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 be a smooth connected complex algebraic variety and let f : U → C * be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C * by f gives rise to an infinite cyclic cover U f of U . The action of the deck group Z on U f induces a Q[t, t −1 ]-module structure on H * (U f ; Q). We show that the torsion parts A * (U f ; Q) of the Alexander modules H * (U f ; Q) carry canonical Qmixed Hodge structures. Furthermore, we compare the resulting mixed Hodge structure on A * (U f ; Q) to the limit mixed Hodge structure on the generic fiber of f . Contents 1. Introduction 2. Preliminaries 2.1. Denotations and Assumptions 2.2. Alexander Modules 2.3. Monodromy action on Alexander Modules 2.4. Differential Graded Algebras 2.5. Mixed Hodge Structures and Complexes 2.6. Real Mixed Hodge Complexes on Smooth Varieties 2.7. Rational Mixed Hodge Complexes on Smooth Varieties 2.8. Limit Mixed Hodge Structure 3. Thickened Complexes 3.1. Thickened Complex of a Differential Graded Algebra

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We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincaré characteristic. Our arguments rely on the construction of circle-valued Morse functions on such spaces, and use in an essential way the stratified Morse theory of Goresky-MacPherson. Our approach also applies (with only minor modifications) for proving similar statements in the analytic context, i.e., for subvarieties of compact complex tori. Alternative proofs of our results can be given by using the general theory of perverse sheaves.Classically, much of the manifold theory, e.g., Morse theory, Lefschetz theorems, Hodge decompositions, and especially Poincaré Duality, is recovered in the singular stratified context if, instead of the usual (co)homology, one uses Goresky-MacPherson's intersection homology groups [12,13]. We recall here the definition of intersection homology of complex analytic (or algebraic) varieties, and discuss some preliminary results concerning the corresponding intersection homology Euler characteristic. For more details on intersection homology, the reader may consult, e.g., [9] and the references therein.Let X be a purely n-dimensional complex analytic (or algebraic) variety with a fixed Whitney stratification. All strata of X are of even real (co)dimension. By [11], X admits a triangulation which is compatible with the stratification, so X can also be viewed as a PL stratified pseudomanifold. Let (C * (X), ∂) denote the complex of finite PL chains on X, with Z-coefficients. The intersection homology groups of X, denoted IH i (X), are the homology groups of a complex of "allowable chains", defined by imposing restrictions on how chains meet the singular strata. Specifically, the chain complex (IC * (X), ∂) of allowable finite PL chains is defined as follows: if ξ ∈ C i (X) has support |ξ|, then ξ ∈ IC i (X) if, and only if, dim(|ξ| ∩ S) < i − s and dim(|∂ξ| ∩ S) < i − s − 1, for each stratum S of complex codimension s > 0. The boundary operator on allowable chains is induced from the usual boundary operator on chains of X. (The second condition above ensures that ∂ restricts to the complex of allowable chains.)

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To any complex algebraic variety endowed with a morphism to a complex affine torus we associate multivariable cohomological Alexander modules, and define natural mixed Hodge structures on their maximal Artinian submodules. The key ingredients of our construction are Gabber-Loeser's Mellin transformation and Hain-Zucker's work on unipotent variations of mixed Hodge structures. As applications, we prove the quasi-unipotence of monodromy, we obtain upper bounds on the sizes of the Jordan blocks of monodromy, and we explore the change in the Alexander modules after removing fibers of the map. We also give an example of a variety whose Alexander module has non-semisimple torsion.