Duke Mathematical Journal

Topology and geometry of cohomology jump loci

Alexandru Dimca, Ştefan Papadima, and Alexander I. Suciu

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Abstract

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic if the group is 1-formal; in particular, the tangent cone to Vk at 1 equals Rk. 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

Article information

Source
Duke Math. J., Volume 148, Number 3 (2009), 405-457.

Dates
First available in Project Euclid: 18 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1245350753

Digital Object Identifier
doi:10.1215/00127094-2009-030

Mathematical Reviews number (MathSciNet)
MR2527322

Zentralblatt MATH identifier
1222.14035

Subjects
Primary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 20F14: Derived series, central series, and generalizations 55N25: Homology with local coefficients, equivariant cohomology
Secondary: 14M12: Determinantal varieties [See also 13C40] 20F36: Braid groups; Artin groups 55P62: Rational homotopy theory

Citation

Dimca, Alexandru; Papadima, Ştefan; Suciu, Alexander I. Topology and geometry of cohomology jump loci. Duke Math. J. 148 (2009), no. 3, 405--457. doi:10.1215/00127094-2009-030. https://projecteuclid.org/euclid.dmj/1245350753


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