Spectral pairs, Alexander modules, and boundary manifolds

Abstract

Let $n>0$ and $f: {\mathbb {C}}^{n+1}\rightarrow {\mathbb {C}}$ be a reduced polynomial map, with $D=f^{-1}(0)$, ${\mathcal {U}}={\mathbb {C}}^{n+1}{\setminus } D$ and boundary manifold $M=\partial {\mathcal {U}}$. Assume that f is transversal at infinity and D has only isolated singularities. Then the only interesting non-trivial Alexander modules of ${\mathcal {U}}$ 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 ${\mathcal {U}}$, 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”.