Lefschetz theorems for tamely ramified coverings

TL;DR

This paper proves that Lefschetz theorems hold for tame coverings of regular projective varieties with normal crossings divisors, establishing equivalences and full faithfulness of restriction functors in certain dimensions.

Contribution

It fills a gap by proving Lefschetz theorems for tame coverings, extending classical results to this specific setting using formal lifting techniques.

Findings

Restriction functor is an equivalence for dimension ≥ 3.

Restriction functor is fully faithful for dimension 2.

Lifting tame coverings from hyperplane sections to the variety.

Abstract

As is well known, the Lefschetz theorems for the \'etale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. 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 \'etale coverings of $X\setminus D$ to those of $Y\setminus D\cap Y$ is an equivalence if dimension $X \ge 3$, and fully faithful if dimension $X=2$. The method is dictated by Grothendieck-Murre ("The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme", Springer LNM 208). The authors 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 $X\setminus D$.