The topology of torus actions on symplectic manifolds:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Basel [u.a.]
Birkhäuser
1991
|
| Series: | Progress in mathematics
93 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 181 S. graph. Darst. |
| ISBN: | 3764326026 0817626026 |
Staff View
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| 240 | 1 | 0 | |a Opérations hamiltoniennes de tores sur les variétés symplectiques |
| 245 | 1 | 0 | |a The topology of torus actions on symplectic manifolds |c Michèle Audin |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 1991 | |
| 300 | |a 181 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804118433108852737 |
|---|---|
| adam_text | Contents
Introduction 7
I Smooth Lie group actions on manifolds 13
1 Generalities 13
1.1 Notation 13
1.2 Orbits and fundamental vector fields 14
1.3 Examples 15
1.4 More definitions 15
1.5 Equivariant maps and orbit spaces 16
2 Equivariant tubular neighborhoods and orbit types decomposition . . 17
2.1 The slice theorem (equivariant tubular neighborhood) 17
2.2 Applications 19
3 Examples: 2 and 3 dimensional 5 1 manifolds 22
3.1 5x actions on surfaces 22
3.2 3 manifolds; the principal case 25
3.3 Seifert manifolds 28
3.4 Associated principal actions and Euler class 34
II Symplectic geometry 37
1 Symplectic manifolds 37
1.1 Symplectic vector spaces 37
1.2 Symplectic manifolds, definition 38
1.3 The Darboux theorem 39
1.4 Examples: t/(n) orbits in hermitian matrices 41
1.5 Calibrated almost complex structures 42
2 Hamiltonian vector fields and Poisson manifolds 43
2.1 Hamiltonian vector fields 43
2.2 The Poisson bracket on a symplectic manifold 45
3 Symplectic and hamiltonian actions 46
3.1 Symplectic actions 46
3.2 Hamiltonian actions 47
3.3 A machine producing examples: the coadjoint representation of
a Lie group 49
3.4 Some properties of moment maps 52
3.5 Noether type theorems 54
3
4 Contents
3.6 Symplectic reduction, examples 57
III Morse theory for hamiltonians 61
1 Critical points of almost periodic hamiltonians 61
1.1 Almost periodic hamiltonians 61
1.2 Critical points 62
2 Morse functions (in the sense of Bott) 63
2.1 Definitions 63
2.2 Frankel s theorem 64
2.3 PerestroTka 64
3 Connectivity of the fibers of the moment map 66
3.1 Connectivity of levels 66
3.2 Case of almost periodic hamiltonians 66
4 Application to convexity theorems 66
4.1 Proof of the Hausdorff Ginsburg theorem 66
4.2 Convexity of the image of the moment map 67
4.3 Application: a theorem of Schur on hermitian matrices 71
4.4 Application: a theorem of Kushnirenko on monomial equations . 72
IV About manifolds of this dimension 75
1 Characterisation of those circle actions which are hamiltonian 76
1.1 Statement of the theorem 76
1.2 Proof 76
2 Symplectic reduction of the regular levels for a periodic hamiltonian . 78
2.1 What happens near an extremum 78
2.2 What happens when going through a critical value 79 t
2.3 First applications 80
2.4 What happens when there are only two critical values 81
3 Blowing up fixed points; creation of index 2 critical points 84
3.1 Blowing up 0 in C2 84
3.2 Extension of an S action 86
3.3 Gradient manifolds and exceptional divisors 87
4 4 manifolds with periodic hamiltonians 88
4.1 Description 88
5 Plumbing 90
5.1 Plumbing of disc bundles 90
5.2 Equivariant plumbing along star shaped graphs 91
5.3 Periodic hamiltonians and plumbing 96
5.4 Description of W up to the maximum: finishing graphs 96
A Appendix: compact symplectic 5t/(2) manifolds of dimension 4 .... 99
A.I A list of examples 99
A.2 Classification 101
B Appendix: 4 dimensional S manifolds with no invariant symplectic
form (examples) 104
B.I Equivariant plumbing on non simply connected graphs, examples 104
Contents 5
B.2 Generalisations Ill
V Equivariant cohomology and the Duistermaat Heckman theorems 113
1 Principal and universal bundles 114
1.1 Principal bundles 114
1.2 Universal bundles 114
2 The Borel construction and equivariant cohomology 119
2.1 The Borel construction 119
2.2 Equivariant cohomology 120
2.3 Generators for de Rham cohomology 121
2.4 Euler classes for fixed point free T actions 123
3 Equivariant cohomology and hamiltonian actions 125
3.1 Relationships between hamiltonian actions and equivariant co¬
homology 125
3.2 Variation of the reduced symplectic forms 126
4 Duistermmat Heckman with singularities 128
4.1 The simple situation 128
4.2 General case of a fixed submanifold of signature (2p, 2^) 129
4.3 Application: The Duistermaat Heckman problem at critical valuesl31
5 Localisation at fixed points 133
5.1 The support of a #*£T module 133
5.2 Supports of HftU), examples 134
5.3 The localisation theorem 136
6 The Duistermaat Heckman formula 139
6.1 The Duistermaat Heckman formula 139
6.2 Examples of applications 140
A Appendix: some algebraic topology 142
A.I The Thorn class of an oriented vector bundle 142
A.2 The Euler class of an oriented bundle, equivariant Euler class . . 143
A.3 The Gysin exact sequence 144
A.4 The cohomology of projective space 144
A.5 The Gysin homomorphism (case of an embedding) 144
A.6 The Gysin homomorphism: integration in the fibers 145
B Appendix: various notions of Euler classes 146
B.I The case of S bundles 147
B.2 Complex line bundles 148
VI Toric manifolds 151
1 The action of Tg and its subgroups onC 152
1.1 Nontrivial stabilizers 153
1.2 Subtori 153
1.3 Real and imaginary parts 154
2 Fans and toric varieties 154
2.1 Fans 154
2.2 Closing a fan, open subsets in C , toric varieties 156
6 Contents
3 Fans, symplectic reduction, convex polyhedra 162
3.1 The moment map for the Abaction 162
3.2 What is left from the T^ action 164
4 Properties of the toric manifolds Xy, 165
4.1 Compacity of X z 165
4.2 Topology of X% and classes of invariant symplectic forms .... 166
4.3 Integral polyhedra and invariant line bundles 167
4.4 The cohomology of X% 171
5 Complex toric surfaces 172
5.1 Graphs associated with dimension 2 fans 172
5.2 Interpretation of the m; 174
5.3 4 manifolds with hamiltonian T2 actions 176
References 177
|
| any_adam_object | 1 |
| author | Audin, Michèle 1954- |
| author_GND | (DE-588)14151552X |
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| ctrlnum | (OCoLC)23140118 (DE-599)BVBBV004236402 |
| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV004236402 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:10:13Z |
| institution | BVB |
| isbn | 3764326026 0817626026 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002635877 |
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| physical | 181 S. graph. Darst. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Audin, Michèle 1954- Verfasser (DE-588)14151552X aut Opérations hamiltoniennes de tores sur les variétés symplectiques The topology of torus actions on symplectic manifolds Michèle Audin Basel [u.a.] Birkhäuser 1991 181 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 93 Hamilton, systèmes de ram Variétés symplectiques ram Hamiltonian systems Symplectic manifolds Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Torische Varietät (DE-588)4786945-8 gnd rswk-swf Torus (DE-588)4185738-0 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Symplektische Mannigfaltigkeit (DE-588)4290704-4 gnd rswk-swf Hamilton-Operator (DE-588)4072278-8 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s Topologie (DE-588)4060425-1 s DE-604 Torus (DE-588)4185738-0 s Hamilton-Operator (DE-588)4072278-8 s Symplektische Mannigfaltigkeit (DE-588)4290704-4 s Lie-Gruppe (DE-588)4035695-4 s Algebraische Topologie (DE-588)4120861-4 s Torische Varietät (DE-588)4786945-8 s 1\p DE-604 Progress in mathematics 93 (DE-604)BV000004120 93 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002635877&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 |
| spellingShingle | Audin, Michèle 1954- The topology of torus actions on symplectic manifolds Progress in mathematics Hamilton, systèmes de ram Variétés symplectiques ram Hamiltonian systems Symplectic manifolds Algebraische Topologie (DE-588)4120861-4 gnd Lie-Gruppe (DE-588)4035695-4 gnd Torische Varietät (DE-588)4786945-8 gnd Torus (DE-588)4185738-0 gnd Topologie (DE-588)4060425-1 gnd Symplektische Mannigfaltigkeit (DE-588)4290704-4 gnd Hamilton-Operator (DE-588)4072278-8 gnd Symplektische Geometrie (DE-588)4194232-2 gnd |
| subject_GND | (DE-588)4120861-4 (DE-588)4035695-4 (DE-588)4786945-8 (DE-588)4185738-0 (DE-588)4060425-1 (DE-588)4290704-4 (DE-588)4072278-8 (DE-588)4194232-2 |
| title | The topology of torus actions on symplectic manifolds |
| title_alt | Opérations hamiltoniennes de tores sur les variétés symplectiques |
| title_auth | The topology of torus actions on symplectic manifolds |
| title_exact_search | The topology of torus actions on symplectic manifolds |
| title_full | The topology of torus actions on symplectic manifolds Michèle Audin |
| title_fullStr | The topology of torus actions on symplectic manifolds Michèle Audin |
| title_full_unstemmed | The topology of torus actions on symplectic manifolds Michèle Audin |
| title_short | The topology of torus actions on symplectic manifolds |
| title_sort | the topology of torus actions on symplectic manifolds |
| topic | Hamilton, systèmes de ram Variétés symplectiques ram Hamiltonian systems Symplectic manifolds Algebraische Topologie (DE-588)4120861-4 gnd Lie-Gruppe (DE-588)4035695-4 gnd Torische Varietät (DE-588)4786945-8 gnd Torus (DE-588)4185738-0 gnd Topologie (DE-588)4060425-1 gnd Symplektische Mannigfaltigkeit (DE-588)4290704-4 gnd Hamilton-Operator (DE-588)4072278-8 gnd Symplektische Geometrie (DE-588)4194232-2 gnd |
| topic_facet | Hamilton, systèmes de Variétés symplectiques Hamiltonian systems Symplectic manifolds Algebraische Topologie Lie-Gruppe Torische Varietät Torus Topologie Symplektische Mannigfaltigkeit Hamilton-Operator Symplektische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002635877&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT audinmichele operationshamiltoniennesdetoressurlesvarietessymplectiques AT audinmichele thetopologyoftorusactionsonsymplecticmanifolds |