Singularity of the varieties of representations of lattices in solvable Lie groups

Abstract

For a lattice $\Gamma$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $\Gamma$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of $\Gamma$ at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds.