Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

TL;DR

This paper proves that the eigenspace decomposition of Alexander modules respects the mixed Hodge structure, extending previous work on the MHS of Milnor fiber cohomology to Alexander modules.

Contribution

It demonstrates that the decomposition of Alexander modules into eigenspaces is compatible with the mixed Hodge structure, and constructs the MHS on the eigenvalue 1 eigenspace without finite covers.

Findings

Eigenspace decomposition of Alexander modules is a MHS decomposition.

MHS on eigenvalue 1 eigenspace can be constructed directly.

Results apply under certain purity conditions.

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.