Admissible local systems for a class of line arrangements

Abstract:

A rank one local system $\mathcal {L}$ on a smooth complex algebraic variety $M$ is admissible if roughly speaking the dimension of the cohomology groups $H^m(M,\mathcal {L})$ can be computed directly from the cohomology algebra $H(M,\mathbb {C})$.

We say that a line arrangement $\mathcal {A}$ is of type $\mathcal {C}_k$ for some $k\ge 0$ if $k$ is the minimal number of lines in $\mathcal {A}$ containing all the points of multiplicity at least 3. We show that if $\mathcal {A}$ is a line arrangement in the classes $\mathcal {C}_k$ for $k\leq 2$, then any rank one local system $\mathcal {L}$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class $\mathcal {C}_3$.