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arXiv · 2110.03058

Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Eva Elduque·Moisés Herradón Cueto
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Abstract

In previous work jointly with Geske, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue 1 can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space.

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Eva Elduque, Moisés Herradón Cueto. 2021-10-06. Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules. https://arxiv.org/abs/2110.03058

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