The deformation theory of representations of fundamental groups of compact Kähler manifolds

Abstract

Let Γ be the fundamental group of a compact Kähler manifold M and let G be a real algebraic Lie group. Let ℜ(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ℜ(Γ, G) it is shown that there exists a neighborhood of ρ in ℜ(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space$Z^1 (\Gamma ,g_{Ad\rho } )$ of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in$H^2 (\Gamma ,g_{Ad\rho } )$. We prove that ℜ(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ℜ(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.