Perverse sheaves over real hyperplane arrangements II

TL;DR

This paper characterizes the category of perverse sheaves on complexified hyperplane arrangements using matrix diagrams, providing a combinatorial framework that relates to the stratification of the arrangement.

Contribution

It introduces a novel description of perverse sheaves over complexified real hyperplane arrangements via matrix diagrams, linking geometric stratification to algebraic data.

Findings

Provides a combinatorial model for Perv(C^n,H)

Relates matrix diagrams to constructible sheaves

Enables algebraic analysis of perverse sheaves

Abstract

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