Admissibility of local systems for some classes of line arrangements

TL;DR

This paper establishes a sufficient condition ensuring that all rank one local systems on the complement of certain complex line arrangements are admissible, simplifying the computation of their cohomology groups.

Contribution

It provides a new criterion for the admissibility of local systems on line arrangement complements, extending understanding of their cohomological properties.

Findings

A sufficient condition for all local systems to be admissible.

Simplification of cohomology computations for these local systems.

Broader applicability to classes of line arrangements.

Abstract

Let $\A$ be a line arrangement in the complex projective plane $\PP^2$. Denote by $M$ its complement and by $\M$ the set of points in $\A$ with multiplicity at least 3. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the dimension of the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{*}(M,\Bbb{C})$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$.