Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics

by Bartocci, Claudio; Bruzzo, Ugo; Ruiperez, Daniel Hernandez; Jardim, Marcos

• ISBN13: 9780817632465
• ISBN10: 0817632468
• Format: Hardcover
• Copyright: 2009-05-29
• Publisher: Birkhauser

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.

Table of Contents

Preface	p. xi
Acknowledgments	p. xv
Integral Functors	p. 1
Notation and preliminary results	p. 2
First properties of integral functors	p. 5
Base change formulas	p. 8
Adjoints	p. 12
Fully faithful integral functors	p. 15
Preliminary results	p. 15
Strongly simple objects	p. 19
The equivariant case	p. 24
Equivariant and linearized derived categories	p. 24
Equivariant integral functors	p. 29
Notes and further reading	p. 30
Fourier-Mukai functors	p. 31
Spanning classes and equivalences	p. 32
Ample sequences	p. 35
Convolutions	p. 40
Orlov's representability theorem	p. 44
Resolution of the diagonal	p. 44
Uniqueness of the kernel	p. 51
Existence of the kernel	p. 54
Fourier-Mukai functors	p. 60
Some geometric applications of Fourier-Mukai functors	p. 61
Characterization of Fourier-Mukai functors	p. 71
Fourier-Mukai functors between moduli spaces	p. 76
Notes and further reading	p. 78
Fourier-Mukai on Abelian varieties	p. 81
Abelian varieties	p. 82
The transform	p. 84
Homogeneous bundles	p. 90
Fourier-Mukai transform and the geometry of Abelian varieties	p. 91
Line bundles and homomorphisms of Abelian varieties	p. 91
Polarizations	p. 94
Picard sheaves	p. 95
Some applications of the Abelian Fourier-Mukai transform	p. 97
Moduli of semistable sheaves on elliptic curves	p. 97
Preservation of stability for Abelian surfaces	p. 102
Symplectic morphisms of moduli spaces	p. 104
Embeddings of moduli spaces	p. 106
Notes and further reading	p. 108
Fourier-Mukai on K3 surfaces	p. 111
K3 surfaces	p. 112
Moduli spaces of sheaves and integral functors	p. 116
Examples of transforms	p. 122
Reflexive K3 surfaces	p. 124
Duality for reflexive K3 surfaces	p. 125
Homogeneous bundles	p. 131
Other Fourier-Mukai transforms on K3 surfaces	p. 133
Preservation of stability	p. 139
Hilbert schemes of points on reflexive K3 surfaces	p. 142
Notes and further reading	p. 145
Nahm transforms	p. 147
Basic notions	p. 148
Connections	p. 148
Instantons	p. 150
The Hitchin-Kobayashi correspondence	p. 153
Dirac operators and index bundles	p. 155
The Nahm transform for instantons	p. 158
Definition of the Nahm transform	p. 158
The topology of the transformed bundle	p. 161
Line bundles on complex tori	p. 161
Nahm transform on flat 4-tori	p. 164
Compatibility between Nahm and Fourier-Mukai	p. 165
Relative differential operators	p. 165
Relative Dolbeault complex	p. 166
Relative Dirac operators	p. 170
Käet;hler Nahm transforms	p. 171
Nahm transform on hyperkäet;hler manifolds	p. 173
Hyperkäet;hler manifolds	p. 173
A generalized Atiyah-Ward correspondence	p. 174
Fourier-Mukai transform of quaternionic instantons	p. 178
Examples	p. 180
Notes and further reading	p. 181
Relative Fourier-Mukai functors	p. 183
Relative integral functors	p. 184
Base change formulas	p. 185
Fourier-Mukai transforms on Abelian schemes	p. 188
Weierstraß fibrations	p. 189
Todd classes	p. 190
Torsion-free rank one sheaves on elliptic curves	p. 192
Relative integral functors for Weierstraß fibrations	p. 193
The compactified relative Jacobian	p. 197
Examples	p. 199
Topological invariants	p. 201
Relatively minimal elliptic surfaces	p. 204
Relative moduli spaces for Weierstraß elliptic fibrations	p. 208
Semistable sheaves on integral genus one curves	p. 208
Characterization of relative moduli spaces	p. 213
Spectral covers	p. 217
Absolutely stable sheaves on Weierstraß fibrations	p. 220
Preservation of absolute stability for elliptic surfaces	p. 221
Characterization of moduli spaces on elliptic surfaces	p. 225
Elliptic Calabi-Yau threefolds	p. 228
Notes and further reading	p. 231
Fourier-Mukai partners and birational geometry	p. 233
Preliminaries	p. 234
Integral functors for quotient varieties	p. 238
Fourier-Mukai partners of algebraic curves	p. 242
Fourier-Mukai partners of algebraic surfaces	p. 242
Surfaces of Kodaira dimension 2	p. 245
Surfaces of Kodaira dimension - &infinity; that are not elliptic	p. 245
Relatively minimal elliptic surfaces	p. 248
K3 surfaces	p. 249
Abelian surfaces	p. 253
Enriques surfaces	p. 254
Nonminimal projective surfaces	p. 256
Derived categories and birational geometry	p. 257
A removable singularity theorem	p. 258
Perverse sheaves	p. 264
Flops and derived equivalences	p. 272
McKay correspondence	p. 275
An equivariant removable singularity theorem	p. 276
The derived McKay correspondence	p. 277
Notes and further reading	p. 279
Derived and triangulated categories	p. 281
Basic notions	p. 281
Additive and Abelian categories	p. 283
Categories of complexes	p. 287
Double complexes	p. 292
Derived categories	p. 295
The derived category of an Abelian categories	p. 295
Other derived categories	p. 300
Triangles and triangulated categories	p. 303
Differential graded categories	p. 307
Derived functors	p. 312
Some remarkable formulas in derived categories	p. 328
Support and homological dimension	p. 335
Lattices	p. 339
Preliminaries	p. 339
The discriminant group	p. 341
Primitive embeddings	p. 342
Miscellaneous results	p. 347
Relative duality	p. 347
Pure sheaves and Simpson stability	p. 351
Fitting ideals	p. 355
Stability conditions for derived categories	p. 359
Introduction	p. 359
Bridgeland's stability conditions	p. 362
Definition and Bridgeland's theorem	p. 363
An example: stability conditions on curves	p. 369
Bridgeland's deformation lemma	p. 371
Stability conditions on K3 surfaces	p. 373
Bridgeland's theorem	p. 374
Construction of stability conditions	p. 375
The covering map property	p. 380
Wall and chamber structure	p. 382
Sketch of the proof of Theorem D.19	p. 383
Moduli stacks and invariants of semistable objects on K3 surfaces	p. 385
Moduli stack of semistable objects	p. 385
Sketch of the proof of Theorem D.35	p. 386
Counting invariants and Joyce's conjecture for K3 surfaces	p. 391
Some ideas from the proof of Theorem D.45	p. 392
References	p. 397
Subject index	p. 419
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