Increasing unions of Stein spaces with singularities

TL;DR

This paper extends classical results on unions of Stein spaces, showing that increasing unions of Stein subsets are Stein under certain conditions, and explores the limitations of such unions in complex spaces with singularities.

Contribution

It generalizes the union theorem for Stein spaces to spaces with singularities and low dimensions, addressing longstanding questions in complex analytic geometry.

Findings

Union of increasing Stein subsets is Stein under certain conditions.

In dimension 2, unions are Stein if contained in a domain of holomorphy.

Counterexamples exist where complex spaces are not holomorphically convex or separate.

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.