Characterization of Linearizability for Holomorphic 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 |$\Phi \colon X\to V$| 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 $| induces a biholomorphism |${\varphi }\colon X/\!\!/G\to V/\!\!/G$| that is stratified, that is, the stratum of |$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$| with |$\dim G\leq 1$|⁠ , the existence of a stratified biholomorphism of |$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. [ABSTRACT FROM AUTHOR]