A criterion for quadraticity of a representation of the fundamental group of an algebraic variety

Abstract

Let $Γ$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $ρ:Γ\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(Γ, G)$. It is known from the work of Goldman and Millson that if $Γ$ is the fundamental group of a compact K{ä}hler manifold and $ρ$ has image contained in a compact subgroup then $ρ$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(Γ, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion holds.