Some remarks on Hodge symmetry

I make some remarks on Hodge symmetry, and prove for instance that if $k$ is a perfect field of characteristic $p>0$ and $X/k$ smooth, proper and Hodge-Witt scheme, and Hodge de Rham sequence of $X$ degenerates at $E_1$ and $X$ has torsion-free crystalline cohomology, then Hodge symmetry holds for $X$.

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