Perverse Sheaves over Real Hyperplane Arrangements II

Abstract

Let $\mathcal{H}$ be an arrangement of hyperplanes in ${\mathbb{R}}^n$ and $\mathrm{Perv}(\mathbb{C}Cn,\mathcal{H})$ be the category of perverse sheaves on ${\mathbb{C}}^n$ smooth with respect to the stratification given by complexified flats of $\mathcal{H}$. We give a description of $\mathrm{Perv}({\mathbb{C}}^n,\mathcal{H})$ in terms of “matrix diagrams”, i.e., diagrams formed by vector spaces $E_{A,B}$ labeled by pairs $(A,B)$ of real faces of $\mathcal{H}$ (of all dimensions) or, equivalently, by the cells $iA+B$ of a natural cell decomposition of $\mathbb{C}$. A matrix diagram is formally similar to a datum describing a constructible (nonperverse) sheaf but with the direction of one-half of the arrows reversed.