Sufficient Conditions for Holomorphic Linearisation

TL;DR

This paper establishes that for most Stein manifolds with reductive group actions, a stratified biholomorphism of their quotients to a linear model suffices to guarantee a holomorphic linearisation of the action.

Contribution

It provides sufficient conditions based on Luna stratification for holomorphic linearisation of reductive group actions on Stein manifolds, extending previous counterexamples.

Findings

Most Stein manifolds with stratified quotient biholomorphic to a linear model are linearisable.

Stratified biholomorphism of quotients is sufficient for linearisation, without requiring the manifold to be biholomorphic to ^n.

Counterexamples lack stratified biholomorphic quotients, explaining their non-linearizability.

Abstract

Let $G$ be a reductive complex Lie group acting holomorphically on $X={\mathbb C}^n$. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ${\mathbb C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $\Phi\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $Q_X$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $\Phi$ as above. Then $\Phi$ induces a biholomorphism $\phi\colon Q_X\to Q_V$ which is stratified, i.e., the stratum of $Q_X$ with a given label is sent isomorphically to the stratum of $Q_V$ with the same label. The counterexamples to the Linearisation Problem construct an action of $G$ such that $Q_X$ is not stratified biholomorphic to any $Q_V$. Our main theorem shows that, for most $X$, a stratified biholomorphism of $Q_X$ to some $Q_V$ is sufficient for linearisation. In fact, we do not have to assume that $X$ is biholomorphic to ${\mathbb C}^n$, only that $X$ is a Stein manifold.