Topology and geometry of cohomology jump loci

Details Topology and geometry of cohomology jump loci Topology and geometry of cohomology jump loci

2009-02-07

arxiv.org/abs/0902.1250v1

Duke Mathematical Journal 148 (2009), no. 3, 405-457

Mathematics

Algebraic Topology

41 pages; this is a major revision of arXiv:math/0512480; accepted for publication in Duke Mathematical Journal

Scientific paper

10.1215/00127094-2009-030

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.

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