论文标题
全态$ \ mathbb {c}^*$ - 动作的线性化性的表征
A characterization of linearizability for holomorphic $\mathbb{C}^*$-actions
论文作者
论文摘要
令$ g $为一个还原的复杂谎言组，在$ x = \ mathbb {c}^n $上进行全体形式。 （HOLOMORTHIC）线性化问题询问$ \ Mathbb {C}^n $上的坐标发生变化，以使$ g $ -Action变为线性。同等地，是否有$ g $ equivariant Biholomormormormormormormormormormormormormormorming $φ\ COLON X \ to V $，其中$ v $是$ g $ -module？分类商$ x /\！ /g $的内在分层称为Luna分层，该地层由$ G $还原亚组的同构表示的同构标记。假设上面有$φ$。然后，$φ$诱导了biholomormormormormormormormormormormorim $ ϕ \ colon x/\！/g \ to v/\！/g $，即分层，即带有给定标签的$ x/\！/g $的阶层是同构发送给$ v/\！/g $带有同一标签的阶层。 线性化问题的反例构建了$ g $的动作，以使$ x/\！/g $均未将Biholomormormormorphic分类为任何$ v/\！/g $。我们的主要定理表明，对于$ g^0 = \ mathbb {c}^*$的还原组$ g $，存在$ x/\！/g $的分层生物形态的存在，不仅是$ v/\！/g $，不仅是必要的，而且还足以进行线性化。实际上，我们不必假设$ x $是biholomorphic to $ \ mathbb {c}^n $，只有$ x $是stein歧管。
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.
