  We prove Zimmer's conjecture for co-compact lattices in ${\rm SL}(n, \mathbb
C)$: for any co-compact lattice in ${\rm SL}(n, \mathbb C)$, $n \geq 3$, any
$\Gamma$-action on a compact manifold $M$ with dimension: (I) less than $2n-2$
if $n \neq 4$, (II) less than $5$ if $n = 4$, by $C^{1+\epsilon}$
diffeomorphisms factors through a finite action.
