Algebraic Geometry Seminar

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 this talk, we will discuss how to equip the torsion part of the Alexander modules (with respect to the Z-actions) with canonical mixed Hodge structures. Since U^f is not an algebraic variety in general, these mixed Hodge structures cannot be obtained from Deligne's theory. The resulting mixed Hodge structures on Alexander modules have some desirable properties. For example, the covering space map U^f -> U induces morphisms of mixed Hodge structures in homology, where the homology of U is equipped with Deligne's mixed Hodge structure. We will explore several consequences/applications of this fact, regarding weights and semisimplicity. We will also compare the mixed Hodge structures on Alexander modules to other well studied mixed Hodge structures in the literature, including the limit mixed Hodge structure on the generic fiber of f. Joint work with C. Geske, M. Herradón Cueto, L. Maxim, and B. Wang. Speaker(s): Eva Elduque (University of Michigan)