A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

TL;DR

This paper proves a general vanishing theorem for the first cohomology of affine smooth complex varieties with rank one local systems and applies it to analyze monodromy actions in specific line arrangements.

Contribution

It introduces a new vanishing result for the first cohomology with local systems and applies it to complex line arrangements including special reflection arrangements.

Findings

Vanishing of the first cohomology for certain affine varieties with local systems.

Determination of monodromy actions on Milnor fiber cohomology for specific arrangements.

Application to monomial and exceptional reflection arrangements.

Abstract

A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.