On the Second Cohomology of Kähler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact Kähler manifold, then virtually . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( -VHS) on the Kähler manifold. We prove the conjecture under some assumption on the -VHS. We also study some related geometric/topological properties of period domains associated to such a -VHS.