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Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the field.

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    $\begingroup$ Variety over $\mathbb C$? Over $\mathbb Q$ (or even $\mathbb Q_p$) it certainly does... $\endgroup$
    – Will Sawin
    Commented Jul 6, 2020 at 1:18
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    $\begingroup$ OK. Over some fields (like any number field or $p$-adic field) one can check that the Galois action on the etale cohomology determines the Hodge numbers using $p$-adic Hodge theory. Since this is determined by the etale topos (considered as a topos over the topos of that field), this gives a positive answer in that case. (Maybe you can even reconstruct this Galois group from the abstract topos.) $\endgroup$
    – Will Sawin
    Commented Jul 6, 2020 at 1:25
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    $\begingroup$ Well, one wants to construct $\mathbb Z/p^n$ for all $n$. Isn't this the colimit of $p^n$ copies of the final object? $\endgroup$
    – Will Sawin
    Commented Jul 6, 2020 at 1:31
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    $\begingroup$ I think this is false in positive characteristic, because of (inseparably) unirational varieties with interesting Hodge numbers (e.g. supersingular K3 surfaces or Fermat surfaces). From these, you should be able to construct a radicial map of smooth projective surfaces where one is rational and the other has nonzero $h^{0,2}$. But topological invariance of the étale site gives an equivalence between the étale topoi. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jul 6, 2020 at 2:02
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    $\begingroup$ Here is another possible construction. Consider a smooth non isotrivial fibration of curves $C\to B$ in positive characteristic, where $B$ is a curve (eg a Kodaira-Parshin fibration). Now let $k>0$ and consider the twisted fibration $C^{p^{(k)}}\to B$ (ie the base change of $C$ by the $k$-th power of the absolute Frobenius on $B$). Then the Hodge numbers of $C^{p^{(k)}}$ (viewed as a surface) can be computed and presumably they will differ from those of $C$. On the other hand $C$ and $C^{p^{(k)}}$ are related by the relative Frobenius morphism, which is radicial. $\endgroup$
    – Damian Rössler
    Commented Jul 6, 2020 at 10:44

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No. See Proposition 2.14 in Canonical models of surfaces of general type in positive characteristic

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