Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics (eBook)

Contents 6
Preface 12
Acknowledgments 16
Integral functors 18
Introduction 18
1.1 Notation and preliminary results 19
1.2 First properties of integral functors 22
1.3 Fully faithful integral functors 32
1.4 The equivariant case 41
1.5 Notes and further reading 47
Fourier-Mukai functors 48
Introduction 48
2.1 Spanning classes and equivalences 49
2.2 Orlov's representability theorem 61
2.3 Fourier-Mukai functors 77
2.4 Notes and further reading 95
Fourier-Mukai on Abelian varieties 97
Introduction 97
3.1 Abelian varieties 98
3.2 The transform 100
3.3 Homogeneous bundles 106
3.4 Fourier-Mukai transform and the geometry of Abelian varieties 107
3.5 Some applications of the Abelian Fourier-Mukai transform 113
3.6 Notes and further reading 124
Fourier-Mukai on K3 surfaces 126
Introduction 126
4.1 K3 surfaces 127
4.2 Moduli spaces of sheaves and integral functors 131
4.3 Examples of transforms 137
4.4 Preservation of stability 154
4.5 Hilbert schemes of points on re 
157 
4.6 Notes and further reading 160
Nahm transforms 162
Introduction 162
5.1 Basic notions 163
5.2 The Nahm transform for instantons 173
5.3 Compatibility between Nahm and Fourier-Mukai 180
5.4 Nahm transforms on hyperkähler manifolds 188
5.5 Notes and further reading 196
Relative Fourier-Mukai functors 198
Introduction 198
6.1 Relative integral functors 199
6.2 Weierstraß fibrations 204
6.3 Relatively minimal elliptic surfaces 219
6.4 Relative moduli spaces for Weierstraß elliptic fibrations 223
6.5 Spectral covers 232
6.6 Absolutely stable sheaves on Weierstraß fibrations 235
6.7 Notes and further reading 246
Fourier-Mukai partners and birational geometry 248
Introduction 248
7.1 Preliminaries 249
7.2 Integral functors for quotient varieties 253
7.3 Fourier-Mukai partners of algebraic curves 257
7.4 Fourier-Mukai partners of algebraic surfaces 257
7.5 Derived categories and birational geometry 272
7.6 McKay correspondence 290
7.7 Notes and further reading 294
Derived and triangulated categories 296
A.1 Basic notions 296
A.2 Additive and Abelian categories 298
A.3 Categories of complexes 302
A.4 Derived categories 310
Lattices 353
B.1 Preliminaries 353
B.2 The discriminant group 355
B.3 Primitive embeddings 356
Miscellaneous results 360
C.1 Relative duality 360
C.2 Pure sheaves and Simpson stability 364
C.3 Fitting ideals 368
Stability conditions for derived categories 372
D.1 Introduction 372
D.2 Bridgeland's stability conditions 375
D.3 Stability conditions on K3 surfaces 386
D.4 Moduli stacks and invariants of semistable objects on K3 surfaces 398
References 409
Subject Index 431