Rank two jump loci for solvmanifolds and Lie algebras

Abstract

We consider representation varieties in $SL_2$ for lattices in solvable Lie groups, and representation varieties in $\mathfrak{sl}_2$ 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.