A remark on the product by the hyperplane class in the Chow ring of some complete intersections

TL;DR

This paper extends a classical result about the hyperplane class product being zero on homologically trivial cycles from smooth hypersurfaces to certain complete intersections in complex projective space.

Contribution

It generalizes the known property of the hyperplane class product to a broader class of complete intersections.

Findings

Product by hyperplane class is zero on homologically trivial cycles for certain complete intersections.

Extends classical hypersurface result to more complex algebraic varieties.

Provides new insights into the Chow ring structure of complete intersections.

Abstract

By classical calculation, for a smooth hypersurface $Y\subset \mathbb P^{n+1}_{\mathbb C}$, the product by the hyperplane class is zero on homologically trivial rational cycles i.e. $H_{|Y}\cdot :{\rm CH}_i(Y)_{hom,\mathbb Q}\rightarrow {\rm CH}_{i-1}(Y)_{hom,\mathbb Q}$ is $0$ for any $i$. This note extends that result to some complete intersections.