Wonderful compactification of character varieties

Indranil Biswas, Sean Lawton and Daniel Ramras

Vol. 302 (2019), No. 2, 413–435

DOI: 10.2140/pjm.2019.302.413

Abstract

Using the wonderful compactification of a semisimple adjoint affine algebraic group $G$ defined over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic, we construct a natural compactification $\overline{{\mathfrak{X}}_{\Gamma }\left(G\right)}$ of the $G$-character variety of any finitely generated group $\Gamma$. When $\Gamma$ is a free group, we show that this compactification is always simply connected with respect to the étale fundamental group, and when $\mathbb{k}=ℂ$ it is also topologically simply connected. For other groups $\Gamma$, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these conditions are satisfied, including closed surface groups and free abelian groups when $G={PGL}_n\left(ℂ\right)$. Additionally, when $\Gamma$ is a free group we identify the boundary divisors of $\overline{{\mathfrak{X}}_{\Gamma }\left(G\right)}$ in terms of previously studied moduli spaces, and we construct a family of Poisson structures on $\overline{{\mathfrak{X}}_{\Gamma }\left(G\right)}$ and its boundary divisors arising from Belavin–Drinfeld splittings of the double of the Lie algebra of $G$. In the appendix, we explain how to put a Poisson structure on a quotient of a Poisson algebraic variety by the action of a reductive Poisson algebraic group.

Keywords

character variety, wonderful compactification, moduli space, fundamental group, Poisson

Mathematical Subject Classification 2010

Primary: 14D20, 14F35, 14L30, 14M27, 53D17

Milestones

Received: 29 October 2017

Revised: 1 February 2019

Accepted: 4 February 2019

Published: 27 November 2019