Invariance of the Parametric Oka Property

Forstnerič, Franc

doi:10.1007/978-3-0346-0009-5_7

Cited by 13 publications

(16 citation statements)

References 25 publications

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“…If PHAP holds over small open subsets of X then sections X → Z satisfy the parametric Oka property (Gromov [7,Theorem 4.5]; the details can be found in [3,5]). The same proof applies in our situation (see [4,Theorem 4.2]) and shows that PHAP implies Theorem 1.1.…”

Section: Reduction Of Theorem 11 To An Approximation Propertysupporting

confidence: 61%

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Oka maps

Forstnerič

2009

Comptes Rendus. Mathématique

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Given a holomorphic submersion of reduced complex spaces, we prove that the basic Oka property of the submersion implies the parametric Oka property. This generalizes the corresponding result for complex manifolds (F. Forstneric, Oka Manifolds, C. R. Acad. Sci. Paris, Ser. I, 347 (2009) 1017-1020). It follows that a stratified elliptic (or subelliptic) holomorphic submersion, or a stratified holomorphic fiber bundle whose fibers are Oka manifolds, enjoys the parametric Oka property. As an application we give a parametric version of the factorization theorem due to Ivarsson and Kutzschebauch (A solution of Gromov's Vaserstein problem, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1239-1243) for holomorphic maps from finite dimensional reduced Stein spaces to the special linear group SL_n(C)

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Section: Reduction Of Theorem 11 To An Approximation Propertysupporting

confidence: 61%

“…A stratified subelliptic holomorphic submersion, or a stratified fiber bundle with Oka fibers, enjoys BOP [3,4]. Hence Theorem 1.1 implies: If π: E → B enjoys the Oka property then by considering liftings of constant maps X → b ∈ B we see that every fiber E b = π −1 (b) is an Oka manifold.…”

Section: Oka Properties Of Holomorphic Mapsmentioning

confidence: 96%

Oka maps

Forstnerič

2009

Comptes Rendus. Mathématique

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“…We conclude the proof (globalization with respect to the parameter p ∈ P ) by following the stepwise extension method as in the Conclusion of the proof of Theorem 4.2 in [19]. Corollary 6.8.…”

Section: For Substep (C) We Applymentioning

confidence: 79%

“…One can dispense with the SDNR condition by using a generalized Tietze extension theorem for maps into Hilbert spaces [19,Proposition 4.1] Remark 6.7. For all compact parameter spaces P 0 ⊂ P contained in a Euclidean space R m we also have the implication CAP =⇒ PCAP (see the note at the end of the paper).…”

Section: The Parametric Oka Principlementioning

confidence: 99%

The Oka Principle for Sections of Stratified Fiber Bundles

Forstnerič¹

2010

Pure and Applied Mathematics Quarterly

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Let Y be a complex manifold with the property that every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C n to Y can be approximated, uniformly on K, by entire maps C n → Y . If X is a reduced Stein space and π : Z → X is a holomorphic fiber bundle with fiber Y then we show that sections X → Z satisfy the Oka principle with approximation and interpolation. The analogous result is proved for sections of stratified fiber bundles, and of submersions with stratified sprays over Stein spaces.

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“…By using a Tietze extension theorem for Banach-valued maps as in [4,Proposition 4.4] and shrinking the set V ⊃ L we can assume that f (p, · ) is holomorphic on V for all p in a neighborhood P 0 ⊂ C m of P 0 .…”

Section: Proof Of Theorem 23mentioning

confidence: 99%

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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Invariance of the Parametric Oka Property

Cited by 13 publications

References 25 publications

Oka maps

Oka maps

The Oka Principle for Sections of Stratified Fiber Bundles

Oka manifolds

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