局部双全纯映射的指定纤维的连续性：一种几何方法
Finiteness of prescribed fibers of local biholomorphisms: a geometric approach
摘要：
设$X$是复维数至少为2的Stein流形，$F : X \rightarrow \mathbb{C}^n$是局部双全纯，$q \in F(X)$ .在本文中，我们制定的充分条件只涉及自然关联到$q$的对象，以纤维$q$是有限的。假设对于包含$q$的通用复杂行$l$，$F^{-1}(l)$是1-连通的，并且每当$l$是通过$q$的异常行时，$F^{-1}(l)$就有有限多个分量。利用拓扑学和微分几何的理论，我们对$F^{-1}(q)$的规模给出了一个精确的估计。因此，对于$n \geq 2$，$X$到$\mathbb{C}^n$的局部双全纯是可逆的当且仅当每个复线的拉回是1-连通的。
英文摘要：
Let $X$ be a Stein manifold of complex dimension at least two, $F : X \rightarrow \mathbb{C}^n$ a local biholomorphism, and $q \in F(X)$. In this paper we formulate sufficient conditions involving only objects naturally associated to $q$, in order for the fiber over $q$ to be finite. Assume that $F^{-1}(l)$ is 1-connected for the generic complex line $l$ containing $q$, and $F^{-1}(l)$ has finitely many components whenever $l$ is an exceptional line through $q$. Using arguments from topology and differential geometry, we establish a sharp estimate on the size of $F^{-1}(q)$. It follows that for $n \geq 2$, a local biholomorphism of $X$ onto $\mathbb{C}^n$ is invertible if and only if the pull-back of every complex line is 1-connected.
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