Admissible local systems for a class of line arrangements

TL;DR

This paper investigates the admissibility of rank one local systems on complements of certain line arrangements, establishing conditions under which these systems are admissible, especially for arrangements with minimal high-multiplicity points.

Contribution

It proves that for line arrangements of types CC_k with k ≤ 2, all rank one local systems are admissible, extending understanding of local system behavior on these arrangements.

Findings

All rank one local systems are admissible for arrangements of type CC_k with k ≤ 2.

Partial results are obtained for arrangements of type CC_3.

Admissibility relates to computability of cohomology dimensions from the cohomology algebra.

Abstract

A rank one local system $\LL$ on a smooth complex algebraic variety $M$ is admissible roughly speaking if the dimension of the cohomology groups $H^m(M,\LL)$ can be computed directly from the cohomology algebra $H^*(M,\C)$. We say that a line arrangement $\A$ is of type $\CC_k$ if $k \ge 0 $ is the minimal number of lines in $\A$ containing all the points of multiplicity at least 3. We show that if $\A$ is a line arrangement in the classes $\CC_k$ for $k\leq 2$, then any rank one local system $\LL$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class $\CC_3$.