Geodesic
Increasing unions of Stein spaces with singularities
Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 1, pp. 5-10 
Y. Alaoui. Increasing unions of Stein spaces with singularities. Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 1, pp. 5-10. http://geodesic.mathdoc.fr/item/VMJ_2021_23_1_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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