JournalscmhVol. 81 , No. 1DOI 10.4171/cmh/46

Intersection homology and Alexander modules of hypersurface complements

  • Laurentiu Maxim

    University of Pennsylvania, Philadelphia, United States
Intersection homology and Alexander modules of hypersurface complements cover

Abstract

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