Let be an open subset of a Stein space . We show that if is locally Stein and the complement is a closed subspace of , then is Stein. We also discuss the applications of the theorem to open subsets whose boundaries in are not closed subspaces of . For example, we show that if for every boundary point , there is a closed subspace of pure codimension 1 in such that , and is locally Stein, then is Stein.
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Jing Zhang 1
@article{CRMATH_2021__359_8_1023_0, author = {Jing Zhang}, title = {Levi {Problem:} {Complement} of a closed subspace in a {Stein} space and its applications}, journal = {Comptes Rendus. Math\'ematique}, pages = {1023--1046}, publisher = {Acad\'emie des sciences, Paris}, volume = {359}, number = {8}, year = {2021}, doi = {10.5802/crmath.244}, language = {en}, }
Jing Zhang. Levi Problem: Complement of a closed subspace in a Stein space and its applications. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 1023-1046. doi : 10.5802/crmath.244. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.244/
[1] Oka’s Heftungslemma and the Levi problem for complex spaces, Trans. Am. Math. Soc., Volume 111 (1964) no. 2, pp. 345-366 | MR | Zbl
[2] Cartier and Weil divisors on varieties with quotient singularities, Int. J. Math., Volume 25 (2014) no. 11, 1450100, 20 pages | MR | Zbl
[3] Elements of mathematics. General topology. Chapters 5-10, Springer, 1989 (translated from the French. 2nd printing) | Zbl
[4] Cohomology of -convex spaces in top degrees, Math. Z., Volume 204 (1990) no. 2, pp. 283-295 | DOI | MR | Zbl
[5] Complex Analytic and Differential Geometry (2012) (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)
[6] A Levi Problem on Two-Dimensional Complex Manifolds, Math. Ann., Volume 261 (1982) no. 2, pp. 255-261 | DOI | MR | Zbl
[7] Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., Volume 140 (1960), pp. 94-123 | DOI | MR | Zbl
[8] The Levi problem on complex spaces with singularities, Math. Ann., Volume 248 (1980), pp. 47-72 | DOI | MR | Zbl
[9] On Levi’s problem and the imbedding of real-analytic manifolds, Ann. Math., Volume 68 (1958), pp. 460-472 | DOI | MR | Zbl
[10] Theory of Stein Spaces, Grundlehren der Mathematischen Wissenschaften, 236, Springer, 1979 (translated by Alan Huckleberry) | MR | Zbl
[11] Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften, 265, Springer, 1984 | MR | Zbl
[12] Introduction to holomorphic functions of several variables. Volume I: Function theory. Volume II: Local theory. Volume III: Homological theory, Wadsworth & Brooks/Cole Mathematics Series, 1990, Wadsworth & Brooks/Cole Advanced Books & Software, 1990 (rev. version and complete rewriting of: Analytic functions of several complex variables by Hugo Rossi and the author) | Zbl
[13] Ample Subvarieties of Algebraic Varieties, Lecture Notes in Mathematics, 156, Springer, 1970 (notes written in collaboration with C. Musili) | MR | Zbl
[14] Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1983 | Zbl
[15] Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math., Volume 79 (1964), pp. 109-326 | DOI | MR | Zbl
[16] Holomorphic functions of several variables, An introduction to the fundamental theory, De Gruyter Studies in Mathematics, 3, Walter de Gruyter, 1983 (with the assistance of Gottfried Barthel. Translated from the German by Michael Bridgland) | Zbl
[17] Algebra, Graduate Texts in Mathematics, 211, Springer, 2002 | Zbl
[18] On holomorphic functions of polynomial growth on a bounded domain, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 21 (1967), pp. 161-166 | Numdam | MR | Zbl
[19] Holomorphic convexity of spaces of analytic cycles, Bull. Soc. Math. Fr., Volume 105 (1977), pp. 191-223 | DOI | Numdam | MR | Zbl
[20] A reduction theorem for cohomology groups of very strongly -convex Kähler manifolds, Invent. Math., Volume 63 (1981), pp. 335-354 | DOI | Zbl
[21] Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur. (French), Jpn. J. Math., Volume 23 (1954), pp. 97-155 | DOI | MR | Zbl
[22] Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics, 108, Springer, 1986 | DOI | MR | Zbl
[23] Weil divisors on normal surfaces, Duke Math. J., Volume 51 (1984) no. 4, pp. 877-887 | MR | Zbl
[24] Generalized divisors and reflexive sheaves,, 2010 (http://www.math.utah.edu/~schwede/Notes/GeneralizedDivisors.pdf)
[25] Prolongement de faiseaux analytique cohérent, Ann. Inst. Fourier, Volume 16 (1966) no. 1, pp. 363-374 | DOI
[26] On the complement of a curve on a stein space of dimension two, Math. Z., Volume 82 (1963), pp. 63-66 | DOI | MR | Zbl
[27] Noetherianness of rings of hololomorphic functions on Stein compact subsets, 21, American Mathematical Society, 1969, pp. 483-489 | Zbl
[28] Holomorphic functions of polynomial growth on bounded domain, Duke Math. J., Volume 37 (1970), pp. 77-84 | MR | Zbl
[29] The Levi problem, Several complex variables. Part 2, Williams Coll., Williamstown, Mass., 1975) (Proceedings of Symposia in Pure Mathematics), Volume 30, American Mathematical Society, 1975, pp. 45-48 | Zbl
[30] Every Stein subvariety in a complex space admits a Stein neighborhood, Invent. Math., Volume 38 (1976), pp. 89-100 | Zbl
[31] Pseudoconvexity and the problem of Levi, Bull. Am. Math. Soc., Volume 84 (1978) no. 4, pp. 481-512 | MR | Zbl
[32] Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math., Volume 7 (1956), pp. 354-361 | DOI | MR | Zbl
[33] Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Mathematics, 439, Springer, 1975 | MR | Zbl
[34] Resolution of singularities of analytic spaces, Proceedings of Gökova Geometry-Topology Conference 2008, 31-63, Gökova Geometry/Topology Conference (GGT) (2009), pp. 31-63 | Zbl
[35] Threefolds with vanishing Hodge cohomology, Trans. Am. Math. Soc., Volume 357 (2005) no. 5, pp. 1977-1994 | DOI | MR | Zbl
[36] There exist nontrivial threefolds with vanishing Hodge cohomology, Mich. Math. J., Volume 54 (2006) no. 2, pp. 447-467 | Zbl
[37] Algebraic Stein varieties, Math. Res. Lett., Volume 15 (2008) no. 4, pp. 801-814 | DOI | MR | Zbl
[38] Stein open subsets with analytic complements in compact complex spaces, Ann. Pol. Math., Volume 113 (2015) no. 1, pp. 43-60 | DOI | MR | Zbl
[39] Complex manifolds with vanishing Hodge cohomology, Houston J. Math., Volume 43 (2017) no. 3, pp. 807-827 | MR | Zbl
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