Rank one local systems and forms of degree one

TL;DR

This paper explores the relationship between cohomology support loci of rank one local systems and the geometry of logarithmic 1-forms on smooth quasiprojective varieties, extending known results from projective to quasi-projective cases.

Contribution

It establishes a new relation connecting cohomology support loci with strata of 1-forms based on vanishing locus dimensions, generalizing previous results.

Findings

Generalizes Green and Lazarsfeld's relation to quasi-projective varieties.

Provides a new geometric interpretation of cohomology support loci.

Extends Dimca's partial relation to a broader setting.

Abstract

Cohomology support loci of rank one local systems of a smooth quasiprojective complex algebraic variety are finite unions of torsion-translated complex subtori of the character variety of the fundamental group. Tangent spaces of the character variety are (partially) represented by logarithmic 1-forms. In this paper, we give a relation between cohomology support loci and the natural strata of 1-forms given by the dimension of the vanishing locus. This relation generalizes the one for the projective case due to Green and Lazarsfeld and also generalizes the partial relation due to Dimca in the quasi-projective case.