Does there exist some subring $k \subset \mathbb{C}$ such that the following assertion holds?

• ($k$-Hodge conjecture) For each nonsingular algebraic variety $X$ over $\mathbb{C}$, and each $q = 0, 1, \ldots, \dim_{\mathbb{C}}(X)$, each class $\mathfrak{z} \in H^{2q}(X; k) \cap H^{q,q}(X)$ is a $k$-linear combination of classes of algebraic cycles.

Atiyah-Hirzebruch proved in 1961 that $k \neq \mathbb{Z}$. As commented by Ben Wieland below, the argument of Atiyah-Hirzebruch also shows that $k \neq \mathbb{Z}_{(p)}$ for any prime $p$.

Remark: If the millenium problem is true, then we may take $k = \mathbb{Q}$.

(The above post has been slightly edited from the original).