Perverse sheaves over real hyperplane arrangements

Abstract

Let $\mathcal{H}$ be an arrangement of real hyperplanes in $\mathbb{R}^n$. The complexification of $\mathcal{H}$ defines a natural stratification of $\mathbb{C}^n$. We denote by $\mathrm{Perv}(\mathbb{C}^n, \mathcal{H})$ the category of perverse sheaves on $\mathbb{C}^n$ smooth with respect to this stratification. We give a description of $\mathrm{Perv}(\mathbb{C}^n, \mathcal{H})$ as the category of representations of an explicit quiver with relations, whose vertices correspond to real faces of $\mathcal{H}$ (of all dimensions). The relations are of monomial nature: they identify some pairs of paths in the quiver. They can be formulated in terms of the oriented matroid associated to $\mathcal{H}$.