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

Kengo Hirachi

Review from Zentralblatt MATH:

The author studies Fefferman's program of expressing the singularity of the Bergman kernel for smoothly bounded strictly pseudoconvex domains $\Omega\subset \Bbb C^n$ in terms of local biholomorphic invariants of the boundary. The Bergman kernel on the diagonal $K(z,\overline{z})$ is written in the form $K=\varphi r^{-n-1}+\psi\log r$ with $\varphi,\psi\in C^\infty(\overline{\Omega}),$ where $r$ is a smooth defining function of $\Omega.$

The purpose of this paper is to give a full invariant expression of the weak singularity $\psi\log r.$

Reviewed by A.V.Chernecky

Keywords: Fefferman's program; Bergman kernel; smoothly bounded strictly pseudoconvex domains; CR invariants; Weyl invariants; defining functions; complex Monge-Ampère equation; asymptotics; biholomorphic invariance; Weyl function

Classification (MSC2000): 32A25

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