Algebraic versus homological equivalence for singular varieties

Let $Y\subseteq {\mathbb{P}}^N$ 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 } $\mathrm{.}Let$Z$beanalgebraiccycleon$Y$ofdimension$d+h$, whose homology class in $H_{2(d+h)}(Y;\mathbb{Q})$ 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.