Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Abstract

In previous work jointly with Geske, Maxim, and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of (one variable) 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. Further, we show a formula relating the Alexander module’s Hodge numbers to those of finite covers of the base space, under some assumptions.Dedicated to the memory of Georgia Benkart