Length and decomposition of the cohomology of the complement to a hyperplane arrangement

Abstract

Let $\mathcal A$ be a hyperplane arrangement in $\mathbb C^n$. We show that the number of decomposition factors as a perverse sheaf of the direct image $Rj_*\mathbb C_U $ of the constant sheaf on the complement $U$ to the arrangement is given by the Poincar\'e polynomial of the arrangement. Furthermore we describe the composition factors of $Rj_*\mathbb C_U $ as certain local cohomology sheaves and give their multiplicity.