Finiteness of prescribed fibers of local biholomorphisms: a geometric approach

الأصل · EN

Let X be a Stein manifold of complex dimension at least two, F: X → Cⁿ a local biholomorphism, and q ∈ 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 ≥ 2, a local biholomorphism of X onto Cⁿ is invertible if and only if the pull-back of every complex line is 1-connected.