# 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 $\rho:\Gamma\rightarrow G(\mathbb{R})$ 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 Kähler 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.
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https://hal.archives-ouvertes.fr/hal-01196355
Contributor : Louis-Clément Lefèvre <>
Submitted on : Wednesday, September 9, 2015 - 4:25:50 PM
Last modification on : Tuesday, May 11, 2021 - 11:36:04 AM
Long-term archiving on: : Monday, December 28, 2015 - 11:17:05 PM

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Louis-Clément Lefèvre. A criterion for quadraticity of a representation of the fundamental group of an algebraic variety. manuscripta mathematica, Springer Verlag, 2017, 152 (3-4), pp.381-397. ⟨10.1007/s00229-016-0866-7⟩. ⟨hal-01196355⟩

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