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Let $k$ be an algebraically closed field of characteristic 0.

I am wondering if there are proofs of the following facts that do not use the analytic topology over $\mathbb{C}$:

  1. Let $X$ be a smooth projective variety over $k$. Then $h^{p,q}(X)=h^{q,p}(X)$ for all $p,q$.
  2. Let $f:X\to Y$ be a smooth proper morphism of varieties over $k$. Then $h^{p,q}(X_y)$ is locally constant on $Y$.

Unlike other facts about Hodge numbers, these are only true over fields of characteristic 0.

More generally, I was wondering if there is a purely algebraic way (i.e. avoiding the analytic topology) of "doing Hodge Theory."

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    $\begingroup$ Yes, there is: Deligne-Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham. Invent. Math. 89 (1987), no. 2, 247–270. $\endgroup$
    – abx
    Commented Dec 13, 2022 at 5:52
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    $\begingroup$ For (1) see Jean-Marc Fontaine and William Messing "p-adic periods and p-adic etale cohomology", section I.4.4. $\endgroup$
    – Piotr Achinger
    Commented Dec 13, 2022 at 7:26
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    $\begingroup$ I agree with the comments above. But I don't know of an algebraic proof of the splitting of the Hodge filtration. For this you seem to need analysis (for now). And that splitting is important for many things that fall under the general umbrella of "doing Hodge theory". $\endgroup$
    – jmc
    Commented Dec 13, 2022 at 7:32
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    $\begingroup$ @jmc what do you mean by "proof of splitting"? Many abstract splittings exist, but the one from Hodge theory cannot come from algebraic geometry (the conjugate filtration is anti-holomorphic). In p-adic Hodge theory however, there is such a splitting (the Hodge-Tate decomposition), though again it does not exist in families. $\endgroup$
    – Piotr Achinger
    Commented Dec 13, 2022 at 7:35
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    $\begingroup$ @PiotrAchinger why not compile your comments into an answer? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Dec 13, 2022 at 20:46

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