Alexander polynomials: Essential variables and multiplicities

Details Alexander polynomials: Essential variables and multiplicities Alexander polynomials: Essential variables and multiplicities

2007-06-17

arxiv.org/abs/0706.2499v1

International Mathematics Research Notices (2008) Vol. 2008: article ID rnm119, 36 pages

Mathematics

Algebraic Geometry

27 pages

Scientific paper

10.1093/imrn/rnm119

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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