Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

Abstract Given a smooth projective variety, Chow–Künneth decomposition is called multiplicative if it compatible with the intersection product. Following works of Beauville and Voisin, Shen Vial conjectured that hyper-Kähler varieties admit decomposition. In this paper, based on mysterious link between Fano cohomology K3 type varieties, we ask whether also decomposition, provide evidence by establishing their existence for cubic fourfolds Küchle c 7. The main input in hypersurface case Franchetta property square variety lines; was established our earlier work fourfold generalized here to arbitrary dimension. On other end spectrum, give ample canonical class might two families Todorov surfaces.