Equivariant Kuznetsov Components of Certain Cubic Fourfolds

TL;DR

This paper proves a new connection between the equivariant Kuznetsov component of a cubic fourfold with a group action and the derived category of a related abelian surface, using derived McKay correspondence.

Contribution

It introduces a novel proof linking the equivariant Kuznetsov component of specific cubic fourfolds to the derived category of an associated abelian surface via derived McKay correspondence.

Findings

Equivariant Kuznetsov component is equivalent to the derived category of an abelian surface.

The abelian surface is naturally derived from the cubic fourfold's defining equation.

The proof employs derived McKay correspondence.

Abstract

Let $M$ denote a specific cubic fourfold that accommodates a group action by $\mathbb{Z}/3\mathbb{Z}$. Through utilization of derived Mckay correspondence, we present a new proof establishing the identification of the equivariant Kuznetsov component in the equivariant derived category of $M$ with the derived category of certain abelian surface. This surface naturally emerges from the defining equation of the cubic fourfold $M$.