no. 17-18
Complex Analysis
Oka manifolds
[Les varietes de Oka]

Présenté par : Mikhaël Gromov

Forstnerič, Franc1
1 Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Comptes Rendus. Mathématique, Tome 347 (2009) no. 17-18, pp. 1017-1020.

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 Cn dans Y peut être approximée par des applications entière de Cn 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 Cn to a certain complex manifold Y is a uniform limit of entire maps Cn→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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.07.005
Forstnerič, Franc 1

1 Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia 
@article{CRMATH_2009__347_17-18_1017_0,
     author = {Forstneri\v{c}, Franc},
     title = {Oka manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1017--1020},
     publisher = {Elsevier},
     volume = {347},
     number = {17-18},
     year = {2009},
     doi = {10.1016/j.crma.2009.07.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.07.005/}
}
TY  - JOUR
AU  - Forstnerič, Franc
TI  - Oka manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 1017
EP  - 1020
VL  - 347
IS  - 17-18
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.07.005/
DO  - 10.1016/j.crma.2009.07.005
LA  - en
ID  - CRMATH_2009__347_17-18_1017_0
ER  - 
%0 Journal Article
%A Forstnerič, Franc
%T Oka manifolds
%J Comptes Rendus. Mathématique
%D 2009
%P 1017-1020
%V 347
%N 17-18
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.07.005/
%R 10.1016/j.crma.2009.07.005
%G en
%F CRMATH_2009__347_17-18_1017_0
Forstnerič, Franc. Oka manifolds. Comptes Rendus. Mathématique, Tome 347 (2009) no. 17-18, pp. 1017-1020. doi : 10.1016/j.crma.2009.07.005. http://www.numdam.org/articles/10.1016/j.crma.2009.07.005/

[1] Forstnerič, F. Extending holomorphic mappings from subvarieties in Stein manifolds, Ann. Inst. Fourier, Volume 55 (2005), pp. 733-751

[2] Forstnerič, F. Runge approximation on convex sets implies Oka's property, Ann. of Math. (2), Volume 163 (2006), pp. 689-707

[3] Forstnerič, F. The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Quarterly (2009) | arXiv

[4] F. Forstnerič, Invariance of the parametric Oka property, in: Proceedings of the Conference in Honor of Linda P. Rothschild, Fribourg, Switzerland, July 2008, Birkhäuser Verlag, in press, | arXiv

[5] F. Forstnerič, E.F. Wold, Fibrations and Stein neighborhoods, preprint, 2009, | arXiv

[6] Grauert, H. Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., Volume 135 (1958), pp. 263-273

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

[8] Lárusson, F. Model structures and the Oka principle, J. Pure Appl. Algebra, Volume 192 (2004), pp. 203-223

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

Cité par Sources :