On the Second Cohomology of Kähler Groups
Résumé
Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact Kähler manifold, then virtually H^2(Γ,ℝ)≠0 . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure (ℂ -VHS) on the Kähler 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 a ℂ -VHS.