Simpson's Theory and Superrigidity of Complex Hyperbolic Lattices

We attack a conjecture of J. Rogawski: any cocompact lattice in $SU(2,1)$ for which the ball quotient $X=B^2\mathrm{/}\mathrm{\Gamma }$ satisfies $b_1(X)=0$ and $H^{1,1}(X)\cap H^2(X,\text{bbq})\approx \text{bbq}$ is arithmetic. We prove the Archimedian suprerigidity for representation of $\mathrm{\Gamma }$ is $SL(3,\text{bbc})$.