  In this paper we prove the Hodge conjecture for products of the form $S_1
\times ... S_n$, where $S_i$ are smooth projective surfaces such that
$p_g(S_i)=1, q(S_i)=2$. We also prove the Hodge conjecture for arbitrary
self-products of a K3 surface $X$ such that $End_{hodge}(T(X))$ is a CM number
field.
