Kutzschebauch, Frank; Larusson, Finnur; Schwarz, Gerald (2015). An Oka principle for equivariant isomorphisms. Journal für die reine und angewandte Mathematik, 2015(706), pp. 193214. de Gruyter 10.1515/crelle20130064

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Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y, which are locally Gbiholomorphic over a common categorical quotient Q. When is there a global Gbiholomorphism 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 localtoglobal 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 Gbiholomorphic if X is Kcontractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a Gdiffeomorphism 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 Ghomeomorphism, but with an assumption about its action on Gfinite 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 Gbiholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a lowdimensional 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 Gaction on ℂn, which is locally Gbiholomorphic 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: 
00754102 
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/crelle20130064 
ArXiv ID: 
1303.4779v3 
BORIS DOI: 
10.7892/boris.82549 
URI: 
https://boris.unibe.ch/id/eprint/82549 