The Bergman kernel and pluripotential theory. (English) Zbl 1140.32003

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 1-9 (2006).
The paper is a survey of recent results relating the notion of the Bergman kernel for a bounded domain \(\Omega\) in \({\mathbb C}^n\) and notions from pluripotential theory, mainly the pluricomplex Green function with logaritmic singularities and the logarithmic capacity. The main part of the paper is a sketch of the main idea of proof of the facts that if \(\Omega\) is hyperconvex then it is Bergman exhaustive and Bergman complete. An example showing that the reverse implications are not true is also given. Various estimates for the Bergman kernel and the Bergman metric are also discussed.
For the entire collection see [Zbl 1102.31001].


32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
31C10 Pluriharmonic and plurisubharmonic functions
32U15 General pluripotential theory