On the geometry of a smooth model of a fibre product of families of K3 surfaces

The Hodge conjecture on algebraic cycles is proved for a smooth projective model $X$ of a fibre product $X_1\times_C X_2$ of nonisotrivial 1-parameter families of K3 surfaces (possibly with degeneracies) $X_{k} \to C$ ($k=1,2$) over a smooth projective curve $C$ under the assumption that, for generic geometric fibres $X_{1s}$ and $ X_{2s}$, the ring $\operatorname{End}_{\operatorname{Hg}(X_{1s})}\operatorname{NS}_{\mathbb Q}(X_{1s})^{\perp}$ is an imaginary quadratic field, $\operatorname{rank}\operatorname{NS}(X_{1s})\neq 18$, and $\operatorname{End}_{\operatorname{Hg}(X_{2s})}\operatorname{NS}_{\mathbb Q}(X_{2s})^{\perp}$ is a totally real field or else $\operatorname{rank}\operatorname{NS}(X_{1s}) < \operatorname{rank}\operatorname{NS}(X_{2s})$.
Bibliography: 10 titles.