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Given a complex quasi-projective variety $X$, let $\alpha$ be an element of the Borel–Moore homology $H_i^\text{BM}(X)$ such that it can be killed by a prime $p$. Under what conditions one can say that there is a closed subvariety of $Z$ such that $\alpha$ is the pushforward of an element in the homology of $Z$? and under what conditions one can say the homology class on $Z$ is torsion?

This question is inspired by the fact that with the assumption of smoothness, if $\alpha$ is a $p$-torsion element in the cohomology of $X$ then there is a divisor $D$ such that pullback of $\alpha$ to $X-D$ is zero. This follows from Bloch–Kato.

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The first question at least can be done as follows. The restriction of $\alpha$ to $X^{\mathrm{sm}}$ vanishes away from a divisor $D$ by the result quoted in your second paragraph and the Poincaré duality isomorphism $H_i^{BM} \cong H^{2d-i}$. Then $\alpha$ is pushed forward from $D \cup X^{\mathrm{sing}}$, by the excision long exact sequence.

The class on $Z=D \cup X^{\mathrm{sing}}$ can not be assumed to be torsion in general. For if it were then one could again find an even smaller subvariety from which it was pushed forward, and repeating the argument indefinitely one would deduce that $\alpha=0$.

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  • $\begingroup$ Yes exactly. I was wondering whether one could put a bound based on $i$, for example something like pushing down torsion in the $i$-th homology to a torsion in a subvariety of dimension $i+1$. $\endgroup$
    – user127776
    Nov 26 '21 at 19:18

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