The Heat Kernel Lefschetz Fixed Point Formula for the Spin c Dirac Operator 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 Kähler 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 Kähler 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 Kähler Case 13 Characteristic Classes 13.1 Weil’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 Cartan’s Theorem 16.4 Proof of Weil’s Theorem 16.5 General Actions. 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 sug­ 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.