等变同构的同伦原理
Homotopy principles for equivariant isomorphisms
摘要：
设$G$是全纯作用在Stein流形$X$和$Y$上的约化复李群.设$p_X\colon X\to Q_X$和$p_Y\colon Y\to Q_Y$为商映射。什么时候存在$X$和$Y$的等变双全纯？一个必要条件是范畴子$Q_X$和$Q_Y$是双全纯的，并且双全纯$ϕ$将$Q_X$的Luna层同构地发送到$Q_Y$的相应Luna层上。修复$ϕ$。在这种情况下，我们证明了两个同伦原理。第一个结果是：如果存在一个$G$ -同构$Φ\colon X\to Y$，诱导$ϕ$，它在商映射的约化纤维上是$G$ -双全纯的，则$Φ$通过满足相同条件的$G$ -同构同伦到一个从$X$到$Y$的$G$ -等变双全纯.第二个结果粗略地说，如果我们有一个$G$ -同胚$Φ\colon X\to Y$，它诱导了纤维$p_X^{-1}(q)$和$p_Y^{-1}(ϕ(q))$对于$q\in Q_X$的连续族$G$ -等变双全纯，并且如果$X$满足一个辅助性质（这对大多数$X$成立），则$Φ$是同伦的，通过满足相同条件的$G$ -同胚，到从$X$到$Y$的$G$ -等变双全纯。我们的结果改进了作者的早期工作，并使用了新的思想和技术。
英文摘要：
Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $ϕ$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $ϕ$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $Φ\colon X\to Y$, inducing $ϕ$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $Φ$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $Φ\colon X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(ϕ(q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $Φ$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon earlier work of the authors and use new ideas and techniques.
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