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Vladikavkazskii Matematicheskii Zhurnal, 2021, Volume 23, Number 1, Pages 5–10
DOI: https://doi.org/10.46698/j5441-9333-1674-x
(Mi vmj750)
 

Increasing unions of Stein spaces with singularities

Y. Alaoui

Department of Fundamental Sciences, Hassan II Institute of Agronomy and Veterinary Sciences, B.P. 6202, Rabat, 10101, Morocco
References:
Abstract: We show that if $X$ is a Stein space and, if $\Omega\subset X$ is exhaustable by a sequence $\Omega_{1}\subset\Omega_{2}\subset\ldots\subset\Omega_{n}\subset\dots$ 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 $X_{1}\subset X_{2}\subset\dots\subset X_{n}\subset\dots$, 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.
Key words: Stein spaces, $q$-complete spaces, $q$-convex functions, strictly plurisubharmonic functions.
Received: 22.05.2020
Document Type: Article
UDC: 517.982
MSC: 32E10, 32E40
Language: English
Citation: Y. Alaoui, “Increasing unions of Stein spaces with singularities”, Vladikavkaz. Mat. Zh., 23:1 (2021), 5–10
Citation in format AMSBIB
\Bibitem{Ala21}
\by Y.~Alaoui
\paper Increasing unions of Stein spaces with singularities
\jour Vladikavkaz. Mat. Zh.
\yr 2021
\vol 23
\issue 1
\pages 5--10
\mathnet{http://mi.mathnet.ru/vmj750}
\crossref{https://doi.org/10.46698/j5441-9333-1674-x}
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