Questions on the Chow ring of complete intersections

TL;DR

This paper explores the structure of the Chow ring of complete intersections, proving specific properties for Calabi-Yau hypersurfaces and quintic threefolds, and discusses conjectures about their decompositions.

Contribution

It establishes new results on the Chow ring structure of Calabi-Yau hypersurfaces and quintic threefolds, including the dimension of certain intersection products and the existence of MCK decompositions.

Findings

The intersection product $A^2(X) imes A^i(X)$ is one-dimensional for general Calabi-Yau hypersurfaces.

Quintic threefolds admit a multiplicative Chow-K"unneth decomposition.

Conditional proof that all Calabi-Yau hypersurfaces have an MCK decomposition, assuming Voisin's conjecture.

Abstract

We state several questions, and prove some partial results, about the Chow ring $A^\ast(X)$ of complete intersections in projective space. For one thing, we prove that if $X$ is a general Calabi-Yau hypersurface, the intersection product $A^2(X)\cdot A^i(X)$ is one-dimensional, for any $i>0$. We also show that quintic threefolds have a multiplicative Chow-K\"unneth (MCK) decomposition. We wonder whether all Calabi-Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.