Increasing sequences of complex manifolds with uniform squeezing constants and their Bergman spaces

Abstract

For $d\geq 2$, we discuss $d$-dimensional complex manifolds $M$ that are the increasing union of bounded open sets $M_n$'s of $\mathbb{C}^d$ with a common uniform squeezing constant. The description of $M$ is given in terms of the corank of the infinitesimal Kobayashi metric of $M$, which is shown to be identically constant on $M$. The main result of this article says that if $M$ has full Kobayashi corank, then $M$ can be written as an increasing union of the unit ball; if $M$ has zero Kobayashi corank, then $M$ has a bounded realization with a uniform squeezing constant; and if $M$ has an intermediate Kobayashi corank, then $M$ has a local weak vector bundle structure. The above description of $M$ is used to show that the dimension of the Bergman space of $M \subseteq \mathbb{C}^d$ is either zero or infinity. This settles Wiegerinck's conjecture for those pseudoconvex domains in higher dimensions that are increasing union of bounded domains with a common uniform squeezing constant.