## Abstract

Let *M* be an almost complex manifold of real dimension *2n*, provided with a Hermitian structure. Furthermore, let *L* be a complex vector bundle over *M*, provided with a Hermitian connection. We also assume that *K*^{*}, the dual bundle of the so-called canonical line bundle *K* of *M*, is provided with a Hermitian connection. We write *E* for the direct sum over *q* of the bundles of (0, *q*)-forms; in it we have the subbundle *E*^{+} and *E*^{−}, where the sum is over the even *q* and odd *q*, respectively. Write Γ and Γ^{±} for the space of smooth sections of *E* ⊗ *L* and *E*^{±} ⊗ *L*, respectively. From these data, one can construct a first order partial differential operator *D*, the spin-c Dirac operator mentioned in the title of this book, which acts on Γ. The restriction *D*^{+} of *D* to Γ^{+} maps into Γ^{−}, and the restriction *D*^{−} of *D* to Γ^{−} maps into Γ^{+}. If *M* is compact, then the fact that *D* is elliptic implies that the kernel *N*^{±} of *D*^{±} is finite-dimensional, and the difference dim *N*^{+} - dim *N*^{−} is equal to the *index* of *D*^{+}.

## Keywords

Dirac Operator Heat Kernel Hermitian Structure Complex Vector Bundle Local Formula## Preview

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