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Franc Forstneric
Extending holomorphic mappings from subvarieties in Stein manifolds
(Prolongements holomorphes dans les variétés de Stein)
Annales de l'institut Fourier, 55 no. 3 (2005), p. 733-751, doi: 10.5802/aif.2112
Article PDF | Reviews MR 2149401 | Zbl 1076.32003 | 2 citations in Cedram
Class. Math.: 32E10, 32E30, 32H02
Keywords: Stein manifold, holomorphic mappings, Oka property

Résumé - Abstract

Suppose that $Y$ is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space ${\Bbb C}^n$ to $Y$ is a uniform limit of entire maps ${\Bbb C}^n\to Y$. We prove that a holomorphic map $X_0 \to Y$ from a closed complex subvariety $X_0$ in a Stein manifold $X$ admits a holomorphic extension $X\to Y$ provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.

Bibliography

[1] R. Brody, “Compact manifolds and hyperbolicity”, Trans. Amer. Math. Soc. 235 (1978), p. 213-219  MR 470252 |  Zbl 0416.32013
[2] G. Buzzard & S.S.Y. Lu, “Algebraic surfaces holomorphically dominable by ${\Bbb C}^2$”, Invent. Math. 139 (2000), p. 617-659 Article |  MR 1738063 |  Zbl 0967.14025
[3] J. Carlson & P. Griffiths, “A defect relation for equidimensional holomorphic mappings between algebraic varieties”, Ann. Math. 95 (1972) no. 2, p. 557-584  MR 311935 |  Zbl 0248.32018
[4] M. Coltoiu, “Complete locally pluripolar sets”, J. Reine Angew. Math. 412 (1990), p. 108-112 Article |  MR 1074376 |  Zbl 0711.32008
[5] M. Coltoiu & N. Mihalache, “On the homology groups of Stein spaces and Runge pairs”, J. Reine Angew. Math. 371 (1986), p. 216-220 Article |  MR 859326 |  Zbl 0587.32026
[6] J.-P. Demailly, “Cohomology of $q$-convex spaces in top degrees”, Math. Z. 204 (1990), p. 283-295 Article |  MR 1055992 |  Zbl 0682.32017
[7] F. Docquier & H. Grauert, “Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten”, Math. Ann. 140 (1960), p. 94-123 Article |  MR 148939 |  Zbl 0095.28004
[8] D. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the Amer. Math. Soc. 96, American Mathematical Society, Providence, R.I., 1970  MR 259165 |  Zbl 0197.05901
[9] F. Forstneric, “The Oka principle for sections of subelliptic submersions”, Math. Z. 241 (2002), p. 527-551 Article |  MR 1938703 |  Zbl 1023.32008
[10] F. Forstneric, “Noncritical holomorphic functions on Stein manifolds”, Acta Math. 191 (2003), p. 143-189 Article |  MR 2051397 |  Zbl 1064.32021
[11] F. Forstneric, The homotopy principle in complex analysis: A survey, Contemporary Mathematics 332, American Mathematical Society, 2003  MR 2016091 |  Zbl 1048.32004
[12] F. Forstneric, “Holomorphic submersions from Stein manifolds”, Ann. Inst. Fourier 54 (2004) no. 6, p. 1913-1942 Cedram |  MR 2134229 |  Zbl 02162446
[13] F. Forstneric, “Runge approximation on convex sets implies Oka's property”, Preprint, http://arxiv.org/abs/math.CV/0402278, 2004 arXiv
[14] F. Forstneric, “Holomorphic flexibility properties of complex manifolds”, Preprint, http://arxiv.org/abs/math.CV/0401439, 2004 arXiv
[15] F. Forstneric & J. Prezelj, “Oka's principle for holomorphic fiber bundles with sprays”, Math. Ann. 317 (2000), p. 117-154 Article |  MR 1760671 |  Zbl 0964.32017
[16] Forstneric & J. Prezelj, “Oka's principle for holomorphic submersions with sprays”, Math. Ann. 322 (2002), p. 633-666 Article |  MR 1905108 |  Zbl 1011.32006
[17] Forstneric & J. Prezelj, “Extending holomorphic sections from complex subvarieties”, Math. Z. 236 (2001), p. 43-68 Article |  MR 1812449 |  Zbl 0968.32005
[18] H. Grauert, “Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen”, Math. Ann. 133 (1957), p. 139-159 Article |  MR 98197 |  Zbl 0080.29201
[19] H. Grauert, “Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen”, Math. Ann. 133 (1957), p. 450-472 Article |  MR 98198 |  Zbl 0080.29202
[20] H. Grauert, “Analytische Faserungen über holomorph-vollständigen Räumen”, Math. Ann. 135 (1958), p. 263-273 Article |  MR 98199 |  Zbl 0081.07401
[21] M. Gromov, “Oka's principle for holomorphic sections of elliptic bundles”, J. Amer. Math. Soc. 2 (1989), p. 851-897  MR 1001851 |  Zbl 0686.32012
[22] R. C. Gunning & H. Rossi, Analytic functions of several complex variables, Prentice--Hall, Englewood Cliffs, 1965  MR 180696 |  Zbl 0141.08601
[23] G. Henkin & J. Leiterer, Andreotti-Grauert Theory by Integral Formulas, Progress in Math. 74, Birkhäuser, Boston, 1988  MR 986248 |  Zbl 0654.32001
[24] G. Henkin & J. Leiterer, “The Oka-Grauert principle without induction over the basis dimension”, Math. Ann. 311 (1998), p. 71-93 Article |  MR 1624267 |  Zbl 0955.32019
[25] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Third ed. North Holland, Amsterdam, 1990  MR 1045639 |  Zbl 0685.32001
[26] S. Kobayashi, “Intrinsic distances, measures and geometric function theory”, 82 (1976) no. Bull. Amer. Math. Soc., p. 357-416 Article |  MR 414940 |  Zbl 0346.32031
[27] S. Kobayashi & T. Ochiai, “Meromorphic mappings onto compact complex spaces of general type”, Invent. Math. 31 (1975), p. 7-16 Article |  MR 402127 |  Zbl 0331.32020
[28] K. Kodaira, “Holomorphic mappings of polydiscs into compact complex manifolds”, J. Diff. Geom. 6 (1971-72), p. 33-46  MR 301228 |  Zbl 0227.32008
[29] F. Lárusson, “Mapping cylinders and the Oka principle”, Preprint, http://www.math.uwo.ca/ larusson/papers/, 2004 arXiv
[30] R. Narasimhan, “The Levi problem for complex spaces”, Math. Ann. 142 (1961), p. 355-365 Article |  MR 148943 |  Zbl 0106.28603
[31] M. Peternell, “Algebraische Varietäten und $q$-vollständige komplexe Räume”, Math. Z. 200 (1989), p. 547-581 Article |  MR 987586 |  Zbl 0675.32014
[32] R. Richberg, “Stetige streng pseudoconvexe Funktionen”, Math. Ann. 175 (1968), p. 257-286  MR 222334 |  Zbl 0153.15401
[33] J.-T. Siu, “Every Stein subvariety admits a Stein neighborhood”, Invent. Math. 38 (1976), p. 89-100 Article |  MR 435447 |  Zbl 0343.32014
[34] G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math 61, Springer-Verlag, 1978  MR 516508 |  Zbl 0406.55001
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