Birationality of étale morphisms via surgery

Details Birationality of étale morphisms via surgery Birationality of étale morphisms via surgery

2007-05-03

arxiv.org/abs/0705.0524v2

J. Reine Angew. Math. 627 (2009), 83--95.

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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