Reidemeister Torsion, Peripheral Complex, and Alexander Polynomials of Hypersurface Complements

TL;DR

This paper investigates the Alexander polynomials of hypersurface complements using Reidemeister torsion and peripheral complexes, providing new estimates, identities, and Hodge structures that deepen understanding of their algebraic and topological properties.

Contribution

It introduces novel estimates and identities for Alexander polynomials of hypersurface complements by leveraging peripheral complexes and Reidemeister torsion, extending previous results to higher dimensions.

Findings

Derived estimates for Alexander polynomials of hypersurface complements.

Established polynomial identities refining previous formulas.

Constructed mixed Hodge structures on Alexander modules.

Abstract

Let $f:\CN \rightarrow \C $ be a polynomial, which is transversal (or regular) at infinity. Let $\U=\CN\setminus f^{-1}(0)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give several estimates for the (infinite cyclic) Alexander polynomials of $\U$ induced by $f$, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of $\U$.