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][1] 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).

  [1]: http://hirzebruch.mpim-bonn.mpg.de/151/1/29_Analytic%20cycles%20on%20complex%20manifolds.pdf