  Carlson and Toledo conjectured that any infinite fundamental group $\Gamma$
of a compact K\"ahler manifold satisfies $H^2(\Gamma,\R)\not =0$. We assume
that $\Gamma$ admits an unbounded reductive rigid linear representation. This
representation necessarily comes from a complex variation of Hodge structure
($\C$-VHS) on the K\"ahler manifold. We prove the conjecture under some
assumption on the $\C$-VHS. We also study some related geometric/topological
properties of period domains associated to such $\C$-VHS.
