Finiteness of prescribed fibers of local biholomorphisms: a geometric approach

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 \(n\ge 2\) 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.