An essay on Bergman completeness (Q363493)

The subject of this article is an investigation of the relationship between Bergman completeness of a Stein mainfold \(X\) and the behavior of the sublevel sets of the pluricomplex Green function \(g_X(\cdot, y)\) of \(X\) with its pole at a point \(y \in X\).NEWLINENEWLINEThe author proves the following relevant criterion for Bergman completeness: NEWLINENEWLINENEWLINENEWLINE Theorem. If a Stein manifold \(X\) possesses the Bergman metric \(B_X\), then \(X\) is complete with respect to \(B_X\), if the following condition is fulfilled.NEWLINENEWLINE(E) For any sequence \((y_k)_k \subset X\) without adherent point in \(X\), there are a subsequence \((y_{k_j})_j \), a number \(a>0\) and a continuous volume form \(dV\) on \(X\) such that, for any compact subset \(K \subset X\), one has NEWLINE\[NEWLINE\int_{\{z \in K\,|\, g_X( z, y_{k_j}) \leq -a\}} dV \longrightarrow 0,\quad \text{as}\quad j \to\infty.NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINE Several applications follow. Here is the first, which improves a result that was obtained before by Siu: NEWLINENEWLINENEWLINENEWLINE Theorem. Every Stein subvariety \(Y\) of a complex manifold \(X\) admits a fundamental family of Bergman complete Stein neighborhoods of \(Y\) in \(X\). NEWLINENEWLINENEWLINENEWLINE The next application is concerned with Galois coverings.NEWLINENEWLINENEWLINENEWLINE Theorem. Let \(X\) be a Stein manifold satisfying condition (E). If \(X\) admits a negative continuous strictly plurisubharmonic function, then every Galois covering of \(X\) is Bergman complete.NEWLINENEWLINEThe article further contains results on the Bergman kernel form and metric and the Poincaré metric on special Riemann surfaces and concludes with an example of a two-dimensional noncompact pseudoconvex manifold that is Bergman complete but does not possess a non-constant negative continuous plurisubharmonic function.