Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f U→ C^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ C^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z Z on U f U^f induces a Q [ t ± 1 ] Q[t± 1] -module structure on H ∗ ( U f ; Q ) H_*(U^f;Q) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;Q) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;Q) carry canonical Q Q -mixed Hodge structures. We also prove that the covering map U f → U U^f → U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;Q) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f U→ C^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;Q) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;Q) with the limit mixed Hodge structure on the generic fiber of f f .