Birationality of étale morphisms via surgery

Abstract

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in ${\mathbb{C}}^n$, then any local diffeomorphism $F:X\to {\mathbb{C}}^n$ 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 étale morphism $F:X\to {\mathbb{C}}^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational.

17 pages. Replaced to add further references and make language more consistent with the literature

References in corpus (1)

• Holomorphic injectivity and the Hopf map

Cited by in corpus (1)

• Conformal geometry, Euler numbers, and global invertibility in higher dimensions