on X, flat over B and of relative degree zero. There exists an open subscheme S(F) ⊆ B which is the largest subscheme of B fulfilling one of the following equivalent conditions: 1. FS(F ) is WIT1 and FbS(F ) is flat over S(F). 2. The sheaves Ft are WIT1 for every point t ∈ S(F). 3. The sheaves Ft are torsion-free and µ-semistable for every point t ∈ S(F).

218

Chapter 6. Relative Fourier-Mukai functors We shall call S(F) the relative semistability locus of F.

Corollary 6.52. Let F be a sheaf on X of fiberwise of degree zero and flat over B. If S(F) is dense, then F is WIT1 . Proof. By Proposition 6.51 FS(F ) is WIT1 ; hence, Φ0S (F)S(F ) = 0, because S(F) → B is a flat base change. The sheaf Φ0S (F) is flat over B and vanishes on an open  dense subset, so that it vanishes everywhere. Thus, F is WIT1 . The notion of spectral cover has been introduced by Friedman, Morgan and Witten [112, 113, 114] (cf. also [10, 145]). Definition 6.53. Let F be a sheaf on X. The spectral cover of F is the modified support C(F) = Supp0 (Φ1 (F)) of Φ1 (F), i.e., the closed subscheme of X defined 4 by the 0-th Fitting ideal F0 (Φ1 (F)). Some fundamental properties of spectral covers are readily established. Lemma 6.54. The restriction of the spectral cover C(F) to a fiber Xt of p is the spectral cover of the restriction Ft of the sheaf to the fiber, C(F)t = C(F) ∩ Xt ' C(Ft ) . Proof. Since Φ1 (F)t ' Φ1t (Ft ) by Corollary 6.3, the result follows from the base change property of the modified support (Lemma C.10).  Lemma 6.55. Let F be a 0-degree torsion-free µ-semistable sheaf of rank n ≥ 1 Lr i be the S-equivalence given by Proposition on a fiber Xt and let F ∼ i=0 mL⊕n i 6.38, where L0 is the unique rank 1 torsion-free sheaf of degree 0 on Xt which is b ≥ n, not locally free (and then one may have n0 = 0). Then, length(OXt /F0 (F)) where equality holds if either n0 = 0 or n0 = 1, that is, if L0 occurs at most once. Proof. As we see in the proof of Proposition 6.38, Eb ' ⊕m i=1 Fi where Fi is a ∗ skyscraper sheaf of length ni supported at the point xi corresponding to Li under the isomorphism X ' J0 (X/B). Since the formation of the 0-th Fitting ideal is multiplicative with respect to direct sums of sheaves (cf. Eq. (C.19)), we have b = Qm F0 (Fi ). Moreover, the sheaf ideals F0 (Fi ) are pairwise coprime, F0 (E) i=1 since they correspond to subschemes supported at different points, and then b ' OXt /F0 (E)

m M

OXt /F0 (Fi ) .

i=1

We now consider the skyscraper sheaves Fi . If the point x∗i is smooth (i.e., if Li is a line bundle), then the local ring OXt ,x∗i is a principal ideals domain, and then Fi is

6.5. Spectral covers

219

a direct sum of skyscraper sheaves of the form OXt /mri , where mi is the ideal of x∗i in Xt . Then F0 (Fi ) = mni i again by Equation (C.19), and length(OXt /F0 (Fi ) = ni . If n0 = 0, that is, if the unique non-locally free rank 1 torsion-free sheaf of degree 0 L0 does not occur in the S-equivalence class of E, we deduce that b = n. length OXt /F0 (E) To conclude, let us assume that n0 ≥ 1. F0 still has filtration whose successive quotients are isomorphic to OXt /m0 , so that Equation (C.20), gives F0 (F0 ) ⊆ mn0 0 and then length(OXt /F0 (F0 )) ≥ length(OXt /mn0 0 ≥ n0 , with equality only if  n0 = 1. The result follows. Proposition 6.56. If F is relatively torsion-free and µ-semistable of rank n and degree zero on X → B, the spectral cover C(F) → B is a finite morphism with fibers of length ≥ n and generic fiber of length n. If in addition C(F) does not meet any singular point of the fibers of p, all the fibers of the spectral cover C(F) → B have length n. Proof. By Lemmas 6.54 and Lemma 6.55, the morphism C(F) → B is finite with fiber of degree ≥ n. The second statement follows from Lemma 6.55 as well.  A useful result is the following. Proposition 6.57. If the relative semistability locus of the sheaf F is dense, the / S(F). spectral cover C(F) contains the whole fiber Xt for every point s ∈ / S(F), we have a destabilizing Proof. By Corollary 6.52, F is WIT1 . Since s ∈ sequence 0 → G → Ft → K → 0 , where K is a µ-semistable sheaf on Xs of negative degree. By Proposition 6.43, K is WIT1 and Φ1 (K) is torsion-free. Since Φ1t (Ft ) → Φ1t (K) is surjective, C(F)t =  C(Ft ) = Xs . We know that Fb = Φ1 (F) = i∗ L , where i : C(F) ,→ X is the immersion of the spectral cover and L is a sheaf on C(F). What can be said about L? A first look at Lemma 6.22 seems to imply that L has rank one at every point (at least on the fibers where F is µ-semistable). This is indeed what happens, though one has to be careful because the spectral cover can be quite singular. For a precise statement we need Simpson’s notions of torsion-free sheaf (Definition C.3) and polarized rank (Definition C.5). Let us consider on X a polarization of the type H = aΘ + bp∗ HB ,

a > 0, b > 0,

220

Chapter 6. Relative Fourier-Mukai functors

where HB is a polarization on B. Proposition 6.58. If F is a relatively torsion-free and µ-semistable sheaf on X → B of relative rank n and degree zero, then the restriction L of Fb to the spectral cover C(F) is a torsion-free sheaf of polarized rank one. Conversely, given a closed subscheme i : C ,→ X such that the projection p ◦ i : C → B is a finite covering of degree n, and a torsion-free sheaf L on C of polarized rank one, then the sheaf b and F = Φ b 0 (i∗ L) is a sheaf on X → B relatively i∗ L is WIT0 with respect to Φ, torsion-free and µ-semistable of rank n and degree zero. Proof. We first prove that L is a torsion-free sheaf on C(F), or equivalently, that Fb is a pure sheaf of dimension equal to dim B on X. Let G be a subsheaf of Fb supported in dimension strictly smaller than dim B. For every point t ∈ B the restriction of the spectral cover to the fiber Xt is a finite set. Hence, the restriction of the modified support of G is finite as well, so that Gt is WIT0 with respect to the b (Corollary b t and G is WIT0 with respect to Φ inverse Fourier-Mukai transform Φ 0 b / p(Supp0 (G)) 6.3). One has that Φ (G) is a subsheaf of F. Since Gt = 0 for s ∈ b 0 (G) is not the whole of X, thus and dim p(Supp0 (G)) < dim B, the support of Φ contradicting that F is torsion-free. Moreover, by Proposition C.12, the polarized rank of Fb|D(F ) is one. To show the converse, we note that, since the restriction of i∗ L to the fiber Xt is supported in dimension zero for every point t ∈ B, (i∗ L)t is WIT0 with b as well. Moreover, b t ; by Corollary 6.3 G is WIT0 with respect to Φ respect to Φ since the polarized rank of L on C is one, C coincides with the modified support Supp0 (i∗ L) by Proposition C.12. Hence, by Proposition C.11, [C] = c1 (i∗ L). Then the relative degree is d(i∗ L) = c1 (i∗ L) · f = C · f = n . b 0 (i∗ L) Since rk(i∗ L) = 0, by Proposition 6.34 the relative Chern character of F = Φ is (n, 0). We have only to check that the restriction Ft of F to every fiber Xt is µ-semistable. Since (i∗ L)t is supported in dimension zero, it is IT0 , and then b 0 ((i∗ L)t ) by Corollary 6.3. Thus it is WIT1 and then µ-semistable by Ft = Φ Corollary 6.39. 

6.6

Absolutely stable sheaves on Weierstraß fibrations

Let p : X → B be a Weierstraß elliptic fibration, and let us assume that X is smooth. We wish to apply the relative Fourier-Mukai transform to the study of moduli spaces of sheaves on X that are µ-stable with respect to certain kinds of polarizations on X. To distinguish them from the moduli spaces of “relatively”

6.6. Absolutely stable sheaves on Weierstraß fibrations

221

semistable sheaves considered in Section 6.4, we have used the terminology “absolutely” stable.

6.6.1

Preservation of absolute stability for elliptic surfaces

We shall use the spectral construction to build µ-stable sheaves on an elliptic surface X admitting a Weierstraß model. Actually, stable sheaves on spectral covers transform to µ-stable sheaves on the surface, and in this way one obtains an open subset of a moduli space of µ-stable sheaves on the surface. To this end we need to study the preservation of stability, and for this, we rely on the computation of the Chern character of the Fourier-Mukai transforms provided by Equation (6.21). The elliptic surface X is polarized by H = aΘ+bf for suitable positive integers a and b (cf. Section C.2). Let F be a sheaf on X with Chern character (n, ∆, s w) with n > 1, and let w be the fundamental class of X. We denote c = ∆ · Θ and d = ∆ · f as in Equation (6.21), and identify the second Chern character s w with the rational number s. By using the formula (6.20) for the Todd class of X, we have the following expressions for the the Hilbert polynomial (cf. Eq. (C.14)) and the Euler characteristic of F: χ(X, F(mH)) = 12 n H 2 m2 + (ac + bd + 12 na(c1 − e))m + χ(X, F) χ(X, F) = s + 12 d(c1 − e) + ne .

(6.32)

Assume now that F is flat over B and that its restrictions to the fibers of X are torsion-free and µ-semistable sheaves of degree d = 0. By Proposition 6.58, the projection C(F) → B of the spectral cover is a finite morphism of degree n. Since B is smooth, C(F) → B is automatically dominant, and hence is flat. We then know that Fb is a torsion-free sheaf of polarized rank one on the spectral cover C(F), as follows from Proposition 6.58. Since C(F) → B is finite, the fiber f induces a polarization on C(F). We can then consider Simpson stability and semistability with respect to f for sheaves on C(F). Proposition 6.59. For any integer a > 0 there is an integer b0 > 0 such that, for any b > b0 , the sheaf F is µ-(semi)stable on X with respect to H = aΘ + bf if and only if L = Fb|C(F ) is µ-(semi)stable with respect to f as a sheaf on the spectral cover C(F). Proof. We can assume a = 1. By Equation (6.21), one has b χ(C(F), L(mf) = χ(X, F(mf)) = n m + c − ne + 12 nc1 . The Simpson slope of Fb is b = µ(F)

c − ne + 12 nc1 . n

(6.33)

222

Chapter 6. Relative Fourier-Mukai functors

Let now 0 → G → Fb → K → 0

(6.34)

be an exact sequence. Then G is supported by C(F), so that it is WIT0 and F 0 = Gb has relative degree 0 and is WIT1 by Proposition 6.58. Reasoning as above, the Simpson slope of G is c0 − n0 e + 12 n0 c1 µ(G) = , n0 where primes denote the topological invariants of F 0 . Moreover one has the exact sequence b → 0. 0 → F0 → F → K Assume that F is µ-semistable with respect to H = Θ + bf for some positive number b. Then, one has c0 c ≤ , n0 n because F and F 0 have degree zero on fibers. On the other hand, if Fb is not µb This is semistable and (6.34) is a destabilizing sequence, we have µ(G) > µ(F). equivalent to n0 c − nc0 > 0 , which is a contradiction. The statement about stability can be proved analogously. For the converse, assume that Fb is µ-semistable on X with respect to f and that 0 → F0 → F → Q → 0

(6.35)

is an exact sequence. The sheaf F 0 is WIT1 so that d0 ≤ 0 by Corollary 6.36. Assume first that d0 < 0 and fix b0 > 0 such that H0 = Θ + b0 f is a polarization. Then the set of the integers nc1 (G) · H0 − rk(G)c, where G ranges over all nonzero subsheaves of F, is bounded; let ρ be its maximum. Thus, n(c0 + bd0 ) − n0 c ≤ ρ + n(b − b0 )d0 is strictly negative for sufficiently large b. This proves that if d0 < 0, the exact sequence (6.35) does not destabilize F with respect to any b sufficiently large. Then, if the sequence in Equation (6.35) is destabilizing with respect to H = Θ + bf for some sufficiently large positive number b, one has d0 = 0. We can assume that n0 < n and that Q is torsion-free and µ-semistable with respect to H; the destabilizing condition is now nc0 − n0 c > 0 . Moreover d(Q) = 0. Since Q is torsion-free, for every t ∈ B there is an exact sequence 0 → Ft0 → Ft → Qt → 0

6.6. Absolutely stable sheaves on Weierstraß fibrations

223

so that Qt is µ-semistable of degree 0. Then Q is WIT1 and one has an exact sequence of Fourier-Mukai transforms: b → 0. 0 → Fb0 → Fb → Q Proceeding as above we see that the µ-semistability of Fb implies nc0 − n0 c ≤ 0, which is a contradiction.  Proposition 6.59 cannot be directly used in the way it is stated to produce isomorphisms between moduli spaces, because for any given a > 0, the polarization H = aΘ + bf depends on the sheaf F. Our next aim is to prove that we can find b depending only on the topological invariants of F. Let us consider a vector v = (n, ∆, s w) with n a natural number, ∆ a divisor, s a rational number and w the fundamental class of X. We set c2 = 12 D2 − s and B(v) = 2nc2 − (n − 1)∆2 . If F is a sheaf with Chern character ch(F) = v, the number B(v) = B(F) is usually called the Bogomolov number of F. If F is torsion-free and µ-semistable with respect to some polarization, the Bogomolov inequality B(F) ≥ 0 holds (cf. [44, 155]). We fix a polarization H0 on X (which can be assumed to be of the form H0 = Θ + b0 f for some b0 > 0), and we write Ht = H0 + tf. By the Nakai-Moishezon criterion (cf. [141, Theorem A.5.1]), this is a polarization for every integer t > 0. By technical reasons we will also consider the real divisors Ht for a real number t. These are elements of the real N´eronSeveri group N S(X) ⊗Z R. The ample cone is the convex subset of N S(X) ⊗Z R generated by the (integral) ample divisors; its elements are called real ample divisors. Stability and semistability with respect to real ample divisors are defined in an analogous way to the ordinary case. Let us assume that F is flat over B and that its restrictions to the fibers are torsion-free and µ-semistable sheaves of degree d = 0. Consider n > 1. The proof of the following result is similar to that of [114, Lemma 7.5]. Lemma 6.60. Assume that L = Fb|C(F ) is µ-stable with respect to f as a sheaf on the spectral cover C(F). If F is not µ-stable with respect to Ht0 , there exists t ≥ t0 and a divisor D such that D · Ht = 0 and p ≤ D2 < 0 , 2

where p = − n4 B(F). Proof. For the sake of simplicity, we write (semi)stable meaning µ-(semi)stable. By Proposition 6.59, F is Ht -stable for t  0. Let t1 be the smallest real number greater or equal to t0 such that F is Ht -stable for every t > t1 . It follows easily that F is strictly Ht1 -semistable, so that there is an exact sequence 0 → F 0 → F → F 00 → 0 ,

224

Chapter 6. Relative Fourier-Mukai functors

where F 0 and F 00 are torsion-free with the same slope as F with respect to Ht1 . Then, if D = n0 c1 (F 00 ) − n00 c1 (F 0 ), where n0 and n00 are the ranks of F 0 and F 00 , respectively, one has D · Ht1 = 0. Moreover, D is not numerically trivial, because otherwise, F would be strictly Ht -semistable for all t, so that D2 < 0 by the Hodge’s index theorem (cf. [141, Theorem A.5.2]). On the other hand, one has B(F) =

n n D2 0 00 B(F ) + B(F ) − . n0 n00 n0 n00

Since F 0 and F 00 are Ht1 -semistable, B(F 0 ) ≥ 0 and B(F 00 ) ≥ 0 by the Bogomolov inequality, and then D2 ≥ −n0 n00 B(F). Furthermore, n = n0 + n00 gives n0 n00 ≤  n2 /4 and we have the inequality of the statement. We can now strengthen Proposition 6.59. Proposition 6.61. For any integer a > 0, there is b0 > 0 depending only on the topological invariants (n, ∆, s w) = ch(F), such that for any b ≥ b0 , the sheaf F is µ-stable on X with respect to H = aΘ + bf if and only if L = Fb|C(F ) is µ-stable with respect to f as a sheaf on the spectral cover C(F). Proof. We can take a = 1. We need to prove that there exists b0 > 0 depending only on (n, ∆, s w) = ch(F), such that if L = Fb|C(F ) is µ-stable with respect to f as a sheaf on the spectral cover C(F), then F is µ-stable with respect to Θ + bf for any b > b0 . This is equivalent to the fact that there exists t0 such that for t ≥ t0 , F is µ-stable with respect to Ht . We divide the proof in two parts. 2

(a) If D is a divisor such that D2 ≥ p = − n4 B(v) and α = D · f > 0, then D · Ht > 0 for every t ≥ −(H0 · f) p/2. Let us consider the divisor E = (Ht · f) D − (D · f) Ht . One has E · f = 0 and f2 = 0, and then E 2 > 0 by the Hodge index theorem; indeed, the divisor ¯ · f = 0 so that ¯ = (H0 · f) E − (H0 · D·) f is not numerically trivial and E E 2 2 2 ¯ 0 > E = (H0 · f) E . It follows that 0 > E 2 = λ2 D2 − 2λβ D · Ht + α2 (H02 + 2λt) , where λ = H0 · f > 0. Since α > 0, one has α2 ≥ 1 and one has 2αλ(D · Ht ) > λ2 D2 + α2 (H02 + 2tλ) ≥ λ2 D2 + (H02 + 2tλ) > λ2 D2 + +2tλ . Since λ > 0, one has α (D · Ht ) > λ D2 + 2t ≥ 0 .

6.6. Absolutely stable sheaves on Weierstraß fibrations

225

2

(b) F is Ht -stable for t ≥ t0 = n8 (H0 · f) B(v) = −(H0 · f) p/2. If F is not Ht0 stable, by Lemma 6.60, there exists t1 ≥ t0 such that F is Ht -stable for t > t1 and strictly µ-semistable with respect to Ht1 . As in the proof of Lemma 6.60, there is an exact sequence 0 → F 0 → F → F 00 → 0 , where F 0 and F 00 are torsion-free and Ht1 -semistable. If D = n0 c1 (F 00 ) − n00 c1 (F 0 ), one has D · Ht1 = 0 and 0 > D2 ≥ p. Using the same notation as in the proof of Proposition 6.59, we have D · f = −nd0 . Since D · f ≤ 0 by part a), whereas the semistability of the restriction of F to the fibers gives d0 ≤ 0, one has d0 = 0. Then D · Ht1 = 0 is equivalent to nc0 − n0 c = 0, which contradicts the stability of b Hence F is Ht -stable, and is also Ht -stable for t ≥ t0 .  F. 0 Proposition 6.61 has an important consequence. Corollary 6.62. Let i : C ,→ be an integral Cartier divisor such that C → B is flat of degree n. Given integers a > 0 and d, there exists an integer b0 depending only on a and d, such that for every b > b0 and every line bundle L on C of degree b 0 (i∗ L) is a locally free sheaf of rank n of d, the Fourier-Mukai transform F = Φ relative degree zero on X and is µ-stable with respect to B = aΘ + bf. This enables us to construct µ-stable locally free sheaves on the elliptic surface X out of the “spectral data” (C, L), and then prove that certain moduli spaces of µ-stable sheaves on X are not empty.

6.6.2

Characterization of moduli spaces on elliptic surfaces

It is convenient to polarize both the elliptic surface and the spectral covers with the same polarization H = aΘ + bf. The advantage is that the (semi)stability of a sheaf L on a spectral cover i : C ,→ X (with respect to the induced polarization H|C ) is equivalent to the (semi)stability of i∗ L with respect to H as a sheaf on X. We have then results analogous to Propositions 6.59 and 6.61. Let (n, ∆, s w) be a cohomology class, with n ∈ Z, s ∈ Z and ∆ ∈ H 2 (X, Z). We write c = ∆ · Θ, d = ∆ · f and χ = s + 12 (c1 − e)d + ne (where c1 = c1 (B) and e = −Θ2 ). Let F be a sheaf on X flat over B with ch(F) = (n, ∆, s w) and whose restrictions to the fibers are torsion-free and semistable sheaves of degree d = 0. b > 0, for any integer a > 0 there Proposition 6.63. Assume that n > 1. If χ(X, F) is b0 > 0 such that for any b > b0 the following conditions are equivalent. 1. The sheaf F is Gieseker-(semi)stable on X with respect to H = aΘ + bf; 2. Fb is µ-(semi)stable on X with respect to the same polarization;

226

Chapter 6. Relative Fourier-Mukai functors

3. L = Fb|C(F ) is µ-(semi)stable as pure sheaf of dimension 1 on the spectral cover C(F), with respect to the restriction of H to C(F).

Proof. The second and third conditions are trivially equivalent, so that it is enough to prove the equivalence between the first and the second. We can take a = 1. We recall from Equations (6.32) and (6.21) that the Hilbert polynomials of F and Fb are given by the formulas χ(X, F(mH)) = 12 n H 2 m2 + (c + 12 n(c1 − e))m + χ ; b χ(X, F(mH)) = (nb − χ)m + χ ˆ,

(6.36)

b = c − ne + 1 nc1 with χ = χ(X, F) = s + ne + 12 c(c1 − e) and χ ˆ = χ(X, F) 2 (c1 = c1 (B)). On the other hand, the Simpson slope of Fb is b = µ(F)

χ ˆ . nb − χ

Let now 0 → G → Fb → K → 0

(6.37)

be an exact sequence. Then G is supported by C(F), so that it is WIT0 and F 0 = Gb has relative degree 0 and it is WIT1 by Proposition 6.58. Reasoning as above, the Simpson slope of G ' Fb0 is χ ˆ0 , µ(G) = 0 n b − χ0 where primes denote the topological invariants of F 0 . Since spectral cover C(F 0 ) is contained in C(F), if we choose an integer b1 > 0 such that H0 = Θ + b1 f is ample, we have 0 < C(F 0 ) · H0 ≤ C(F) · H0 , that is: 0 < b1 n0 − χ0 ≤ b1 n − χ .

(6.38)

Here n0 is the rank of F 0 , so that there are only a finite number of possible values for n0 , and Equation (6.38) implies that there are only a finite number of possible values for χ0 as well. Moreover these values of n0 and χ0 only depend on the topological invariants of F. Let us consider the exact sequence of Fourier-Mukai transforms b → 0, 0 → F0 → F → K and assume that F is Gieseker-stable with respect to H = Θ + bf for some positive b. Since F and F 0 have degree zero on fibers, this is equivalent to saying that either nc0 − n0 c < 0 or one simultaneously has nc0 − n0 c = 0 and nχ0 − n0 χ < 0.

6.6. Absolutely stable sheaves on Weierstraß fibrations

227

We now consider b = (nb − χ)χ ˆ0 − (nb − χ0 )χ ˆ ∆(Fb0 , F) = (nc0 − n0 c)b + cχ0 − c0 χ + n0 eχ − neχ0 + 12 (nc1 χ0 − n0 c1 χ)

(6.39)

nχ0 − n0 χ = (nc − n c)(b − χ/n) + χ . ˆ n 0

0

On the one hand, since there are only a finite number of values for n0 χ − nχ0 , and these values depend only on the topological invariants of F, if nc0 − n0 c < 0, there b < 0, which proves the is b0 = b0 (n, ∆, s w) such that for b > b0 one has ∆(Fb0 , F) 0 0 0 b On the other hand, if nc − n c = 0 and nχ − n0 χ < 0, the condition stability of F. b < 0. χ ˆ > 0 also implies ∆(Fb0 , F) The corresponding semistability statement is proved analogously. For the converse, assume that Fb is µ-semistable with respect to Θ + bf for b  0. Proceeding as in the proof of Proposition 6.59, we see that there exists b0 depending on the sheaf F such that if 0 → F0 → F → Q → 0 is a destabilizing sequence (in the sense of Gieseker) with respect to H = Θ + bf for b ≥ b0 with n0 < n and Q torsion-free and Gieseker-semistable with respect to, then d0 = 0. Then, the fact that F 0 destabilizes F is equivalent either to nc0 − n0 c > 0, or nc0 − n0 c = 0 and nχ0 − n0 χ > 0. Again as in the proof of Proposition 6.59, we prove that Q is WIT1 and that there is an exact sequence b → 0. 0 → Fb0 → Fb → Q b we see as before that since there are From the expression (6.39) for ∆(Fb0 , F) 0 only a finite number of values for n χ − nχ0 , and these values depend only on the topological invariants of F, if nc0 − n0 c > 0, then there is b0 = b0 (n, ∆, s w) such b > 0. Moreover if nc0 − n0 c = 0 and nχ0 − n0 χ > 0, that for b > b0 , one has ∆(Fb0 , F) b > 0 also in this case. This contradicts the the condition χ ˆ > 0 gives that ∆(Fb0 , F) b semistability of F. The statement about stability can be proved in a completely similar way.  b < 0, absolute Proposition 6.63 implies that, if we assume that χ(X, F) Gieseker-stability with respect to aΘ + bf is preserved by the Fourier-Mukai transform for values of b  0 that depend on a and on the Chern character (n, ∆, s w). This was proved in a different way in [145]; similar results can be found in [166, 294].

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Chapter 6. Relative Fourier-Mukai functors

6.6.3

Elliptic Calabi-Yau threefolds

In this section the base B of the elliptic fibration is a smooth projective surface and we assume that X is a smooth Calabi-Yau threefold. As we mentioned in Section 6.2.1, the existence of a section implies that B has to be a surface of one of the following types: a del Pezzo surface, a Hirzebruch surface, an Enriques surface or a blowup of a Hirzebruch surface. As we did in Section 6.5, we consider on X a polarization of the type H = aΘ + b p∗ HB ,

a > 0, b > 0,

where HB is a polarization on B. Our aim is to prove that also for elliptic CalabiYau threefolds, the (absolute) µ-stability with respect to H of sheaves on X of degree zero on the fibers is preserved by the relative Fourier-Mukai transform for suitable values of a and b. This was first proved by Friedman, Morgan and Witten [113, 114] for sheaves constructed from spectral data (C, L), where C is an irreducible surface in X, of finite degree n over B, and L is a line bundle on C. In this way they produced instances of µ-stable bundles on elliptic CalabiYau threefolds, an issue of great interest in string theory and mirror symmetry, especially in constructing compactifications of the heterotic string. The proof of the preservation of the absolute µ-stability for arbitrary spectral covers was obtained in [4]. As for elliptic surfaces, the proof relies on the computation of the Chern character of the Fourier-Mukai transforms provided by (6.26). We have the following expressions for the the Hilbert polynomial (cf. Eq. (C.14)) and the Euler characteristic of F: χ(X, F(mH)) = 16 n H 3 m3 +

1 2

ch1 (F) · H 2 m2

+ (ch2 (F) · H + nH · td2 (X))m + χ(X, F)

(6.40)

χ(X, F) = ch3 (F) + nH · td2 (X) , where n = ch0 (F), c1 = p∗ c1 (B) and c2 = p∗ c2 (B). Moreover, by Equation (6.24) 1 (c2 + 11c21 + the second Todd class of the Calabi-Yau threefold X is td2 (X) = 12 12Θ · c1 ). In the following, we shall assume that there is a decomposition H 2i (X, Q) = Θp∗ H 2i−2 (B, Q) ⊕ p∗ H 2i (B, Q).

(6.41)

Let us consider a torsion-free sheaf F on X of rank n and degree zero on ˜f, s), where η fibers and write its Chern characters as ch(F) = (n, p∗ S, Θp∗ η + a and S are classes in A1 (B) ⊗Z Q, s ∈ A3 (X) ⊗Z Q ' Q anf f ∈ A2 (X) ⊗Z Q is the class of a fiber of p, in agreement with Equation (6.25).

6.6. Absolutely stable sheaves on Weierstraß fibrations

229

Assume that F is flat over B and that its restrictions to the fibers are semistable. Then, F is WIT1 , the spectral cover C(F) is finite of degree n over B, and by Proposition 6.58, Fb is a pure sheaf of dimension 1 of polarized rank one on C(F). We know from Equation (6.26) that the topological invariants of Fb are b =0 ch0 (F) b = nΘ − p∗ η ch1 (F) b = −( 1 nc1 − p∗ S)Θ − (s − 1 p∗ ηc1 Θ)f ch2 (F) 2 2 1 1 2 ∗ b = nΘc + a ˜ − Θc1 p S . ch3 (F) 1 6 2 As in the case of elliptic surfaces, we polarize the spectral cover C(F) with the restriction HC(F ) = p∗ HB |C(F ) of the pullback of the polarization we have fixed on the base surface B. Since Fb is supported in codimension 1, by Equation b and d(F) b with respect to HC(F ) are (C.16) the Simpson invariants r(F) 2 b = nHB , r(F)

b = S · HS − 1 bc1 · HB d(F) 2

(6.42)

(recall that td1 (X) = 0 because X is a Calabi-Yau threefold). Proposition 6.64. For any integer a > 0, there is an integer b0 > 0, such that for any b > b0 , the following holds true: 1. if F is µ-semistable on X with respect to H = aΘ + bp∗ HB , then L = Fb|C(F ) is µ-semistable with respect to HC(F ) as a pure sheaf of dimension 1 on the spectral cover C(F). 2. If L = Fb|C(F ) is µ-stable with respect to HC(F ) as pure sheaf of dimension 1 on C(F), then F is µ-stable on X with respect to H = aΘ + bp∗ HB . Proof. We can take a = 1. Since Fb is supported by the spectral cover, the support of every subsheaf G of Fb is contained in C(F) as well. Thus, Fb is WIT0 with respect to the inverse Fourier-Mukai transform and its transform is a WIT1 subsheaf F¯ of F. Moreover, F¯ has degree zero on fibers, again by (6.27), so that (6.42) remains ¯ true, mutatis mutandis, for F. 1. Assume that F is µ-semistable with respect to H = Θ + bp∗ HB for b  0 and that Fb is destabilized, as a sheaf on the spectral cover, by a subsheaf G. Then, b¯ ¯ as we said before, G = F |C(F ) for certain subsheaf F of V of degree zero on fibers, and we have b¯ ) > µ(F) b , µ(F

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Chapter 6. Relative Fourier-Mukai functors

which is equivalent to ¯ S · HB > 0 . nS¯ · HB − n On the other hand, the µ-semistability of F with respect to H = Θ + bp∗ HB gives ¯ · c1 ≤ 0 , ¯ S · HB ) + (¯ nS − nS) 2b(nS¯ · HB − n for an arbitrarily large b, which is a contradiction. 2. Assume now that L = Fb|C(F ) is µ-stable with respect to HC(F ) as a sheaf on the spectral cover, and that 0 → F¯ → F → Q → 0 is a destabilizing sequence with respect to H = Θ + bp∗ HB . We can assume that n ¯ < n and that Q is torsion-free and µ-stable with respect to H. Moreover, we have ¯ · H2 > n ¯ c1 (F) · H 2 . nc1 (F) The sheaf F¯ is WIT1 , so that d¯ ≤ 0 by Corollary 6.36. Assume first that d¯ < 0 and fix b0 > 0 such that H0 = Θ+b0 p∗ HB is a polarization. Then the set of the integers (nc1 (G)−n(G)c1 (F))·H02 for all nonzero subsheaves G of F is bounded from above; let ρ be its maximum. If Y ,→ X is a surface in the linear system p∗ HB , the set of integers (nc1 (K) − n(K)c1 (F|Y )) · H0|Y for all nonzero subsheaves K of F|Y has a lower bound. Hence, the set of the integers (nc1 (G) − n(G)c1 (F)) · H0 · p∗ HB for all nonzero subsheaves G of F is bounded from above; let ρ0 be its maximum. Now, for b ≥ b0 , one has ¯ 2H 2 , ¯ −n ¯ c1 (F)) · H 2 ≤ ρ + 2(b − b0 )ρ0 − ndb (nc1 (F) B which is strictly negative for b sufficiently large, which is a contradiction. Then d¯ = 0 and the destabilizing condition is (nS¯ − n ¯ S) · (2bHB − c1 ) ≥ 0 . One has (nS¯ − n ¯ S) · (2b0 HB − c1 ) ≤ ρ0 as before, and then one has ¯ S) · HB . 0 ≤ (nS¯ − n ¯ S) · (2bHB − c1 ) ≤ ρ0 + 2(b − b0 )(nS¯ − n

(6.43)

Moreover, since Q is torsion-free, for every t ∈ B there is an exact sequence 0 → F¯t → Ft → Qt → 0 , so that Qt is µ-semistable of degree 0. Then Q is WIT1 and one has an exact sequence of Fourier-Mukai transforms b¯ → Fb → Q b → 0. 0→F Proceeding as above we see that the µ-stability of Fb for b  0 implies that (nS¯ − n ¯ S) · HB < 0. Then the right-hand side of the inequality (6.43) is strictly negative for b  0, which is a contradiction. 

6.7. Notes and further reading

231

As in the case of elliptic surfaces, one can use Proposition 6.64 to construct µ-stable locally free sheaves on the elliptic Calabi-Yau threefold X out of the “spectral data” (C, L), and to prove that certain moduli spaces of µ-stable sheaves on X are not empty. Corollary 6.65. Let i : C ,→ X be an integral Cartier divisor such that C → B is flat of degree n. Given integers a > 0 and d, there exists an integer b0 such that for every b > b0 and every line bundle L on C of degree d, the Fourier-Mukai b 0 (i∗ L) is a locally free sheaf of rank n of relative degree zero on transform F = Φ X and is µ-stable with respect to B = aΘ + b p∗ HB . The integer b depends on the spectral data (C, L). It is not known whether it can be chosen as a function only of the topological invariants of (C, L) as it is the case for elliptic surfaces.

6.7

Notes and further reading

Relative integral functors on singular fibrations. Many of the results stated in this chapter for elliptic fibrations X → B when the total space X is smooth, still hold true if the smoothness condition is weakened. In particular, the relative integral functors defined in Section 6.2.3 are equivalences of categories for arbitrary Weierstraß fibrations. One can prove this statement using the generalization to Gorenstein varieties of the characterization of fully faithful integral functors in terms of the kernel (see [144]); another proof using spherical objects can be found in [80]. Compactified Jacobians. Though the compactified relative Jacobians introduced in Section 6.2.4 can be defined for elliptic fibrations X → B of any dimension, their geometry is not known in the general situation. A particularly interesting case is that of the compactified relative Jacobian J¯0 (X/B) = M (X/B, 1, 0) parameterizing S-equivalence classes of relatively semistable sheaves on the fibers having relative rank 1 and relative degree 0. One finds out that J¯0 (X/B) → B is independent of the polarization and is an elliptic fibration as well. Moreover, if X → B has reducible fibers (hence J¯0 (X/B) contains strictly semistable points as pointed out at the end of Section 6.3), the elliptic surface J¯0 (X/B) → B may even be singular. In particular, it may fail to be isomorphic to : X → B (cf. [82]). A fairly complete investigation of a class of compactified relative Jacobians, including those associated to relatively minimal elliptic surfaces, has been carried out by L´opez Mart´ın in [198, 199]. Stable sheaves on Calabi-Yau threefolds and string theory. The preservation of the absolute stability provides a method for constructing stable locally free sheaves on minimal elliptic surfaces or elliptic Calabi-Yau threefolds (cf. Propositions 6.62 and

232

Chapter 6. Relative Fourier-Mukai functors

6.65). This method, known as “spectral data construction,” is due to Friedman, Morgan and Witten [113, 114]. This construction of stable bundles on Calabi-Yau threefolds is of great importance in string theory, since a stable bundle F, satisfying suitable properties, provides a compactification of the heterotic string. From a physical viewpoint, the Calabi-Yau condition and the holomorphicity of the bundle follow from supersymmetry in the 10-dimensional space-time, and one has also to specify a B-field. Cancellation of anomalies forces the structure group of the bundle to be E8 × E8 , SO(32) or one of their subgroups. Because of all these constraints, the bundle F has to satisfy the topological condition ch2 (F) = ch2 (X) + W , where W is an effective algebraic class. Moreover, the Euler characteristic χ(F) must be small (physically, the Euler characteristic corresponds to the number of generations of fermionic particles). This appears to be a quite strong restriction. Actually, when X is the quintic in P4 and F is the tangent bundle, one has χ(F) = 100; similarly, the Euler characteristic is quite big in other examples. Spectral covers. This construction has received many applications in physics. An extensive bibliography can be found in the review paper [5]. Gerby transforms. One can also study a kind of transform associated with elliptic fibrations X → B which do not admit a section. To deal with this case, one needs to specify some additional data (a B-field from the physical viewpoint, an ∗ -gerbe on X in mathematical terms). This “gerby” transform establishes a OX correspondence between some bundle data on X and spectral data on a gerbe associated with the compactified relative Jacobian. This is treated by Donagi and Pantev in [97] and also relates to work of C˘ ald˘ araru [82, 83]. Fourier-Mukai transform for real families of tori. The homological mirror symmetry conjecture by Kontsevich (see, e.g., [188]) postulates (very loosely speaking) an equivalence between the derived category of coherent sheaves on a Calabi-Yau manifold X, and the so-called Fukaya category of the mirror Calabi-Yau maniˆ In a celebrated paper, Strominger, Yau and Zaslow conjectured that any fold X. Calabi-Yau 3-manifold which admits a mirror partner contains an open, dense subset which is a fibration in real 3-tori (the SYZ conjecture). Substantial work toward a proof of a (possibily modified) SYZ conjecture has been made by M. Gross [132]. Building on this, it has been conjectured (see, e.g., Fukaya in [118]) that homological mirror symmetry is described by a “real” Fourier-Mukai transform. Such a transform was introduced in [74, 75], see also Arinkin and Polishchuk [8] and Leung, Yau and Zaslow [195]; assuming that X is a symplectic manifold fibered in ˆ has a natural complex strucLagrangian (real) tori, the fiberwise dual fibration X ture, and a suitably defined transform maps local systems supported on Lagrangian ˆ and vice versa. The construction of submanifolds of X to coherent sheaves on X, this transform was further developed in [123].

Chapter 7

Fourier-Mukai partners and birational geometry Introduction In this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform. We have already seen some facts about Fourier-Mukai partners. Two projective varieties X and Y are Fourier-Mukai partners if there is an exact equivalence of triangulated categories between their bounded derived categories (Definition 2.36). By Orlov’s representability theorem 2.15, when X and Y are smooth they are Fourier-Mukai partners if and only if there is a Fourier-Mukai functor • ∼ b b ΦK X→Y : D (X) → D (Y ) (Lemma 2.37). As a consequence of this fundamental result, in Chapter 2 we were able to prove that a Fourier-Mukai partner of a smooth variety is also smooth (Lemma 2.37) of the same dimension and the canonical bundles ωX and ωY have the same order (Theorem 2.38). We also know that if two smooth projective varieties X and Y are Fourier-Mukai partners, and ωX is either ample or anti-ample, there is an isomorphism X ' Y (Theorem 2.51), that is, the only Fourier-Mukai partner of X is X itself. In this chapter we shall complete this picture by developing a classification of all Fourier-Mukai partners Y of a smooth projective surface X. Section 7.1 contains some introductory material. We shall show that FourierMukai partners have the same Kodaira dimension, and that K-equivalent FourierMukai partners are isomorphic. We shall provide a characterization of crepant C. Bartocci et al., Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progress in Mathematics 276, DOI: 10.1007/b11801_7, © Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2009

233

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Chapter 7. Fourier-Mukai partners and birational geometry

morphisms in terms of derived categories. Again as a preliminary tool for studying Fourier-Mukai partners, in Section 7.2 we study integral functors for quotient varieties. After quickly showing that an algebraic curve has no nontrivial Fourier-Mukai partner, we devote Section 7.4 to the study of Fourier-Mukai partners of algebraic surfaces. We basically follow Bridgeland and Maciocia [70], but we obtain significant simplifications by using results by Kawamata [175]. We also include the nonmimimal case. In the case of K3 and Abelian surfaces we stay closer to Orlov’s approach. We also include the computation of the number of Fourier-Mukai partners of a K3 surface. A question which is naturally related to the classification of Fourier-Mukai partners is the characterization of the autoequivalences of the derived category of coherent sheaves. We shall not deal with this problem, just recalling that Bondal and Orlov [49] gave a first contribution to the solution of this question by showing that if X is a smooth projective variety such that ωX is either ample or anti-ample, then all autoequivalences of Db (X) are generated by shifts, automorphisms of X and twisting by line bundles. Moreover, the autoequivalences of Db (X) have been characterized by Orlov when X is an Abelian variety [243]. In Section 7.5 we shall approach the second topic of this chapter, i.e., the relationship between derived categories and birational geometry, in particular, Bridgeland’s theorem about bounded derived categories of different crepant resolutions of singularities. This will allow us to show that birational Calabi-Yau threefolds are Fourier-Mukai partners. The machinery we introduce will include perverse sheaves and flops. Section 7.6 is about the McKay correspondence and basically draws from Bridgeland-King-Reid [68]. Let X be a smooth projective variety, acted on by a finite group G such that the canonical bundle of X/G is locally trivial as a G-equivariant sheaf. Then there is a crepant resolution of X/G whose bounded derived category is equivalent to the G-linearized derived category of X, provided that some dimensional conditions are satisfied (cf. Theorem 7.76).

7.1

Preliminaries

Before approaching the problem of classifying all Fourier-Mukai partners of smooth complex projective surfaces, we state some properties of Fourier-Mukai partners that hold true in arbitrary dimension. This complements Section 2.3.1. Specific properties for surfaces or threefolds will be considered later in this chapter. Proposition 7.1. Let X and Y be smooth Fourier-Mukai partners. There is an i ) ' H 0 (Y, ωYi ) for every integer i, so that X and Y have isomorphism H 0 (X, ωX

7.1. Preliminaries

235

the same Kodaira dimension. Proof. Let n be the dimension of both X and Y (cf. Theorem 2.38). Let K• a • K• kernel such that ΦK X→Y is a Fourier-Mukai functor. Since the right adjoint of ΦX→Y is also a quasi-inverse, and equivalences intertwine the Serre functors (Corollary •

∗ K•∨ ⊗πY ωY [n]

k 1.18), one has SYk ' ΦK X→Y ◦ SX ◦ ΦX→Y •∨ ∗ • • K ⊗π ω [n] k Y Y ΦK ' ΦW X→Y ◦ SX ◦ ΦX→Y Y→Y ; here

δ

k ωY [n]

Y∗ . We have SYk ' ΦY→ Y

and

•

M k W • = ΦX×X→ Y ×Y (δX∗ ωX [n]) L

with M• = K•  K•∨ ⊗πY∗ ωY [n] by Proposition 1.3, and δX and δY are the diagonal immersions X ,→ X × X and Y ,→ Y × Y , respectively. The uniqueness of the M• kernel (Theorem 2.25) yields δY ∗ ωYk ' W • . Finally, ΦX×X→ Y ×Y is an equivalence by Corollary 2.60, so that j i HomD(X×X) (δX∗ ωX , δX∗ ωX ) ' HomD(Y ×Y ) (δY ∗ ωYj , δY ∗ ωYi ) , i for any pair of integers i and j. Taking j = 0 one finds that H 0 (X, ωX ) ' 0 i  H (Y, ωY ) as claimed.

For completeness’ sake we recall some definitions and results about singularities and their resolutions. A more detailed exposition may be found in [187, 257]. Definition 7.2. A quasi-projective normal variety X has only canonical singularities if it satisfies the following conditions: 1. the canonical divisor KX is a Q-Cartier divisor (i.e., for some integer r > 0 the Weil divisor rKX is a Cartier divisor). 2. If p : Z → X is a resolution of singularities of X and {Ei } are all the excepP tional prime divisors of p, then rKZ = p∗ (rKX ) + i ai Ei , with ai ≥ 0. If ai > 0 for all the prime divisors Ei , then we say that X has only terminal singularities. 4 The minimum of the integers r such that rKX is a Cartier divisor is the index of X. Definition 7.3. A morphism f : X → Y of Gorenstein varieties is crepant if f ∗ ωY ' 4 ωY . A weaker notion, which is useful in dealing with Q-divisors, is the following:

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Definition 7.4. Let X and Y be quasi-projective normal varieties whose canonical divisors are Q-Cartier. A proper birational map α : X 99K Y is crepant if there are birational morphisms p¯X : Z¯ → X and p¯Y : Z¯ → Y such that p¯Y = α ◦ p¯X and 4 p¯∗X KX is Q-linearly equivalent to p¯∗Y KY . In particular, if X and Y are K-equivalent smooth projective varieties (cf. Definition 2.47), then there exists a crepant birational map α : X 99K Y . We start by recalling a preliminary result. Lemma 7.5. [175, Lemma 4.2] Let X, Y be quasi-projective normal varieties with only terminal singularities, and α : X 99K Y a crepant birational map. Then α is an isomorphism in codimension 1, i.e., there exist closed subvarieties X 0 ,→ X and Y 0 ,→ Y of codimension at least 2, such that α induces an isomorphism X − X 0 ' Y − Y 0. Since the total transform of any fundamental point of a birational map has dimension at least 1 [141, V.5.2], we obtain the following result. Proposition 7.6. Let X and Y be smooth projective surfaces. Any crepant birational map α : X 99K Y is an isomorphism. Then, if two smooth projective surfaces X and Y are K-equivalent, they are isomorphic. Crepant morphisms can be characterized in terms of the derived category. In order to describe such characterization we introduce some notation. If f : X → Y is a morphism of algebraic varieties and y is a (closed) point of Y , we denote by Df −1 (y) (X) the full triangulated subcategory of Db (X) of complexes topologically supported on the fiber f −1 (y) (cf. Definition A.90). We note that Df −1 (y) (X) contains the structure sheaves of all the infinitesimal neighborhoods m · f −1 (y) = X ×Y Spec(OY,y /mm y ) of the fiber, where my denotes the maximal ideal of the local ring OY,y (the stalk of OY at y). The formal fiber of f over y is the fiber product bY,y ) , fb−1 (y) = X ×X FSpec(O bY,y = proj limm OY,y /mm and FSpec stands for its formal spectrum (i.e., where O y the formal completion of Spec OY,y at its closed point, cf. [141, §II.9]). The formal fiber is a formal scheme and can be viewed as the projective limit of the infinitesimal neighborhoods m · f −1 (y) of the fiber. Lemma 7.7. Assume that Rf∗ OX ' OY . A line bundle L on X is the pullback of some line bundle N on Y if and only if its restriction to the formal fiber fb−1 (y) is trivial for every (closed) point y ∈ Y . Moreover in this case one has N ' f∗ L.

7.1. Preliminaries

237

Proof. If L ' f ∗ N , its restriction to each formal fiber is trivial, because the stalk of N at y is isomorphic to OY,y . For the converse, if the restriction of L to each formal fiber is trivial, by the theorem on formal functions (cf. [141, Thm. III.11.1]), i f L and Ri\ f∗ OXy of the stalks at any point y of the higher the completions R\ ∗ y bY,y -modules. Since Rf∗ OX ' OY , direct images of L and OX are isomorphic as O i one has R f∗ L = 0 for i > 0 and N = f∗ L is a line bundle. One then has an exact sequence η → L → Q → 0, (7.1) 0 → f ∗N − where η : f ∗ f∗ L → L is the natural map, which is injective because f ∗ N is a line bundle. By the projection formula one has Ri f∗ (f ∗ N ) ' N ⊗ Ri f∗ OX = 0 for i > 0. Moreover, since η is the adjunction map between f ∗ and f∗ , the morphism f∗ η has a left inverse, and then it is surjective, which implies that f∗ Q = 0. Applying again the theorem on formal functions we also see that f∗ (Q ⊗ L−1 ) = 0 and thus, H 0 (X, f∗ (Q ⊗ L−1 ) = 0. Let us prove that this implies Q = 0, thus finishing the proof. First, from the exact sequence (7.1) we get an injective morphism α : f ∗ N ⊗L−1 ,→ OX . Secondly, H 0 (X, f∗ (Q⊗L−1 )) = 0 gives H 0 (X, f ∗ N ⊗ L−1 ) ' H 0 (X, OX ) = k, and then f ∗ N ⊗ L−1 has a nowhere zero section. It follows that f ∗ N ⊗ L−1 is a trivial line bundle, and then the immersion α is an isomorphism.  Let f : X → Y be a resolution of singularities of a quasi-projective normal variety Y (that is, X is smooth and f is birational). We assume that Rf∗ OX ' OY (i.e., Y has rational singularities) and that f is proper. Proposition 7.8. If the triangulated category Df −1 (y) (X) has trivial Serre functor for every (closed) point y ∈ Y , then Y is Gorenstein and f : X → Y is crepant. Proof. The restriction of the Serre functor SX of X is a Serre functor for Df −1 (y) (X). Then, twisting by ωX is the identity on Df −1 (y) (X). It follows that the restriction of ωX to each infinitesimal neighborhood m · f −1 (y) of the fiber is trivial, and then its restriction to the formal fiber fb−1 (y) is trivial. By Lemma 7.7, f∗ ωX is a line bundle, Rf∗ ωX ' f∗ ωX , and ωX ' f ∗ f∗ ωX . We may prove at the same time that Y is Gorenstein and f is crepant, by checking that Rf∗ ωX [m], where m = dim X = dim Y , is a dualizing complex for Y . We denote by ΓX and ΓY the functors of global sections on X and Y respectively. Then ΓX ' ΓY ◦ f∗ . For every object F • in the derived category of Y one has HomD(Y ) (F • , Rf∗ ωX [m]) ' HomD(Y ) (Lf ∗ F • , ωX [m]) ' Homk (RΓX (Lf ∗ F • ), k) ' Homk (RΓY (Rf∗ (Lf ∗ F • )), k) .

238

Chapter 7. Fourier-Mukai partners and birational geometry L

By the projection formula, Rf∗ (Lf ∗ F • ) ' F • ⊗ Rf∗ OX ' F • , so that HomD(Y ) (F • , Rf∗ ωX [m]) ' Homk (RΓY (Rf∗ (Lf ∗ F • )), k) . This proves that Rf∗ ωX [m] is a dualizing complex for Y as claimed.

7.2



Integral functors for quotient varieties

˜ carries a free action a finite group G, the derived category When a variety X ˜ of the quotient variety X/G is equivalent to the equivariant derived category of the original variety (cf. Example 1.38). Moreover, in some cases, Fourier-Mukai ˜ functors between the derived categories of such two quotients X/G and Y˜ /G lift ˜ and to equivariant Fourier-Mukai functors between the derived categories of X ˜ Y . In this section we discuss these issues, following [69] with some modifications (cf. Remark 7.15), which will be useful in Section 7.4.6 when we characterize the Fourier-Mukai partners of the Enriques surfaces. Let X be a smooth projective variety whose canonical line bundle ωX has n ' OX and n is the first exponent such that this property finite order n, that is ωX n−1 has a natural structure of algebra over holds. Then AX = OX ⊕ ωX ⊕ · · · ⊕ ωX OX and defines a finite ´etale covering of degree n ˜ = Spec AX → X , ρX : X such that AX ' ρX∗ OX˜ . Moreover ωX˜ ' ρ∗X ωX is trivial. There is a free action ˜ defined by letting the generator n of G act as of the cyclic group G = Zn on X, ˜ is naturally isomorphic to X. the twist by ωX on AX . The quotient variety X/G We shall call ρX the canonical cover of X. By Example 1.38, one has equivalences of categories ˜ , Lρ∗X : Db (X) ' DG,b (X)

G,b ˜ RρG (X) ' Db (X) . X∗ : D

This characterizes the image of the derived inverse image functor. We can also char˜ → Db (X). acterize the essential image of the direct image functor ρX∗ : Db (X) To do so, let us denote by Spcl(X) the category whose objects are pairs (F, ϕ) where F is a quasi-coherent sheaf on X and ϕ : ωX ⊗ F ' F is an isomorphism; a morphism (F, ϕ) → (F 0 , ϕ0 ) is a morphism f : F → F 0 of OX -modules such that f ◦ ϕ = ϕ0 ◦ (1 ⊗ f ). Since F is a special sheaf for any object (F, ϕ) ∈ Spcl(X), we shall denominate Spcl(X) the category of special sheaves on X. One easily checks that Spcl(X) is an Abelian category. The forgetful functor (F, ϕ) 7→ F induces a functor Db (Spcl(X)) → Db (Qco(X)), which induces a functor Dcb (Spcl(X)) → Db (X) between the corresponding subcategories of complexes with coherent cohomology sheaves. An object of Db (X) is in the image of the above functor if it is

7.2. Integral functors for quotient varieties

239

special, or equivalently, if its cohomology sheaves are all special (cf. Proposition 2.55). ˜ F = ρX∗ E is a quasi-coherent OX module If E is a quasi-coherent sheaf on X, n−1 . Such endowed with a structure of module over ρX∗ OX˜ ' OX ⊕ ωX ⊕ · · · ⊕ ωX a module structure is equivalent to the existence of a morphism ϕ : ωX ⊗ F → F, n−1

which on the other hand is an isomorphism because (1 ⊗ ϕ) ◦ · · · ◦ (1 ⊗ · · · ⊗ 1 ⊗ n ⊗ F → ωX ⊗ F is its inverse. Then (F = ρX∗ E, ϕ) is an object of ϕ) : F ' ωX ˜ and an Spcl(X) and any such object defines in a quasi-coherent sheaf E on X isomorphism F ' ρX∗ E. Moreover, E is coherent if and only if F is coherent. The following proposition is then straightforwardly checked. Proposition 7.9. The direct image functor induces a functor ˜ → Spcl(X) , ρX∗ : Qco(X) which is an equivalence of Abelian categories and induces an equivalence of derived categories ˜ ' Db (Spcl(X)) . ρX∗ : Db (X) c Let Y be another smooth projective variety with canonical line bundle of order n, ρY : Y˜ → Y its canonical cover, and Φ : Db (X) → Db (Y ) an integral functor. ˜ → Db (Y˜ ) is a lift of Φ if the ˜ : Db (X) Definition 7.10. An integral functor Φ diagram of functors ˜ Db (X)

˜ Φ

ρY ∗

ρX∗

 Db (X) ˜ ◦ ρ∗ ' ρ∗ ◦ Φ. commutes, that is, Φ X Y

/ Db (Y˜ )

Φ

 / Db (Y ) , 4

˜ : Db (X) ˜ → Lemma 7.11. Let Φ : Db (X) → Db (X) be an integral functor and Φ b ˜ D (X) is a lift of Φ. ˜ is a ˜ ' g∗ for some element g ∈ G. In particular, Φ 1. If Φ = Id, then Φ Fourier-Mukai transform. ˜ = Id, then Φ is a Fourier-Mukai transform and is isomorphic to the 2. If Φ m twisting by ωX for some m ≤ n − 1. ˜ we have that Oρ (˜x) ' ρX∗ (Φ(O ˜ x˜ )) for every point Proof. 1. Since ρX∗ ' ρX∗ ◦ Φ, X ˜ Then Φ(O ˜ x˜ ) ' Og(˜x) for a point g(˜ x ˜ ∈ X. x) ∈ X which is in the fiber ρ−1 x)). X (ρX (˜

240

Chapter 7. Fourier-Mukai partners and birational geometry

˜ →X ˜ and a line bundle L on X such By Corollary 1.12 there is a morphism g : X • • • b ˜ Moreover since f = f ◦ g, the ˜ ) ' g∗ (E ⊗ L) for every E ∈ D (X). that Φ(E moprhism g is the multiplication by an element of G that we still denote by g. Let us prove that L ' OX˜ , which finishes the proof of the first part. One ∗ n ˜ has ρX∗ OX˜ ' ρX∗ (Φ(O ˜ ) ' ρX∗ g∗ L ' ρX∗ L and then ρX (ρX∗ L) ' OX ˜ since X ∗ ∗ n ρX ωX ' OX˜ . The natural morphism L → ρX (ρX∗ L) ' OX˜ is nonzero and induces ˜ ∗ ) ' ρX∗ g∗ O ˜ ' ρX∗ O ˜ , a nonzero morphism L → OX˜ . Since ρX∗ L∗ ' ρX∗ (Φ(L X X we prove in the same way the existence of a nonzero morphism L∗ → OX˜ . Thus L is trivial. 2. If x ∈ X and x ˜ ∈ ρ−1 X (x), then ρX∗ ' Φ ◦ ρX∗ gives Φ(Ox ) ' Ox . By Corollary 1.12, there is a line bundle L on X such that Φ(E • ) ' E • ⊗ L for every bounded complex E • in Db (X), which proves that Φ is an equivalence of categories. n−1 , Moreover, ρX∗ OX˜ Φ(ρX∗ OX˜ ) ' ρX∗ OX˜ ⊗L. Since ρX∗ OX˜ ' OX ⊕ωX ⊕· · ·⊕ωX m  one has L ' ωX for some m ≤ n − 1. • f• b b b ˜ b ˜ K ˜ Lemma 7.12. Let Φ = ΦK ˜ Y ˜ : D (X) → D (Y ) X→Y : D (X) → D (Y ) and Φ = ΦX→ be integral functors. Assume that

f• . (ρX × 1)∗ K• ' (1 × ρY )∗ K ˜ is a lift of Φ. Then Φ Proof. Let us consider the diagram ˜ ˜ hhhuuX × YIIVIVVVVVV h h h h VVVV πY˜ πX ˜ II u hh VVVV II hhhh uu h h VVVV u h u 1×ρY ρX ×1 II hhh VVVV u h h z $ u h h pX˜ VVV+ pX h ˜ h s h o / Y˜ ˜ ˜ ˜ X ×Y X × YJ X ?? JJ ρ ×1 t t ?? ρX 1×ρY t ρY JJX ?? JJ tt ?? JJ tt t % yt   πX πY /Y X ×Y Xo ˜ one has For any object E • in Db (X) L

∗ Φ(ρX∗ E • ) ' πY ∗ (πX (ρX∗ E • ) ⊗ K• ) L

L

' πY ∗ ((ρX × 1)∗ p∗X˜ E • ) ⊗ K• ) ' πY ∗ (ρX × 1)∗ (p∗X˜ E • ⊗ (ρX × 1)∗ K• ) L

f• ) ' πY ∗ (ρX × 1)∗ (p∗X˜ E • ⊗ (1 × ρY )∗ K L

f• ) ' πY ∗ (ρX × 1)∗ (1 × ρY )∗ ((1 × ρY )∗ p∗X˜ E • ⊗ K L

∗ • f• ˜ • ' ρY˜ ∗ RπY˜ ∗ (πX ˜ E ⊗ K ) ' ρY˜ ∗ Φ(E ) ,

7.2. Integral functors for quotient varieties

241

where we have used several times the projection formula in derived category (Proposition A.83).  Let Y be a Fourier-Mukai partner of X and Φ : Db (X) → Db (Y ) an equivalence of triangulated categories. By Theorem 2.38, ωY has also order n, and we can consider its canonical cover ρY : Y˜ → Y . ˜ of Φ. Moreover Φ ˜ is an equivalence of categories Proposition 7.13. There is a lift Φ ˜ ' Db (Y˜ ) , ˜ : Db (X) Φ and is equivariant under the natural action of G on the derived categories in the ˜ = following sense: there is an automorphism τ of the group G such that g∗ ◦ Φ ˜ Φ ◦ τ (g)∗ for every g ∈ G. •

•∨

K Proof. Since Φ = ΦK X→Y is an equivalence, the integral functor ΦY→X is an equiva•∨ lence as well by Theorem 2.38. Then, by applying Theorem 2.38 to ΦK Y→X , we obtain ∗ ωX . Let us write now M• = (ρX × 1)∗ K• ∈ an isomorphism K• ⊗ πY∗ ωY ' K• ⊗ πX b ˜ ∗ ' pX˜ ωX˜ ⊗ p∗Y ωY ' p∗Y ωY where pX˜ and pY are the D (X × Y ). Since ωX×Y ˜ ˜ × Y onto its factors, one has projections of X

M• ⊗ ωX×Y ' ((ρX × 1)∗ K• ) ⊗ p∗Y ωY ' (ρX × 1)∗ (K• ⊗ p∗Y ωY ) ˜ ' (ρX × 1)∗ (K• ⊗ p∗X ωX ) ' M• . f• ∈ Db (X ˜ × Y˜ ) by Proposition 7.9 f• for some object K Thus, M• ' (1 × ρY )∗ K ˜ ˜ applied to the canonical covering X × Y → X × Y . By Lemma 7.12, the integral • b ˜ b ˜ ˜ = ΦKf functor Φ ˜ ˜ : D (X) → D (Y ) is a lift of Φ. X→Y

If Ψ is a quasi-inverse of Φ, proceeding as above we prove that there is a lift e is a lift of the identity and then Ψ0 ◦ Φ e ' g∗ for some element Ψ0 of Ψ. Thus Ψ0 ◦ Φ e = g −1 ◦ Ψ0 is still a lift of Ψ and g ∈ G by Lemma 7.11. The integral functor Ψ ∗ e ◦Φ e ' Id. Moreover Φ e ◦Ψ e also lifts the identity, so that Φ e ◦Ψ e ' h∗ verifies that Ψ n n e e e Ψ, e for some h ∈ G again by Lemma 7.11. Since h = 1, one has Id ' (Φ◦ Ψ) ' Φ◦ e e e where the latter isomorphism follows from Ψ ◦ Φ ' Id. Thus Ψ is a quasi-inverse e so that Φ e is an equivalence. of Φ, e is a e is equivariant. If g is an element of G, g∗ ◦ Φ We now prove that Φ e e lift of Φ and then Ψ ◦ g∗ ◦ Φ ' τ (g)∗ for some τ (g) ∈ G by Lemma 7.11. Thus e ' Φ e ◦ τ (g)∗ and this defines a morphism of groups τ : G → G. One has g∗ ◦ Φ e ' Φ e ◦ h∗ defines a morphism of groups $ : G → G analogously that $(h)∗ ◦ Φ which is inverse to τ . Hence τ is an isomorphism.



e is equivariant if and only if there is an automorRemark 7.14. Since g ∗ ' g∗−1 , Φ ˜ =Φ ˜ ◦ τ (g)∗ for every g ∈ G. 4 phism τ of the group G such that g ∗ ◦ Φ

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Chapter 7. Fourier-Mukai partners and birational geometry

Remark 7.15. The definition of a lift given in [69] is stronger than ours because e ◦ ρ∗ ' ρ∗ ◦ Φ is also required. Thus our Lemma 7.11 there the compatibility Φ X Y is stronger than the corresponding statement [69, Lemma 4.3]. It seems that [69, Lemma 4.4] may fail to hold true unless the additional condition (ρY × 1)∗ K• ' f• is imposed. But if one does so, the proof of the existence of the lift (1 × ρX )∗ K given in [69] is not complete. 4

7.3

Fourier-Mukai partners of algebraic curves

It is quite simple to show that the only Fourier-Mukai partner of a smooth projective curve is the curve itself. We give an easy proof which uses general results and the structure of moduli spaces of stable sheaves on a elliptic curve, which we described in Section 3.5.1. Theorem 7.16. A smooth projective curve has no Fourier-Mukai partners but itself. Proof. Let Y be a Fourier-Mukai partner of a smooth projective curve X of genus g. The canonical line bundle ωX of X is either ample (if g > 1), anti-ample (if g = 0) or trivial (if g = 1). In the first two cases, X ' Y by Theorem 2.51. Let X be elliptic and take a Fourier-Mukai functor Φ : Db (Y ) → Db (X). If y ∈ Y P is a point, by Proposition 2.35 one has i dim Hom1D(X) (Φi (Oy ), Φi (Oy )) ≤ 1, and then there is a unique value of i for which Φi (Oy ) 6= 0, that is, Oy is WITi . The integer i is independent of y because of Proposition 6.5. Then Y is a fine moduli space of simple sheaves over X by Corollary 2.64. If the sheaves Φi (Oy ) have torsion, they are skyscraper sheaves of length 1, so that Y ' X in this case. If the sheaves Φi (Oy ) are torsion-free, they are stable by Corollary 3.33 so that Y ' X by Corollary 3.34. 

7.4

Fourier-Mukai partners of algebraic surfaces

In this section we prove the result expressed by the following theorem. We shall assume that the ground field k is the field C of the complex numbers and all surfaces are smooth and projective. Theorem 7.17. A smooth projective surface has a finite number of Fourier-Mukai partners. Note that by Lemma 2.37 a Fourier-Mukai partner of a smooth projective surface is a smooth projective surface as well. Since Fourier-Mukai partners have the same Kodaira dimension (Proposition 7.1), in order to count the number of partners of projective surfaces it seems natural to adopt a case by case approach

7.4. Fourier-Mukai partners of algebraic surfaces

243

essentially based on the Enriques-Kodaira classification. The treatment we offer follows Bridgeland and Maciocia [70] with some simplifications due to Kawamata [175]. The ideas taken from [175] allow us to consider also surfaces with (−1)curves, which are not treated in [70]. Let X be a smooth projective surface. We denote by ai the dimension of the rational Chow group Ai (X) ⊗ Q. The number a1 is the Picard number ρ(X) of X. If we assume that X is connected, then a0 = a2 = 1. The topological P Euler characteristic of X is given by χ(X, Q) = i≥0 (−1)i bi (X) where bi (X) = dim H i (X, Q) are the Betti numbers. For X connected one has b0 (X) = 1 and also b4 (X) = 1, b3 (X) = b1 (X) by Poincar´e duality. Moreover, the topological Euler characteristic of X equals the Euler class of the tangent bundle, that is, χ(X, Q) = c2 (X) .

(7.2)

Lemma 7.18. Two smooth projective surfaces X and Y that are Fourier-Mukai partners have the same Picard number, the same Betti numbers, and then, the same topological Euler characteristic. Proof. By Corollary 2.40, the rational Chow groups of X and Y are isomorphic, A• (X) ⊗ Q ' A• (Y ) ⊗ Q, and there are isomorphisms of Q-vector spaces H • (X, Q) ' H • (Y, Q) and H 2• (X, Q) ' H 2• (Y, Q). We may assume that X is connected. Then Y is also connected by Proposition 2.53. Thus, ρ(X) = dim A• (X) ⊗ Q − 2 = dim A• (Y ) ⊗ Q = ρ(Y ) − 2 and b2 (X) = dim H 2• (X, Q) − 2 = dim H 2• (Y, Q) − 2 = b2 (Y ) . Since dim H • (X, Q) = 2 + b2 (X) − 2b1 (X), we deduce that b1 (X) = b1 (Y ) as well.  A useful criterion to establish whether two smooth projective surfaces X and Y that are Fourier-Mukai partners are isomorphic can be deduced from the results • b b in Section 2.3.1. Let ΦK Y→X : D (Y ) → D (X) be a Fourier-Mukai functor. Proposition 2.48 and Theorem 2.49 suggest the importance of the irreducible component ZY (K• ) of the support of the kernel K• introduced in Lemma 2.46. Recall that pY = πX |ZY (K• ) : ZY (K• ) → Y is dominant, and if ZeY (K• ) → ZY (K• ) is the normalization of ZY (K• ), then the composition morphism p˜Y : ZeY (K• ) → Y is dominant as well. Moreover, one has r p˜∗X ωX ' p˜∗Y ωYr

for some r > 0 ,

where pX = πX |ZY (K• ) : ZY (K• ) → X and p˜X : ZeY (K• ) → Y denote the induced morphisms.

244

Chapter 7. Fourier-Mukai partners and birational geometry

Proposition 7.19. One has: 1. If dim ZY (K• ) = 2, then Y and X are isomorphic. 2. If no multiple of KX is zero (that is, if κ(X) 6= 0), then any special sheaf E has rank zero and dim ZY (K• ) ≤ 3. Proof. 1. By Proposition 2.48, Y and X are K-equivalent, hence they are isomorphic by Proposition 7.6. 2. Since E ' E ⊗ ωX , the special sheaf E has r = 0, hence E is supported on curves or points. Moreover, for every point y ∈ Y , p−1 Y (y) is nonempty and ∗ • K• contained in the support of Ljy K ' ΦY→X (Oy ), which is special by Proposition 2.56. Then all its cohomology sheaves are also special, so that dim p−1 Y (y) ≤ 1. It  follows that dim ZY (K• ) ≤ 3, as claimed. The dimension of the variety ZY (K• ) takes any possible value, as the following examples show: • if X = Y and K• = O∆ (so that the corresponding integral functor is the identity), then ZY (O∆ ) = ∆ which has dimension 2. We shall see nontrivial examples in Section 7.4.1, among other places. • If p : X → B is a relatively minimal elliptic surface, Y = JX/B → B is the P compactified relative Jacobian and Φ = ΦY→ X is the relative Fourier-Mukai e is supported on the fiber product functor defined in Section 6.3, then P Y ×B X, and dim ZY (P) = 3. e

• If the kernel of the Fourier-Mukai functor is a locally free sheaf P on X × Y , then dim ZY (P) = 4. Examples of this situation, which can only occur when the Kodaira dimension of X (and then of Y ) is zero, are the Abelian FourierMukai transform (Chapter 3) or the various Fourier-Mukai transforms for K3 surfaces described in Chapter 4. We recall that a smooth algebraic surface X is called minimal if it contains no exceptional curves of the first kind, namely, smooth rational curves with self-intersection equal to −1. Exceptional curves of the first kind are also called (−1)-curves. We shall prove Theorem 7.17 by considering several particular cases. We start with the minimal surfaces of Kodaira dimension 2. The case of Kodaira dimension κ = −∞ is covered by Section 7.4.2 for the nonelliptic case, and by Section 7.4.3 for the elliptic case; in both sections some nonminimal surfaces are also considered. In the case κ = 0 we treat separately the K3 surfaces (Section 7.4.4), the Abelian surfaces (Section 7.4.5), and the Enriques surfaces (Section 7.4.6); the other cases of minimal surfaces with κ = 0, e.g., hyperelliptic surfaces, are again

7.4. Fourier-Mukai partners of algebraic surfaces

245

covered by Section 7.4.3, which includes also the case of Kodaira dimension 1. The only remaining surfaces are the nonminimal ones that are not relatively minimal elliptic; this case is eventually treated in Section 7.4.7. We shall freely use [141, §V.2] and [22, Ch. VI], where the reader can find all the results about the classification of surfaces that we use here.

7.4.1

Surfaces of Kodaira dimension 2

Proposition 7.20. Let X be a minimal surface of Kodaira dimension 2. If Y is a Fourier-Mukai partner of X, then X ' Y . Proof. By Theorem 2.49, Y and X are K-equivalent, so that they are isomorphic by Proposition 7.6.  In this case, dim ZY (K• ) = 2. A slightly different proof of Proposition 7.20 may be obtained by noting that a projective surface of Kodaira dimension 2 is minimal if and only if its canonical divisor is nef [111, p. 282]. Indeed, by Theorem 2.49, the canonical divisor of Y is nef as well, so that Y is also minimal, and the birational morphism Y → X is actually an isomorphism.

7.4.2

Surfaces of Kodaira dimension −∞ that are not elliptic

By the classification theory of surfaces, a minimal (algebraic) surface X with κ(X) = −∞ which is not elliptic is either the projective plane P2 or a ruled surface π : X → C over a smooth projective curve C of genus g. The geometry of ruled surfaces is studied in [141, §V.2], where the reader is referred for additional information. Recall that a ruled surface π : X → C is isomorphic to a projective bundle P(E ∗ ) = Proj S • (E) → C, where E is a rank 2 locally free sheaf on C. The bundle E can be normalized so that H 0 (C, E) 6= 0 but H 0 (C, E ⊗ L) = 0 for any line bundle L of negative degree [141, V.2.8]. If E is normalized in this way, the integer e = − deg(E) is an invariant of the surface. There is a section of π, whose image we denote by C0 , such that OX (C0 ) is the relative line bundle OX/C (1). Notice that a ruled surface may fail to be minimal. This is important because in Section 7.4.7 we shall need to resort to the computation of the Fourier-Mukai partners of a (nonminimal) ruled surface we do in this section. A ruled surface π : X → C has Picard number 2, and its N´eron-Severi group is generated by C0 and the class f of the fiber; they satisfy the relations C02 = −e ,

f2 = 0 ,

C0 · f = 1 .

246

Chapter 7. Fourier-Mukai partners and birational geometry

The canonical divisor of X and the topological Euler characteristic are KX = −2C0 + (2g − 2 − e)f ,

c2 (X) = 4(1 − g) .

(7.3)

If F is a special sheaf on X of rank r, one has rKX = 0, so that r = 0 since no multiple of KX is zero , as follows from Equation (7.3). In other words, F is a torsion sheaf and we can apply Equation (1.7) to obtain χ(F, F) = −c1 (F)2 . Moreover, Ext2X (F, F)∗ ' HomX (F, F) by Serre duality because F is special, and one has dim Ext1X (F, F) = 2 dim HomX (F, F) + c1 (F)2 ≥ 2 + c1 (F)2 .

(7.4)

Lemma 7.21. Let π : X → C be a ruled surface. If D is an irreducible curve in X such that D · KX = 0, then D2 ≥ 0 unless g = 0 and e = 2. Moreover, if g = 0 and e = 2, then D = C0 . Proof. Since D · KX = 0, if D and KX are linearly dependent over Q, one has D2 = 0. Otherwise, (KX , D) is a basis for the vector space A1 (X) ⊗ Q. If D2 < 0, 2 > 0. Thus g = 0, that is, X is rational, the Hodge index theorem implies that KX and then e ≥ 0. If D = aC0 + bf , one has 0 ≤ D · f = a because D is irreducible. By the same reason, 0 ≤ D ·C0 = b−ae unless D = C0 . Since 0 > D2 = a(2b−ae), one cannot have 0 ≤ b − ae, and then D = C0 , so that 0 = C0 · KX = e − 2. This proves the first part. The second claim follows easily from [141, V.2.18].  Proposition 7.22. Let X be minimal surface with κ(X) = −∞ that admits no elliptic fibration. The unique Fourier-Mukai partner of X is X itself. •

b b 2 Proof. Let Φ = ΦK Y→X : D (Y ) → D (X) be a Fourier-Mukai functor. If X ' P , then ωX is anti-ample, so that X ' Y by Bondal and Orlov’s reconstruction theorem (Theorem 2.51). Assume then that X is a ruled surface over a curve C of genus g with invariant e. By Proposition 7.19, dim ZY (K• ) ≤ 3 and if dim ZY (K• ) = 2 then X ' Y , so that we can assume that dim ZY (K• ) = 3. This excludes the situation g = 0 and e = 2: indeed, in this case C0 is the unique irreducible curve D such that D · KX = 0 by Lemma 7.21, and then the support of Φ(Oy ) must consist of a point for generic y ∈ Y ; this implies dim ZY (K• ) = 2.

Our first aim is to prove that Oy is WITi for some integer i for every point y ∈ Y . Indeed Lemma 7.21 implies one has c1 (Φi (Oy ))2 ≥ 0 for every i, and then if more than one of the sheaves Φi (Oy ) is nonzero, one has X i

dim Hom1D(X) (Φi (Oy ), Φi (Oy )) ≥ 4

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247

by Equation (7.4); this contradicts Proposition 2.35. In principle, the integer i might depend on the point y, but it does not due to Proposition 6.5. Thus, Φ(Oy ) ' Φi (Oy )[−i], so that c1 (Φi (Oy ))2 = c1 (Φ(Oy ))2 = −χ(Φ(Oy ), Φ(Oy )) = −χ(Oy , Oy ) = 0 by the Parseval formula (Proposition 1.34). It follows that all fibers of πY : ZY (K• ) → Y are curves Dy = Supp Φ(Oy ) such that Dy2 = 0 and Dy · KX = 0. By adjunction all curves Dy have arithmetic genus 1, so that the smooth ones are elliptic. Moreover, Dy1 · Dy2 = 0 when y1 6= y2 due to Equation (1.7). Now, we fix a very ample divisor H on Y with H · KY 6= 0; we may assume that H is smooth. If Ψ : Db (X) → Db (Y ) is a quasi-inverse of Φ, for any point x ∈ X the support of Ψ(Ox ) meets H at a finite number of points, because Ψ(Ox ) is special. We are going to prove that this induces a morphism X → Symd (H) for some integer d which is an elliptic fibration onto its image. Let us consider the integral functor ΨH = Lj ∗ ◦ Ψ : Db (X) → Db (H) where j : H ,→ Y is the immersion. If Q• is the kernel of Ψ, then ΨH has kernel Q• |X×H = L(1×j)∗ Q• . We now prove that Q• |X×H [−1] reduces to a single sheaf which is flat over X. This is equivalent to proving that all the sheaves Ox are WIT0 with respect ∨ ' OH (H)[−1], the Parseval isomorphism (Proposition 1.34) to ΨH [−1]. Since OH gives L

HomiDb (X) (L, ΨH (Ox )[−1]) ' HomiDb (X) (L, OH ⊗ Ψ(Ox )[−1]) ∨ ' HomiDb (Y ) (L ⊗ OH , Ψ(Ox )[−1])

' HomiDb (Y ) (L ⊗ OH (H), Ψ(Ox )) ' HomiDb (X) (Φ(L ⊗ OH (H)), Ox ) for any line bundle L on Y . If we take L to be very ample, Φ(Ln ⊗ OH (H)) = RπX∗ (πY∗ (Ln ⊗ OH (H)) ⊗ P) reduces to a locally free sheaf in degree 0 for n big enough. By Lemma 2.8, ΨH (Ox )[−1] reduces to a single sheaf in degree 0, that is, Ox is WIT0 with respect to ΨH [−1] for every x. Since the support of any Ψ(Ox ) meets H at a finite number of points, we have proved that the sheaf Q• |X×H [−1] defines a flat family of skyscraper sheaves on H, so that it defines a morphism f : X → Sd (H) for some integer d > 0, where Sd (H) is the (coarse) moduli space of skyscraper sheaves on H of length d (cf. Theorem C.6). Given a point ξ ∈ Sd (H) defined by P mi , the fiber the equivalence class of the sheaf Oym11 ⊕ · · · ⊕ Oymss , where d = −1 f (ξ) is the intersection of the curves Dy1 , . . . , Dys . Since Dyi · Dyj = 0 for i 6= j,

248

Chapter 7. Fourier-Mukai partners and birational geometry

we see that the schematic image of f is isomorphic to the curve H embedded into Sd (H) by y 7→ Oyd . Therefore, the generic fiber of f : X → H is an elliptic curve, contradicting our hypothesis. 

7.4.3

Relatively minimal elliptic surfaces

We prove now that the partners of relatively minimal elliptic surfaces (cf. Definition 6.9) are precisely the relative compactified Jacobians JX/B (d) = JX/B (1, d) introduced in Section 6.3. Our treatment in this section includes all surfaces with κ = 1, the only case with κ = −∞ that is still to be covered (i.e., the elliptic case), and some cases with κ = 0. Proposition 7.23. Let p : X → B be a relatively minimal elliptic surface (Definition 6.8). Any Fourier-Mukai partner Y of X is isomorphic to the relative compactified Jacobian JX/B (d) for some integer d coprime to λX/B . Proof. Recall that by Equation (6.28), KX is a rational multiple of the elliptic fiber f. Then, the support of any nonzero special sheaf is contained in a finite number of fibers of p. •

Now, let Y be a Fourier-Mukai partner of X and Φ = ΦK Y→X a Fourier- Mukai functor. Take a (closed) point x in a smooth fiber of p and a point y ∈ Y such that the support of Φ(Oy ) contains x; they exist by Propositions A.91 and 2.52. Since HomD(X) (Φ(Oy ), Φ(Oy )) = C, the support of Φ(Oy ) is connected; the object Φ(Oy ) being special, the support is either the point x or the fiber p−1 (p(x)). In the first case, Φ(Oy ) ' Ox [i] for some integer i, and Proposition 6.6 implies that X and Y are K-equivalent; by Proposition 7.6, X and Y are isomorphic. Thus we assume that the support of Φ(Oy ) is the fiber p−1 (p(x)), so that the Chern character of Φ(Oy ) is (0, rf, d) for some integers r > 0 and d. By Equation (1.4) and Parseval formula (Proposition 1.34), one has 1 = χ(Oy , OY ) = χ(Φ(Oy ), Φ(OY )) = rc1 · f − c0 d where c0 and c1 are the zeroth and first Chern character of Φ(OY ). Since c1 · f is an integer multiple of λX/B by definition of λX/B , one has that rλX/B is coprime to d. Then, the relative compactified Jacobian JX/B (r, d) (Definition 6.27) exists and is a smooth surface equipped with a relatively minimal elliptic fibration q : JX/B (r, d) ' M (X, r, d) → B (Proposition 6.28 and Corollary 6.30). We are going to see that actually Y ' JX/B (r, d), which concludes the proof by Proposition 6.32. Since the sheaf Φ(Oy ) is supported on a elliptic curve, one has Hom1X (Φi (Oy ), Φ (Oy )) 6= 0 for every i and then Proposition 2.35 implies that actually there is only one nonzero cohomology sheaf. In other words, the sheaf Oy is WITi for some i

7.4. Fourier-Mukai partners of algebraic surfaces

249

i for every y ∈ Y . The integer i is independent of the point y by Proposition 6.5. Furthermore, Φi (Oy ) is simple and then it is stable after restriction to its support since the latter is an elliptic curve (see Corollary 3.33). Thus, Φi (Oy ) defines a closed point of Ye = JX/B (r, d). By Corollary 2.64, Φ induces an isomorphism between Y and a fine moduli space for the sheaves Φ(Oy ). Since JX/B (r, d) is also a fine moduli space, Φ induces an immersion Y ,→ Ye = JX/B (r, d) which has to be an isomorphism because both varieties are projective surfaces, and JX/B (r, d) is irreducible since it is an elliptic surface. 

7.4.4

K3 surfaces

By a result of Orlov, the Fourier-Mukai partners of a K3 surface are completely characterized in terms of isometries of the transcendental lattice (see Definition 4.9). Theorem 7.24. [242, Thm. 3.3] Let X, Y be two projective K3 surfaces. X and Y are Fourier-Mukai partners if and only if the transcendental lattices T(X) and T(Y ) are Hodge isometric. Proof. Let Ψ = Db (X) → Db (Y ) be an equivalence of categories. By Theorem • 2.15, there exists a kernel K• such that ΦK X→Y ' Ψ. The proofs of the first assertion in Proposition 4.27 and of Corollary 4.28 apply to this situation, showing that the induced map f : T(X) → T(Y ) is a Hodge isometry. Conversely, assume that a Hodge isometry g : T(Y ) → T(X) is given. Since ˜ • (X, Z) contains the hyperbolic sublattice the orthogonal complement of T(X) in H ˜ • (Y, Z) → U , by Proposition B.8 the map g extends to a Hodge isometry g˜ : H • ∗ 4 ˜ g ($)) ˆ , where $ ˆ is the fundamental class in H (Y, Z), and H (X, Z). Let v = (˜ write v = (r, H, t). We may recast v into a standard form where r > 0 and H is ˜ • (X, Z) of the form ample by using the isometries of H v

7→

v · (1, `, 12 `2 )

(s, `, z)

7→

(z, `, s) ,

with ` ∈ Pic(X)

(7.5) (7.6)

which restrict to the identity on T(X). Indeed we may note that v cannot be of the form (0, L, 0), so that, possibly by applying the isometry (7.6) and changing sign to v, we may assume that r > 0. Moreover, by repeatedly applying the isometry (7.5) with ` an ample class, we may assume that H is ample, and may be taken as a polarization in X. Since $ ˆ is primitive and isotropic, so is v. By [227, Thm. 5.4] the moduli space MH (v) of H-stable bundles on X is nonempty. Moreover, there ˜ 1,1 (X, Z) such that v · u = 1 (take u = −˜ g (1)). Then by exists an element u ∈ H Theorem 4.20 there is a universal family E on X × MH (v), while by Proposition

250

Chapter 7. Fourier-Mukai partners and birational geometry

4.21 the integral functor ΦQ X→MH (v) is a Fourier-Mukai transform, and by Theorem 4.25 MH (v) is a K3 surface. Therefore one has a Hodge isometry ˜ • (X, Z) → H ˜ • (MH (v), Z) . f = fQ : H Now, for any K3 surface S, one has H 2 (S, Z) ' $⊥ /Z$, where $ is the fundamental class in H 4 (S, Z). As a consequence, the composition g˜−1 ◦ f −1 establishes a Hodge isometry between H 2 (MH (v), Z) and H 2 (Y, Z). By the weak Torelli theorem (Corollary 4.11), the K3 surfaces MH (v) and Y are isomorphic, so that  Db (X) ' Db (Y ). Remark 7.25. By Proposition B.8, Theorem 7.24 can be alternatively stated as follows: the derived categories Db (X) and Db (Y ) are equivalent if and only if the ˜ • (Y, Z) are Hodge isometric. ˜ • (X, Z) and H 4 full Mukai lattices H The proof of Theorem 7.24 implies the following remarkable result, which shows that, to a certain extent, the geometry of the moduli space of stable sheaves on a K3 surface X is encoded into the derived category Db (X). (Again, in the remainder of this section by “(semi)stable” we shall mean “Gieseker-(semi)stable.”) Theorem 7.26. Let X, Y be two projective K3 surfaces. Y is a Fourier-Mukai partner of X if and only if it is isomorphic to a 2-dimensional fine compact moduli space of stable sheaves on X (and vice versa). The examples in Section 4.3 realize indeed this situation. A natural question is whether the number of Fourier-Mukai partners of a projective K3 surface X is finite or not, and in the former case, to write down a counting formula for it. The first result in this direction was already contained in [227], Proposition 6.2. Proposition 7.27. If the Picard number of a projective K3 surface X is greater than 11, then X has no other Fourier-Mukai partner than itself. Proof. Let Y be a Fourier-Mukai partner, and let g : T(X) → T(Y ) be a Hodge isometry, which exists by Theorem 7.24. Thanks to Proposition B.8, g extends to an isometry g˜ : H 2 (X, Z) → H 2 (Y, Z). Hence X and Y are isomorphic by the weak Torelli theorem (Corollary 4.11).  Since the Picard number of a Kummer surface (see Example 4.5) is at least 17, this implies that a Kummer surface has no Fourier-Mukai partner other than itself. Another simple instance is provided by an elliptic K3 surface X with a section.

7.4. Fourier-Mukai partners of algebraic surfaces

251

Proposition 7.28. A projective elliptic K3 surface X with a section has no other Fourier-Mukai partner than itself. Proof. The fiber of the projection X → P1 and the section generate a hyperbolic lattice sitting in Pic(X). Then the claim follows from Proposition B.8 by reasoning as in Proposition 7.27.  The finiteness of the number of Fourier-Mukai partners of any K3 surface X was first proved by Bridgeland and Maciocia [70]. Here we shall follow [150], where an explicit counting formula is also given. We shall denote by FM(X) the set of nonisomorphic Fourier-Mukai partners of X. Given a K3 surface X, a primitive embedding i : T(X) → Σ (where Σ is the standard K3 lattice) determines a point in the period domain ∆ ⊂ P(Σ ⊗ C). Note indeed that T(X) has a natural Hodge structure induced by the Hodge structure ˜ • (X, Z). The complexification T(X) ⊗ C contains the line H 2,0 (X, C), and the of H image of this line via i provides the aforementioned point in ∆. By the surjectivity of the period map there is a marked K3 surface (Y, φ) corresponding to this point. Now, the trascendental lattice T(S) of a K3 surface S has the property of being the minimal primitive sublattice of H 2 (S, Z) whose complexification cointains H 2,0 (S, C). This implies the existence of a commutative diagram T(Y ) _  H 2 (Y, Z)

g

/ T(X) _ i

φ

 /Σ

where g is a Hodge isometry. In view of Theorem 7.24, Y is a Fourier-Mukai partner of X. Lemma 7.29. Two primitive embeddings i, i0 : T(X) → Σ determine isomorphic K3 surfaces Y , Y 0 if and only if there is an α ∈ O(Σ) and a Hodge isometry h of T(X) such that the diagram T(X) _

h

i0

i

 Σ

/ T(X) _

α

(7.7)

 /Σ

commutes. Proof. Follows from the (weak) Torelli theorem, Corollary 4.11.



252

Chapter 7. Fourier-Mukai partners and birational geometry

Let J be the subgroup of O(T(X)) formed by Hodge isometries. For later use we provide the following characterization of J; for a proof see [150, Prop. B1]. Proposition 7.30. J is isomorphic to Z/2mZ for some m ∈ N. Moreover, the order of the subgroup J × of the invertibles of J divides the rank of T(X). According to the terminology of Section B.3, we say that two primitive embeddings i, i0 : T(X) ,→ Σ are J-equivalent if they fit into a diagram such as Equation (7.7). Corollary 7.31. Let EJ (T(X), Σ) = {i : T(X) ,→ Σ | primitive embedding}/J-equivalence . There is a bijective map µ : EJ (T(X), Σ) → FM(X) . Theorem 7.32. The set FM(X) is finite. Proof. In view of the previous discussion, in particular Corollary 7.31, the result follows from Lemma B.9.  Remark 7.33. This result combined with Theorem 7.26 implies that the number of nonisomorphic 2-dimensional fine compact components of the moduli space of stable sheaves on a K3 surface is finite. 4 One can write a formula computing the number of elements in FM(X), which follows straightforwardly from Theorem B.10. Let g(Pic(X)) = {K1 , . . . , Ks } be the genus of the lattice Pic(X), with K1 ' Pic(X) (for the notion of genus see Section B.1). Theorem 7.34. [150] The cardinality of the set FM(X) is given by ](FM(X)) =

s X

](O(Ki )\O(AKi )/J)

i=1

where AKi is the discriminant group of the lattice Ki (see Section B.2). The counting formula takes a particularly elegant form when ρ(X) = 1. Corollary 7.35. [239, 150] Let X be a projective K3 surface of Picard number 1, with a generator H such that H 2 = 2d. Then ](FM(X)) = 2ω(d)−1 , where ω(1) = 1 and ω(d) is the number of prime factors in d for d ≥ 2.

7.4. Fourier-Mukai partners of algebraic surfaces

253

Proof. One has APic(X) ' Z/2d Z with the quadratic form q(1) = 1/2d. Then by Theorem B.4, we have g(Pic(X)) = {Pic(X)}. Moreover, O(Pic(X)) ' Z/2Z. By Proposition 7.30, the group J is Z/2Z as well. By the counting formula, ](FM(X)) =

1 2

](O(Z/2dZ)) .

It is now easy to show that O(Z/2dZ) ' (Z/2Z)ω(d) , cf. e.g. [271].



In the case of Picard number ρ(X) equal to 2, the counting of the FourierMukai partners of X appears to be related to issues of a number-theoretic nature. One has for example the following result [150]. We denote by d(Pic(X)) the discriminant of the Picard lattice Pic(X). Proposition 7.36. Let X be a projective K3 surface with ρ(X) = 2 and d(Pic(X)) = −p for some prime number p 6= 2. Then ](FM(X)) = 12 (h(p) + 1)) , √ where h(p) is the class number of the extension Q( p). When the Picard number of X is greater than 2, the set FM(X) consists of just one element. Corollary 7.37. If the Picard number of X is at least 3, and the discriminant d(Pic(X)) of Pic(X) is square-free, then X has no other Fourier-Mukai partner than itself. Proof. Since the order of the discriminant group APic(X) equals the absolute value of d(Pic(X)), the group APic(X) is cyclic, so that l(APic(X) ) = 1, where l(G) is the minimal number of generators of a finite group G. By applying [235, Theorem 1.14.2] one sees that ](FM(X)) = ](O(Pic(X))\O(APic(X) )) = 1. 

7.4.5

Abelian surfaces

Fourier-Mukai partners of Abelian surfaces may be characterized exactly as in the case of K3 surfaces. Before analyzing this problems, let us notice that we know ˆ are Fourier-Mukai from Chapter 3 that an Abelian variety X and its dual X partners. For an Abelian surface X, the transcendental lattice T(X) is defined as for K3 surfaces as the orthogonal lattice to Pic(X) in H 2 (X, Z).

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Chapter 7. Fourier-Mukai partners and birational geometry

Theorem 7.38. Let X, Y be two Abelian surfaces. X and Y are Fourier-Mukai partners if and only if the transcendental lattices T(X) and T(Y ) are Hodge isometric. Proof. The proof is the same as for Theorem 7.24 up to the point where we get a Hodge isometry f : H 2 (X, Z) → H 2 (Y, Z). In this case, Theorem 1 of [268] implies ˆ In either case, that Y is isomorphic either to X or to the dual Abelian surface X. we get a Fourier-Mukai partner of X.  Again as in the case of K3 surfaces, one proves the finiteness of the number of Fourier-Mukai partners of Abelian surfaces. Corollary 7.39. An Abelian surface has only finitely many Fourier-Mukai partners. Another characterization of the Fourier-Mukai partners of an Abelian variety was given by Orlov in [243] (note that this characterization works in any dimension, not just for surfaces, and for every ground field). For the sake of completeness we ∼ Y ×Yˆ ˆ→ state it here without proof. Let X, Y be Abelian varieties, and let f : X×X be an isomorphism, which we represent in the matrix form   a b f= . c d We say that f is isometric if the inverse f −1 has the matrix form f −1 =

 dˆ −ˆb . −ˆ c a ˆ



Theorem 7.40. [243, Thm. 2.19] Two Abelian varieties X and Y are Fourier-Mukai ∼ Y × Yˆ . ˆ→ partners if and only if there is an isometric isomorphism X × X Orlov used this to prove that an Abelian variety (of arbitrary dimension, defined on any ground field) has only finitely many Fourier-Mukai partners.

7.4.6

Enriques surfaces

Fourier-Mukai partners of Enriques surfaces have been computed by Bridgeland and Maciocia in [70] building on their previous work about Fourier-Mukai transform for quotient varieties [69] and on some results by Nikulin about lattices [235], also used in the study of Fourier-Mukai partners of K3 and Abelian surfaces. We recall that an Enriques surface is a projective smooth minimal surface 2 ' OX and ωX is X whose canonical line bundle ωX is of order 2, that is, ωX

7.4. Fourier-Mukai partners of algebraic surfaces

255

nontrivial. Then A = OX ⊕ ωX has a natural structure of OX -algebra and defines a finite ´etale covering of degree 2 ˜ = Spec AX → X , ρX : X such that AX ' ρX∗ OX˜ . We call ρX the canonical cover of X. The twist of AX ˜ and one has X/G ˜ by ωX defines a free action of G = Z/(2) on X, ' X. Moreover, ˜ ωX˜ is trivial so that X is a K3 surface. Since ρX is finite and X is a minimal ˜ is minimal as well. surface, X ˜ Z) and gives rise The generator  of G acts on the integer cohomology H • (X, to a an orthogonal decomposition ˜ Z) ' H • (X, ˜ Z) ⊥ H • (X, ˜ Z) H • (X, + − as a direct sum of the sublattices where  acts as the identity and as the multi• ˜ Z) ⊂ H 2 (X, ˜ Z) and H 0.2 (X, ˜ C) ⊂ plication by −1, respectively. One has H− (X, • ˜ H− (X, Z) ⊗Z C. One also has an orthogonal decomposition ˜ Z) ' H 2 (X, ˜ Z) ⊥ H • (X, ˜ Z) H 2 (X, − + 2 ˜ which proves that H+ (X, Z) is even and unimodular and is the sublattice orthog• ˜ Z) in H 2 (X, ˜ Z). Moreover, the pullback by ρX gives an immersion onal to H− (X, 2 ˜ ρ∗X : H 2 (X, Z)/ Torsion ,→ H+ (X, Z) . 2 ˜ Since H 2 (X, Z)/ Torsion is indefinite by [22, VIII.15.1], H+ (X, Z) is indefinite as well. In particular, it is 2-elementary in the sense of [235, Def. 3.6.1 ].

Let Y be a Fourier-Mukai partner of X and Φ : Db (X) → Db (Y ) an equivalence of triangulated categories. By Theorem 2.38, Y is also an Enriques surface and there is a canonical cover ρY : Y˜ → Y which identifies Y with the quotient of a free action of G on a K3 minimal surface Y˜ . By Proposition 7.13 (cf. also Remark 7.14), there is a lift of Φ, that is, a G˜ → Db (Y˜ ) such that ˜ : Db (X) equivariant equivalence of triangulated categories Φ ∗ ∗ ˜ ˜  ◦ Φ = Φ ◦  (because the unique automorphism of Za is the identity). Thus, the induced Hodge isometry ˜ • (X, ˜ Z) → H ˜ • (X, ˜ Z) f: H (cf. Proposition 4.27) is G-equivariant. Proposition 7.41. The surface Y is isomorphic to X, that is, an Enriques surface has no Fourier-Mukai partners but itself.

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Proof. The G-equivariant Hodge isometry f induces a G-equivariant isometry • ˜ Z) → H • (Y˜ , Z). Let us prove that it extends to an isometry (X, f − : H− − ˜ Z) → H 2 (Y˜ , Z) . f˜: H 2 (X, ˜ Z) and H 2 (Y˜ , Z) are isometric, we need to check that any isomeSince H 2 (X, • ˜ Z) is induced by an isometry of H 2 (X, ˜ Z). There is an orthogonal try of H− (X, 2 ˜ 2 ˜ • ˜ decomposition H (X, Z) ' H+ (X, Z) ⊥ H− (X, Z), and then the orthogonal to • ˜ Z) in H 2 (X, ˜ Z) is H 2 (X, ˜ Z). As we have already shown, the latter is an (X, H− + even, indefinite and 2-elementary lattice; then [235, Thm. 3.6.2, Thm. 3.6.3] imply that we are in the hypotheses of [235, Prop. 1.14.1]. This enables us to conclude. ˜ Z) → H 2 (Y˜ , Z). We thus have proved that f− extends to an isometry f˜: H 2 (X, Moreover, f˜ is automatically a G-equivariant Hodge isometry. By the Torelli theorem for Enriques surfaces [22, VIII.21.2], X and Y are isomorphic.

7.4.7



Nonminimal projective surfaces

In this section we complete the study of the Fourier-Mukai partners of a smooth algebraic surface. The case which still remains open is that of some nonminimal surfaces. We have however already proved in Proposition 7.23 that relatively minimal elliptic surfaces (Definition 6.9) have a finite number of Fourier-Mukai partners, even if they may fail to be minimal. Here we study the remaining cases of nonminimal surfaces following Kawamata’s treatment [175]. Let us recall that if two smooth projective surfaces X and Y are Fourier• b b Mukai partners and ΦK X→Y : D (X) → D (Y ) is a Fourier-Mukai functor, there exist an irreducible component Z of the support W of K• (cf. Definition A.90), such that the projection pX = πX |Z : Z → X is surjective. By Lemma 2.46, if Ze → Z is the normalization morphism, and p˜X : Ze → X, p˜Y : Ze → Y are the r ' p˜∗Y ωYr for some r > 0. projections, then p˜X is dominant and p˜∗X ωX We need a preliminary result about the numerical Kodaira dimension which we introduced in Chapter 2. Lemma 7.42. Let L be a nef line bundle on a smooth projective surface X wtih ν(X, L) = 1. Then κ(X, L∗ ) = −∞. Proof. Assume that H 0 (X, Lk ) 6= 0 for some k < 0. Then either Lk is trivial or kc1 (L) is represented by an effective divisor. The first case contradicts that ν(X, L) = 1, while the second contradicts that L is nef. The statement follows.  Theorem 7.43. [175, Thm. 1.6] Let X be a nonminimal smooth projective surface. If X is not a relatively minimal elliptic surface (Definition 6.9), any Fourier-Mukai partner Y of X is isomorphic to X.

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257

Proof. We use similar arguments as in the proof of Theorem 2.49. Let C be a (−1)∗ curve in X, that is, a smooth rational curve with C 2 = −1. Then ωX |C ' OC (1) is −1 e ample. Let us write T = p˜ (C) ,→ Z. Since no curve can be contracted by the two X

projections p˜X and p˜Y , if D ,→ T is contracted by p˜Y , the induced map D → C ∗ r ˜∗X ωX ' p˜∗Y ωYr implies is finite. It follows that p˜∗X ωX |D is ample and the equality p that p˜∗Y ωY∗ |D is ample as well, which is impossible. Thus, p˜Y |T : T → Y is a finite morphism, so that dim T is either 1 or 2. If dim T = 1, one has dim Z = dim Ze = 2 and X ' Y by Proposition 7.19. If dim T = 2, one has projections p˜X |T : T → C −r ∗ and p˜Y |T : T → Y . Since ωX |C is ample, ωY is nef by Lemma 2.43. Moreover, ∗ Lemma 2.44 yields ν(Y, ωY ) = ν(T, p˜Y |T ωY ) = ν(T, p˜X ∗|T ωX ) = ν(C, ωX |C ) = 1, ∗ is nef and ν(X) = 1 by Theorem 2.49. By because ωX |C is anti-ample. Then ωX 2 = 0 since ν(X) = 1. Thus, the Lemma 7.42, κ(X) = −∞. Moreover, one has KX 2 ≤ 0, with equality only if minimal model X0 of X verifies κ(X0 ) = −∞ and KX 0 X is minimal. The classification of surfaces implies thus that X is either a minimal elliptic ruled surface or a rational ruled surface with invariant e = 2. Since X is not minimal, only the second possibility may occur. Then X ' Y by Proposition 7.22.  We know that there are only a finite number of nonisomorphic relative compatified Jacobians JX/B (r, d) (Proposition 6.32). Since we also know that there are a finite number of Fourier-Mukai partners of a K3 surface (Theorem 7.32) or of an Abelian surface (Corollary 7.39), this completes the proof of Theorem 7.17.

7.5

Derived categories and birational geometry

The condition that the derived categories of two surfaces are equivalent allows one to characterize very precisely the relationship between the surfaces, basically because minimal models of algebraic surfaces are completely classified. In higher dimensions, the situation is more involved. The so-called “minimal model program” has been completed in dimension three thanks to the work of Mori [221], who built on previous results by Reid, Kawamata, Koll´ar, Shokurov and others. A fundamental ingredient in Mori’s work is the reduction of the existence of flips, a rather difficult result, to the easier question of the existence of flops. These may regarded as birational analogues of surgery in algebraic topology (one should note that topological invariants of smooth projective varieties, such as Hodge numbers, are invariant under flops). The crux of Bridgeland’s contribution to the topic is a moduli space interpretation of flops of smooth threefolds. This is accomplished by introducing fine moduli spaces of perverse sheaves; the integral functor associated with the relevant universal object is a Fourier-Mukai transform, i.e., it is an equivalence of derived

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Chapter 7. Fourier-Mukai partners and birational geometry

categories. This is the key to proving the following important theorems, due to Bridgeland. Theorem 7.44. [62, Thm. 1.1] Let X be a (complex) projective threefold with terminal singularities and f1 : Y1 → X, f2 : Y2 → X crepant resolutions of singularities. Then there is an equivalence of triangulated categories Db (Y1 ) ' Db (Y2 ). Theorem 7.45. [62] Let X and Y be two birational smooth Calabi-Yau threefolds. Then X and Y are Fourier-Mukai partners, that is, there is an equivalence of triangulated categories Db (X) ' Db (Y ). In the course of this section we shall build proofs of these results.

7.5.1

A removable singularity theorem

We now generalize Bridgeland’s criterion for an integral functor to be an equivalence of categories. We assume that the base field k has characteristic zero. Let us consider two projective varieties X and Y . We know from Proposition 1.11 that a flat family of sheaves on Y parameterized by X may be characterized as an object K• of the derived category Db (X × Y ) such that the object Ljx∗ K• is a sheaf on Y for every (closed) point x ∈ X (where jx : {x} × Y ,→ X × Y is as usual the natural immersion). More generally, any object K• of Db (X × Y ) of finite Tor-dimension over X can be thought of as a parameterization of objects Ljx∗ K• of the derived category Db (Y ); finite Tor-dimension over X is needed to guarantee that the objects Ljx∗ K• have bounded cohomology. Moduli problems can be also formulated in terms of derived categories. Let D be a full subcategory of Db (Y ) changes with the property that any object of Db (Y ) isomorphic to an object of D is also an object of D. Then a family of objects of D parameterized by an algebraic variety S is an object E • of Db (S×Y ) of finite Tor-dimension over S such that for every (closed) point s ∈ S, the complex Ljs∗ E • is an object of D. Two such families E • , F • are considered equivalent if E • ' F • ⊗ πS∗ (L) for a line bundle L on S. We can define a functor FD on the category of schemes by associating to an algebraic variety S the set of equivalence classes of families of objects of D parameterized by S. An algebraic variety M (FD ) is a fine moduli space for D if there exists a family (a relative universal family) P • of objects of D parameterized by M (FD ) such that for any algebraic variety S and any family E • of objects of D parameterized by S, there exists a unique morphism φ : S → M (FD ) such that E • ' L(φ × Id)∗ (P • ) ⊗ πS∗ (L) for a line bundle L on S. In other words, the variety M (FD ) represents the functor FD . Thus, there is an equivalence Hom(S, M (FD )) ' FD (S) (7.8) φ 7→ L(φ × Id)∗ (P • ) .

7.5. Derived categories and birational geometry

259

We will see in Section 7.5.2 an example of a functor defined by triangulated subcategories of the bounded derived categories — the functor of relative perverse point sheaves — which is representable. There is a Kodaira-Spencer map for families E • of objects of D parametrized by an algebraic variety S. The definition is based on the tangent space to the functor, which we are going to describe. Let D = Spec k[ε]/ε2 be the double point scheme and j0 : {s0 } ,→ D the immersion of its unique closed point. We recall that the tangent space Ts S to an algebraic variety S at a point s can be defined as the space Hom(D, S)s of the scheme morphisms v : D → S mapping s0 to s. In the same vein, given an object G • of D, the tangent space to the functor FD at the “point” [G • ] is the space T[G • ] FD of classes of families F • ∈ FD (D) endowed with an isomorphism ∼ G • in the derived category (cf. [197, Def. 3.2.1]). Moreover, such $ : Lj0∗ F • → • families (F , ϕ) are in a one-to-one correspondence with the extensions of G • by itself in the sense of the triangulated categories (cf. Definition A.69). This can be easily seen by representing F • as a bounded complex of sheaves L• on D × Y ∼ G • induces an isomorphism j ∗ L• → ∼ G• which are flat over D; then $ : Lj0∗ F • → 0 Moreover, if p denotes the projection D ×Y → Y , the exact sequence of complexes 0 → j0∗ L• → p∗ L• → j0∗ L• → 0 , ∼ G • define an extension of G • by itself as an object of and the morphism j0∗ L• → the triangulated category Db (Y ). Taking into account the identification between extensions and Hom1 groups given by Proposition A.70, we have a one-to-one correspondence T[G • ] FD ' Hom1Db (Y ) (G • , G • ) .

(7.9)

Let now E • be a family of objects of D parametrized by an algebraic variety S. Any tangent vector v ∈ Ts S ' Hom(D, S)s defines a morphism (v × Id) : D × Y → S × Y , whose composition with the immersion Y ,→ D × Y , y 7→ (s0 , y), is the inclusion js : Y ' {s} × Y ,→ S × Y . Thus, L(v × Id)∗ E • is a family of objects of D parameterized by the double point scheme D equipped with an isomorphism ∼ Lj ∗ E • . Therefore it defines as a point in T ∗ • F . According Lj0∗ (L(v×Id)∗ E • )→ Ljs E D s to Equation (7.9), we identify the latter space with Hom1Db (Y ) (Ljs∗ E • , Ljs∗ E • ). In this way we have a map KSs (E • ) : Ts S → Hom1Db (Y ) (Ljs∗ E • , Ljs∗ E • ) v 7→ L(v × Id)∗ E • , called the Kodaira-Spencer map for the family E • .

(7.10)

260

Chapter 7. Fourier-Mukai partners and birational geometry

Lemma 7.46. If there exists a fine moduli space M = M (FD ) for D, and P • M is a relative universal family, the Kodaira-Spencer map KSx (P • M ) : Tx M → Hom1D(Y ) (Ljx∗ P • M , Ljx∗ P • M ) , is an isomorphism for every (closed) point x ∈ M . Proof. Since M is a fine moduli space, we have an equivalence ∗ • ∗ • ∼ T ∗ • F ' Hom1 Tx M ' Hom(D, M )x → Ljx P M D D(Y ) (Ljx P M , Ljx P M )

v 7→ L(v × Id)∗ (P • M ) , which is precisely the Kodaira-Spencer map for the universal family P • M .



Proceeding as in the proof of Lemma 1.24, one has the following result. Lemma 7.47. Let X be a projective variety and E • an object of Db (X × Y ) which is a family of objects of D parameterized by X. The morphism Hom1D(X) (Ox , Ox ) → Hom1D(Y ) (Ljx∗ E • , Ljx∗ E • ) •

E induced by the integral functor ΦX→ Y coincides with the Kodaira-Spencer morphism • for the family E .

In the rest of this section we make the following assumptions: 1. Y is a smooth projective algebraic variety of dimension m. 2. D is a full subcategory of Db (Y ) whose objects fulfil the following properties: • They are simple, that is, one has HomD(Y ) (E • , E • ) = k for any object E • of D. • They are special (Definition 2.54), that is, E • ' E • ⊗ωY for every object E • of D. • HomiD(Y ) (E • , E • ) = 0 for i < 0 and for every object E • of D. • HomD(Y ) (E • , F • ) = 0 if E • and F • are non-isomorphic objects of D. 3. There is a projective algebraic variety M which is a fine moduli space M = M (FD ) for D and an irreducible component j : X ,→ M of dimension m. Since Y is smooth and the objects E • are special, the third property listed in / [0, m]. Condition 2 is equivalent by Serre duality to HomiD(Y ) (E • , E • ) = 0 for i ∈ We give now a criterion for deciding when such a moduli space is smooth. This is based on a corollary of an “intersection theorem” in commutative algebra (cf. Corollary A.99). Let P • M be the universal family. We denote by P • = L(j × IdY )∗ (P • M ) its restriction to X × Y .

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261

Theorem 7.48. If there is a closed subscheme Z ,→ X × X of dimension d ≤ m + 1 / Z one has such that for every (closed) point (x1 , x2 ) ∈ HomiD(Y ) (Ljx∗1 P • , Ljx∗2 P • ) = 0

for every i ∈ Z ,

•

b b then X is smooth and the integral functor ΦP X→Y : D (X) → D (Y ) is an equivalence of categories. P •∨ ⊗π ∗ ω [m]

is a left adProof. By Proposition 1.13, the integral functor Ψ = ΦY→X Y Y • b b : D (X) → D (Y ). The composition Ψ ◦ Φ is then an integral joint to Φ = ΦP X→Y functor whose kernel is the convolution M• = (P •∨ ⊗ πY∗ ωY [m]) ∗ P • of the two kernels. Given two points x1 , x2 of X, the adjunction between Φ and Ψ gives rise to isomorphisms HomiD(X×X) (M• , O(x1 ,x2 ) ) ' HomiD(X) (Ljx∗1 M• , Ox2 ) ' HomiD(Y ) (Φ(Ox1 ), Φ(Ox2 )) ' HomiD(Y ) (Ljx∗1 P • , Ljx∗2 P • ) for every integer i. By Proposition A.91, the support of the restriction of M• to the complement U = (X × X) − ∆ of the diagonal is contained in (X × X) − Z and so that it has codimension greater than or equal to m − 1. We can now apply Corollary A.97 to obtain that hd(M• |U ) ≥ m − 1 .

(7.11)

Since Y is smooth and the objects Ljx∗ P • are special, by Serre duality, one has that ∗ • ∗ • ∗ • ∗ • ∗ Homm D(Y ) (Ljx1 P , Ljx2 P ) ' HomD(Y ) (Ljx2 P , Ljx1 P ) . By the last property in our second assumption, the latter group vanishes for x1 6= x2 , and then hd(M• |U ) ≤ m − 2; thus Equation (7.11) implies that M• |U = 0, so that M• is (topologically) supported on the diagonal. Then one has HomiD(Y ) (Ljx∗2 P • , Ljx∗1 P • ) = 0 unless x1 = x2 and i ∈ [0, m]. We now apply Corollary A.99 to prove that X is smooth. To do so, we consider for every (closed) point x ∈ X the complex E • (x) = (Ψ ◦ Φ)(Ox ) ' i M• • ΦX→ X (Ox ). Since HomD(X) (E (x1 ), Ox2 ) = 0 unless x1 = x2 and 0 ≤ i ≤ m, we 0 • have to prove that H (E (x)) ' Ox . This will prove that X is smooth and that E • (x) = (Ψ ◦ Φ)(Ox ) ' Ox . The proof follows the same idea of that of Theorem 1.27 with some modifications (cf. [71, Thm. 6.1]): we consider an exact triangle α

x Ox → C • [1] , C • → (Ψ ◦ Φ)(Ox ) −−→

(7.12)

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Chapter 7. Fourier-Mukai partners and birational geometry

where αx is the adjunction morphism and C • [−1] is a cone of αx . By taking homomorphisms in Ox and using that Ψ is a left adjoint to Φ, we get an exact sequence 0 → Hom0D(X) (Ox , Ox ) → Hom0D(X) (Φ(Ox ), Φ(Ox )) → Hom0D(X) (C • , Ox ) τ

→ Hom1D(X) (Ox , Ox ) − → Hom1D(X) (Φ(Ox ), Φ(Ox )) → . . .

(7.13)

and isomorphisms HomiD(X) (Φ(Ox ), Φ(Ox )) ' HomiD(X) (C • , Ox ), for i < 0, so that HomiD(X) (C • , Ox ) = 0 for i < 0. Moreover, by Lemma 7.47, the morphism τ is the Kodaira-Spencer map for the universal family P • . Furthermore, the KodairaSpencer map for P • is the composition of the tangent map Tx X → Tx M with the Kodaira-Spencer map for the universal family P • M . Since the tangent map is injective because k has characteristic zero and the Kodaira-Spencer map for the universal family P • M is an isomorphism by Lemma 7.46, we have that τ is injective. It follows that HomiD(X) (C • , Ox ) = 0 for i < 0 so that Hi (C • ) = 0 for i < 0 by Remark A.92. Taking cohomology in the exact triangle (7.12) one obtains that H0 ((Ψ ◦ Φ)(Ox )) ' Ox . Now we know that X is smooth and that (Ψ◦Φ)(Ox ) ' Ox for every point x. We can apply Theorem 2.6 to prove that Ψ◦Φ is fully faithful, since the skyscraper sheaves Ox form a spanning class for Db (X) (Proposition 2.52). By Remark 1.21, this implies that Φ is fully faithful as well so that Φ is an equivalence of categories by Corollary 2.56.  Flops Flops are very simple instances of birational transformations. The precise definitions is the following: Definition 7.49. A flop is a diagram X@ @@ @@ @ f @@ 

Y

X+ | | || || f + | } |

where Y is a projective Gorenstein variety and f and f + are crepant resolutions of singularities (Definition 7.3) whose exceptional loci have codimension equal or greater than 2. We also require the existence of a divisor D in X such that −D is relatively f -ample in X and the strict transform D+ of D in X + is relatively 4 f + -ample. Since D+ is f + -ample, we have that f + is isomorphic to the projective morL phism X + ' Proj s≥0 f∗+ (OX + (sD+ )) → Y which is actually isomorphic to

7.5. Derived categories and birational geometry

263

L Proj s≥0 f∗ (OX (sD)) → Y . This proves that f + is completely determined by f and D. By Lemma 7.5, if Y has terminal singularities, the condition on the exceptional locus is automatically fulfilled. The importance of flops is evident from the following result. Proposition 7.50. [175, Lemma 4.6] Let α : X 99K Y be a crepant birational map between projective threefolds with only terminal singularities. Then α may be decomposed into a sequence of flops. t-structures We now give some notions about triangulated categories that generalize what we saw in Section 2.1. Definition 7.51. Let A be a triangulated category. A full subcategory B ⊆ A is right admissible if the inclusion functor B ,→ A has a right adjoint. 4 If B ⊆ A is a full subcategory, one can define the right orthogonal subcategory as the full subcategory B⊥ of A whose objects are the objects a in A such that HomiA (b, a) = 0 ,

for all objects b in B and all i ∈ Z .

One easily sees that if B ⊆ A is a right admissible triangulated subcategory, then for every object a in A there is an exact triangle b → a → c → b[1] where b is an object in B and c is an object in B⊥ . We then say that A admits a semi-orthogonal decomposition in terms of B and B⊥ and write A ' (B⊥ , B). The first example of a right admissible triangulated subcategory is given in the next proposition. This example will be of great importance in this section. Proposition 7.52. Let f : X → Y be a projective morphism of varieties such that Rf∗ OX ' OY . Then the inverse image Lf ∗ : D(Y ) → D(X) is fully faithful and makes D(Y ) into a right admissible triangulated subcategory of D(X). Proof. The functor Lf ∗ has Rf∗ as a right adjoint. By the projection formula, the composition Rf∗ ◦ Lf ∗ is the twist by Rf∗ OX so that it is the identity by the condition Rf∗ OX ' OY . Then Lf ∗ is fully faithful (see Section 1.3) and we can identify D(Y ) with this image by Lf ∗ , and this is a right admissible triangulated category. 

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Chapter 7. Fourier-Mukai partners and birational geometry

If either Y is smooth or f is of finite Tor-dimension, then Lf ∗ maps Db (X) to D (Y ) and we can identify Db (X) with a right admissible triangulated category of Db (Y ). b

Let A be a triangulated category. Definition 7.53. A t-structure on A is a right admissible subcategory A≤0 of A 4 which is preserved by the shift functor, that is A≤0 [1] ⊂ A≤0 . We will use the notations A≤i = A≤0 [−i], A≥i = (A≤i−1 )⊥ , Ai = A≥i+1 . Definition 7.54. The heart (or core) of a t-structure A≤0 ⊂ A is the full subcategory 4 H = A≤0 ∩ A≥0 . One can prove [35] that the heart H of a t-structure is an Abelian category. An exact sequence 0→a→b→c→0 in H is by definition an exact triangle a → b → c → a[1] in A whose vertices are objects of H. The first example of a t-structure is the standard t-structure on the derived category D(A) of an Abelian category A. This is defined by taking D(A)≤0 as the full subcategory defined by all complexes E • with no strictly positive cohomology objects, namely Hi (E • ) = 0 for all i > 0. The heart of the standard t-structure is the subcategory H of complexes E • such that Hi (E • ) = 0 for all i 6= 0. There is a natural equivalence of Abelian categories A ' H. The same applies to the bounded derived category Db (A). Assume that B is another Abelian category and that there is an equivalence ∼ D(B). The image by Φ of the standard of triangulated categories Φ : D(A) → t-structure on D(A) is a t-structure on D(B) and we can identify A with the full subcategory of D(B) defined as the heart of this t-structure. This suggests that the study of the t-structures on a derived category D(B) is the right tool to determine all the Abelian categories A whose derived category is equivalent to D(B). This applies in particular for the derived category D(X) (or Db (X)) of an algebraic variety: any Fourier-Mukai partner Y defines a t-structure in D(X) and the study of such t-structures is the natural tool for finding Fourier-Mukai partners.

7.5.2

Perverse sheaves

In this section, f : X → Y will be a birational morphism of projective varieties such that Rf∗ OX ' OY , and f has relative dimension 1, that is, no subvarieties

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of dimension greater than one are contracted by f . The exceptional locus of f will be denoted by E. It is a subscheme of X of dimension not exceeding one. We shall write V = Y − f (E) and U = f −1 (U ) = X − Y so that f induces an isomorphism ∼ V. f|U : U → By Proposition 7.52, the inverse image Lf ∗ makes D(Y ) into a right admissible triangulated subcategory of D(X). Then we have a semi-orthogonal decomposition D(X) ' (C, D(Y )) where C = D(Y )⊥ . The objects of C are the complexes E • in D(X) such that Rf∗ E • = 0. They are supported on the exceptional locus of f. Lemma 7.55. An object E • of D(X) is in C if and only if all its cohomology sheaves Hi (E • ) are objects of C. p+q Proof. There is a spectral sequence E2p,q = Rp f∗ (Hq (E • )) converging to E∞ = p+q • • q • H Rf∗ (E ). Then Rf∗ (E ) = 0 if all the sheaves H (E ) belong to C. For the converse, since f has relative dimension 1, one has E2p,q = 0 for p 6= 0, 1. If there is a nonzero element in E2p,q , it defines a cycle that survives to infinity and gives p+q = 0. It follows that E2p,q = 0 for all p and q.  a nonzero element of E∞

Let D(X)≤0 be the standard t-structure on the derived category D(X). It induces a t-structure C≤0 = D(X)≤0 ∩ C on C. We also denote by D(Y )≤0 the standard t-structure on D(Y ). Following [35] we can define for every integer p a t-structure p D(X)≤0 in D(X) by letting E • ∈ Ob(p D(X)≤0 ) if Rf∗ (E • ) ∈ Ob(D(Y )≤0 ) and HomD(X) (E • , F • ) = 0 for all F • ∈ Ob(C>p ) ; as a consequence, one has E • ∈ Ob(p D(X)≥0 ) if Rf∗ (E • ) ∈ Ob(D(Y )≥0 ) and HomD(X) (F • , E • ) = 0 for all F • ∈ Ob(C

Definition 7.56. The category of p perverse sheaves for f is the heart p

Per(X/Y ) = p D(X)≤0 ∩ p D(X)≥0

of the above t-structure. When p = −1 we simply write Per(X/Y ) = −1 Per(X/Y ) and call it the category of perverse sheaves.

4

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As it often happens in mathematics, this terminology is actually a misnomer: perverse sheaves may fail to be sheaves. By its very definition, a complex E • in D(X) is a perverse sheaf if and only if it verifies three conditions: 1. Rf∗ (E • ) is a sheaf, i.e., it is a complex concentrated in degree zero. 2. HomD(X) (E • , F • ) = 0 for all complexes F • with Rf∗ F • = 0 and Hi (F • ) = 0 for i < 0. 3. HomD(X) (F • , E • ) = 0 for all complexes F • with Rf∗ F • = 0 and Hi (F • ) = 0 for i ≥ −1. One should be aware that these are completely different objects from the “usual” constructible perverse sheaves (for these, see, e.g., [173, 35]). Especially in the literature about stability conditions for derived categories (see Appendix D), it is somewhat standard to use “perverse (coherent) sheaf” to refer to an object in the heart of a fixed nonstandard t-structure, regardless of the origin of that t-structure. The following result describes explicitly what perverse sheaves look like. Lemma 7.57. A complex E • in D(X) is a perverse sheaf if and only if it satisfies the following conditions: 1. Hi (E • ) = 0 unless i = −1 or i = 0. 2. R1 f∗ (H0 (E • )) = 0 and f∗ (H−1 (E • )) = 0. 3. HomX (H0 (E • ), F) = 0 for any sheaf F in C. Proof. If E • is a perverse sheaf, Rf∗ (E • ) is a sheaf. From the spectral sequence p+q = Hp+q Rf∗ (E • ), and the fact that E2p,q = 0 for E2p,q = Rp f∗ (Hq (E • )) =⇒ E∞ p 6= 0, 1 (see the proof of Lemma 7.55), we deduce 2 and that Rf∗ (Hi (E • )) = 0 unless i = −1 or i = 0. Then, by Lemma 7.55, the truncated complex (E • )>0 is an object of C>0 and the truncated complex (E • )<−1 is an object of C<−1 . Since there exist morphisms (E • )<−1 → E • and E • → (E • )>0 , one has that (E • )<−1 = 0 and (E • )>0 = 0, which proves 1. Part 3 follows because any nonzero morphism of sheaves H0 (E • ) → F gives a nonzero morphism E • → F in D(X). For the converse, assume that the three conditions of the statement are fulfilled. Conditions 1 and 2 and the above spectral sequence prove that Rf∗ E • is a sheaf. If F • is an object of C≥0 and E • → F • is a nonzero morphism, then all the cohomology morphisms Hi (E • ) → Hi (F • ) for i 6= 0 are automatically zero due to condition 1, and then H0 (E • ) → H0 (F • ) cannot be zero. This contradicts condition 3 because H0 (F • ) is an object of C by Lemma 7.55; then HomD(X) (E • , F • ) = 0.  Finally, if F • is an object of C<−1 , HomD(X) (F • , E • ) = 0 by condition 1.

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Corollary 7.58. Let E • be a perverse sheaf. If Rf∗ (E • ) = 0, then E • [−1] is a sheaf. Proof. E • is an object of C, so that H0 (E • ) is also an object of C by Lemma 7.55. Condition 3 of Lemma 7.57 implies then that H0 (E • ) = 0; thus E • [−1] is a sheaf.  An elementary consequence of Lemma 7.57 is that the structure sheaf OX is a perverse sheaf. Now we consider exact sequences 0 → I • → OX → E • → 0

(7.14)

in the Abelian category Per(X/Y ) of perverse sheaves. We adopt the following definition. Definition 7.59. The perverse sheaf E • is called a perverse structure sheaf and the 4 perverse sheaf I • is called the corresponding perverse ideal sheaf. Lemma 7.60. Perverse ideal sheaves are actually sheaves, that is, Hi (I • ) = 0 for i 6= 0. Proof. The result is obtained by taking cohomology in Equation (7.14).



The Euler characteristic of two objects of Db (X) was defined (cf. Eq. (1.5)) when X is a smooth projective variety. However, the formula X χ(L, E • ) = (−1)i dim HomiD(X) (L, E • ) i

makes sense for any projective variety if L is a locally free sheaf and E • is an object of Db (X). Definition 7.61. Let X be a projective variety. Two objects E • and F • of Db (X) are derived numerically equivalent if for any locally free sheaf L on X the Euler 4 characteristics χ(L, E • ) and χ(L, F • ) coincide. When X is smooth, E • and F • are derived numerically equivalent if and only if they have the same Chern characters. For sheaves, derived numerical equivalence has some elementary properties. Lemma 7.62. Let X be a projective variety and E a coherent sheaf on X. 1. E is derived numerically equivalent to zero if and only if E = 0. 2. E is derived numerically equivalent to the structure sheaf of a closed point if and only if E ' Ox for some closed point x ∈ X.

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Proof. For n  0, HomiD(X) (OX (−n), E) = 0 for i 6= 0, so that χ(O(−n), E) = dim Hom0D(X) (OX (−n), E); moreover, there is a surjective morphism Hom0D(X) (OX (−n), E) ⊗k OX → E(n) → 0 . Thus, if E is derived numerically equivalent to zero, one has E = 0. If E is derived numerically equivalent to the structure sheaf of a closed point, one has χ(L, E) = rk(L) for any locally free sheaf L. Reasoning as above, there is a surjective morphism OX → E(n) → 0, and then E(n) is the structure sheaf of a closed subscheme Y of X. Moreover E(n) is also derived numerically equivalent to the structure sheaf of a closed point. If x ∈ Y is a closed point, the kernel of the surjection OY → Ox → 0 is a sheaf derived numerically equivalent to zero, and  then OY ' Ox . We now give the following definition: Definition 7.63. A perverse point sheaf is a perverse structure sheaf which is de4 rived numerically equivalent to the structure sheaf Ox of a closed point. Example 7.64. If x ∈ U is a point in the open subset where f is an isomorphism, then Ox is a perverse point sheaf. However if x ∈ E is a point of the exceptional locus, this may fail to be true. Assume for instance that the exceptional locus of f is a smooth rational curve i : E ,→ X defined by an ideal sheaf JE . For every point x ∈ E there is an exact sequence 0 → JE → mx → i∗ mE,x → 0 , where mE,x denotes the ideal of x in the curve E. Since mE,x ' OE (−1), we see that Rf∗ i∗ mE,x = 0 and then i∗ mE,x is an object of C. By Lemma 7.57, the existence of the nonzero morphism mx → i∗ mE,x implies that mx is not a perverse 4 sheaf. Thus, the skyscraper sheaf Ox is not a perverse point sheaf. Lemma 7.65. If E • is a perverse point sheaf, then Rf∗ E • ' Oy for a (closed) point y ∈ Y . Proof. By adjunction, the sheaf Rf∗ Ox is derived numerically equivalent to the structure sheaf of a closed point, so that it is isomorphic to the structure sheaf of some closed point y ∈ Y by Lemma 7.62.  Lemma 7.66. Let E • and F • be perverse point sheaves on X. Then ( k if E • ' F • HomD(X) (E • , F • ) ' 0 otherwise.

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Moreover, one has HomiD(X) (E • , E • ) = 0 for i < 0 and every perverse point sheaf E •. Proof. If E • is a perverse point sheaf, Rf∗ (E • ) ' Oy for a point y ∈ Y by Lemma 7.65. Then HomD(X) (OX , E • ) ' HomD(Y ) (OY , Oy ) ' k and HomiD(X) (OX , E • ) ' HomiD(Y ) (OY , Oy ) = 0 for i < 0. Taking homomorphisms of the exact sequence (7.14) of perverse sheaves into F • gives 0 → HomD(X) (E • , F • ) → HomD(X) (OX , F • ) ' k . Then, if HomD(X) (E • , F • ) 6= 0, the surjective morphism OX → F • of perverse sheaves factors trough a surjective morphism φ : E • → F • of perverse sheaves. The kernel ker φ in the category Per(X/Y ) is derived numerically equivalent to zero, so that Rf∗ ker φ is a sheaf of Y numerically equivalent to zero. By Lemma 7.62, Rf∗ ker φ = 0. By Corollary 7.58, ker φ[−1] is a sheaf. Since it is numerically equivalent to zero, it has to be zero by Lemma 7.62, so that φ is an isomorphism. This proves the first part. For the second, taking derived homomorphisms of I • into the exact sequence (7.14) of perverse sheaves, and using the fact that I • is a sheaf (Lemma 7.60), we obtain Homi (I • , E • ) = 0 for i < 1. Moreover, since HomiD(X) (OX , E • ) = 0 for i < 0, by taking homomorphisms of the exact sequence (7.14) into E • one has HomiD(X) (E • , E • ) = 0 for i < 0 as claimed.  Our next aim is to construct an algebraic variety W and a projective morphism f + : W → Y (which under suitable assumptions will be a flop of f ) such that the closed points of W parameterize perverse point sheaves on X. We will not enter into the details of the proof of the existence of the moduli space W (see [62]); however we need to state some of the natural base change properties of perverse sheaves which makes the construction of this moduli space possible, because they are important on their own and will be useful for other purposes. Let S be a scheme, which will play the role of the base scheme of the space of parameters for a family. We have for every point s ∈ S a commutative diagram X



js

f

  Y

/ S×X fS

js

 / S×Y

where fS = 1 × f . Let us denote by πS the projection S × X → S as in Section 1.2.1. Definition 7.67. A family of perverse sheaves (or a relative perverse sheaf) for f : X → Y over S is an object E • in D(S × X), flat over S, such that for every point s, the complex Ljs∗ E • of D(X) is a perverse sheaf for f . Two such families 4 E • , F • are equivalent if F • ' E • ⊗ πS∗ L for some line bundle L on S.

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If φ : T → S is a morphism of schemes, and E • is a relative perverse sheaf for f over S, the derived pullback L(φ × 1)∗ (E • ) is a relative perverse sheaf for f over T . Proposition 7.68. Let E • ∈ Ob(D(S × X)) be a family of perverse sheaves for f over S. Then the complex RfS∗ E • is a sheaf G on S ×Y flat over S and one has an isomorphism of sheaves Gs ' Rf∗ (Ljs∗ E • ) for every point s ∈ S, where Gs = js∗ G. Proof. One has LfS∗ (js∗ (OY )) ' js∗ (OX ). Then, the projection formula gives L

js∗ (Rf∗ (Ljs∗ E • )) ' RfS∗ (js∗ (Ljs∗ E • )) ' RfS∗ (js∗ (OX ) ⊗ E • )) L

' js∗ (OY ) ⊗ RfS∗ (E • ) ' js∗ (Ljs∗ (RfS∗ (E • ))) . Since Ljs∗ E • is a perverse sheaf on X for any s ∈ S, Rf∗ (Ljs∗ E • ) is a sheaf on Y for any s ∈ S by Lemma 7.57. It follows that Ljs∗ (RfS∗ (E • )) is also a sheaf for any s ∈ S, and then RfS∗ (E • ) is a sheaf G on S × Y flat over S such that  Gs ' Rf∗ (Ljs∗ E • ) by Proposition 1.11. We define the functor of relative perverse point sheaves as the functor which assigns to a scheme S the set of exact triangles I • → OS×X → E • → I • [1] , where for every point s ∈ S, the restriction Ljs∗ E • is a perverse point sheaf. By abuse of language we refer to E • as a relative perverse point sheaf, or a family of perverse point sheaves. Notice that the restriction Ljs∗ I • is a perverse ideal sheaf; since perverse ideal sheaves are sheaves by Lemma 7.60, by Proposition 1.11, I • ' I is a sheaf flat over S. The following result then follows. Lemma 7.69. A relative perverse point sheaf E • is of finite Tor-dimension over S. There is an existence theorem for the moduli space. Theorem 7.70. [62, Theorem 3.8] The functor which assigns to a scheme S the set of equivalence classes of families of perverse point sheaves for f over S is representable by a projective scheme M(X/Y ). So M(X/Y ) is a fine moduli space of perverse point sheaves and there is a universal perverse point sheaf P • in D(M(X/Y ) × X) for f over M(X/Y ), of finite Tor-dimension over M(X/Y ), such that the perverse point sheaf on X ∗ • P , where jw : X ' corresponding to a point w ∈ M(X/Y ) is the object Ljw {w} × X ,→ M(X/Y ) × X is the natural embedding. More generally, given a relative perverse point sheaf E • ∈ Ob(D(S × X)) for f over S, there exists a

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unique scheme morphism φ : S → M(X/Y ) such that E • ' L(φ × 1)∗ P • ⊗ πS∗ (L) for a line bundle L on S. By Proposition 7.68, if E • ∈ Ob(D(S × X)) is a family of perverse point sheaves for f over S, then RfS∗ E • is a sheaf G on S × Y flat over S and one has an isomorphism of sheaves Gs ' Rf∗ (Ljs∗ E • ) for every point s ∈ S, where Gs = js∗ G. Since Rf∗ (Ljs∗ E • ) ' Oy for a point y ∈ Y , we see that, up to twisting by a line bundle coming from S, the sheaf G is the structure sheaf of the graph of a unique morphism of schemes g : S → Y , that is, one has RfS∗ E • ' Γg∗ OS ⊗ p∗S L for some line bundle L on S, where Γg : S → S × Y is the graph of g and pS : S × Y → S is the projection. In particular, the universal perverse point sheaf P • gives rise to a morphism f + : M(X/Y ) → Y , uniquely characterized by the condition RfM(X/Y )∗ P • ' Γf + ∗ OM(X/Y ) ⊗ p∗M(X/Y ) L

(7.15)

for some line bundle L on M(X/Y ). The effect of f + on closed points is simply that of taking the derived direct image, that is, if E • is a perverse point sheaf on X and [E • ] ∈ M(X/Y ) is the closed point determined by it, then f + ([E • ]) = y ,

with Oy ' Rf∗ E • .

∗ −1 we can normalize the universal relative perNow, twisting P • by πM(X/Y ) (L) • verse point sheaf P so that

RfM(X/Y )∗ P • ' Γf + ∗ OM(X/Y ) .

(7.16)

The fact that f is birational implies that f + is birational as well. Actually, since ∼ V = f (U ) on the complement U of the exceptional f induces an isomorphism U → locus, the skyscraper sheaf Ox is a perverse point sheaf (Example 7.64) and it is the unique perverse point sheaf E • such that f + ([E • ]) = f (x), so that f + gives an isomorphism ∼V. f + |(f + )−1 (V ) : (f + )−1 (V ) → We can make this construction more algebraic by noting that the inverse of f + over V is given by the morphism φ : V → M(X/Y ), y 7→ Of −1 (y) , corresponding by the universal property of M(X/Y ) (Theorem 7.70) to the structure sheaf Γg˜∗ (OV ) of ∼U the graph of the morphism g˜ : V ,→ X given as the composition of f −1 : V → and the immersion i : U ,→ X. Let W be the irreducible component of M(X/Y ) containing (f + )−1 (V ). One can prove that actually W = M(X/Y ) [62], though we do not need this result.

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If we still denote by f + the restriction of f + : M(X/Y ) → Y to W , we have a commutative diagram of birational morphisms X@ @@ @@ f @@ Y

}} }} } } + ~}} f

W (7.17)

Remark 7.71. Let us denote again by P • the restriction to W × X of the universal perverse point sheaf. There is a universal exact triangle I → OW ×X → P • → I[1] which defines the universal perverse ideal sheaf I. As we have already observed, I is flat over W . Moreover, Chen has proved in [87, Prop. 4.2] that I → OW ×X is injective, and that I is actually the ideal of the fiber product X ×Y X ,→ W × X. Thus, P • is isomorphic to the structure sheaf OW ×Y X and there is a universal exact sequence of coherent sheaves 0 → I → OW ×X → P • ' OW ×Y X → 0 . Though I and OW ×X are flat over W , the sheaf OW ×Y X is not, and this explains why for some points w ∈ W , the corresponding perverse point sheaf P • w ' ∗ OW ×Y X is a complex and not a sheaf. 4 Ljw

7.5.3

Flops and derived equivalences

We now apply the results and notation of Section 7.5.2 to a crepant resolution of singularities f : X → Y of a projective threefold with Gorenstein terminal singularities (Definition 7.2). The morphism f satisfies Rf∗ (OX ) ' OY and contracts only a finite number of curves. If Y 0 is the image of the exceptional locus, the anti-image Z˜ of Y 0 by the fiber product morphism W ×Y W → Y is a surface. Let us denote by P • the restriction to D(W × X) of the relative universal perverse sheaf. Since it is of finite Tor-dimension over W by Lemma 7.69, we can consider the associated integral functor •

P b b Φ = ΦW→ Y : D (W ) → D (X) , L

∗ (F • ) ⊗ P • ). described as Φ(F • ) = RπX∗ (πW

Theorem 7.72. W is smooth, f + is crepant and Φ is a Fourier-Mukai functor. Proof. We use the removable singularity Theorem 7.48. By Lemma 7.65, the object ∗ • P • w = Ljw P is simple and then its support is connected. Since Rf∗ (P • w ) ' Oy

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273

with y = f + (w), the support of P • w is contained in the fiber f −1 (y). Then P • w ' P • w ⊗ ωX because f is crepant. Moreover HomD(X) (P • w1 , P • w2 ) = 0 for w1 6= w2 and HomiD(X) (P • w , P • w ) = 0 for i < 0 and every point w, by Lemma 7.66. Now, if w1 and w2 are distinct points of W , one sees that HomiD(X) (P • w1 , P • w2 ) = 0 for all i if f + (w1 ) 6= f + (w2 ), that is, if (w1 , w2 ) ∈ / Z. Since dim Z˜ = 2 ≤ 4, Theorem 7.48 implies that W is smooth and Φ is an equivalence. Let us now see that f + is crepant. For each point y ∈ Y , Φ induces an equivalence of categories between the full subcategory Df −1 (y) (X) ⊂ Db (X) of objects topologically supported on the fiber f −1 (y) and the full sucategory D(f + )−1 (y) (W ) ⊂ Db (W ) of complexes topologically supported on the fiber (f + )−1 (y). Since ωX is trivial on an open neighborhood of f −1 (y), the triangulated category Df −1 (y) (X) has trivial Serre functor, and then D(f + )−1 (y) (W ) has trivial Serre functor as well.  By Proposition 7.8, f + is crepant. Consider the diagram W ×X v πW vvv fW v v v  {vv πW W ×Y W o

πX

/X

πY

 /Y

f

Then, by the projection formula and Equation (7.16), we have L

L

∗ ∗ Rf∗ Φ(F • ) ' RπY ∗ RfW ∗ (πW (F • ) ⊗ P • ) ' RπY ∗ (πW (F • ) ⊗ RfW ∗ P • )) L

∗ (F • ) ⊗ Γf + ∗ OW )) ' Rf∗+ F • , ' RπY ∗ (πW

for any object F • of D(W ). Thus, there is a commutative diagram of exact functors Φ / D(X) D(W ) HH v HH v HH vv v H v v Rf∗+ HH$ {vv Rf∗ D(Y )

(7.18)

We are now going to prove that the diagram (7.17) is a flop. We need a preliminary result. Lemma 7.73. Let F • be an object of D(W ) such that Rf∗+ (F • ) = 0. Then F • is WIT−1 with respect to Φ if and only if it is a sheaf F • ' F on W .

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Proof. Assume first that F • is WIT−1 so that Φ(F • ) ' E[1] for a sheaf E on X. Then Rf∗ (E[1]) = 0 by Equation (7.18). This implies that E is supported on the exceptional locus of f , so that E ⊗ ωX ' E because f is crepant. It follows that 3−i ∗ ∗ HomiD(X) (E[1], OX ) ' Hom3−i D(X) (OX , E[1]) ' HomD(Y ) (OY , Rf∗ (E[1])) = 0

for every integer i. Thus, for any (closed) point w ∈ W , from the exact sequence 0 → Iw → OX → P • w → 0 in the category of perverse sheaves, one obtains HomiD(X) (E[1], P • w ) ' HomiX (E[1], Iw ) = 0

unless 0 ≤ i ≤ 3 ,

together with 0 → HomD(X) (P • w , E[1]) → HomD(X) (OX , E[1]) = 0. Then one has the following vanishing result: Hom3D(X) (E[1], P • w ) ' Hom0D(X) (P • w , E[1]) = 0 . The Parseval formula HomiD(W ) (F • , Ow ) ' HomiD(X) (E[1], P • w ) (cf. Proposition 1.34) implies that F • has homological dimension smaller than or equal to 2, and it is then isomorphic in D(W ) to a complex 0 → Lm−2 → Lm−1 → Lm → 0 of coherent locally free sheaves (cf. Definition A.93). The cohomology sheaves Hj (F • ) are supported on curves, actually contracted by f ; if Z is any irreducible component of one of these curves and z0 is its generic point, the stalk OW,z0 is a local regular Noetherian ring of dimension 2. Applying the acyclicity Lemma → Lm−1 → Lm A.95 to the complex Lm−2 z0 z0 z0 of free OW,z0 -modules, we have that m−j • m−j (F )z0 = 0 for j > 0. Then H (F • ) = 0 for j > 0 and F • ' H0 (F • ) is a H sheaf. We now prove the converse. Suppose that F • is a sheaf F on W . As above Rf∗ (Φ(F)) = 0 by Equation (7.18) and then Rf∗ (Φi (F)) = 0 for every i by Lemma 7.55. Let us denote by n0 and n1 the minimum and the maximum of the integers n such that the n-th cohomology sheaf Hn = Φn−1 (F) of Φ(F)[−1] is not zero. We then have nonzero groups n1 n1 1 Hom−n D(X) (Φ(F)[−1], H ) ' HomD(X) (Φ(F)[−1], H [−n1 ]) 0 (Hn0 , Φ(F)[−1]) ' HomD(X) (Hn0 [−n0 ], Φ(F)[−1]) . HomnD(X)

If Φ(F)[−1] is not a sheaf, one has either n0 < 0 or 0 < n1 , and we can find a positive integer n and a sheaf H verifying Rf∗ H = 0 and such that either −n Hom−n D(X) (H, Φ(F)[−1]) 6= 0 or HomD(X) (Φ(F)[−1], H) 6= 0. By the first part, the object Φ−1 (H[1]) is a sheaf G on W . Then either Hom−n D(W ) (G, F) 6= 0 or (F, G) = 6 0 which is absurd because F and G are sheaves.  Hom−n D(X) Proposition 7.74. The diagram (7.17) is a flop, or in other words, f + : W → Y is a flop of f : X → Y .

7.6. McKay correspondence

275

Proof. We have to prove that if D+ is a divisor in W such that −D+ is relatively f + -nef, then the the strict transform D in X is relatively f -nef. Let Z be a rational curve contracted by f + . If E • = Φ(OW (D+ )) and F • = Φ(OZ (−1)), we have χ(E • , F • ) = χ(OW (D+ ), OZ (−1)) = χ(OZ (−1 − D+ · Z)) = −D+ · Z ≥ 0 , by the Parseval formula (Proposition 1.34). On the complementary U of the exceptional locus, E • is isomorphic to OX (D) and then c1 (E • ) = [D]. By Lemma 7.73, F • [−1] is a sheaf G and since Rf∗ G = 0, the support of G is curve Z 0 contracted by f . Thus the Chern characters of G are ch0 (G) = 0, ch1 (G) = 0, ch2 (G) = Z 0 and ch3 (G) = 0, where the last equality follows from Riemann-Roch for f after taking into account that Rf∗ G = 0 and that f is crepant. Moreover Z 0 · c1 (X) = 0 because Z 0 is contracted by f and f is crepant, so that Riemann-Roch gives χ(E • , F • ) = −χ(E • , G) = D · Z 0 Then D · Z 0 ≥ 0 and D is f -nef.



Proof of Theorem 7.44. By Proposition 7.50, the crepant birational map X1 A AA AA A f1 AA Y

X2 } } }} }} ~ } f2 }

is decomposed into a sequence of flops. By applying Proposition 7.74 and Theorem  7.72 we see that the derived categories D b (Y1 ) and Db (Y2 ) are equivalent. Proof of Theorem 7.45. We proceed as above, taking into account that if X and Y are Calabi-Yau threefolds, any birational map between them is crepant. 

7.6

McKay correspondence

The classical McKay correspondence relates representations of a finite subgroup G ⊂ SL(2, C) to the cohomology of the minimal resolution of the Kleinian singularity C2 /G, cf. [127]. An important application of the theory of integral functors is a deep generalization of this correspondence due to Bridgeland, King and Reid [68]. In this section we review this result. Part of the material regarding equivariant derived categories has been taken from [250]. The starting point for the derived McKay correspondence is a smooth projective complex variety Y acted on by a finite group G of automorphisms in such a way that the canonical line bundle ωY is locally trivial as a G-sheaf, that is, there is a covering of Y by G-invariant open neighborhoods each of which carry a

276

Chapter 7. Fourier-Mukai partners and birational geometry

nonvanishing G-invariant section of ωY . Then the quotient variety Y /G has only Gorenstein singularities and the McKay correspondence can be generalized as an equivalence between the G-equivariant derived category of Y , and the ordinary derived category of a crepant resolution of singularities W → Y /G, Db (W ) ' DG (Y ). This equivalence is a G-equivariant version of an integral functor in a sense we are going to describe. We restrict ourselves to the projective case, while the reader is referred to [68] for the quasi-projective situation.

7.6.1

An equivariant removable singularity theorem

The removable singularity theorem given in Section 7.5.1 can be straightforwardly extended to the linearized case. We outline here the main results, whose proofs are completely analogous to the corresponding ones in the ordinary case. We now consider an algebraic variety Y acted on by a finite group G and a full subcategory D of DG,b (Y ) with the property that any object of DG,b (Y ) isomorphic to an object of D is also an object of D. A family of objects of D parameterized by S is an object E • of D{e}×G,b (S × Y ) of finite Tor-dimension over S such that for every (closed) point s ∈ S, the complex Ljs∗ E • is an object of D. As in the ordinary case, two such families E • , F • are considered equivalent if E • ' F • ⊗ πS∗ (L) in D{e}×G,b (S × Y ) for a line bundle L on S. We can define a functor FD on the category schemes by associating to an algebraic variety S the set of equivalence classes of families of objects of D parameterized by S. The tangent space to FD at an object [G • ] of D is now given by T[G • ] FD ' Hom1DG,b (Y ) (G • , G • ) . We also have a linearized Kodaira-Spencer map KSsG (E • ) : Ts S → Hom1DG,b (Y ) (Ljs∗ E • , Ljs∗ E • ) , where s is a point in a scheme S parameterizing a family E • of objects of D. If there exists a fine moduli space M = M (FD ) for D, and P • M is a relative universal family, by proceeding as in the proof of Lemma 7.46, we see that the linearized Kodaira-Spencer map KSxG (P • M ) is an isomorphism for every (closed) point x ∈ M (FD ). As in Section 7.5.1, we assume that this is indeed the case, and consider an irreducible component j : X ,→ M = M (FD ) of the fine moduli space for D. We denote by P • = L(j ×IdY )∗ (P • M ) the restriction to X ×Y of the relative universal family P • M . P • is an object of D{e}×G,b (X × Y ) flat over X. One can consider the associated equivariant integral functor P • ,{e}×G

ΦX→Y

: Db (X) → DG,b (Y ) .

7.6. McKay correspondence

277

Proceeding as in the proof of Proposition 7.48, one has the following equivariant removable singularity result. Theorem 7.75. Suppose that Y is a smooth projective algebraic variety of dimension m = dim X and that the following properties are fulfilled: 1. One has

( HomDG,b (Y ) (Ljx∗1 P • , Ljx∗2 P • )

'

0

for x1 6= x2

k

for x1 = x2

for every pair of (closed) points x1 and x2 in X. 2. Ljx∗ P • ' Ljx∗ P • ⊗ ωY in DG,b (Y ) for every (closed) point x ∈ X, where ωY is equipped with its natural linearization (cf. Example 1.35). 3. HomiDG,b (Y ) (Ljx∗ P • , Ljx∗ P • ) = 0 for i < 0 and any (closed) point x ∈ X. If there is a closed subscheme Z˜ ,→ X × X of dimension d ≤ m + 1 such that for / Z˜ one has every (closed) point (x1 , x2 ) ∈ HomiDG,b (Y ) (Ljx∗1 P • , Ljx∗2 P • ) = 0

for every i ∈ Z ,

then X is smooth and the equivariant integral functor P • ,{e}×G

ΦX→Y

: Db (X) → DG,b (Y )

is an equivalence of categories.

7.6.2

The derived McKay correspondence

Let G be a finite group acting on a smooth complex projective variety Y . We assume that the canonical line bundle ωY is locally trivial as an equivariant sheaf (cf. Example 1.35), that is, every point of Y has an open G-invariant neighborhood U such that there is an equivariant isomorphism OU ' ωU . This implies that the quotient variety Y /G has only Gorenstein singularities. We are going to construct a crepant resolution of Y /G whose bounded derived category is equivalent to the G-linearized derived category DG,b (Y ) of Y , using equivariant integral functors. A candidate for such a resolution is Nakamura’s GHilbert scheme (cf. [232]), which parameterizes G-clusters on Y ; by a G-cluster on Y we mean a zero dimensional G-invariant subscheme Z ,→ Y such that Γ(Z, OZ ) is isomorphic, as a G-vector space, to the regular representation C[G] of G. The length of a G-cluster is then the order ](G) of the group G, and any free orbit of G is a G-cluster. The construction of this G-Hilbert scheme can be outlined as follows.

278

Chapter 7. Fourier-Mukai partners and birational geometry

We consider the functor associating to a scheme S the set of closed G ' {e} × G-invariant subschemes Z ,→ S × Y , flat over S, such that for every closed point s ∈ S, the fiber Zs ,→ Y is a G-cluster. Here S is equipped with the trivial action of {e}. This functor is representable by a closed subscheme HilbG (Y ) of the Hilbert scheme of zero dimensional subschemes of Y of length ](G). There is a universal relative G-invariant closed subscheme ZY ,→ HilbG (Y ) × Y , which is flat over HilbG (Y ). There is a Hilbert-Chow morphism τ : HilbG (Y ) → Y /G which sends a closed point z ∈ HilbG (Y ) to the orbit supporting the corresponding G-cluster Zz . It is a surjective projective morphism and it is birational on one component. Let W be the irreducible component of HilbG (Y ) containing the points that correspond to the free orbits of G. We consider this component because it is not known in general whether HilbG (Y ) is irreducible. We denote by Z = Z|W ×Y the restriction of the universal relative G- invariant closed subscheme, which is finite and flat over W . If we consider the trivial group {e} acting on W , the structure sheaf OZ of Z defines an equivariant kernel in D{e}×G,b (W × Y ), flat over W . We then have an equivariant integral functor O ,{e}×G

Z ΦW→ Y

: Db (W ) → DG,b (Y ) O ,{e}×G

Z E • 7→ ΦW→ Y

G ∗ (E • ) = RπY,∗ (πW E • ⊗ OZ ) .

We denote again by τ : W → Y /G the restriction to W of the Hilbert-Chow morphism, which is a birational morphism. Let W ×Y /G W be the fiber product with respect to τ . The derived McKay correspondence is the following result [68, Thm. 1.1]. Theorem 7.76. Suppose that dim(W ×Y /G W ) ≤ 1+dim Y . The equivariant integral OZ ,{e}×G : Db (W ) → DG,b (Y ) is an equivalence of categories. Moreover, functor ΦW→ Y W is smooth and τ : W → Y /G is a crepant resolution of singularities of the quotient variety Y /G. O ,{e}×G

Z Proof. We prove that ΦW→ is an equivalence of categories and that W is Y smooth by applying Theorem 7.75 for the triangulated subcategory D ⊂ DG,b (Y ) of all G-clusters in Y . In our situation P • = OZ , which is flat over W , and then ∗ • ∗ P ' jw OZ ' OZw for every point w ∈ W . Ljw

/ W ×Y /G W , Since HomiDG,b (Y ) (OZw1 , OZw2 ) = 0 for all integers i if (w1 , w2 ) ∈ the dimension condition of Theorem 7.75 is satisfied by hypothesis.

7.7. Notes and further reading

279

We then need only to check that G-clusters in Y fulfil all the requirements of Theorem 7.75. Condition 3 is automatic. For Condition 1, one first notices that G since Γ(Zw , OZw ) ' C[G] as G-vector spaces, then HomG Y (OZw , OZw ) ' C[G] ' C. If w1 and w2 are distinct points of W , the G-clusters Zw1 and Zw2 are different, and then any equivariant morphism ϕ : Zw1 → Zw2 must vanish at the points of Zx1 which are not in Zw2 ; since the equivariant sections are constant, this forces ϕ = 0. For Condition 2, we have to see that OZw ⊗ ωY ' OZw as G-linearized sheaves for every G-cluster OZw , and this follows because ωY is trivial as a Glinearized sheaf on an open neighborhood of every orbit of G. To finish the proof, we have only to see that τ : W → Y /G is crepant. The proof is similar to the proof that f + is crepant in Theorem 7.72. For each point OZ ,{e}×G gives an equivalence of categories between x ∈ Y /G, the equivalence ΦW→ Y the full subcategory Dτ −1 (x) (W ) ⊂ Db (W ) of objects topologically supported on the fiber τ −1 (x) and the full subcategory DπG−1 (x) (Y ) ⊂ DG,b (Y ) of G-linearized complexes topologically supported on the fiber π −1 (x) of the quotient morphism π : Y → Y /G. Since ωY is trivial as a G-linearized sheaf on an open neighborhood of π −1 (x), the triangulated category DπG−1 (x) (Y ) has trivial Serre functor, and then Dτ −1 (x) (W ) also has trivial Serre functor. Proposition 7.8 implies that τ is crepant.  Remark 7.77. When dim Y ≤ 3, the condition on the dimension of the fiber product in the statement of this theorem holds true because the dimension of the exceptional locus of W → Y /G is less than or equal to 2. 4

7.7

Notes and further reading

Fourier-Mukai partners of K3 surfaces. If X and Y are K3 surfaces which are Fourier-Mukai partners, the respective Hilbert schemes of n points, X [n] and Y [n] , are Fourier-Mukai partners as well [250]. It is interesting to note the existence of Hilbert schemes X [n] and Y [n] , where X and Y are surfaces, which are FourierMukai partners but are not birational. An example is given by Markman [210]. Theorem 7.26 says that a Fourier-Mukai partner Y of a K3 surface X is a moduli space of stable sheaves on X. Huybrechts [154] proved a stronger statement, namely, that Y is isomorphic to a moduli space of µ-stable sheaves. Fourier-Mukai partners of Abelian surfaces. Let Kum(A) be the Kummer surface associated to the Abelian surface A. One defines the set K(Kum(A)) of isomorphism classes of Abelian surfaces B such that Kum(B) ' Kum(A). Hosono, Lian, Oguiso and Yau [149] proved that FM(A) = K(Kum(A)). Fourier-Mukai partners of bielliptic surfaces. Bridgeland and Maciocia have proved in this case a stronger statement than Theorem 7.17, namely, that a bielliptic

280

Chapter 7. Fourier-Mukai partners and birational geometry

surface has no other partner than itself [70]. Derived categories and birational geometry. Kawamata, using a different approach, has proved Bridgeland’s Theorem 7.44 for orbifolds [175]. Other significant contributions are due to Chen [87] and Van den Bergh [289]. For a survey, we refer to Bridgeland’s ICM address [63] and to Rouquier [260]. Derived categories, integral functors and string theory. In recent years, integral functors have found several applications in string theory. The most notable example is Kontsevich’s homological mirror symmetry conjecture [188]. This predicts an equivalence between the bounded derived category of coherent sheaves on a CalabiYau manifold and the Fukaya category of the mirror dual manifold [115, 116, 117]. The conjecture implies a correspondence between self-equivalences of the derived category and certain symplectic self-equivalences of the mirror manifold. Evidence in this direction has been provided in [96, 6, 148], among others.

Appendix A

Derived and triangulated categories by Fernando Sancho

A.1

Basic notions

We assume that the reader is familiar with the basics of category theory, as expounded for instance in Mac Lane’s standard textbook [201]. Nevertheless, mainly in order to fix notation and terminology, we recall here a few notions. A category C consists of the following set of data: 1. a class Ob(C), whose elements are called the objects of C; 2. for each ordered pair of objects A, B ∈ Ob(C), a class HomC (A, B), whose elements are called morphisms from A to B and denoted f : A → B; 3. for each ordered triple of objects A, B, C, a map HomC (B, C) × HomC (A, B) → HomC (A, C) (f, g) 7→ f ◦ g , called the composition map. One requires that the composition is associative and that for any object A there exists the identity morphism IdA ∈ HomC (A, A), satisfying f = IdA ◦ f for any f ∈ HomC (B, A) and g = g ◦ IdA for any g ∈ HomC (A, B).

282

Appendix A. Derived and triangulated categories

A category is said to be small if the classes of both its objects and its morphisms are sets. A category that is not small is said to be large. A category C is locally small if for any pair of objects A and B of C the class HomC (A, B) is a set. Many of the categories we will consider in this book (the categories of sets, groups, rings, modules over a ring, sheaves on a topological spaces, etc.) are locally small. Given two categories C and D, a functor F : C → D consists of the following set of data: 1. a map Ob(C) → Ob(D), A 7→ F (A); 2. a map HomC (A, B) → HomD (F (A), F (B)) for any pair A, B ∈ Ob(C), such that F (f ◦ g) = F (f ) ◦ F (g). Let F and G be two functors from C to D. A morphism of functors θ : F → G is a family of morphisms θA ∈ HomD (F (A), G(A)), one for each object A of C, such that the diagram F (A)

F (f )

θA

 G(A)

/ F (B) θB

G(f )

 / G(B)

commutes for any morphism f ∈ HomC (A, B). A morphism of functors θ : F → G is an isomorphism if and only if θA is an isomorphism for any object A. A functor F : C → D is said to be an equivalence of categories if there exists a functor G : D → C such that the composition F G is isomorphic to the identity functor IdC and the composition GF is isomorphic to the identity functor IdD . The functor G is said to be a quasi-inverse to F . Given functors F : C → D and G : D → C, one says that G is left adjoint to F (and that F is right adjoint to G) if there are functorial isomorphisms HomD (B, F (A)) ' HomC (G(B), A) for all objects A in C and B in D. If F is an equivalence of categories, its quasi-inverse is both right and left adjoint to F . Indeed, a functor F that admits a left adjoint G and a right adjoint H is an equivalence of categories if and only if G and H are isomorphic. If C is a category, the opposite category C◦ is the category whose objects are the same as those of C and and whose morphisms are HomC◦ (A, B) = HomC (B, A) . A functor F : C → D◦ is often called a contravariant functor from C to D. A fundamental notion is that of representable functor.

A.2. Additive and Abelian categories

283

Definition A.1. A contravariant functor F : C → Sets (where Sets is the category of sets) is said to be representable if there exists an object A in C such that F is 4 isomorphic to the homomorphism functor HomC (•, A). We recall that the homomorphism functor hA = HomC (•, A) is defined by letting hA (C) = HomC (C, A) for any object C, while for any morphism η : C → D one defines hA (η) : HomC (D, A) → HomC (C, A) as the composition with η, i.e., hA (η)(ϕ) = ϕ ◦ η. Lemma A.2 (Yoneda’s lemma). Any isomorphism of functors Φ : HomC (•, A) ≡ HomC (•, B) is induced by an isomorphism A ' B. Proof. For every object C we write ΦC : hA (C) → hB (C) for the induced map. We have a morphism ϕ = ΦA (IdA ) : A → B, and for every morphism η : C → A a commutative diagram ΦA / hB (A) hA (A) hA (η)

 hA (C)

ΦC

hB (η)

 / hB (C)

.

so that ΦA (η) = hB (η)(ϕ) = ϕ ◦ η. This proves that Φ is induced by ϕ : A → B. A similar argument proves that the inverse Φ−1 is induced by a morphism ψ : B → A. By functoriality, ϕ◦ψ induces the identity hB → hB and ϕ◦ψ induces the identity hA → hA , so that ψ and φ are isomorphisms, and one is the inverse of the other.  As a consequence of Yoneda’s Lemma A.2, a representable contravariant functor F is represented by an object which is unique up to isomorphisms.

A.2

Additive and Abelian categories

The most basic environment suitable to develop the machinery of homological algebra is provided by additive categories. Definition A.3. A category C is additive if the following conditions are satisfied: 1. for any A, B ∈ Ob(C), there is an Abelian group structure on HomC (A, B) such that all the composition maps HomC (A, B) × HomC (B, C) → HomC (A, C) are bilinear;

284

Appendix A. Derived and triangulated categories

2. there exists a zero object 0 such that HomC (0, 0) is the trivial group; 3. for any A, B ∈ Ob(C), there are an object Z ∈ Ob(C) and morphisms i

pA

i

A A −→ Z ←B− B ,

pB

A ←−− Z −−→ B

such that pA ◦ iA = IdA , pB ◦ iB = IdB , pA ◦ iB = 0, pB ◦ iA = 0 and pA ◦ iA + pB ◦ iB = IdZ (hence, Z is the direct sum and the direct product of A and B). 4 A functor F : C → D between two additive categories is said to be additive if for any pair of morphisms f, g : A → B one has F (f + g) = F (f ) + F (g). The kernel of a morphism f : A → B is a morphism i : K → A such that for any object M , the sequence of Abelian groups 0 → Hom(M, K) → Hom(M, A) → Hom(M, B) is exact. It easy to see that the kernel of a morphism, if it exists, is unique up to a unique isomorphism. Analogously, the cokernel of a morphism f : A → B is a morphism p : B → C such that for any for any object M , the sequence of Abelian groups 0 → Hom(C, M ) → Hom(B, M ) → Hom(A, M ) is exact. Assume that the morphism f : A → B has a cokernel p : B → coker f and that p has a kernel. That kernel is called the image of f and is denoted by im f ; one has natural morphisms p

u

im f = ker p − →B− → coker f . Suppose that f has a kernel i : ker f → A and that i has a cokernel. That cokernel is called the coimage of f and is denoted by coim f ; one has natural morphisms q

i

− A− → coker i = coim f . ker f → Theorem A.4. Let C be an additive category. Assume that f : A → B is a morphism having a kernel, a cokernel, an image and a coimage. There exists a unique morphism f¯: coim f → im f such that the diagram A

f

q

 coim f is commutative.

/B O u

f¯

/ im f

A.2. Additive and Abelian categories

285

Definition A.5. Let k be a field. An additive category C is said to be k-linear if for any objects A, B, the Abelian groups HomC (A, B) are k-vector spaces and the composition maps are k-bilinear. 4 An additive functor F : C → D between two k-linear categories is said to be klinear if for any morphism f : A → B and any scalar λ ∈ k one has F (λf ) = λF (f ). In C is a k-linear category, a contravariant k-linear functor F : C → Vectk (where Vectk is the category of k-vector spaces) is said to be representable if there exists an object A in C such that F is isomorphic, as a k-linear functor, to the homomorphism functor HomC (•, A). We also assume that all additive functors F between two (k-linear) categories A and B are k-linear. Definition A.6. An additive category is said to be Abelian if the following additional conditions are satisfied: 1. any morphism has kernel and cokernel (consequently, any morphism has image and coimage); 2. for any morphism f : A → B, the morphism f¯: coim f → im f defined in Theorem A.4 is an isomorphism. 4 Remark A.7. A category is additive (resp. Abelian) if and only if its opposite category is additive (resp. Abelian). 4 Example A.8.

• The category of R-modules, where R is any ring, is Abelian.

• The category of free R-modules, where R is any commutative ring, is additive but not Abelian. • The category Mod(X) of sheaves of OX -modules on a ringed space (X, OX ) is Abelian. This category will appear mostly in the case when X is an algebraic variety, i.e., a separated scheme of finite type over a field. Another case is the category of sheaves of Abelian groups on a topological space. • The category Qco(X) of quasi-coherent sheaves of OX -modules on an algebraic variety X is Abelian. • The category Coh(X) of coherent sheaves of OX -modules on an algebraic variety X is also Abelian. 4 fn−2

fn−1

fn

fn+1

Definition A.9. A sequence of morphisms . . . −−−→ An−1 −−−→ An −→ An+1 −−−→ . . . in an Abelian category A is said to be exact at An if ker fn = im fn−1 . The sequence is exact if it is exact at every term. 4

286

Appendix A. Derived and triangulated categories

It is straightforward to show that a sequence 0 → A → B → C is exact if and only if the sequence of Abelian groups 0 → Hom(M, A) → Hom(M, B) → Hom(M, C) is exact for any object M ∈ Ob(A). Let F : A → B be an additive functor between two Abelian categories. We say that F is • left exact if for any exact sequence 0 → A0 → A → A00 → 0 in A, the sequence 0 → F (A0 ) → F (A) → F (A00 ) is exact in B; • right exact if for any exact sequence 0 → A0 → A → A00 → 0 in A, the sequence F (A0 ) → F (A) → F (A00 ) → 0 is exact in B; • exact if it is both right and left exact. Though we shall not go into details, it is worth mentioning that we can define Artinian, Noetherian and finite-length Abelian categories in terms of monomorphisms, mimicking what one does for modules. However, to do so one does not need all the conditions that Abelian categories fulfil. We now give for future use the more general notion of quasi-Abelian category. Let C be an additive category. A morphism f : A → B in D having a kernel, a cokernel, an image and a coimage is strict if the natural morphism f¯: coim f → im f given by Theorem A.4 is an isomorphism. Note that an Abelian category is precisely an additive category with kernels and cokernels (i.e., every morphism has kernel and cokernel) in which all morphisms are strict. If C is an additive category with kernels and cokernels, it also has pull-backs and push-outs. The pull-back of a morphism f : A → B by a morphism g : C → B is the induced morphism g ∗ (f ) : ker(f + g) → C where f + g : A ⊕ C → B is the sum morphism; the push-out of f : A → B by a morphism h : A → C is the induced morphism h∗ (f ) : C → im(f × h) where f × h : A → B × C is the product morphism. Definition A.10. A quasi-Abelian category is an additive category with kernels and cokernels such that every pull-back of a strict epimorphism is a strict epimorphism, and every push-out of a strict monomorphism is a strict monomorphism. 4 An example is the category of torsion-free sheaves on a smooth projective variety. A strict short exact sequence in a quasi-Abelian category is a sequence i

j

− B− →C→0 0→A→ in which i is the kernel of j and j is the cokernel of i (in particular, i is a strict monomorphism and j a strict epimorphism).

A.3. Categories of complexes

287

Formulating the corresponding notions in terms of strict monomorphisms, it makes perfect sense to speak of Artinian, Noetherian and finite-length quasiAbelian categories, as one does with Abelian categories.

A.3

Categories of complexes

Let A be an additive category. A complex (K• , dK• ) in A is a sequence dn−1 •

dn •

K K −→ Kn −− → Kn+1 → · · · · · · → Kn−1 −−

where the K n are objects in A and the morphisms dnK• are morphisms in A satisn fying the condition dn+1 K• ◦ dK• = 0 for all n ∈ Z. We say that dK• is the differential of the complex K• . Definition A.11. The category of complexes C(A) is the category whose objects are complexes (K• , dK• ) in A and whose morphisms f : (K• , dK• ) → (L• , dL• ) are collections of morphisms f n : Kn → Ln , n ∈ Z, in A such that the diagrams ···

··· are commutative.

/ Kn−1 

dn−1

f n−1

/ Ln−1

n−1

d

/ Kn

dn

fn

 / Ln

d

n

/ Kn+1 

dn+1

/ ···

f n+1

/ Ln+1

dn+1

/ ··· 4

Here and sometimes later on, when no ambiguity can arise, we omit the subscripts in the symbols of the differentials. Given two complexes K• and L• , their direct sum K• ⊕L• is defined by setting (K ⊕ L)n = Kn ⊕ Ln and dnK• ⊕L• = dnK• ⊕ dnL• . If A has kernels and cokernels, and f : K• → L• is a morphism of complexes, its kernel is the complex ker f , such that (ker f )n = ker f n , endowed with the differential induced by dK• . In an analogous fashion one defines the cokernel of f . Hence, the following result holds true. Proposition A.12. The category of complexes C(A) of an Abelian (resp. additive) category A is Abelian (resp. additive). Remark A.13. The category A can be considered as a faithful subcategory of C(A). Indeed, any object A of A defines the complex A0 = A and An = 0 for n 6= 0, having the zero morphisms as differentials. 4 Remark A.14. Assume that the Abelian category A has arbitrary direct sums, i.e., direct sums labeled by arbitrary sets (this is the case, for example, for the category of modules over a ring). Then, a complex K• in A can be regarded as a

288

Appendix A. Derived and triangulated categories

L L L graded object n Kn and the differential dK• as a morphism n Kn → n Kn of degree 1. It follows that the morphisms of complexes from K• to L• form a L L subgroup of HomA ( n Kn , n Ln ). Hence, the category of complexes C(A) is itself an Abelian category with arbitrary direct sums. 4 Let K• and L• be complexes; for each n ∈ Z, we set Y HomA (Ki , Li+n ) . Hom(K• , L• )n = i

These groups form a complex of Abelian groups M Hom(K• , L• )n Hom• (K• , L• ) =

(A.1)

n

endowed with the differential given by dn : Hom(K• , L• )n → Hom(K• , L• )n+1 i n+1 i+1 f i 7→ di+n f ◦ diK• , . L• ◦ f + (−1)

(A.2)

Definition A.15. For any integer n, one defines the shift functor [n] : C(A) → C(A) by letting K[n]p = Kp+n with the differential dK• [n] = (−1)n dK• , while a morphism of complexes f : K• → L• is mapped to the morphism f [n] : K• [n] → L• [n] given 4 by f [n]p = f p+n . The shift functor turns out to be additive and exact. Sometimes we shall denote by τ the functor [1]. One has canonical isomorphisms Hom• (K• , L• [n]) ' Hom• (K• , L• )[n] ' Hom• (K• [−n], L• ) . Definition A.16. The n-th cohomology object of a complex K• is the object Hn (K• ) = ker dn / im dn−1 . We say that Z n (K• ) = ker dn is the n-cycle object of K• , and B n (K• ) = im dn−1 is the n-boundary object of K• . 4 The cohomology objects of a complex may be assembled into a complex H(K ) whose differentials are all set to zero. •

A morphism of complexes f : K• → L• induces morphisms between cycles and the boundaries, and passes to cohomology yielding morphisms Hn (f ) : Hn (K• ) → Hn (L• ) , for every n. One has Hn (K• [m]) ' Hn+m (K• ) and Hn (f [m]) ' Hn+m (f ).

A.3. Categories of complexes

289

We say that a complex K• is acyclic or exact if H(K• ) = 0; we also say that a morphism of complexes f : K• → L• is a quasi-isomorphism if H(f ) : H(K• ) → H(L• ) is an isomorphism. The composition of two quasi-isomorphisms is a quasiisomorphism. We now introduce the important notion of homotopy equivalence, which will allow us to build a new category — the homotopy category — out of the category of complexes. Let f : K• → L• a morphism of complexes. We say that f is homotopic to zero if there is a collection of morphisms hn : Kn → Ln−1 such that f n = n • hn+1 ◦ dnK• + dn−1 L• ◦ h for every n. A complex K is said to be homotopic to zero if its identity morphism is homotopic to zero. Finally, two morphisms f, g : K• → L• are said to be homotopic if f − g is homotopic to zero. It is clear that the sum of two morphisms homotopic to zero is homotopic to zero. Moreover, the composition f ◦ g is homotopic to zero whenever either f or g is homotopic to zero. Let us denote by Ht(K• , L• ) the subgroup of the morphisms of complexes f : K• → L• which are homotopic to zero. Definition A.17. The homotopy category K(A) is the category whose objects are the objects of C(A) and whose morphisms are HomK(A) (K• , L• ) = HomC(A) (K• , L• )/ Ht(K• , L• ) . 4 From Equation (A.2) we see that the n-cycles of the complex of homomorphisms Hom• (K• , L• ) coincide with the morphisms of complexes K• → L• [n], while the n-boundaries coincide with morphisms homotopic to zero. Therefore, Hn (Hom• (K• , L• )) = HomK(A) (K• , L• [n]) .

(A.3)

A morphism of complexes f : K• → L• which is homotopic to zero induces in cohomology the zero morphism, H(f ) = 0; hence, two homotopic morphisms induce the same morphism in cohomology. In particular, if a complex K• is homotopic to zero, then it is acyclic. We now define the cone of a morphism; this notion comes from classical homotopy theory, and is a way of overcoming the fact that the homotopy category K(A) does not have kernels and cokernels. Definition A.18. The cone of a morphism of complexes f : K• → L• is the complex Cone(f ) such that Cone(f )n = Kn+1 ⊕ Ln and the differential is defined as   n+1 0 −dK• dnCone(f ) = f n+1 dnL• 4

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Although one has Cone(f )n = (K• [1])n ⊕ Ln for every n, Cone(f ) is not isomorphic as a complex with the direct sum K• [1] ⊕ L• , because the differential of the latter is the direct sum of the differentials of the summands. There are functorial morphisms β : Cone(f ) → K• [1],

α : L• → Cone(f )

(k, l) 7→ k ,

l 7→ (0, l) ,

and an exact sequence of complexes 0 → L• → Cone(f ) → K• [1] → 0. Let us consider the sequence f

β

α

→ L• − → Cone(f ) − → K• [1] . K• −

(A.4)

The composition α ◦ f is homotopic to zero. If we consider the sequence (A.4) in the homotopy category K(A), the composition of any two consecutive morphisms is zero. As we shall see in Section A.4.3, the sequence (A.4) will be the model for exact triangles in the homotopy and derived categories. The following property is readily checked: Proposition A.19. For every integer n there is an exact sequence of cohomology groups Hn (f )

Hn (α)

Hn (β)

Hn (K• ) −−−−→ Hn (L• ) −−−−→ Hn (Cone(f )) −−−−→ Hn (K• [1]) ' Hn+1 (K• ) . Gathering all these exact sequences together we have the so-called cohomology long exact sequence: Hn−1 (β)

Hn (f )

Hn (α)

. . . −−−−−−→ Hn (K• ) −−−−→ Hn (L• ) −−−−→ Hn (β)

Hn (Cone(f )) −−−−→ Hn+1 (K• ) . . .

(A.5)

Proposition A.19 tells us that the functors Hn : K(A) → A are cohomological, in the following sense: if A and B are Abelian categories, an additive functor f β α → L• − → Cone(f ) − → F : K(A) → B is cohomological if for every sequence K• − F (f )

α

F (β)

→ F (Cone(f )) −−−→ F (K• )[1] is exact. K• [1] the sequence F (K• ) −−−→ F (L• ) − An important consequence of the cohomology long exact sequence (A.5) is the following: Corollary A.20. A morphism of complexes f : K• → L• is a quasi-isomorphism if and only if Cone(f ) is acyclic. Cones are good substitutes for exact sequences of complexes, as the following proposition shows (we omit the simple proof).

A.3. Categories of complexes

291 f

g

→ L• − → N • → 0 be an exact sequence of complexes Proposition A.21. Let 0 → K• − • in C(A), and let γ : Cone(f ) → N be the morphism of complexes defined in degree n by Kn+1 ⊕ Ln → N n (an+1 , bn ) 7→ g(bn ) . The morphism γ is a quasi-isomorphism. Combining this with the cohomology long exact sequence (A.5), we obtain the more usual cohomology sequence: there exist functorial morphisms δ n : H n (N • ) → H n+1 (L• ) and an exact sequence δ n−1

δn

· · · −−−→ H n (L• ) → H n (M• ) → H n (N • ) −→ H n+1 (L• ) → δ n+1

→ H n+1 (M• ) → H n+1 (N • ) −−−→ · · · Truncated complexes Let K• be a complex. The truncations K• ≤n and K• ≥n are defined as the complexes · · · → Kn−2 → Kn−1 → ker dn → 0 · · · · · · → 0 → coker dn−1 → Kn+1 → Kn+2 → · · · , respectively. One has natural morphisms of complexes K• ≤n → K• ,

K• → K• ≥n ,

and for any m ≤ n, K• ≤m → K• ≤n ,

K• ≥m → K• ≥n .

A morphism of complexes K• → N • induces morphisms between the corresponding truncations K• ≤n → N • ≤n and K• ≥n → N • ≥n . Remark A.22. It is straightforward to check that ( ( Hj (K• ) for j ≤ n Hj (K• ) for j ≥ n j • j • H (K ≥n ) = H (K ≤n ) = 0 for j > n 0 for j < n Thus, the truncation functors K• → K• ≤n and K• → K• ≥n preserve quasiisomorphisms. Moreover, if we consider the natural morphism i : K• ≤n−1 → K• ≤n , 4 the induced morphism Cone(i) → Hn (K• )[−n] is a quasi-isomorphism.

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A.3.1

Double complexes

A double complex K•• in an Abelian category A is a diagram .. .O ···

.. .O

/ Kp,q+1 O

d1

/ Kp+1,q+1 O

/ Kp,q O

(A.6)

d2

d2

···

/ ...

d1

/ Kp+1,q O

/ ...

.. .

.. . where the Kp,q are objects of A, and one has d21 = 0 ,

f

p,q

d22 = 0

and d1 ◦ d2 = d2 ◦ d1 .

A morphism of double complexes f : K•• → N •• is a collection of morphisms : Kp,q → N p,q commuting with the differentials d1 and d2 .

A double complex K•• can be thought of as a complex of complexes in two different ways. If we regard the columns Kp,• as complexes with differential d2 , we get a complex d1 d1 d1 K• I = . . . −→ Kp,• −→ Kp+1,• −→ ... in the category C(A). Analogously, if we consider the rows K•,q as complexes with differential d1 , we get a complex K• II in C(A) with differential d2 . For any integer n the cohomology Hn (K• I ) is a complex with respect to d2 , and Hn (K• II ) is a complex with respect to d1 . We shall use the notation Hdn1 (K•• ) = Hn (K• I ) and Hdn2 (K•• ) = Hn (K• II ). In many cases, it is useful to associate a complex with a double complex L K•• . Let us assume that for any n the direct sum p+q=n Kp,q exists. This is the case for example when A admits infinite direct sums or when all antidiagonals in diagram (A.6) have only a finite number of nonzero terms. The simple complex S • (K•• ) associated to K•• consists of the objects M Kp,q S n (K•• ) = p+q=n

and the differentials dn : S n (K•• ) → S n+1 (K•• ) such that dn = d1 + (−1)p d2 over Kp,q .

A.3. Categories of complexes

293

Example A.23. We shall need to consider categories which admit tensor products. For an introduction to such categories we refer the reader, for instance, to Chapter VII of Mac Lane’s book [201]. Actually, for the concrete categories we shall deal with, the tensor product structure is quite evident. So, let A be an Abelian category which admits tensor products, and, in addition, infinite direct sums. If K• and L• are two complexes Kp ⊗N q defines a double complex with differentials d1 = dK• ⊗1, d2 = 1 ⊗ dL• . We can define the tensor product K• ⊗ L• of the complexes K• and L• as the simple complex associated with this double complex. That is, one sets M (Kp ⊗ Lq ) (K• ⊗ L• )n = p+q=n

equipping this complex with the differential which acts as dK• ⊗ Id + (−1)p Id ⊗ dL• over Kp ⊗ Lq . If A has no infinite direct sums (as the category Coh(X)), then K• ⊗ L• is defined only if for every n there are only a finite number of summands L in p+q=n (Kp ⊗ Lq ). The tensor product is compatible with the shift functor, i.e., one has canonical isomorphisms K• [n] ⊗ L• ' (K• ⊗ L• )[n] ' K• ⊗ L• [n] . 4 Example A.24. The double complex of homomorphisms of two complexes K and L N • is defined by the objects p,q Hom(K−q , N p ) with the differentials •

d1 : Hom(K−q , N p ) → Hom(K−q , N p+1 ), −q

−q−1

f 7→ dN • ◦ f

, N ) → Hom(K , N ), f 7→ (−1)q+1 f ◦ dK• . L When the direct sum p+q=n Hom(K−q , N p ) is isomorphic with the direct prodQ uct p+q=n Hom(K−q , N p ), the associated simple complex is isomorphic with the complex Hom• (K• , N • ) as defined in (A.1). This happens, for instance, when K• is bounded above and N • is bounded below. 4 • • Example A.25. A morphism of complexes f : K → L may be regarded as a double complex with two columns. One sets K−1,• = K• , K0,• = L• , while the other columns are zero. The horizontal differential d1 : K−1,• → K0,• is f , and the vertical differential is given by the differentials of K• and L• . The simple complex associated with this double complex is the cone of f . 4 d2 : Hom(K

p

p

Truncated double complexes Let K•• be a double complex. The truncations K≤n,• and K≥n,• are defined as the double complexes · · · → Kn−2,• → Kn−1,• → ker[d1 : Kn,• → Kn+1,• ] → 0 · · · · · · → 0 → coker[d1 : Kn−1,• → Kn,• ] → Kn+1,• → Kn+2,• → · · · ,

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respectively. In a similar way by replacing rows by columns, one defines the truncated double complexes K•,≤n and K•,≥n . Proposition A.21 and the content of Example A.25 imply the following result. Proposition A.26. Consider the natural morphism i : S • (K≤n−1,• ) → S • (K≤n,• ). The induced morphism Cone(i) → Hdn1 (K•• )[−n] is a quasi-isomorphism. Therefore, one has a long exact sequence · · · → Hi (S • (K≤n−1,• )) → Hi (S • (K≤n,• )) → → Hdi 2 (Hdn1 (K•• )[−n]) → Hi+1 (S • (K≤n−1,• )) → · · · An analogous result holds true for the truncated complex K•,≤n . We say that a double complex K•• has bounded below antidiagonals (respectively, bounded above antidiagonals) if for each n one has Kp,n−p = 0 for p  0 (respectively, Kn−q,q = 0 for q  0). Proposition A.27. Let f : K•• → N •• be a morphism of double complexes. Assume that both K•• and N •• have bounded below antidiagonals. If the induced morphism Hd2 (Hd1 (K•• )) → Hd2 (Hd1 (N •• )) is an isomorphism, the morphism S • (K•• ) → S • (N •• ) between the associated simple complexes is a quasi-isomorphism. Analogously, assume that K•• and N •• have bounded above antidiagonals. If the induced morphism Hd1 (Hd2 (K•• )) → Hd1 (Hd2 (N •• )) is an isomorphism, the morphism S • (K•• ) → S • (N •• ) between the associated simple complexes is a quasi-isomorphism. Proof. Assume first that Kp,• = N p,• = 0 for p  0. The complexes S(K≤n−1,• ) and S(N ≤n−1,• ) vanish for n  0, so the result holds for n small enough, and we may use induction on n. We have a morphism of exact sequences ...

/ Hi (S • (K≤n−1,• ))

/ Hi (S • (K≤n,• ))

/ Hi (Hn (K•• )[−n]) d2 d1

/ ...

...

 / Hi (S • (N ≤n−1,• ))

 / Hi (S • (N ≤n,• ))

 / Hi (Hn (N •• )[−n]) d2 d1

/ ... .

Since by assumption the vertical morphism between the cohomology complexes is a quasi-isomorphism for every n, by Proposition A.26 the morphism S • (K≤n,• ) → S • (N ≤n,• ) is a quasi-isomorphism for all n. This proves that S • (K•• ) → S • (N •• ) is

A.4. Derived categories

295

a quasi-isomorphism as well, since for each k one has Hk (S • (K•• )) ' Hk (S • (K≤n,• )) and Hk (S • (N • )) ' Hk (S • (N ≤n,• )) when n is big enough. To deal with the general case of double complexes with bounded below antidiagonals, it is enough to apply the previous argument to the induced morphism K≥n,• → N ≥n,• . Indeed, for each k one has Hk (S • (K•• )) ' Hk (S • (K≥n,• )) and Hk (S • (N • )) ' Hk (S • (N ≥n,• )) provided n is small enough. The case of double complexes with bounded above antidiagonals is completely analogous.  Corollary A.28. Let K•• be a double complex with bounded below antidiagonals (respectively, bounded above antidiagonals). If there exists n such that Hdi 1 (K•• ) = 0 for i 6= n (respectively, Hdi 2 (K•• ) = 0 for i 6= n), then Hi+n (S • (K•• )) ' Hdi 2 (Hdn1 (K•• )) (respectively, Hi+n (S • (K•• )) ' Hdi 1 (Hdn2 (K•• ))). Proof. By Proposition A.27, the morphisms Hdn1 (K•• )[−n] ← S • (K≤n,• ) → S • (K•• ) are quasi-isomorphisms, so that Hi (S • (K•• )) ' Hi (Hdn1 (K•• )[−n]) ' Hdi−n (Hdn1 (K•• )) . 2 

A.4

Derived categories

The basic idea behind the introduction of the derived category is to replace quasi-isomorphism with isomorphisms. The first step consists in identifying homotopic morphisms, thus moving from the category of complexes C(A) to the homotopy category K(A). A second step consists in “localizing” by (classes of) quasi-isomorphisms. This localization is a fractional calculus for categories if we just think of the composition of morphisms as a product. Recall that if one has a ring A and we want to make the elements s in a part S of A invertible, so that a fraction a/s makes sense, this can be done if S is a multiplicative system, namely, if it contains the unity and is closed under products. Then one can define the localized ring S −1 A whose elements are equivalence classes a/s of pairs (a, s) ∈ A × S where (a, s) ∼ (a0 , s0 ) (or a/s = a0 /s0 ) if there is t ∈ S such that t(as0 − a0 s) = 0. Any element s ∈ S becomes invertible in the fractions ring S −1 A because s/1 · 1/s = 1.

A.4.1

The derived category of an Abelian category

The localization process can be also done for morphisms of complexes, since quasiisomorphisms verify the conditions for being a nice set of denominators (or a multiplicative system as before), namely, the identity is a quasi-isomorphism and

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the composition of two quasi-isomorphisms is a quasi-isomorphism. We now take a fraction as a diagram of morphisms of complexes R• { CCC f { CC {{ CC {{ C! }{ { φ

K•

L•

where φ is a quasi-isomorphism. We denote such a diagram by f /φ. A second diagram g/ψ S• B BB g { ψ {{ BB { BB { { B! { }{ • L• K is said to be equivalent to the former, if there are quasi-isomorphisms R• ← T • → S • such that the diagram

K

T •C CC { { CC {{ CC { { C! { } { R• VVVVV h S• B BB g VVhVhVhhhhh { φ {{ h V BB VVVV { hhh h { h V h V { hhhh ψ VVVV BBB f { VVV* ! }{ hhh • th

L•

is commutative in K(A). One can prove that equivalence of fractions is actually an equivalence relation using the following result. Lemma A.29. Given a diagram R• g

M•

f

 / N• g0

f0

in C(A), there are morphisms of complexes M• ←− Z • −→ R• such that the diagram Z•

f0

g0

 M•

/ R• g

f

 / N•

is commutative in K(A). Moreover, f 0 (respectively, g 0 ) is a quasi-isomorphism if and only if f (respectively, g) is so.

A.4. Derived categories

297 f −g

Proof. Let us consider the morphism M• ⊕ R• −−−→ N • and set Z • = Cone(f − g)[−1]. The natural morphism Z • → M• ⊕ R• induces morphisms g 0 : Z • → M• and f 0 : Z • → R• . The natural projection hn : Z n ' Mn ⊕ Rn ⊕ N n−1 → N n−1 defines a homotopy h between g ◦ f 0 and f ◦ g 0 . Finally, the commutative diagram of exact sequences 0

/ M•

0

 / N•

/ M• ⊕ R•

/ R•

f −g

f Id

 / N•

 /0

/0

0

induces an exact sequence of complexes f0

0 → Cone(f )[−1] → Z • −→ R• → 0 . Hence, f 0 is a quasi-isomorphism if and only if Cone(f )[−1] is acyclic, and this is equivalent by Corollary A.20 to f being a quasi-isomorphism. An analogous  argument holds in the case of g and g 0 . Definition A.30. The derived category D(A) of A is the category whose objects are the objects of K(A) (that is, they are complexes of objects of A), and whose morphisms are equivalence classes [f /φ] of diagrams. 4 To make full sense of this definition we need to specify how to compose morphisms. This can be done thanks to Lemma A.29. Given two morphisms [f /φ] and [g/ψ] in D(A), corresponding to diagrams

K•

R• B BB | | φ | BBf | BB | | B! | }|

L•

| ψ || | | | | ~|

S• E EE EEg EE E" M• ,

their composition is defined through the diagram { ψ {{ { { { }{{ • R { CCC f φ {{ CC { CC {{ C! { }{ 0

K•

T •C CC f 0 CC CC C!

L•

{ ψ {{ {{ { { }{

S• E EE g EE EE E" M• .

Hence, we set [g/ψ] ◦ [f /φ] = [(g ◦ f 0 )/(φ ◦ ψ 0 )], which makes sense because the above construction is independent of the representatives.

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Appendix A. Derived and triangulated categories

A result similar to Lemma A.29 holds true by inverting all the arrows. A diagram / N• M•  R• in C(A) can be completed to a diagram M•

/ N•

 R•

 / Z•

which is commutative in K(A). Moreover, each arrow of this diagram is a quasiisomorphism if and only if its parallel arrow is so. Putting altogether, a morphism from K• to L• in the derived category is also defined by an equivalence class [ψ\g] of diagrams L• K• C CC g { { ψ { CC CC {{ C! }{{{ R• where ψ a quasi-isomorphism. The composition is defined in a completely analogous way. By a slight abuse of notation, we shall simply denote by f /φ : K• → L• the morphism in the derived category defined by the equivalence class [f /φ] and by φ\f : K• → L• the morphism defined by the equivalence class [φ\f ]. Useful descriptions of morphisms in the derived category are provided by the following isomorphisms: HomD(A) (K• , M• ) ' lim HomK(A) (K• , I • ) ' lim HomK(A) (P • , M• ) , −→ −→ • • I

(A.7)

P

where the first limit runs over all the quasi-isomorphisms of complexes M• → I • , while the second runs over all the quasi-isomorphisms of complexes P • → K• . A morphism f : K• → L• in C(A) defines a morphism f /IdK• : K• → L• in the derived category, which we shall denote simply by f . Moreover, if f is homotopic to zero, f /IdK• = 0. Hence, we have a functor K(A) → D(A). Proposition A.31. The derived category D(A) is an additive category and the functor K(A) → D(A) is additive. Proof. Two morphisms f /φ : K• → M• and g/ψ : L• → M• yield a morphism (f + g)/(φ ⊕ ψ) : K• ⊕ L• → M• . Conversely, a morphism K• ⊕ L• → M• in the

A.4. Derived categories

299

derived category, being represented by a morphism of complexes f : K• ⊕ L• → R• and a quasi-isomorphism φ : M• → R• , defines morphisms φ\fK• : K• → R• and φ\fL• : L• → M• in the derived category, where fK• : K• → R• and fL• : L• → R• are the morphisms induced by f . This shows that K• ⊕ L• is the direct sum in the derived category. In a similar way, one proves that K• ⊕ L• is the direct product in the derived category. The sum of two morphisms f /φ : K• → L• and g/ψ : K• → L• is defined as the composition of the diagonal morphism K• → K• ⊕ K• and the direct sum (f + g)/(φ ⊕ ψ) : K• ⊕ K• → L• . The additivity of the functor K(A) → D(A) is easily checked.



Definition A.32. Two complexes K• and L• are quasi-isomorphic if there are a 4 complex Z • and quasi-isomorphisms K• ← Z • → L• . It follows from Lemma A.29 that the notion of quasi-isomorphism induces an equivalence relation between complexes. One can also prove that two complexes K• and L• are quasi-isomorphic if and only if there are a complex Z • and quasiisomorphisms K• → Z • ← L• . We now have the result we were looking for: Proposition A.33. A morphism of complexes f : K• → L• is a quasi-isomorphism if and only if the induced morphism in the derived category is an isomorphism. Moreover, two complexes are quasi-isomorphic if and only if they are isomorphic in the derived category. Proof. If f is a quasi-isomorphism, we can define a morphism IdK• /f : L• → K• in the derived category, which is precisely the inverse of f . Conversely, if f is an isomorphism in the derived category and g/ψ is its inverse, then H(g/ψ) is the inverse of H(f ). The second statement follows straightforwardly.  The derived category can be also defined by means of a universal property. Proposition A.34. Let C be an additive category. An additive functor F : K(A) → C factors through an additive functor D(A) → C if and only if it maps quasiisomorphisms to isomorphisms. If B is an Abelian category, an additive functor G : K(A) → K(B) mapping quasi-isomorphisms into quasi-isomorphisms induces an additive functor G : D(A) → D(B) such that the diagram K(A)  D(A) is commutative.

G

/ K(B)

G

 / D(B)

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Appendix A. Derived and triangulated categories

Proof. If F factors through D(A), it maps quasi-isomorphisms to isomorphisms by Proposition A.33. Conversely, if F maps quasi-isomorphisms to isomorphisms, one defines a functor D(F ) : D(A) → C by letting D(F )(M• ) = F (M• ) for any object M• and D(F )(f /φ) = F (f ) ◦ F (φ)−1 for any morphism f /φ : K• → M• G in D(A). By applying this to the composition F : K(A) → K(B) − → D(B), the second statement follows.  Since a quasi-isomorphism of complexes induces a quasi-isomorphism between the truncated complexes (Remark A.22), the truncations functors pass to the derived category. Corollary A.35. There exist additive functors (−)≥n : D(A) → D(A) ,

A.4.2

(−)≤n : D(A) → D(A) .

Other derived categories

We can also build derived categories out of some subcategories of C(A), as long as all the operations we have done so far can be reproduced; namely, we need to construct the corresponding homotopy categories, and localize by quasi-isomorphims. In particular, we must define the cone of a morphism inside the new category (cf. for instance Lemma A.29, whose proof requires the cone construction). The most natural examples are the following. Example A.36. Let C+ (A) be the category of bounded below complexes in A, that is, complexes K• for which there is n0 such that Kn = 0 for all n ≤ n0 . We can define the homotopy category K + (A) and a derived category D+ (A) by following an analogous procedure as for arbitrary complexes. Due to Proposition A.34, the natural functor K + (A) → D(A) induces a functor γ : D+ (A) → D(A). This is fully faithful and its essential image is the faithful subcategory of D(A) consisting of complexes in A with bounded below cohomology. (The essential image of the functor γ is the subcategory of the objects which are isomorphic to objects of the form γ(K• ) for some K• in D+ (A)). In a similar way, the categories C− (A) of bounded above complexes (i.e., complexes for which there is n0 such that Kn = 0 for all n ≥ n0 ) and Cb (A)) of bounded (on both sides) complexes, give rise to derived categories D− (A) and Db (A), which are characterized as faithful subcategories of D(A) as above. 4 Example A.37. Let A0 be a thick Abelian subcategory of A, that is, any extension in A of two objects of A0 is also in A0 . If CA0 (A) is the category of complexes whose cohomology objects are in A0 , we can construct its homotopy category KA0 (A) and

A.4. Derived categories

301

its derived category DA0 (A). The functor KA0 (A) → D(A) induces by Proposition A.34 a functor DA0 (A) → D(A), which is fully faithful; its essential image is the subcategory of D(A) whose objects are all complexes with cohomology objects in 4 A0 . Example A.38. Combining the two procedures, we also have the homotopy cat+ − b egories KA 0 (A), KA0 (A) and KA0 (A) of complexes bounded below, above and on both sides, respectively, and whose cohomology objects are in the subcategory + − A0 of A. We also have the corresponding derived categories DA 0 (A), DA0 (A) and b 4 DA 0 (A). We have special notations for the Abelian categories we are most interested in. • If A is the category of modules over a commutative ring A, we use the notations D(A), D+ (A), D− (A), and Db (A). • If A = Mod(X) is the category of sheaves of OX -modules on an algebraic variety X and A0 = Qco(X) is the category of quasi-coherent sheaves of OX -modules on X, the derived category DA0 (A) of complexes of OX modules with quasi-coherent cohomology sheaves is denoted Dqc (Mod(X)). + − (Mod(X)), Dqc (Mod(X)) and In a similar way we have the categories Dqc b Dqc (Mod(X)). • If A = Mod(X) as above and A0 = Coh(X) is the category of coherent sheaves of OX -modules on X, the derived category DA0 (A) of complexes of OX -modules with coherent cohomology sheaves is denoted Dc (Mod(X)). One can also introduce the derived categories Dc+ (Mod(X)), Dc− (Mod(X)) and Dcb (Mod(X)). • If A = Qco(X) and A0 = Coh(X), the derived category DA0 (A) of complexes of quasi-coherent OX -modules with coherent cohomology sheaves will be denoted by D(X) for simplicity. Also, the corresponding derived categories of bounded below, bounded above and bounded complexes are denoted by D+ (X), D− (X), and Db (X). Let us write ? for any of the symbols +, −, b or for no symbol at all. Since the natural functors K ? (A0 ) → D(A) map quasi-isomorphisms to isomorphisms, they ? yield functors D? (A0 ) → DA 0 (A), which in general may fail to be equivalences of categories. However, if A0 has enough injectives in A, that is, if for every object K of A0 there is an immersion 0 → K → I where I is an object of A0 which is injective in A, one can prove that every bounded below complex K• in K + (A) whose cohomology objects are in A0 admits a quasi-isomorphism K• → I • ,

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where I • is a complex of objects of A0 which are injectives in A (cf. [139]). Then one has: Proposition A.39. If A0 has enough A-injectives, the functor D+ (A0 ) → D+ (A) + is fully faithful and induces an equivalence of categories D+ (A0 ) ' DA 0 (A).

Proof. Let K• and L• be objects of D+ (A0 ). A morphism K• → L• in D+ (A) can be represented by a diagram L• K• C CC g { { ψ { CC CC {{ C! }{{{ R• + where R• is in D+ (A) and ψ a quasi-isomorphism. Then R• is in DA 0 (A), so that, • • • as have seen, there is a quasi-isomorphism γ : R → I with I ∈ K + (A0 ). Thus, K• → L• is also represented by the diagram

K• B BB γ◦g BB BB B!

| γ◦ψ || | | | }||

L• ,

I•

which defines a morphism K• → L• in D+ (A0 ). It follows that HomD+ (A0 ) (K• , L• ) = + HomD+ (A) (K• , L• ). Moreover, since any object of DA 0 (A) is quasi-isomorphic to a + + 0 + 0  complex in D (A ), the essential image of D (A ) → D+ (A) is DA 0 (A). − • − 0 If for any complex K• in KA 0 (A) there is complex L in K (A ) and a quasi• • isomorphism L → K , proceeding as above (but using the representation of morphisms in the derived category given by Definition A.30), we may check that there is an equivalence of categories − D− (A0 ) ' DA 0 (A) ,

(A.8)

and similarly for bounded complexes. For the derived categories associated with an algebraic variety X, one has the following result. Proposition A.40. Let X be an algebraic variety. There is an equivalence of categories + D+ (Qco(X)) ' Dqc (Mod(X)) .

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303

Proof. This follows from Proposition A.39, as every quasi-coherent sheaf K on an algebraic variety can be embedded as a subsheaf of an injective quasi-coherent sheaf.  Corollary A.41. If X is an algebraic variety, one has an equivalence of categories Db (Coh(X)) ' Db (X) ' Dcb (Mod(X)) . Proof. In order to prove the first isomorphism, the only nontrivial thing to show is that the natural functor Db (Coh(X)) → Db (X) is essentially surjective, i.e., that every bounded complex E • of quasi-coherent sheaves with coherent cohomology sheaves is quasi-isomorphic to a bounded complex G • of coherent sheaves. A quasiisomorphism G • → E • , where G • is a bounded complex of coherent sheaves, is constructed by standard techniques (as in [229, Lemma II.1]) by exploiting the following fact: given a surjection of quasi-coherent sheaves E → H → 0 where H• is coherent, there exists a coherent subsheaf G of E such that the composition E → H is surjective as well (cf. [141, Exercise II.5.15]. The second isomorphism follows from Proposition A.40.  Definition A.42. A complex F • of OX -modules is of finite homological dimension, or a perfect complex, if it is locally quasi-isomorphic to a bounded complex of locally free sheaves of finite rank. That is, every point x has an open neighborhood U such that F • |U is quasi-isomorphic to a complex 0 → E n → · · · → E n+m → 0 4 of locally free OU -modules of finite rank. When X is smooth, every bounded complex of OX -modules with coherent cohomology sheaves is a perfect complex.

A.4.3

Triangles and triangulated categories

The notion of triangle is a replacement of that of exact sequence which is well suited for derived categories, which are not Abelian. Indeed, the derived category of an Abelian category has a natural structure of triangulated category. Definition A.43. A triangle in D(A) is a sequence of morphisms u

v

w

A• − → B• − → C• − → A• [1] . 4 A triangle is also written in the form A• aB

u

B w

B

B C•

/ B• | | || || v | ~|

,

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where the dashed arrow stands for a morphism C • → A• [1]. For any morphism of complexes f : K• → L• one has a triangle f

→ L• → Cone(f ) → K• [1] . K• − A morphism of triangles is a commutative diagram A•

u

/ B•

/ C•

g

f

 • A0

v

u

0



/ B0 •

v

0



w

/ A• [1]

h

/ C0•

w

0



f [1]

.

/ A0 • [1] f

→ L• → A triangle is called exact if it is isomorphic to a triangle of the type K• − • • i • • Cone(f ) → K [1]. For example, if 0 → K → − L → M → 0 is an exact sequence of complexes, then i

K• → − L• → M• → K• [1] is an exact triangle, where M• → K• [1] is the morphism in D(A) given by the diagram Cone(i) II II vv v II v v . II v v I$ {vv K• [1] M• A triangle in D(A) induces a long sequence in cohomology H i (u)

H i (v)

H i (w)

· · · → H i (A• ) −−−−→ H i (B • ) −−−−→ H i (C • ) −−−−→ H i+1 (u)

H i+1 (v)

H i+1 (w)

H i+1 (A• ) −−−−−→ H i+1 (B • ) −−−−−→ H i+1 (C • ) −−−−−→ · · · . If the triangle is exact, this long sequence is exact. Another interesting example of an exact triangle is constructed out of the truncation functors: since for any integer n the natural morphism i : K• ≤n−1 → K• ≤n induces a quasi-isomorphism Cone(i) → Hn (K• )[−n], we have an exact triangle in Db (A): K• ≤n−1 → K• ≤n → Hn (K• )[−n] → K• ≤n−1 [1] . We enumerate here the principal properties of exact triangles. Proposition A.44. The collection of exact triangles in D(A) satisfies the following properties:

A.4. Derived categories

305

(TR0) A triangle isomorphic to an exact triangle is exact. Id

(TR1) For any object A• in D(A), the triangle A• −→ A• → 0 → A• [1] is exact. (TR2) Any morphism f : A• → B • in D(A) can be embedded into an exact triangle f

A• − → B • → C • → A[1]. u

v

w

v

w

(TR3) A triangle A• − → B• − → C• − → A• [1] is exact if and only if B• − → C• − → u[1]

A• [1] −−→ B • [1] is an exact triangle. u

v

w

u0

v0

→ B• − → C• − → A• [1] and A0 −→ B 0 −→ (TR4) Given two exact triangles A• − w0

•

•

C 0 −→ A0 [1], a commutative diagram •

•

A•

/ B•

u

g

f

 • A0



u0

/ B0 •

can be embedded into a morphism between the two triangles (not necessarily unique). (TR5) (Octahedral axiom). Given exact triangles u

/ B•

/ C0•

/ A• [1],

v

/ C•

/ A0 •

/ B • [1],

w

/ C•

/ B0 •

/ A• [1],

A• B• A•

there exists an exact triangle C 0 → B 0 → A0 → C 0 [1] such that the following diagram is commutative: •

A•

u

/ B•

w

 / C•

v



C0

•

•

/ C0•

/ A• [1]





Id

/ B0 •

/ A• [1]

 / C•

 / A0 •





 / A0 •

u

 B•

•

v

Id

 A•

•

Id

/ B0 •

Id

u[1]

/ B • [1]  / C 0 • [1] .

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Appendix A. Derived and triangulated categories

These properties of exact triangles are the model for the notion of triangulated category, which we proceed to define. Let A be an additive category with an automorphism τ : A → A, which we call the translation or shift functor. We use the notation a[1] = τ (a) for any object a in A. A triangle in A is a sequence of morphisms u

v

w

→b− →c− → a[1] . a− A morphism of triangles is a commutative diagram u

a

v

/c

0

 / c0

g

f

 a0

/b

u

0

 / b0

w

f [1]

h

v

/ a[1]

w

0

 / a0 [1] .

Definition A.45. An additive category A is a triangulated category if it has a shift functor τ : A → A as above, and a collection of triangles, called the exact or distinguished triangles of A, which fulfil the axioms TR0 to TR5 listed in Proposition A.44. 4 It is then clear from Proposition A.44 that the various derived categories D? (A) associated with an Abelian category A are triangulated categories. The homotopy categories K ? (A) are triangulated as well, provided one decrees that a triangle is exact when it is isomorphic, in the obvious sense, to the triangle defined by the cone of a morphism (cf. Eq. (A.4)). A full subcategory A0 of A is a triangulated subcategory if for any object a of A0 , its shift a[1] is also an object of A0 , and for any exact triangle in A such that two of its vertices are in A0 , the third vertex is in A0 as well. Definition A.46. Let A and B be triangulated categories. A covariant functor F : A → B is said to be exact if it satisfies: 1. F is additive and commutes with the shift functor, F (a[1]) ' F (a)[1]. 2. For any exact triangle a → b → c → a[1] in A, the triangle in B F (a) → F (b) → F (c) → F (a[1]) ' F (a)[1] is exact. 4 In particular, the shift functor is an exact functor. The notion of cohomological functor that we have already met in this appendix makes sense in the general context of triangulated categories.

A.4. Derived categories

307

Definition A.47. Let A be a triangulated category. A functor H from A to the category of Abelian groups is said to be cohomological if it maps exact triangles to long exact sequences, that is, for every exact triangle a → b → c → a[1] the sequence · · · → H(a[i]) → H(b[i]) → H(c[i]) → H(a[i + 1]) → . . . is exact. One analogously defines the notion of contravariant cohomological functor by reversing the arrows in the last sequence. 4

A.4.4

Differential graded categories

Most triangulated categories that appear in algebra and in algebraic geometry can be obtained as “homotopy categories” associated to certain more highly structured categories: the differential graded categories. The idea of enhancing triangulated categories was originally proposed by Bondal and Kapranov [46]. In this section, we give a brief account of the definition and basic properties of dg-categories, following quite closely the excellent review paper [178]. Definition A.48. A k-linear category C is said to be a differential graded category (or dg-category for short) if the morphism spaces are differential graded k-vector spaces (i.e., complexes of k-vector spaces) and the composition maps HomC (A, B) ⊗ HomC (B, C) → HomA (A, C) are morphisms of differential graded k-vector spaces.

4

The opposite category C◦ of a dg-category C is also a dg-category, if one defines the composition of two morphisms f ∈ HomC◦ (A, B)p = HomC (B, A)p and g ∈ HomC◦ (B, C)q = HomC (C, B)q as (−1)pq f ◦ g. A functor F : C → D between two dg-categories is said to be a dg-functor if it induces, for any A, B ∈ Ob(C), morphisms of differential graded k-vector spaces F (A, B) : HomC (B, C) → HomD (F (A), F (C)) which are compatible with the composition maps and the units. One can also define the notions of dg-subcategory and full dg-subcategory in the obvious way. A graded category is a dg-category such that for any A, B ∈ Ob(C), the differentials of the complex HomC (A, B) are zero. We denote by Cgr the graded category naturally associated to the dg-category C. Example A.49. 1. Any dg-algebra A is nothing but a dg-category with one object. Thus dg-categories can be thought of as “dg-algebras with many objects.” 2. The category of complexes C(A) of a k-linear Abelian category A can be made into a dg-category, Cdg (A), by setting M Hom(K• , L• )n (A.9) HomCdg (A) (K• , L• ) = Hom• (K• , L• ) = n

308

Appendix A. Derived and triangulated categories

(cf. (A.1)) with the differential given by (A.2). 3. The subcategories C+ (A), C− (A) and Cb (A) carry natural structures of 4 full dg-subcategories of Cdg (A). The homotopy category H 0 (C) has the same objects as C, while the morphism spaces are the zero-cohomology groups of the complex HomC (A, B), namely HomH 0 (C) (A, B) = H 0 (HomC (A, B)) . A dg-functor F : C → D between two dg-categories induces a homotopy functor H 0 (F ) : H 0 (C) → H 0 (D). We can also define the category Z 0 (C) with the same objects as C and morphism spaces given by the zero-cycles: HomZ 0 (C) (A, B) = Z 0 (HomC (A, B)) . A dg-functor F : C → D is a quasi-equivalence if the following conditions are satisfied: 1. F (A, B) is a quasi-isomorphism for all objects A, B ∈ Ob(C); 2. H 0 (F ) : H 0 (C) → H 0 (D) is an equivalence. The category of dg-categories One important feature of dg-categories is that they form a dg-category as well (provided one takes the precaution to deal only with small categories). Definition A.50. The category of small dg-categories is the category dgcatk with the dg-categories as objects and the dg-functors as morphisms. 4 The dg-category dgcatk has an initial object, the empty category, and a final object, the dg-category with one object whose endomorphism ring is zero. The tensor product of two dg-categories C, D is the category C ⊗ D whose objects are pairs (A, B), where A is an object of C and B is an object of D, and whose morphism spaces are the graded tensor products HomC⊗D ((A, B), (A0 , B 0 )) = HomC (A, A0 ) ⊗ HomD (B, B 0 ) . If F : C → D and G : C → D are dg-functors, one defines the complex of graded homomorphisms Hom(F, G) = ⊕n Homn (F, G), where Homn (F, G) is the family of morphisms φA ∈ HomnD (F (A), G(A)) such that G(f ) ◦ φA = φB ◦ F (f ) for all morphisms f ∈ HomC (A, B). The differential is induced by the differential of HomnD (F (A), G(A)). Notice that the set of morphisms F → G is Z 0 Hom(F, G).

A.4. Derived categories

309

One defines the dg-category Hom(C, D) whose objects are the dg-functors from C to D and whose graded spaces of morphisms are Hom(F, G). The category dgcatk , equipped with the tensor product, becomes a symmetric tensor category admitting an internal Hom-functor, namely, Hom(B ⊗ C, D) = Hom(B, Hom(C, D)) for any B, C, D ∈ Ob(dgcatk ). By relying on the techniques developed by Drinfeld in [105], Tabuada proved the following fundamental result. Theorem A.51. [274] The category dgcatk admits a structure of cofibrantly generated model category, whose weak equivalences are the quasi-equivalences. Quillen’s notion of model category (introduced in [256]) provides a general setting where it is possible to develop the basic machinery of homotopy theories. One denotes by Ho(dgcatk ) the “homotopy category” associated to dgcatk , which can be realized as the localization of dgcatk with respect to quasi-equivalences (cf. [106, § 5, 6]). The derived category of a dg-category If A is a differential graded algebra, any right differential graded module M over A can be viewed as a dg-functor A◦ → Cdg (k), where A is the category with one object associated to A and Cdg (k) is the dg-category of complexes of k-vector spaces. In the same vein, a right dg-module over a small dg-category C is defined as a dg-functor M : C◦ → Cdg (k) and a morphism of right dg-modules is defined as a morphism of dg-functors. Equivalently, a right dg-C-module M is specified by giving for each object C of C a complex M (C) of k-vector spaces, and for each pair of objects C and D, a morphism of complexes M (D) ⊗ HomC (C, D) → M (C) in a compatible way with compositions and units. The shift M [n] (n ∈ Z) of a right dg-C-module is defined by M [n](C) = M (C)[n] for any object C in C. The category C(C) of right dg-C-modules is an Abelian category. A morphism M → N of dg-C-modules is an epimorphism (resp. a monomorphism, a quasi-isomorphism) if for every object C in C the morphism of complexes of kvector spaces M (C) → N (C) is an epimorphism (resp. a monomorphism, a quasiisomorphism). Right dg-modules can be used to construct the derived category of a dgcategory C. We define the dg-category of complexes of right dg-C-modules as the dg-category Cdg (C) = Hom(C◦ , Cdg (k)) ,

310

Appendix A. Derived and triangulated categories

whose objects are the right dg-C-modules and whose morphism spaces are the complexes of graded homomorphisms of dg-functors. Notice that there is an equiv∼ Z 0 (C (C)). alence of categories C(C) → dg The homotopy category of the category Cdg (C) is denoted by H(C) = H 0 (Cdg (C)) . It is a triangulated category [177, (2.2)]. It is worth mentioning that there is a dg-functor C → Cdg (C), given by A 7→ hA , where hA is the right dg-C-module defined by hA (B) = HomC (A, B)). It is fully faithful, that is, one has HomC (A, B) = HomCdg (C) (hA , hB ) (Yoneda’s formula). A right dg-C-module M is said to be representable (resp. representable up to homotopy) if it is isomorphic in Cdg (C) (resp. in H(C)) to hA for some object A ∈ Ob(C). In complete analogy to the procedure used to introduce the usual notion of derived category for an Abelian category (cf. Definition A.30), one gives the following definition. Definition A.52. The derived category D(C) of the dg-category C is the localization of H(C) with respect to the class of quasi-isomorphisms. 4 One says that a right dg-C-module M is quasi-representable if it is isomorphic in D(C) to hA for some A ∈ Ob(C). Remark A.53. When C is an Abelian category, so that HomC (A, B) = Hom0C (A, B) for each pair of objects A, B in C, it turns out that D(C) is the usual derived category. 4 To¨en’s results Let us recall the notions of cofibrant and fibrant dg-modules, referring the reader to [177] and [284] for further details. A dg-C-module P is cofibrant if for every surjective quasi-isomorphism M → N of dg-C-modules, the morphism HomC(C) (P, M ) → HomC(C) (P, N ) is an epimorphism. A dg-C-module I is fibrant if for every injective quasi-isomorphism M → N of dg-C-modules, the morphism HomC(C) (N, I) → HomC(C) (M, I) is an epimorphism. It can be shown that for each C-module M , there are quasi-isomorphisms pM → M and M → iM where pM is cofibrant and iM is fibrant [177]. Moreover, pM and iM are unique up to homotopy, and the natural functor H(C) → D(C) admits a fully faithful left adjoint given by M 7→ pM and a fully faithful right adjoint given by M 7→ iM . When C is the dg-category associated to a k-algebra A and M is a right

A.4. Derived categories

311

A-module, pM → M is a projective resolution and M → iM is an injective resolution. Definition A.54. The dg-derived category of a dg-category C is the full dg-subcategory Ddg (C) of Cdg (C) whose objects are all the cofibrant right dg C-modules. 4 Remark A.55. The previous definition is taken from [178], in the spirit of [284]. For a different, but equivalent definition of the category Ddg (C) based on a construction which generalizes Verdier’s quotient to the differential graded setting, see [177] and [105]. 4 The homotopy category H 0 (Ddg (C)) is a triangulated subcategory of H(C) = H (Cdg (C)) and the functor Cdg (C) → H(C) → D(C) induces an exact equivalence of triangulated categories 0

∼ D(C) . H 0 (Ddg (C)) → Let us consider two dg-categories C, D. We can form the dg-category of CD-bimodules (i.e., right dg-(C◦ ⊗ D)-modules, according to our notation). Given an object A in C, one defines the natural dg-functor iA : D◦ → C ⊗ D◦ given by B 7→ A ⊗ B. A C-D-bimodule N is called quasi-representable if for any A ∈ Ob(C) the right dg-D-module i∗A (N ) is quasi-representable (cf. [284, Def. 4.1]). Quasirepresentable C-D-bimodules are also called quasi-functors because they induce functors H(C) → H(D). As we have noticed, the category dgcatk is a tensor category admitting an internal Hom-functor Hom. The tensor product ⊗ induces a tensor product, deL

noted ⊗ , on the category Ho(dgcatk ), since the functor C ⊗ • preserves weak equivalences when C is cofibrant. On the other hand, even in the case when C is cofibrant, the functor Hom(C, •) does not preserve weak equivalences. In spite of that, the following result can be proved. L

Theorem A.56. [284, Theorem 6.1] The monoidal category (Ho(dgcatk ), ⊗ ) admits an internal Hom-functor RHom. For any dg-categories C, D, the dg-category RHom(C, D) is naturally isomorphic in Ho(dgcatk ) to the dg-category of cofibrant right quasi-representable C-D-bimodules, i.e., to the category of quasi-functors from C to D. Let us now take an algebraic variety X and the Abelian category Qco(X) of quasi-coherent sheaves on it. We can form the dg-category Cdg (Qco(X)) (cf. Example A.49), and construct the dg-derived category Ddg (Cdg (Qco(X))), which we denote simply by Ddg (X). As we already remarked, one has H 0 (Ddg (X)) ' D(X). The following result is particularly relevant to the purposes of this book (see Section 2.4).

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Appendix A. Derived and triangulated categories

Theorem A.57. [284, Theorem 8.9] Let X and Y be algebraic varieties over k. There is a natural isomorphism in Ho(dgcatk ) Ddg (X × Y ) ' RHomc (Ddg (X), Ddg (Y )) , where RHomc denotes the full subcategory of RHom consisting of coproduct preserving quasi-functors. In particular, the set of isomorphism classes of objects in the derived category D(X ×k Y ) is isomorphic to the set of direct sum preserving morphisms between Ddg (X) and Ddg (Y ) in Ho(dgcatk ). A more easily handled version of the previous theorem can be proved in the case when X and Y are smooth projective varieties. Let us denote by parf dg (X) the full sub-dg-category of Ddg (X) whose objects are the perfect complexes (cf. Definition A.42). Then, one has [284, Theorem 8.15] parf dg (X ×k Y ) ' RHom(parf dg (X), parf dg (Y )) .

A.4.5

(A.10)

Derived functors

We know that the cohomology groups of a sheaf F on an algebraic variety X are the cohomology objects of the complex of global sections Γ(X, I • ) of a resolution I • of F by injective sheaves (two different injective resolutions I • and J • give rise to the same cohomology groups). Indeed, the complexes Γ(X, I • ) and Γ(X, J • ) are quasi-isomorphic, so that they define isomorphic objects in the derived category of the category of Abelian groups. We can then associate with F a single object RΓ(X, F) = Γ(X, I • ) ' Γ(X, J • ). This is the procedure we mimic to define derived functors on the derived category. To simplify the construction we strengthen the notion of “having enough injectives”: this will mean that there is a functor I : A → A (where A is an Abelian category) such that I(M) is injective for any object M ∈ A, and that there is an immersion 0 → M → I(M) which depends functorially on M. It can be easily checked that if X is an algebraic variety, the categories Mod(X) and Qco(X) have enough injectives also in this stronger sense. It follows that any object M in A has an injective resolution M → I 0 (M) → I 1 (M) → . . . which is functorial in M. If M• is a bounded below complex, we consider the double complex I • (M• ) obtained by associating to any term in the complex M• its injective resolution. The associated simple complex is denoted by I(M• ). This defines a functor I : K + (A) → K + (A). Moreover, by Proposition A.27, the natural morphism M• → I(M• )

A.4. Derived categories

313

is a quasi-isomorphism. Let now B be another Abelian category and F : A → B a left exact functor. Then F induces a functor RF : K + (A) → D+ (B) by RF (M• ) = F (I(M• )). If J • is an acyclic complex of injective objects, then F (J • ) is acyclic, because J • splits. Since a morphism of complexes is a quasi-isomorphism if and only if its cone is acyclic (cf. Corollary A.20), we deduce that RF maps quasi-isomorphisms to isomorphisms, and then by Proposition A.34 it yields a functor RF : D+ (A) → D+ (B) , which is the right derived functor of F . We shall denote Ri F (M• ) = H i (RF (M• )). The restriction of the functor R F : D+ (A) → B to A is the “classical” i-th right derived functor of F . i

The right derived functor RF is exact, that is, it maps exact triangles to exact triangles. In particular, an exact triangle in D+ (A) M0 → M• → M00 → M0 [1] •

•

•

induces a long exact sequence · · · → Ri F (M0 ) → Ri F (M• ) → Ri F (M00 ) → •

•

Ri+1 F (M0 ) → Ri+1 F (M• ) → Ri+1 F (M00 ) → · · · . •

•

For any bounded below complex M• there is a natural morphism F (M• ) → RF (M• ) in the derived category. The complex M• is said to be F -acyclic if this morphism is an isomorphism, that is, M• ' RF (M• ) in D+ (B). One can develop in a similar way a theory for deriving left exact functors on the left if one assumes that A has enough projectives, so that to any object M one can functorially associate a projective resolution . . . P 1 (M) → P 0 (M) → M → 0 . Then for every bounded above complex M• there exists a bounded above complex P (M• ) of projective objects which defines a functor P : K − (A) → K − (A). Then the functor LF : K − (A) → K − (B) given by LF (M• ) = F (P (M• )) defines as above a left derived functor LF : D− (A) → D− (B) . Analogous properties to those proved for right derived functors hold for left derived functors.

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We can also derive on the right functors F : K(A) → K(B) that are not induced by a left exact functor A → B. The theory requires the substitution of the injective resolution I(K• ) of a complex with a resolution by complexes that are F -acyclic in some suitable sense. Let A0 be a thick Abelian subcategory of A. ? We consider the homotopy categories KA 0 (A) that we have already introduced. ? Definition A.58. An exact functor F : KA 0 (A) → K(B) has enough acyclics if ? there exists a triangulated subcategory K F (A) of KA 0 (A) such that the following conditions are fulfilled: ? F 1. There is a functor I : KA 0 (A) → K (A); ? • • 2. for every object M• of KA 0 (A) there is a quasi-isomorphism M → I(M ) • which is functorial on M ;

3. if J • is an object of K F (A) which is acyclic as a complex, then F(J • ) is an acyclic complex of objects of B. 4 ? Example A.59. If A has enough injectives, the functor F : KA 0 (A) → K(B) induced by a right exact functor F : A → B has enough F-acyclics. One simply takes K F (A) as the category I + (A) of bounded below complexes of injective objects of A. 4 ? We shall denote by RF : KA 0 (A) → K(B) the composition of the functors I • and F , that is, RF(M ) = F(I(M• )).

Lemma A.60. RF transforms quasi-isomorphisms into quasi-isomorphisms. ? Proof. Since M• → I(M• ) is a quasi-isomorphism, the functor I : KA 0 (A) → F K (A) transforms quasi-isomorphisms into quasi-isomorphisms. Hence, it suffices to show that F : K F (A) → K(B) transforms quasi-isomorphisms into quasiisomorphisms as well. Let f : I • → J • be a quasi-isomorphism in K F (A). Since F F(f )

is exact, F(I • ) −−−→ J • → F(Cone(f )) → F(I • )[1] is an exact triangle in K(B) and then F(Cone(f )) ' Cone(F(f )). Moreover the cone of f is an acyclic complex by Corollary A.20 and is an object of K F (A), and F(Cone(f )) ' Cone(F(f )) is acyclic so that F(f ) is a quasi-isomorphism again by Corollary A.20.  ? By Proposition A.34, RF : KA 0 (A) → K(B) induces a functor ∗ RF : DA 0 (A) → D(B)

which is called right derived functor of F . We shall write as before Ri F(M• ) = H i (RF(M• )), the i-th right derived functor of F applied to M• . This is an object of the Abelian category B. The notion of F-acyclic complex is defined as before.

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315

Definition A.61. A complex M• is F-acyclic if the natural morphism F(M• ) → 4 RF(M• ) induced by M• → I(M• ) is a quasi-isomorphism. One then may check that any complex in K F (A) is F-acyclic by using Corollary A.20. Indeed, if M• is an object of K F (A), the cone of j : M• → I(M• ) is acyclic and is also in K F (A). Thus, F(Cone(j)) is mapped into an acyclic complex in K(B); moreover, F(Cone(j)) ' Cone(F(j)) because F is exact, an this implies that F (M• ) → RF (M• ) is a quasi-isomorphism. The right derived functor RF satisfies a derived version of de Rham’s the? • ' J • an isomorphism in the orem. Let M• be a complex in DA 0 (A) and M • derived category where J is F-acyclic. De Rham’s theorem in its derived version is just the existence of isomorphisms in the derived category RF(M• ) ' RF(J • ) ' F(J • ) ,

(A.11)

where the second isomorphism holds because J • is F-acyclic. ? Proposition A.62. The right derived functor RF : DA 0 (A) → D(B) is exact. Moreover, if

M0 → M• → M00 → M0 [1] •

•

•

∗ is an exact triangle in DA 0 (A), we have a long exact sequence of derived functors

· · · → Ri F(M0 ) → Ri F(M• ) → Ri F(M00 ) → •

•

Ri+1 F(M0 ) → Ri+1 F(M• ) → Ri+1 F(M00 ) → · · · •

•

Proof. One readily checks that RF commutes with the shift functor. Let us prove ? • = that it is additive. If M• and N • are complexes in KA 0 (A), the complex J • • • • • I(M ) ⊕ I(N ) is F-acyclic and the sum morphism M ⊕ N → J is a quasiisomorphism and then an isomorphism in the derived category. By de Rham’s theorem (A.11), one has RF(M• ⊕ N • ) ' RF(M• ) ⊕ RF (N • ). It follows that ? RF(f +g) = RF(f )+RF(g) for any pair of morphisms f, g : M• → N • in DA 0 (A). f

→ N• → We now take an exact triangle, which we may assume of the form M• − • Cone(f ) → M [1] where f is a morphism of complexes. By Axiom TR2 of the definition of a triangulated category, the commutative diagram M•  I(M• )

f

I(f )

/ N•  / I(N • )

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can be embedded into a morphism of exact triangles M•

f

iM •

 I(M• )

/ N• iN •

I(f )

 / I(N • )

/ Cone(f )

/ M• [1]

 / Cone(I(f ))

 / I(M• )[1]

iM• [1]

.

The morphisms iM• and iN • are isomorphisms in the derived category, and then ? Cone(f ) → Cone(I(f )) is an isomorphism in DA 0 (A) as well. Since F is exact, one concludes.  ∗ 0 Assume now that F takes values in a full subcategory KB 0 (B), where B is a thick Abelian subcategory of B and ∗ is any of the superscripts +, −, b or none. ∗ Suppose also that C is a third Abelian category, and that G : KB 0 (B) → K(C) is another exact functor with enough acyclics (cf. Definition A.58).

Proposition A.63 (Grothendieck’s composite functor theorem). If F transforms F-acyclic objects into G-acyclic objects, one has: 1. G ◦ F has enough acyclics, so that its right derived functor ? R(G ◦ F) : DA 0 (A) → D(C)

exists; ∼ RG ◦ RF. 2. one has a natural isomorphism of derived functors R(G ◦ F) → Proof. For the first part, one simply takes K G◦F (A) = K F (A). The second part follows from R(G ◦ F)(M• ) ' (G ◦ F)(I(M• )) = G(RF(M• )) ' RG(RF(M• )) , where the last isomorphism is due to de Rham’s theorem (A.11) because RF(M• )  ' F(I(M• )) is G-acyclic. One can similarly develop a theory of derived functors on the left. To do so, one has to replace the second condition in Definition A.58 with the following: there is quasi-isomorphism P (M• ) → M• which is functorial in M• . The left derived functor is then defined by LF(M• ) = F(P (M• )) and it has similar properties to those proved for the right derived functors.

A.4. Derived categories

317

Derived direct image Let f : X → Y be a morphism of algebraic varieties (i.e., a morphism of schemes between algebraic varieties). The direct image functor f∗ : Mod(X) → Mod(Y ) is left exact, so that it induces a right derived functor Rf∗ : D+ (Mod(X)) → D+ (Mod(Y )) described as Rf∗ M• ' f∗ (I • ), where I • is a complex of injective OX -modules quasi-isomorphic to M• . Under very mild conditions, the direct image of a quasicoherent sheaf is also quasi-coherent (f has to be quasi-compact and locally of finite type); in this case, Rf∗ maps complexes with quasi-coherent cohomology to complexes with quasi-coherent cohomology, thus defining a functor + + Rf∗ : Dqc (Mod(X)) → Dqc (Mod(Y ))

that we denote with the same symbol. We shall assume that f satisfies these conditions. Then, the category Qco(X) has enough injectives as well; we can derive f∗ : Qco(X) → Qco(Y ) and obtain a derived functor Rf∗ : D+ (Qco(X)) → D+ (Qco(Y )), which is naturally identified with the previous one under the equiv+ + (Mod(X)) and D+ (Qco(Y )) ' Dqc (Mod(Y )). alences D+ (Qco(X)) ' Dqc When f is proper, so that the higher direct images of a coherent sheaf are coherent as well (cf. [134, Thm.3.2.1] or [141, Thm. 5.2] in the projective case), we also have a functor Rf∗ : D+ (X) → D+ (Y ) between the derived categories of complexes of quasi-coherent sheaves with coherent cohomology sheaves. Finally, since the dimension of X bounds the number of higher direct images of a sheaf of OX -modules, Rf∗ maps complexes with bounded cohomology to complexes with bounded cohomology, thus defining a functor b b Rf∗ : Dqc (Mod(X)) → Dqc (Mod(Y )) .

We check now that Rf∗ can be extended to a functor Rf∗ : D(Mod(X)) → D(Mod(Y )) between the whole derived categories. If M• is a complex of OX -modules, we denote by C • (M• ) the double complex obtained by associating to each term in M• its canonical Godement resolution (cf. [124]). Moreover, we write C(M• ) for the associated simple complex. Since for any x ∈ X the cone of M• x → C(M• )x is homotopic to zero, the natural map M• → C(M• ) is a quasi-isomorphism. Furthermore, C(M• ) is a complex of flabby sheaves, since the infinite direct sum

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of flabby sheaves is flabby on Noetherian spaces. Finally, if K• is an acyclic complex of flabby sheaves, the complex f∗ K• is acyclic. This follows easily from the long exact sequence of higher direct images and the fact that the dimension of X bounds the number of higher direct images of a sheaf of OX -modules. In conclusion, we can take the category of complexes of flabby OX -modules as a category of f∗ acyclic complexes (Definition A.58), and define the right derived functor Rf∗ by Rf∗ (M• ) = f∗ (C(M• )). This functor maps Dqc (Mod(X)) to Dqc (Mod(Y )) b b (Mod(X)) to Dqc (Mod(Y )). If f is proper, it maps Dcb (Mod(X)) to and also Dqc b b b Dc (Mod(Y )) and D (X) to D (Y ). Proposition A.64. Let f : X → Y be a morphism of algebraic varieties and M• ∈ D(Mod(X)) a complex of OX -modules. Then Rf∗ M• ' lim Rf∗ (M• ≤n ) −→ n Proof. By construction, C(M• ) ' limn C(M• ≤n ). One concludes as on Noethe−→  rian spaces f∗ commutes with direct limits [124]. If g : Y → Z is another morphism of algebraic varieties, then (g ◦f )∗ = g∗ ◦f∗ and one may apply Grothendieck’s composite functor theorem (since f∗ transforms flabby sheaves into flabby sheaves), obtaining R(g ◦ f )∗ ' Rg∗ ◦ Rf∗ . If Y is a point, Mod(Y ) is the category of k-vector spaces and f∗ is the functor of global sections Γ(X, ). In this case, Rf∗ M• = RΓ(X, M• ) and Ri f∗ M• is called the i-th hypercohomology group Hi (X, M• ) of the complex M• . It coincides with the cohomology group H i (X, M) when the complex reduces to a single sheaf. Derived homomorphism functor Let A be an Abelian category with enough injectives (in the strong sense required in Section A.4.5). We wish to construct a “derived functor” of the complex of homomorphisms L• 7→ F(L• ) = Hom• (K• , L• ) for a fixed complex K• . Since this functor is not induced by a left-exact functor A → B, we have to find a suitable category of F-acyclics. Definition A.65. A complex I • is injective if the functor Hom• ( isomorphisms into quasi-isomorphisms.

, I • ) takes quasi4

Since the functor Hom• ( , I • ) transforms the cone of a morphism R• → M• into the cone of Hom• (M• , I • ) → Hom• (R• , I • ), a complex I • is injective if and only if it transforms acyclic complexes into acyclic complexes.

A.4. Derived categories

319

Lemma A.66. 1. Any bounded below complex of injective OX -modules is injective. 2. If I • is injective and acyclic, then Hom• (M• , I • ) is acyclic for any complex M• . Proof. 1. Let J • be a bounded below complex of injective OX -modules and M• an acyclic complex. One has to prove that any morphism of complexes f : M• → J • is homotopic to zero. We construct a homotopy operator proceeding by recurrence. Assume that for every i ≤ r we have constructed a morphism hi : Mi → J i−1 satisfying di−1 ◦ hi + hi+1 ◦ di = f i for any i < r. The morphism g = f r − dr−1 ◦ hr : Mr → J r vanishes on im dr−1 = ker dr , and induces a morphism Mr / ker dr → J r . Since J r is injective, this morphism lifts to a morphism hr+1 : Mr+1 → J r , and one has dr−1 ◦ hr + hr+1 ◦ dr = f r . 2. The complex Hom• (I • , I • ) is acyclic, and in particular, the identity morphism is homotopic to zero, i.e., I • is homotopic to zero. Hence Hom• (M• , I • ) is also homotopic to zero.  Lemma A.67. Let I • be an injective complex. For any complex M• the natural morphism HomK(A) (M• , I • ) → HomD(A) (M• , I • ) is an isomorphism. Proof. It is enough to see that if R• → M• is a quasi-isomorphism, any morphism of complexes R• → I • lifts (up to homotopies) to a morphism of complexes M• → I • . Since Hom• (M• , I • ) → Hom• (R• , I • ) is a quasi-isomorphism, taking H0 one concludes.  Let I + (A) be the full subcategory of K + (A) formed by the bounded above complexes of injective objects. Recall that there is a functor I : K + (A) → I + (A) and a natural quasi-isomorphism M• → I(M• ), which depends functorially on M• . It follows from Lemma A.66 that for any complex K• , the functor F(L• ) = Hom• (K• , L• ) has enough injectives and one can take for K F (A) the category I + (A) of bounded above complexes of injective objects of A. Therefore, there exists a right derived functor RII Hom• (M• , ) : D+ (A) → D(Ab) . (The subscript “II” reflects the fact that we are deriving with respect to the second variable.) By using Lemma A.66, one proves that for any fixed object L• ∈ D+ (X), the functor RII Hom• ( , N • ) : K(A)0 → D(Ab)

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Appendix A. Derived and triangulated categories

maps quasi-isomorphisms to isomorphisms, so that it induces, again by Proposition A.34, a functor RI RII Hom• ( , N • ) : D(A)0 → D(Ab). Hence one obtains a bifunctor RHom•X : D(A)0 × D+ (A) → D(Ab) defined as RHom• (M• , N • ) = RI RII Hom• (M• , N • ) ' Hom• (M• , I • )), where I • is complex of injective sheaves quasi-isomorphic to N • . We shall denote Exti (M• , N • ) = Ri Hom• (M• , N • ) = Hi (RHom• (M• , N • )) . If A has enough projectives, one can derive the homomorphisms in the reverse order than before, so that for any complex N • we have a right derived functor RI Hom• ( , N • ) : D− (A)0 → D(Ab) given by RI Hom• (M• , N • ) ' Hom• (P (M• ), N • ), where P (M• ) → M• is a projective resolution. Moreover, this functor induces a bifunctor RII RI Hom• : D− (A)0 × D(A) → D(Ab) . If A has both enough injectives and projectives, the functors RI RII Hom• and RII RI Hom• coincide over D− (A)0 × D+ (A). The following property, known as Yoneda’s formula, holds true. Proposition A.68. Let M• be a complex and N • a bounded below complex. One has Exti (M• , N • ) ' HomiD(A) (M• , N • ) , where we have written HomiD(A) (M• , N • ) = HomD(A) (M• , N • [i]). Proof. One has Exti (M• , N • ) ' Hi (RHom• (M• , N • )) ' Hi (Hom• (M• , I(N • ))) ' HomK(A) (M• , I(N • )[i]) ' HomD(A) (M• , I(N • [i])) ' HomD(A) (M• , N • [i]) where the fourth isomorphism is due to Lemma A.67.



We shall also use the notation HomiB (a, b) = HomB (a, b[i]) .

(A.12)

for objects a and b of any triangulated category B. Eventually, we consider the case when A is one of the categories Mod(X) or Qco(X) associated with an algebraic variety X over an algebraically closed field

A.4. Derived categories

321

k. In these cases we write HomX (M, N ) for HomA (M, N ) and the same for the corresponding complexes of homomorphisms. We have a bifunctor RHom•X : D(Mod(X))0 × D+ (Mod(X)) → D(Ab) , which now takes values in the derived category D(k) of k-vector spaces. One can also consider the complex of sheaves of homomorphisms, which we denote by Hom•OX (M• , N • ). This is given by Homn (M• , N • ) =

Y

HomOX (Mi , N i+n )

i

with the differential df = f ◦ dM• + (−1)n+1 dN • ◦ f . Proceeding as above we can define a derived sheaf homomorphism RHom•OX = RI RII Hom•OX : D(Mod(X))0 × D+ (Mod(X)) → D(Mod(X)) described as RHom•OX (K• , L• ) ' RI RII Hom•OX (K• , L• ) ' Hom•OX (K• , I • ) , where I • is a bounded below complex of injective objects quasi-isomorphic to L• . The total derived bifunctor RHom•OX induces bifunctors RHom•OX : D(X)0 × D+ (Qco(X)) → D(Qco(X)) RHom•OX : D− (X)0 × D+ (X) → D(X) . We can apply Grothendieck’s composite functor theorem to the composition Γ(U, Hom•OX (K• , L• )) ' Hom•OU (K• |U , L• |U ), to obtain an isomorphism in the derived category D(Mod(U )): RΓ(U, RHom•OX (K• , L• )) ' RHom•OU (K• |U , L• |U ) . The categories Mod(X), Qco(X) and Coh(X) do not have enough projectives. However, in some situations one can derive the local homomorphisms first with respect to the first argument and then with respect to the second. One such case occurs when X has the resolution property, that is, every coherent sheaf admits a resolution by locally free sheaves (possibly of infinite rank). This happens, for instance, in the following cases: • X is smooth; • X is quasi-projective.

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The problem is now circumvented by considering complexes P • of locally free sheaves. For every bounded above complex M• of OX -modules with coherent cohomology there exist a bounded above complex P (M• ) of locally free sheaves of finite rank and a quasi-isomorphism P (M• ) → M• . Then, for any complex N • we have a right derived functor RI Hom•OX ( , N • ) : Dc− (Mod(X))0 → D(Mod(X)) given by RI Hom•OX (M• , N • ) ' Hom•OX (P (M• ), N • ). Moreover, this induces a bifunctor RII RI Hom•OX : Dc− (Mod(X))0 × D(Mod(X)) → D(Mod(X)) . The derived functors RI RII Hom•OX and RII RI Hom•OX coincide on the product Dc− (Mod(X))0 × D(Mod(X)). Yoneda product We describe here the Yoneda product between Hom groups in a triangulated category B. Let a, b, c be objects in B. The composition of Hom groups gives morphisms HomB (a, b[i]) × HomB (b[i], c[k]) → HomB (a, c[k]) . The shift functor yields an isomorphism HomB (b[i], c[k]) ' HomB (b, c[k − i]); therefore we get the Yoneda product Yij : HomB (a, b[i]) × HomB (b, c[j]) → HomB (a, c[i + j]) or, equivalently, Yij : HomiB (a, b) × HomjB (b, c) → Homi+j B (a, c) .

(A.13)

When B is a k-linear category, this product is actually bilinear. Whenever B is the derived category of an Abelian category A, by Yoneda’s formula (see Proposition A.68) we may write this product in the more usual form as Exti (M• , N • ) × Extj (N • , Q• ) → Exti+j (M• , Q• ) where M• , N • , Q• are objects in the derived category D(A), with N • and Q• bounded below. One should notice that any exact functor of triangulated categories F : B → C, being compatible with compositions and shifts, yields a commutative diagram HomiB (a, b) × HomjB (b, c)

Yij

B

F ×F

 HomiC (F (a), F (b)) × HomjC (F (b), F (c))

/ Homi+j (a, c) F

Yij

 / Homi+j (F (a), F (c)) . C

(A.14)

A.4. Derived categories

323

Extensions in triangulated categories If M and N are objects of an Abelian category A, the elements of the group Ext1 (M, N ) can be identified with the extensions of M by N , that is, with the exact sequences 0 → N → F → M → 0, where two extensions 0 → N → F → M → 0 and 0 → N → F 0 → M → 0 are equivalent if there is an isomorphism φ : F → F 0 fitting into a commutative diagram /F /M /0 /N 0 .

φ

 / F0

/N

0

/M

/0

Taking Proposition A.68 into account, one has Ext1 (M, N ) ' Hom1D(A) (M, N ) so that Hom1D(A) (M, N ) is identified with the group of extensions of M by N . We are going to check that this identification still holds true if we take an arbitrary triangulated category B instead of D(A). Let a, b be objects of B. Definition A.69. An extension of b by a is an exact triangle f

→ a[1] . a→c→b− f0

f

→ a[1], a → c0 → b −→ a[1] are said to be equivalent if Two extensions a → c → b − there is an isomorphism of triangles a

/c

/b

f

/ a[1]

/b

f0

/ a[1] .

' φ

a

 / c0

4 So f 0 = f , and an equivalence class of extensions gives rise to a unique morphism f ∈ HomB (b, a[1]) = Hom1B (b, a), cf. (A.12). Conversely, by axioms TR1 and TR2 any morphism f : b → a[1] can be embedded in an exact triangle f

a→c→b− → a[1] , and two such triangles give rise to isomorphic extensions due to axiom TR4. So we have the following result. Proposition A.70. Given two objects a and b of a triangulated category B, the set of equivalence classes of extensions of b by a can be naturally identified with the group Hom1B (b, a).

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Derived tensor product and inverse image Let X be an algebraic variety. We want to derive on the left the functor “tensor product of complexes.” The lack of projectives in Mod(X) is here circumvented by considering flat sheaves. We say that a complex P • of OX -modules is flat if for any acyclic complex N • , the tensor product complex N • ⊗ P • is also acyclic. This amounts to saying that the functor ⊗P • transforms quasi-isomorphisms to quasi-isomorphisms. One can readily check that a bounded complex P • is flat if and only if every sheaf P n is a flat OX -module. For unbounded complexes, we have the following result. Lemma A.71. If P • is a bounded above complex of flat OX -modules, then P • is flat. Proof. Let N • be an acyclic complex. One has N • ' limn N • ≤n and an isomor−→ phism of bicomplexes N • ⊗ P • ' limn (N • ≤n ⊗ P • ) (cf. Example A.23). Since −→ cohomology commutes with direct limits, we may assume that N • is bounded above. Now, Hd1 (N • ⊗ P • ) = Hd1 (N • ) ⊗ P • = 0 and by Proposition A.27, the  (simple) complex N • ⊗ P • is acyclic. Lemma A.72. Any OX -module M is a quotient of a flat OX -module P (M), which depends functorially on M. Proof. For each open subset U , let OX,U be the OX -module defined by letting OX,U (V ) be the subgroup of sections s ∈ OX (V ) with support contained in U ∩ V for any open subset V ⊆ X. The sheaf OX,U is flat for every open subset U , because the stalk of OX,U at a point x is the local ring OXx if x ∈ U and zero otherwise. Moreover HomX (OX,U , M) = Γ(U, M). Then, taking a copy of OX,U for each nonzero section in Γ(U, M) and the direct sum P (M) = ⊕OX,U over all the open sets U and all such sections, we obtain an epimorphism P (M) → M → 0. We also set P (0) = 0. The functoriality follows from the construction.  It follows that M admits a resolution P • (M) → M → 0 by flat OX -modules. By Lemma A.71, the complex P • (M) is flat. For any bounded above complex N • , let P (N • ) be the simple complex associated with the double complex P • (N q ), and for any complex N • let us define P (N • ) = limn P (N • ≤n ). We then have a −→ functor P : C(A) → C(A) . Lemma A.73. The natural morphism P (N • ) → N • is a quasi-isomorphism, and the complex P (N • ) is flat. Proof. Assume at first that N • is bounded above. By Lemma A.71, P (N • ) is flat, and by Proposition A.27, P (N • ) → N • is a quasi-isomorphism as claimed.

A.4. Derived categories

325

In the general situation, the morphism P (N • ) → N • is a quasi-isomorphism, since cohomology commutes with direct limits. Finally, P (N • ) is flat because both tensor product and cohomology commute with direct limits.  Corollary A.74. Let P • be a flat and acyclic complex. Then N • ⊗ P • is acyclic for any complex N • . Proof. Since P (N • ) → N • is a quasi-isomorphism and P • is flat, P (N • ) ⊗ P • → N • ⊗ P • is a quasi-isomorphism as well. One concludes because P (N • ) is flat and  P • is acyclic. It follows that for a fixed M• ∈ K(Mod(X)) the functor “tensor product of complexes” M• ⊗ : K(Mod(X)) → K(Mod(X)) •

has enough acyclics (one has to take for K M ⊗ the category of flat complexes). Thus, there exists a left derived functor, denoted by M• ⊗ : D(Mod(X)) → D(Mod(X)). Now, for a fixed N • , the functor ⊗N • : K(Mod(X)) → D(Mod(X)), induces a bifunctor, called derived tensor product L

⊗ : D(Mod(X)) × D(Mod(X)) → D(Mod(X)) L

whose description is M• ⊗ N • ' M• ⊗ P (N • ), where P (N • ) → N • is a quasiisomophism and P (N • ) is a complex of flat sheaves. One can derive the tensor product reversing the sense of the derivations and L

L

obtaining the same result, i.e., M• ⊗ N • ' N • ⊗ M• . It is also easy to check that L

L

L

L

(M• ⊗ N • ) ⊗ P • ' M• ⊗ (N • ⊗ P • ) . The derived tensor product induces functors L

⊗

Dqc (Mod(X)) × Dqc (Mod(X)) −→ Dqc (Mod(X)) L

⊗

Dc− (Mod(X)) × Dc− (Mod(X)) −→ Dc− (Mod(X)) . In order to get a derived tensor product for the category D(Qco(X)), one has to modify the previous approach, as for a complex M• of quasi-coherent sheaves the flat resolution P (M• ) may fail to be a complex of quasi-coherent sheaves. Nonetheless it is possibly to construct a different flat resolution Q(M• ) which is a complex of quasi-coherent sheaves. In the affine case, one sets Q(M• ) = limn Q(M• ≤n ), where Q(M• ≤n ) → M• ≤n is a resolution by free (possibly of −→

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ˇ infinite rank) sheaves. The general case can be handled by using Cech resolutions associated with an affine covering. In this way the functor L

⊗

D(Qco(X)) × D(Qco(X)) −→ D(Qco(X)) is defined. Moreover, it induces a functor L

⊗

D− (X) × D− (X) −→ D− (X) . Proposition A.75. Let K• , L• , M• be complexes of OX -modules with M• bounded below. One has an isomorphism of k-vector spaces L

HomD(Mod(X)) (K• ⊗ L• , M• ) ' HomD(Mod(X)) (K• , RHom•OX (L• , M• )) . Moreover, if L• is bounded above, one has an isomorphism in the derived category D(Mod(X)) L

RHom•OX (K• ⊗ L• , M• ) ' RHom•OX (K• , RHom•OX (L• , M• )) . Proof. One has an isomorphism between the homomorphism complexes Hom• (K• ⊗ L• , M• ) ' Hom• (K• , Hom•OX (L• , M• )) . In particular, if L• is flat and M• is injective, Hom• (L• , M• ) is injective. Now, let P • → L• be a flat resolution and M• → I • an injective resolution. By Lemma A.67 and Equation A.3, L

HomD(Mod(X)) (K• ⊗ L• , M• ) ' H0 (Hom• (K• ⊗ P • , I • ) ' H0 (Hom• (K• , Hom•OX (P • , I • ))) . Since P • is flat and I • is injective, Hom•OX (P • , I • ) is an injective resolution of RHom•OX (L• , M• ). By applying again Lemma A.67, one concludes. For the second part, it is enough to prove that for any complex R• of OX -modules, one has L

HomD(Mod(X)) (R• , RHom•OX (K• ⊗ L• , M• )) ' HomD(Mod(X)) (R• , RHom•OX (K• , RHom•OX (L• , M• ))) which follows easily from the first statement.



Definition A.76. A complex of OX -modules M• is of finite Tor-dimension if there exist a bounded and flat complex P (M• ) and a quasi-isomorphism P (M• ) → M• . 4

A.4. Derived categories

327

Complexes of finite Tor-dimension are characterized by the following result, whose proof is straightforward. Proposition A.77. A complex of OX -modules M• is of finite Tor-dimension if and only if the functor M• ⊗ maps Db (Mod(X)) to Db (Mod(X)). Moreover for complexes with coherent cohomology, finite Tor-dimension is equivalent to finite homological dimension. Proposition A.78. Let M• be an object in Db (X). The following conditions are equivalent: 1. M• is of finite homological dimension. 2. M• is of finite Tor-dimension. 3. RHom•OX (M• , G • ) is in Db (X) for every G • in Db (X). Proof. The three conditions are local so that we can assume that X is affine. It is clear that (1) implies (2) (by Proposition A.77) and (3). We check that (3) implies (1). Let us consider a quasi-isomorphism L• → M• , where L• is a bounded above complex of finitely generated free modules. If Kn is the kernel of the differential Ln → Ln+1 , for n small enough the truncated complex Kn → Ln → . . . is quasi-isomorphic to M• because M• is an object of Db (X). Let x be a point and Ox its residual field. Since RHom•OX (M• , Ox ) has bounded homology, one has Ext1OX (Kn , Ox ) = 0 for n small enough. For such an n the module Kn is free in a neighborhood of x and one concludes. To prove that (2) implies (1), one proceeds  analogously by replacing Ext1 with Tor1 . Let f : X → Y be a morphism of algebraic varieties. The inverse image left exact functor f ∗ : Mod(Y ) → Mod(X) can be derived on the left in a similar way ∗ to the tensor product of complexes. We take K f (Mod(Y )) as the category of flat complexes of OY -modules, and obtain a left derived functor Lf ∗ : D(Mod(Y )) → D(Mod(X)) given by Lf ∗ (M• ) = f ∗ (P (M• )), provided that P (M• ) is a flat complex of OY modules and P (M• ) → M• a quasi-isomorphism. One readily checks that Lf ∗ induces a functor Lf ∗ : Dqc (Mod(Y )) → Dqc (Mod(X)). To define the functor Lf ∗ : D(Qco(Y )) → D(Qco(X)), one has to proceed as in the case of the derived tensor product for complexes of quasi-coherent sheaves. In this way we get a functor Lf ∗ : D(Y ) → D(X). All these morphisms map bounded above complexes to bounded above complexes. When f is a flat morphism, there is a natural isomorphism of functors f ∗ ' Lf ∗ .

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If g : Z → X is another morphism, g ∗ transforms flat complexes to flat complexes. By Proposition A.63, one has an isomorphism of functors L(f ◦ g)∗ ' Lg ∗ ◦ Lf ∗ . Compatibility between the derived tensor product and the derived inverse image is easily checked. Proposition A.79. Let f : X → Y be a morphism of algebraic varieties. If M• and N • are complexes in D(Mod(Y )), there is a functorial isomorphism L L ∼ Lf ∗ (M• ⊗ (Lf ∗ M• ) ⊗ (Lf ∗ N • ) → N •) .

Proof. P (M• ) ⊗ P (N • ) → M• ⊗ N • is a flat resolution. Since f ∗ takes flat complexes to flat complexes, the natural isomorphism ∼ f ∗ P (M• ) ⊗ f ∗ P (N • ) f ∗ (P (M• ) ⊗ P (N • )) → L L ∼ (Lf ∗ M• ) ⊗ induces an isomorphism Lf ∗ (M• ⊗ N • ) → (Lf ∗ N • ).



In some situations the derived inverse image induces a functor Lf ∗ : Db (Y ) → Db (X) . This is the case, for instance, when every coherent sheaf G on Y admits a finite resolution by coherent locally free sheaves, a condition which, by Serre’s criterion, is equivalent to the smoothness of Y . In this hypothesis, every object M• in Db (Y ) can be represented as a bounded complex L• of coherent locally free sheaves so that Lf ∗ M• ' f ∗ L• is bounded. Another example occurs when f has finite Tordimension, that is, when for every coherent sheaf G on Y there are only a finite number of nonzero derived inverse images Lj f ∗ (G) = H−j (Lf ∗ (G)); in particular, flat morphisms have finite Tor-dimension.

A.4.6

Some remarkable formulas in derived categories

In this section we gather together some formulas in the derived category which are used throughout this book. We recall that if f : X → Y is a morphism of algebraic varieties, the dimension of X bounds the number of higher direct images of a sheaf of OX -modules; in this case we say that f has finite cohomological dimension. It follows that Rf∗ maps complexes with bounded cohomology to complexes with bounded cohomology. The first result gives the adjunction formula between the derived inverse and the derived direct images.

A.4. Derived categories

329

Proposition A.80. Let f : X → Y be a morphism of algebraic varieties. One has an isomorphism of k-vector spaces HomD(Mod(X)) (Lf ∗ M• , N • ) ' HomD(Mod(Y )) (M• , Rf∗ N • ) , where M• is a complex of OY -modules and N • is a complex of OX -modules. Proof. Every quasi-isomorphism N • → R• of complexes of OX -modules is dominated by an f∗ -acyclic complex. To see this it is enough to consider an f∗ -acyclic resolution of R• . As a consequence of Equation (A.7), Lemma A.67 and Equation (A.3), it follows that HomD(Mod(X)) (L• , N • ) ' lim H0 (Hom• (L• , I • )) , −→ • I

where I • runs over the f∗ -acyclic resolutions of N • . Let P • → M• be a flat resolution. In particular we have, HomD(Mod(X)) (Lf ∗ M• , N • ) ' lim H0 (Hom• (f ∗ P • , I • )) . −→ • I

Since this isomorphism holds for any flat resolution of M• , one has HomD(Mod(X)) (Lf ∗ M• , N • ) ' lim H0 Hom• (f ∗ P • , I • ) . − →• • I ,P

The usual adjunction formula for sheaves yields Hom• (f ∗ P • , I • ) ' Hom• (P • , f∗ I • ). By remarking that every resolution R• → M• is dominated by a flat one, for example by P (R• ), one can similarly conclude that lim H0 Hom• (P • , f∗ I • ) ' HomD(Mod(Y )) (M• , Rf∗ N • ) . − →• •

I ,P

 Corollary A.81. Let f : X → Y be a morphism of algebraic varieties and N • a bounded below complex of OY -modules. One has a functorial isomorphism τ : Rf∗ RHom•OX (Lf ∗ M• , N • ) ' RHom•OY (M• , Rf∗ N • ) . Proof. For any K• ∈ D(Mod(Y )), one has isomorphisms of k-vector spaces HomD(Mod(Y )) (K• ,Rf∗ RHom•OX (Lf ∗ M• , N • ) ' HomD(Mod(X)) (Lf ∗ K• , RHom•OX (Lf ∗ M• , N • ) L

' HomD(Mod(X)) (Lf ∗ K• ⊗ Lf ∗ M• , N • ) L

' HomD(Mod(X)) (Lf ∗ (K• ⊗ M• ), N • ) L

' HomD(Mod(Y ))) (K• ⊗ M• , Rf∗ N • ) ' HomD(Mod(Y )) (K• , RHom•OY (M• , Rf∗ N • )) .

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One concludes by Yoneda’s Lemma A.2.



Corollary A.82. If f has finite Tor-dimension, the functor Lf ∗ : Db (Y ) → Db (X) is left adjoint to Rf∗ : Db (X) → Db (Y ). Proposition A.83 (Projection formula). Let f : X → Y be a morphism of algebraic varieties. For every complex M• of OX -modules and every complex N • of OY modules, there is a functorial morphism in D(Mod(Y )) L

L

Rf∗ (M• ) ⊗ N • → Rf∗ (M• ⊗ Lf ∗ N • ) , which is an isomorphism if N • has quasi-coherent cohomology. Proof. Let C(M• ) ⊗ f ∗ P (N • ) → J • be a quasi-isomorphism, where J • is a f∗ acyclic complex. One has morphisms f∗ (C(M• )) ⊗ P (N • ) → f∗ (C(M• ) ⊗ f ∗ P (N • )) → f∗ J • , where the first morphism is the projection formula for sheaves. Hence one has a morphism L

L

Rf∗ (M• ) ⊗ N • → Rf∗ (M• ⊗ Lf ∗ N • ) in the derived category. We prove that this is an isomorphism if N • has quasicoherent cohomology. Since the question is local on Y and N • ' limn N • ≤n , −→ we may assume that Y is affine and that N • is bounded above. There is a quasiisomorphism L• → N • , where L• is a bounded above complex of free OY -modules. L

Then C(M• ) ⊗ f ∗ L• is a f∗ -acyclic resolution of M• ⊗ Lf ∗ N • ; we need to check that f∗ (C(M• )) ⊗ L• → f∗ (C(M• ) ⊗ f ∗ L• ) is a quasi-isomorphism, which follows straightforwardly from the fact that L• is a complex of free OY -modules (actually, this is an isomorphism).  Proposition A.84. Let f : X → Y be a flat morphism of algebraic varieties. If M• ∈ Dc− (Mod(Y )) and N • ∈ Dc+ (Mod(Y )), one has a functorial isomorphism ∼ RHom• (f ∗ M• , f ∗ N • ) f ∗ RHom•OY (M• , N • ) → OX in the derived category. Proof. Let N • → I • be an injective resolution. One has natural morphisms f ∗ Hom• (M• , I(N • )) → Hom• (f ∗ M• , f ∗ I(N • )) → Hom• (f ∗ M• , J • ) , where f ∗ I(N • ) → J • is a quasi-isomorphism and J • is a complex of injective OX -modules. Hence we have a morphism f ∗ RHom• (M• , N • ) → RHom• (f ∗ M• , f ∗ N • ) .

A.4. Derived categories

331

To prove that this is an isomorphism, we may assume that Y is affine. There is a quasi-isomorphism L• → M• , where L• is a bounded above complex of free modules of finite rank, and we have only to check that f ∗ Hom• (L• , N • ) → Hom• (f ∗ L• , f ∗ N • ) is a quasi-isomorphism. This follows from a direct computation (which indeed proves that this is an isomorphism).  One of the most useful formulas in connection with the Fourier-Mukai transform is the base change formula in derived category. Proposition A.85. Let us consider a Cartesian diagram of morphisms of algebraic varieties g ˜ /X X × Ye Y

 Ye

f˜

f g

.

 /Y

For any complex M• of OX -modules there is a natural morphism g ∗ M• . Lg ∗ Rf∗ M• → Rf˜∗ L˜ Moreover, if M• has quasi-coherent cohomology and either f or g is flat, the above morphism is an isomorphism. g∗ L˜ g ∗ M• induces a morphism Proof. The natural morphism M• → R˜ g∗ L˜ g ∗ M• ' Rg∗ Rf˜∗ L˜ g ∗ M• Rf∗ M• → Rf∗ R˜ and, by adjunction, a morphism in the derived category g ∗ M• . Lg ∗ Rf∗ M• → Rf˜∗ L˜ We can prove that this is an isomorphism if M• has quasi-coherent cohomology. The question is local, so that we can assume that both Y and Ye are affine, and then g and g˜ are affine morphisms. It is enough to check that the induced morphism g ∗ M• ) ' Rf∗ (R˜ g∗ L˜ g ∗ M• ) Rg∗ (Lg ∗ Rf∗ M• ) → Rg∗ (Rf˜∗ L˜ is an isomorphism. By the projection formula (Proposition A.83), this is equivalent to proving that the morphism L

L

Rf∗ M• ⊗ g∗ OYe → Rf∗ (M• ⊗ g˜∗ OXe )

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Appendix A. Derived and triangulated categories

is an isomorphism. Again by the projection formula, the latter morphism is an isomorphism whenever g˜∗ (OXe ) ' Lf ∗ (g∗ OYe ) . This happens if either f or g is flat, since in both cases the last equation reduces  to the isomorphism g˜∗ (OXe ) ' f ∗ (g∗ OYe ). Proposition A.86. Let X be an algebraic variety, M• a bounded above complex of OX -modules with coherent cohomology, and N • and H• bounded below complexes of OX -modules. Assume either that H• has finite homological dimension or that M• has finite homological dimension and X has the resolution property (for instance, X smooth or quasi-projective). Then one has a functorial isomorphism L

L

RHom•OX (M• , N • ) ⊗ H• ' RHom•OX (M• , N • ⊗ H• ) in the derived category. Proof. Note that in both cases, the two sides of the isomorphism are well defined. We prove first that there exists a morphism L

L

RHom•OX (M• , N • ) ⊗ H• → RHom•OX (M• , N • ⊗ H• ) in the derived category. Let us consider the case when H• has finite homological dimension. Let P • → H• be a quasi-isomorphism where P • is a bounded complex of flat sheaves. If I(N • )⊗P • → J • is a quasi-isomorphism, where J • is a bounded below complex of injective sheaves, one has natural morphisms Hom•OX (M• , I(N • )) ⊗ P • → Hom• (M• , I(N • ) ⊗ P • ) → Hom•OX (M• , J • ) and hence a morphism in the derived category L

L

RHom•OX (M• , N • ) ⊗ H• → RHom•OX (M• , N • ⊗ H• ) . The proof of the existence of such a morphism when M• has finite homological dimension and X has the resolution property is done in a similar way. To check that this is an isomorphism we can proceed locally, so that we can assume that X is affine. Then there is a quasi-isomorphism L• → M• , where L• is a bounded above complex of free modules of finite rank, and we can use L• to derive the homomorphisms on the first variable. We are thus reduced to check that there is an isomorphism of complexes Hom•OX (L• , N • ) ⊗ P • → Hom•OX (L• , N • ⊗ P • ) , which is a direct computation.



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333

In the remainder of this section we assume that X has the resolution property, i.e., any coherent sheaf on X is a quotient of a locally free sheaf of finite rank (for instance, X smooth or quasi-projective). Proposition A.87. Let K• be a complex of finite homological dimension. The derived dual K•∨ = RHom•OX (K• , OX ) has finite homological dimension and one has functorial isomorphisms K• ' K•∨ ∨ L

and K•∨ ⊗ N • ' RHom•OX (K• , N • ) in the derived category, where N • is a bounded below complex of OX -modules. Moreover L

L

RHom•OX (M• , N • ) ⊗ K• ' RHom•OX (M• , N • ⊗ K• ) L

' RHom•OX (M• ⊗ K•∨ , N • ) for every bounded above complex M• of OX -modules. Proof. By hypothesis, every point x has an open neighborhood U such that there exists a quasi-isomorphism L• → K• |U , where L• is a bounded complex of finite locally free OU -modules. Then K•∨ |U ' Hom• (L• , OU ), and the natural morphism K• |U → K•∨ ∨ |U is an isomorphism. This proves that K•∨ has finite homological L

dimension and that K• ' K•∨ ∨ . The isomorphism K•∨ ⊗ N • ' RHom•OX (K• , N • ) follows from Proposition A.86. Finally, one has L

RHom•OX (M• , N • ) ⊗ K• ' RHom•OX (K•∨ , RHom•OX (M• , N • )) L

' RHom•OX (K•∨ ⊗ M• , N • )) ' RHom•OX (M• , RHom•OX (K•∨ , N • )) L

' RHom•OX (M• , N • ⊗ K• ) .  A straightforward consequence is the following result. Corollary A.88. Let K• be a complex of finite homological dimension. The functor L

L

⊗ K•∨ : Db (X) → Db (X) is both left and right adjoint to the functor ⊗ K• : Db (X) → Db (X), that is, there are functorial isomorphisms L

L

HomDb (X) (M• , N • ⊗ K• ) ' HomDb (X) (M• ⊗ K•∨ , N • ) L

L

HomDb (X) (M• , N • ⊗ K•∨ ) ' HomDb (X) (M• ⊗ K• , N • ) where M• and N • are bounded complexes of quasi-coherent sheaves with coherent cohomology.

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Appendix A. Derived and triangulated categories

We now prove a K¨ unneth formula in the derived category. Let X and Y be algebraic varieties. We denote by πX and πY the projections of X × Y onto its factors. Given complexes M• of OX -modules and N • of OY -modules, we can consider their box product L

L

∗ M•  N • = πX M• ⊗ πY∗ N • .

in the derived category D(Mod(X × Y )). Theorem A.89. If X and Y are smooth and projective, one has an isomorphism L

L

HomhDb (X×Y ) (M•  N • , E •  F • ) ' M

HomiDb (X) (M• , E • ) ⊗ HomjDb (Y ) (N • , F • )

i+j=h

in the derived category. Proof. Let us denote by p : X → Spec k and q : Y → Spec k the projections onto a point, so that we have a base change diagram X ×Y

πY

/Y

p

 / Spec k

q

πX

 X

.

One has L

L

L

L

RHom•X×Y (M•  N • , E •  F • ) ' Rp∗ RπX∗ RHom•OX×Y (M•  N • , E •  F • ) . We now compute the right-hand side of this formula. L

L

∗ ∗ • M• ⊗ πY∗ N • , πX E ⊗ πY∗ F • ) RπX∗ RHom•OX×Y (πX L

∗ ∗ • M• , RHom•OX×Y (πY∗ N • , πX E ⊗ πY∗ F • )) ' RπX∗ RHom•OX×Y (πX L

∗ ∗ • M• , RHom•OX×Y (πY∗ N • , πY∗ F • ) ⊗ πX E ) ' RπX∗ RHom•OX×Y (πX L

∗ ∗ • M• , πY∗ RHom•OY (N • , F • ) ⊗ πX E ) ' RπX∗ RHom•OX×Y (πX L

∗ • E ]) ' RHom•OX (M• , RπX∗ [πY∗ RHom•OY (N • , F • ) ⊗ πX L

' RπX∗ RHom•OX (M• , RπX∗ [πY∗ RHom•OY (N • , F • )] ⊗ E • )

A.4. Derived categories

335

by Propositions A.75, A.86, A.80, and the projection formula (Proposition A.83). We may use these results in view of the smoothness hypothesis. Moreover one has L

RHom•OX (M• , RπX∗ [πY∗ RHom•OY (N • , F • )] ⊗ E • ) L

' RHom•OX (M• , E • ) ⊗ p∗ [Rq∗ RHom•OY (N • , F • )] by Proposition A.86, and then L

L

RHom•Db (X×Y ) (M•  N • , E •  F • ) L

' Rp∗ (RHom•OX (M• , E • ) ⊗ p∗ [Rq∗ RHom•OY (N • , F • )]) L

' Rp∗ RHom•OX (M• , E • ) ⊗ Rq∗ RHom•OY (N • , F • ) L

' RHom•Db (X) (M• , E • ) ⊗ RHom•Db (Y ) (N • , F • )) , again by the projection formula. We finish by applying Yoneda’s formula (Proposition A.68). 

A.4.7

Support and homological dimension

In this section we recall results about the support and the homological dimension of an object of the bounded derived category taken from [68] and [71]. Let X be an algebraic variety which has the resolution property. Definition A.90. The support Supp(F • ) of a complex F • in Db (X) is the union 4 of the supports of all its cohomology sheaves Hi (F • ). The support is easily characterized in terms of morphisms to the structure sheaves Ox of the (closed) points. Proposition A.91. A (closed) point x ∈ X is in the support of an object F • of Db (X) if and only if HomiD(X) (F • , Ox ) 6= 0 for some integer i. Proof. There is a spectral sequence E2p,q = ExtpX (H−q (F • ), Ox ) converging to p+q • • = Homp+q E∞ D(X) (F , Ox ). If x ∈ Supp(F ), let q0 be the maximum of the q’s −q such that x belongs to the support of H (F • ). Then there is a nonzero morphism H−q0 (F • ) → Ox which gives an element of E20,q0 that survives to infinity; thus 0 •  Hom−q D(X) (F , Ox ) 6= 0. The converse is evident. Remark A.92. An analogous argument shows that if there is an integer i0 such that HomiD(X) (F • , Ox ) = 0 for i < i0 , then Hi (F • ) = 0 at the point x for i < i0 . 4

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Appendix A. Derived and triangulated categories

The length of a bounded complex 0 → E n → · · · → E n+m → 0 (n ∈ Z) is the number m. Since every coherent sheaf on X is a quotient of a locally free sheaf of finite rank, every complex F • in Db (X) is quasi-isomorphic to a bounded above complex E • of locally free sheaves of finite rank. Definition A.93. If F • is of finite homological dimension (Definition A.42), the minimum hd(F • ) of the lengths m of the complexes of locally free sheaves isomorphic with it in the derived category is called the homological dimension of F •. 4 By Remark A.92, one has the following interpretation of the homological dimension. Lemma A.94. If F • is an object of Db (X) and m ≥ 0 is a nonnegative integer, then hd(F • ) ≤ m if and only if there is an integer j such that for any (closed) point x ∈ one has HomiD(X) (F • , Ox ) = 0

unless j ≤ i ≤ j + m.

The relationship between support and homological dimension is a consequence of some deep results in commutative algebra. One is the following acyclicity lemma of Peskine and Szpiro. Lemma A.95. [246, Lemma 1.8], [107, Lemma 1.3] Let O be a local Noetherian ring. Suppose that 0 → M −s → · · · → M 0 → 0 is a complex of O-modules whose cohomology modules H −i (M • ) have finite length. If depthO (M −i ) ≥ i for all 0 ≤ i ≤ s, then H −i (M • ) = 0 for all i > 0. Moreover, depthO (M −i ) > i for all 0 ≤ i ≤ s implies that H −i (M • ) = 0 for all i. Another important result is the so-called “new intersection theorem” [259] (cf. also [107, Theorem 1.13] for a proof). Theorem A.96. Let O be a local Noetherian ring of dimension d and m its maximal ideal. Suppose that 0 → M −s → · · · → M 0 → 0 is a nonexact complex of free O-modules whose cohomology modules H −i (M • ) have finite length. Then s ≥ d. Corollary A.97. [71, Cor. 5.5] or [68, Cor. 5.2] Let F • be a nonzero object of Db (X). Then codim(Supp(F • )) ≤ hd(F • ) .

A.4. Derived categories

337

Proof. We can assume that F • is represented be a finite complex E • ≡ E m−s → · · · → E m of locally free sheaves, with s = hd(F • ). If Y is an irreducible component of Supp(F • ) and y0 is the generic point of Y , by localization at y0 we obtain a → · · · → Eym0 of free modules over the local ring OX,y0 (the stalk complex Eym−s 0 of OX at y0 ). The latter is a local Noetherian ring of dimension codim(Y ), and all the cohomology modules H j (E • y0 ) ' (Hj (E • ))y0 are supported on the unique closed point y0 ∈ Spec OX,y0 . The result follows directly from Theorem A.96.  A refinement of the “new intersection theorem” has been proved by Bridgeland and Iyengar, who have fixed a gap in the proof of the similar statement [71, Thm. 4.3]. Let O be a local Noetherian ring of dimension d and m its maximal ideal. Suppose that 0 → M −s → · · · → M 0 → 0 is a nonexact complex of free O-modules whose cohomology modules H −i (M • ) have finite length. We now that s ≥ d by Theorem A.96. Theorem A.98. [67, Thm. 1.1] Assume that O contains a field or dim O ≤ 3. If s = d and the residue field O/m is a direct summand of H 0 (M • ), then O is regular and H −i (M • ) = 0 for i > 0. We can also characterize smooth schemes by means of the “new intersection theorem.” Corollary A.99. [67, Cor. 1.2] Let Z be an irreducible algebraic variety and fix a closed point x ∈ Z. Assume that there exists an object E • (x) in Db (Z) such that for any closed point z ∈ Z and any integer i, one has HomiD(Z) (E • (x), Oz ) = 0

unless z = x and 0 ≤ i ≤ dim Z.

Assume also that Ox is a direct summand of H0 (E • (x)). Then Z is smoooth at x and E • (x) ' H0 (E • (x)) in D(Z). The proof of Corollary A.99 is analogous to that of the similar statement [71, Cor. 5.6] or [68, Cor. 5.3], by taking the care of replacing [71, Thm. 4.3] by [67, Thm. 1.1].

Appendix B

Lattices In this appendix we gather together some results about integral lattices. In particular, we state a counting formula, due to S. Hosono, B.H. Lian, K. Oguiso and S.-T. Yau [150], which is needed in Chapter 7 to compute the number of FourierMukai partners of a K3 surface. Our basic references are classical monographs such as [267] and [84], and Nikulin’s seminal paper [235].

B.1

Preliminaries

A lattice Λ is a free Z-module of finite rank equipped with an integral nondegenerate symmetric bilinear form h·, ·iΛ : Λ × Λ → Z. Some comments about this definition are perhaps not out of place. It is clear that, after having fixed a basis {e1 , . . . , er }, a lattice L can be regarded as a discrete subgroup of the real vector space Rr that generates all of it, i.e., as a “lattice,” according to Serre’s terminology. However, we would like to adopt here a more abstract point of view, since the main issue we are interested in is the classification of embeddings of one given lattice into another. For the same reason, the language of lattices we opt for seems to be more convenient than the (otherwise equivalent) language of integral quadratic forms. Homomorphisms of lattices are homomorphisms of Z-modules preserving the bilinear form. An injective homomorphism is called an embedding and an isomorphism an isometry. We denote by O(Λ) the group of isometries of Λ with itself. The direct sum of two lattices Λ1 , Λ2 is the Z-module Λ1 ⊕ Λ2 endowed with the bilinear form h(x1 , y1 ), (x2 , y2 )i = hx1 , x2 iΛ1 + hy1 , y2 iΛ2 . The rank r(Λ) of the lattice Λ is its rank as a Z-module. If we fix a basis {e1 , . . . , er }, the determinant of the matrix hei , ej i does not depend on the choice

340

Appendix B. Lattices

of this basis; it is called the discriminant of the lattice and denoted by d(Λ). The signature (τ + , τ − ) of the lattice Λ is the signature of the R-extension of h·, ·iΛ to Λ ⊗ R. The integer τ (Λ) = τ + − τ − is called the index of Λ. We say that the lattice Λ is even (or of type II) if hλ, λiΛ ∈ 2Z for all λ ∈ Λ, and that is odd (or of type I) if it is not even. The lattice is called unimodular if its discriminant is ±1. Example B.1. For any integer n, we denote by Ihni the rank 1 lattice generated by a vector e such that he, ei = n. The hyperbolic lattice U is the rank 2 (even unimodular) lattice with a basis {e1 , e2 } such that he1 , e1 i = he2 , e2 i = 0 and he1 , e2 i = 1. Another important example is provided by the rank 8 lattice whose bilinear form with respect to the canonical basis coincides with the Cartan matrix associated to the exceptional Lie algebra e8 , namely:   2 0 −1 0 0 0 0 0  0 2 0 −1 0 0 0 0     −1 0 2 −1 0 0 0 0     0 −1 −1 2 −1 0 0 0  .  (B.1)  0 0 0 −1 2 −1 0 0     0 0 0 0 −1 2 −1 0     0 0 0 0 0 −1 2 −1  0 0 0 0 0 0 −1 2 This lattice — which is denoted by E8 — is even, unimodular and positive definite. 4 Given a lattice Λ and an integer n, we denote by Λhni the lattice obtained by multiplying its bilinear form by n. When its bilinear form is indefinite, a unimodular lattice is determined up to isomorphism by its rank, index and type. For a proof of the following classical result see, e.g., [267, Chap. V]. Theorem B.2. Let Λ be an indefinite unimodular lattice of rank r and index τ = τ + − τ − . Then, +

−

• if Λ is odd, it is isomorphic to the lattice (⊕τ1 Ih1i) ⊕ (⊕τ1 Ih−1i); • if Λ is even and τ ≥ 0, it is isomorphic to the lattice (⊕p1 U ) ⊕ (⊕q1 E8 ), where p = 12 (r − τ ) and q = 18 τ ; • if Λ is even and τ < 0, it is isomorphic to the lattice (⊕p1 U ) ⊕ (⊕q1 E8 h−1i), where p = 12 (r + τ ) and q = − 18 τ . Given two rational quadratic forms, the weak Hasse principle [267, Theorem 9, Chap. IV] states that they are equivalent if and only if they are equivalent over

B.2. The discriminant group

341

Qp for all primes p and over R (here Qp is the field of p-adic numbers). Although this is no longer true for integral quadratic forms (see [84, p. 129] for an example), the number of nonequivalent integral quadratic forms which are equivalent over Zp for all primes p and over R is finite. Let Λ be an even lattice. The genus g of Λ is defined as the set of isomorphism classes of lattices Λ0 such that Λ⊗Zp ' Λ0 ⊗Zp for all primes p and Λ⊗R ' Λ0 ⊗R. Theorem B.3. [84, Theorem 1.1, Chap. 9] The genus g is a finite set.

B.2

The discriminant group

Let Λ be a lattice. The dual Z-module Λ∗ = Hom(Λ, Z) is endowed with the dual Q-valued bilinear form h·, ·iΛ∗ , which is naturally induced by h·, ·iΛ . If we fix a basis for Λ and the dual basis for Λ∗ , the matrix defining h·, ·iΛ∗ is just the inverse matrix of that defining h·, ·iΛ . The bilinear form h·, ·iΛ gives an immersion of Zmodules Λ ,→ Λ∗ ; we shall identify Λ with its image in Λ∗ . The restriction of h·, ·iΛ∗ to Λ coincides with h·, ·iΛ . Since Λ and Λ∗ are of the same dimension as Z-modules, their quotient AΛ = Λ∗ /Λ is finite. Actually, the order of AΛ is equal to the absolute value of the discriminant of Λ. In particular, when Λ is unimodular, AΛ = 0, and indeed the immersion Λ ,→ Λ∗ is an isometry. On AΛ is naturally defined a nondegenerate symmetric Q/Z-valued bilinear form bΛ : AΛ × AΛ → Q/Z given by bΛ ([x], [y]) ≡ hx, yiΛ∗ mod Z ,

(B.2)

for any [x], [y] in AΛ . If Λ is even, the bilinear form bΛ : AΛ × AΛ → Q/Z induces a Q/2Z-valued quadratic form qΛ on AΛ . One has 2bΛ ([x], [y]) ≡ qΛ ([x] + [y]) − qΛ ([x]) − qΛ ([y]) mod 2Z .

(B.3)

We shall call the pair (AΛ , qΛ ) the discriminant group associated with the lattice Λ. There is a canonical homomorphism O(Λ) → O(AΛ , qΛ ). For example, it is an easy exercise to show that, for any integer n 6= 0, the discriminant group of U hni is AU hni = U hni∗ /U hni = Z/nZ ⊕ Z/nZ endowed with the quadratic form qU hni ([x], [y]) ≡

2 xy mod 2Z . n

342

Appendix B. Lattices

The following result shows that the genus of an even lattice is uniquely determined by its discriminant group and signature. Theorem B.4. [235, Cor. 1.9.4] Two even lattices Λ, Λ0 of the same rank are in 0 ). the same genus if and only if τ (Λ) = τ (Λ0 ) and (AΛ , qΛ ) ' (A0Λ , qΛ We shall make use of the following “stability criterion” in the sequel. Theorem B.5. [235, Corollary 1.13.4] Let Λ be an even lattice having signature (τ + , τ − ) and discriminant form q. The lattice Λ ⊕ U is the unique even lattice having signature (τ + + 1, τ − + 1) and discriminant form q. Let Λ be an even lattice. A subgroup G ⊂ AΛ is called isotropic if qΛ |G = 0. Any isotropic group G determines an even lattice Σ together with an embedding Λ ,→ Σ such that the quotient Σ/Λ is the group G itself. Indeed, for any [x], [y] in G, the formula (B.3) shows that 2bΛ ([x], [y]) ≡ 0 mod 2Z. Hence, the bilinear form bΛ induces a well-defined Z-valued even nondegenerate symmetric bilinear form on the Z-module Σ = {x ∈ Λ∗ |[x] ∈ G}. It is clear that the image of Λ in Λ∗ is contained in Σ since G ⊂ AΛ . An even lattice Σ together with an embedding Λ ,→ Σ such that the quotient is a finite group is called an overlattice of Λ. Given such an overlattice, the natural immersion Λ ,→ Λ∗ factors through the three immersions Λ ,→ Σ ,→ Σ∗ ,→ Λ∗ (B.4) so that the group GΣ = Σ/Λ ⊂ Λ∗ /Λ = AΛ turns out to be isotropic. So we have proved the following useful result [235, Prop. 1.4.1]. Proposition B.6. Let Λ be an even lattice. There is a bijection between isotropic subgroups GΣ of the discriminant group AΛ and overlattices Σ of Λ. The quadratic form qΣ on the discriminant group AΣ can be obtained from qΛ . By taking the quotient of (B.4) by Λ, we get the inclusions of Abelian groups ⊥ GΣ ⊂ Σ∗ /Λ ⊂ AΛ . It is easy to check that Σ∗ /Λ = G⊥ Σ , where GΣ is the orthogonal complement of GΣ in AΛ w.r.t. the bilinear form bΛ . Finally, G⊥ Σ /GΣ = AΣ and (qΛ |G⊥ Σ )/GΣ = qΣ .

B.3

(B.5)

Primitive embeddings

An embedding i : L ,→ Σ of lattices is primitive if the quotient Σ/i(L) is a free Z-module. Notice that this condition is equivalent to the requirement that the adjoint homomorphism i∗ : Σ∗ → L∗ is surjective. A vector x ∈ Σ is primitive if the sublattice generated by x is primitive.

B.3. Primitive embeddings

343

The orthogonal complement of a sublattice M ,→ Σ in Σ is the sublattice M ⊥ = {x ∈ Σ|hx, yiΣ = 0 for all y ∈ M } endowed with the naturally induced bilinear form. It is straightforward to check that M ⊥ ⊂ Σ is actually a primitive sublattice. Let us consider a primitive embedding i : L ,→ Σ, where both L and Σ are even lattices, and a lattice K isomorphic to the orthogonal complement i(L)⊥ of i(L) in Σ. Then L ⊕ K ,→ Σ, and Σ is an overlattice of L ⊕ K. By Proposition B.6, this overlattice is uniquely determined by the isotropic subgroup GΣ = Σ/(L ⊕ K) of the discriminant group AL⊕K = AL ⊕ AK . The group GΣ is embedded in AL ⊕ AK by the composition of homomorphisms φΣ : GΣ = Σ/(L ⊕ K) ,→ Σ∗ /(L ⊕ K) → (L ⊕ K)∗ /(L ⊕ K) = AL ⊕ AK . (B.6) Using Equation (B.5), we can express the quadratic form qΣ on the discriminant group AΣ in the following way:
(B.7) qΣ = (qL ⊕ qK )|G⊥ Σ /GΣ . The composition of φΣ with the canonical projection p1 : AL ⊕ AK → AL yields a homomorphism φΣ,L : GΣ → AL , which is injective due to the primitivity of L in Σ (actually, the injectivity of φΣ,L is equivalent to the primitivity condition). Analogously, we obtain an injective homomorphism φΣ,K = φΣ ◦ p2 : GΣ → AK . When Σ is unimodular, by composing with the lattice isomorphism Σ ' Σ∗ , we obtain a surjective homomorphism Σ ' Σ∗ → L∗

(B.8)

and similarly for K ∗ . Hence, the maps φΣ,L , φΣ,K are isomorphisms. The composition hΣ = φΣ,K ◦ φ−1 Σ,L : AL → AK satisfies the condition qK ◦ hΣ = −qL (the minus sign depends on the fact that GΣ is to be isotropic in AL ⊕ AK ); in other ∼ words, hΣ is an isometry (AL , qL ) → (AK , −qK ) [235, Prop. 1.6.1]. This isometry uniquely determines the primitive embedding i : L ,→ Σ, where Σ is unimodular and the orthogonal complement of i(L) is isomorphic to K. We can rephrase this result in the following way. Theorem B.7. Let L, K be even lattices. Any isometry h : (AL , qL ) → (AK , −qK ) uniquely determines an even, unimodular overlattice Σh of L ⊕ K, together with a primitive embedding i : L ,→ Σh whose orthogonal complement is isomorphic to K. Let us now fix two even lattices L and Σ, assuming that Σ is unimodular. Fix a subgroup of isometries J ⊂ O(L). Two primitive embeddings i : L ,→ Σ,

344

Appendix B. Lattices

i0 : L ,→ Σ are J-equivalent if there are isometries α ∈ O(Σ) and β ∈ J making commutative the following diagram: L _

β

i0

i

 Σ

/ L _

α

 /Σ

When J reduces to the identity, we simply say that two embeddings are are equivalent. We are interested in studying the set EJ (L, Σ) = {i : L ,→ Σ | primitive embedding}/J-equivalence and in showing it is finite (we will follow the treatment given in [150]). Given any two primitive embeddings i : L ,→ Σ, i0 : L ,→ Σ, let us consider K ' i(L)⊥ and K 0 ' i0 (L)⊥ . Both K and K 0 are even, and by Theorem B.7 one has (B.9) (AK , qK ) ' (AL , −qL ) ' (AK 0 , qK 0 ) . Moreover, since Σ is an overlattice of both L ⊕ K and L ⊕ K 0 , by tensoring by R we get isomorphisms of R-vector spaces Σ ⊗ R ' (L ⊕ K) ⊗ R ' (L ⊕ K 0 ) ⊗ R. So, τ (K) = τ (Σ) − τ (L) = τ (K 0 ). In view of Theorem B.4, we get that K and K 0 are in the same genus g. As we have noticed at the end of Section B.1, the genus is a finite set, so we let g = {K1 , . . . , Ks }, where the Ki ’s are nonisometric lattices. In some cases, the genus consists of just one element. The following result is a straightforward consequence of Theorem B.5. Proposition B.8. Let i, i0 : L ,→ Σ be primitive embeddings. Assume that both K and K 0 contain the hyperbolic U lattice as a direct summand. Then K ' K 0 , and the two embeddings are equivalent. On the other hand, if the primitive embeddings i : L ,→ Σ, i0 : L ,→ Σ are Jequivalent, we know that there are α ∈ O(Σ) and β ∈ J such that α◦i = i0 ◦β; this implies that the restriction α|i(L) : i(L) → i0 (L) is an isometry. As a consequence, α|i(L)⊥ : i(L)⊥ → i0 (L)⊥ is an isometry as well. Hence, K ' K 0 ' Ki for some Ki ∈ g. Let E(L, Σ, Ki ) be the subset of EJ (L, Σ) constituted by J-equivalence classes of primitive embeddings i : L ,→ Σ such that i(L)⊥ ' Ki . By the previous discussion, it is clear that [ EJ (L, Σ, Ki ) . EJ (L, Σ) = Ki ∈g

B.3. Primitive embeddings

345

Lemma B.9. The set EJ (L, Σ) is finite. Proof. Let Oi the set of even unimodular overlattices of L ⊕ Ki . The discussion before Theorem B.7 shows that there is a surjective map Oi → EJ (L, Σ, Ki ). By  Proposition B.6, Oi is finite. More precisely, the cardinality of EJ (L, Σ) can be computed by a counting formula proved in [150]. Theorem B.10. In the previous notation, fix Ki ∈ g. One has ](EJ (L, Σ, Ki )) = ](O(Ki )\O(AKi )/J) .

Appendix C

Miscellaneous results For the reader’s convenience, in this appendix we collect several standard results that are used throughout this book. These concern relative duality, Simpson’s notion of stability for pure sheaves, and Fitting ideals.

C.1

Relative duality

We review here the foundations of relative duality. A more exhaustive treatment may be found in many standard references, see, e.g., [139, 291, 2, 234, 89]. Let f : X → Y be a proper morphism of algebraic varieties. The derived direct image functor Rf∗ is defined over the entire category D(X) and has a right adjoint f ! : D(Y ) → D(X) (see Appendix A). There is a functorial isomorphism HomD(Y ) (RfX∗ F • , G • ) ' HomD(X) (F • , f ! G • ) .

(C.1)

This also has a local form: RHom•OY (Rf∗ F • , G • ) ' Rf∗ RHom•OX (F • , f ! G • ) . The complex f ! OY is called the dualizing complex of f . The functor f ! is sometimes determined by the dualizing complex; indeed, whenever G • or f ! OY is of finite Tordimension, there is an isomorphism L

f ! G • ' Lf ∗ G • ⊗ f ! OY .

(C.2)

Another important property of dualizing complexes is their compatibility with base change. If φ : T → Y is a morphism of finite Tor-dimension and fT : XT =

348

Appendix C. Miscellaneous results

X ×Y T → T , φX : XT → X are the induced morphisms, there is an isomorphism Lφ∗X f ! G • ' fT! Lφ∗ G • ,

(C.3)

provided that at least one of the morphisms fT and φX is flat (cf. Lipman’s article in [2]). A key feature is compatibility with the composition of morphisms. Assume that in the next diagram all morphisms are proper: /Y X@ @@ @@ h ; g @@@  T f

(C.4)

since Rg∗ ' Rh∗ ◦ Rf∗ we have a functor isomorphism g ! ' f ! ◦ h!

(C.5)

between the right adjoints. If one of the dualizing complexes h! OT or f ! OY is of finite Tor-dimension, then (C.2) yields an isomorphism L

g ! OT ' Lf ∗ h! OT ⊗ f ! OY .

(C.6)

For particular morphisms we have more concrete expressions for the functor f and the dualizing complex. For example, whenever f is finite the isomorphism !

f∗ f ! G • ' RHomOY (f∗ OX , G • )

(C.7)

holds. This is the case, for instance, when f is a closed immersion. Assume that in addition f is a local complete intersection of codimension d (in the sense of [119, 6.6]) defined by an ideal sheaf J ; that is, J is locally generated by a regular sequence of length d. Then all the cohomology sheaves of f ! OY vanish but the d-th Vd (J /J 2 )∗ . one, which turns out to be isomorphic to the normal bundle NY /X ' So one has d ^ f ! OY ' NY /X [−d] ' (J /J 2 )∗ [−d] (C.8) and f ! G • ' Lf ∗ G • ⊗

d ^ (J /J 2 )∗ [−d]

(C.9)

for every complex G • in D(X). A Cohen-Macaulay morphism is a flat morphism whose fibers are CohenMacaulay varieties. When f is flat of relative dimension n, the condition that f is Cohen-Macaulay is equivalent to the fact that all the cohomology sheaves

C.1. Relative duality

349

Hi (f ! OY ) vanish for i 6= −n. In this case we call the sheaf ωX/Y = H−n (f ! OY ) the dualizing sheaf of f , and we have f ! OY ' ωX/Y [n] . The relative dualizing complex can be also used to characterize Gorenstein morphisms, that is, flat morphisms whose fibers are Gorenstein varieties. A flat morphism of relative dimension n is Gorenstein if and only if Hi (f ! OY ) = 0 for i 6= −n, (so that it is Cohen-Macualay) and the relative dualizing sheaf ωX/Y = Hi (f ! OY ) is a line bundle. The relative canonical divisors are the divisors KX/Y such that ωX/Y ' OX (KX/Y ). Smooth morphisms are Gorenstein and the relative dualizing sheaf in that case is the sheaf ωX/Y = ∧n ΩX/Y of relative n-differentials. When X is a proper Cohen-Macaulay variety X of dimension n and f is the projection of X onto a point, Equation (C.1) yields RΓ(X, F • )∗ ' RHomD(X) (F • , ωX [n]) Hi (X, F • )∗ '

• Hom−i D(X) (F , ωX [n])

or

• ' Homn−i D(X) (F , ωX )

(C.10)

where Hi (X, F • ) = H i (RΓ(X, F • )) is the hypercohomology of the complex F • and ωX is the dualizing sheaf of X. For a complex F concentrated in degree zero, one obtains the usual Serre duality theorem for coherent sheaves on a Cohen-Macaulay proper variety [139, 141] H i (X, F) ' Extn−i (F, ωX )∗ . If f : X → Y is a proper Gorenstein morphism, one has f ! G • ' f ∗ G • ⊗ ωX/Y [n]

(C.11)

for G • in D(Y ). In this situation, the duality isomorphisms (C.1) and (C.10) take the more familiar form RHom•OY (Rf∗ F • , G • ) ' Rf∗ RHom•OX (F • , f ∗ G • ⊗ ωX/Y [n]) HomD(Y ) (RfX∗ F • , G • ) ' HomD(X) (F • , f ∗ G • ⊗ ωX/Y [n])

(C.12)

for arbitrary complexes G • in D(Y ) and F • in D(X). If X is a proper Gorenstein variety of dimension n, the isomorphism HomD(X) (F • , G • ) ' HomD(X) (G • F • ⊗ ωX [n])∗

(C.13)

holds when the complex F • has finite homological dimension; indeed in this case we can use Proposition A.87, and since all Hom groups in (C.10) are finitedimensional, by taking duals one has L

L

HomD(X) (F • , G • ) ' H 0 (RΓ(X, F •∨ ⊗ G • )) ' HomD(X) (F •∨ ⊗ G • , ωX [n])∗ ' HomD(X) (G • , F • ⊗ ωX [n])∗ .

350

Appendix C. Miscellaneous results

In particular, given two coherent sheaves F and G, with F of finite Tordimension, there are duality isomorphisms ∗ ExtiX (F, G) ' Extn−i X (G, F ⊗ ωX )

where n = dim X and ωX is the dualizing line bundle. We conclude this section with some results on adjunction. Let us consider proper morphisms f : X → Y , h : Y → T and g = h ◦ f : X → T . Proposition C.1. 1. Assume that one of the two morphisms f , h is Gorenstein. If the other morphism is Cohen-Macaulay (resp. Gorenstein), then g is also Cohen-Macaulay (resp. Gorenstein) and L

ωX/T ' Lf ∗ ωY /T ⊗ ωX/Y . 2. If f and g are Gorenstein and h is flat and surjective, then h is CohenMacaulay and ωX/T ' f ∗ ωY /T ⊗ ωX/Y . 3. Assume that f is a closed immersion and a local complete intersection. If h is Cohen-Macaulay (resp. Gorenstein), then g is Cohen-Macaulay (resp. Gorenstein) as well and ωX/T ' f ∗ ωY /T ⊗ NX/Y , where NX/Y is the normal bundle. Proof. 1. If n, m are the relative dimensions of f and h, one has f ! OY ' ωX/Y [n], h! OT ' ωY /T [m]. By Equation (C.6), one has g ! OY ' f ∗ ωY /T ⊗ ωX/Y [m + n]. 2. Again by (C.6), ωX/T [m+n] ' f ∗ h! OT ⊗ωX/Y [n] and then f ∗ Hi (h! OT ) = 0 for i 6= −m. Since f is flat and surjective it is faithfully flat, so that Hi (h! OT ) = 0 for i 6= −m. 3. The proof is similar once we know that f ! OY ' NY /X [−d], which follows from Equation (C.8).  In particular, if f is the immersion of a Cartier divisor Y in X, one has an isomorphism ωY /T ' ωX/T |Y ⊗ OY (Y ) and an induced linear equivalence of Cartier divisors KY /T ≡ KX/T · Y + Y · Y

(adjunction formula) .

C.2. Pure sheaves and Simpson stability

C.2

351

Pure sheaves and Simpson stability

A notion of stability for pure sheaves, generalizing the ordinary definition for torsion-free sheaves on irreducible varieties, is due to Simpson [269]. Let X be a projective variety of dimension n, and fix a polarization H in X. If E is a coherent sheaf on X, the Euler characteristic χ(X, E(sH)) = P i i i≥0 (−1) dim H (X, E(sH)) of the twisted sheaf E(sH) = E ⊗ OX (sH) can be written as a polynomial in s with rational coefficients; it is called the Hilbert polynomial of E, and has the form P (E, s) = χ(X, E(sH)) =

d(E) m−1 r(E) m s + s + ... m! (m − 1)!

(C.14)

where r(E) > 0 and d(E) are integer numbers and m ≤ n is the dimension of the support of E. Definition C.2. [269] A coherent sheaf E is pure of dimension m if the support of E has dimension m and the support of any nonzero subsheaf 0 → F → E has dimension m as well. 4 When X is integral, the pure sheaves of dimension n = dim X are precisely the torsion-free sheaves. We can then adopt the following definition. Definition C.3. A coherent sheaf E is torsion-free if it is pure of dimension n = dim X. 4 Simpson also defined the reduced Hilbert polynomial and the (Simpson) slope of E by the following formulas: pS (E, s) =

P (E, s) , r(E)

µS (E) =

d(E) . r(E)

This enables us to define (Simpson) Gieseker stability, µ-stability, Gieseker semistability and µ-semistability for pure sheaves as in the usual case. Definition C.4. A pure sheaf E on X is µS -stable (resp. µS -semistable) with respect to H, if for every proper subsheaf F ,→ E one has µS (F) < µS (E)

(resp. µS (F) ≤ µS (E)).

Analogously, a pure sheaf E on X is (Simpson) Gieseker-stable (resp. Giesekersemistable) with respect to H, if for every proper subsheaf F ,→ E there is an integer s0 such that pS (F, s) < pS (E, s) for all s > s0 .

(resp. pS (F, s) ≤ pS (E, s)) 4

352

Appendix C. Miscellaneous results

If the sheaf E is of finite homological dimension, so that its Chern classes are defined, we can apply the Riemann-Roch theorem for singular varieties [119, Cor. 18.3.1] to prove that the Hilbert polynomial is given by Z (C.15) χ(X, E(s)) = ch(E) · ch(OX (sH)) · Td(X) , where Td(X) is the Todd class of X; for a local complete intersection scheme, the class Td(X) is the usual Todd class td(X) of the virtual tangent bundle. Since the degree of the Hilbert polynomial of E is the dimension of the support of E, one has r(E) = chn−m (E) · H m , (C.16) d(E) = (chn−m+1 (E) + chn−m (E) · td1 (X)) · H m−1 . If X is integral and E is torsion-free, the numbers r(E) and d(E) are closely related to the usual rank rk(E) and degree deg(E) (with respect to H), because in that case Equation (C.16) reads r(E) = rk(E) · deg(X) ,

d(E) = deg(E) + rk(E)C

(C.17)

where deg(X) is the degree of X defined in terms of H and C is a constant. One then sees that for a torsion-free sheaf E on an irreducible projective variety X, the Simpson notions of µS -stability and semistability (and Gieseker stability and semistability) are equivalent to the usual ones. Equation (C.17) also suggests a sensible definition of the rank of a sheaf on an arbitrary projective variety. Definition C.5. Let X be a projective scheme and H polarization in X. The polarized rank of a coherent sheaf E on X is the rational number rk(X,H) (E) =

r(E) . deg(X) 4

Note that if X is irreducible and Supp(E) is different from X the ordinary rank of E is zero, but the polarized rank is not. Simpson constructed moduli spaces for (Gieseker) stable and semistable sheaves on the fibers of a projective morphism by fixing the Hilbert polynomial (or reduced Hilbert polynomial) of the sheaves. Let us state the relevant existence theorem [269, Theorem 1.21]. Let X → B be a projective morphism with a fixed relative polarization H, P a rational polynomial and Mss X/B,P the moduli functor of relatively pure semistable sheaves on the fibers (with respect to the induced polarization) which have Hilbert polynomial P . To be more precise, the functor Mss X/B,P associates to any variety f : T → B over B the set of equivalence classes of coherent sheaves E on T ×B X flat over T and whose restriction Et = jt∗ E to

C.2. Pure sheaves and Simpson stability

353

the fiber Xf (t) is pure and semistable with respect to the induced polarization, and has Hilbert polynomial P . Here two such sheaves E and F are considered to be equivalent if E ' F ⊗ πT∗ L for a line bundle L on T . We say that a morphism of schemes π : M ss (X/B, P ) → B is a coarse moduli space for the moduli functor Mss X/B,P if there is a morphism of functors φ : Mss X/B,P → HomB (

•

, M ss (X/B, P ))

ss which universally corepresents Mss X/B,P . We recall that φ corepresents MX/B,P if the following universal property holds: if Z → B is another B-scheme, for every morphism of functors ψ : Mss X/B,P → HomB ( • , Z) there is a unique morphism of B-schemes g : M ss (X/B, P ) → Z such that the diagram

Mss X/B,P

φ / HomB ( • , M ss (X/B, P )) RRR RRR RRR g RRR ψ RR)  HomB ( • , Z)

commutes. The property that φ universally corepresents Mss X/B,P means that for every base change T → B the fiber product T ×B M ss (X/B, P ) corepresents the fiber product functor HomB ( • , T ) ×HomB ( • ,B) Mss X/B,P . One easily sees that a coarse moduli space, if it exists, is unique up to isomorphisms in the category of B-schemes. Given a morphism T → B and a sheaf E on T ×B X flat over T , which is relatively pure and semistable with Hilbert polynomial P , we denote by φE : T → M ss (X/B, P ) the induced morphism to the coarse moduli space. We recall the definition of S-equivalence. As we shall see in Theorem C.6, this notion is needed to ensure the existence a coarse moduli space of semistable sheaves with prescribed topological invariants (actually, points in the moduli space parameterize S-equivalence classes of semistable sheaves rather than semistable sheaves themselves). This definition requires the notion of Jordan-H¨older filtration: every semistable sheaf F has a filtration F = Fm ⊃ Fm−1 ⊃ · · · ⊃ F0 = 0 whose quotients Fi /Fi−1 are stable with the same slope as F. The Jordan-H¨older filtration is not unique but the associated graded sheaf G(F) = ⊕i Fi /Fi−1 is uniquely determined. Two semistable sheaves F, G are then called S-equivalent if G(F) ' G(G). Thus two stable sheaves are S-equivalent if and only if they are isomorphic. Theorem C.6. 1. There exists a coarse moduli scheme π : M ss (X/B, P ) → B for the moduli functor Mss X/B,P of relatively semistable pure sheaves with fixed Hilbert polynomial P .

354

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2. The morphism π : M ss (X/B, P ) → B is projective. 3. The closed points of the fiber M ss (Xt , p) = π −1 (t) of π : M ss (X/B, P ) → B over a closed point t ∈ B represent S-equivalence classes of semistable sheaves on the fiber Xt with Hilbert polynomial P . 4. There exists an open subscheme M s (X/B, P ) ⊆ M ss (X/B, P ) whose closed points represent the isomorphism classes of stable sheaves on the fibers of X → B. 5. Locally for the ´etale topology on M s (X/B, P ), there exists a universal sheaf E univ on M s (X/B, P ) ×B X → M s (X/B, P ). If a universal sheaf exists globally (and not only locally) on M s (X/B, P ) ×B X → M s (X/B, P ), we say that M s (X/B, P ) is a fine moduli space, a terminology that we have already used in some parts of this book. In this case, M s (X/B, P ) → B represents the moduli functor MsX/B,P of relatively stable sheaves, namely, ss the morphism of functors φ : Mss X/B,P → HomB ( • , M (X/B, P )) induces an isomorphism ∼ Hom ( • , M s (X/B, P )) . φ : MsX/B,P → B This implies that, given a morphism T → B and sheaf E on T ×B X flat over T , which is relatively pure and stable with Hilbert polynomial P , then there is a unique morphism φE : T → M s (X/B, P ) such that E ' (φE × IdX )∗ E univ ⊗ πT∗ L for a line bundle L on T . In the general situation, a universal sheaf E univ only exists locally in the ´etale topology (also in the analytic topology, if the base field is the field of the complex ¯ → M s (X/B, P ) and numbers). This means that there exist an ´etale covering γ : M univ ¯ ¯ ¯ on M ×B X → M , flat over M and relatively pure and stable with a sheaf E Hilbert polynomial P , fulfilling the following universal property: take a morphism T → X and a sheaf E on T ×B X flat over T , which is relatively pure and stable with Hilbert polynomial P . We then have an induced morphism φE : T → M s (X/B, P ), ¯ and the which enables us to consider the fiber product T¯ = T ×M s (X/B,P ) M ¯ induced ´etale covering γ × IdX : T ×B X → X. One has an isomorphism (γ × IdX )∗ E ' (φ¯E × IdX )∗ E univ ⊗ πT∗¯ L of sheaves on T¯ ×T X for a line bundle L on T¯. In words, though E is not the pullback of the universal sheaf, it is so “locally in the ´etale topology,” that is, it becomes the pullback of the universal sheaf after a base change by an ´etale covering. Analogous properties hold true for Gieseker semistability.

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355

The following result is very useful. Let P (s) = rs + d a rational polynomial of degree 1, where r and d are integer numbers, r > 0. We consider relatively pure semistable sheaves with Hilbert polynomial P on the fibers of a projective morphism X → B. This means that the relevant sheaves on the fibers are supported on curves. Proposition C.7. If r and d are coprime, every pure semistable sheaf on a fiber Xt is stable and M s (X/B, P ) → B is a projective morphism and a fine moduli space, so that there exists a universal family E univ on M s (X/B, P )×B X → M s (X/B, P ). Proof. Assume E is a semistable sheaf on a fiber Xt with Hilbert polynomial χ(Xt , E(sHt )) = rs + d (here Ht is the polarization induced by H on the fiber). If E is not stable, there is an exact sequence 0→F →E →G→0 of coherent sheaves on Xt such that µS (F) = µS (E) = d/r. The sheaf E is pure of dimension one, so that F is pure of dimension one too and its Hilbert polynomial is of the form r(F)s + d(F). Since Hilbert polynomials are additive and r(F) is positive, we have r(F) ≤ r. From r(F)d = rd(F) and the coprimality of r and d we obtain r(F) = r and d(F) = d. Thus the Hilbert polynomial of G is zero, so that G = 0, proving that E is stable. The existence of a universal family can be seen using the arguments of [227, Theorem A.6]. 

C.3

Fitting ideals

We review here some elementary properties of the Fitting ideals. Further information can be found, for instance, in [258]. Let A be a ring that we assume Noetherian for simplicity. If M is a finitely generated A-module, we can think of M as the cokernel of a morphism between free modules of finite rank, φ

L1 − → L0 → M → 0 . This is called a finite presentation of M . If we choose bases of L0 and L1 , the morphism φ is determined by a n × m matrix (aij ) where n and m are the ranks of L0 and L1 . Definition C.8. The i-th Fitting ideal of M is the ideal Fi (M ) of A generated by 4 the minors of order n − i of the matrix (aij ). Fitting ideals are well defined, in the sense that they depend only on the module M and not on the matrix expression of a finite presentation φ, nor on the choice of the finite presentation itself. We now list a few more properties of the

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Fitting ideals. The first is that the ring A/Fi (M ) is compatible with base change, in the following sense: if A → B is a ring morphism, one has Fi (M ⊗A B) = Fi (M )·B, so that B/Fi (M ⊗A B) ' A/Fi (M ) ⊗A B . (C.18) Since A/Fi (M ) is the ring corresponding to the closed subscheme of Spec A defined by the Fitting ideal Fi (M ), Equation (C.18) means that this subscheme is compatible with base change. The second fact is that Fitting ideals are multiplicative over direct sums of modules [258, 5.1], that is, X Fh (M ) · Fj (N ) , Fi (M ⊕ N ) = h+j=i

and in particular F0 (M ⊕ N ) = F0 (M ) · F0 (N ) .

(C.19)

Though in general the Fitting ideals are not multiplicative over exact sequences, the following property is true: if 0→M →P →N →0 is an exact sequence of modules, then F0 (P ) ⊆ F0 (M ) · F0 (N ) .

(C.20)

Fitting ideals can be defined for coherent sheaves F on algebraic varieties X, since the local Fitting ideals Fi (F(U )) constructed for the OX (U )-modules F(U ) for every affine open subset U ⊆ X coincide on the intersections, thus gluing together to give an ideal sheaf Fi (F). Let us denote by Zi (F) the closed subscheme defined by the Fitting ideal Fi (F). Since the 0-th Fitting ideal F0 (F) is contained in the annihilator of F, one has Z0 (F) ⊇ Supp(F). These two closed subschemes are very similar: they have the same isolated components, though these can be counted more times in Z0 (F) than in the support, while the embedded components may be different. We then define: Definition C.9. The modified support of F is the closed subscheme Supp0 (F) = Z0 (F) ⊇ Supp(F) defined by 0-th Fitting ideal F0 (F). The base change property (C.18) now gives

4

C.3. Fitting ideals

357

Lemma C.10. Let p : X → B a morphism of algebraic varieties and F a coherent sheaf on X. For every point s ∈ B, the restriction Zi (F)s = Zi (F) ∩ Xs to the fiber Xs of the closed subscheme Zi (F) defined by the Fitting ideal Fi (F) is the closed subscheme Zi (Fs ) defined by the Fitting ideal Fi (Fs ) of the restriction Fs of F to the fiber, Zi (F)s ' Zi (Fs ) . In particular, the modified support Supp0 (F) is compatible with base changes, i.e., Supp0 (Fs ) ' Supp0 (F)s = Supp0 (F) ∩ Xs . The latter property justifies the introduction of the modified support, since the ordinary support Supp(F) does not enjoy this property. Our next aim is to compute the cohomology class of the modified support Supp0 (F) = Z0 (F) of a coherent sheaf. We start with a description of the 0-th Fitting ideal of a coherent sheaf F in terms of a presentation of F as the cokernel of a morphism between locally free sheaves of finite rank φ

→ E0 → F → 0 . E1 − One has an exact sequence s ^

∧s φ⊗1

E1 ⊗ det E0−1 −−−−→ OX → OZ0 (F ) → 0

(C.21)

where s = rk E0 . Proposition C.11. Let X be a smooth projective variety X of dimension n > 0 and F a torsion coherent sheaf on it, that is, rk(F) = 0. Then one has det OZ0 (F ) ' det F , so that c1 (OZ0 (F ) ) = c1 (F). φ

Proof. Let us write F as a cokernel E1 − → E0 → F → 0 as above and let N be the image of φ, so that there is an exact sequence τ

→ E0 → F → 0 . 0→N −

(C.22)

Now (C.21) implies that the sequence n ^

∧n τ ⊗1

N ⊗ (det E0 )−1 −−−−→ OX → OZ0 (F ) → 0

(C.23)

is exact. The sheaf N has rank s = rk(E0 ) as rk(F) = 0 and is locally free in the complementary U of the n − 1-singularity set Sn−1 (N ). The exact sequence (C.21)

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implies that Sn−1 (N ) ⊂ Sn−2 (F), so that Sn−1 (N ) has codimension at least 2 (cf. for instance [144, Prop. 1.13], or [184, Thm. 5.8] in the complex case). Moreover Vs Vs ∧s τ ⊗1 N ⊗ (det E0 )−1 |U is a line bundle and N ⊗ (det E0 )−1 |U −−−−→ F0 (F)|U is Vs an isomorphism by [140, Theorem 3.8]). Thus det F0 (F) ' det( N ⊗(det E0 )−1 ). Vs Vs N coincides with (N )∗∗ ' det N on U , and then det(F0 (F)) ' Moreover, −1 det N ⊗ (det E0 ) . By the multiplicativity of the determinant bundle det F ' det E0 ⊗ (det N )−1 ' (det(F0 (F)))−1 ' det(OZ0 (F ) ) .  Proposition C.12. Let E be a coherent sheaf on a smooth projective variety X with Supp(E) of codimension 1. The polarized rank of E as a sheaf on the modified support Supp0 (E) of E (Definitions C.5 and C.9) is one, rk(Supp0 (E),H) (E) = 1 . Moreover, if the polarized rank of E as a sheaf on its ordinary support is also one, then the modified support coincides with the ordinary one, Supp0 (E) = Supp(E). Proof. If m = dim X, then the numerator of the leading coefficient of the Hilbert polynomial (Eq. (C.14)) is r(E) = c1 (E) · H m−1 = c1 (OZ0 (E) ) · H m−1 = [OZ0 (E) ] · H m−1 = [Supp0 (E)] · H m−1 by Equation (C.16), where the second equality is due to Proposition C.11. This proves the first part by Definition C.5. For the second, if rk(Supp(E),H) (E) = 1, then deg(Supp0 (E)) = deg(Supp(E)) and the closed immersion Supp(E) ,→ Supp0 (E) is an isomorphism.  This result deserves a comment, since the polarized rank of a sheaf E on its ordinary support Y = Supp(E) equals the rank (understood as the 0-th Chern character) of the restriction E|Y because of Equation (C.16). Take for instance an integral Cartier divisor j : Y ,→ X, a locally free sheaf F of rank r on X and let E = F ⊗ j∗ OY . Then Y = Supp(E) and the rank of E on Y is r. However if Y0 = Supp0 (E) is the modified support, the polarized rank rkY0 ,H (E) is one, that is, E has rank one on Y0 . This is not contradictory because the modified support of E is Y0 = rY ; to see this, notice that φ

→F →E →0 0 → F(−Y ) − is a finite presentation of E as the quotient of a morphism of locally free OX modules. If f is a local equation of Y , we see that locally the matrix of φ is f times the identity r × r matrix. Then the 0-th Fitting ideal is locally (f r ) so that Y0 = rY as claimed.

Appendix D

Stability conditions for derived categories by Emanuele Macr`ı

D.1

Introduction

The notion of stability condition on a triangulated category has been introduced by Bridgeland in [65], following ideas from physics by Douglas [104] on π-stability for D-branes. A stability condition on a triangulated category T is given by abstracting the usual properties of µ-stability for sheaves on complex projective varieties; one introduces the notion of slope, using a group homomorphism from the Grothendieck group K(T) of T to C, and requires that a stability condition has generalized Harder-Narasimhan filtrations and is compatible with the shift functor. The main property is that there exists a parameter space Stab(T) for stability conditions, endowed with a natural topology, which is a (possibly infinitedimensional) complex manifold. The space of stability conditions Stab(T) thus yields a geometric invariant naturally attached to a triangulated category T. For motivations and interpretations of stability conditions from the physics viewpoint we refer the reader to the original papers by Douglas, Aspinwall, and others (see, e.g., [104, 103, 9, 58] and references therein). Here we shall concentrate on the mathematical aspects of the definition. From the mathematical viewpoint, one of the main motivations for introducing stability conditions on a triangulated category T is to single out subsets of objects of T which can be classified via some sort of “well-behaved” moduli space. The basic example to keep in mind is Bridgeland’s construction [62] of the three-dimensional flop as a moduli space of

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“stable” objects in the derived category; we already met this in Section 7.5. More generally, the idea is that fixing a stability condition provides the data necessary to reconstruct a variety from its bounded derived category. Another motivation comes from the fact that the space of stability conditions gives a geometric object which is useful in studying algebraic structures, e.g., t-structures and groups of autoequivalences. More precisely, hard combinatorial questions (like the structure of spherical objects on the derived category of a K3 surface) can be reduced to more manageable geometric questions. In this direction, the conjecture stated in [66], and explained in Section D.3.1 of this appendix, is the main example. Unfortunately it is not easy to construct examples of stability conditions. In particular, up to now, there is no example of stability conditions on derived categories of smooth and projective Calabi-Yau threefolds: hence Bridgeland’s construction of the three-dimensional flop cannot be interpreted as a moduli space of stable objects. At the same time, stability conditions on the derived category of the local model of the three-dimensional flop can be described and the flop interpreted as a moduli space of stable point-like objects (see Section 7.5 and [282]). The following cases have been so far studied, at different levels of detail: • smooth projective curves [65, 205, 241]; • singular elliptic curves [79]; • local resolutions of surface singularities of ADE type [278, 59, 161, 240]. In particular in [161] the case of singularities of type A has been completely described (for the A1 case see also [240, 206]); • smooth and projective K3 and Abelian surfaces [66] (for Abelian surfaces and generic twisted K3 surfaces see also [157]); • Enriques surfaces (or more generally equivariant derived categories) [253, 206]; • projective spaces and Del Pezzo surfaces [12, 204]; • the total space of the canonical bundle over the projective plane [64] (for canonical bundles over Del Pezzo surfaces see also [38]); • local three-dimensional crepant resolutions [282]; • local fibrations in elliptic curves or in K3 and Abelian surfaces [283]; • some graded matrix factorizations arising from regular weight systems [275, 170];

D.1. Introduction

361

• generic analytic tori is in [216]. A subject which is receiving a growing interest is the definition of invariants of a (say smooth and projective) variety X from moduli spaces of “stable” objects in the derived category of X (see, for example, [169, 167, 72, 244, 245, 280, 31, 279, 172]). In a series of papers (see in particular [169]), Joyce has started a program for studying invariants constructed in this way and, in particular, for understanding how they vary under a change of the stability condition. The idea is that these invariants should be encoded in holomorphic generating functions that are globally well defined on the space of stability conditions (for an interpretation in terms of families of isomonodromic irregular connections on P1 see [72]). This part of the theory is still largely conjectural, starting from the choice of the definition of stability condition. Some clarifications (at least for the Calabi-Yau three-dimensional case) are provided in [172]. For the case of K3 surfaces the situation is somehow more clear, especially after Toda’s paper [281] (see Section D.4 of this appendix), where moduli space of semistable objects with respect to Bridgeland’s stability conditions are constructed (at least as Artin stacks of finite type), and a conjecture by Joyce on how invariants vary is completely solved. These various conjectural aspects show how the theory of stability conditions is still at its very beginning, and quite likely, the most interesting developments are still to be discovered. The definition itself might need to be modified to some extent, at least in the non-Calabi-Yau case. Some other definitions have been indeed proposed after Bridgeland. In [128] a definition is given by modeling Rudakov’s stability for Abelian categories (see [261]), but in that generality the construction of the space of stability conditions is missing. In [31, 279] a sort of “limit” of Bridgeland stability conditions is defined, which is quite useful for constructing examples in any dimension. A more refined definition of Bridgeland stability condition is in [172]. Finally, in [159] a definition of stability is given having in mind mostly the construction of “good” moduli space of stable objects. In this appendix we shall explain in some detail the basic properties of stability conditions. In Section D.2 we shall essentially follow Bridgeland’s foundational paper [65]. In the first part, we shall give the definition of stability condition and state Bridgeland’s main theorem (Theorem D.8) on the existence of a natural topology on the space of stability condition giving it the structure of a complex manifold. Then we will examine the easiest examples in which the space of stability condition can be explicitly described: bounded derived categories of smooth projective curves. Finally, in the last part, we shall sketch the proof of Theorem D.8. In Section D.3 we shall describe — following [66] — an important example, in which the space of stability conditions can be (at least partly) described:

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the bounded derived category of a smooth projective K3 surface. After stating Bridgeland’s result (Theorem D.19), we shall outline its proof, first by constructing stability conditions on K3 surfaces, proving a covering property for certain connected components of the space of stability conditions and then, after recalling the important wall and chamber structure, singling out a particular connected component. All the results of this section hold also for Abelian surfaces. Actually in this case Conjecture D.20 can be proved to be true (see [243, 66, 157]). In Section D.4 we shall consider the question of constructing moduli spaces of “semistable” objects and shall have a first glimpse at counting invariants for derived categories of K3 surfaces, following [281]. The main results of this section are Theorem D.35, which says that, after fixing a stability condition and numerical invariants, the set of semistable objects of a given phase is an Artin stack of finite type over C, and Theorem D.45, which is the solution of a conjecture by Joyce on how certain invariants of semistable objects vary on the space of stability conditions. A word on notation. All our schemes and varieties will be always over the complex numbers, and most categories will be linear over C. Derived functors will often be denoted as their underived counterparts (i.e., we shall often omit R and L in front of them). This appendix is a survey of the main ideas that have been developed about stability conditions by a number of authors. Among other survey papers existing in the literature we shall cite [12] for an introduction from the physical viewpoint and [63, 58] for an account of Bridgeland’s work, more examples, explanations, and further directions of study. Acknowledgments The structure and presentation of this appendix has been inspired by a series of lectures given by Sukhendu Mehrotra, Paolo Stellari, and Yukinobu Toda at the “First CTS Conference on Vector Bundles” at the Tata Institute for Fundamental Research in Mumbai. I would like to thank them for their beautiful lectures and the organizers of the conference for making this possible. Many thanks are due to Stefano Guerra and Paolo Stellari for comments, suggestions, and for going through a preliminary version of the manuscript.

D.2

Bridgeland’s stability conditions

In this section we outline Bridgeland’s paper [65]. In the first part we give the definition of stability condition on a triangulated category and state Bridgeland’s main theorem (Theorem D.8) showing that stability conditions are parameterized

D.2. Bridgeland’s stability conditions

363

by a (possibly infinite-dimensional) manifold. Then we show the first examples: stability conditions on the derived categories of curves, following [65, 205, 241]. Finally we present the proof of Theorem D.8.

D.2.1

Definition and Bridgeland’s theorem

Let T be an essentially small triangulated category (i.e., T is equivalent to a small category, i.e., a category whose class of objects is a set). Denote by K(T) the Grothendieck group of T. Definition D.1. A stability condition on T is a pair σ = (Z, P) where Z : K(T) → C is a group homomorphism (the central charge) and P(φ) ⊂ T are full additive subcategories, φ ∈ R, satisfying the following conditions: 1. If 0 6= E ∈ P(φ), then Z(E) = m(E) exp(iπφ) for some m(E) ∈ R>0 . 2. P(φ + 1) = P(φ)[1] for all φ ∈ R. 3. If φ1 > φ2 and Ei ∈ P(φi ) (i = 1, 2), then HomT (E1 , E2 ) = 0. 4. Any 0 6= E ∈ T admits a Harder-Narasimhan filtration (HN-filtration for short) given by a collection of distinguished triangles Ei−1 → Ei → Ai → Ei−1 [1] with E0 = 0 and En = E such that Ai ∈ P(φi ) with φ1 > . . . > φn . 4 The nonzero objects in the category P(φ) are said to be (σ-)semistable of phase φ, while the objects Ai in (4) are the semistable factors of E. Note that an HN-filtration of a nonzero object E is unique up to a unique isomorphism. We P − write φ+ σ (E) = φ1 , φσ (E) = φn , and mσ (E) = j |Z(Aj )|. For any interval I ⊂ R, one defines P(I) as the extension-closed subcategory of T generated by the subcategories P(φ), for φ ∈ I. Note that the definition is consistent when I is the interval consisting of only one point, i.e., P(φ) is extension-closed. Remark D.2. (i) For all φ ∈ R, P((φ, φ + 1]) is the heart of a bounded t-structure on T (we gave the notion of heart of a t-structure in Definition 7.54). This is shown quite easily by considering the subcategory P(> φ) ⊂ T and using the HN-filtrations to find its right orthogonal in T. The category P((0, 1]) is called the heart of σ. (ii) As a corollary of (i), each subcategory P(φ) is Abelian. The simple objects of P(φ) are called (σ-)stable of phase φ. 4 Let A be a small Abelian category. A stability function on A is a group homomorphism Z : K(A) → C such that Z(E) ∈ H for all 0 6= E ∈ A, where H =

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{0 6= z ∈ C : z/|z| = exp(iπφ), with 0 < φ ≤ 1}. Given a stability function Z, the phase of a nonzero object E ∈ A is defined to be φ(E) = (1/π) arg Z(E) ∈ (0, 1]. An object 0 6= E ∈ A is called semistable (with respect to Z) if φ(A) ≤ φ(E) for all nonzero subobjects A ,→ E. A stability function is said to have the HN-property if every nonzero object of A admits a finite (Harder-Narasimhan) filtration with semistable quotients of decreasing phases. Using Remark D.2, it is easy to see that a stability condition on a triangulated category T induces a stability function on the Abelian category P((0, 1]) for all φ ∈ R. The existence of HN-filtrations implies that the induced stability function has the HN-property, the semistable objects of phase φ being precisely the nonzero objects of P(φ), for φ ∈ (0, 1]. An important feature of Bridgeland’s stability conditions is that the converse of this is also true. Proposition D.3. Giving a stability condition on T is equivalent to giving a bounded t-structure on T and a stability function on its heart with the HN-property. Proof. Let A be the heart of a bounded t-structure on T and let Z : K(A) → C be a stability function with the HN-property. By using the fact that K(A) can be identified with K(T), we can define a central charge Z : K(T) → C. Define now P(φ), for each φ ∈ (0, 1], as the subcategory of T consisting of the semistable objects of A with phase φ, together with the zero object. For φ ∈ R, set P(φ) = P(ψ)[k], where ψ ∈ (0, 1], k ∈ Z, and φ = ψ + k. It is not difficult to check that the pair (Z, P) thus defined yields a stability condition on T.  In view of this result, sometimes we shall denote a stability condition as a pair (Z, A), where A is the heart of a bounded t-structure and Z is a stability function on it having the HN-property. Example D.4. Let C be a smooth projective curve over C, let Coh(C) be the category of coherent sheaves on it, and Db (C) = Db (Coh(C)) its bounded derived category. Define Z : K(C) → C by Z(E • ) = − deg(E • )+i rk(E • ) for all E • ∈ K(C). It is easy to see that this defines a stability function on Coh(C), whose semistable objects are either µ-semistable vector bundles or torsion sheaves on C. The HNproperty follows from the existence of Harder-Narasimhan filtrations for sheaves. 4 By Proposition D.3 this induces a stability condition on Db (C). Remark D.5. (i) If A ⊂ T is the heart of a bounded t-structure and moreover it is an Abelian category of finite length (i.e., Artinian and Noetherian), then a group homomorphism Z : K(A) → C with Z(S) ∈ H, for all simple objects S ∈ A, extends to a unique stability condition on T. This follows from the fact that A is generated by taking extensions of its simple objects. (ii) Let A be a finite-dimensional associative algebra over C and let Db (A) = Db (Mod-A) the bounded derived category of (right) finitely generated A-modules.

D.2. Bridgeland’s stability conditions

365

As Mod-A is of finite length, by using the previous remark one can construct 4 examples of stability conditions on Db (A). By using Remark D.2 one can prove (see [65, Lemma 4.3]) that, given a stability condition (Z, P) on a triangulated category T, for all intervals I of length less than 1, the subcategory P(I) ⊂ T is quasi-Abelian, the strict exact sequences being triangles of T all of whose vertices are in P(I) (see Definition A.10). A stability condition is called locally finite if there exists some  > 0 such that for all φ ∈ R each quasi-Abelian subcategory P((φ − , φ + )) is of finite length. In this case P(φ) has finite length as well and so every object in P(φ) has a finite Jordan-H¨ older filtration into stable factors of the same phase. The set of locally finite stability conditions will be denoted by Stab(T). Example D.6. (i) If C a smooth projective curve over C, the stability condition constructed in Example D.4 is locally finite. To show this one uses the fact that the image of the central charge is a discrete subgroup of C. (ii) Let A be a finite-dimensional associative algebra over C. Any of the stability conditions constructed in Remark D.5 is locally finite. This follows from the fact that P((0, 1]) = Mod-A is of finite length. 4 We want to define a topology on Stab(T). Let σ = (Z, P) ∈ Stab(T). Define a map k − kσ : HomZ (K(T), C) → [0, +∞] by letting   |U (E)| kU kσ = sup : E is σ-semistable . |Z(E)| Moreover, for τ = (W, Q) ∈ Stab(T) we define  + − − f (σ, τ ) = sup |φ+ σ (E) − φτ (E)|, |φσ (E) − φτ (E)| ∈ [0, +∞]. 06=E∈T

We may note that f depends only on the slicings P and Q; indeed f defines a generalized metric on the set of slicings, i.e., it has all the properties of a metric but it may be infinite (see [65, Sect. 6] for definitions and details). Finally, for  ∈ (0, 1/4), define B (σ) = {τ = (W, Q) : kW − Zkσ < sin(π) and f (σ, τ ) < } ⊂ Stab(T). Lemma D.7. The set {B (σ) : σ ∈ Stab(T),  ∈ (0, 1/4)} yields a basis for a topology on Stab(T). Proof. We have to show that given 1 , 2 ∈ (0, 1/4) and σ1 , σ2 ∈ Stab(T), for all τ ∈ B1 (σ1 ) ∩ B2 (σ2 ) there exists an η > 0 such that Bη (τ ) ⊂ B1 (σ1 ) ∩ B2 (σ2 ). From the definition, it will be sufficient to prove that, given  ∈ (0, 1/4) and σ ∈ Stab(T) such that τ ∈ B (σ), there exists an η > 0 for which Bη (τ ) ⊂ B (σ).

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This follows easily from the following inequality k1 kU kσ < kU kτ < k2 kU kσ

(D.1)

for some constants k1 , k2 > 0 and for all U ∈ HomZ (K(T, C). We leave the proof of (D.1) to the reader (for details see [65, Lemma 6.2]).  We endow Stab(T) with the topology generated by the basis of open subsets B (σ). By [65, Prop. 8.1], this topology can be equivalently described as the topology induced by the generalized metric d(σ1 , σ2 ) =   mσ2 (E) + + − − ∈ [0, ∞], |φσ2 (E) − φσ1 (E)|, |φσ2 (E) − φσ1 (E)|, log sup mσ1 (E) 06=E∈T

(D.2)

for σ1 , σ2 ∈ Stab(T). Let now Σ ⊂ Stab(T) be a connected component. By (D.1) the subspace {U ∈ HomZ (K(T), C) : kU kσ < +∞} ⊂ HomZ (K(T), C) is locally constant on Stab(T) and hence constant on Σ. Denote it by V (Σ). Note that k − kσ , for σ ∈ Σ, defines a norm on V Σ. Moreover, by (D.1), all norms k − kσ on V (Σ) are equivalent, and therefore they define the same topology on V (Σ). By its own definition, the map Z : Σ → V (Σ) which associates to a stability condition its central charge is continuous. Bridgeland’s main theorem asserts that this map is actually a local homeomorphism. Theorem D.8. (Bridgeland) For all connected components Σ ⊂ Stab(T), the map Z : Σ → V (Σ) which associates to a stability condition its central charge is a local homeomorphism. In particular, Σ has a manifold structure, locally modeled on the topological vector space V (Σ). We shall sketch a proof of this theorem in Section D.2.3. If K(T) ⊗ C is finite-dimensional over C, then Stab(T) is a finite-dimensional complex manifold. This is the case, for example, when T = Db (A) for a finitedimensional associative algebra A over C. On the contrary, a smooth projective curve of positive genus has infinite-dimensional Grothendieck group. In general, to get spaces parameterizing stability conditions that are finite dimensional complex manifolds, it is sufficient to take finite-dimensional slices of HomZ (K(T), C). An example, which is well suited for derived categories of smooth projective varieties, is obtained by asking that the central charge factorizes through the singular cohomology of X. This can be formalized as follows.

D.2. Bridgeland’s stability conditions

367

Suppose that the category T is C-linear and of finite type, that is, for any pair of objects E and F the space ⊕i HomT (E, F [i]) is a finite-dimensional C-vector space. The Euler-Poincar´e form on K(T) χ(E, F ) =

X (−1)i dimC HomT (E, F [i]) i∈Z

allows us to define the numerical Grothendieck group N (T) = K(T)/K(T)⊥ (where orthogonality is with respect to χ). If N (T) has finite rank, then T is said to be numerically finite. For example, T = Db (X) = Db (Coh(X)), for X a smooth projective variety over C, is numerically finite by the Riemann-Roch theorem. A stability condition σ = (Z, P) such that Z factors through the epimorphism K(T) → N (T) is called numerical. We denote by StabN (T) the set of locally finite numerical stability conditions. An immediate consequence of Theorem D.8 is that if T is numerically finite, then StabN (T) becomes a finite-dimensional complex manifold, locally modeled over V (Σ) ∩ HomZ (N (T), C). The next result provides an important property of the space of stability conditions. Proposition D.9. The space of stability conditions Stab(T) carries a right action + f (R), the universal cover of Gl+ (R), and a left action of the group of the group Gl 2 2 Aut(T) of exact autoequivalences of T. These two actions commute. +

f (R) acts in the following way. Consider a pair (G, f ), with Proof. The group Gl 2 + G ∈ Gl2 (R) while f : R → R is an increasing map, such that f (φ + 1) = f (φ) + 1 and G exp(iπφ)/|G exp(iπφ)| = exp(2iπf (φ)), for all φ ∈ R. It is easy to see that + f (R) can be thought as the set of such a pairs. Then (G, f ) maps (Z, P) ∈ Gl 2

Stab(T) to (G−1 ◦ Z, P ◦ f ), where P ◦ f (φ) = P(f (φ)). For the second action, Φ ∈ Aut(T) maps (Z, P) to (Z ◦ φ−1 , Φ(P)), where φ is the automorphism of K(T) induced by Φ.  Note that the action of Aut(T) preserves the generalized metric d of (D.2), i.e., Aut(T) acts on Stab(T) by isometries. Clearly, a result analogous to Proposition D.9 holds for StabN (T) when T is numerically finite. We shall see in Section D.3 that this can be applied to obtain information on the group of autoequivalences of derived categories of smooth projective varieties from the topology of StabN (Db (X)). We conclude this section by spending a few words on the moduli problem, which could be roughly summarized in the following two questions:

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(1) given a (numerical) stability condition σ on T, does there exist a good notion of “moduli space” of σ-semistable objects? (2) If (1) is true, then how do moduli spaces vary under changes of the stability condition? In Section D.4 we shall examine these two questions when T = Db (X), for X a smooth and projective K3 surface over C. Here we consider a simple example where the answer to these questions is quite straightforward. Example D.10. Let A be a finite-dimensional associative algebra over C and let Db (A) be as in Remark D.5(ii). Then A = Mod-A is an Abelian category of finite length with a finite number of simple objects S1 , . . . , Sn . As we saw in Remark D.5, given z1 , . . . , zn ∈ H, we can define a stability condition σ with heart A and stability function Z(Si ) = zi , for all i, where K(A) = K(A) ' ⊕i Z[Si ]. In this case, an object V of Db (A) is (semi)stable with respect to σ if and only if it is a shift of a θ-(semi)stable A-module in the sense of King [182], where θ is defined by   Z(U ) θ(U ) = −= ∈ R, Z(V ) for all U ∈ K(A). (We shall denote by < and = the real and imaginary part of a complex number.) In particular, by the results in [182], one can construct moduli spaces M v (σ) of (S-equivalence classes of) semistable objects in A having fixed class v ∈ K(A) using geometric invariant theory techniques. M v (σ) is then a projective variety over C. This gives a complete answer to question (1) for this example. 4 Example D.11. In the situation of Example D.10, take A = C[Q] as the pathalgebra associated to the quiver Q : • → • with two vertices and one arrow from the first to the second vertex. Then A = Mod-A ' Rep(Q) =   Vi finite-dimensional C-vector spaces (V1 , V2 , φ) : . φ : V1 → V2 a linear map There are only three indecomposable objects in A: the two simple objects S1 = (C, 0, 0) and S2 = (0, C, 0) and their unique nontrivial extension E = (C, C, Id), 0 → S2 → E → S1 → 0. One may show that, fixing v = [S1 ] + [S2 ], one has ( Spec(C) if arg(Z(S2 )) ≤ arg(Z(S1 )), v M (σ) ' 0 if arg(Z(S2 )) > arg(Z(S1 )). (This follows from the fact that S2 is the only proper subobject of E.) This gives an example related to question (2) above. 4

D.2. Bridgeland’s stability conditions

D.2.2

369

An example: stability conditions on curves

We give here some examples of spaces of stability conditions. We consider the case of smooth projective curves over C of positive genus, where stability conditions are substantially equivalent to µ-stability for sheaves. Then, for completeness, we sketch the case of P1 . The results of this section, motivated by open problems in an early version of [65], are contained in [65, 205, 241]. A study of stability conditions on singular elliptic curves, not included here, is in [79]. Let C be a smooth projective curve over C of positive genus. We will denote by Stab(C) the space of locally finite numerical stability conditions on D b (C). The Riemann-Roch theorem shows that N (C) = N (Db (C)) can be identified with Z ⊕ Z, with the quotient map K(C) → N (C) sending a class E • ∈ K(C) to the pair consisting of its rank and degree. In Example D.4 we constructed a stability condition on Db (C) which is numerical and locally finite (see Example D.6(i)). We want to prove that, up to the + f (R), this is the only one ([205, Thm. 2.7], however the case of elliptic action of Gl 2

curves had been previously treated by Bridgeland). Proposition D.12. Let C be a smooth projective curve over C of genus g(C) ≥ 1. + f (R) on Stab(C) is free and transitive, so that The action of Gl 2 +

f (R) ' H × C, Stab(C) ' Gl 2 where H ⊂ C is the complex upper half plane. Proof. First of all recall the following technical fact (see [128, Lemma 7.2]): (*) If E ∈ Coh(C) is included in a triangle F • → E → G • , with F • , G • ∈ Db (C) and (F • , G • ) = 0, then F • , G • ∈ Coh(C). Hom≤0 D b (C) Now it is not difficult to prove that the skyscraper sheaf Ox , x ∈ C, is stable in all stability conditions in Stab(C). Indeed, an easy consequence of (*) is that Ox is semistable and moreover all its stable factors are isomorphic to a single object K• ∈ Db (C). But this implies that K• ∈ Coh(C) and so that K• ' Ox . In the same way it can be shown that all line bundles are stable in all stability conditions. Take σ = (Z, P) ∈ Stab(C) and a line bundle L on C. By what we have seen above, L and Ox are stable in σ with phases φL and ψx respectively. The existence of maps L → Ox and Ox → L[1] gives inequalities ψx − 1 ≤ φL ≤ ψx , which implies that if Z is an isomorphism (seen as a map from H ∗ (C, R) ' R2 to C ' R2 ) then it must be orientation preserving. But Z is an isomorphism: indeed if not, there exist stable objects with the same phase having nontrivial morphisms, + f (R), one can assume that which is impossible. Hence, acting by an element of Gl 2 Z(E • ) = − deg(E • ) + i rk(E • ) and that for some x ∈ C, the skyscraper sheaf Ox

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has phase 1. Then all line bundles on C are stable in σ with phases in the interval (0, 1) and, as a consequence, all skyscraper sheaves are stable of phase 1. But this implies that P((0, 1]) = Coh(C) so that the stability condition σ is precisely the one induced by µ-stability on C.  Remark D.13. Two remarks on the previous proposition are in order. Let us note that although the stability conditions on a curve of positive genus are all in the + f (R) orbit, their hearts are not at all trivial. For example, in the case same Gl 2 of elliptic curves, it is possible to prove that the choice of a stability condition is equivalent to the choice of a noncommutative structure on C in the sense of Polishchuk and Schwarz [254, 252]: the heart of a stability condition corresponds to the Abelian category of vector bundles with respect a noncommutative structure on C. Secondly, already in the case of elliptic curves, the quotient Stab(C)/ Aut(Db (C)) is of some interest. Indeed it can be proved (e.g., from the study of their action on Stab(C)) that the autoequivalences of Db (C) are generated by shifts, automorphisms of C and twists by line bundles together with the Fourier-Mukai transform associated to the Poincar´e sheaf (see [228]). Automorphisms of C and twists by line bundles of degree zero act trivially on Stab(C) and one obtains Stab(C)/ Aut(Db (C)) ' Gl+ 2 (R)/Sl2 (Z), which is a C∗ -bundle over the moduli space of elliptic curves.

4

The case of the projective line over C is slightly more involved, due to the presence of “degenerate” stability conditions, i.e., stability conditions with very few stable objects. This purports some evidence that the definition of stability condition for categories with nontrivial Serre functor may need some modification. The basic idea for studying Stab(P1 ) = Stab(Db (P1 )) ' StabN (Db (P1 )) is to use the well-known Be˘ılinson equivalence Db (P1 ) ' Db (A), where A is the path 2 → • consisting of two vertices and algebra associated to the Kronecker quiver • − two arrows from the first to the second vertex. In this way, examples of stability conditions can be constructed using both Example D.4 and Remark D.5(ii). In + f (R) is neither free nor transitive. Nevertheless, we can this case the action of Gl 2

+

f (R) given by z = x + iy 7→ (exp(x)Ry , fy ), where look at the subgroup C ,→ Gl 2 + Ry ∈ Gl2 (R) is the rotation by the angle −πy and fy (φ) = φ + y, for φ ∈ R. This action of C on Stab(P1 ) is free and the quotient Stab(P1 )/C is isomorphic to C (see [241, Sect. 4]). Hence we deduce the following result [241, Sec. 4]. Proposition D.14. Stab(P1 ) ' C2 . The fact that Stab(P1 ) is connected and simply connected was proved independently in [205].

D.2. Bridgeland’s stability conditions

D.2.3

371

Bridgeland’s deformation lemma

In this section we sketch a proof of Theorem D.8. We need to show that, for all connected components Σ ⊂ Stab(T), the continuous map Z : Σ → V (Σ) which associates to a stability condition its central charge is a local homeomorphism. For more details see [65, Sect. 7]. We start with a useful lemma which shows that Z is locally injective. Lemma D.15. Let σ = (Z, P), τ = (Z, Q) ∈ Stab(T) be stability conditions with the same central charge Z such that f (σ, τ ) < 1. Then σ = τ . Proof. Suppose that σ 6= τ . There exists E ∈ P(φ) such that E ∈ / Q(φ). Since f (σ, τ ) < 1, there is a triangle A → E → B such that A ∈ Q((φ, φ + 1)) and B ∈ Q((φ − 1, φ]). We claim that both A and B are nonzero. Indeed, assume A = 0. Then E ∈ Q((φ − 1, φ]), which is a contradiction since σ and τ have the same central charge. In the same way, B 6= 0. Now, by the same argument, A ∈ / P(≤ φ). Hence there exists an object h

→ A. Since E ∈ P(φ), C ∈ P(ψ), with ψ > φ and a nonzero morphism C − h

→ A → E must be zero, and so h factorizes through the composite map C − g → B[−1] → A. But, since f (σ, τ ) < 1, C ∈ Q((ψ − 1, ψ + 1)) ⊂ Q(> (φ − 1)) C− and B[−1] ∈ Q((φ − 2, φ − 1]), and so g = 0, a contradiction.  To conclude the proof of the theorem we need to show the following deformation lemma, due to Bridgeland. Lemma D.16. Let σ ∈ Stab(T). Fix 0 > 0 be such that 0 < 1/10 and, for all φ ∈ R, each of the quasi-Abelian categories P((φ − 40 , φ + 40 )) ⊂ T is of finite length. If 0 <  < 0 and W : K(T) → C is a group homomorphism satisfying |W (E) − Z(E)| < sin(π)|Z(E)|

(D.3)

for all σ-stable E ∈ T, there is a locally finite stability condition τ = (W, Q) on T with f (σ, τ ) < . Notice that, by Lemma D.15, the stability condition τ is unique. Moreover, asking that Equation (D.3) is satisfied for all σ-stable objects in T clearly implies, by local finiteness, that it is satisfied by all σ-semistable objects. Hence Theorem D.8 follows. To prove Bridgeland’s deformation lemma D.16 we need some preparatory results. Let W : K(T) → C and  be as in the statement. Call a subcategory of T thin if it is of the form P((a, b)), for a and b real numbers such that 0 < b − a < 1 − 2. A thin subcategory is quasi-Abelian.

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By Equation (D.16), if E ∈ T is σ-semistable, the phases of W (E) and Z(E) differ at most by . Hence, if A = P((a, b)) is thin, W defines a skewed stability function on A, that is a group homomorphism W : K(A) → K(T) → C (the Grothendieck group of A being the quotient of the free group generated by the isomorphism classes of the objects of A with respect of strict short exact sequences) such that W (E) ∈ Ha,b for all objects 0 6= E ∈ A, where Ha,b = {0 6= z ∈ C : z/|z| = exp(iπψ), with a −  < ψ < b + }. Exactly in the same way as for Abelian categories, we can define a notion of semistability, by looking at the phase ψ(−) = (1/π) arg W (−) ∈ (a − , b + ) and at the strict subobjects. An object of A which is semistable with respect to this skewed stability function will be called W -semistable. Example D.17. Suppose E is W -semistable in some thin subcategory A ⊂ T and set ψ = ψ(E). Then E ∈ P((ψ − , ψ + )). To show this, one uses the fact that the phases of the points W (A) and Z(A) differ by at most , for A ∈ T semistable in σ. 4

Let A be a thin subcategory of T. A nonzero object E ∈ A is said to be enveloped by A if a +  ≤ ψ(E) ≤ b − . It can be checked (see [65, Lemma 7.5]) that, if E ∈ T is enveloped by two thin subcategories B and C, then E is W semistable in B if and only if it is W -semistable in C. More precisely, the notion of W -semistability for enveloped object is independent of the choice of the thin subcategory containing it. For all ψ ∈ R, define Q(ψ) ⊂ T as the full additive subcategory of T consisting of the zero object together with those E ∈ T that are W -semistable of phase ψ in some thin enveloping subcategory P((a, b)). The pair τ = (W, Q) defines a stability condition on T. Indeed, the conditions (1) and (2) of Definition D.1 are automatically satisfied by definition. Condition (3) can be proved by using Example D.17 (see [65, Lemma 7.6]). To prove the existence of HN-filtrations is more delicate. First of all it can be shown (see [65, Lemma 7.7]) that if A = P((a, b)) ⊂ T is a thin subcategory of finite length, every nonzero object of P((a + 2, b − 4)) has a finite HN-filtration, whose factors are W -semistable objects of A that are enveloped by A. Using this and Example D.17 it is not difficult to show that, if for any t ∈ R we define a subcategory Q(> t) as the full extension-closed subcategory of T generated by the subcategories Q(ψ), for ψ > t, then Q(> t) is a bounded t-structure on T. Define similarly Q(I) for any interval I ⊂ R. Let 0 6= E ∈ T. We need to construct a finite filtration by objects of the categories Q(ψ). Since Q(> t) is a bounded t-structure, we can assume E ∈ Q((t, t + 1]), for some t ∈ R. But using the t-structure Q(> (t + δ)) we can further

D.3. Stability conditions on K3 surfaces

373

assume E ∈ Q((t, t + δ]), for δ = 50 − 5 fixed. But now Q((t, t + δ]) ⊂ P((t − , t +  + δ)) ⊂ P((t − 3, t + 5 + δ)) ⊂ P((t − 30 , t + 50 )). Since P((t − 30 , t + 50 )) is thin and of finite length, by [65, Lemma 7.7] we have HN-filtrations. Finally, Example D.17 shows that f (σ, τ ) < . Hence τ is locally finite since, for all t ∈ R, one has Q((t − , t + )) ⊂ P((t − 2, t + 2)). This concludes the proof of Lemma D.16. Remark D.18. Let X be a smooth projective variety over C. Set Stab(X) = StabN (X) as the set of locally finite numerical stability conditions on Db (X). (i) Let σ = (Z, P) be a numerical stability condition such that the image of Z is a discrete subgroup of C (i.e., Z is discrete). Fix some 0 <  < 1/2. Then, for all φ ∈ R, the quasi-Abelian category P((φ − , φ + )) is of finite length. In particular σ is locally finite. This follows from the fact that for a given object E ∈ P((φ − , φ + )) the central charges of all sub- and quotient objects of E lie in a certain bounded region, and from the discreteness assumption. See also Remark D.5. (ii) A connected component Σ ⊂ Stab(X) is called full if the subspace V (Σ)∩ HomZ (N (T), C) is equal to HomZ (N (T), C). A stability condition σ ∈ Stab(X) is called full if it belongs to a full connected component. Take σ = (Z, P) ∈ Stab(X) a full stability condition and fix 0 <  < 1/2. Then, for all φ ∈ R, the quasi-Abelian category P((φ − , φ + )) is of finite length. Indeed, Theorem D.8 and the fact that σ is full allow one to find a stability condition τ = (W, Q) ∈ Stab(X) with discrete central charge such that f (σ, τ ) < η, for η > 0 sufficiently small. Then one uses (i) and the fact that P((φ − , φ + )) ⊂ Q((φ −  − η, φ +  + η)). 4

D.3

Stability conditions on K3 surfaces

The aim of this section is to describe in detail a connected component of the space of (numerical, locally finite) stability conditions on smooth and projective K3 surfaces, following [66]. After stating Theorem D.19, we sketch a proof and construct examples of stability conditions on K3 surfaces. As an interlude, we introduce the wall and chamber structure, which will be very important in Section D.4 to study how moduli spaces of semistable objects behave under a change of the stability condition.

374

D.3.1

Appendix D. Stability conditions for derived categories

Bridgeland’s theorem

Let X be a projective K3 surface over the complex numbers and denote by Stab(X) = StabN (Db (X)) the space of numerical locally finite stability conditions on Db (X). Recall that the numerical Grothendieck group in this case is isomorphic to the lattice ˜ 1,1 (X, Z) = H 0 (X, Z) ⊕ NS(X) ⊕ H 4 (X, Z) ⊂ H ∗ (X, Z), H where NS(X) is the N´eron-Severi group of X (see Section 4.1). For a Z-module R, ˜ 1,1 (X, Z)R = H ˜ 1,1 (X, Z) ⊗Z R and NS(X)R = NS(X) ⊗Z R. we set H ∼ By Orlov’s representability theorem 2.15, every autoequivalence Φ : Db (X)→ ∼ b Φ e e D (X) induces a Hodge isometry f : H(X, Z) → H(X, Z) (see Section 4.2). Let ˜ 1,1 (X, Z)C , we can write Z(E • ) = σ = (Z, P) ∈ Stab(X). Since π(σ) is in H p • • b hπ(σ), v(E )i, for all E ∈ D (X), where v(E • ) = ch(E • ) · td(X) is the Mukai vector of E • , and h·, ·i is the Mukai pairing (see Section 1.1). By Theorem D.8, we ˜ 1,1 (X, Z)C . Denote by P(X) the subset of get a continuous map π : Stab(X) → H 1,1 ˜ H (X, Z)C defined as ( ) ˜ 1,1 (X, Z)R are linearly independent w , w ∈ H 1 2 ˜ 1,1 (X, Z)C : w1 + iw2 ∈ H (aw1 + bw2 )2 > 0 for all a, b ∈ R, (a, b) 6= (0, 0) It is easy to see that P(X) has two connected components, P + (X) and P (X), which are exchanged by conjugation. P + (X) is defined as the connected component containing the vector (1, iω, −ω 2 /2), for ω ∈ H 1,1 (X, Z) the class of ˜ 1,1 (X, Z) : δ 2 = −2} and, for an ample divisor. Furthermore, set ∆(X) = {δ ∈ H δ ∈ ∆, ˜ 1,1 (X, Z)C : hΩ, δi = 0}. δ ⊥ = {Ω ∈ H −

Theorem D.19. There is a connected component Stab† (X) ⊂ Stab(X) which is mapped by π onto the open subset [ P0+ (X) = P + (X) \ δ ⊥ ⊂ NS(X)C . δ∈∆(X)

Moreover, the induced map π : Stab† (X) → P0+ (X) is a covering map and the group ( ) Φ(Stab† (X)) = Stab† (X) † b b Aut0 (D (X)) = Φ ∈ Aut(D (X)) : e e Z) → H(X, Z) f Φ = Id : H(X, acts freely on Stab† (X) and is the group of deck transformation of π.

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375

Before starting the proof of Theorem D.19, we observe that a more detailed topological study of the connected component Stab† (X) would yield a description of the group of autoequivalences of the derived category of a K3 surface. Indeed, as remarked in [66], Theorem D.19 is not enough to determine the structure of Aut(Db (X)). Bridgeland conjectured the following. Conjecture D.20. The action of Aut(Db (X)) on Stab(X) preserves the connected 4 component Stab† (X). Moreover Stab† (X) is simply connected. e e Denote by O(H(X, Z)) the group of Hodge isometries of H(X, Z) and by e O+ (H(X, Z)) the subgroup consisting of the isometries which preserve the oriene tation of the positive four-space in H(X, R). The following conjecture is due to Bridgeland and Szendr˝ oi [66, 273]. Conjecture D.21. There is a short exact sequence of groups f (−)

e 1 → π1 (P0+ (X)) → Aut(Db (X)) −−−→ O+ (H(X, Z)) → 0 .

(D.4) 4

By Theorem D.19, a proof of Conjecture D.20 would imply Conjecture D.21. The fact that every autoequivalence of Db (X) induces an orientation-preserving Hodge isometry in cohomology was conjectured by Szendr˝oi in [273], by seeing it as a “mirror-symmetric” version of a result of Donaldson [101, 55] about the orientation preserving property for diffeomorphisms of K3 surfaces. Szendr˝oi’s conjecture has been proved in [157, 156], by demonstrating Conjectures D.20 and D.21 in the easier case of analytic K3 surfaces with trivial Picard group and then using the deformation theory of Fourier-Mukai kernels. At this point, the surjectivity in Equation (D.4) follows easily from the theory of moduli spaces on K3 surfaces. e Z)) induces a surTheorem D.22. The morphism f (−) : Aut(Db (X)) → O(H(X, e Z)). jective morphism f (−) : Aut(Db (X)) → O+ (H(X,

D.3.2

Construction of stability conditions

In this section we show that Stab(X) is nonempty by exhibiting a collection of stability conditions σβ,ω on X. We also describe some special semistable objects in σβ,ω . We begin by observing that the category of coherent sheaves cannot be the heart of a stability condition on a variety Y if the dimension of Y is greater than 1 (the proof below is taken from [279, Lemma 2.7]). Proposition D.23. Let Y be a smooth projective variety over C of dimension d ≥ 2. There is no numerical stability condition σ ∈ Stab(Y ) with heart Coh(Y ).

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Proof. Assume that there exists σ = (Z, P) ∈ Stab(Y ) with heart Coh(Y ). Write Pd Z(E) = j=0 (uj + ivj ) · chj (E), for all E ∈ Coh(Y ), where uj , vj ∈ H 2d−2j (Y, R) and chj (E) ∈ H 2j (Y, Q) is the j-th component of the Chern character of E. Since ι − Y of dimension 2. The composition d ≥ 2, there exists a smooth subvariety S → ι∗ Z K(Y ) − → C induces a numerical stability function on Coh(S). Hence, we K(S) −→ can assume d = 2. Let C ⊂ Y be a smooth curve and take a divisor D on C. Then, by assumption, we have =(Z(OC (D))) = v2 (deg(D) + ch2 (OC )) + v1 · [C] ≥ 0. Since D can be of arbitrary degree, we must have v2 = 0. Similarly, using OY (mC) for m sufficiently small, we have v1 · [C] = 0. Therefore =(Z(OC (D))) = 0, and so <(Z(OC (D))) = u2 (deg(D) + ch2 (OC )) + u1 · [C] ≤ 0. By repeating the same argument, we have u2 = 0. But then Z(Ox ) = u2 + iv2 = 0,  for all skyscraper sheaves Oy , y ∈ Y , a contradiction. So it is clear that to find an example of a stability condition on a K3 surface X we need first to indentify a good heart of a t-structure. Let β, ω ∈ NS(X)R with ω ˜ 1,1 (X, Z) → C an ample R-divisor (we write ω ∈ Amp(X)). Moreover, let Zβ,ω : H be the morphism induced by b

Zβ,ω (E • ) = (exp(β + iω), v(E • ))M ,

for all E • ∈ Db (X),

where exp(β + iω) = 1 + β + iω + 12 (β 2 − ω 2 + 2iβ · ω), and furthermore define ( ) either E = Etor or , Tβ,ω = E ∈ Coh(X) : µ− ω (E/Etor ) > β · ω  Fβ,ω = E ∈ Coh(X) : E torsion-free and µ+ ω (E) ≤ β · ω ,   Hi (E • ) = 0 if i ∈ / {−1, 0}     , Aβ,ω = E • ∈ Db (X) : H−1 (E • ) ∈ Fβ,ω     H0 (E • ) ∈ Tβ,ω + where µ− ω (resp. µω ) denotes the smallest (resp. largest) slope of the HarderNarasimhan filtration of a torsion-free sheaf on X with respect to µω -stability and Etor denotes the torsion part of a sheaf E ∈ Coh(X).

By [137, Prop. 2.1] and the properties of µ-stability, Aβ,ω is the heart of a bounded t-structure on Db (X). Let us note that since the canonical bundle of a K3 surface is trivial, an object E • ∈ Db (X) is spherical if HomDb (X) (E • , E • [i]) ' C when i = 0, 2 and HomDb (X) (E • , E • [i]) = 0 for all other indices (cf. Section 2.4).

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Lemma D.24. The morphism Z = Zβ,ω defines a stability function on Aβ,ω if and only if Z(E) ∈ / R≤0 , for all E ∈ Coh(X) spherical. In particular, this holds if ω 2 > 2. Proof. The only case we need to check is when E ∈ Fβ,ω and =(Z(E)) = (c − rβ) · ω = 0, where we have set v(E) = (r, c, s). We have to prove that <(Z(E)) > 0. By taking a filtration with respect to µω -stability, we can assume that E is µω -stable. Then, by the Riemann-Roch theorem, we have c2 − 2rs = v(E)2 ≥ −2, with equality if and only if E is spherical. Moreover, by the Hodge index theorem, we have (c − rβ)2 ≤ 0. Hence, writing <(Z(E)) explicitly, we have <(Z(E)) =


1 (c2 − 2rs) + rω 2 − (c − rβ)2 . 2r

The claim now is clear.



Lemma D.25. Assume that β, ω ∈ NS(X)Q and ω ∈ Amp(X) are such that / R≤0 for all spherical E ∈ Coh(X). The stability function Zβ,ω on Zβ,ω (E) ∈ Aβ,ω has the HN property, hence it defines a numerical stability condition σβ,ω on Db (X). Moreover, this stability condition is locally finite. Proof. We only delineate the main argument. Set φ(−) = (1/π) arg(Zβ,ω (−)) ∈ [0, 1). As in the classical existence results for Harder-Narasimhan filtrations (see [261, Thm. 2] for a general approach, and [65, Prop. 2.4] for this case), for Zβ,ω to have the HN property the following two conditions are to be satisfied: (a) There are no infinite chains of subobjects in Aβ,ω . . . ⊂ E • j+1 ⊂ E • j ⊂ . . . ⊂ E • 2 ⊂ E • 1 with φ(E • j+1 ) > φ(E • j ), for all j. (b) There are no infinite chains of quotients in Aβ,ω E • 1  E • 2  . . .  E • j  E • j+1  . . . with φ(E • j ) > φ(E • j+1 ), for all j. Assume for a contradiction that an infinite chain as in (a) does exist. Since =(Zβ,ω ) is discrete, then there exists a positive integer N ∈ N such that 0 ≥ =(Zβ,ω (E • n )) = =(Zβ,ω (E • n+1 )), for all n ≥ N . Let F • n = E • n /E • n+1 ∈ Aβ,ω . Then, by additivity, =(Zβ,ω (Fn )) = 0, for all n ≥ N . Hence φ(F • n ) = 1, for all n ≥ N and φ(E • n ) ≥ φ(E • n+1 , a contradiction. Assume now that an infinite chain as in (b) exists. As before, 0 ≥ =(Zβ,ω (E • n )) = =(Zβ,ω (E • n+1 )), for all n ≥ N . Let G • n = ker(E • N  E • n ) ∈ Aβ,ω . Then =(G • n ) = 0 and φ(G • n ) = 1 for all n ≥ N .

378

Appendix D. Stability conditions for derived categories We need the following facts.

(i) Let P(1) be the full Abelian subcategory of Aβ,ω whose objects are those P • ∈ Aβ,ω having =(Zβ,ω (P • )) = 0. Then P(1) is of finite length. To show this, one uses the fact that the real part of Zβ,ω is discrete. (ii) By using this, and the explicit description of objects in Aβ,ω , one may show that for all such objects E • there exists an exact sequence 0 → A• → E • → B • → 0 in Aβ,ω such that A• ∈ P(1) and HomAβ,ω (P • , B • ) = 0 for all P ∈ P(1). Then there exists an exact sequence 0 → A• → E • N → B • → 0 in Aβ,ω such that A• ∈ P(1) and HomAβ,ω (P • , B • ) = 0, for all P • ∈ P(1). Since G • n ∈ P(1) then the inclusion G • n ⊂ E • N factorizes through G • n ⊂ A• ⊂ E • N . Hence we get an infinite chain of subobjects of A• 0 = G • N ⊂ G • N +1 ⊂ . . . ⊂ A• , which contradicts fact (i) above. The fact that σβ,ω is locally finite follows from Remark D.18(i) since Zβ,ω is discrete.  By Lemma D.25, σβ,ω is in Stab(X) if β and ω are rational. We shall see that HN filtrations and the local finiteness hold also for general real β, ω, provided the condition of Lemma D.24 is satisfied. Now we examine some particular semistable objects in σβ,ω . Example D.26. All skyscraper sheaves Ox , for x ∈ X are stable in σβ,ω of phase 1, for β and ω as in Lemma D.25. This follows from the fact that they are simple 4 in Aβ,ω . Proposition D.27. (“large volume limit”) Assume β = 0 and ω ∈ NS(X) ample. Let E • ∈ Db (X) be such that r > 0 and c · ω > 0, where v(E • ) = (r, c, s). Then E • is semistable in σn = (Z0,nω , A0,nω ) for a sufficiently large n ∈ N if and only if E • is a shift of a ω-Gieseker semistable (torsion-free) sheaf on X. Proof. Observe that A0,nω = A0,ω = A and that Zn (r, c, s) = Z0,nω (r, c, s) = (rn2

ω2 − s) + inc · ω. 2

Assume E • σn -semistable for all n  0. Up to shift, we can assume E • ∈ A. The first step is to show that E • = E ∈ T0,ω and that it is torsion-free (hence µ− ω (E) > 0). For this, consider the exact sequence in A 0 → H−1 (E • )[1] → E • → H0 (E • ) → 0,

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with H−1 (E • ) ∈ F0,ω and H0 (E • ) ∈ T0,ω . Now H−1 (E • ) being torsion-free implies that lim (1/π) arg(Zn (H−1 (E • )[1])) = 1 > 0 = lim φn (E • ). n→∞

n→∞

Hence E • is not σn -semistable for n  0 unless H−1 (E • ) = 0. So, E • = E ∈ T0,ω . The proof that E is torsion-free is analogous. For the second step, assume that E is not ω-Gieseker semistable. Then there exists an exact sequence in Coh(X) 0 → G → E → B → 0, with G Gieseker semistable destabilizing E. Note that, since µ− ω (E) > 0, G, B ∈ A. Writing explicitly the condition that E is σn -semistable for n  0 gives, for v(G) = (r(G), c(G), s(G)) and v(E) = (r(E), c(E), s(E)),   ω2 c(E) · ω c(G) · ω − + (s(E)(c(G) · ω) − s(G)(c(E) · ω)) > 0 n2 r(G)r(E) 2 r(E) r(G) for n  0. As a consequence, either µω (G) < µω (E) or µω (G) = µω (E) and s(E) s(G) ≥ . r(E) r(G) Both cases lead to a contradiction. For the converse implication, assume E ∈ Coh(X) torsion-free and ω-Gieseker semistable. Let 0 → A• → E → B • → 0 be an exact sequence in A. Then we have an exact sequence in Coh(X) 0 → H−1 (B • ) → A• ' H0 (A• ) → E → H0 (B • ) → 0. • Hence µω (A• ) ≤ µ+ ω (A ) ≤ µω (E). We use now the following boundedness result (see [66, Lemma 14.3]): if F ∈ Coh(X) is a torsion-free ω-Gieseker semistable sheaf with µω (F) > 0, there exists an integer P ∈ N such that, for all proper subobjects 0 6= G • ,→ F in A, we have s(G • )/r(G • ) ≤ P .

After setting φn (−) = (1/π) arg(Zn (−)) ∈ [0, 1) from this we have, for all n,  ω2 • (r(A• )(c(E) · ω) − r(E)(c(A• ) · ω)) φn (E) − φn (A ) = n n2 2   c(A• ) · ω s(E) c(E) · ω s(A• ) − +r(A• )r(E) r(A• ) r(E) r(E) r(A• ) ( 0 if µω (A• ) = µω (E) ≥ n − µω (E)P if µω (A• ) < µω (E). Therefore, for n  0, φn (A• ) ≤ φn (E) for all subobjects 0 6= A• ,→ E in A and so  E is σn -semistable for all n  0.

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Remark D.28. The categories Aβ,ω , with ω, β ∈ NS(X)Q just introduced, are useful also in other circumstances. Indeed it has been shown in [154] that they behave “quite naturally” with respect to the category of coherent sheaves. Theorem 0.1 in [154] shows that two smooth projective K3 surfaces X and X 0 have equivalent derived categories if and only if there exists β, ω ∈ NS(X)Q , ω ∈ Amp(X) and β 0 , ω 0 ∈ NS(X 0 )Q , ω 0 ∈ Amp(X 0 ) such that Aβ,ω and Aβ 0 ,ω0 are equivalent (as Abelian categories). This fact is then applied to the question of preservation of stability under a Fourier-Mukai equivalence associated to a locally free universal family of µ-stable sheaves ([154, Thm. 0.3]). More precisely, let ω ∈ NS(X) be an ample divisor and assume X 0 is isomorphic to a fine moduli of µω -stable vector bundles on X with Mukai vector v = (r, c, s). Denote the universal family by E• : Db (X) → Db (X 0 ). E • ∈ Db (X × X 0 ) and the induced equivalence by Φ = ΦX→ X0 0 0 Then, there exists an ample divisor ω ∈ NS(X ) such that, for any µω -stable vector bundle E on X with µω (E) = −(c · ω)/r, one has either Φ(E) ' C(y)[−2] if [E ∨ ] = y ∈ X 0 or otherwise Φ(E) ' F[−1], for F a µω0 -stable vector bundle on X 0. These results have been generalized in [296] to obtain a general asymptotical theorem on preservation of stability. Unfortunately we cannot summarize here the complete result. We simply observe that, roughly, [296, Thm. 3.13] yields a complete result on preservation of stability under a Fourier-Mukai transform, provided the notion of µ-stability is replaced by that of twisted stability and the degree is sufficiently large (but universally bounded). The reader should compare this with the results in Section 4.4 4

D.3.3

The covering map property

In this section we prove that the map π : Stab(X) → P0+ (X) is a covering on the preimage of P0+ (X) in Stab(X). This will follow from Bridgeland’s deformation lemma and will be the key to proving Theorem D.19. ˜ 1,1 (X, Z)C and let f ∈ P + (X). Then there Lemma D.29. Let k − k be a norm on H 0 exists a real number r = r(f) > 0 such that |hu, vi| ≤ rkuk · |hf, vi|, ˜ 1,1 (X, Z)C and for all v ∈ H ˜ 1,1 (X, Z) ⊗ R with either v 2 ≥ 0 or for all u ∈ H v ∈ ∆(X). Proof. See [66, Lemma 8.1].



˜ 1,1 (X, Z)C is open and the restriction Proposition D.30. The subset P0+ (X) ⊂ H π : π −1 (P0+ (X)) ⊂ Stab(X) → P0+ (X)

D.3. Stability conditions on K3 surfaces

381

is a covering map (and, in particular, is surjective). Proof. Fix a norm k − k on NC (X). Take f ∈ P0+ (X) and let r = r(f) be as in Lemma D.29. For η > 0, define n o ˜ 1,1 (X, Z)C : kf0 − fk < η/r ⊂ H ˜ 1,1 (X, Z)C . Dη (f) = f0 ∈ H open

By Lemma D.29 (and some linear algebra), if η < 1 then Dη (f) ⊂ P0+ (X). Hence ˜ 1,1 (X, Z)C . Now, for all σ ∈ Stab(X) with π(σ) = f, define P0+ (X) is open in H  Cη (σ) = τ ∈ π −1 (Dη (f)) : f (σ, τ ) < 1/2 ⊂ Stab(X). open

By Bridgeland’s deformation lemma, for η > 0 sufficiently small π|Cη (σ) : Cη (σ) → Dη (f) is an homeomorphism. Indeed, for all E • ∈ Db (X) that are σ-stable, an easy application of the Riemann-Roch theorem shows that v(E • )2 ≥ −2, v(E • ) ∈ ˜ 1,1 (X, Z). By Lemma D.29 we have H |hf0 , v(E • )i − hf, v(E • )i| ≤ rkf0 − fk · |hf, v(E • )i| < η|hf, v(E • )i|, for f0 ∈ Dη (f) and for all E • ∈ Db (X) σ-stable, which is precisely (D.16). This implies that π −1 (P0+ (X)) is the union of full connected components of Stab(X) (in the sense of Remark D.18(ii)). By using in addition Lemma D.16, π|Cη (σ) is a homeomorphism for any η < sin(π) if  < 1/10. As a consequence of the uniform choice for η, we have G Cη (σ), π −1 (Dη (f)) = σ∈π −1 (f)

where the union is disjoint by Lemma D.15. Eventually we need to show that π is surjective on P0+ (X). By Lemma D.25, π −1 (P0+ (X)) is not empty. Indeed, π(σβ,ω ) ∈ P0+ (X), whenever β, ω ∈ NS(X)Q , ˜ 1,1 (X, Z)C . We have just proved that with ω ample. Set Γ = π(π −1 (P0+ (X))) ⊂ H Γ is open. We need to show that Γ is closed in P0+ (X). Take a convergent sequence fn → f in P0+ (X) with fn ∈ Γ, n ∈ N. Consider Dη (f), with η < (1/2) sin(π). Then there exist N ∈ N and σN ∈ Stab(X) such that π(σN ) = fN ∈ Dη (f). Hence, for all E • ∈ Db (X) σN -stable, we have |hf, v(E • )i − hfN , v(E • )i| < η|hf, v(E • )i| < η |hfN , v(E • )i| < 2η|hfN , v(E • )i|. 1−η Again by Lemma D.16 there exists σ ∈ Stab(X) such that π(σ) = f. Therefore Γ is a nonempty, open and closed subset of P0+ (X) and so, since P0+ (X) is connected,  Γ = P0+ (X).

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The proof of Proposition D.30 works also for the subset P0− (defined in the same way as P0+ but using the other connected component P − (X) of P(X)). However, there is no known example of stability condition whose central charge takes values in P0− (X). A connected component of Stab(X) is called good if it contains a point σ with π(σ) ∈ P0 (X) = P0+ (X) ∪ P0− (X). As we saw in the proof of the previous proposition, a good connected component is full. A stability condition will be called good if it lies in a good connected component.

D.3.4

Wall and chamber structure

As an interlude we briefly show that a good connected component of the space of stability conditions on a K3 surface has a wall and chamber structure for semistable objects very similar to the one of the ample cone. Let Stab∗ (X) be a good connected component of Stab(X). Definition D.31. A set of objects S ⊂ Db (X) has bounded mass with respect to Stab∗ (X) if sup {mσ (E • ) : E • ∈ S} < ∞ for some (and hence for all) σ ∈ Stab∗ (X).

4

An easy consequence of the definition is that the set of Mukai vectors of a set of objects with bounded mass is finite (see [66, Lemma 9.2]). Proposition D.32. Let S ⊂ Db (X) be a subset with bounded mass and let B ⊂ Stab∗ (X) be compact. There exists a finite collection {Wγ }γ∈Γ of (not necessarily closed) real codimension 1 submanifolds of Stab∗ (X) such that any connected S component C ⊂ B \ Wγ has the following property: if E • ∈ S is σ-semistable for some σ ∈ C, then E • is τ -semistable for all τ ∈ C. Moreover, if v(E • ) is primitive, then E • is τ -stable, for all τ ∈ C. Proof. We only show how to construct the walls. Define T as the set of nonzero objects G • in Db (X) for which there exist σ ∈ B and E • ∈ S such that mσ (G • ) ≤ mσ (E • ). For example, if G • is a σ-semistable HN factor of E • for some σ ∈ B, then G • ∈ T . Since B is compact, T has bounded mass. Then the set of Mukai vectors of T is finite. Let us denote it by {vi }i∈I . Set Γ = {(i, j) ∈ I × I : vi 6= αvj , for all α ∈ R} . Finally, for γ = (i, j) ∈ Γ, define Wγ = {σ = (Z, P) ∈ Stab∗ (X) : Z(vi )/Z(vj ) ∈ R>0 } . For the conclusion of the proof, see [66, Prop. 9.3].



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383

Proposition D.33. Let S ⊂ Db (X) be a subset with bounded mass and assume that for all E ∈ S, v(E) is primitive. Then the subset {σ ∈ Stab∗ (X) : all E • ∈ S are σ-stable} ⊂ Stab∗ (X) is open. Proof. See [66, Prop. 9.4].

D.3.5



Sketch of the proof of Theorem D.19

Define a subset U (X) ⊂ Stab(X) by U (X) = {σ ∈ Stab∗ (X) : σ is good and {Ox }x∈X are all σ-stable of the same phase} . We now list, mostly without proof, some important properties of the subset U (X). (i) By Proposition D.33, U (X) is open. By Example D.26, it is nonempty. (ii) By [66, Prop. 10.3], if σ ∈ U (X) there exists a unique element G ∈ + f (R) and there exist β, ω ∈ NS(X)R , ω ∈ Amp(X) such that σ · G = Gl 2 + f (R) is free on U (X). A section (Zβ,ω , Aβ,ω ). In particular, the action of Gl 2

is given by V (X) = (

˜ 1,1 (X, Z)C , ω ∈ Amp(X) π(σ) = exp(β + iω) ∈ H σ ∈ U (X) : φσ (Ox ) = 1 for all x ∈ X

) .

(iii) By [66, Prop. 11.2], the map π : V (X) → L(X), where L(X) = {exp(β + iω)} ˜ 1,1 (X, Z)R and ω ∈ Amp(X) as in Lemma D.24, is a homeowith β, ω ∈ H morphism. In particular, all stability conditions arising from the construction of Lemma D.24 admit HN-filtrations and are locally finite. Moreover, an easy check shows that π(U (X)) ⊂ P0+ (X) and that σ ∈ U (X) is uniquely determined by π(σ) up to even shifts. By (iii), U (X) is connected. Let Stab† (X) be the connected component of Stab(X) which contains U (X). By definition, it is a good connected component. The important fact about U (X) is that a “general” point of its boundary in Stab† (X) can be explicitly described (see [66, Sec. 12,13]). From this, we can deduce the following additional two facts:

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Appendix D. Stability conditions for derived categories

(iv) For all τ ∈ Stab† (X), there exists Φ ∈ Aut(Db (X)) such that f Φ ((0, 0, 1)) = ˜ 1,1 (X, Z) and Φ(τ ) ∈ U (X), where U (X) denotes the closure of (0, 0, 1) in H U (X) in Stab† (X). (v) π(Stab† (X)) = P0+ (X). We can now complete the proof of Theorem D.19. Sketch of the proof of Theorem D.19. The first part of the claim follows from Proposition D.30 and (v) above. Moreover, clearly Aut†0 (Db (X)) preserves π. Hence we only have to show that, if σ, τ ∈ Stab† (X) are such that π(σ) = π(τ ), there exists a b unique Φ ∈ Aut+ 0 (D (X)) such that Φ(σ) = τ . Since π is a covering, it is sufficient to prove this for σ = σβ,ω as in Section D.3.2. Uniqueness: assume Φ(σ) = σ. Then, since Ox is σ-stable of phase 1, for all x ∈ X, Φ(Ox ) is σ-stable of phase 1, too. But an easy check shows that the only stable objects in σ of phase 1 and Mukai vector (0, 0, 1) are the skyscraper sheaves. Hence, for all x ∈ X, there exists y ∈ X such that Φ(Ox ) ' Oy . By Corollary 1.12, Φ ' (L ⊗ −) ◦ f ∗ , for f ∈ Aut(X) an automorphism of X and L ∈ Pic(X). Since, by assumption, f Φ = Id, then L ' OX . But then, by the Torelli theorem for K3 surfaces (Theorem 4.10), Φ = Id, as wanted. Existence: assume that τ ∈ Stab† (X) satisfies π(τ ) = π(σ). By (iv), there ˜ 1,1 (X, Z) and exists Φ ∈ Aut(Db (X)) such that f Φ ((0, 0, 1)) = (0, 0, 1) in H Φ(τ ) ∈ U (X). By suitably modifying σ we can assume that both σ and Φ(τ ) are in U (X). Now, since f Φ is an isometry with respect to the Mukai pairing, it can be easily checked that f Φ ((1, 0, 0)) = exp(c1 (L)), for some L ∈ Pic(X). Hence, by composing Φ with the functor L ⊗ − (which preserves U (X)), we can e Z) preserves assume f Φ ((1, 0, 0)) = (1, 0, 0). Therefore, the action of Φ on H(X, 0 2 4 the grading H (X, Z) ⊕ H (X, Z) ⊕ H (X, Z). Since σ and Φ(τ ) are in U (X), the induced Hodge isometry on H 2 (X, Z) is effective, i.e., it preserves the K¨ahler cone of X. Again, by the Torelli theorem, there exists an automorphism f ∈ Aut(X) such that f∗ = f Φ on H 2 (X, Z). By composing with f∗ , we can eventually assume f Φ = Id. But so π(σ) = π(Φ(τ )). By (iii), σ = Φ(τ )[2k], for some integer k ∈ Z. Since the shift by 2 functor induces the identity in cohomology, we are done.  We conclude this section by presenting, without proof, an example (which is a particular case of a more general construction given by Meinhardt and Partsch in [217]) of a space of stability conditions which is not connected. Example D.34. Let X be a smooth projective K3 surface over C. Consider the b (X) defined as the Verdier quotient of Db (X) by its thick derived category D(1) subcategory consisting of complexes whose cohomologies are supported in codib (X) is equivalent mension greater than 1. In [217, Thm. 1.2] it is shown that D(1) to the bounded derived category of Coh(1) (X), which is an Abelian category of dimension 1 obtained as a quotient of Coh(X) by sheaves supported in codimension

D.4. Moduli stacks and invariants of semistable objects on K3 surfaces

385

greater than 1. Let Stab(X(1) ) be the space of locally finite numerical stability b (X). Then, by [217, Thm. 1.3], Stab(X(1) ) is isomorphic to a conditions on D(1) +

f (R)-orbits disjoint union of free Gl 2 Stab(X(1) ) '

G

+

f (R), σω · Gl 2

ω∈C(X)/R>0

where C(X) = {ω ∈ NS(X)R : inf{ω · D : D ⊂ X effective divisor on X} > 0} , R>0 acts on C(X) by multiplication, and σω has heart Coh(1) (X) and stability function Zω (E) = −ω · c(E) + ir(E), for v(E) = (r(E), c(E), s(E)), E ∈ Coh(X). 4

D.4

Moduli stacks and invariants of semistable objects on K3 surfaces

In this section we examine a further problem related to stability conditions: the construction of moduli spaces of stable objects. This will require some notions and techniques, such as those of stacks, 2-categories and 2-functors, that we have not used in this book. For quite readable introductions to these topics, the reader may refer to [193, 126]. We concentrate on the case of K3 surfaces described in the previous section, where the situation is reasonably well understood, thanks to the beautiful paper [281] (but see also [7], where some specific moduli spaces are studied in more detail). In the final part we also sketch the problem of studying invariants by counting semistable objects. The main results about K3 surfaces presented here were conjectured by D. Joyce in [169] and have been proved by Y. Toda in [281]. Some computations of such invariants are contained in [218]. For very recent developments about the very interesting subject of counting invariants and wall-crossing formulas see, for example, [169, 167, 72, 244, 245, 280, 31, 279, 172].

D.4.1

Moduli stack of semistable objects

Let X be a smooth projective K3 surface over C. Let SchC be the site of locally Noetherian schemes over C (endowed with the ´etale topology). Define a 2-functor, with values in the category Grp of groupoids, M : SchC → Grp, by mapping a Cscheme S to the groupoid M(S) whose objects consist of those E • ∈ DS-perf (X ×S) which satisfy Exti (E • s , E • s ) = 0, for all i < 0 and all s ∈ S. Here DS-perf (X × S) ⊂ D(OX×S -Mod)

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Appendix D. Stability conditions for derived categories

denotes the derived category of S-perfect complexes (an S-perfect complex is a complex of OX×S -modules which, locally over S, is quasi-isomorphic to a bounded complex of coherent sheaves which are flat over S) and E • s is the derived restriction of E • to X × {s} ,→ X × S. The main theorem in [197] (generalizing results in [160]) shows that M is an Artin stack, locally of finite type over C. ˜ 1,1 (X, Z). We define a sub Fix σ = (Z, P) ∈ Stab† (X), φ ∈ R, and v ∈ H 2-functor M(v,φ) (σ) ⊂ M by considering the objects E • ∈ DS-perf (X × S) whose restrictions E • s belong to P(φ) and have Mukai vector v for all s ∈ S. The main result of this section, [281, Thm. 1.4], is a first step in understanding the first question at page 368. Theorem D.35. The 2-functor M(v,φ) (σ) is an Artin stack of finite type over C. We shall presently describe a proof of this theorem. Example D.36. Take σ ∈ U (X), v = (0, 0, 1), and φ = 1. By what we have seen in Section D.3, the only semistable objects verifying these conditions are the skyscraper sheaves, which are σ-stable and have as automorphism group the torus Gm ' C∗ . Hence M((0,0,1),1) (σ) ' [X/Gm ] , 4 1,1 ˜ Example D.37. Let ω ∈ NS(X) be an ample divisor. Take v = (r, c, s) ∈ H (X, Z) with r > 0, c · ω > 0 and gcd{(exphnω), vi : n ∈ Z} = 1. Proposition 6.4 and Lemma 6.5 in [281] show the existence of a universal bound for the situation described in Example D.27. Namely, there exists a positive integer N such that the semistable objects in σN = σ0,N ω that are in A0,ω with Mukai vector v are precisely the ω-Gieseker stable (torsion-free) sheaves on X with Mukai vector v. Denote by M v (ω) their fine moduli space. Notice that M v (ω) is a smooth, projective, symplectic variety. We have where Gm acts trivially on X.

M(v,φ) (σN ) ' [M v (ω)/Gm ] , where again Gm acts trivially on M v (ω) and φ ∈ (0, 1] is such that Z0,N ω (v)/|Z0,N ω (v)| = exp(iπφ) . 4

D.4.2

Sketch of the proof of Theorem D.35

We divide the proof in a few steps. Fix σ = (Z, P) ∈ Stab† (X), φ ∈ R, and ˜ 1,1 (X, Z). Denote by M (v,φ) (σ) the subset of objects in Db (X) consisting of v∈H the semistable objects in σ of phase φ and Mukai vector v.

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387

Step 1. (Understanding the problem.) Let us recall the definition of boundedness for a set of objects in Db (X). Let S ⊂ Db (X) be a set of objects in Db (X). We say that S is bounded if there exists a C-scheme Q of finite type and F • ∈ DQ-perf (X × Q) such that any object E • ∈ S is isomorphic to F • q for some closed point q ∈ Q. Lemma D.38. Assume that the following two conditions hold: (i) M(v,φ) (σ) ⊂ M is an open substack. (ii) M (v,φ) (σ) ⊂ Db (X) is bounded. Then M(v,φ) (σ) is an Artin stack of finite type over C. Proof. Let M → M be an atlas of M. The openness of M(v,φ) (σ) implies that there is an open subset M 0 ⊂ M which gives a surjective smooth morphism M 0 → M(v,φ) (σ). Hence M(v,φ) (σ) is an Artin stack, locally of finite type over C. Moreover, the boundedness of M (v,φ) (σ) yields a surjection Q → M 0 , where Q is a scheme of finite type over C. But then M 0 is of finite type, too, and then  M(v,φ) (σ) becomes an Artin stack of finite type over C. Hence, by Lemma D.38, to prove Theorem D.35 we only need to show that conditions (i) and (ii) in the statement hold. The first step is to give a sufficient condition for (i). Lemma D.39. Assume that the following is true: (i’) if, for a smooth quasi-projective variety S and E ∈ M(S), the locus S 0 = {s ∈ S : v(E • s ) = v and E • s ∈ P(φ)}

(D.5)

is nonempty, then it contains a nonempty open subset of S. Then condition (i) above holds. Proof. Since M is an Artin stack of locally finite type over C, it is sufficient to prove that, for any affine scheme S of finite type over C and for any E ∈ M(S), the locus S 0 , defined analogously as in (D.5), is open in S. Assume such S 0 is nonempty. By using resolution of singularities and (a0 ), it is not difficult to see that there exists a nonempty open subset U1 ⊂ S such that U1 ⊂ S 0 . Let Z1 = S \ U1 . If Z1 ∩ S 0 is empty, we have S 0 = U1 and we are done. Assume Z1 ∩ S 0 6= ∅. Take the pull-back EZ1 ∈ M(Z1 ) of E to Z1 and apply the same argument. Then we obtain an open subset U2 ⊂ Z1 such that U2 ⊂ (Z1 ∩ S 0 )

388

Appendix D. Stability conditions for derived categories

and a closed subset Z2 = Z1 \ U2 . Repeating the same argument again, we get a sequence of closed subsets in S . . . ⊂ Zn ⊂ Zn−1 ⊂ . . . ⊂ Z1 which must terminate since S is Noetherian. Then Z = ∩i Zi is a closed subset of  S and we have S 0 = S \ Z, which is open as wanted. Step 2. (Reduction of (i’) to generic flatness for algebraic stability conditions which verify (ii).) A stability condition σ ∈ Stab(X) is called algebraic if the image of its central charge is contained in Q + iQ ⊂ C. For example, if β, ω ∈ NS(X)Q , with ω ample, then σβ,ω (when it exists, e.g., under the assumption of Lemma D.24) is algebraic. Note that the heart of an algebraic stability condition is a Noetherian Abelian category by [1, Thm. 5.0.1]. Let S be a smooth projective variety over C and let L be an ample line bundle (for generalizations to arbitrary schemes, but giving up some results that will be important for us, see [253]). The authors of [1], given any Noetherian heart A of a bounded t-structure on Db (X), construct a heart of a bounded t-structure on Db (X × S) which is Noetherian and independent of the choice of L:  AS = F ∈ Db (X × S) : (pX )∗ (F ⊗ p∗S (Ln )) ∈ A, for all n  0 , where pS and pX denote the two projections from X × S to S and X, respectively. Moreover, if S is smooth and quasi-projective, we can define AS as the essential image of AS under the functor (Id × j)∗ : Db (X × S) → Db (X × S), where S is a smooth compactification of S and j : S ,→ S denotes the natural inclusion. It can be proved (see [1], Theorem 2.7.2 and Lemma 3.2.1) that AS is the heart of a bounded t-structure and that its definition is independent of the chosen smooth compactification. Lemma D.40. Let σ = (Z, P) ∈ Stab† (X) be an algebraic stability condition such that M (v,φ) (σ) is nonempty. Assume that generic flatness holds for Aφ = P((φ − 1, φ]), that is, that for a smooth quasi-projective variety S and E • ∈ AφS , there is an open subset U ⊂ S such that, for each s ∈ U , E • s ∈ Aφ . Then condition (i’) holds. Proof. Note that since σ algebraic and M (v,φ) (σ) is nonempty, Aφ is a Noetherian Abelian category. In particular, it makes sense to consider AφS for a smooth quasiprojective variety S. Let E • ∈ M(S) and assume the locus S 0 in (D.5) nonempty. Take s ∈ S 0 . Then E • s ∈ P(φ) ∈ Aφ . By the open heart property (see [1, Thm. 3.3.2]), there

D.4. Moduli stacks and invariants of semistable objects on K3 surfaces

389

exists an open neighborhood s ∈ U ⊂ S such that E • U ∈ AφU . By applying the generic flatness condition to U , we have the existence of an open subset U 0 ⊂ U ⊂ S such that for each s0 ∈ U 0 , E • s0 ∈ Aφ . Since v(E • s0 ) = v(E • s ), we have Z(E • s0 )/|Z(E • s0 )| = exp(iπφ). But then E • s0 is semistable, i.e., E • s0 ∈ P(φ). This  shows that U 0 ⊂ S 0 . We now want to reduce the condition in this lemma to generic flatness for P((0, 1]). This is the main technical result in [281]. Lemma D.41. Let σ = (Z, P) ∈ Stab† (X) be an algebraic stability condition such that M (v,φ) (σ) is nonempty. Assume the following two conditions are satisfied: (a) Generic flatness holds for A = P((0, 1]), i.e., for a smooth quasi-projective variety S and E ∈ AS , there is an open subset U ⊂ S such that, for each s ∈ U , Es ∈ A. (b) M (v,φ) (σ) ⊂ Db (X) is bounded. Then generic flatness holds for Aφ = P((φ − 1, φ]). Proof. See [281], Proposition 3.18.



Hence, in particular, if (a) and (b) hold for an algebraic stability condition σ, then M(v,φ) (σ) is an Artin stack of finite type over C. Step 3. Here we state, without proving, the main reduction step: Theorem D.42. Assume (a) and (b) hold for all stability conditions in Stab† (X) of the form σβ,ω with β, ω ∈ NS(X)Q . Then M(v,φ) (σ) is an Artin stack of finite ˜ 1,1 (X, Z). type over C, for all σ ∈ Stab† (X), φ ∈ R and v ∈ H Proof. See[281], Theorem 1.3



At this point, to complete the proof of Theorem D.35 we only need to show that boundedness and generic flatness for Aβ,ω hold for σβ,ω , β and ω rational. One should note that the previous theorem is stated in [281] in a greater generality, namely, for any smooth projective varieties X; the set of stability conditions of the form σβ,ω with β and ω rational is replaced by a subset of algebraic stability conditions which is dense in a fundamental domain for the action of the autoequivalence group. However a further assumption on subsets of bounded mass must be added (see [281], Assumption 3.1). Step 4. (Proof of generic flatness for σβ,ω , with β, ω ∈ NS(X)Q .) We first sketch the proof of generic flatness for A = Aβ,ω , β, ω ∈ NS(X)Q , ω ∈ Amp(X).

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Appendix D. Stability conditions for derived categories

Lemma D.43. For a smooth quasi-projective variety S and E • ∈ AS , there is an open subset U ⊂ S such that E • s ∈ A for all s ∈ U . Proof. Clearly we can assume that S is projective. Pick an ample line bundle L ∈ Pic(X) and assume E • ∈ AS . By definition, we have (pX )∗ (E • ⊗ p∗S (Ln )) ∈ A, for all n  0. By using the spectral sequence E2p,q = Ri pX∗ (Hq (E • ) ⊗ p∗S (Ln )) =⇒ Rp+q pX∗ (E • ⊗ p∗S (Ln )), the fact that pX∗ (E ⊗ p∗S (Ln )) has only nonzero cohomologies in degree −1 and 0 implies that the same holds true for E • . The existence of relative Harder-Narasimhan filtrations (see, e.g.,Thm. 2.3.2 in [155]) gives an open subset U ⊂ S and filtrations by coherent sheaves 0 = F 0 ⊂ F 1 ⊂ . . . ⊂ F k−1 ⊂ F k = H−1 (E • )U , 0 = T 0 ⊂ T 1 ⊂ . . . ⊂ T l−1 ⊂ T l = H0 (E • )U

(D.6)

such that all F i and T i are flat sheaves on U . Moreover, for all s ∈ U , the filtrations in (D.6) give the Harder-Narasimhan filtrations (with respect to ωGieseker stability) of H−1 (E • )s and of H0 (E • )s , respectively. Now, using the definition of Aβ,ω and Proposition 3.5.3 in [1] (which roughly says that (a) holds for a dense subset in S), it is easy to see that E • s ∈ Aβ,ω for all s ∈ U .  Step 5. (Proof of (b) for σβ,ω , with β, ω ∈ NS(X)Q .) Finally, we are reduced to show boundedness for σβ,ω ∈ Stab† (X), with β, ω ∈ NS(X)Q . Lemma D.44. The subset M (v,φ) (σβ,ω ) ⊂ Db (X) is bounded. Proof. We only give the basic ideas of the main argument. For the details we refer to Section 4.5 of [281]. First of all, by shifting, it is sufficient to show the boundedness of S = {E • ∈ Aβ,ω : v(E • ) = v and E • semistable in σβ,ω } . Consider the following three sets of objects:  T 0 = H0 (E • )tor : E • ∈ S ,  T = H0 (E • )/H0 (E • )tor : E • ∈ S ,  F = H1 (E • ) : E • ∈ S .

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391

Clearly it is sufficient to show that each of the previous subsets is bounded. This is achieved by showing that the sets of Mukai vectors of possible µω -semistable factors of every object in either T or F are finite and similarly that the same is true for (β, ω)-twisted semistable factors of every object in T 0 (for the notion of twisted stability, see [213]). Hence the boundedness of T 0 , T , and F follows from the corresponding one for the usual notions of stability for sheaves.  By Theorem D.42, this completes the proof of Theorem D.35.

D.4.3

Counting invariants and Joyce’s conjecture for K3 surfaces

In this section we try to understand the second question at page 368. First we briefly recall the work of Joyce in [169]. Let K(VarC ) be the Grothendieck ring of quasi-projective varieties over C, i.e., the Z-module generated by the isomorphism classes of quasi-projective varieties with relations generated by X − Y − (X \ Y ) for closed subschemes Y ⊂ X. The product is induced by the formula X · X 0 = X × X 0 . Suppose Λ is a commutative associative Q-algebra (with identity 1) and γ : K(Var) → Λ a motivic invariant of varieties, i.e., a ring homomorphism. Set L = γ(A1C ). We assume that both L and Lk − 1 are invertible, for k ≥ 1. An example is given by Λ = Q(z) with γ given by the virtual Poincar´e polynomial of X (for X smooth and projective this is nothing but the usual Poincar´e polynomial). Let X be a smooth and projective K3 surface over C. Fix ω ∈ NS(X) an ˜ 1,1 (X, Z) and fix a motivic invariant of varieties γ. In [169, ample divisor and v ∈ H Sect. 6] is constructed a (weighted) system of invariants Jˆv (ω) ∈ Λ “counting” ωGieseker semistable sheaves E ∈ Coh(X) with Mukai vector v(E) = v. For example, if we are in the situation as in Example D.37, then Jˆv (ω) = γ(M v (ω)). In [169, Thm. 6.24] it is shown that Jˆv (ω) does not depend on ω. Denote it by Jˆv . Now the natural question [169, Conj. 6.25] is whether Joyce’s theorem generalizes to Bridgeland’s stability conditions. The answer is yes: Theorem D.45. (Toda) Fix a motivic invariant γ : K(Var) → Λ for some commu˜ 1,1 (X, Z), there exists tative associative Q-algebra Λ. For σ ∈ Stab† (X) and v ∈ H v a weighted system of invariants J (σ) ∈ Λ “counting” semistable objects in σ with Mukai vector v, such that (i) J v (σ) does not depend on the choice of σ.   v ˜ 1,1 →H (X, Z) , then J v (σ) = Jˆv (σ). (ii) If v ∈ C(X) = im Coh(X) − In the next section we shall sketch how to construct the invariants J v (σ) and prove Theorem D.45. Here we only make a few comments. Denote by Aut† (Db (X))

392

Appendix D. Stability conditions for derived categories

the subgroup of the autoequivalence group of Db (X) consisting of the autoequivalences which preserve the connected component Stab† (X). Then, by Theorem D.45, for all Φ ∈ Aut† (Db (X)), we have J v (σ) = J f

Φ

(v)

(Φ(σ)) = J f

Φ

(v)

for some (and then any) σ ∈ Stab† (X). In particular this may be useful for constructing some interesting automorphic functions on Stab† (X), i.e., functions which are invariant under autoequivalences. An example, as pointed out in [169], is provided by the map fk : Stab† (X) → Λ ⊗Z C (k ∈ Z) defined by (ignoring convergence problems) σ = (Z, P) 7→

X ˜ 1,1 (X,Z)\{0} v∈H

J v (σ) . Z(v)k

As pointed out in the Introduction to this appendix, it seems quite interesting to understand how this generalizes to higher dimensions. More precisely, one is interested in defining invariants for Calabi-Yau threefolds arising from moduli spaces of “stable” objects in Db (X) (see Conjecture 6.30 and Section 7 in [169]). Here the word “invariant” refers to invariance with respect to deformations (i.e., these objects should behave like generalized Donaldson-Thomas invariants) and with respect to the change of stability conditions (i.e., there should exist wall-crossing formulas which explain how they vary under a change of the stability condition). At the moment these constructions turn out to be quite problematic. Most such problems are described in detail in Sections 6 and 7 of [169]. Here we just point to some recent literature. For examples, explanations, and relevant conjectures the reader is referred to [244, 245]. For the “stability condition” interpretation of these results see [31, 279] (where a different notion of stability condition on derived categories is introduced: a sort of limit of the notion of Bridgeland stability condition as presented in this note). For an attempt in understanding Joyce’s work via certain geometric structures on the space of stability conditions (but only for Abelian categories and under certain finiteness conditions) see [167, 72]. A more general approach which also covers derived categories of Calabi-Yau threefolds is given in [172].

D.4.4

Some ideas from the proof of Theorem D.45

For the construction of the invariants J v (σ) we need to extend a motivic invariant of varieties to stacks. Here we shall be quite sketchy; the reader is referred to [168, 281] for full details. Consider the ring SF(StC ) defined as the Q-vector spaces generated by the isomorphism classes of Artin stack of finite type over C with affine stabilizers modulo the ideal generated by the relations X − Y − (X \ Y) for closed

D.4. Moduli stacks and invariants of semistable objects on K3 surfaces

393

substacks Y ⊂ X . The product is induced by the Cartesian product of stacks over C. An algebraic C-group G will be called special if every principal G-bundle is locally trivial. In Theorem 4.9 in [168] it is shown that, given a motivic invariant of varieties γ : K(VarC ) → Λ, there exists a unique morphism of Q-algebras γ 0 : SF(StC ) → Λ such that, if G is a special group (and so γ(G) is invertible in Λ) acting on a quasi-projective variety X, then γ 0 ([X/G]) = γ(X)/γ(G). ˜ 1,1 (X, Z), φ ∈ R, and σ = (Z, P) ∈ Stab† (X). By Theorem Take now v ∈ H D.35, M(v,φ) (σ) is an Artin stack of finite type over C. Definition D.46. Given a motivic invariant γ, define I v (σ) ∈ Λ as follows: ( γ(M(v,φ) (σ)), if Z(v) 6= 0 and Z(v)/|Z(v)| = exp(iπφ), I v (σ) = 0, if Z(v) = 0. 4 Notice that, in the previous definition, I v (σ) is independent of the choice of φ. To introduce the invariants J v (σ) we still need to introduce some definitions. ˜ 1,1 (X, Z)) \ {0}. Set Cσ (φ) = im(P(φ) → H Definition D.47. Let {Cv }v∈H˜ 1,1 (X,Z) be a set of formal variables parameterized ˜ 1,1 (X, Z). Define: by H L

(i) a ring H =

Λ · Cv with product ∗ induced by

˜ 1,1 (X,Z) v∈H 0

Cv ∗ Cv0 = Lhv,v i · Cv+v0 , ˜ 1,1 (X, Z), where L = γ(A1 ). for all v, v 0 ∈ H C (ii) δ v (σ) = I v (σ) · Cv . (iii)

v

 (σ) =

  

P v1 +...+vm

(−1)m−1 m

δ v1 (σ) ∗ . . . ∗ δ vm (σ),

if Z(v)/|Z(v)| = exp(iπφ),

vi ∈Cσ (φ)

  0,

otherwise. 4

It is not too difficult to check that the sum in the definition of v (σ) is finite (see Lemma 5.12 in [281]). Then v (σ) = B · Cv , for some B ∈ Λ. Define J v (σ) ∈ Λ

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Appendix D. Stability conditions for derived categories

by J v (σ) = B · (L − 1). More explicitly, we have m (−1)m−1 (L − 1) j>ihvj ,vi i Y vi L I (σ). m i=1 P

J v (σ) =

X v1 +...+vm vi ∈Cσ (φ)

The definition of Jˆv (ω) in [169, Def. 6.22] is analogous (replacing M(v,φ) (σ) with the stack Mv (ω) of ω-Gieseker semistable sheaves of Mukai vector v and the condition vi ∈ Cσ (φ) with the condition that the Hilbert polynomials are the same). To prove Theorem D.45, we first need to check that the previous definition of J v (σ) is indeed independent from σ. Take σ and τ in Stab† (X). Choose a path α : [0, 1] → Stab† (X) such that α(0) = σ and α(1) = τ . Consider a open subset B 0 ⊂ Stab† (X) such that α([0, 1]) ⊂ B 0 and its closure B is compact. Define a subset S ⊂ Db (X) by  S = E • ∈ Db (X) there exists σ 0 = (Z 0 , P 0 ) ∈ B such that E • is σ-semistable with |Z 0 (E • )| ≤ |Z 0 (v)|} . Since B is compact, then S has bounded mass. By Proposition D.32, there exists a wall and chamber structure {Wl } on B with respect to S. We may assume that the set of points K ⊂ [0, 1] on which σt = (Zt , Pt ) = α(t) is algebraic and Pt ((ψ − 1, ψ]) satisfies generic flatness for any ψ such that P(ψ) is nonempty, is dense in [0, 1]. Take s0 , s1 , . . . , sN +1 ∈ [0, 1] and t± i ∈ (si , si+1 ) ∩ K such that • For 1 ≤ i ≤ N , si ∈ Wl for some Wl and s0 = 0, sN +1 = 1. • For any t ∈ (si , si+1 ), we have α(t) ∈ / Wl , for all l. • σt+ ∈ B (σsi+1 ), σt− ∈ B (σsi ), with  > 0 fixed sufficiently small. i

i

Hence we only need to show the following two cases: (i) σ and τ are in the same chamber. (ii) σ is contained in a chamber and τ is in a boundary of that chamber. Case (i) is proved in [281, Prop. 5.17]. The basic idea is that an object in S is semistable in σ if and only if it is semistable in τ . Case (ii) is proved in [281, Prop. 5.23], by translating a combinatorial argument of Joyce in [169] to the algebra H previously defined, which shows that v (σ) = v (τ ) if σ and τ are sufficiently close (which we may assume by case (i)). ˜ 1,1 (X, Z), define J v = J v (σ) for some (any) stability condition For v ∈ H

σ ∈ Stab† (X). To conclude the proof of Theorem D.45 we only need to show that

D.4. Moduli stacks and invariants of semistable objects on K3 surfaces

395

if v ∈ C(X), then J v = Jˆv . This is proved in [281, Sect. 6]. The main point is essentially a generalization of Example D.37. Indeed, if v ∈ C(X), then it is easy to see that, up to tensoring by a line bundle (operation which does not change J v , i.e., for a line bundle L ∈ Pic(X), J v = J v·ch(L) and Jˆv = Jˆv·ch(L) ), we can reduce to the case where v = (r, c, s) with either ω · c > 0 (ω an ample divisor) or r = c = 0. Then to prove the theorem, it is enough to compare J v (σ0,kω ) and Jˆv (ω), for k  0. This is done in Proposition 6.4 and Lemma 6.5 of [281], by showing first that, in the above situation, if φk ∈ (0, 1] is such that Z0,kω (v)/|Z0,kω (v)| = exp(iπφk ), then there exists an integer N > 0 such that for all k ≥ N and all 0 v 0 ∈ Cσ0,kω (φk ) with |=(Z0,ω (v 0 ))| ≤ |=(Z0,ω (v))|, any E ∈ M (v ,φk ) (σ0,kω ) is a ωGieseker semistable sheaf. Then, vice versa, if v 0 has the same Hilbert polynomial as v and |=(Z0,ω (v 0 ))| ≤ |=(Z0,ω (v))|, any ω-Gieseker semistable sheaf of Mukai vector v 0 is σ0,kω -semistable. A short technical computation yields the desired equality between J v (σ0,kω ) and Jˆv (ω).

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Subject Index 1-irreducible, 159 Abelian Fourier-Mukai transform, 85 schemes, 188 variety, 82 Adjoint bundle, 149 functors, 12, 282 ALE spaces, 182 Algebraic group, 82 Ample sequences, 35 Associated bundle, 149 Atiyah-Ward correspondence, 176 Automorphy factor, 162 Base change for integral functors, 8, 186 in derived category, 331 Be˘ılinson resolution of the diagonal, 45 Bogomolov number, 223 Category, 281 k-bilinear, 284 Abelian, 285 additive, 283 derived, 297 of a dg-catgeory, 310 dg-derived of a dg-category, 311 differential graded, 307 essentially small, 363 homotopy, 289 large, 281 locally small, 281 numerically finite, 367 of complexes, 287

of dg-categories, 308 of finite length, 364 of finite type, 366 quasi-Abelian, 286 saturated, 15 small, 281, 363 triangulated, 306 Chern character in derived category, 3 relative, 190 connection, 153 Clifford bundle, 155 Cohomology long exact sequence, 290 of a complex, 288 Complex, 287 double, 292 of finite homological dimension, 303 of finite Tor-dimension, 6, 326 of homomorphisms, 288 perfect, 3, 303 simple (associated with a double complex), 292 tensor product, 293 Cone of a morphism of complexes, 289 Connection on a principal bundle, 149 flat, 148 irreducible, 149 on a vector bundle, 148 reducible, 149 Convolution of complexes, 40 of kernels, 5

420 Crepant birational map, 235 morphism, 235 Curvature, 148 D-equivalence implies K-equivalence, 69 Decomposable triangulated category, 32 Derived direct image, 317 homomorphism functor, 318 inverse image, 327 tensor product, 324 dg-category, 307 dg-derived, 311 dg-functor, 307 dg-module, 309 cofibrant, 310 fibrant, 310 Dirac Laplacian, 159 operator, 155, 159 Discriminant group of a lattice, 252, 341 Dual Abelian variety, 83 Duality in derived categories, 63, 347 Elliptic fibration, 190 surface, relatively minimal, 190 Equivalence of categories, 14, 282 Essential image of a functor, 300 Essentially small category, 363 Euler characteristic of two complexes, 4 Exact triangle in derived category, 303 Exponent, of an isogeny, 82 Finite homological dimension of a complex, 303 Tor-dimension of a complex, 6, 326 of a morphism, 328 Finiteness, of the number of Fourier-Mukai partners of an algebraic surface, 242 Fitting ideal, 217, 355 Flat base change in derived category, 331

Subject Index Fourier-Mukai functor, 60 partners, 61 of a curve, 242 of a K3 surface, 249 of a Kummer surface, 250 of a nonminimal projective surface, 256 of a surface of Kodaira dimension −∞ and not elliptic, 246 of a surface of Kodaira dimension 2, 245 of a surface of Kodaira dimension 1, 248 of an elliptic surface, 248 transform, 60 on Abelian schemes, 188 on Abelian varieties, 85 on K3 surfaces, 120 Fully faithful integral functors, 15 Functor, 282 k-bilinear, 285 additive, 284 cohomological, 290, 306 of finite type, 15 derived, 312 exact full faithfulness of, 33 of Abelian categories, 286 of triangulated categories, 306 left derived, 313 of dg-categories, 307 representable, 282 right derived, 313, 314 Gauge group, 150 Genus, of a lattice, 341 Green function, 153 Harder-Narasimhan filtration, 363 Heart of a stability condition, 363 of a t-structure, 264 Hilbert scheme of points, 142 Hitchin-Kobayashi correspondence, 153 Hodge

Subject Index duality, 150 isometry, 115 Homogeneous bundles on Abelian varieties, 90 sheaf (on a K3 surface), 132 Homological dimension of a complex, 336 finite, 303 Homomorphisms of Abelian varieties, 82 Homotopy category of a dg-category, 308 Hyperk¨ ahler manifold, 173 Indecomposable triangulated category, 32 Index of a singular variety, 235 Index theorem (Atiyah-Singer), 156 for families, 157 Instanton, 150 Integral functor, 5 full faithfulness of, 20 relative, 8, 184 relative, for Weierstraß fibrations, 193 Isogeny, 82 Isometric isomorphism (of Abelian varieties), 254 Isotropic embeddings of moduli spaces, 105 IT condition, 7, 168, 179 Jacobian of a Weierstraß fibration, 197 Jordan-H¨ older filtration, 365 K-equivalence, 68 K3 surface, 112 branched over a sextic, 113 K¨ ahlerian, 114 Kummer, 113 quartic, 113 reflexive, 124 strongly reflexive, 125 K¨ unneth formula in derived category, 334 Kernel of an integral functor, 5 convolution of, 5 relative, 64 strongly simple, 19 uniqueness of, 51 Kodaira

421 conjecture, 114 dimension, 65 numerical dimension, 66 Kodaira-Spencer map, 17 for families of complexes, 259 Koszul algebra, 46 line bundle, 47 Kummer surface, see K3 surface, Kummer Fourier-Mukai partners of a, 250 Lagrangian embeddings of moduli spaces, 105 Lattice, 339 E8 , 340 genus of, 341 unimodular, 340 Length of a bounded complex, 336 Marking of a K3 surface, 115 Minimal surface, 244 Modified support, 214, 356 Moduli space of instantons, 151 of K3 surfaces, 116 of semistable pure sheaves, 353 Morphism of finite Tor-dimension, 328 of functors, 282 Motivic invariant, 391 Mukai pairing, 4 vector, 3 N´eron-Severi group, 114, 162 Nahm transform, 158 Nodal curves, 114 Nondegenerate line bundles (on an Abelian variety), 92 Normalization, of the Poincar´e bundle, 84 Numerical Kodaira dimension, 66 Numerically effective line bundle, 65

422 equivalent, 267 Orbit space, 151 Orlov’s representability theorem, 44 Overlattice, 342 Parseval formula (preservation of the Ext groups), 23 Period domain, for K3 surfaces, 115 map, for K3 surfaces, 115 Perverse point sheaf, 268 ideal sheaf, 267 sheaf, 265 structure sheaf, 267 Picard functor, 83 lattice, 114 number, 114 sheaves, 95 Poincar´e bundle, 83, 163 relative sheaf, 193 sheaf, 95 Polarization of the dual Abelian variety, 94 on Abelian varieties, 94 principal, 94 Polarized rank, 352 Polystable sheaf, 154 Pontrjagin product, 89 Preservation of the Ext groups, 23 Preservation of stability absolute, for elliptic Calabi-Yau threefolds, 228 absolute, for elliptic surfaces, 221 for Abelian surfaces, 102, 108 for elliptic curves, 98 for K3 surfaces, 139, 145, 380 for Weierstraß curves, 210 relative, for elliptic fibrations, 213 Primitive embedding, 342 vector, 342

Subject Index Principal bundle, 149 Projection formula in derived category, 330 Pure sheaves, 351 Quasi-Abelian category, 286 Quasi-universal sheaf, 118 Quaternionic instanton, 175 K¨ ahler manifold, 173 Reconstruction theorem, 70 Reduction of the structure group, 149 Reflexive K3 surfaces, 124 Relative connection, 167 differential operator, 165 Dirac operators, 170 Dolbeault complex, 166 dualizing complex, 347 Fourier-Mukai functors, 64 Fourier-Mukai transforms, 64 on elliptic fibrations, 197 kernel, 64 Todd class of an elliptic fibration, 191 Relatively semi-stable sheaf, 190 torsion-free sheaf, 190 Resolution of the diagonal Be˘ılinson’s, 45 Kawamata’s, 46 Resolution property, of an algebraic variety, 321 S-equivalence, 353 Semi-orthogonal decomposition, 263 Semicharacter, 162 Semistable bundles on elliptic curves, 97 object in a category, 364 Serre functor, 13 Shift functor, 288, 306 Signature of the intersection form, 114 Simpson stability, 351 Singularity canonical, 235

Subject Index terminal, 235 Skyscraper sheaves, 5 Small category, 363 Spanning class, 32 Special sheaf, 72 complex, 72 Lagrangian submanifolds, 108 Spectral cover, 218 Spherical object, 78 Spin structure, 155 Stability condition, 363 algebraic, 388 good, 382 locally finite, 365 numerical, 367 Strict short exact sequence, 286 Strongly reflexive K3 surfaces, 125 Strongly simple kernel, 19 sheaf, 22 Subcategory right admissible, 263 right orthogonal, 263 thick (of an Abelian category), 300 thin, 371 triangulated, 306 Support of a complex, 335 Symplectic form (on the moduli space of stable sheaves), 104, 142 morphisms of moduli spaces, 105, 142 t-structure, 264 Theta divisor, 94 Tor-dimension, finite of a complex, 6, 326 of a morphism, 328 Torelli theorem for K3 surfaces, 115 weak, for K3 surfaces, 115 Transcendental lattice, 114 Triangulated category, 306 subcategory, 306

423 Truncation in derived category, 300 of a complex, 291 of a double complex, 293 Twist functor, 78 Twisted Gieseker stability, 108, 145 Twistor space, 174 Unipotent bundles, 86 Uniqueness of the kernel, 51 Universal bundle on the instanton moduli space, 152 connection on the instanton moduli space, 152 family, relative, 258 line bundle, 84 perverse point sheaf, 270 sheaf, 17, 75, 118, 119, 353 Walls of a space of stability conditions, 382 Weak Torelli theorem, for K3 surfaces, 115 Weierstraß fibration, 190 Weil-Petersson metric, 151 Weitzenb¨ ock formula, 159 WIT condition, 7, 168 Without flat factors, 164 Yoneda product, 322 Yoneda’s formula, 320 lemma, 283