Increasing Unions of Stein Spaces with Singularities

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

Home Editorial board Publication ethics Peer review guidelines Current Archive Rules for authors Contacts Address: Markusa st. 22, Vladikavkaz, 362027, RNO-A, Russia Phone: (8672)50-18-06 E-mail: rio@smath.ru	DOI: 10.46698/j5441-9333-1674-x Increasing Unions of Stein Spaces with Singularities Alaoui, Y. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 1. 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. 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 + References 1. Behnke, H. and Stein, K. Konvergente Folgen Von Regularitatsbereichen and die Meromorphiekonvexitat, Mathematische Annalen, 1939, vol.116, pp. 204--216. DOI: 10.1007/BF01597355. 2. Markoe, A. Runge Families and Inductive Limits of Stein Spaces, Annales de l'Institut Fourier, 1977, vol. 27, no. 3, pp. 117-127. DOI: 10.5802/aif.663. 3. Silva, A. Rungescher Satz and a Condition for Steiness for the Limit of an Increasing Sequence of Stein Spaces, Annales de l'Institut Fourier, 1978, vol. 28, no. 2, pp. 187-200. DOI: 10.5802/aif.695. 4. Fornaess, J. E. An Increasing Sequence of Stein Manifolds whose Limit is not Stein, Mathematische Annalen, 1976, vol. 223, pp. 275-277. DOI: 10.1007/BF01360958. 5. Fornaess, J. E. \(2\)-Dimensional Counterexamples to Generalizations of the Levi Problem, Mathematische Annalen, 1977, vol. 230, pp. 169-173. DOI: 10.1007/BF01370661. 6. Fornaess, J. E. and Stout, E. L. Polydiscs in Complex Manifolds, Mathematische Annalen, 1977, vol. 227, pp. 145-153. DOI: 10.1007/BF01350191. 7. Coltoiu, M. Remarques sur les Reunions Croissantes d'Ouverts de Stein, Comptes Rendus de l'Academie des Sciences. Ser. I, 1988, vol. 307, pp. 91-94. 8. Vajaitu, V. \(q\)-Completeness and \(q\)-Concavity of the Union of Open Subspaces, Mathematische Zeitschrift, 1996, vol. 221, pp. 217-229. DOI: 10.1007/PL00022735. 9. Andreotti, A. and Narasimhan, R. Oka’s Heftungslemma and the Levi Problem for Complex Spaces, Transactions of the American Mathematical Society, 1964, vol. 111, no. 2, pp. 345-366. DOI: 10.1090/S0002-9947-1964-0159961-3. 10. Simha, R. R. On the Complement of a Curve on a Stein Space of Dimension Two, Mathematische Zeitschrift, 1963, vol. 82, pp. 63-66. DOI: 10.1007/BF01112823. ← Contents of issue


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