The cohomology groups of finite cyclic covers of complexified real arrangement complements

Abstract

In this paper, we study the cohomology groups of finite cyclic covers of complexified real arrangement complements. An open problem is whether the torsion in the (co)homology of finite covering spaces of hyperplane arrangement complements, including the classical Milnor fiber, is combinatorially determined. Using the chamber cochain complex constructed by Yoshinaga, we obtain explicit upper bounds for the Betti numbers of these covers over arbitrary fields. Furthermore, we introduce a combinatorial condition analogous to the Cohen-Dimca-Orlik (CDO) condition. Under this condition, we prove that the integral cohomology groups of the finite cyclic covers are torsion-free. This partially generalizes the recent results on complex line arrangements obtained by the author and Liu \cite[Theorem 1.3]{LX26}.