WONDERFUL COMPACTIFICATION OF CHARACTER VARIETIES

摘要

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 X_Γ(G) of the Gcharacter variety of any finitely generated group 0. When 0 is a free group, we show that this compactification is always simply connected with respect to the étale fundamental group, and when k = C it is also topologically simply connected. For other groups 0, 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(C). Additionally, when 0 is a free group we identify the boundary divisors of X_Γ(G) in terms of previously studied moduli spaces, and we construct a family of Poisson structures on X_Γ(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.