Increasing unions of Stein spaces with singularities
Youssef Alaoui
TL;DR
This work addresses the union problem for Stein spaces: whether a Stein space Ω that is the increasing union of Stein open subsets remains Stein, generalizing the Behnke-Stein result beyond X = C^n. The authors develop a constructive method that produces, for Ω = ∪Ω_ν, a sequence of open subsets Ω'_ν with continuous strictly plurisubharmonic exhaustion functions whose limits give a global exhaustion, thereby establishing Steinness of Ω. A key step is a dimension-2 reduction to domains of holomorphy in 2-dimensional normal Stein spaces, using local normalization arguments to prove local Steinness near singularities, and then gluing these local pieces into a global exhaustion in the general setting. The results confirm that unions of Stein open sets in Stein spaces are Stein and illuminate the role of exhaustion functions and local holomorphic properties in controlling holomorphic convexity and extension. This contributes to the broader understanding of holomorphic convexity, cohomology vanishing criteria, and the structure of complex spaces with singularities.
Abstract
We show that if $X$ is a Stein space and, if $Ω\subset X$ is exhaustable by a sequence $Ω_1 \subset Ω_2 \subset \ldots \subset Ω_n \subset \ldots$ of open Stein subsets of $X$, then $Ω$ 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 $Ω\subset \subset X$ 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.