An equivariant parametric Oka principle for bundles of homogeneous spaces

Kutzschebauch, Frank; Lárusson, Finnur; Schwarz, Gerald W.

doi:10.1007/s00208-017-1588-1

Cited by 6 publications

(11 citation statements)

References 14 publications

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“…Classical Oka theory and geometric invariant theory were first brought together in the work of Heinzner and Kutzschebauch [12]. The present authors have continued this development, most recently with an equivariant parametric Oka principle for bundles of homogeneous spaces [16]. The present paper is the first step towards an equivariant version of modern, Gromov-style Oka theory.…”

Section: Introductionmentioning

confidence: 71%

“…The most basic result of Oka theory says that if X is a Stein manifold and Y is an Oka manifold, then every continuous map f : X → Y can be deformed to a holomorphic map. In [16], we proved that if f is equivariant with respect to holomorphic actions of a reductive complex Lie group G on X and Y , then f can be deformed through equivariant maps to an equivariant holomorphic map -provided that the G-action on Y factors through a transitive action of some other complex Lie group (not necessarily reductive) on Y , so Y is in particular homogeneous. The main result of [16] says much more, but the homogeneity assumption is essential.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 71%

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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“…The other special case is the “uncoupled” case. It is a parametric Oka principle for equivariant maps from a Stein -space to a complex homogeneous -space G / H , where the -action on G / H can be quite general (see the introduction to [ 23 ]). Namely, the theorem covers -actions on G / H by Lie automorphisms of G that preserve H followed by left multiplication by elements of G .…”

Section: Introductionmentioning

confidence: 99%

Equivariant Oka theory: survey of recent progress

Kutzschebauch

Lárusson

Schwarz

2022

Complex Anal Synerg

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We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle $${{\mathscr {G}}}$$ G , all over a reduced Stein space X with compatible actions of a reductive complex group on E, $${{\mathscr {G}}}$$ G , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a G-manifold being G-Oka.

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“…In [16], we proved that if f is equivariant with respect to holomorphic actions of a reductive complex Lie group G on X and Y , then f can be deformed through equivariant maps to an equivariant holomorphic map-provided that the G-action on Y factors through a transitive action of some other complex Lie group (not necessarily reductive) on Y , so Y is in particular homogeneous. The main result of [16] says much more, but the homogeneity assumption is essential. Our goal here, in the spirit of Gromov, is to remove it.…”

Section: Introductionmentioning

confidence: 99%

Gromov’s Oka Principle for Equivariant Maps

2020

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We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group G acts on a Stein manifold X and another manifold Y in such a way that Y is G-Oka, then every G-equivariant continuous map X → Y can be deformed, through such maps, to a G-equivariant holomorphic map. Approximation on a G-invariant holomorphically convex compact subset of X and jet interpolation along a G-invariant subvariety of X can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.

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An equivariant parametric Oka principle for bundles of homogeneous spaces

Cited by 6 publications

References 14 publications

An Oka principle for equivariant isomorphisms

An Oka principle for equivariant isomorphisms

Equivariant Oka theory: survey of recent progress

Gromov’s Oka Principle for Equivariant Maps

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