K3 surfaces associated to a cubic fourfold

Abstract

Let $X\subset P^5$ 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 $A_X\simeq D^b(S,\alpha )$ where $A_X$ is the Kuznetsov component of $D^b\left(X\right)$ and $\alpha$ is a Brauer class, or if there is an isomorphism between the transcendental motive $t\left(X\right)$ 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.