Algebraic versus homological equivalence for 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.