Extending holomorphic mappings from subvarieties in Stein manifolds
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, p. 733-751
Suppose that Y is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space n to Y is a uniform limit of entire maps 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 XY 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 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 : https://doi.org/10.5802/aif.2112
Classification:  32E10,  32E30,  32H02
Keywords: Stein manifold, holomorphic mappings, Oka property
@article{AIF_2005__55_3_733_0,
     author = {Forstneric, Franc},
     title = {Extending holomorphic mappings from subvarieties in Stein manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     pages = {733-751},
     doi = {10.5802/aif.2112},
     zbl = {1076.32003},
     mrnumber = {2149401},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_3_733_0}
}
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. http://www.numdam.org/item/AIF_2005__55_3_733_0/

[1] R. Brody Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., Tome 235 (1978), pp. 213-219 | MR 470252 | Zbl 0416.32013

[2] G. Buzzard; S.S.Y. Lu Algebraic surfaces holomorphically dominable by 2 , Invent. Math., Tome 139 (2000), pp. 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., Tome 95 (1972) no. 2, pp. 557-584 | MR 311935 | Zbl 0248.32018

[4] M. Coltoiu Complete locally pluripolar sets, J. Reine Angew. Math., Tome 412 (1990), pp. 108-112 | MR 1074376 | Zbl 0711.32008

[5] M. Coltoiu; N. Mihalache On the homology groups of Stein spaces and Runge pairs, J. Reine Angew. Math., Tome 371 (1986), pp. 216-220 | MR 859326 | Zbl 0587.32026

[6] J.-P. Demailly Cohomology of q-convex spaces in top degrees, Math. Z., Tome 204 (1990), pp. 283-295 | Article | MR 1055992 | Zbl 0682.32017

[7] F. Docquier; H. Grauert Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., Tome 140 (1960), pp. 94-123 | Article | MR 148939 | Zbl 0095.28004

[8] D. Eisenman Intrinsic measures on complex manifolds and holomorphic mappings, American Mathematical Society, Providence, R.I., Memoirs of the Amer. Math. Soc., Tome 96 (1970) | MR 259165 | Zbl 0197.05901

[9] F. Forstneric The Oka principle for sections of subelliptic submersions, Math. Z., Tome 241 (2002), pp. 527-551 | Article | MR 1938703 | Zbl 1023.32008

[10] F. Forstneric Noncritical holomorphic functions on Stein manifolds, Acta Math., Tome 191 (2003), pp. 143-189 | Article | MR 2051397 | Zbl 1064.32021

[11] F. Forstneric The homotopy principle in complex analysis: A survey, American Mathematical Society, Contemporary Mathematics, Tome 332 (2003) | MR 2016091 | Zbl 1048.32004

[12] F. Forstneric Holomorphic submersions from Stein manifolds, Ann. Inst. Fourier, Tome 54 (2004) no. 6, pp. 1913-1942 | Article | Numdam | MR 2134229 | Zbl 02162446

[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., Tome 317 (2000), pp. 117-154 | Article | MR 1760671 | Zbl 0964.32017

[16] Forstneric; J. Prezelj Oka's principle for holomorphic submersions with sprays, Math. Ann., Tome 322 (2002), pp. 633-666 | Article | MR 1905108 | Zbl 1011.32006

[17] Forstneric; J. Prezelj Extending holomorphic sections from complex subvarieties, Math. Z., Tome 236 (2001), pp. 43-68 | Article | MR 1812449 | Zbl 0968.32005

[18] H. Grauert Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen, Math. Ann., Tome 133 (1957), pp. 139-159 | Article | MR 98197 | Zbl 0080.29201

[19] H. Grauert Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann., Tome 133 (1957), pp. 450-472 | Article | MR 98198 | Zbl 0080.29202

[20] H. Grauert Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., Tome 135 (1958), pp. 263-273 | Article | MR 98199 | Zbl 0081.07401

[21] M. Gromov Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., Tome 2 (1989), pp. 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, Birkhäuser, Boston, Progress in Math., Tome 74 (1988) | MR 986248 | Zbl 0654.32001

[24] G. Henkin; J. Leiterer The Oka-Grauert principle without induction over the basis dimension, Math. Ann., Tome 311 (1998), pp. 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, Tome 82 (1976) no. Bull. Amer. Math. Soc., pp. 357-416 | MR 414940 | Zbl 0346.32031

[27] S. Kobayashi; T. Ochiai Meromorphic mappings onto compact complex spaces of general type, Invent. Math., Tome 31 (1975), pp. 7-16 | Article | MR 402127 | Zbl 0331.32020

[28] K. Kodaira Holomorphic mappings of polydiscs into compact complex manifolds, J. Diff. Geom., Tome 6 (1971-72), pp. 33-46 | MR 301228 | Zbl 0227.32008

[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., Tome 142 (1961), pp. 355-365 | Article | MR 148943 | Zbl 0106.28603

[31] M. Peternell Algebraische Varietäten und q-vollständige komplexe Räume, Math. Z., Tome 200 (1989), pp. 547-581 | Article | MR 987586 | Zbl 0675.32014

[32] R. Richberg Stetige streng pseudoconvexe Funktionen, Math. Ann., Tome 175 (1968), pp. 257-286 | MR 222334 | Zbl 0153.15401

[33] J.-T. Siu Every Stein subvariety admits a Stein neighborhood, Invent. Math., Tome 38 (1976), pp. 89-100 | Article | MR 435447 | Zbl 0343.32014

[34] G. W. Whitehead Elements of Homotopy Theory, Springer-Verlag, Graduate Texts in Math, Tome 61 (1978) | MR 516508 | Zbl 0406.55001