Localization of $\frak{u}$-modules. I. Intersection cohomology of real arrangements

TL;DR

This paper investigates the intersection cohomology of real hyperplane arrangements using perverse sheaves and combinatorial complexes, laying groundwork for geometric constructions related to quantum groups at roots of unity.

Contribution

It provides explicit combinatorial complexes for computing cohomology of Goresky-MacPherson extensions of local systems over hyperplane complements.

Findings

Explicit complexes for cohomology computation

Connection between perverse sheaves and hyperplane arrangements

Foundations for geometric tensor categories related to quantum groups

Abstract

This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a root of unity. In the present paper we study perverse sheaves over a complex affine space which are smooth along the stratification determined by a finite arrangement of complex affine hyperplanes defined by real equations. In particular, we construct explicitely (in terms of combinatorial data) complexes computing cohomology of Goresky-MacPherson extensions of one-dimensional local systems over the complement of hyperplanes.