An Oka principle for equivariant isomorphisms

Kutzschebauch, Frank; Larusson, Finnur; Schwarz, Gerald (2015). An Oka principle for equivariant isomorphisms. Journal für die reine und angewandte Mathematik, 2015(706), pp. 193-214. de Gruyter 10.1515/crelle-2013-0064

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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 ψ : X → Y over Q, which is holomorphic when restricted to each fibre of the quotient map X → Q. We prove a similar theorem when ψ 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 SL2(ℂ). Our work is in part motivated by the linearisation problem for actions on ℂn. It follows from one of our main results that a holomorphic G-action on ℂn, which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Kutzschebauch, Frank; Larusson, Finnur and Schwarz, Gerald

Subjects:

500 Science > 510 Mathematics

ISSN:

0075-4102

Publisher:

de Gruyter

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

24 Jun 2016 10:15

Last Modified:

14 May 2019 15:40

Publisher DOI:

10.1515/crelle-2013-0064

ArXiv ID:

1303.4779v3

BORIS DOI:

10.7892/boris.82549

URI:

https://boris.unibe.ch/id/eprint/82549

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