Kuznetsov components ans transcendental motives of cubic fourfolds

TL;DR

This paper explores the relationship between Kuznetsov components and transcendental motives of cubic fourfolds, especially focusing on Fourier-Mukai partners and special cases with automorphisms.

Contribution

It provides explicit descriptions of transcendental motive isomorphisms for Fourier-Mukai partners and establishes new relations for special cubic fourfolds with automorphisms.

Findings

Isomorphism of transcendental motives for Fourier-Mukai partners.

Explicit description of motive isomorphisms in special cases.

Existence of special cubic fourfolds with automorphisms related to other fourfolds.

Abstract

Let $X \subset \P^5_{\C}$ be a smooth cubic fourfold.The Kuznetsov component $\sA_X$ is contained in the derived category $D^b(X)$ and the transcendental motive $t(X)$ is contained in the category of Chow motives $\sM_{rat}(\C))$. If $X$ and $Y$ are {\it Fourier -Mukai partners} and hence the categories $\sA_X$ and $\sA_Y$ are equivalent, then their transcendental motives $t(X)$ and $t(Y)$ are isomorphic. The aim of this note is to consider families of special cubic fourfolds $X$ with their FM-partners $Y$ and to give an explicit description of the isomorphism between the transcendental motives, in the case $X$ and $Y$ are rational and when they are conjecturally irrational. We also prove that ,for special cubic fourfolds $X $ in countably many Hassett divisors, with a symplectic automorphism of order 3, there exists another special cubic fourfold $Y$, an equivalence of categories $\sA^G_X \simeq \sA_{Y}$, where $\sA^G_X$ is the equivariant Kuznetsov component, and an isomorphism $t(X) \simeq t(Y)$.