On the Hodge conjecture for products of certain surfaces

Resum

In this paper we prove the Hodge conjecture for all products of surfaces $S_1\times\cdots\times S_n$ with $q(S_i) = 2, p_g(S_i) = 1$. We also prove the Hodge conjecture for arbitrary self-products of a $K3$ surface $X$ such that the field $E = End_{hg}T(X)$ is a $CM$-field.