On the second cohomology of K\"ahler groups

Carlson and Toledo conjectured that any infinite fundamental group $\mathrm{\Gamma }$ of a compact K"ahler manifold satisfies $H^2(\mathrm{\Gamma },\mathbb{R})≠0$. We assume that $\mathrm{\Gamma }$ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ($\text{C}$-VHS) on the K"ahler manifold. We prove the conjecture under some assumption on the $\text{C}$-VHS. We also study some related geometric/topological properties of period domains associated to such $\text{C}$-VHS.