Torus Actions on Symplectic Manifolds 对Symplectic多头管的花托行动
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作者： Torus Actions on Symplectic Manifolds著
出 版 社：
出版时间： 1994-12-1字数：版次： 1页数： 325印刷时间： 1991/12/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783764321765包装： 精装内容简介
The action of a compact Lie group，G，on a compact sympletic manifold gives rise to some remarkable combinatorial invariants。 The simplest and most interesting of these is the moment polytope，a convex polyhedron which sits inside the dual of the Lie algebra of G，One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope。For instance，the first chapter is largely devoted to the Delzant theorem， which says that there is a one-one correspondence between certain types of moment polytopes and certain types of symplectic G-spaces。（One of the most challenging unsolved problems in symplectic geometry is to determine to what extent Delzant’s theorem is true of every compact symplectic G-Space。）
The moment polytope also encodes quantum information about the actions of G。 Using the methods of geometric quantization，one can frequently convert this action into a representations，p ，of G on a Hilbert space， and in some sense the moment polytope is a diagrammatic picture of the irreducible representations of G which occur as subrepresentations of Precise versions of this item of folklore are discussed in Chapters 3 and 4，Also，midway through Chapter 2 a more complicated object is discussed: the Duistermaat-Heckman measure，and the author explains in Chapter 4 how one can read off from this measure the approximate multiplicities with which the irreducible representations of G occur in p，This gives an excuse to touch on some results which are in themselves of great current interest: the Duistermaat-Heckman theorem，the localization theorems in equivariant cohomology of Atiyah-Bott and Berline-Vergne and the recent extremely exciting generalizations of these results by Witten，Jeffrey-Kirwan，Lalkman，and others。
The last two chapters of this book are a self-contained and somewhat unorthodox treatment of the theory of toric varieties in which the usual hierarchal relation of complex to symplectic is reversed。 This book is addressed to researchers。
目录
Introductory preface
How I have （re-）written this
Acknowledgements
What I have written in this
Ⅰ. Smooth Lie group
Ⅰ.1. Generalities
Ⅰ.2. Equivariant tubular neighborhoods and orbit types decomposition
Ⅰ.3. Examples： S1-actions on manifolds of dimension 2 and 3
Ⅰ.4. Appendix： Lie groups， Lie algebras， homogeneous spacesExercises
Ⅱ. Symplectie manifolds
Ⅱ.1. What is a symplectic manifold?
Ⅱ.2. Calibrated almost complex structures
Ⅱ.3. Hamiltonian vector fields and Poisson brackets Exercises
Ⅲ. Symplectic and Hamiltonian group actions
Ⅲ.1. Hamiltonian group actions
Ⅲ.2. Properties of momentum mappings
Ⅲ.3. Torus actions and integrable systems Exercises
Ⅳ. Morse theory for Hamiltonians
Ⅳ.1. Critical points of almost periodic Hamiltonians
Ⅳ.2. Morse functions （in the sense of Bott）
Ⅳ.3. Connectedness of the fibers of the momentum mapp ing
Ⅳ.4. Application to convexity theorems
Ⅳ.5. Appendix： compact symplectic SU（2）-manifolds of dimension 4 Exercises
Ⅴ. Moduli spaces of flat connections
Ⅴ.1. The moduli space of fiat connections
Ⅴ.2. A Poisson structure on the moduli space of fiat connections
Ⅴ.3. Construction of commuting functions on M
Ⅴ.4. Appendix： connections on principal bundles Exercises
Ⅵ. Equivariant cohomology and the Duistermaat-tteckman theorem
Ⅵ. 1. Milnor joins， Borel construction and equivariant cohomology
Ⅵ.2. Hamiltonian actions and the Duistermaat Heckman theorem
Ⅵ.3. Localization at fixed points and the Duistermaat Heckmanformula
Ⅵ.4. Appendix：some algebraic topology
Ⅵ.5. Appendix：various notions of Euler classes Exercises
Ⅶ. Toric manifolds
Ⅶ.1. Fans and toric varieties
Ⅷ.2. Symplectic reduction and convex polyhedra
Ⅶ.3. Cohomology of
Ⅶ.4. Complex toric surfaces
Exercises
Ⅷ. Hamiltonian circle actions on manifolds of dimension
Ⅷ.1. Symplectic S-actions， generalities
Ⅷ.2. Periodic Hamiltonians on 4-dimensional manifolds Exercises
Bibliography
Index
