Local coordinates for $operatorname{SL}(n,mathbf{C})$-character varieties of finite-volume hyperbolic 3-manifolds

摘要

Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in $operatorname{SL}(2,mathbf{C})$ with the $n$-dimensional irreducible representation of $operatorname{SL}(2,mathbf{C})$ in $operatorname{SL}(n,mathbf{C})$. In this paper we give local coordinates of the $operatorname{SL}(n,mathbf{C})$-character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.