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

TL;DR

This paper investigates the structure of the direct image sheaf of the constant sheaf on the complement of a hyperplane arrangement, revealing its decomposition factors relate to the arrangement's Poincaré polynomial and describing their local cohomology structure.

Contribution

It provides a detailed description of the decomposition factors of the direct image sheaf in terms of local cohomology sheaves and connects these to the Poincaré polynomial of the arrangement.

Findings

Number of decomposition factors equals the Poincaré polynomial

Decomposition factors are described as local cohomology sheaves

Multiplicity of factors is explicitly given

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.