On algebraic cycles on a fibre product of families of K3-surfaces

We prove the Hodge conjecture and the standard conjecture of Lefschetz type for fibre squares of smooth projective non-isotrivial families of $\mathrm K3$-surfaces over a smooth projective curve under the assumption that the rank of the lattice of transcendental cycles on a generic geometric fibre of the family is an odd prime. We prove the Hodge conjecture for a fibre product of two non-isotrivial families of $\mathrm K3$-surfaces (possibly with degenerations) under the condition that, for every point of the curve, at least one family has non-singular fibre over this point, and the rank of the lattice of transcendental cycles on a generic geometric fibre of one family is odd and not equal to the corresponding rank for the other.