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Mixed Hodge Structures on Alexander Modules
About this Title
Eva Elduque, Christian Geske, Moisés Herradón Cueto, Laurenţiu George Maxim and Botong Wang
Publication: Memoirs of the American Mathematical Society
Publication Year:
2024; Volume 296, Number 1479
ISBNs: 978-1-4704-6967-2 (print); 978-1-4704-7816-2 (online)
DOI: https://doi.org/10.1090/memo/1479
Published electronically: April 4, 2024
Keywords: Infinite cyclic cover,
Alexander module,
mixed Hodge structure,
thickened complex,
limit mixed Hodge structure,
semisimplicity
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Thickened complexes
- 4. Thickened complexes and mixed Hodge complexes
- 5. Mixed Hodge structures on Alexander modules
- 6. The geometric map is a morphism of MHS
- 7. The geometric map is an MHS morphism: Consequences
- 8. Semisimplicity for proper maps
- 9. Relation to the limit MHS
- 10. Examples and open questions
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 $f\colon U\to \mathbb {C}^*$ be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of $\mathbb {C}^*$ by $f$ gives rise to an infinite cyclic cover $U^f$ of $U$. The action of the deck group $\mathbb {Z}$ on $U^f$ induces a $\mathbb {Q}[t^{\pm 1}]$-module structure on $H_*(U^f;\mathbb {Q})$. We show that the torsion parts $A_*(U^f;\mathbb {Q})$ of the Alexander modules $H_*(U^f;\mathbb {Q})$ carry canonical $\mathbb {Q}$-mixed Hodge structures. We also prove that the covering map $U^f \to U$ induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of $A_*(U^f;\mathbb {Q})$, as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when $f\colon U\to \mathbb {C}^*$ is proper, we prove the semisimplicity and purity of $A_*(U^f;\mathbb {Q})$, and we compare our mixed Hodge structure on $A_*(U^f;\mathbb {Q})$ with the limit mixed Hodge structure on the generic fiber of $f$.- Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 354657
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