  There is a natural stratification of the character variety of a finitely
presented group coming from the jumping loci of the first cohomology of
one-dimensional representations. Equations defining the jumping loci can be
effectively computed using Fox calculus. In this paper, we give an exposition
of Fox calculus in the language of group cohomology and in the language of
finite abelian coverings of CW complexes. Work of Simpson, Arapura and others
show that if $\Gamma$ is the fundamental group of a compact K\"ahler manifold,
then the strata are finite unions of translated affine subtori. If follows that
for K\"ahler groups the jumping loci must be defined by binomial ideals. We
discuss properties of the jumping loci of general finitely presented groups and
apply the ``binomial criterion" to obtain new obstructions for one-relator
groups to be K\"ahler.
