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Given a smooth projective variety, a Chow-K\"unneth decomposition is calledmultiplicative if it is compatible with the intersection product. Followingworks of Beauville and Voisin, Shen and Vial conjectured that hyper-K\"ahlervarieties admit a multiplicative Chow-K\"unneth decomposition. In this paper,based on the mysterious link between Fano varieties with cohomology of K3 typeand hyper-K\"ahler varieties, we ask whether Fano varieties with cohomology ofK3 type also admit a multiplicative Chow-K\"unneth decomposition, and provideevidence by establishing their existence for cubic fourfolds and K\"uchlefourfolds of type c7. The main input in the cubic hypersurface case is theFranchetta property for the square of the Fano variety of lines; this wasestablished in our earlier work in the fourfold case and is generalized here toarbitrary dimension. On the other end of the spectrum, we also give evidencethat varieties with ample canonical class and with cohomology of K3 type mightadmit a multiplicative Chow-K\"unneth decomposition, by establishing this fortwo families of Todorov surfaces.

Multiplicative Chow-K\"unneth decompositions and varieties of cohomological K3 type