Table of Contents
Preface to the second edition ix
Preface to the first edition xi
Introduction xiii
Part I General Theory 1
1 Preliminaries 3
1.1 Some Homological Algebra 3
1.2 Semistable Sheaves 10
1.3 The Harder-Narasimhan Filtration 15
1.4 An Example 20
1.5 Jordan-Hölder Filtration and S-Equivalence 23
1.6 μ-Semistability 25
1.7 Boundedness I 29
2 Families of Sheaves 34
2.1 Flat Families and Determinants 34
2.2 Grothendieck's Quot-Scheme 40
2.3 The Relative Harder-Narasimhan Filtration 48
Appendix
2.A Flag-Schemes and Deformation Theory 51
2.B A Result of Langton 58
2.C Further comments (second edition) 61
3 The Grauert-Mülich Theorem 63
3.1 Statement and Proof 64
3.2 Finite Coverings and Tensor Products 69
3.3 Boundedness II 75
3.4 The Bogomolov Inequality 79
Appendix
3.A e-Stability and Some Estimates 82
3.B Further comments (second edition) 87
4 Moduli Spaces 89
4.1 The Moduli Functor 90
4.2 Group Actions 91
4.3 The Construction---Results 98
4.4 The Construction---Proofs 104
4.5 Local Properties and Dimension Estimates 112
4.6 Universal Families 117
Appendix
4.A Gieseker's Construction 121
4.B Decorated Sheaves 122
4.C Change of Polarization 126
4.D Further comments (second edition) 132
Part II Sheaves on Surfaces 141
5 Construction Methods 143
5.1 The Serre Correspondence 145
5.2 Elementary Transformations 151
5.3 Examples of Moduli Spaces 154
Appendix
5.A Further comments (second edition) 164
6 Moduli Spaces on K3 Surfaces 166
6.1 Low-Dimensional... 167
6.2 ...and Higher-Dimensional Moduli Spaces 175
Appendix
6.A The Irreducibility of the Quot-scheme 184
6.B Further comments (second edition) 187
7 Restriction of Sheaves to Curves 193
7.1 Flenner's Theorem 193
7.2 The Theorems of Mehta and Ramanathan 197
7.3 Bogomolov's Theorems 204
Appendix
7.A Further comments (second edition) 212
8 Line Bundles on the Moduli Space 213
8.1 Construction of Determinant Line Bundles 213
8.2 A Moduli Space for μ-Semistable Sheaves 220
8.3 The Canonical Class of the Moduli Space 232
8.4 Further comments (second edition) 236
9 Irreducibility and Smoothness 239
9.1 Preparations 239
9.2 The Boundary 241
9.3 Generic Smoothness 242
9.4 Irreducibility 243
9.5 Proof of Theorem 9.2.2 245
9.6 Proof of Theorem 9.3.2 251
10 Symplectic Structures 255
10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map 256
10.2 The Tangent Bundle 262
10.3 Forms on the Moduli Space 264
10.4 Non-Degeneracy of Two-Forms 267
Appendix
10.A Further comments (second edition) 271
11 Birational properties 272
11.1 Kodaira Dimension of Moduli Spaces 272
11.2 More Results 277
11.3 Examples 281
Appendix
11.A Further comments (second edition) 287
References 290
Glossary of Notations 316
Index 321
