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

Abstract

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

arXiv comment13 pages