A characterization of linearizability for holomorphic $\mathbb{C}^*$-actions

TL;DR

This paper characterizes when holomorphic actions of reductive complex Lie groups with a one-dimensional connected component are linearizable, linking linearization to the existence of a stratified biholomorphism of their quotients.

Contribution

It establishes that for such groups, the presence of a stratified biholomorphism of quotients is both necessary and sufficient for linearization, extending previous understanding beyond affine spaces.

Findings

Stratified biholomorphism of quotients characterizes linearizability.

Linearization is equivalent to quotient stratification preservation.

Main theorem applies to Stein manifolds, not just affine spaces.

Abstract

Let $G$ be a reductive complex Lie group acting holomorphically on $X=\mathbb{C}^n$. The (holomorphic) Linearization 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 $X /\!/G$, 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 $\phi\colon X/\!/G\to V/\!/G$ which is stratified, i.e., the stratum of $ X/\!/G$ with a given label is sent isomorphically to the stratum of $V/\!/G$ with the same label. The counterexamples to the Linearization Problem construct an action of $G$ such that $X/\!/G$ is not stratified biholomorphic to any $V/\!/G$. Our main theorem shows that, for a reductive group $G$ with $G^0=\mathbb{C}^*$, the existence of a stratified biholomorphism of $X/\!/G$ to some $V/\!/G$ is not only necessary but also sufficient for linearization. In fact, we do not have to assume that $X$ is biholomorphic to $\mathbb{C}^n$, only that $X$ is a Stein manifold.