On the Hodge conjecture for products of certain surfaces

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.