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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 EL 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • J. J. Duistermaat
    • 1
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands

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