Moduli Spaces and Vector Bundles

Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
Acknowledgments......Page 10
Part I Lecture Notes......Page 12
1 Introduction......Page 13
2.1 Generalities on principal bundles......Page 14
3 Principal bundles, basic properties......Page 21
4 Moduli spaces of principal bundles......Page 29
4.1 Construction of the moduli space......Page 30
4.2 The construction of the moduli space for principal bundles......Page 31
4.3 Points of the moduli......Page 34
1 Introduction......Page 40
2 Scheme structure on Bkn,d......Page 41
3 Classical Results and Brill-Noether geography......Page 44
4 Problem in higher ranks (and genus)......Page 46
5 Non-emptiness of Brill-Noether loci for large number of sections......Page 49
6 Further results......Page 52
7 Generalized Clifford bounds......Page 55
8 The case of rank two and canonical determinant......Page 56
3 Introduction to Fourier-Mukai and Nahm Transforms with an Application to Coherent Systems on Elliptic Curves......Page 62
Introduction......Page 63
1.1 Categories of complexes......Page 64
1.2 Derived Category......Page 67
1.2.1 The derived category as a triangulated category......Page 71
1.3 Derived Functors......Page 72
1.3.1 Derived Direct Image......Page 74
1.3.3 Derived homomorphism functor and derived tensor product......Page 75
1.3.4 Base change in the derived category......Page 76
2.1 Definitions......Page 77
2.1.2 Orlov’s representation theorem......Page 79
2.2.1 Action of integral functors on cohomology......Page 80
2.2.2 Fully faithful integral functors and Fourier-Mukai functors......Page 81
2.2.3 The abelian Fourier-Mukai transform revisited......Page 82
2.2.4 Fourier-Mukai functors on K3 and abelian surfaces......Page 83
2.2.5 Relative integral functors and base change......Page 84
2.2.6 Fourier-Mukai functors between moduli spaces......Page 85
2.3 Fourier-Mukai partners......Page 86
2.3.1 D-equivalence implies K-equivalence......Page 87
2.3.3 Fourier-Mukai partners of surfaces......Page 88
2.3.4 Fourier-Mukai partners for threefolds......Page 89
3.1 Line bundles on complex tori......Page 90
3.2 Nahm transform......Page 92
3.3 Fourier-Mukai vs. Nahm......Page 93
3.4 Hyperkahler Fourier-Mukai transform......Page 94
4 Moduli spaces of sheaves and coherent systems on elliptic curves......Page 96
4.1.1 (Semi)stable sheaves on an elliptic curve......Page 97
4.1.2 Geometry of the moduli spaces of stable sheaves on elliptic curves......Page 99
4.1.3 Autoequivalences of the derived category of an elliptic curve......Page 100
4.2 Coherent systems......Page 101
4.3 Moduli spaces of coherent systems......Page 102
4.3.2 Large values of the parameter......Page 103
4.4 Fourier-Mukai transforms of coherent systems on elliptic curves......Page 104
4.4.2 Preservation of stability. Large α......Page 105
4.4.3 Birational type of the moduli spaces G(α; r, d, k)......Page 106
4 Geometric Invariant Theory......Page 110
1.1 Types of quotient......Page 111
1.2 Rings of invariants......Page 112
1.3 Affine Quotients......Page 113
1.4 Projective quotients......Page 114
2 Lecture 2 – The Hilbert-Mumford criterion......Page 115
2.1 Affine definition of stability......Page 116
2.2 Actions of k......Page 117
2.3 The criterion......Page 118
3.1 Reductivity and finite generation......Page 119
3.2 Proof of Theorem 1.1......Page 121
3.3 Proof of Corollaries 1.2, 1.3, 1.4, Proposition 1.6......Page 122
3.4 Proof of Theorem 1.7......Page 123
3.5 Linearisation......Page 124
4.2 Binary forms......Page 126
4.3 Cubic curves......Page 127
5.1 Moduli and quotients......Page 129
5.3 Vector bundles on a curve......Page 131
5.4 Sheaves and cohomology......Page 132
5.5 Stable bundles as quotient sheaves......Page 133
5.6 Linearisation of Q and computation of stability......Page 134
5.7 Consequences......Page 136
Basic example 1: Deformations of a point on a scheme......Page 139
Basic example 2: Deformations of a coherent sheaf......Page 140
Basic example 3: Deformations of a quotient......Page 141
Relation with moduli functors......Page 142
Tangent space to a functor......Page 143
Artin local algebras......Page 144
Versal, miniversal, universal families......Page 146
Grothendieck’s pro-representability theorem......Page 148
Schlessinger’s conditions and the resulting group action......Page 149
Schlessinger’s theorem......Page 151
Obstruction theory......Page 156
3 Calculations for basic examples......Page 158
Deformations of a coherent sheaf......Page 162
Pro-Representability for a simple sheaf......Page 165
Homological preliminaries for the Quot functor......Page 168
Pro-representability and tangent space for the Quot functor......Page 169
Obstruction theory for Q......Page 173
Some suggestions for further reading......Page 174
1 Introduction......Page 176
2.1 Periods of holomorphic differentials......Page 177
2.2 The Albanese variety......Page 178
2.4 Invertible sheaves and line bundles......Page 180
2.5 Cohomological interpretation......Page 181
2.6 Linear systems......Page 182
2.7 Polarisation......Page 183
2.8 Abelian varieties......Page 184
2.9 Theta functions......Page 185
3.1 Families of line bundles......Page 186
4.1 Locally free sheaves and vector bundles......Page 187
4.2 Duality and Riemann-Roch theorems......Page 188
4.3 Extensions......Page 189
4.5 Direct images......Page 190
4.6 Representations of the fundamental group......Page 191
5 Moduli space of vector bundles......Page 193
5.2 Elementary properties......Page 194
5.3 Theorem of Narasimhan and Seshadri......Page 196
5.4 Moduli space of vector bundles......Page 197
5.5 Connectedness......Page 198
5.6 Universal property......Page 199
6.1 The space SU(n, d)......Page 200
7.2 Quadratic complexes and vector bundles......Page 202
8 Study of SU(2,O) for higher genera......Page 204
8.2 Theorem of Brivio and Verra......Page 205
9 Moduli of curves and principally polarised abelian varieties......Page 206
9.1 Generalized Theta divisor......Page 207
9.2 Schottky relation......Page 208
10.1 Spectral cover and the moduli of vector bundles......Page 209
11.1 Fundamental groups and Higgs pairs......Page 211
12.1 Heisenberg extensions......Page 213
13 Another application: Syzygies of the canonical curve......Page 217
14 Opers and quantization......Page 218
Part II Survey Articles......Page 222
7 Moduli of Sheaves from Moduli of Kronecker Modules......Page 223
1 Simpson’s construction revisited......Page 226
2 The functorial point of view......Page 227
3 Semistability......Page 229
4 Moduli spaces......Page 231
5 Theta functions......Page 234
1 Introduction......Page 240
2 Definitions, basic facts and general features......Page 242
2.1 Transformations at critical values of α......Page 246
2.2 Relation to Brill-Noether theor......Page 247
3.1 Boundary critical values and related structure results......Page 248
3.2 Generic α-stable objects with small s......Page 250
4 Structure results for destabilizing patterns and flip loci......Page 251
4.1 Destabilizing patterns......Page 252
4.2 Counting codimensions: good, better, and best flips......Page 254
5.1 Homotopy groups......Page 257
5.2 Poincare polynomial......Page 258
6.1 Analytic description of Coherent Systems......Page 260
6.2 Equations......Page 262
6.4 Symplectic interpretation......Page 263
6.5 Holomorphic k-pairs......Page 264
6.6 Elliptic cu......Page 266
6.7 The projective line......Page 268
1 Introduction......Page 276
2.2 Stability of G-Higgs bundles......Page 277
2.3 Deformation theory of G-Higgs bundles......Page 280
2.4 G-Higgs bundles and Hitchin equations......Page 282
3.1 Surface group representations......Page 284
3.2 Representations and G-Higgs bundles......Page 285
4.1 Symplectic and K¨ahler quotients......Page 286
4.2 Moduli spaces of Higgs bundles as K¨ahler quotients......Page 289
5.1 Stability......Page 291
5.2 HyperKahler quotients and moduli spaces......Page 292
5.3 The Hitchin system......Page 294
6.1 Some general facts......Page 295
6.2 SL(n,R)-Higgs bundles......Page 296
6.3 SO0 (p, q)-Higgs bundles......Page 299
6.4 Hitchin component for SO0 (n, n)......Page 301
6.5 Hitchin component for SO0 (n, n + 1)......Page 303
6.6 Hitchin component for Sp(2n,R)......Page 304
7.1 G-Higgs bundles for groups of Hermitian......Page 305
7.2 SU(p, q)-Higgs bundles......Page 307
7.3 Sp(2n,R)-Higgs bundles......Page 311
8.1 Involution (E,ϕ) → (E,−ϕ) in M(SL(n,C))......Page 315
8.2 Involution (E,ϕ) → (E∗, ϕt ) in M(SL(n,C))......Page 316
8.3 Involutions in M(G)......Page 317
1 Introduction......Page 322
2 Mumford’s geometric invariant theory......Page 325
2.1 Classical geometric invariant theory......Page 326
2.2 Partial desingularisations of quotients......Page 329
2.3 Variation of GIT......Page 331
3 Quotients by non-reductive actions......Page 334
4 Choosing reductive envelopes......Page 341
4.1 Actions of (C+)r which extend to SL(r + 1;C)......Page 342
4.2 General (C+)r actions......Page 351
4.3 Naturality properties......Page 359
5 Hypersurfaces in P(1, 1, 2)......Page 365
5.1 The action of U = (C+)3......Page 366
5.2 The action of ˆU = C3  C∗......Page 368
5.3 The action of H......Page 371
5.4 Symplectic descriptions......Page 372
§0. Introduction......Page 378
§1. ACI......Page 380
§2. What is this duality?......Page 384
§3. Lax-pair representations; other dualities?......Page 390
1 Introduction......Page 399
Conventions......Page 402
2.1 Preliminaries......Page 403
2.2 Moduli spaces over the field of complex numbers......Page 404
2.2.2 The work of Faltings......Page 405
2.2.4 The work of G´omez and Sols......Page 406
2.3.1 The work of Balaji and Parameswaran......Page 407
2.3.3 The work of Heinloth......Page 408
3.1 Principal bundles as decorated vector bundles......Page 409
3.2 The notion of a pseudo bundle......Page 410
3.3.1 Some GIT-considerations......Page 411
3.3.2 Definition of semistability and moduli spaces......Page 413
3.4.1 Semistability for singular principal G-bundles (see [Sch03], Introduction)......Page 415
3.4.2 The semistable reduction theorem......Page 418
3.4.3 The work of Balaji......Page 419
3.5.1 Semistability ([Sch05], Section 1.1)......Page 420
3.5.2 Degenerations......Page 421
4 Quasi-parabolic principal bundles and the theorem of Atiyah and Bott......Page 422
4.1.2 Semistability......Page 423
4.1.3 Moduli stacks of semistable quasi-parabolic principal G-bundles......Page 425
4.3 The theorem of Atiyah and Bott......Page 426
5.1 Loop groups......Page 428
5.3 Picard groups and the Verlinde formula......Page 429
Appendix: One parameter subgroups......Page 430
Part III Research Articles......Page 436
1 Introduction......Page 437
2 Extensions of sections on vector bundles.......Page 438
3 Resolutions of diagonals and Beilinson spectral sequences......Page 439
4.1 General scrolls.......Page 440
4.2 Rational scrolls.......Page 443
4.3 Relative differentials.......Page 444
1 Introduction......Page 448
2.1 Preliminaries......Page 449
2.2 Extensions of Coherent systems......Page 451
3 The moduli space G(α; n, d, k)......Page 454
3.1 Variation of α......Page 457
4 The moduli space GL for k ≤ n......Page 460
4.1 The space BGNs (n, d, k)......Page 461
4.2 Moduli space GL for k = n......Page 465
1 Introduction......Page 467
2 General results......Page 472
3 Proof of Theorems 1.1 and 1.2......Page 474
4 Dual span......Page 477
5 Coherent systems over special curves......Page 480
1 Introduction......Page 484
2 The equations......Page 485
3 Existence......Page 487
4 The moduli space......Page 489
§1. Introduction......Page 495
§1. Distinguished G-spaces......Page 497
§2 Connection with parabolic structures......Page 501