Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 3

Bergman kernel and Kähler tensor calculus

Pages: 507 – 546

DOI: https://dx.doi.org/10.4310/PAMQ.2013.v9.n3.a7

Author

Hao Xu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China; Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman’s asymptotic expansion building partly on Graham’s work on CR invariants and Nakazawa’s work on the asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains. In this paper, we prove a formula for coefficients in Nakazawa’s asymptotic expansion as explicit summations over strongly connected graphs, and a formula expressing partial derivatives of Kähler metrics (resp. functions) as summations over rooted trees encoding covariant derivatives of curvature tensors (resp. functions). These formulae shall be useful in studying general patterns of Fefferman’s asymptotic expansion.

Keywords

Bergman kernel, asymptotic expansion, Laplace integral

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Published 13 November 2013