Cubic fourfolds, Kuznetsov components, and Chow motives

Abstract

We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier–Mukai equivalent are isomorphic as Frobenius algebra objects. As a corollary, there exists a Galois-equivariant isomorphism between their $\ell$-adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is equivalent to the derived category of a K3 surface.