Decomposition of Perverse Sheaves on Plane Line Arrangements

Abstract

On the complement $X= {\mathbb C}^2 - \bigcup_{i=1}^n L_i$ to a central plane line arrangement $\bigcup_{i=1}^n L_i \subset {\mathbb C}^2$, a locally constant sheaf of complex vector spaces $\mathcal L_a$ is associated to any multi-index $a \in {\mathbb C}^n$. Using the description of MacPherson and Vilonen of the category of perverse sheaves (\cite{MV2} and \cite {MV3}) we obtain a criterion for the irreducibility and number of decomposition factors of the direct image $Rj_* \mathcal L_a$ as a perverse sheaf, where $j: X \rightarrow {\mathbb C}^2$ is the canonical inclusion.