  We describe locally the representation varieties of fundamental groups for
smooth complex varieties at representations coming from the monodromy of a
variation of mixed Hodge structure. Given such a manifold $X$ and such a linear
representation $\rho$ of its fundamental group $\pi_1(X,x)$, we use the theory
of Goldman-Millson and pursue our previous work that combines mixed Hodge
theory with derived deformation theory to construct a mixed Hodge structure on
the formal local ring $\widehat{\mathcal{O}}_\rho$ to the representation
variety of $\pi_1(X,x)$ at $\rho$. Then we show how a weighted-homogeneous
presentation of $\widehat{\mathcal{O}}_\rho$ is induced directly from a
splitting of the weight filtration of its mixed Hodge structure. In this way we
recover and generalize theorems of Eyssidieux-Simpson ($X$ compact) and of
Kapovich-Millson ($\rho$ finite).
