Fundamental groups and cohomology jumping loci

The characteristic varieties of a space $X$ are the support
loci for cohomology with coefficients in rank one local systems.
These sets, which generalize the Alexander polynomial of a
knot complement, provide precise information on the homology
of abelian covers of the given space.  Closely related are
the resonance varieties associated to the cohomology ring
of $X$.  The interplay between these two types of cohomology
jump loci leads to a deeper understanding of the nature of
the fundamental group $\pi_1(X)$, and of its homological
and geometric finiteness properties.