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

1. Front Matter

   Pages i-viii

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2. Introduction

   • J. J. Duistermaat

   Pages 1-5

3. The Dolbeault-Dirac Operator

   • J. J. Duistermaat

   Pages 7-17

4. Clifford Modules

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   Pages 19-33

5. The Spin Group and the Spin-c Group

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   Pages 35-40

6. The Spin-c Dirac Operator

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   Pages 41-51

7. Its Square

   • J. J. Duistermaat

   Pages 53-68

8. The Heat Kernel Method

   • J. J. Duistermaat

   Pages 69-75

9. The Heat Kernel Expansion

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   Pages 77-97

10. The Heat Kernel on a Principal Bundle

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    Pages 99-115

11. The Automorphism

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    Pages 117-129

12. The Hirzebruch-Riemann-Roch Integrand

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    Pages 131-145

13. The Local Lefschetz Fixed Point Formula

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    Pages 147-156

14. Characteristic Classes

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    Pages 157-170

15. The Orbifold Version

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    Pages 171-186

16. Application to Symplectic Geometry

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    Pages 187-219

17. Appendix: Equivariant Forms

    • J. J. Duistermaat

    Pages 221-235

18. Back Matter

    Pages 237-247

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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.

• diffusion
• Dirac operator
• geometry
• Hamiltonian
• Laplace operator
• manifold
• operator

• Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands

  J. J. Duistermaat

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.