Intersection homology and Alexander modules of hypersurface complements

Abstract

Let $V$ be a degree $d$, reduced hypersurface in ${\mathbb{CP}}^{n+1}$, $n\ge 1$, and fix a generic hyperplane, $H$. Denote by $\mathcal{U}$ the (affine) hypersurface complement, ${\mathbb{CP}}^{n+1}-V\cup H$, and let ${\mathcal{U}}^c$ be the infinite cyclic covering of $\mathcal{U}$ corresponding to the kernel of the total linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules $H_i({\mathcal{U}}^c;\mathbb{Q})$ of the hypersurface complement and show that, if $i\le n$, these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials: their zeros are roots of unity of order $d$ and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair $({\mathbb{CP}}^{n+1},V)$. As an application, we give obstructions on the eigenvalues of monodromy operators associated to