  We study the germs at the origin of $G$-representation varieties and the
degree 1 cohomology jump loci of fundamental groups of quasi-projective
manifolds. Using the Morgan-Dupont model associated to a convenient
compactification of such a manifold, we relate these germs to those of their
infinitesimal counterparts, defined in terms of flat connections on those
models. When the linear algebraic group $G$ is either
$\textrm{SL}_2(\mathbb{C})$ or its standard Borel subgroup and the depth of the
jump locus is 1, this dictionary works perfectly, allowing us to describe in
this way explicit irreducible decompositions for the germs of these embedded
jump loci. On the other hand, if either $G=\textrm{SL}_n(\mathbb{C})$ for some
$n\ge 3$, or the depth is greater than 1, then certain natural inclusions of
germs are strict.
