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Does there exist a simply connected smooth proper complex variety that is not rationally homotopy equivalent to a simply connected smooth proper complex variety defined over $\mathbb{Q}$?

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    $\begingroup$ Certainly, any such variety should be homotopic to one over $\overline{\mathbb{Q}}$. Concerning your actual question, my guess is no. Take a look at the variety constructed by Serre in "Examples de varieties projective conjuguees non homemorphes". It seems that it wouldn't descend to $\mathbb{Q}$. $\endgroup$
    – Donu Arapura
    Commented Jul 8, 2020 at 14:33
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    $\begingroup$ I think fake projective planes should be a counterexample to this, since they are characterized by their homotopy type and can't be defined over any real field. $\endgroup$
    – Will Sawin
    Commented Jul 8, 2020 at 15:56
  • $\begingroup$ @WillSawin you are right, I got confused $\endgroup$
    – user145520
    Commented Jul 8, 2020 at 16:15

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