Fourier-Mukai and Nahm transforms in geometry and mathematical physics / Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez.

Format:

Book

Author/Creator:

Bartocci, C. (Claudio), 1962-

Contributor:

Bruzzo, U. (Ugo)

Hernández-Ruipérez, Daniel, 1954-

Series:

Progress in mathematics (Boston, Mass.) ; v. 276.

Progress in mathematics ; v. 276

Language:

English

Subjects (All):

Fourier transformations.

Fourier analysis.

Physical Description:

xvi, 423 pages : illustrations ; 25 cm.

Place of Publication:

Boston : Birkhäuser, [2009]

Summary:

Integral transforms, such as the Laplace and Fourier transforms, have been major tools in mathematics for at least two centuries. In the last three decades the development of a number of novel ideas in algebraic geometry, category theory, gauge theory, and string theory has been closely related to generalizations of integral transforms of a more geometric character.

Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics examines the algebro-geometric approach (Fourier-Mukai functors) as well as the differential-geometric constructions (Nahm). Also included is a considerable amount of material from existing literature which has not been systematically organized into a monograph.

Basic constructions and definitions are presented in preliminary background chapters

Presentation explores applications and suggests several open questions

Extensive bibliography and index

This self-contained monograph provides an introduction to current research in geometry and mathematical physics and is intended for graduate students and researchers just entering this field.

Contents:

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

1.3 Fully faithful integral functors 15

1.3.1 Preliminary results 15

1.3.2 Strongly simple objects 19

1.4 The equivariant case 24

1.4.1 Equivariant and linearized derived categories 24

1.4.2 Equivariant integral functors 29

1.5 Notes and further reading 30

2 Fourier-Mukai functors 31

2.1 Spanning classes and equivalences 32

2.1.1 Ample sequences 35

2.1.2 Convolutions 40

2.2 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

2.2 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

2.4 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

3.5 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

3.6 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 116

4.3 Examples of transforms 122

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 147

5.1 Basic notions 148

5.1.1 Connections 148

5.1.2 Instantons 150

5.1.3 The Hitchin-Kobayashi correspondence 153

5.1.4 Dirac operators and index bundles 155

5.2 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

5.3 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 Kähler Nahm transforms 171

5.4 Nahm transform on hyperkähler manifolds 173

5.4.1 Hyperkähler 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

5.5 Notes and further reading 181

6 Relative Fourier-Mukai functors 183

6.1 Relative integral functors 184

6.1.1 Base change formulas 185

6.1.2 Fourier-Mukai transforms on Abelian schemes 188

6.2 Weierstraβ 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 Weierstraβ 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 Weierstraβ elliptic fibrations 208

6.4.1 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 Weierstraβ fibrations 220

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

6.7 Notes and further reading 231

7 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 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

7.5 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

7.6 McKay correspondence 275

7.6.1 An equivariant removable singularity theorem 276

7.6.2 The derived McKay correspondence 277

7.7 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

A.3.1 Double complexes 292

A.4 Derived categories 295

A.4.1 The derived category of an Abelian categories 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

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.

Notes:

Includes bibliographical references (pages [397]-418) and index.

ISBN:

9780817632465

0817632468

9780817646639

0817646639

OCLC:

312626395