Homology of Local Systems on Real Line Arrangement Complements

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.