Rank two jump loci for solvmanifolds and Lie algebras

摘要

We consider representation varieties in SL_2 for lattices in solvable Lie groups, and representation varieties in sl_2 for finite-dimensional Lie algebras. Inside them, we examine depth 1 characteristic varieties for solvmanifolds, respectively resonance varieties for cochain Differential Graded Algebras of Lie algebras. We prove a general result that leads, in both cases, to the complete description of the analytic germs at the origin, for the corresponding embedded rank 2 jump loci.