Birationality of \'etale morphisms via surgery

TL;DR

The paper proves that certain local diffeomorphisms and étale morphisms in complex algebraic geometry are either birational or have infinite degree, using counting and surgery theory techniques.

Contribution

It introduces a novel approach combining counting arguments and surgery theory to establish birationality of étale morphisms under specific conditions.

Findings

Étale morphisms covering away from a general hypersurface are birational.

Local diffeomorphisms with finite degree are restricted to degree 1 or infinity.

All degrees greater than 1 are possible if the map fails to be a local diffeomorphism at some point.

Abstract

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (however all degrees $d > 1$ are possible if $F$ fails to be a local diffeomorphism at even a single point). In particular, any \'etale morphism $F:X \to \Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational.