Increasing unions of Stein spaces with singularities

Abstract

We show that if $X$ is a Stein space and, if $\Omega \subset X$ is exhaustable by a sequence $\Omega_1 \subset \Omega_2 \subset \ldots \subset \Omega_n \subset \ldots$ of open Stein subsets of $X$, then $\Omega$ is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for $X=\mathbb{C}^n$ and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension 2, we prove that the same result follows if we assume only that $\Omega \subset \subset X$ is a domain of holomorphy in a Stein normal space. It is known, however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets $X_1 \subset X_2 \subset \cdots \subset X_n \subset \cdots$, it does not follow in general that $X$ is holomorphically-convex or holomorphically-separate (even if $X$ has no singularities). One can even obtain 2-dimensional complex manifolds on which all holomorphic functions are constant.