Moduli spaces:
A graduate-level introduction to some of the important contemporary ideas and problems in the theory of moduli spaces
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2014
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society Lecture Note Series
411 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | A graduate-level introduction to some of the important contemporary ideas and problems in the theory of moduli spaces |
| Physical Description: | XI, 333 S. graph. Darst. |
| ISBN: | 9781107636385 |
Staff View
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Record in the Search Index
| _version_ | 1804152056291786752 |
|---|---|
| adam_text | Titel: Moduli spaces
Autor: Brambila-Paz, Leticia
Jahr: 2014
Contents
Preface page ix
List of contributors xi
1 Introduction to algebraic Stacks 1
K. Behrend
Introduction 3
1.1 Topological Stacks: triangles 8
1.1.1 Families and their symmetry groupoids 8
1.1.2 Continuous families 13
1.1.3 Classification 16
1.1.4 Scalene triangles 23
1.1.5 Isosceles triangles 24
1.1.6 Equilateral triangles 26
1.1.7 Oriented triangles 26
1.1.8 Stacks 32
1.1.9 Versal families 34
1.1.10 Degenerate triangles 43
1.1.11 Change of versal family 58
1.1.12 Weierstrass compactification 66
1.2 Formalism 74
1.2.1 Objects in continuous families: categories fibered
in groupoids 74
1.2.2 Families characterized locally: prestacks 79
1.2.3 Families which can be glued: Stacks 82
1.2.4 Topological Stacks 82
1.2.5 Deligne-Mumford topological Stacks 89
1.2.6 Lattices up to homothety 96
1.2.7 Fundamental groups of topological Stacks 99
vi Contents
1.3 Algebraic Stacks 104
1.3.1 Groupoid fibrations 104
1.3.2 Prestacks 107
1.3.3 Algebraic Stacks 113
1.3.4 The coarse moduli space 117
1.3.5 Bundles on Stacks 120
1.3.6 Stacky curves: the Riemann-Roch theorem 124
BPS states and the P = W conjecture 132
W.-Y. Chuang, D.-E. Diaconescu, andG. Pan
2.1 Introduction 132
2.2 Hausel-Rodriguez-Villegas formula and P = W 138
2.2.1 Hausel-Rodriguez-Villegas formula 139
2.2.2 Hitchin System and P = W 140
2.3 Refined stable pair invariants of local curves 141
2.3.1 TQFT formalism 142
2.3.2 Refined invariants from instanten sums 142
2.4 HRV formula as a refined GV expansion 145
Representations of surface groups and Higgs bundles 151
Peter B. Gothen
3.1 Introduction 151
3.2 Lecture 1: Character varieties for surface groups
and harmonic maps 152
3.2.1 Surface group representations and character varieties 152
3.2.2 Review of connections and curvature in principal
bundles 153
3.2.3 Surface group representations and flat bundles 155
3.2.4 Fiat bundles and gauge equivalence 156
3.2.5 Harmonic metrics in flat bundles 157
3.2.6 The Corlette-Donaldson theorem 159
3.3 Lecture 2: G-Higgs bundles and the Hitchin-Kobayashi
correspondence 160
3.3.1 Lie theoretic prelirninaries 160
3.3.2 The Hitchin equations 161
3.3.3 G-Higgs bundles, stability and the
Hitchin-Kobayashi correspondence 163
3.3.4 The Hitchin map 166
3.3.5 The moduli space of SU(p, ?)-Higgs bundles 166
Contents vii
3.4 Lecture 3: Morse-Bott theory of the moduli space
of G-Higgs bundles 168
3.4.1 Simple and infinitesimally simple G-Higgs bundles 168
3.4.2 Deformation theory of G-Higgs bundles 169
3.4.3 The C*-action and topology of moduli spaces 170
3.4.4 Calculation of Morse indices 172
3.4.5 The moduli space of Sp(2n, R)-Higgs bundles 174
4 Introduction to stability conditions 179
D. Huybrechts
4.1 Torsion theories and t-structures 181
4.1.1 /x-stability on curves (and surfaces): recollections 181
4.1.2 Torsion theories in abehan categories 186
4.1.3 t-structures on triangulated categories 187
4.1.4 Torsion theories versus t-structures 190
4.2 Stability conditions: definition and examples 192
4.2.1 Slicings 192
4.2.2 Stability conditions 194
4.2.3 Aut(P)-actionandGL+(2,E)-action 198
4.2.4 Stability conditions on curves 200
4.3 Stability conditions on surfaces 203
4.3.1 Classification ofhearts 204
4.3.2 Construction ofhearts 208
4.4 The topological space of stability conditions 210
4.4.1 Topology ofSlice(P) 210
4.4.2 Topology ofStab(r ) 213
4.4.3 Main result 214
4.5 Stability conditions on K3 surfaces 216
4.5.1 Main theorem and conjecture 216
4.5.2 Autoequivalences 219
4.5.3 Building up Stab(X)0 220
4.5.4 Moduli space rephrasing 224
4.6 Further results 226
4.6.1 Non-compact cases 226
4.6.2 Compact cases 227
5 An introduction to d-manifolds and derived differential
geometry 230
Dominic Joyce
5.1 Introduction 231
viii Contents
5.2 C°°-rings and C°°-schemes 234
5.2.1 C°°-rings 234
5.2.2 C°°-schemes 236
5.2.3 Modules over C00-rings, and cotangent modules 239
5.2.4 Quasicoherent sheaves on C°°-schemes 240
5.3 The 2-category of d-spaces 243
5.3.1 The definitionof d-spaces 243
5.3.2 Gluing d-spaces by equivalences 246
5.3.3 Fibre producta in dSpa 249
5.4 The 2-category of d-manifolds 250
5.4.1 The definitionof d-manifolds 250
5.4.2 Standard model d-manifolds, 1-and 2-morphisms 251
5.4.3 The 2-category of Virtual vector bundles 255
5.4.4 Equivalences in dMan, and gluing by equivalences 257
5.4.5 Submersions, immersions and embeddings 259
5.4.6 D-transversality and fibre products 262
5.4.7 Embedding d-manifolds into manifolds 264
5.4.8 Orientations on d-manifolds 266
5.4.9 D-manifolds with boundary and corners, d-orbifolds 269
5.4.10 D-manifold bordism, and Virtual cycles 272
5.4.11 Relation to other classes of Spaces in mathematics 274
A Basics of 2-categories 277
13/2 ways of counting curves 282
R. Pandharipande and R. P. Thomas
0 Introduction 283
5 Naive counting of curves 287
li Gromov-Witten theory 289
2 Gopakumar-Vafa/BPS invariants 294
3* Donaldson-Thomas theory 298
4i Stable pairs 306
sl Stable unramified maps 314
6| Stable quotients 319
Appendix: Virtual classes 324
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| spelling | Moduli spaces ed. by Leticia Brambila-Paz ... 1. publ. Cambridge Cambridge Univ. Press 2014 XI, 333 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society Lecture Note Series 411 A graduate-level introduction to some of the important contemporary ideas and problems in the theory of moduli spaces Modulraum (DE-588)4183462-8 gnd rswk-swf 1\p (DE-588)1071861417 Konferenzschrift gnd-content Modulraum (DE-588)4183462-8 s DE-604 Brambila-Paz, Leticia Sonstige oth London Mathematical Society Lecture Note Series 411 (DE-604)BV000000130 411 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027201977&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
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| title | Moduli spaces |
| title_auth | Moduli spaces |
| title_exact_search | Moduli spaces |
| title_full | Moduli spaces ed. by Leticia Brambila-Paz ... |
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| title_full_unstemmed | Moduli spaces ed. by Leticia Brambila-Paz ... |
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