Twisted Alexander Polynomials of Hypersurface Complements

TL;DR

This paper introduces twisted Alexander polynomials for complex hypersurfaces with singularities, extending classical invariants and establishing torsion and divisibility properties related to local topological data.

Contribution

It generalizes classical Alexander polynomials to hypersurfaces with arbitrary singularities and proves key torsionness and divisibility results.

Findings

Twisted Alexander modules are torsion modules under certain conditions.

The twisted Alexander polynomials divide products of local polynomials.

The framework extends classical invariants to more complex hypersurface singularities.

Abstract

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, under certain assumptions, the twisted Alexander modules of a complex hypersurface are torsion modules, and that their orders, the twisted Alexander polynomials, divide the product of certain `local polynomials' defined in terms of the topology near singularities.