PUB - Publikationen an der Universität Bielefeld

Given a smooth projective variety, a Chow-Kunneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kahler varieties admit a multiplicative Chow-Kunneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kahler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow-Kunneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Kuchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow-Kunneth decomposition, by establishing this for two families of Todorov surfaces.