Nearby Cycles and Alexander Modules of Hypersurface Complements

TL;DR

This paper uses Sabbah's specialization complex to analyze Alexander modules of hypersurface complements, proving divisibility results, confirming a conjecture, and connecting nearby cycles with mixed Hodge structures.

Contribution

It provides a new description of Alexander modules via specialization complexes, proves a conjecture on peripheral complexes, and links nearby cycles to mixed Hodge structures.

Findings

Derived a new description of Alexander modules using Sabbah's specialization complex.

Proved a divisibility result for Alexander polynomials of hypersurface complements.

Confirmed Maxim's conjecture on the decomposition of the Cappell-Shaneson peripheral complex.

Abstract

Let $f:\CN \rightarrow \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 $\CN\setminus f^{-1}(0)$, 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.