Hodge Theory of Abelian Covers

Abstract

Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge structure (MHS). More precisely, we focus on covers which arise algebraically in the following way: if $U$ is a smooth connected complex algebraic variety and $G$ is a complex semiabelian variety, the pullback of the exponential map by an algebraic morphism $f:U\to G$ yields a covering space $\pi:U^f\to U$ whose group of deck transformations is $\pi_1(G)$. The new MHSs are compatible with Deligne's MHS on the homology of $U$ through the covering map $\pi$ and satisfy a direct sum decomposition as MHSs into generalized eigenspaces by the action of deck transformations. This provides a vast generalization of the previous results regarding univariable Alexander modules by Geske, Maxim, Wang and the authors. Lastly, we reduce the problem of whether the first Betti number of the Milnor fiber of a central hyperplane arrangement complement is combinatorial to a question about the Hodge filtration of certain MHSs defined in this paper, providing evidence that the new structures contain interesting information.