Characteristic forms for holomorphic families of local systems

Abstract

Let Γ be a finitely generated group and G a complex Lie group with Lie algebra g. The complex variety of group homomorphisms Hom(Γ, G) is equipped with an action of G by inner automorphism and thus it defines a complex-analytic stack. We associate to each Φ ∈ (Symn g)G a closed holomorphic n-form on this stack with values in Hn (Γ, C). This construction is multiplicative so that we get a graded algebra homomorphism. We obtain this as a universal refinement of a Chern-Weil homomorphism. If we take n = 2 and let Γ be the fundamental group of a closed orientable surface, then we recover a classical construction due to Goldman.