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

Abstract

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 $\psi\log r$.