Does Poincaré duality preserve algebraic cycles?

Question

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 $2k > 2n-2k$.

Answer 1

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. 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).

Answer 2

I don't think the question makes sense as stated. What is the definition of the integration of $\int_{W_i} \colon H^{2n-2k}(X,\mathbb C) \to \mathbb C$ for $W_i$ a variety of codimension $n-k$? This map is defined for $W_i$ a variety of dimension $n-k$, for example by integrating $2n-2k$-forms over a space of real dimension $2n-2k$.

If you ask the question with integration over varieties of dimension $n-k$, the answer is positive.

Since $Z$ is a subvariety of codimension $k$ in a variety of dimension $n-k$, $Z$ itself is a variety of dimension $n-k$. So we can take $W_i=Z$, and ask whether $\phi_Z = \int_Z$.

This is true by the definition of the cohomology class of an algebraic cycle. If we define $[Z]$ as the Poincare dual to the homology class of $Z$, then $\phi[Z]$ is the linear form on cohomology classes given by pairing with the homology class of $Z$, which for differential forms is the same as integration over $Z$.

For the hard Lefschetz theorem or Lefschetz type standard conjecture to appear, the hyperplane class would have to be involved in some way. It's possible to introduce this into the definition of $\int_{W_i}$, although, I think, rather strangely.

For $k> n/2$ you can define $\int_{W_i}$ by first cupping with the $2k-n$th power of the hyperplane class. In this case, the claim is equivalent to the existence of a cycle whose class, cupped with $[H]^{2k-n}$, is $[Z]$. This would indeed be a form of the Standard Conjecture A as described in Donu Arapura's answer.

For $k< n/2$ you can define $\int_{W_i}$ using the Hard Lefschetz theorem, i.e. by composing the Hard Lefschetz map $H^{2n-2k}(X, \mathbb C) \to H^{2k}(X, \mathbb C)$ with the usual integral of cohomology classes over cycles. In this case, we just need to take $W_i$ to be components of a cycle with cycle class $[Z] \cup [H]^{n-2k}$, which exists by the construction of the intersection product on cycles - we just take the intersection of $Z$ with $n-2k$ copies of $H$, or more concretely the literal intersection with $n-2k$ generic hyperplanes.