Nearby Cycles and Alexander Modules of Hypersurface Complements

Let $f:\text{CN}\to \text{C}$ be a polynomial map, which is transversal at infinity. Using Sabbah’s specialization complex, we give a new description of the Alexander modules of the hypersurface complement $\text{CN}\setminus f^{-1}(0)$, and obtain a general divisibility result for the associated Alexander polynomials. As a byproduct, we prove a conjecture of Maxim on the decomposition of the Cappell-Shaneson peripheral complex of the hypersurface. Moreover, as an application, we use nearby cycles to recover the mixed Hodge structure on the torsion Alexander modules, as defined by Dimca and Libgober. We also explore the relation between the generic fibre of $f$ and the nearby cycles.