Homology of Local Systems on Real Line Arrangement Complements

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.

Revised version: coefficient field extended from C to an arbitrary field. 21 pages, 9 figures. Comments welcome!