UC Santa Barbara

There are two ways that certain Fano fourfolds (for example, cubic fourfoldsand Gushel-Mukai fourfolds) can be associated with K3 surfaces. On the one hand, we can associate a K3 surface to the fourfold Hodge-theoretically, meaning the middle cohomology of the fourfold contains the middle cohomology of the K3 as a sub-Hodge structure; on the other hand, we can homologically associate a K3 to the fourfold, which is to require the Kuznetsov component of the fourfolds to be equivalent to the bounded derived of the K3. Conjecturally, one expect such K3 associations detect rationality of the fourfolds. It has been proved that for cubic fourfolds, these two types of K3 associations are equivalent, whereas for Gushel-Mukai fourfolds, Hodge-association of a K3 strictly implies homological association of a K3. We continue this line of study, instead of comparing fourfolds- K3 association we consider fourfolds-fourfolds association. We prove that, at least for a generic Gushel-Mukai fourfold in the Hodge-special loci, if it admits Hodgeassociated cubic fourfolds, then it admits a homological-associated one, and vice versa.