Duke Mathematical Journal
- Duke Math. J.
- Volume 148, Number 3 (2009), 405-457.
Topology and geometry of cohomology jump loci
We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, and , related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of and are analytically isomorphic if the group is -formal; in particular, the tangent cone to at equals . These new obstructions to -formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 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
Duke Math. J., Volume 148, Number 3 (2009), 405-457.
First available in Project Euclid: 18 June 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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