Let $X$ be a smooth projective variety over a field $k$. Let $h^{p,q}=dim\_k H^q(X,\Omega\_{X/k}^p)$ be the Hodge numbers.


If $k$ is of char $0$, by Lefschetz principle, we always have Hodge symmetry, i.e. $h^{p,q}=h^{q,p}$, for the following reason: we can regard $X$ as a base change of a smooth projective variety $X\_0$ over a finitely generated field $k\_0$, but fix an arbitrary embedding $k\_0\rightarrow \mathbb{C}$, by base change to $\mathbb{C}$, the flat base change theorem and Hodge symmetry for complex smooth projective variety will give us the reason.


Do we always have Hodge symmetry for char $p$ ? If we do, how do we prove it?


My motivation might be very trivial for experts...I was computing the Hodge numbers for abelian varieties. Let X be an abelian variety over $k$ of dim $g$, then I want to compute all the hodge numbers (for any character). I crucially need Hodge symmetry:


The relative differential is trivial: $\Omega\_{X/k}=\mathcal{O}\_X^{\oplus g}$. If we have Hodge symmetry, then the Hodge numbers $h^{p,q}=dim H^q(X,\Omega^p)=dim H^q(X,\mathcal{O}\_X^{\oplus \binom {g}{p}})= \binom{g}{p} dim H^q(X, \mathcal{O}\_X)=\binom{g}{p} h^{q,0}=\binom{g}{p}h^{0,q}=\binom{g}{p}\binom{g}{q} h^{0,0}=\binom{g}{p}\binom{g}{q}$.


Is there any proof that doesn't depend on Hodge symmetry? Any references are welcome!



Do we have Hodge symmetry for char $p$?
Hodge symmetry fails in positive characteristic in general; see Serre's Mexico paper [Ser58, Prop. 16] (for a more modern/conceptual version of that argument, see e.g. [vDdB21, Prop. 1.4]).


However, it is true for abelian varieties; see for example [Mum08, §13, Cor. 2]. As you note, this boils down to the computation that
$$h^i(A,\mathcal O\_A) = {g \choose i}.$$




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**References.**


[vDdB21] R. van Dobben de Bruyn, *The Hodge ring of varieties in positive characteristic*. Algebra Number Theory **15**.3, p. 729–745 (2021).


[Mum08] D. Mumford, *Abelian varieties*. Second edition, corrected reprint. With appendices by C. P. Ramanujam and Yuri Manin. New Delhi: Hindustan Book Agency/distrib. by American Mathematical Society (AMS); Bombay: Tata Institute of Fundamental Research (ISBN 978-81-85931-86-9), 2008. [ZBL1177.14001](https://zbmath.org/?q=an:1177.14001).


[Ser58] J.-P. Serre, *Sur la topologie des variétés algébriques en caractéristique $p$*. Sympos. internac. Topología algebraica 24–53 (1958). [ZBL0098.13103](https://zbmath.org/?q=an:0098.13103).


