Author	Duistermaat, J. J. author
Title	The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator [electronic resource] / by J. J. Duistermaat
Imprint	Boston, MA : Birkhรคuser Boston, 1996
Connect to	http://dx.doi.org/10.1007/978-1-4612-5344-0
Descript	VIII, 247 p. online resource

When visiting M.I.T. for two weeks in October 1994, Victor Guillemin made me enthusiastic about a problem in symplectic geometry which involved the use of the so-called spin-c Dirac operator. Back in Berkeley, where I had l spent a sabbatical semester , I tried to understand the basic facts about this operator: its definition, the main theorems about it, and their proofs. This book is an outgrowth of the notes in which I worked this out. For me this was a great learning experience because of the many beautiful mathematical structures which are involved. I thank the Editorial Board of Birkhauser, especially Haim Brezis, for sugยญ gesting the publication of these notes as a book. I am also very grateful for the suggestions by the referees, which have led to substantial improvements in the presentation. Finally I would like to express special thanks to Ann Kostant for her help and her prodding me, in her charming way, into the right direction. J.J. Duistermaat Utrecht, October 16, 1995

1 Introduction -- 1.1 The Holomorphic Lefschetz Fixed Point Formula -- 1.2 The Heat Kernel -- 1.3 The Results -- 2 The Dolbeault-Dirac Operator -- 2.1 The Dolbeault Complex -- 2.2 The Dolbeault-Dirac Operator -- 3 Clifford Modules -- 3.1 The Non-Kรคhler Case -- 3.2 The Clifford Algebra -- 3.3 The Supertrace -- 3.4 The Clifford Bundle -- 4 The Spin Group and the Spin-c Group -- 4.1 The Spin Group -- 4.2 The Spin-c Group -- 4.3 Proof of a Formula for the Supertrace -- 5 The Spin-c Dirac Operator -- 5.1 The Spin-c Frame Bundle and Connections -- 5.2 Definition of the Spin-c Dirac Operator -- 6 Its Square -- 6.1 Its Square -- 6.2 Dirac Operators on Spinor Bundles -- 6.3 The Kรคhler Case -- 7 The Heat Kernel Method -- 7.1 Traces -- 7.2 The Heat Diffusion Operator -- 8 The Heat Kernel Expansion -- 8.1 The Laplace Operator -- 8.2 Construction of the Heat Kernel -- 8.3 The Square of the Geodesic Distance -- 8.4 The Expansion -- 9 The Heat Kernel on a Principal Bundle -- 9.1 Introduction -- 9.2 The Laplace Operator on P -- 9.3 The Zero Order Term -- 9.4 The Heat Kernel -- 9.5 The Expansion -- 10 The Automorphism -- 10.1 Assumptions -- 10.2 An Estimate for Geodesies in P -- 10.3 The Expansion -- 11 The Hirzebruch-Riemann-Roch Integrand -- 11.1 Introduction -- 11.2 Computations in the Exterior Algebra -- 11.3 The Short Time Limit of the Supertrace -- 12 The Local Lefschetz Fixed Point Formula -- 12.1 The Element g0 of the Structure Group -- 12.2 The Short Time Limit -- 12.3 The Kรคhler Case -- 13 Characteristic Classes -- 13.1 Weilโ{128}{153}s Homomorphism -- 13.2 The Chern Matrix and the Riemann-Roch Formula -- 13.3 The Lefschetz Formula -- 13.4 A Simple Example -- 14 The Orbifold Version -- 14.1 Orbifolds -- 14.2 The Virtual Character -- 14.3 The Heat Kernel Method -- 14.4 The Fixed Point Orbifolds -- 14.5 The Normal Eigenbundles -- 14.6 The Lefschetz Formula -- 15 Application to Symplectic Geometry -- 15.1 Symplectic Manifolds -- 15.2 Hamiltonian Group Actions and Reduction -- 15.3 The Complex Line Bundle -- 15.4 Lifting the Action -- 15.5 The Spin-c Dirac Operator -- 16 Appendix: Equivariant Forms -- 16.1 Equivariant Cohomology -- 16.2 Existence of a Connection Form -- 16.3 Henri Cartanโ{128}{153}s Theorem -- 16.4 Proof of Weilโ{128}{153}s Theorem -- 16.5 General Actions