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Generalized Bergman kernels on symplectic manifolds

Xiaonan Ma·George Marinescu
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Abstract

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut-Lebeau.

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BibTeXRIS

Xiaonan Ma, George Marinescu. 2005-11-15. Generalized Bergman kernels on symplectic manifolds. https://doi.org/10.1016/j.aim.2007.10.008

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