Let $X$ ba a smooth projective variety of dimension $n$. Hodge Conjecture states that every Hodge cycle in $Hdg^k(X,\mathbb{Q})$ comes from a Chern class of codimension $k$ in $CH^k(X,\mathbb{Q})$. Now the $k$-th Chern class of holomorphic vector bundles generates a subgroup $CH^k_{vec}(X,\mathbb{Q})$. Is it possible that every Hodge cycle in $Hdg^k(X,\mathbb{Q})$ comes from $CH^k_{vec}(X,\mathbb{Q})$? Is there any counterexamples or results?
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4$\begingroup$ Aren't Chern classes of holomorphic vector bundles the same as cycle classes (rationally)? To go from cycles to vector bundles, you can take a finite locally free resolution of the ideal sheaf. $\endgroup$– Will SawinCommented Jun 26, 2019 at 23:32
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$\begingroup$ @WillSawin Thanks! In the Noetherian casse the ideal sheaf of a closed subvariety is coherent and hence have a finite locally free resolution of coherent sheaves, and then the result follows. $\endgroup$– BonbonCommented Jun 27, 2019 at 6:41
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