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Let $X$ be a singular, projective (complex) variety of dimension $n$ with at worst isolated singularities. We know that taking cup-product with the cohomology class of $X$ induces a morphism from the cohomology group to the Borel-Moore homology: $$ \cap [X]: H^{2p}(X) \to H_{2n-2p}^{\mathrm{BM}}(X)$$ Is this map necessarily injective? We know that if $X$ is non-singular then this map is an isomorphism.

Any idea/reference will be most welcome.

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This map isn't always injective for singular $X$ with isolated singularities, consider the $3$ dimensional schubert variety of the Grassmannian $G(2,4)$. Explicitly, this is $2$ planes in four space with nontrivial intersection with a given reference $2$ plane $V$.

This has an affine stratification, so it's easy to count it's Betti numbers as $1,0,1,0,2,0,1$, and thus the cap product map from $H^4$ to $H_{6-4}=H_2$ can't be injective. I'm ignoring the distinction between Borel moore and usual homology since the space is compact.

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  • $\begingroup$ Thank you so much for the answer! I am not quite familiar with the Betti number computations using affine stratification. Could you please add a reference in the comment so that I can understand the computation better? $\endgroup$
    – user45397
    Commented Apr 27 at 6:49
  • $\begingroup$ I'm not sure of a specific reference for this, but any source on Schubert varieties explains their stratification into affine spaces. For computing the cohomology, it's just the computation of cellular cohomology when all (CW) cells are even dimensional, this is explained in any algebraic topology text, eg, Hatcher's Algebraic topology. $\endgroup$
    – Chris H
    Commented Apr 27 at 10:47
  • $\begingroup$ Thank you for the extra hints. $\endgroup$
    – user45397
    Commented Apr 27 at 11:10

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