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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm119, 36 pages, doi:10.1093/imrn/rnm119 published on January 15, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Alexander Polynomials: Essential Variables and Multiplicities

Alexandru Dimca1, Stefan Papadima2 and Alexander I. Suciu3,*

1 Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France
2 Institute of Mathematics Simion Stoilow, P.O. Box 1-764, RO-014700 Bucharest, Romania
3 Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA

Correspondence: * Correspondence to be sent to: a.suciu{at}neu.ed

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.


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