Alexander polynomials: Essential variables and multiplicities

TL;DR

This paper investigates the role of Alexander polynomials in understanding the structure of fundamental groups of complex varieties and boundary manifolds, providing new criteria and bounds related to their cohomology and quasi-projectivity.

Contribution

It introduces new criteria based on Alexander polynomials for fundamental groups of complex varieties and boundary manifolds, and establishes explicit bounds on twisted Betti ranks.

Findings

Criteria for fundamental groups of complex varieties derived from Alexander polynomials.

Sharp upper bounds for twisted Betti ranks in terms of polynomial multiplicities.

Explicit formulas linking Alexander polynomials to characteristic varieties in Seifert links.

Abstract

We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give sharp upper bounds for the twisted Betti ranks of a group, in terms of multiplicities constructed from the Alexander polynomial. For Seifert links in homology 3-spheres, these bounds become equalities, and our formula shows explicitly how the Alexander polynomial determines all the characteristic varieties.