On the second cohomology of Kähler groups

Abstract

Carlson and Toledo conjectured that any infinite fundamental group $\Gamma$ of a compact Kähler manifold satisfies $H^2(\Gamma ,\mathbb{R})\ne 0$. 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 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 $\C$-VHS.

21 pages. Exposition improved. Final version