Spis treści : Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics


Contents
Preface, s. xi
Acknowledgments, s. xv

1. Integral functors, s. 1

1.1     Notation and preliminary results, s. 2
1.2    First properties of integral functors, s. 5
1.2.1     Base change formulas, s. 8
1.2.2     Adjoints, s. 12
1.3    Fully faithful integral functors, s. 15
1.3.1     Preliminary results, s. 15
1.3.2     Strongly simple objects, s. 19
1.4    The equivariant case, s. 24
1.4.1     Equivariant and linearized derived categories, s. 24
1.4.2     Equivariant integral functors, s. 29
1.5    Notes and further reading, s. 30

2. Fourier-Mukai functors, s. 31

2.1     Spanning classes and equivalences, s. 32
2.1.1     Ample sequences, s. 35
2.1.2     Convolutions, s. 40
2.2    Orlov's representability theorem, s. 44
2.2.1     Resolution of the diagonal, s. 44
2.2.2     Uniqueness of the kernel, s. 51
2.2.3     Existence of the kernel, s. 54
2.3    Fourier-Mukai functors, s. 60
2.3.1     Some geometric applications of Fourier-Mukai functors, s. 61
2.3.2     Characterization of Fourier-Mukai functors, s. 71
2.3.3     Fourier-Mukai functors between moduli spaces, s. 76
2.4    Notes and further reading, s. 78

3. Fourier-Mukai on Abelian varieties, s. 81

3.1     Abelian varieties, s. 82
3.2    The transform, s. 84
3.3    Homogeneous bundles, s. 90
3.4    Fourier-Mukai transform and the geometry of Abelian varieties, s. 91
3.4.1     Line bundles and homomorphisms of Abelian varietie, s. 91
3.4.2     Polarizations, s. 94
3.4.3     Picard sheaves, s. 95
3.5     Some applications of the Abelian Fourier-Mukai transform, s. 97
3.5.1     Moduli of semistable sheaves on elliptic curves, s. 97
3.5.2     Preservation of stability for Abelian surfaces, s. 102
3.5.3     Symplectic morphisms of moduli spaces, s. 104
3.5.4     Embeddings of moduli spaces, s. 106
3.6    Notes and further reading, s. 108

4. Fourier-Mukai on K3 surfaces, s. 111

4.1     K3 surfaces, s. 112
4.2     Moduli spaces of sheaves and integral functors, s. 116
4.3    Examples of transforms, s. 122
4.3.1     Reflexive K3 surfaces, s. 124
4.3.2     Duality for reflexive K3 surfaces, s. 125
4.3.3     Homogeneous bundles, s. 131
4.3.4     Other Fourier-Mukai transforms on K3 surfaces, s. 133
4.4    Preservation of stability, s. 139
4.5    Hilbert schemes of points on reflexive K3 surfaces, s. 142
4.6    Notes and further reading, s. 145

5. Nahm transforms, s. 147

5.1    Basic notions, s. 148
5.1.1     Connections, s. 148
5.1.2     Instantons, s. 150
5.1.3     The Hitchin-Kobayashi correspondence, s. 153
5.1.4     Dirac operators and index bundles, s. 155
5.2    The Nahm transform for instantons, s. 158
5.2.1     Definition of the Nahm transform, s. 158
5.2.2     The topology of the transformed bundle, s. 161
5.2.3     Line bundles on complex tori, s. 161
5.2.4     Nahm transform on flat 4-tori, s. 164
5.3    Compatibility between Nahm and Fourier-Mukai, s. 165
5.3.1     Relative differential operators, s. 165
5.3.2     Relative Dolbeault complex, s. 166
5.3.3     Relative Dirac operators, s. 170
5.3.4     Kahler Nahm transforms, s. 171
5.4    Nahm transforms on hyperkahler manifolds, s. 173
5.4.1     Hyperkahler manifolds, s. 173
5.4.2     A generalized Atiyah-Ward correspondence, s. 174
5.4.3     Fourier-Mukai transform of quaternionic instantons, s. 178
5.4.4     Examples, s. 180
5.5    Notes and further reading, s. 181

6. Relative Fourier-Mukai functors, s. 183

6.1     Relative integral functors, s. 184
6.1.1     Base change formulas, s. 185
6.1.2     Fourier-Mukai transforms on Abelian schemes, s. 188
6.2    Weierstrafi fibrations, s. 189
6.2.1     Todd classes, s. 190
6.2.2     Torsion-free rank one sheaves on elliptic curves, s. 192
6.2.3     Relative integral functors for Weierstrafi fibrations, s. 193
6.2.4     The compactified relative Jacobian, s. 197
6.2.5     Examples, s. 199
6.2.6     Topological invariants, s. 201
6.3    Relatively minimal elliptic surfaces, s. 204
6.4    Relative moduli spaces for Weierstrafi elliptic fibrations, s. 208
6.4.1     Semistable sheaves on integral genus one curves, s. 208
6.4.2     Characterization of relative moduli spaces, s. 213
6.5     Spectral covers, s. 217
6.6    Absolutely stable sheaves on Weierstrafi fibrations, s. 220
6.6.1     Preservation of absolute stability for elliptic surfaces, s. 221
6.6.2     Characterization of moduli spaces on elliptic surfaces, s. 225
6.6.3     Elliptic Calabi-Yau threefolds, s. 228
6.7    Notes and further reading, s. 231

7. Fourier-Mukai partners and birational geometry, s. 233

7.1     Preliminaries, s. 234
7.2    Integral functors for quotient varieties, s. 238
7.3    Fourier-Mukai partners of algebraic curves, s. 242
7.4    Fourier-Mukai partners of algebraic surfaces, s. 242
7.4.1     Surfaces of Kodaira dimension 2, s. 245
7.4.2     Surfaces of Kodaira dimension — oo that are not elliptic, s. 245
7.4.3     Relatively minimal elliptic surfaces, s. 248
7.4.4     K3 surfaces, s. 249
7.4.5     Abelian surfaces, s. 253
7.4.6     Enriques surfaces, s. 254
7.4.7     Nonminimal projective surfaces, s. 256
7.5     Derived categories and birational geometry, s. 257
7.5.1     A removable singularity theorem, s. 258
7.5.2     Perverse sheaves, s. 264
7.5.3     Flops and derived equivalences, s. 272
7.6     McKay correspondence, s. 275
7.6.1     An equivariant removable singularity theorem, s. 276
7.6.2     The derived McKay correspondence, s. 277
7.7    Notes and further reading, s. 279

A. Derived and triangulated categories, s. 281

A.1   Basic notions., s. 281
A.2   Additive and Abelian categories, s. 283
A.3   Categories of complexes, s. 287
A.3.1    Double complexes, s. 292
A.4   Derived categories, s. 295
A.4.1    The derived category of an Abelian category, s. 295
A.4.2   Other derived categories, s. 300
A.4.3   Triangles and triangulated categories, s. 303
A.4.4   Differential graded categories, s. 307
A.4.5    Derived functors, s. 312
A.4.6   Some remarkable formulas in derived categories, s. 328
A.4.7   Support and homological dimension, s. 335

B. Lattices, s. 339

B.1   Preliminaries, s. 339
B.2   The discriminant group, s. 341
B.3   Primitive embeddings, s. 342

C. Miscellaneous results, s. 347

C.1   Relative duality, s. 347
C.2   Pure sheaves and Simpson stability, s. 351
C.3   Fitting ideals, s. 355

D. Stability conditions for derived categories, s. 359

D.1   Introduction, s. 359
D.2   Bridgeland's stability conditions, s. 362
D.2.1    Definition and Bridgeland's theorem, s. 363
D.2.2   An example: stability conditions on curves, s. 369
D.2.3   Bridgeland's deformation lemma, s. 371
D.3   Stability conditions on K3 surfaces, s. 373
D.3.1    Bridgeland's theorem, s. 374
D.3.2   Construction of stability conditions, s. 375
D.3.3   The covering map property, s. 380
D.3.4   Wall and chamber structure, s. 382
D.3.5   Sketch of the proof of Theorem D.19, s. 383
D.4   Moduli stacks and invariants of semistable objects on K3 surfaces, s. 385
D.4.1    Moduli stack of semistable objects, s. 385
D.4.2   Sketch of the proof of Theorem D.35, s. 386
D.4.3   Counting invariants and Joyce's conjecture for K3 surfaces, s. 391
D.4.4   Some ideas from the proof of Theorem D.45, s. 392

References, s. 397
Subject index, s. 419

 

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