  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.
