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

L-infinity pairs and applications to singularities