The Geometry of Moduli Spaces of Sheaves

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the second edition......Page 11
Preface to the first edition......Page 13
Introduction......Page 15
Part I General Theory......Page 21
1.1 Some Homological Algebra......Page 23
1.2 Semistable Sheaves......Page 30
1.3 The Harder-Narasimhan Filtration......Page 35
1.4 An Example......Page 40
1.5 Jordan-Hölder Filtration and S-Equivalence......Page 43
1.6  mu-Semistability......Page 45
1.7 Boundedness I......Page 49
2.1 Flat Families and Determinants......Page 54
2.2 Grothendieck's Quot-Scheme......Page 60
2.3 The Relative Harder-Narasimhan Filtration......Page 68
2.A Flag-Schemes and Deformation Theory......Page 71
2.B A Result of Langton......Page 78
II. The cotangent sheaf of the Quot-scheme......Page 81
III. Explicit calculations of Quot-schemes......Page 82
3 The Grauert-Mülich Theorem......Page 83
3.1 Statement and Proof......Page 84
3.2 Finite Coverings and Tensor Products......Page 89
3.3 Boundedness II......Page 95
3.4 The Bogomolov Inequality......Page 99
3.A e-Stability and Some Estimates......Page 102
II. Algebraic holonomy and splitting of tensor products......Page 107
4 Moduli Spaces......Page 109
4.1 The Moduli Functor......Page 110
Group Actions and Linearizations......Page 111
Geometric Invariant Theory (GIT)......Page 115
Some Descent Results......Page 117
4.3 The Construction — Results......Page 118
4.4 The Construction — Proofs......Page 124
4.5 Local Properties and Dimension Estimates......Page 132
4.6 Universal Families......Page 137
4.A Gieseker's Construction......Page 141
4.B Decorated Sheaves......Page 142
4.C Change of Polarization......Page 146
I. Moduli space constructions via quiver representations......Page 152
III. Moduli spaces of principal bundles......Page 153
IV. Moduli spaces of decorated sheaves......Page 155
V. Moduli spaces of complexes......Page 156
VI. Moduli of twisted sheaves......Page 158
VII. Moduli spaces on non-reduced schemes......Page 160
Part II Sheaves on Surfaces......Page 161
5 Construction Methods......Page 163
5.1 The Serre Correspondence......Page 165
5.2 Elementary Transformations......Page 171
5.3 Examples of Moduli Spaces......Page 174
II. Moduli spaces on higher dimensional varieties......Page 184
6 Moduli Spaces on K3 Surfaces......Page 186
6.1 Low-Dimensional
.........Page 187
6.2 ... and Higher-Dimensional Moduli Spaces......Page 195
6.A The Irreducibility of the Quot-scheme......Page 204
I. Derived categories......Page 207
II. Hilbert schemes of K3 surfaces......Page 208
III. Moduli spaces of sheaves on K3 surfaces......Page 210
7.1 Flenner's Theorem......Page 213
7.2 The Theorems of Mehta and Ramanathan......Page 217
7.3 Bogomolov's Theorems......Page 224
I. Restriction theorems in positive characteristic......Page 232
8.1 Construction of Determinant Line Bundles......Page 233
8.2 A Moduli Space for mu-Semistable Sheaves......Page 240
8.3 The Canonical Class of the Moduli Space......Page 252
I. Theta functions......Page 256
II. Strange duality......Page 257
9.1 Preparations......Page 259
9.2 The Boundary......Page 261
9.3 Generic Smoothness......Page 262
9.4 Irreducibility......Page 263
9.5 Proof of Theorem 9.2.2......Page 265
9.6 Proof of Theorem 9.3.2......Page 271
10 Symplectic Structures......Page 275
10.1 Trace Map, Atiyah Class and Kodaira-Spencer Map......Page 276
10.2 The Tangent Bundle......Page 282
10.3 Forms on the Moduli Space......Page 284
10.4 Non-Degeneracy of Two-Forms......Page 287
I. Differential forms on moduli spaces......Page 291
11.1 Kodaira Dimension of Moduli Spaces......Page 292
11.2 More Results......Page 297
11.3 Examples......Page 301
I. Birational properties of moduli spaces......Page 307
II. Numerical invariants of moduli spaces......Page 308
References......Page 310
New References (second edition)......Page 325
Glossary of Notations......Page 336
Index......Page 341