Rank two jump loci for solvmanifolds and Lie algebras

TL;DR

This paper characterizes the local structure of rank two jump loci for solvmanifolds and Lie algebras by analyzing representation and resonance varieties, providing a comprehensive description of their analytic germs at the origin.

Contribution

It offers a unified approach to describe the analytic germs of rank two jump loci in both solvmanifolds and Lie algebra contexts, extending previous partial results.

Findings

Complete description of the analytic germs at the origin for rank two jump loci.

Unified framework for solvmanifolds and Lie algebras.

Explicit characterization of depth 1 characteristic and resonance varieties.

Abstract

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.