The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator

Description

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 Front Matter....Pages i-viii Introduction....Pages 1-5 The Dolbeault-Dirac Operator....Pages 7-17 Clifford Modules....Pages 19-33 The Spin Group and the Spin-c Group....Pages 35-40 The Spin-c Dirac Operator....Pages 41-51 Its Square....Pages 53-68 The Heat Kernel Method....Pages 69-75 The Heat Kernel Expansion....Pages 77-97 The Heat Kernel on a Principal Bundle....Pages 99-115 The Automorphism....Pages 117-129 The Hirzebruch-Riemann-Roch Integrand....Pages 131-145 The Local Lefschetz Fixed Point Formula....Pages 147-156 Characteristic Classes....Pages 157-170 The Orbifold Version....Pages 171-186 Application to Symplectic Geometry....Pages 187-219 Appendix: Equivariant Forms....Pages 221-235 Back Matter....Pages 237-247