Non-abelian cohomology jump loci from an analytic viewpoint

Abstract

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...

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

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