An Oka principle for equivariant isomorphisms
-
Frank Kutzschebauch
,
Finnur Lárusson
Abstract
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.
Funding source: Schweizerischer Nationalfond
Award Identifier / Grant number: 200021-140235/1
Funding source: Australian Research Council
Award Identifier / Grant number: DP120104110
We thank G. Tomassini for pointing us to the reference [`Algebraic methods in the global theory of complex spaces', Editura Academiei, Bucuresti, and John Wiley & Sons, London 1976], which helped us complete the proof of Theorem 4.4. We also thank the referees for valuable comments that helped us improve the exposition.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Cosmetic surgeries on knots in S3
- Integral points in two-parameter orbits
- Arithmetic moduli and lifting of Enriques surfaces
- Analytic varieties with finite volume amoebas are algebraic
- Vanishing resonance and representations of Lie algebras
- A new notion of angle between three points in a metric space
- Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit
- An Oka principle for equivariant isomorphisms
- Birkhoff matrices, residues and rigidity for q-difference equations
Articles in the same Issue
- Frontmatter
- Cosmetic surgeries on knots in S3
- Integral points in two-parameter orbits
- Arithmetic moduli and lifting of Enriques surfaces
- Analytic varieties with finite volume amoebas are algebraic
- Vanishing resonance and representations of Lie algebras
- A new notion of angle between three points in a metric space
- Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit
- An Oka principle for equivariant isomorphisms
- Birkhoff matrices, residues and rigidity for q-difference equations
