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

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 $Φ\colon X\to V$ 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 $Φ$ as above. Then $Φ$ induces a biholomorphism $ϕ\colon X/\!/G\to V/\!/G$ 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.

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