The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator: (18 Progress in Nonlinear Differential Equations and Their Applications)

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.

Table of Contents:
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 TheLefschetz 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.

About the Author :
Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was (co-)author of eleven books. Duistermaat was affiliated to the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.