Local system homology of arrangement complements

Abstract

We use the critical points of a multivalued holomorphic function and Morse theory to find a basis for a local system homology group defined on the complement of an arrangement of hyperplanes. This generalizes results of Kohno [4] and Douai–Terao [2] from complexified real arrangements to all arrangements. We also show that the set of critical points satisfies the same recursion with respect to deletion and restriction as the $\beta$nbc set of the arrangement.