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+.
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© 1996 Birkhäuser Boston
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Duistermaat, J.J. (1996). Introduction. In: The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator. Progress in Nonlinear Differential Equations and their Applications, vol 18. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-5344-0_1
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DOI: https://doi.org/10.1007/978-1-4612-5344-0_1
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-5346-4
Online ISBN: 978-1-4612-5344-0
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