Some general \'etale Weak Lefschetz-type theorems
Abstract
We establish new general etale versions of theorems of Barth and Sommese. Respectively, we compute the lower etale cohomology of closed subvarieties of of small codimensions and of their preimages with respect to proper morphisms (that are not necessarily finite; this statement is completely new), and also of the zero loci of sections of ample vector bundles; all these statements are valid over fields of arbitrary characteristics. To obtain these results, we use a new 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties that is related to a result of Goresky and MacPherson (over complex numbers).
Cite
@article{arxiv.2507.06816,
title = {Some general \'etale Weak Lefschetz-type theorems},
author = {Sergei I. Arkhipov and Mikhail V. Bondarko},
journal= {arXiv preprint arXiv:2507.06816},
year = {2025}
}
Comments
This is a major update of arXiv:1203.2595. Lots of new results (including a Sommese-type theorem, several statements related to singularities, l-adic and singular cohomology formulations) and remarks (in particular, on the relation to the literature) were added. Exposition was modified drastically, and notation was changed