Vol. 302, No. 2, 2019

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Wonderful compactification of character varieties

Indranil Biswas, Sean Lawton and Daniel Ramras

Appendix: Arlo Caine and Samuel Evens

Vol. 302 (2019), No. 2, 413–435
Abstract

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification XΓ(G)¯ of the G-character variety of any finitely generated group Γ. When Γ is a free group, we show that this compactification is always simply connected with respect to the étale fundamental group, and when k = it is also topologically simply connected. For other groups Γ, 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 = PGLn(). Additionally, when Γ is a free group we identify the boundary divisors of XΓ(G)¯ in terms of previously studied moduli spaces, and we construct a family of Poisson structures on XΓ(G)¯ 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
Authors
Indranil Biswas
Tata Institute of Fundamental Research
Mumbai
India
Sean Lawton
George Mason University
Fairfax, VA
United States
Daniel Ramras
Indiana University-Purdue University Indianapolis
Indianapolis, IN
United States
Arlo Caine
California State Polytechnic University Pomona
Pomona, CA
United States
Samuel Evens
University of Notre Dame
Notre Dame, IN
United States