Guillemin，Ginzburg和Karshon的研究表明，从隐含的拓扑脉络来看， G 流形不变量的计算是涉及同变配边的线性化定理的结果。本书呈现了这一当前极受关注的快速发展领域中的许多新的成果，采用了新颖的方法，并展示了令人激动的新研究。 在过去的几十年中，“局部化”一直是同变微分几何学领域的重要主题之一。典型的结果是Duistermaat-Heckman理论、同变de Rham理论中的Berline-Vergne-Atiyah-Bott局部化定理以及“量化与约化交换”定理及其各种推论。为了阐述这些定理都是涉及同变配边的单个结论的结果这一想法，作者开发了允许对象是非紧致流形的配边理论。这种非紧致配边的关键要素是同变几何对象，他们称其为“抽象矩映射”。这是在Hamilton动力学理论中出现的矩映射的自然而重要的推广。本书还包含了多个附录，内容包括流形上正常群作用、同变上同调、Spinc结构和稳定复结构的介绍。 本书适合于对微分几何感兴趣的研究生和相关研究人员阅读，也可供拓扑学家、Lie理论学家、组合学家和理论物理学家参考。阅读本书需要流形上的微积分和基础研究生水平的微分几何方面的一些专业知识。

前辅文
Chapter 1.Introduction
  1.Topological aspects of Hamiltonian group actions
  2.Hamiltonian cobordism
  3.The linearization theorem and non-compact cobordisms
  4.Abstract moment maps and non-degeneracy
  5.The quantum linearization theorem and its applications
  6.Acknowledgements
Chapter 2.Hamiltonian cobordism
  1.Hamiltonian group actions
  2.Hamiltonian geometry
  3.Compact Hamiltonian cobordisms
  4.Proper Hamiltonian cobordisms
  5.Hamiltonian complex cobordisms
Chapter 3.Abstract moment maps
  1.Abstract moment maps：definitions and examples
  2.Proper abstract moment maps
  3.Cobordism
  4.First examples of proper cobordisms
  5.Cobordism s of surfaces
  6.Cobordism s of linear actions
Chapter 4.The linearization theorem
  1.The simplest case of the linearization theorem
  2.The Hamiltonian linearization theorem
  3.The linearization theorem for abstract moment maps
  4.Linear torus actions
  5.The right-hand side of the linearization theorems
  6.The Duistermaat-Heckman and Guillemin-Lerman-Sternberg formulas
Chapter 5.Reduction and applications
  1.(Pre-) symplectic reduction
  2.Reduction for abstract moment maps
  3.The Duis term a at-Heckman theorem
  4.Kahler reduction
  5.The complex Del zant construction
  6.Cobordism of reduced spaces
Chapter 6.Geometric quantization
  1.Quantization and group actions
  2.Pre-quantization
  3.Pre-quantization of reduced spaces
  4.Kirillov-Kosta nt pre-quantization
  5.Polarizations， complex structures，and geometric quantization
  6.Dol be ault Quantization and the Riemann-Roch formula
  7.Stable complex quantization and Spin°quantization
  8.Geometric quantization as a push-forward
Chapter 7.The quantum version of the linearization theorem
  1.The quantization of Cd
  2.Partition functions
  3.The character of Q(Cd
  4.A quantum version of the linearization theorem
Chapter 8.Quantization commutes with reduction
  1.Quantization and reduction commute
  2.Quantization of stable complex toric varieties
  3.Linearization of[Q， R] = 0
  4.Straightening the symplectic and complex structures
  5.Passing to holomorphic sheaf cohomology
  6.Computing global sections； the lit set
  7.The Cech complex
  8.The higher cohomology
  9.Singular[Q， R] = 0 for non-symplectic Hamiltonian G-manifolds
  10.Overview of the literature
Appendix A.Signs and normalization conventions
  1.The representation of GonC°(M)
  2.The integral weight lattice
  3.Connection and curvature for principal torus bundles
  4.Curvature and Chern classes
  5.Equivariant curvature； integral equivariant cohomology
Appendix B.Proper actions of Lie groups
  1.Basic definitions
  2.The slice theorem
  3.Corollaries of the slice therrem
  4.The Mostow-Palais embedding theorem
  5.Rigidity of compact group actions
Appendix C.Equivariant cohomology
  1.The definition and basic properties of equivariant cohomology
  2.Reduction and cohomology
  3.Additivity and localization
  4.Formality
  5.The relation between H*G and H*
  6.Equivariant vector bundles and characteristic classes
  7.The Atiyah-Bott-Berline-Vergne localization formula
  8.Applications of the Atiyah-Bott-Berline-Vergne localization formula
  9.Equivariant homology
Appendix D.Stable complex and Spin°-structures
  1.Stable complex structures
  2.Spin°-structures
  3.Spin-structures and stable complex structures
Appendix E.Assignments and abstract moment maps
  1.Existence of abstract moment maps
  2.Exact moment maps
  3.Hamiltonian moment maps
  4.Abstract moment maps on linear spaces are exact
  5.Formal cobordism of Hamiltonian spaces
Appendix F.Assignment cohomology
  1.Construction of assignment cohomology
  2.Assignments with other coefficients
  3.Assignment cohomology for pairs
  4.Examples of calculations of assignment cohomology
  5.Generalizations of assignment cohomology
Appendix G.Non-degenerate abstract moment maps
  1.Definitions and basic examples
  2.Global properties of non-degenerate abstract moment maps
  3.Existence of non-degenerate two-forms
Appendix H.Characteristic numbers,non-degenerate cobordisms, and non-virtual quantization
  1.The Hamiltonian cobordism ring and characteristic classes
  2.Characteristic numbers
  3.Characteristic numbers as a full system of invariants
  4.Non-degenerate cobordisms
  5.Geometric quantization
Appendix I.The Kawasaki Riemann-Roch formula
  1.Todd classes
  2.The Equivariant Riemann-Roch Theorem
  3.The KawasakiRiemann-RochformulaI：finiteabelianquotients
  4.The KawasakiRiemann-RochformulaII：torusquotients
Appendix J.Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman
  1.The SpinC-Dirac operator and the SpinC-quantization
  2.The summary of the results
  3.Transversally elliptic operators and their indexes
  4.Index of the operator Ba
  5.The model operator
  6.Proof of Theorem 1
Bibliography
index