Let $k$ be aDoes there exist some subring of $\mathbb{C}$. By the $k$-Hodge conjecture, we mean the statement$k \subset \mathbb{C}$ such that, for each nonsingular algebraic variety $X$ over $\mathbb{C}$, and each $p = 0, 1, \ldots, \dim_{\mathbb{C}}(X)$, each class $\mathfrak{z} \in H^{2p}(X; k) \cap H^{p,p}(X)$ is a $k$-linear combination of classes of algebraic cycles. The millenium problem is the $\mathbb{Q}$-Hodge conjecture.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 the $\mathbb{Z}$-Hodge conjecture is false$k \neq \mathbb{Z}$.
Given $\mathbb{Z} \subset k \subset K \subset \mathbb{C}$, the $k$-Hodge conjecture implies the $K$-Hodge conjecture. (The strikethrourgh assertion is false, as As commented by Ben Wieland commented below.)
For which $\mathbb{Z} \subset k \subset \mathbb{C}$ is the status of the $k$-Hodge conjecture known?
More specifically, is the $k$argument of Atiyah-Hodge conjecture known to be true for someHirzebruch also shows that $\mathbb{Q} \subset k \subset \mathbb{C}$? In particular, is the$k \neq \mathbb{Z}_{(p)}$ for any prime $\overline{\mathbb{Q}}$-Hodge conjecture true?$p$.
In another direction, isRemark: If the $k$-Hodge conjecture known to be false for somemillenium problem is true, then we may take $\mathbb{Z} \subset k \subset \mathbb{Q}$?$k = \mathbb{Q}$.
This will be community wiki, so there can be one answer for each $k$(The above post has been slightly edited from the original).