Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures

Let $X$ be a compact Kähler manifold, $x\in X$ be a base point and $\rho: \pi_1(X,x) \to GL_N(\mathbb C)$ be the monodromy representation of a $\mathbb C$-VHS. Building on Goldman–Millson’s classical work, we construct a mixed Hodge structure on the complete local ring of the representation variety at $\rho$ and a variation of mixed Hodge structures whose monodromy is the universal deformation of $\rho$ .