Extending holomorphic mappings from subvarieties in Stein manifolds

Forstnerič, Franc

doi:10.5802/aif.2112

Cited by 31 publications

(56 citation statements)

References 26 publications

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“…In the absence of topological obstructions, CAP of Y also implies extendibility of holomorphic maps from closed complex subvarieties in Stein manifolds to Y [11,35]. Among the conditions implying CAP are (from strongest to weakest):…”

Section: Introductionmentioning

confidence: 99%

Approximation of holomorphic mappings on strongly pseudoconvex domains

Drinovec-Drnovšek¹,

Forstnerič²

2008

Forum Mathematicum

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Section: Introductionmentioning

confidence: 99%

Approximation of holomorphic mappings on strongly pseudoconvex domains

Drinovec-Drnovšek¹,

Forstnerič²

2008

Forum Mathematicum

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“…This property was introduced in [2,1] where it was shown to be equivalent to the basic Oka property of Y .…”

Section: The Oka-grauert-gromov Principlementioning

confidence: 93%

Oka manifolds

Forstnerič

2009

Comptes Rendus. Mathématique

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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 → 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. RésuméLes varietes de Oka. 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.

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“…Classical results of Oka [49], Grauert [30][31][32] and Gromov [37] give an affirmative answer when Y is a complex homogeneous manifold or, more generally, if it admits a dominating spray (see also [21,23]). Recently this Oka property of Y has been characterized in terms of a Runge approximation property for entire maps C n → Y on certain special compact convex subset of C n [19,20]. The Oka property holds only rarely as it implies in particular that Y is dominated by a complex Euclidean space, and this fails for any compact complex manifold of Kodaira general type.…”

Section: Theorem 11 Let X Be a Stein Manifold With The Complex Strucmentioning

confidence: 99%

Stein structures and holomorphic mappings

Forstnerič

Slapar

2007

Math. Z.

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We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).Comment: The original publication is available at http://www.springerlink.co

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Extending holomorphic mappings from subvarieties in Stein manifolds

Cited by 31 publications

References 26 publications

Approximation of holomorphic mappings on strongly pseudoconvex domains

Approximation of holomorphic mappings on strongly pseudoconvex domains

Oka manifolds

Stein structures and holomorphic mappings

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