Oka manifolds

Forstnerič, Franc

doi:10.1016/j.crma.2009.07.005

Cited by 26 publications

(24 citation statements)

References 8 publications

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“…A particular reason for looking at this problem is that a question of this type, for compact sets that are laminated by holomorphic leaves, appears in the recent work by the first author [8]; the relevant result is provided by Corollary 2.2 below.…”

Section: The Analogous Results Holds If π(S) Belongs To a Totally Realmentioning

confidence: 99%

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Fibrations and Stein neighborhoods

Forstnerič

Wold

2009

Proc. Amer. Math. Soc.

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Abstract. Let Z be a complex space and let S be a compact set in C n × Z which is fibered over R n . We give a necessary and sufficient condition for S to be a Stein compactum. Fibrations over totally real setsWe denote by O(Z) the algebra of all holomorphic functions on a (reduced, paracompact) complex space Z, endowed with the compact-open topology. A compact subset K of Z is said to be a Stein compactum if K has a basis of Stein open neighborhoods.For the theory of Stein spaces we refer to [10,11].Let Z be a complex space, and consider the product C n × Z with the projection π : C n × Z → C n . Let S be a compact set in C n × Z. For every point ζ ∈ C n we let S ζ = {z ∈ Z : (ζ, z) ∈ S} denote the fiber of S over ζ. We are interested in the following:Question: Under what conditions on the projection π(S) ⊂ C n and on the fibers S ζ is S a Stein compactum in C n × Z?We give the following precise answer under the assumption that the projection π(S) is contained in R n , the real subspace of C n . Theorem 1.1. Let S be a compact set in and only if for every open neighborhood The analogous result holds if π(S) belongs to a totally real submanifold of C n .A particular reason for looking at this problem is that a question of this type, for compact sets that are laminated by holomorphic leaves, appears in the recent work by the first author [8]; the relevant result is provided by Corollary 2.2 below.The following simple example (a thin version of Hartog's figure) shows that it is not enough to assume in Theorem 1.1 that each fiber of S is a Stein compactum.

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Section: The Analogous Results Holds If π(S) Belongs To a Totally Realmentioning

confidence: 99%

“…The following corollary is used in an essential way in the proof of the main result in [8] to the effect that all Oka properties of a complex manifold are equivalent to each other.…”

Section: Theorem 21 Let X Be a Closed Stein Subvariety Of Complex Smentioning

confidence: 99%

Fibrations and Stein neighborhoods

Forstnerič

Wold

2009

Proc. Amer. Math. Soc.

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“…This is apparently a stronger condition than the BOP. However, it turns out that the POP with approximation and interpolation is equivalent to the BOP with approximation and interpolation; either condition can be taken as the definition of an Oka manifold (see [5,Section 1]).…”

Section: Definition 221 (Parametric Oka Property (Pop) Simple Versmentioning

confidence: 99%

Holomorphic Flexibility Properties of the Space of Cubic Rational Maps

Hanysz

2014

J Geom Anal

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Abstract. For each natural number d, the space R d of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the cases of degree 2 and 3, studying a double action of the Möbius group on R d . The action on R 2 is transitive, implying that R 2 is an Oka manifold. The action on R 3 has C as a categorical quotient; we give an explicit formula for the quotient map and describe its structure in some detail. We also show that R 3 enjoys the holomorphic flexibility properties of strong dominability and C-connectedness. Contents

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“…Taking S to be a contractible submanifold of X = C n gives an ostensibly much weaker property called the convex interpolation property. It is a deep theorem of Forstnerič that the two properties are equivalent [3]. In fact, by Forstnerič's work, a dozen or more properties of complex manifolds having to do with interpolation or approximation or both are mutually equivalent.…”

Section: Introduction 1 Introductionmentioning

confidence: 99%

Extending holomorphic maps from Stein manifolds into affine toric varieties

Lärkäng

Lárusson

2016

Proc. Amer. Math. Soc.

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Abstract.A complex manifold Y is said to have the interpolation property if a holomorphic map to Y from a subvariety S of a reduced Stein space X has a holomorphic extension to X if it has a continuous extension. Taking S to be a contractible submanifold of X = C n gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular targets. The targets that we study are affine toric varieties, not necessarily normal. We prove that every affine toric variety satisfies a weakening of the interpolation property that is much stronger than the convex interpolation property, but the full interpolation property fails for most affine toric varieties, even for a source as simple as the product of two annuli embedded in C 4 .

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Oka manifolds

Cited by 26 publications

References 8 publications

Fibrations and Stein neighborhoods

Fibrations and Stein neighborhoods

Holomorphic Flexibility Properties of the Space of Cubic Rational Maps

Extending holomorphic maps from Stein manifolds into affine toric varieties

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