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arXiv · 0902.1250

Topology and geometry of cohomology jump loci

Alexandru Dimca·Stefan Papadima·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, 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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Alexandru Dimca, Stefan Papadima, Alexander I. Suciu. 2009-02-07. Topology and geometry of cohomology jump loci. https://doi.org/10.1215/00127094-2009-030

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