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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?

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    $\begingroup$ You do not really need Artin-Popescu to run the dévissage you are hinting to, of which I have been aware for a while. You can construct an affine smooth connected $S$ with the desired property by hand using the soft EGAIV, $\S11.2.6$ as input, and Serre vanishing to chose the algebra generators defining the $\mathbf{C}$-algebra corepresenting $S$ well. $\endgroup$ – user87684 Nov 23 '17 at 3:28
  • $\begingroup$ It is a minor reduction that surely everybody is aware of, but has never pointed out in the literature because there's really no Hodge theory content in it, and, on the other hand, nobody knows how to show that on smooth projective $\mathbf{Q}$-varieties every Hodge cycle is absolute in the sense of Deligne. $\endgroup$ – user87684 Nov 23 '17 at 3:30
  • $\begingroup$ I believe the Variational Hodge Conjecture is mostly as open as the actual Hodge Conjecture, and work of Morrow on its "deformational part" really can't shed much light on a solution to (VHC) anyway, unfortunately. $\endgroup$ – user87684 Nov 23 '17 at 3:33
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    $\begingroup$ The interesting question here is: for any smooth projective variety $X$ over $\mathbf{C}$, does there exist a smooth proper map $\pi : \mathcal{X}\to S$ of smooth varieties over $\mathbf{C}$, with $S$ connected, such that $X$ is isomorphic to a fiber of $\pi$ over some $\mathbf{C}$-point of $S$, and some fiber of $\pi$ over some $\mathbf{C}$-point of $S$ has the property that every Hodge cycle is absolute? The reason why this question is wide open is that the only examples of varieties over $\mathbf{C}$ for which every Hodge cycle is absolute are either low dimensional or abelian varieties. $\endgroup$ – user87684 Nov 23 '17 at 3:37
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    $\begingroup$ OK. So the upshot is: nobody knows how to prove Hodge cycles are absolutely Hodge, beyond the low-dimensional cases in which the Hodge Conjecture is known, and the case of abelian varieties, where Deligne proved this. Also, this is the only obstruction for the Variational Hodge Conjecture of Grothendieck to imply the Hodge Conjecture. $\endgroup$ – user113453 Nov 23 '17 at 5:36

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