Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections

TL;DR

This paper proves new decompositions of small diagonals in Calabi-Yau complete intersections and hypersurfaces, leading to insights into their Chow rings and rational equivalence of 0-cycles.

Contribution

It introduces novel decompositions of small diagonals that reveal the structure of Chow rings and rational equivalence in Calabi-Yau and hypersurface varieties.

Findings

Decomposition of the small diagonal in Calabi-Yau complete intersections.

Rational equivalence of degree 0 0-cycles up to torsion.

Decomposition of the smallest diagonal in hypersurfaces' powers.

Abstract

On one hand, for a general Calabi-Yau complete intersection X, we establish a decomposition, up to rational equivalence, of the small diagonal in X^3, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally equivalent to 0, up to torsion. On the other hand, we find a similar decomposition of the smallest diagonal in a higher power of a hypersurface, which provides us an analogous result on the multiplicative structure of its Chow ring.