K3 surfaces associated to a cubic fourfold

TL;DR

This paper explores the conditions under which smooth cubic fourfolds have associated K3 surfaces, focusing on categorical and motive-based correspondences, especially in families with automorphisms.

Contribution

It characterizes when cubic fourfolds with automorphisms have associated K3 surfaces via categorical equivalences or motive isomorphisms.

Findings

Identification of cases with associated K3 surfaces in automorphism families

Conditions for categorical equivalence between Kuznetsov component and K3 derived category

Criteria for isomorphism of transcendental motives between fourfolds and K3 surfaces

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 an 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 $\sA_X \simeq D^b(S,\alpha)$ where $\sA_X$ is the Kuznetsov component of $D^b(X)$ and $\alpha$ 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.