A 'fat hyperplane section' weak Lefschetz (in arbitrary characteristic), and Barth-type theorems

TL;DR

This paper establishes a generalized Weak Lefschetz theorem for etale cohomology of non-projective varieties in arbitrary characteristic, extending classical results like Barth's theorems without relying on stratified Morse theory.

Contribution

It introduces a 'fat hyperplane section' Weak Lefschetz-type theorem applicable in arbitrary characteristic, broadening the scope of classical Lefschetz and Barth-type theorems.

Findings

Proves a Weak Lefschetz-type theorem for etale cohomology in non-projective varieties.

Provides generalizations of Barth's theorems in arbitrary characteristic.

Eliminates the need for stratified Morse theory in these proofs.

Abstract

We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).