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arXiv · 0705.0524

Birationality of \'etale morphisms via surgery

Scott Nollet·Laurence R. Taylor·Frederico Xavier
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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.

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Scott Nollet, Laurence R. Taylor, Frederico Xavier. 2007-05-03. Birationality of \'etale morphisms via surgery. https://doi.org/10.1515/crelle.2009.012

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