Bergman kernel and pluripotential theory (Q2786811)

The article is a survey on results concerning questions on the relationship between Bergman geometry of a bounded pseudoconvex domain \(\Omega \subset \mathbf{C}^n\) and the properties of certain maximal plurisubharmonic functions, especially the pluricomplex Green function with a pole at a point \(w \in \Omega\).NEWLINENEWLINEIn a first part the crucial properties of the pluricomplex Green function \(G_\Omega ( \cdot ,w)\) with a pole at a point \(w \in \Omega\) are discussed, in particular its behavior under approach of the pole \(w\) towards the boundary of \(\Omega\).NEWLINENEWLINEThen the case where \(\Omega\) is hyperconvex is treated, and the completeness of the Bergman distance and also lower bounds on the Bergman kernel and metric are mentioned that were obtained by means of the Green function.NEWLINENEWLINEFinally the survey article reports on the Suita conjecture in planar domains and its solution.NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].