Intersection homology and Alexander modules of hypersurface complements

  • Laurentiu Maxim

    University of Pennsylvania, Philadelphia, United States


Let VV be a degree dd, reduced hypersurface in CPn+1\mathbb{CP}^{n+1}, n1n \geq 1, and fix a generic hyperplane, HH. Denote by U\mathcal{U} the (affine) hypersurface complement, CPn+1VH\mathbb{CP}^{n+1}-V \cup H, and let Uc\mathcal{U}^c be the infinite cyclic covering of U\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 Hi(Uc;Q)H_i(\mathcal{U}^c;\mathbb{Q}) of the hypersurface complement and show that, if ini \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 dd and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair (CPn+1,V)(\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