On Hodge polynomials of Singular Character Varieties

Abstract

Let X Γ G := Hom(Γ , G)//G be the G-character variety of Γ , where G is a complex reductive group and Γ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of X Γ G, and present some applications, focusing on the cases when Γ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere. IntroductionLet G be a connected reductive complex algebraic group, and Γ be a finitely presented group. The G-character variety of Γ is defined to be the (affine) geometric invariant theory (GIT) quotientThe most well studied families of character varieties include the cases when the group Γ is the fundamental group of a Riemann surface Σ , and its "twisted" variants. In these cases, the non-abelian Hodge correspondence (see, for example [Si]) shows that (components of) X Γ G are homeomorphic to certain moduli spaces of G-Higgs bundles which appear in connection to important problems in Mathematical-Physics: for example, these spaces play an important role in the quantum field theory interpretation of the geometric Langlands correspondence, in the context of mirror symmetry ([KW]).The study of geometric and topological properties of character varieties is an active topic and there are many recent advances in the computation of their Poincaré polynomials and other invariants. For the surface group case (Γ = π 1 (Σ ) and related groups) the calculations of Poincaré polynomials started with Hitchin and Gothen, and have been pursued more recently by Hausel, Lettelier, Mellit, Rodriguez-Villegas, Schiffmann and others, who also considered the parabolic version of these character varieties (see [HRV, Me, Sc]). Those recent results use arithmetic methods: it is shown that the number of points of the corresponding moduli space over finite fields is given by a polynomial,