\'Etale cohomology of rank one $\ell$-adic local systems in positive characteristic

TL;DR

This paper studies the structure of rank one $ ext{ell}$-adic local systems in positive characteristic, proving they have quasilinear deformation loci and deriving key theorems like Hard Lefschetz and generic vanishing.

Contribution

It establishes the quasilinear nature of special loci in deformation spaces of rank one $ ext{ell}$-adic local systems in positive characteristic, leading to new geometric theorems.

Findings

Deformation loci are quasilinear in positive characteristic.

Hard Lefschetz theorem holds for rank one $ ext{ell}$-adic local systems.

A generic vanishing theorem is proven.

Abstract

We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem. Last version: a few typos corrected