In general, if the manifold is not Kähler, then the Dolbeault-Dirac operator D = 2 \((\bar{\partial} + \bar{\partial}^{*})\) is not the most suitable one for getting explicit formulas for (2.39) and (2.40). For instance, if M is a complex analytic manifold and n = 2, then Gilkey [29, Thm. 3.7] proved that the difference
$$\rm{trace}_{\bf{c}}\,\it{K}^{+}_{1} (\it{x}) - \rm{trace}_{\bf{c}}\,\it{K}^{-}_{1} (\it{x})$$
of the traces of the coefficients of t−1 in the asymptotic expansion (1.2) is equal to a universal constant times
$$\rm{d}\,\bar{\partial} \sigma\,=\,\partial \bar{\partial} \sigma\,=\,\partial\,\rm{d}\,\sigma.$$


Clifford Algebra Principal Symbol Leibniz Rule Unitary Frame Clifford Multiplication 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

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

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