Alexander polynomials: Essential variables and multiplicities
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.
Cite
@article{arxiv.0706.2499,
title = {Alexander polynomials: Essential variables and multiplicities},
author = {Alexandru Dimca and Stefan Papadima and Alexander I. Suciu},
journal= {arXiv preprint arXiv:0706.2499},
year = {2008}
}