Cohomology of the Orlik-Solomon algebras and local systems

TL;DR

This paper develops a combinatorial approach to identify when the space of local systems with non-zero first cohomology on line arrangements has positive-dimensional irreducible components, linking algebraic and geometric properties.

Contribution

It introduces a new combinatorial method for classifying arrangements with positive-dimensional cohomology components and compares cohomology of Orlik-Solomon algebras with local system cohomology.

Findings

Partial classification of arrangements with positive-dimensional cohomology components

A comparison theorem for cohomology of Orlik-Solomon algebra and local systems

Use of Vinberg-Kac classification and algebraic curve pencils

Abstract

The paper provides a combinatorial method to decide when the space of local systems with non vanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik-Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg-Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.