# Algebraic cycles and very special cubic fourfolds

## TL;DR

This paper explores variants of Voisin's conjecture related to algebraic cycles on special cubic fourfolds and hyperk"ahler varieties, providing verified examples through specific geometric constructions.

## Contribution

It introduces variant versions of Voisin's conjecture for cubic fourfolds and hyperk"ahler varieties and verifies these conjectures for certain special cases.

## Key findings

- Verified conjectures for specific very special cubic fourfolds
- Identified Fano varieties of lines as key examples
- Provided new insights into algebraic cycles on Calabi-Yau related varieties

## Abstract

Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for hyperk\"ahler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.