Spectral pairs, Alexander modules, and boundary manifolds

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".