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

We give an example of a regular dessin d'enfant whose field of moduli is not an abelian extension of the rational numbers, namely it is the field generated by a cubic root of 2. This answers a previous question. We also prove that the underlying curve has non-abelian field of moduli itself, giving an explicit example of a quasiplatonic curve with non-abelian field of moduli. In the last section, we note that two examples in previous literature can be used to find other examples of regular dessins d'enfants with non-abelian field of moduli.

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.

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We survey recent developments in the study of Hodge theoretic aspects of Alexander-type invariants associated with smooth complex algebraic varieties.