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Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.

If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply that $Y$ satisfies the Hodge conjecture?

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    $\begingroup$ Kang and I (arxiv.org/abs/math/0506210) proved that if two smooth projective varieties $X_i$ have the same class in the Grothendieck group of varieties with $\mathbb{A}^1$ inverted, then HC for $X_1$ is equivalent to HC for $X_2$. This is not exactly what you asked but certainly related. $\endgroup$ – Donu Arapura Jan 27 '17 at 17:54
  • $\begingroup$ Thank you for your reference, Donu. Does being in the same class in the Grothendieck group a kind of weak homotopy equivalence, in the sense that (1) two varieties which are A^1-homotopy equivalence lie in the same class, (2) if two varieties lie in the same class, then they are not necessarily A^1-homotopy equivalent? Is there some condition which guarantees that two varieties lying in the same class in the Grothendieck group is in fact A^1-homotopy equivalent? $\endgroup$ – user94803 Jan 28 '17 at 1:53
  • $\begingroup$ Colin, I believe (1) should be true, but it might take some effort to flesh out since the formalism is pretty sophisticated, cf Berner's answer. $\endgroup$ – Donu Arapura Jan 28 '17 at 14:36
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Isn't this very easy? If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram commutative). So it remains to note that all the ingredients of your question are "motivic". For this purpose one may recall that Chow motives embed into Voevodsky one and note that both the cycle classes and the Hodge classes are determined by Chow motives of the corresponding varieties.

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    $\begingroup$ I think that this should be the answer. $\endgroup$ – Donu Arapura Jan 30 '17 at 15:04
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This is more of a very long comment, but I believe so.

A modern description of the motivic homotopy category is $sPre(Sm/k)[N^{-1}][L^{-1}]$ where $N^{-1}$ is localizing with respect to the topology and $L^{-1}$ is localizing with respect to all the maps $X \times \mathbb{A}^1 \rightarrow X$. Chow groups of course satisfy Nisnevich descent, and are $\mathbb{A}^1$ invariant, and so they give a motivic sheaf. The question is then if the Hodge classes can be extended to a motivic sheaf. Since the definition of Hodge classes is only for smooth and projective $X$, it's not totally obvious what to use for a general smooth variety.

Here we should use the fact that Betti realization $X \mapsto X^{an}$ is a motivic sheaf of spaces, and that both the weight-filtered , I'll write $\hat{\mathcal{W}_k}A_{\log}$, and also $\mathcal{F}^p \cap \hat{\mathcal{W}_k}A_{\log}$ the weight-and-Hodge-filtered de Rham complexes are perfectly good motivic sheaves of cochain complexes (up to q.iso). Here I'm using the notation of Bunke and Tamme from "Regulators and cycle maps...", where they define them carefully. Bunke and Tamme prove Zariski descent for the latter two in Lemma 3.6, but their method actually shows etale descent. $\mathbb{A}^1$-invariance can be checked on the level of cohomology. We can just take the set-theoretic conjugate to get a sheaf $\overline{\mathcal{F}^p} \cap \hat{\mathcal{W}_k}A_{\log}$. To just work with sheaves of cochain complexes (up to q.iso), let's look at the singular cochains functor instead of the Betti realization functor. That is, let's look at $X \mapsto Sing_\mathbb{Q}^\bullet(X)$. We have a diagram

$\require{AMScd}$ \begin{CD} \mathcal{F}^p \cap \hat{\mathcal{W}_{2p}}A_{\log} @>>> A_{\log} @<<< \overline{\mathcal{F}^p} \cap \hat{\mathcal{W}_{2p}}A_{\log} \\ @. @AAA \\ @. Sing_\mathbb{Q}^\bullet(X) \end{CD}

Whose homotopy limit sheaf should output the $p$-Hodge classes for input a smooth projective $X$.

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    $\begingroup$ To address your question in paragraph 2, if $X$ is any complex variety take the space of Hodge cycles to mean $Hom(\mathbb{Q}(-p), H^{2p}(X,\mathbb{Q})$ in the category of mixed Hodge structures. $\endgroup$ – Donu Arapura Jan 28 '17 at 14:33
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I was initially reluctant to contribute an answer because I'm not an expert on $\mathbb{A}^1$ homotopy theory. But let me flesh out my first comment a little, and you can decide whether this is close to what you wanted. Let $K_0(Var)$ be the Grothendieck ring of all complex algebraic varieties. Let $\mathbb{L}$ denote the class of the affine line. The quotient $K_0(Var)/(\mathbb{L}-1)$ might be thought of as the Grothendieck ring of naive $\mathbb{A}^1$-homotopy classes. This maps to $K_0(Var)[\mathbb{L}^{-1}]$ and the Hodge conjecture "factors through" the second group by http://arxiv.org/abs/math/0506210

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