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.