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Zbl pre05578928
Dimca, Alexandru; Papadima, Ştefan; Suciu, Alexander I.
Topology and geometry of cohomology jump loci. (English)
[J] Duke Math. J. 148, No. 3, 405-457 (2009). ISSN 0012-7094

Summary: We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, $\cal{V}_k$ and $\cal{R}_k$, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of $\cal{V}_k$ and $\cal{R}_k$ are analytically isomorphic if the group is 1-formal; in particular, the tangent cone to $\cal{V}_k$ at 1 equals $\cal{R}_k$. 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

MSC 2000:: *14F35 Homotopy theory (algebraic geometry)
20F14 Series of groups and generalizations
55N25 Homology with local coefficients, equivariant cohomology
14M12 Determinantal varieties
20F36 Braid groups; Artin groups
55P62 Rational homotopy theory

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