A criterion for quadraticity of a representation of the fundamental group of an algebraic variety (Q512817)

Let \(X\) be a smooth complex quasi-projective variety and \(\Gamma\) be its fundamental group. Let \(G\) be a real linear algebraic group and \(\rho: \Gamma \to G\) a representation with finite image. The author shows that there is a finite étale Galois cover \(Y \to X\), corresponding to the kernel of \(\rho\), such that if \(Y\) has a smooth compactification \(Z\) with trivial first Betti number \(b_1(Z)=0\), then the germ \((\mathrm{Hom}(\Gamma,G), \rho)\) is a quadratic singularity.