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Compatibility of Hodge Theory on Alexander Modules

Eva Elduque, Moisés Herradón Cueto

Abstract

Let $U$ be a smooth connected complex algebraic variety, and let $f\colon U\to \mathbb C^*$ be an algebraic map. To the pair $(U,f)$ one can associate an infinite cyclic cover $U^f$, and (homology) Alexander modules are defined as the homology groups of this cover. In two recent works, the first of which is joint with Geske, Maxim and Wang, we developed two different ways to put a mixed Hodge structure on Alexander modules. Since they are not finite dimensional in general, each approach replaces the Alexander module by a different finite dimensional module: one of them takes the torsion submodule, the other takes finite dimensional quotients, and the constructions are not directly comparable. In this note, we show that both constructions are compatible, in the sense that the map from the torsion to the quotients is a mixed Hodge structure morphism.

Compatibility of Hodge Theory on Alexander Modules

Abstract

Let be a smooth connected complex algebraic variety, and let be an algebraic map. To the pair one can associate an infinite cyclic cover , and (homology) Alexander modules are defined as the homology groups of this cover. In two recent works, the first of which is joint with Geske, Maxim and Wang, we developed two different ways to put a mixed Hodge structure on Alexander modules. Since they are not finite dimensional in general, each approach replaces the Alexander module by a different finite dimensional module: one of them takes the torsion submodule, the other takes finite dimensional quotients, and the constructions are not directly comparable. In this note, we show that both constructions are compatible, in the sense that the map from the torsion to the quotients is a mixed Hodge structure morphism.
Paper Structure (13 sections, 22 theorems, 66 equations)

This paper contains 13 sections, 22 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Alexander modules
  4. The group ring and its quotients
  5. Alexander modules and local systems
  6. Relation between homology and cohomology
  7. Mixed Hodge complexes
  8. Summary of previous work
  9. Thickened Mixed Hodge complexes
  10. The MHS on the quotients of Alexander modules
  11. The MHS on the torsion part of Alexander modules
  12. Auxiliary mixed Hodge complexes and structures
  13. Proof of the main theorem

Key Result

Theorem 1.2

Let $U$ be a smooth connected complex algebraic variety with an algebraic map $f\colon U\to {\mathbb C}^*$, let $U^f$ be the infinite cyclic covering map in Diagram eq:fiberProductIntro. Let $m, N\ge 1$. Let $R = {\mathbb Q}[\pi_1({\mathbb C}^*)]\cong {\mathbb Q}[t^{\pm 1}]$. Consider the MHS on $\m This composition map is a MHS morphism for all $m, N\geq 1$. Moreover, there exists $N\in {\mathbb

Theorems & Definitions (69)

  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • Definition 2.11
  • ...and 59 more