Scalar boundary invariants and the Bergman kernel. (English) Zbl 0626.32027

Complex analysis II, Proc. Spec. Year, College Park/Md. 1985-86, Lect. Notes Math. 1276, 108-135 (1987).
[For the entire collection see Zbl 0615.00005.]
The author continues the program of C. Fefferman [Ann. Math., II. Ser. 103, 395-416 (1976; Zbl 0322.32012)] to invariantly express the asymptotic expansion of the Bergman kernel of a strictly pseudoconvex domain \(\Omega\) in \({\mathbb{C}}^ n\), \(n\geq 2\), in terms of the local biholomorphic geometry of the boundary. If \(\rho\) is the defining function for \(\Omega\), then on the diagonal the Bergman kernel may be written in the form \(K= \phi /\rho^{n+1}+ \psi \log \rho,\) where \(\phi,\psi \in C^{\infty}({\bar \Omega})\). The goal is to obtain expansions of \(\psi mod \rho^{n+1}\) and \(\psi\) in powers of \(\rho\) with coefficients which can be identified as invariants of the biholomorphic geometry of \(b\Omega\).
Reviewer: M.Stoll


32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32T99 Pseudoconvex domains