Oka manifolds: From Oka to Stein and back

Franc Forstnerič

doi:10.5802/afst.1388

Franc Forstnerič

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809.

Résumé
Abstract

La théorie d’Oka tire ses origines du principe classique d’Oka-Grauert, dont la principale application est la classification par Grauert des fibrés holomorphes principaux sur les espaces de Stein. La théorie d’Oka moderne traite des applications holomorphes depuis des variétés ou des espaces de Stein vers des variétés d’Oka. Elle est devenue un sous-domaine à part entière de la géométrie complexe depuis la parution d’un article fondateur de M. Gromov en 1989.

Nous présentons ici les variétés et les applications d’Oka. Nous décrivons les caractérisations équivalentes des variétés d’Oka, les propriétés fonctorielles de cette classe, et des conditions suffisantes géométriques pour qu’une variété soit d’Oka, dont la plus importante est l’ellipticité de Gromov. Nous donnons un panorama de l’état actuel de la théorie en ce qui concerne les exemples connus de variétés d’Oka, mentionnons les problèmes ouverts et esquissons les démonstrations des résultats principaux. Dans l’appendice, dû à F. Lárusson, on explique comment les variétés d’Oka, et les applications d’Oka, s’inscrivent dans le cadre d’une théorie homotopique abstraite.

Le présent article est une version augmentée des exposés de l’auteur lors de l’Ecole d’Hiver KAWA 4 à Toulouse, France, en janvier 2013. On trouvera une présentation exhaustive de la théorie d’Oka dans la monographie [32].

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989.

In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations of Oka manifolds, the functorial properties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov’s ellipticity. We survey the current status of the theory in terms of known examples of Oka manifolds, mention open problems and outline the proofs of the main results. In the appendix by F. Lárusson it is explained how Oka manifolds and Oka maps, along with Stein manifolds, fit into an abstract homotopy-theoretic framework.

The article is an expanded version of lectures given by the author at Winter School KAWA 4 in Toulouse, France, in January 2013. A comprehensive exposition of Oka theory is available in the monograph [32].

MR Zbl

DOI : 10.5802/afst.1388

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Franc Forstnerič. Oka manifolds: From Oka to Stein and back. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809. doi : 10.5802/afst.1388. https://afst.centre-mersenne.org/articles/10.5802/afst.1388/

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