K3 surfaces associated to a cubic fourfold

Abstract

Let X⊂P5 be a smooth cubic fourfold. A well known conjecture asserts that X is rational if and only if there a Hodge theoretically associated K3 surface S. The surface S can be associated to X in two other different ways. If there is an equivalence of categories AX≃Db(S,α) where AX is the Kuznetsov component of Db(X) and α is a Brauer class, or if there is an isomorphism between the transcendental motive t(X) and the (twisted ) transcendental motive of a K3 surface S. In this note we consider families of cubic fourfolds with a finite group of automorphisms and describe the cases where there is an associated K3 surface in one of the above senses.