**Motivation.**
Let us state the following version of Grothendieck's variational Hodge Conjecture:

**Conjecture (VHC).** Let $\mathcal{X}\to S$ be a proper smooth map of smooth algebraic varieties over $\mathbf{C}$, with $S$ connected.
Let $\omega$ be a flat section of $R^{2p}\pi_*\Omega^{\bullet}_{\mathcal{X}/S}$.
For $s\in S(\mathbf{C})$, we say $\omega(s)$ is *algebraic* if and only if:
$$\omega(s)\in (R^{2p}\pi_*\Omega^{\bullet}_{\mathcal{X}/S})_s\otimes_{\mathcal{O}_{S,s}}\kappa(s)$$ is in the image of the rational cycle class map: $$\text{cl}_s : \text{CH}^p(\mathcal{X}_s)_{\mathbf{Q}}\to H^{2p}_{\rm dR}(\mathcal{X}_s/\kappa(s)) \simeq (R^{2p}\pi_*\Omega^{\bullet}_{\mathcal{X}/S})_s\otimes_{\mathcal{O}_{S,s}}\kappa(s).$$

Then, $\omega(s)$ is algebraic for *every* $s\in S(\mathbf{C})$ if and only if $\omega(s)$ is algebraic for *some* $s\in S(\mathbf{C})$.

In other words, algebraicity spreads out in proper smooth families with smooth connected base.

**Remark.** The property of Hodge cycles being *absolute* in the sense of Deligne ("Algebraic Cycles on Abelian Varieties", notes by Milne) likewise spreads out in smooth proper families with smooth connected base: a reformulation of Deligne's Principle B (Thm. 2.12, 2.15 and clear generalizations). In particular, one can prove (VHC) follows from the Tate Conjecture on Algebraic Cycles for smooth projective geometrically irreducible varieties over finitely generated fields of characteristic zero.

Despite not knowing whether the full Hodge Conjecture, abbr. (HC), for smooth projective varieties over $\mathbf{C}$ follows from (VHC) (at least to the best of my knowledge), it would indeed be the case that (VHC) implies (HC) if, on every smooth projective variety over $\mathbf{C}$, every Hodge cycle was absolute.

**Questions.**

(1) Is it known that for every rational variety $X$ defined over $\mathbf{Q}$, every Hodge cycle of $X_{\mathbf{C}}(\mathbf{C})$ (ie. relative to the *unique* field embedding $\mathbf{Q}\to\mathbf{C}$) is an absolute Hodge cycle?

The answer to (1) should be no: indeed, by elementary spreading out techniques and Artin-Popescu, we can always find a proper smooth map $\pi : \mathcal{X}\to S$ of smooth varieties over $\mathbf{C}$ with $S$ connected, such that there exist $s,s'\in S(\mathbf{C})$ with $\mathcal{X}_s\simeq X$ as $\mathbf{C}$-schemes, and $\mathcal{X}_{s'}$ is isomorphic as a $\mathbf{C}$-scheme to the base change along $\mathbf{Q}\to\mathbf{C}$ of a smooth projective $\mathbf{Q}$-scheme. By Deligne's Principle B we would deduce every Hodge cycle on any smooth projective variety over $\mathbf{C}$, is an absolute Hodge cycle, and I am obviously assuming nobody missed this. The question, therefore, asks whether I am missing a reference for this fact, and it's only me who thinks this is not known yet.

(2) Any meaningful references on the variational Hodge Conjecture in any sense?