Abstract: Let f : U → C* be an algebraic map from a smooth complex connected algebraic variety U to the punctured complex line C*. Using f to pull back the exponential map C→C*, one obtains an infinite cyclic cover U^f of the variety U. The homology groups of this infinite cyclic cover, which are endowed with Z-actions by deck transformations, determine the family of Alexander modules associated to the map f. In previous work jointly with Geske, Herradón Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules. In this talk, we will talk about work in progress aimed at generalizing this theory to abelian covering spaces of algebraic varieties which arise in an algebraic way, i.e. from maps f : U → G, where G is a semiabelian variety. Joint work with Moisés Herradón Cueto.