Oka maps

Forstnerič, Franc

doi:10.1016/j.crma.2009.12.004

Cited by 11 publications

(11 citation statements)

References 15 publications

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“…• The theory of Oka maps can to a large extent be formulated for holomorphic submersions between reduced complex spaces. Also, for some purposes, P and Q in Definition 6.1 may be taken to be arbitrary compact subsets of R m or even arbitrary compact Hausdorff spaces, and S and T may be taken to be reduced Stein spaces (see [18,20]).…”

Section: From Manifolds To Mapsmentioning

confidence: 99%

Survey of Oka theory

Forstnerič¹,

Lárusson²

2010

Preprint

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Oka theory has its roots in the classical Oka principle in complex analysis. 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. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the firstnamed author at the conference RAFROT 2010 in Rincón, Puerto Rico, on 22 March 2010, and of a recent survey article by the second-named author [51].

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Section: From Manifolds To Mapsmentioning

confidence: 99%

Survey of Oka theory

Forstnerič¹,

Lárusson²

2010

Preprint

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“…In a seminal paper of 1989, Gromov showed that the structure group is immaterial, so Grauert's theorem holds for any holomorphic fibre bundle whose fibre is a complex Lie group [10]. And recently, Forstnerič has shown that Grauert's theorem holds even more generally for sections of any holomorphic submersion over a Stein space with the structure of a stratified holomorphic fibre bundle with complex Lie groups as fibres [6]. (We should say that we have not stated the theorems of Grauert, Gromov, and Forstnerič in their full strength.)…”

Section: Introductionmentioning

confidence: 99%

An Oka principle for equivariant isomorphisms

Kutzschebauch

Lárusson

Schwarz

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y , which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X ! Y ?If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch.We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism W X ! Y over Q, which is holomorphic when restricted to each fibre of the quotient map X ! Q. We prove a similar theorem when is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL 2 .C/.Our work is in part motivated by the linearisation problem for actions on C n . It follows from one of our main results that a holomorphic G-action on C n , which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.

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“…Clearly, the next question is how to recognise Oka maps. A major theorem of Forstnerič provides the best available answer [17,Corollary 1.3]. The following is a simple version of the theorem, originating from [23, §3.3.C'].…”

Section: Oka Manifoldsmentioning

confidence: 99%

“…• Dominability. 17 The only published survey on surfaces of class VII is [37]. In the following, take all surfaces to be minimal.…”

Section: Deformations Of Oka Manifoldsmentioning

confidence: 99%

Eight lectures on Oka manifolds

Larusson

2014

Preprint

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Oka maps

Cited by 11 publications

References 15 publications

Survey of Oka theory

Survey of Oka theory

An Oka principle for equivariant isomorphisms

Eight lectures on Oka manifolds

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