Oka manifolds

Franc Forstnerič

doi:10.1016/j.crma.2009.07.005

Complex Analysis

Oka manifolds
[Les varietes de Oka]

Présenté par : Mikhaël Gromov

Franc Forstnerič ¹

¹ Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia

Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1017-1020.

Résumé
Abstract

Nous donnons une réponse positive à la question suivante posée par Gromov [Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897, 3.4.(D), p. 881] : Si une variété analytique complexe Y est telle que toute application holomorphe d'un voisinage d'un sous-ensemble compact convexe de l'espace euclidien $C^{n}$ dans Y peut être approximée par des applications entière de $C^{n}$ dans Y, alors les applications holomorphes d'un espace de Stein réduit X dans Y possèdent la propriété d'Oka paramétrique.

We give a positive answer to Gromov's question [Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897, 3.4.(D), p. 881]: If every holomorphic map from a compact convex set in a Euclidean space $C^{n}$ to a certain complex manifold Y is a uniform limit of entire maps $C^{n} \to Y$ , then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion $O (X, Y) ↪ C (X, Y)$ of the space of holomorphic maps into the space of continuous maps is a weak homotopy equivalence.

Reçu le : 2009-06-08
Accepté le : 2009-07-05
Publié le : 2009-08-05

DOI : 10.1016/j.crma.2009.07.005

Affiliations des auteurs :

Franc Forstnerič ¹

¹ Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia

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     title = {Oka manifolds},
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     pages = {1017--1020},
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Franc Forstnerič. Oka manifolds. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1017-1020. doi : 10.1016/j.crma.2009.07.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.07.005/

Bibliographie
Cité par

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[9] F. Lárusson Mapping cylinders and the Oka principle, Indiana Univ. Math. J., Volume 54 (2005), pp. 1145-1159

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