  Over a field of characteristic zero, every deformation problem with
cohomology constraints is controlled by a pair consisting of a differential
graded Lie algebra together with a module. Unfortunately, these pairs are
usually infinite-dimensional. We show that every deformation problem with
cohomology constraints is controlled by a typically finite-dimensional
L-infinity pair. As a first application, we show that for complex algebraic
varieties with no weight-zero 1-cohomology classes, the components of the
cohomology jump loci of rank one local systems containing the constant sheaf
are tori. This imposes restrictions on the fundamental groups. The same holds
for links and Milnor fibers.
