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DOI: 10.46698/j5441-9333-1674-x

Increasing Unions of Stein Spaces with Singularities

Alaoui, Y.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 1.
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.
Keywords: Stein spaces, \(q\)-complete spaces, \(q\)-convex functions, strictly plurisubharmonic functions
Language: English Download the full text  
For citation: Alaoui, Y. Increasing Unions of Stein Spaces with Singularities, Vladikavkaz Math. J., 2021, vol. 23, no. 1, pp. 5-10. DOI 10.46698/j5441-9333-1674-x
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