On the second cohomology of K\"ahler groups

Abstract

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.Comment: 21 pages. Exposition improved. Final versio