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Franc Forstnerič
Holomorphic submersions from Stein manifolds
(Submersions holomorphes d'une variété de Stein)
Annales de l'institut Fourier, 54 no. 6 (2004), p. 1913-1942, doi: 10.5802/aif.2071
Article PDF | Reviews MR 2134229 | Zbl 1093.32003 | 1 citation in Cedram
Class. Math.: 32E10, 32E30, 32H02
Keywords: Stein manifolds, holomorphic submersions, Oka principle

Résumé - Abstract

We establish the homotopy classification of holomorphic submersions from Stein manifolds to Complex manifolds satisfying an analytic property introduced in the paper. The result is a holomorphic analogue of the Gromov--Phillips theorem on smooth submersions.

Bibliography

[A] R. Abraham, “Transversality in manifolds of mappings.”, Bull. Amer. Math. Soc. 69 (1963), p. 470-474 Article |  MR 149495 |  Zbl 0171.44501
[B] W. Barth, C. Peters & A. Van De Ven, Compact Complex Surfaces, Springer, Berlin--Heidelberg--New Zork--Tokyo, 1984  MR 749574 |  Zbl 0718.14023
[De] J.-P Demailly, “Cohomology of $q$-convex spaces in top degrees”, Math. Z. 204 (1990), p. 283-295  MR 1055992 |  Zbl 0682.32017
[DG] F. Docquier & H. Grauert, “Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. (German)”, Math. Ann. 140 (1960), p. 94-123  MR 148939 |  Zbl 0095.28004
[E] Y. Eliashberg, “Topological characterization of Stein manifolds of dimension $>2$”, Internat. J. Math 1 (1990), p. 29-46  MR 1044658 |  Zbl 0699.58002
[EM] Y. Eliashberg & N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Math 48, Amer. Math. Soc., Providence, RI, 2002  MR 1909245 |  Zbl 1008.58001
[Fo] O. Forster, “Plongements des variétés de Stein”, Comment. Math. Helv 45 (1970), p. 170-184  MR 269880 |  Zbl 0184.31403
[F1] F. Forstnerič, “Noncritical holomorphic functions on Stein manifolds”, Acta Math. 191 (2003), p. 143-189  MR 2051397 |  Zbl 1064.32021
[F2] F. Forstnerič, The homotopy principle in complex analysis: A survey, Contemporary Mathematics, American Mathematical Society, 2003, p. 73-99  Zbl 1048.32004
[F3] F. Forstnerič, “The Oka principle for sections of subelliptic submersions”, Math. Z. 241 (2002), p. 527-551  MR 1938703 |  Zbl 1023.32008
[F4] F. Forstnerič, “Runge approximation on convex sets implies the Oka property”, e-print, arXiv: math.CV/0402278, February 2004 arXiv
[FK] F. Forstnerič & J. Kozak, “Strongly pseudoconvex handlebodies.”, J. Korean Math. Soc. 40 (2003), p. 727-746  MR 1995074 |  Zbl 1044.32025
[FL\O] F. Forstnerič, E. L$/\kern-0.65em$ow & N. Øvrelid, “Solving the $d$- and $\overline\partial$-equations in thin tubes and applications to mappings.”, Michigan Math. J. 49 (2001), p. 369-416 Article |  MR 1852309 |  Zbl 1016.32018
[FP1] F. Forstnerič & J. Prezelj, “Oka's principle for holomorphic fiber bundles with sprays.”, Math. Ann. 317 (2000), p. 117-154  MR 1760671 |  Zbl 0964.32017
[FP2] F. Forstnerič & J. Prezelj, “Oka's principle for holomorphic submersions with sprays.”, Math. Ann. 322 (2002), p. 633-666  MR 1905108 |  Zbl 1011.32006
[FP3] F. Forstnerič & J. Prezelj, “Extending holomorphic sections from complex subvarieties.”, Math. Z. 236 (2001), p. 43-68  MR 1812449 |  Zbl 0968.32005
[FR] F. Forstnerič & J.-P. Rosay, “Approximation of biholomorphic mappings by automorphisms of $\C^n$.”, Invent. Math. 112 (1993), p. 323-349  MR 1213106 |  Zbl 0792.32011
[G1] H. Grauert, “Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen.”, Math. Ann. 133 (1957), p. 139-159  MR 98197 |  Zbl 0080.29201
[G2] H. Grauert, “Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen.”, Math. Ann. 133 (1957), p. 450-472  MR 98198 |  Zbl 0080.29202
[G3] H. Grauert, “Analytische Faserungen über holomorph-vollständigen Räumen.”, Math. Ann. 135 (1958), p. 263-273  MR 98199 |  Zbl 0081.07401
[Gr1] M. Gromov, “Stable maps of foliations into manifolds.”, Izv. Akad. Nauk, S.S.S.R 33 (1969), p. 707-734  MR 263103 |  Zbl 0197.20404
[Gr2] M. Gromov, “Convex integration of differential relations, I”, Izv. Akad. Nauk SSSR Ser. Mat (Russian) 37 (1973), p. 329-343  MR 413206 |  Zbl 0254.58001
[Gr3] M. Gromov, Partial Differential Relations., Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 9, Springer, Berlin--New York, 1986  MR 864505 |  Zbl 0651.53001
[Gr4] 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
[GN] R. C. Gunning & R. Narasimhan, “Immersion of open Riemann surfaces.”, Math. Ann. 174 (1967), p. 103-108  MR 223560 |  Zbl 0179.11402
[GR] R. C. Gunning & H. Rossi, Analytic functions of several complex variables., Prentice--Hall, Englewood Cliffs, 1965  MR 180696 |  Zbl 0141.08601
[HL1] G. M. Henkin & J. Leiterer, Andreotti-Grauert Theory by Integral Formulas., Progress in Math. 74, Birkhäuser, Boston, 1988  MR 986248 |  Zbl 0654.32002
[HL2] G. M. Henkin & J. Leiterer, “The Oka-Grauert principle without induction over the basis dimension.”, Math. Ann. 311 (1998), p. 71-93  MR 1624267 |  Zbl 0955.32019
[Hö1] L. Hörmander, “$L\sp2$ estimates and existence theorems for the $\bar \partial$ operator.”, Acta Math. 113 (1965), p. 89-152  MR 179443 |  Zbl 0158.11002
[Hö2] L. Hörmander, An Introduction to Complex Analysis in Several Variables, North Holland, Amsterdam, 1990  MR 1045639 |  Zbl 0685.32001
[HW] L. Hörmander & J. Wermer, “Uniform approximations on compact sets in $\C^n$.”, Math. Scand 23 (1968), p. 5-21  MR 254275 |  Zbl 0181.36201
[O] K. Oka, “Sur les fonctions des plusieurs variables. III: Deuxième problème de Cousin.”, J. Sc. Hiroshima Univ. 9 (1939), p. 7-19  Zbl 0020.24002 |  JFM 65.0361.01
[P] A. Phillips, “Submersions of open manifolds.”, Topology 6 (1967), p. 170-206  MR 208611 |  Zbl 0204.23701
[RS] R. M. Range & Y. T. Siu, “$\cC^k$ approximation by holomorphic functions and $\bar \partial$-closed forms on $\cC^k$ submanifolds of a complex manifold.”, Math. Ann. 210 (1974), p. 105-122  MR 350068 |  Zbl 0275.32008
[Ro] J.-P. Rosay, “A counterexample related to Hartog's phenomenon (a question by E.\ Chirka).”, Michigan Math. J. 45 (1998), p. 529-535 Article |  MR 1653267 |  Zbl 0960.32020
[S] J.-T. Siu, “Every Stein subvariety admits a Stein neighborhood.”, Invent. Math. 38 (1976), p. 89-100  MR 435447 |  Zbl 0343.32014
[W] J. Winkelman, “The Oka-principle for mappings between Riemann surfaces.”, Enseign. Math. (2) 39 (1993), p. 143-151  MR 1225261 |  Zbl 0783.30031
[<L>FR</L>] F. Forstnerič & J.-P. Rosay, “Approximation of biholomorphic mappings by automorphisms of $\C^n$”, Invent. Math. (Erratum) 118 (1994), p. 573-574  MR 1296357 |  Zbl 0808.32017
[<L>Gr2</L>] M. Gromov, “Convex integration of differential relations, I”, Math. USSR--Izv. (English translation) 7 (1973), p. 329-343  MR 413206 |  Zbl 0281.58004
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