Algebraic versus homological equivalence for singular varieties

TL;DR

This paper demonstrates that for certain singular projective varieties, algebraic and homological equivalence of cycles differ, extending Nori's smooth case result to singular varieties and providing explicit examples.

Contribution

It generalizes Nori's result to singular varieties, showing non-zero homology classes restrict to non-algebraically equivalent zero cycles on intersections.

Findings

Homological and algebraic equivalence differ in certain singular varieties.

Provides explicit examples illustrating the difference between homological and algebraic equivalence.

Extends known smooth case results to singular varieties.

Abstract

Let $ Y \subseteq \Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that $\dim Y=n+h$ and $ \dim Y_{sing} \le \min\{ d+h-1 , n-1 \} $. Let $Z$ be an algebraic cycle on $Y$ of dimension $d+h$, whose homology class in $H_{2(d+h)}(Y; \Bbb Q)$ is non-zero. In the present paper we prove that the restriction of $Z$ to $X$ is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case $Y$ is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.