Geometric invariant theory and decorated principal bundles

Contents......Page 5
Introduction......Page 9
1.1.1 Linear Algebraic Groups I......Page 25
1.1.2 Representations......Page 27
1.1.3 Linear Algebraic Groups II......Page 29
1.1.4 Reductive Affine Algebraic Groups......Page 31
1.1.5 Homogeneous Representations of the General Linear Group......Page 32
1.1.6 Faithful Representations and Extensions of Representations......Page 34
1.1.7 Appendix. The Reductivity of the Classical Groups via Weyl’s Unitarian Trick......Page 35
1.2.1 The Categorical Quotient of a Vector Space by a Representation......Page 38
1.3.1 Algebraic Forms......Page 42
1.3.2 Examples......Page 43
1.3.3 The Invariant Theory of Matrices......Page 51
1.4.1 Good and Geometric Quotients......Page 57
1.4.2 Quotients of Affine Varieties......Page 59
1.4.3 Linearizations......Page 63
1.5.1 The Hilbert–Mumford Criterion......Page 74
1.5.2 Semistability for Direct Sums of Representations......Page 94
1.5.3 Semistability for Actions of Direct Products of Groups......Page 97
1.6.1 The Finiteness of the Number of GIT-Quotients......Page 100
1.6.2 Variation of the GIT-Quotients......Page 101
1.7 The Analysis of Unstable Points......Page 103
1.7.1 AFewWords about GIT on Non-Algebraically Closed Fields......Page 104
1.7.2 The Theory of the Instability Flag......Page 105
1.7.3 The Instability Flag in a Product......Page 110
2.1.1 Principal Bundles - Definitions and First Properties......Page 111
2.1.2 The Classification Problem for Decorated Principal Bundles......Page 123
2.2.1 The Topology of Vector Bundles......Page 126
2.2.2 The Riemann–Roch Theorem......Page 127
2.2.3 Bounded Families of Vector Bundles......Page 129
2.2.4 The Moduli Space of Semistable Vector Bundles......Page 132
2.3.1 Set-Up of the Moduli Problem......Page 143
2.3.2 Semistability of Swamps......Page 146
2.3.3 Examples......Page 151
2.3.4 Boundedness......Page 152
2.3.5 The Parameter Space......Page 154
2.3.6 On the Geometry of the Moduli Spaces......Page 171
2.3.7 The Chain of Moduli Spaces......Page 175
2.4.1 Principal Bundles and Associated Vector Bundles......Page 182
2.4.2 Back to Some GIT......Page 187
2.4.3 Pseudo G-Bundles......Page 193
2.4.4 Semistable Reduction for Principal Bundles and the Proof of Theorem 2.4.1.8......Page 196
2.4.5 The Proof of Theorem 2.4.3.3......Page 198
2.4.7 Appendix I: Some Remarks Concerning the Moduli Stack of Principal Bundles......Page 203
2.4.8 Appendix II: Moduli Spaces for Principal Bundles with Reductive Structure Groupvia the Ramanathan–Gómez–Sols Method......Page 207
2.4.9 Appendix III: Anti-Dominant Characters......Page 216
2.5.1 Homogeneous Representations......Page 218
2.5.2 More on One Parameter Subgroups......Page 219
2.5.3 The Moduli Problem of Tumps......Page 225
2.5.4 Proof of Theorem 2.5.3.7......Page 231
2.5.5 Properties of the Semistability Concept......Page 237
2.5.6 Quiver Representations......Page 245
2.6 Principal Bundles as Tumps......Page 265
2.6.1 Principal Bundles and Associated Tuples of Vector Bundles......Page 266
2.6.2 The Relevant GIT-Quotients......Page 267
2.6.3 Pseudo G-Bundles......Page 271
2.7.1 The Moduli Spaces......Page 274
2.7.2 Decorated Pseudo G-Bundles......Page 278
2.7.3 Asymptotic Semistability......Page 287
2.7.4 Hitchin Pairs......Page 291
2.7.5 Fine Tuning of the Theory......Page 294
2.8.1 The Moduli Functors......Page 296
2.8.2 Comparison with Projective \\rho-Bumps......Page 300
2.8.3 Construction of the Moduli Spaces......Page 303
2.8.4 Further Properties and Examples......Page 311
2.9.1 Positive Characteristic......Page 329
2.9.2 Higher Dimensional Base Varieties......Page 333
2.9.3 Parabolic Structures......Page 362
Bibliography......Page 371
Index......Page 387