No abstract

No abstract

Let ρ be a maximal representation of a uniform lattice Γ ⊂ SU(n, 1), n ≥ 2, in a classical Lie group of Hermitian type G. We prove that necessarily G = SU(p, q) with p ≥ qn and there exists a holomorphic or antiholomorphic ρ-equivariant map from the complex hyperbolic space to the symmetric space associated to SU(p, q). This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of SU(p, q), the representation ρ extends to a representation of SU(n, 1) in SU(p, q).

International audienceCarlson 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