For a topological space, we investigate its cohomology support loci, sitting
inside varieties of (nonabelian) representations of the fundamental group. To
do this, for a CDG (commutative differential graded) algebra, we define its
cohomology jump loci, sitting inside varieties of (algebraic) flat connections.
We prove that the analytic germs at the origin 1 of representation varieties
are determined by the Sullivan 1-minimal model of the space. Under mild
finiteness assumptions, we show that, up to a degree q, the two types of jump
loci 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 (for which we establish 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 jump loci: up
to degree q, all their irreducible components passing through the origin are
connected affine subtori, respectively rational linear subspaces. Furthermore,
the global exponential map sends all algebraic cohomology jump loci, up to
degree q, into their topological counterpart.Comment: New Corollary 1.7 added and Theorem D. strengthened. Final version,
to appear in Communications in Contemporary Mathematic