The results on the Chow ring of K3 surfaces and of Calabi-Yau hypersurfaces are obtained by decomposing the
small diagonal in the Chow group of the triple product $X^3$ . In the case of a K3 surface, this decomposition has the following consequence on families $f : S\to B$ of projective K3 surfaces parametrized by a quasi-projective basis $B$: Up to shrinking $B$ to a dense Zariski open set, there is a multiplicative decomposition of $Rf_*Q$, that is a decomposition as the direct sum of its cohomology sheaves, which is compatible with cup-product on both sides. This is reminiscent to what happens with families of abelian varieties, and is very restrictive on the topology of the family.