On representations of complex hyperbolic lattices

The following superrigidity type theorem for complex hyperbolic lattices is proved in this paper. Let X = Gamma\B-n be a compact complex ball quotient, n = 2 or 3. Suppose H-1,H-1(X, C) boolean AND H-2(X, Z) is generated by the Kahler class of X. Then any representation of Gamma in GL(n + 1, C) can either be deformed to a unitary representation or be extended to a homomorphism from SU(n, 1) into GL(n + 1, C).