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.Comment: 41 pages; this is a major revision of arXiv:math/0512480; accepted for publication in Duke Mathematical Journa

No abstract

Abstract. We investigate the relationship between the geometric Bieri-NeumannStrebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as Kähler and quasi-Kähler manifolds.

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber-Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart. question of computing the corresponding twisted Betti numbers, i.e. describing the so-called jump subvarieties. Our main results give both local answers, concerning homomorphisms near the trivial representation, and global information on these jump loci. We approach the problem via flat connections and algebraic jump loci coming from the associated covariant derivative. Relative jump lociWe work over the field k = R or C. We consider, up to homotopy, a connected CW complex X, and we want to analyze twisted Betti numbers of X up to a fixed degree q (1 ≤ q ≤ ∞). We also fix a homomorphism of k-linear algebraic groups,When ι is an inclusion, this allows us to treat simultaneously various types of local systems (general, volume-preserving, unitary, unipotent, etc.). Following [19], we are interested in the relative characteristic varieties V i r (X, ι), defined for i, r ≥ 0 by 1350025-2 Commun. Contemp. Math. 2014.16. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/03/15. For personal use only. Cohomology jump loci from an analytic viewpoint Rational homotopy theoryThe space X and the CDGA A • may be related via homotopical algebra, in the sense of Sullivan [54]. The starting point is the notion of q-equivalence: a CDGA map inducing in cohomology an isomorphism up to degree q, and a monomorphism in degree q + 1. We say that two CDGA's have the same q-type (notation: A q B) if they can be connected by a zigzag of q-equivalences. In particular, A • is q-formal (in the sense from [41,18]. For q = ∞, we recover Sullivan's celebrated notion of formality. For a space X, we denote by Ω • (X, k) Sullivan's CDGA of piecewise C ∞ k-forms on X, with cohomology algebra H • (X, k), the untwisted singular cohomology ring. When X is path-connected, it is known that Ω • (X, k) q A • if and only if X and A • have the same q-minimal model. In particular, we may speak about q-formal spaces, and groups (by considering X = K(G, 1)). Local resultsWe...