Introduction

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 E and E^± 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 E. The restriction D⁺ of D to E⁺ maps into E^-, and the restriction D^- of D to E^- maps into E⁺. 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⁺.