On Algebraic Cohomology Classes on a Smooth Model of a Fiber Product of Families of K3 surfaces

Hodge's conjecture on algebraic cycles is proved for a smooth projective model $X$ of the fiber product $X_1\times_CX_2$ of nonisotrivial one-parameter families of K3 surfaces (possibly with degeneracies) under certain constraints on the ranks of the transcendental cycle lattices of the general geometric fibers $X_{ks}$ and representations of the Hodge groups $\operatorname{Hg}(X_{ks})$.