An Oka principle for equivariant isomorphisms

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 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.

An equivariant parametric Oka principle for bundles of homogeneous spaces

“…Equivariant Oka theory started with the 1995 paper of Heinzner and Kutzschebauch [HK95]. The present paper and our previous paper [KLS15] rely heavily on [HK95]. As a corollary of Theorem 1.1, we obtain the following strengthening of Heinzner and Kutzschebauch's main result on the classification of principal bundles with a group action (Theorem 2.3 below).…”

“…The full definition is somewhat involved, so we shall not recall it, but it is natural, whereas special G-homeomorphisms only play an auxiliary role. One of our Oka principles for equivariant isomorphisms [KLS15,Theorem 22] states that every strong G-homeomorphism X → Y can be deformed, through strong G-homeomorphisms, to a special strong G-homeomorphism. We then argued that the existence of a special G-homeomorphism, or merely a special K-homeomorphism, implies the existence of a G-biholomorphism, without establishing that the former can be deformed to the latter.…”

Homotopy principles for equivariant isomorphisms

“…The proof of Theorem 1.4 is along the lines of Grauert's Oka principle for principal bundles of complex Lie groups (Section 10). A main result of our previous paper [KLS15] is a weaker version of Theorem 1.4. In (1) and (2) we were only able to state the existence of a G-biholomorphism, but not that it was homotopic to Φ.…”

“…A strict G-diffeomorphism is not necessarily a strong G-homeomorphism (Example 3.2). Our definition of strict is more general than in [KLS15]; see Remark 5.9.…”

An Oka principle for Stein G-manifolds

“…Let Y be another Stein G-manifold. In [KLS15,KLS] we determined sufficient conditions for X and Y to be equivariantly G-biholomorphic. Clearly we need that Q Y is biholomorphic to Q X , so let us assume that we have fixed an isomorphism of Q Y with Q = Q X .…”