  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.
