Torus Actions On Symplectic Manifolds

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Springer Science & Business Media, 27 sept. 2004 - 325 pages
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How I have (re-)written this book The book the reader has in hand was supposed to be a new edition of [14]. I have hesitated quite a long time before deciding to do the re-writing work-the first edition has been sold out for a few years. There was absolutely no question of just correcting numerous misprints and a few mathematical errors. When I wrote the first edition, in 1989, the convexity and Duistermaat-Heckman theorems together with the irruption of toric varieties on the scene of symplectic geometry, due to Delzant, around which the book was organized, were still rather recent (less than ten years). I myself was rather happy with a small contribution I had made to the subject. I was giving a post-graduate course on all that and, well, these were lecture notes, just lecture notes. By chance, the book turned out to be rather popular: during the years since then, I had the opportunity to meet quite a few people(1) who kindly pretended to have learnt the subject in this book. However, the older book does not satisfy at all the idea I have now of what a good book should be. So that this "new edition" is, indeed, another book.
 

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Table des matières

Smooth Lie group actions on manifolds
9
12 Equivariant tubular neighborhoods and orbit types decomposition
13
S¹actions on manifolds of dimension 2 and 3
18
Lie groups Lie algebras homogeneous spaces
32
Exercises
37
Symplectic manifolds
43
22 Calibrated almost complex structures
52
23 Hamiltonian vector fields and Poisson brackets
58
connections on principal bundles
170
Exercises
175
Equivariant cohomology and the DuistermaatHeckman theorem
177
61 Milnor joins Borel construction and equivariant cohomology
178
62 Hamiltonian actions and the DuistermaatHeckman theorem
189
63 Localization at fixed points and the Duistermaat Heckman formula
201
some algebraic topology
212
various notions of Euler classes
218

Exercises
62
Symplectic and Hamiltonian group actions
71
32 Properties of momentum mappings
77
33 Torus actions and integrable systems
87
Exercises
97
Morse theory for Hamiltonians
105
42 Morse functions in the sense of Bott
108
43 Connectedness of the fibers of the momentum mapping
111
44 Application to convexity theorems
113
compact symplectic SU2manifolds of dimension 4
131
Exercises
136
Moduli spaces of flat connections
147
52 A Poisson structure on the moduli space of flat connections
154
53 Construction of commuting functions on M
162
Exercises
220
Toric manifolds
225
71 Fans and toric varieties
226
72 Symplectic reduction and convex polyhedra
244
73 Cohomology of X𝛴
257
74 Complex toric surfaces
262
Exercises
266
Hamiltonian circle actions on manifolds of dimension 4
271
81 Symplectic S¹actions generalities
272
82 Periodic Hamiltonians on 4dimensional manifolds
279
Exercises
305
Bibliography
311
Index
321
Droits d'auteur

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Fréquemment cités

Page 318 - The Clarendon Press Oxford University Press, New York, 1995, Oxford Science Publications.

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