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. Language: English