## Torus Actions On Symplectic ManifoldsHow 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. |

### Avis des internautes - Rédiger un commentaire

Aucun commentaire n'a été trouvé aux emplacements habituels.

### 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 |

311 | |

321 | |

### Autres éditions - Tout afficher

### Expressions et termes fréquents

4-manifolds associated Assume bilinear form Check coadjoint commutative complex structure components cone consider coordinates Corollary corresponding critical points critical submanifold Deduce defined denoted diffeomorphism differential element equivariant Euler class Example exceptional orbit Exercise fiber fibration fixed points function fundamental vector field G-action gradient spheres graph group action group G Hamiltonian action Hamiltonian vector fields Hence Hermitian Hirzebruch surface homotopy integral invariant IS8N isomorphism Kahler Lemma Lie algebra Lie group line bundle manifold of dimension matrix metric moduli space momentum mapping morphism nondegenerate normal bundle Notice open subset oriented periodic Hamiltonian Pn(C point of index Poisson bracket polyhedron principal G-bundle projective space Proof Proposition Prove quotient real number regular levels regular values Remark S^action S1-action Seifert shown in Figure skew-symmetric smooth stabilizer subgroup submanifold subspace symplectic form symplectic manifold theorem toric manifold torus action trivial vector bundle vector space zero

### Fréquemment cités

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