The Dolbeault-Dirac Operator
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In this chapter we set the stage, by introducing complex and almost structures, the Dolbeault complex and Hermitian structures. The holomorphic Lefschetz number, defined as the alternating sum of the trace of the automorphism acting on the cohomology of the sheaf of holomorphic sections, will be expressed in terms of a selfadjoint operator, which is built out of the Dolbeault operator and its adjoint; the Dolbeault-Dirac operator in the title of this chapter. This material is very well-known but, also in order to fix the notations, we have taken our time for the description of these structures. Just for convenience, we will assume that all objects are smooth (infinitely differentiable).
KeywordsVector Bundle Dirac Operator Complex Linear Selfadjoint Operator Holomorphic Section
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