no. 3
Extending holomorphic mappings from subvarieties in Stein manifolds
Forstneric, Franc1
1 Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana (Slovenia)
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 733-751.

Suppose that Y is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space ℂ n to Y is a uniform limit of entire maps ℂ n →Y. We prove that a holomorphic map X 0 →Y from a closed complex subvariety X 0 in a Stein manifold X admits a holomorphic extension X→Y provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.

Soit Y une variété analytique complexe telle que toute application holomorphe d’une partie convexe compacte de l’espace euclidien ℂ n à valeurs dans Y est limite uniforme d’applications entières à valeurs dans Y. On prouve que toute application holomorphe d’un sous ensemble analytique complexe fermé X 0 d’une variété de Stein X à valeurs dans Y possède un prolongement holomorphe à X à condition qu’elle admette un prolongement continu. On établit ensuite l’équivalence entre quatre propriétés de type Oka pour une variété analytique complexe.

DOI: 10.5802/aif.2112
Classification: 32E10, 32E30, 32H02
Keywords: Stein manifold, holomorphic mappings, Oka property
Mot clés : variétés de Stein, applications holomorphes, propriété d'Oka
Forstneric, Franc 1

1 Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana (Slovenia) 
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Forstneric, Franc. Extending holomorphic mappings from subvarieties in Stein manifolds. Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 733-751. doi : 10.5802/aif.2112. https://aif.centre-mersenne.org/articles/10.5802/aif.2112/

[1] R. Brody Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., Volume 235 (1978), pp. 213-219 | MR | Zbl

[2] G. Buzzard; S.S.Y. Lu Algebraic surfaces holomorphically dominable by ℂ 2 , Invent. Math., Volume 139 (2000), pp. 617-659 | DOI | MR | Zbl

[3] J. Carlson; P. Griffiths A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math., Volume 95 (1972) no. 2, pp. 557-584 | MR | Zbl

[4] M. Coltoiu Complete locally pluripolar sets, J. Reine Angew. Math., Volume 412 (1990), pp. 108-112 | MR | Zbl

[5] M. Coltoiu; N. Mihalache On the homology groups of Stein spaces and Runge pairs, J. Reine Angew. Math., Volume 371 (1986), pp. 216-220 | MR | Zbl

[6] J.-P. Demailly Cohomology of q-convex spaces in top degrees, Math. Z., Volume 204 (1990), pp. 283-295 | DOI | MR | Zbl

[7] F. Docquier; H. Grauert Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., Volume 140 (1960), pp. 94-123 | DOI | MR | Zbl

[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 | Zbl

[9] F. Forstneric The Oka principle for sections of subelliptic submersions, Math. Z., Volume 241 (2002), pp. 527-551 | DOI | MR | Zbl

[10] F. Forstneric Noncritical holomorphic functions on Stein manifolds, Acta Math., Volume 191 (2003), pp. 143-189 | DOI | MR | Zbl

[11] F. Forstneric The homotopy principle in complex analysis: A survey, Contemporary Mathematics, 332, American Mathematical Society, 2003 | MR | Zbl

[12] F. Forstneric Holomorphic submersions from Stein manifolds, Ann. Inst. Fourier, Volume 54 (2004) no. 6, pp. 1913-1942 | DOI | Numdam | MR | Zbl

[13] F. Forstneric Runge approximation on convex sets implies Oka's property (2004) (Preprint, http://arxiv.org/abs/math.CV/0402278)

[14] F. Forstneric Holomorphic flexibility properties of complex manifolds (2004) (Preprint, http://arxiv.org/abs/math.CV/0401439)

[15] F. Forstneric; J. Prezelj Oka's principle for holomorphic fiber bundles with sprays, Math. Ann., Volume 317 (2000), pp. 117-154 | DOI | MR | Zbl

[16] Forstneric; J. Prezelj Oka's principle for holomorphic submersions with sprays, Math. Ann., Volume 322 (2002), pp. 633-666 | DOI | MR | Zbl

[17] Forstneric; J. Prezelj Extending holomorphic sections from complex subvarieties, Math. Z., Volume 236 (2001), pp. 43-68 | DOI | MR | Zbl

[18] H. Grauert Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen, Math. Ann., Volume 133 (1957), pp. 139-159 | DOI | MR | Zbl

[19] H. Grauert Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann., Volume 133 (1957), pp. 450-472 | DOI | MR | Zbl

[20] H. Grauert Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., Volume 135 (1958), pp. 263-273 | DOI | MR | Zbl

[21] M. Gromov Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., Volume 2 (1989), pp. 851-897 | MR | Zbl

[22] R. C. Gunning; H. Rossi Analytic functions of several complex variables, Prentice--Hall, Englewood Cliffs, 1965 | MR | Zbl

[23] G. Henkin; J. Leiterer Andreotti-Grauert Theory by Integral Formulas, Progress in Math., 74, Birkhäuser, Boston, 1988 | MR | Zbl

[24] G. Henkin; J. Leiterer The Oka-Grauert principle without induction over the basis dimension, Math. Ann., Volume 311 (1998), pp. 71-93 | DOI | MR | Zbl

[25] L. Hörmander An Introduction to Complex Analysis in Several Variables, Third ed. North Holland, Amsterdam, 1990 | MR | Zbl

[26] S. Kobayashi Intrinsic distances, measures and geometric function theory, Volume 82 (1976) no. Bull. Amer. Math. Soc., pp. 357-416 | MR | Zbl

[27] S. Kobayashi; T. Ochiai Meromorphic mappings onto compact complex spaces of general type, Invent. Math., Volume 31 (1975), pp. 7-16 | DOI | MR | Zbl

[28] K. Kodaira Holomorphic mappings of polydiscs into compact complex manifolds, J. Diff. Geom., Volume 6 (1971-72), pp. 33-46 | MR | Zbl

[29] F. Lárusson Mapping cylinders and the Oka principle (2004) (Preprint, http://www.math.uwo.ca/ larusson/papers/)

[30] R. Narasimhan The Levi problem for complex spaces, Math. Ann., Volume 142 (1961), pp. 355-365 | DOI | MR | Zbl

[31] M. Peternell Algebraische Varietäten und q-vollständige komplexe Räume, Math. Z., Volume 200 (1989), pp. 547-581 | DOI | MR | Zbl

[32] R. Richberg Stetige streng pseudoconvexe Funktionen, Math. Ann., Volume 175 (1968), pp. 257-286 | MR | Zbl

[33] J.-T. Siu Every Stein subvariety admits a Stein neighborhood, Invent. Math., Volume 38 (1976), pp. 89-100 | DOI | MR | Zbl

[34] G. W. Whitehead Elements of Homotopy Theory, Graduate Texts in Math, 61, Springer-Verlag, 1978 | MR | Zbl

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