On the Chow ring of some special Calabi-Yau varieties

TL;DR

This paper investigates the Chow ring structure of special Calabi-Yau varieties derived from hyperplane arrangements, demonstrating injectivity into cohomology and confirming several conjectures for these varieties.

Contribution

It applies Fu-Vial's theory to show the injectivity of a subring of the Chow ring and proves Voisin's and Voevodsky's conjectures for these Calabi-Yau varieties.

Findings

Subring of Chow ring injects into cohomology

Voisin's conjecture holds for these varieties

Voevodsky's smash-nilpotence conjecture proven for odd-dimensional cases

Abstract

We consider Calabi-Yau $n$-folds $X$ arising from certain hyperplane arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of $X$ generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin's conjecture for $X$, and Voevodsky's smash-nilpotence conjecture for odd-dimensional $X$.