Lefschetz theorems for tamely ramified coverings

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 $X\setminus D$.