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

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