Spectral pairs, Alexander modules, and boundary manifolds

TL;DR

This paper investigates the mixed Hodge structures and spectral pairs of Alexander modules associated with complex hypersurface complements, providing bounds and computational methods that refine existing divisibility results.

Contribution

It introduces bounds for spectral pairs of Alexander modules and relates boundary manifold spectral pairs to local singularity data and infinity contributions.

Findings

Upper bounds for spectral pairs of the Alexander modules of the complement.

Spectral pairs of the boundary manifold's Alexander module can be computed from local and infinity data.

Refinement of Libgober's divisibility result through Hodge-theoretic bounds.

Abstract

Let $f: \CN \rightarrow \C $ be a reduced polynomial map, with $D=f^{-1}(0)$, $\U=\CN \setminus D$ and boundary manifold $M=\partial \U$. Assume that $f$ is transversal at infinity and $D$ has only isolated singularities. Then the only interesting non-trivial Alexander modules of $\U$ and resp. $M$ appear in the middle degree $n$. We revisit the mixed Hodge structures on these Alexander modules and study their associated spectral pairs (or equivariant mixed Hodge numbers). We obtain upper bounds for the spectral pairs of the $n$-th Alexander module of $\U$, which can be viewed as a Hodge-theoretic refinement of Libgober's divisibility result for the corresponding Alexander polynomials. For the boundary manifold $M$, we show that the spectral pairs associated to the non-unipotent part of the $n$-th Alexander module of $M$ can be computed in terms of local contributions (coming from the singularities of $D$) and contributions from "infinity".