# On the Chow groups of certain cubic fourfolds

## TL;DR

This paper investigates the Chow groups of a special family of smooth cubic fourfolds with symplectic involutions, establishing a relation with associated K3 surfaces and proving finite-dimensionality of their motives.

## Contribution

It introduces a new family of cubic fourfolds with symplectic involutions and relates their Chow motives to those of K3 surfaces, showing finite-dimensionality.

## Key findings

- Relation between cubic fourfolds and K3 surfaces on Chow motives
- Finite-dimensionality of motives for certain fourfolds
- Symplectic involution induces fixed K3 surface

## Abstract

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold $X$ in the family has an involution such that the induced involution on the Fano variety $F$ of lines in $X$ is symplectic and has a $K3$ surface $S$ in the fixed locus. The main result establishes a relation between $X$ and $S$ on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.