Finiteness of prescribed fibers of local biholomorphisms: a geometric approach

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.