Length of Perverse Sheaves on Hyperplane Arrangements

TL;DR

This paper investigates the length of perverse sheaves associated with hyperplane arrangements, providing combinatorial formulas for specific cases and deriving related intersection cohomology Betti numbers.

Contribution

It introduces combinatorial formulas for the length of perverse sheaves on certain hyperplane arrangements, a novel result in this area.

Findings

Combinatorial formulas for perverse sheaf lengths in specific hyperplane arrangements.

Formulas for intersection cohomology Betti numbers of rank one local systems.

Applicable to arrangements with at most triple points.

Abstract

In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with same monodromy around the planes.