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, V-k and Vk, 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.