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

TL;DR

This paper investigates conditions under which representations of the fundamental group of a complex algebraic variety are characterized by quadratic equations, extending known results from Kähler manifolds to algebraic varieties.

Contribution

It introduces a criterion that determines when such representations are defined by quadratic equations, generalizing previous work on Kähler manifolds.

Findings

Established a criterion for quadraticity of representations

Extended Goldman-Millson results to algebraic varieties

Provided conditions for algebraic variety fundamental groups

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 K{\"a}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.