Mixed Hodge Structures on Alexander Modules

Abstract

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

Hodge theory on Alexander invariants -- a survey

“…Moreover, we showed in [10] that, if U and f are as in case (2) above, the mixed Hodge structure of Theorem 1.1 recovers the mixed Hodge structure obtained by different means in both [8] and [16].…”

“…In this expository note, we survey recent developments in the study of Hodge theoretic aspects of Alexander-type invariants of complex algebraic manifolds. Our main goal is to provide the reader with a down-to-earth introduction and multiple access points to the circle of ideas discussed in detail in our paper [10].…”

“…In [10], we investigated a question of S. Papadima about the existence of mixed Hodge structures on the torsion parts A i (U ξ ; k) of the Alexander modules of a complex algebraic manifold U endowed with an epimorphism ξ : π 1 (U) ։ Z. Note that the infinite cyclic cover U ξ is not in general a complex algebraic variety, so Deligne's classical mixed Hodge theory does not apply.…”

“…In this note, we indicate the main steps in the proof of Theorem 1.1, and describe several geometric applications. For complete details, the interested reader may consult our paper [10]. We also rely heavily on terminology and constructions from [20 Let U be a smooth connected complex algebraic variety, and let f : U → C * be an algebraic map inducing an epimorphism f * : π 1 (U) ։ Z on fundamental groups.…”

Alexander modules, Mellin transformation and variations of mixed Hodge structures

“…The motivation for terminology comes from the fact that the corresponding homology modules H i (X, L X ) can be identified with the multivariable homology Alexander modules H i ( X, Q) of the pair (X, f ), with the module structure induced by the deck group action. Up to replacing L X by its A-dual, the cohomological Alexander modules may be related to the homological ones by the Universal Coefficient spectral sequence (see, e.g., [DM07, Section 2.3]), which in the case n = 1 simplifies into a short exact sequence, see [E+20,Remark 2.3.4]. We note that the modules H * (X, L X ) are not isomorphic to the cohomology of X in general.…”