# Voisin's Conjecture for Zero--cycles on Calabi--Yau Varieties and their Mirrors

## TL;DR

This paper investigates Voisin's conjecture on zero-cycles for Calabi-Yau varieties with geometric genus one, providing a criterion for verification and applying it to examples up to dimension five using mirror symmetry and cohomological methods.

## Contribution

It introduces a new criterion for verifying Voisin's conjecture on Calabi-Yau manifolds and demonstrates its effectiveness through examples up to five dimensions.

## Key findings

- Criterion successfully applied to various Calabi-Yau examples
- Mirror symmetry aids in cohomological computations
- Voisin's conjecture verified for multiple cases up to dimension five

## Abstract

We study a conjecture, due to Voisin, on 0-cycles on varieties with $p_g=1$. Using Kimura's finite dimensional motives and recent results of Vial's on the refined (Chow-)K\"unneth decomposition, we provide a general criterion for Calabi-Yau manifolds of dimension at most $5$ to verify Voisin's conjecture. We then check, using in most cases some cohomological computations on the mirror partners, that the criterion can be successfully applied to various examples in each dimension up to $5$.