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| Main Author: | |
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| Format: | Preprint |
| Published: |
2011
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1112.6292 |
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Table of Contents:
- 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.