Resonant bands and local system cohomology groups for real line arrangements

TL;DR

This paper introduces a new algorithm for computing local system cohomology groups in complexified real line arrangements, providing conditions for their vanishing and low-dimensionality, and applies it to specific arrangements.

Contribution

The paper presents a novel algorithm for local system cohomology computation and generalizes existing results by Cohen-Dimca-Orlik using geometric conditions.

Findings

Conditions for vanishing of first local system cohomology

Conditions for it to be at most one-dimensional

Computed the characteristic variety of the deleted B3-arrangement

Abstract

We give a new algorithm computing local system cohomology groups for complexified real line arrangements. Using it, we obtain several conditions for the first local system cohomology to vanish and to be at most one-dimensional, which generalize a result by Cohen-Dimca-Orlik. The conditions are described in terms of discrete geometric structures of real figures. The proof is based on a recent study on minimal cell structures. We also compute the characteristic variety of the deleted $B_3$-arrangement.