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.

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全纯 ℂ*-Action 的线性化表征

令$G$ 是一个全纯地作用在$X=\mathbb{C}^n$ 上的约简复李群。（全纯）线性化问题询问 $\mathbb{C}^n$ 上的坐标是否存在全纯变化，使得 $G$-动作变为线性。等效地，是否存在 $G$-等变双全同态 $\Phi \colon X\to V$，其中 $V$ 是 $G$-module？分类商 $X/\!\!/G$ 有一个内在分层，称为 Luna 分层，其中各层由 $G$ 的约简子群表示的同构类标记。假设有一个如上所述的 $\Phi $。然后$\Phi $ 诱导出一个分层的双全同态${\varphi }\colon X/\!\!/G\to V/\!\!/G$，即$X/\!\ 的层具有给定标签的 !/G$ 被同构地发送到具有相同标签的 $V/\!\!/G$ 的层。线性化问题的反例构造了 $G$ 的动作，使得 $X/\!\!/G$ 不分层双全纯到任何 $V/\!\!/G$。我们的主要定理表明，对于具有 $\dim G\leq 1$ 的还原群 $G$，存在 $X/\!\!/G$ 到某个 $V/\!\!/ 的分层双全同态G$ 不仅是必要的，而且对于线性化也是足够的。事实上，我们不必假设$X$ 是$\mathbb{C}^n$ 双全纯的，只需假设$X$ 是一个斯坦流形。