Homology of Local Systems on Real Line Arrangement Complements

TL;DR

This paper investigates the homology of complements of complexified real line arrangements with local system coefficients, introducing an algorithm for dimension computation and exploring special cases related to a conjecture.

Contribution

It presents a new algorithm for computing homology dimensions and provides bounds and partial results for arrangements with sharp pairs, advancing understanding of Yoshinaga's conjecture.

Findings

Developed an algorithm to compute homology dimensions from real figures.

Established a new upper bound for homology dimensions.

Made partial progress on Yoshinaga's conjecture for arrangements with sharp pairs.

Abstract

We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real figures of the arrangement. It enables us to give a new upper bound. We further consider the case where the arrangement contains a sharp pair and make partial progress on a conjecture proposed by Yoshinaga.