  To study infinitesimal deformation problems with cohomology constraints, we
introduce and study cohomology jump functors for differential graded Lie
algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs
bundles, and representations of fundamental groups. The results obtained
describe the analytic germs of the cohomology jump loci inside the
corresponding moduli space, extending previous results of Goldman-Millson,
Green-Lazarsfeld, Nadel, Simpson, Dimca-Papadima, and of the second author.
