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+.
KeywordsDirac Operator Heat Kernel Hermitian Structure Complex Vector Bundle Local Formula
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