Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of $Z$ is an element in $H^{2k}(X,\mathbb{C})$. Denote by $\phi\_Z \in H^{2n-2k}(X,\mathbb{C})^{\vee}$ the element corresponding to $[Z]$ under the Poincaré duality $H^{2k}(X,\mathbb{C}) \cong H^{2n-2k}(X,\mathbb{C})^{\vee}$. Is it true that $\phi\_Z$ is a linear combination of integrations of the form $\int\_{W\_i}$ where $W\_i$ is a subvariety in $X$ of codimension $n-k$? Of course, if the Hodge conjecture is true then this is true as well. I wanted to know if this statement holds without the Hodge conjecture? Any hint / reference will be most welcome.


**EDIT** Assume $2k > 2n-2k$.



Does Poincaré duality preserve algebraic cycles?
A positive answer to your question\* would imply one of Grothendieck's standard conjectures (conjecture A) for complex varieties. See his note: [Standard conjectures on algebraic cycles](https://web.archive.org/web/20210822125536/http://www.math.tifr.res.in/~publ/studies/SM_04-Algebraic-Geometry.pdf). It's safe to say it's open!


\* As interpreted it. See Will Sawin's answer for clarifying remarks.


Also as I mentioned in the comments. The standard conjectures over $\mathbb{C}$ are in some sense weaker than the Hodge conjecture. They are known in many more cases (e.g. abelian varieties, Hilbert schemes of points on a surface, and certain hyper Kähler varieties).


