The heat kernel Lefschetz fixed point formula for the spin-c dirac operator /

Interest in the spin-c Dirac operator originally came about from the study of complex analytic manifolds, where in the non-Kähler casethe Dolbeault operator is no longer suitable for getting local formulas for the Riemann-Roch number or the holomorphic Lefschetz number. However, every symplectic man...

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Bibliographic Details
Main Author:	Duistermaat, J. J. (Johannes Jisse), 1942-2010
Corporate Author:	SpringerLink (Online service)
Format:	eBook
Language:	English
Published:	Boston : Birkhäuser, [2011]
Series:	Modern Birkhäuser classics.
Subjects:	Almost complex manifolds. Operator theory. Dirac equation. Differential topology. Mathematical physics. Mathematics. Electronic books.
Online Access:	Connect to the full text of this electronic book

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-Khler 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 Khler 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 Khler Case
13. Characteristic Classes
13.1. Weils 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 Cartans Theorem
16.4. Proof of Weils Theorem
16.5. General Actions.