Finiteness of prescribed fibers of local biholomorphisms: a geometric approach

TL;DR

This paper provides geometric conditions under which the fibers of local biholomorphisms from Stein manifolds to complex Euclidean spaces are finite, leading to criteria for invertibility based on the topology of preimages of lines.

Contribution

It introduces new topological criteria involving the connectivity of preimages of lines to determine finiteness of fibers and invertibility of local biholomorphisms.

Findings

Finiteness of fibers is guaranteed under specific topological conditions.

A sharp estimate on the size of fibers is established.

Invertibility of local biholomorphisms is characterized by the connectivity of pull-backs of lines.

Abstract

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.