Some general \'etale Weak Lefschetz-type theorems

TL;DR

This paper develops new etale cohomology theorems generalizing classical Lefschetz results, applicable over arbitrary fields, and introduces a novel 'fat hyperplane section' technique for non-projective varieties.

Contribution

It provides the first etale cohomology bounds for preimages under proper morphisms and zero loci of sections of ample vector bundles, extending classical results to broader contexts.

Findings

Computed lower etale cohomology for small codimension subvarieties of projective space

Established etale cohomology bounds for preimages under proper morphisms

Introduced a 'fat hyperplane section' Lefschetz-type theorem for non-projective varieties

Abstract

We establish new general etale versions of theorems of Barth and Sommese. Respectively, we compute the lower etale cohomology of closed subvarieties of $P^N$ 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).