Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let $Gamma$ be a torsion-free lattice of $text{PU}(p,1)$ with $p geq 2$ and let $(X,mu_X)$ be an ergodic standard Borel probability $Gamma$-space. We prove that any maximal Zariski dense measurable cocycle $sigma: Gamma times X longrightarrow text{SU}(m,n)$ is cohomologous to a cocycle associated to a representation of $text{PU}(p,1)$ into $text{SU}(m,n)$, with $1 < m leq n$. The proof follows the line of Zimmer Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, it cannot exist a maximal measurable cocycle with the above properties when $n eq m$.