Construction of boundary invariants and the logarithmic singularity of the Bergman kernel

Details Construction of boundary invariants and the logarithmic singularity of the Bergman kernel Construction of boundary invariants and the logarithmic singularity of the Bergman kernel

2000-10-02

arxiv.org/abs/math/0010014v1

Ann. of Math. (2) 151 (2000), no. 1, 151--191

Mathematics

Complex Variables

41 pages

Scientific paper

This paper studies Fefferman's program \cite{F3} of expressing the singularity of the Bergman kernel, for smoothly bounded strictly pseudoconvex domains $\Omega\subset\C^n$, in terms of local biholomorphic invariants of the boundary. By \cite{F1}, the Bergman kernel on the diagonal $K(z,\c{z})$ is written in the form $$ K=\phi r^{-n-1}+\psi \log r \qtext{with} \phi,\psi\in C^\infty(\c\Omega), $$ where $r$ is a (smooth) defining function of $\Omega$. Recently, Bailey, Eastwood and Graham \cite{BEG}, building on Fefferman's earlier work \cite{F3}, obtained a full invariant expression of the strong singularity $\phi r^{-n-1}$. The purpose of this paper is to give a full invariant expression of the weak singularity $\psi\log r$.

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