# Intersection homology and Alexander modules of hypersurface complements

### Laurentiu Maxim

University of Pennsylvania, Philadelphia, United States

## Abstract

Let $V$ be a degree $d$, reduced hypersurface in $\mathbb{CP}^{n+1}$, $n \geq 1$, 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 $i \leq n$, 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

## Cite this article

Laurentiu Maxim, Intersection homology and Alexander modules of hypersurface complements. Comment. Math. Helv. 81 (2006), no. 1, pp. 123–155

DOI 10.4171/CMH/46