Ebook: 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.