Spectral pairs, Alexander modules, and boundary manifolds

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".Comment: comments are very welcom