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Lefschetz theorems for tamely ramified coverings

Authors: Hélène Esnault and Lars Kindler
Journal: Proc. Amer. Math. Soc. 144 (2016), 5071-5080
MSC (2010): Primary 14E20, 14E22
Published electronically: June 3, 2016
MathSciNet review: 3556253
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Abstract: As is well known, the Lefschetz theorems for the étale fundamental group of quasi-projective varieties do not hold. We fill a small gap in the literature showing they do for the tame fundamental group. Let $ X$ be a regular projective variety over a field $ k$, and let $ D\hookrightarrow X$ be a strict normal crossings divisor. Then, if $ Y$ is an ample regular hyperplane intersecting $ D$ transversally, the restriction functor from tame étale coverings of $ X\setminus D$ to those of $ Y\setminus D\cap Y$ is an equivalence if dimension $ X \ge 3$, and is fully faithful if dimension $ X=2$. The method is dictated by work of Grothendieck and Murre (1971). They showed that one can lift tame coverings from $ Y\setminus D\cap Y$ to the complement of $ D\cap Y$ in the formal completion of $ X$ along $ Y$. One has then to further lift to $ X\setminus D$.

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Additional Information

Hélène Esnault
Affiliation: FB Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

Lars Kindler
Affiliation: Department of Mathematics, Harvard University, Science Center, One Oxford Street, Cambridge, Massachusetts 02138

Received by editor(s): October 3, 2015
Received by editor(s) in revised form: February 2, 2016
Published electronically: June 3, 2016
Additional Notes: The first author was supported by the Einstein Program.
The second author was supported by a research scholarship of the DFG (“Deutsche Forschungsgemeinschaft”).
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2016 H. Esnault and L. Kindler

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