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Let $X$ be a smooth projective variety over $\mathbb{Z}_p$, with $X_0$ its reduction mod $p$. Assume that $X_0$ is ordinary.

For any lift $X'$ over $\mathbb{Z}_p$ of $X_0$, there is a canonical isomorphism $$H^i_\text{cris}(X_0 / \mathbb{Z}_p) \cong H^i_\text{dR}(X'/\mathbb{Z}_p)$$

The left side of this isomorphism has a slope filtration, while the right side has a Hodge filtration, but I don't think this isomorphism is filtered in general.

Is there an example of $X_0$ as above such that no $X'$ can be chosen to make the isomorphism above filtered?

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    $\begingroup$ What if $X_0$ is a supersingular elliptic curve? $\endgroup$ – R. van Dobben de Bruyn Aug 8 at 2:30
  • $\begingroup$ @R.vanDobbendeBruyn I have amended the question. $\endgroup$ – rj7k8 Aug 8 at 3:04
  • $\begingroup$ Just a comment, but the general result is that the Hodge Filtration is opposed to the filtration by increasing slopes (a result of Hyodo, explained by Illusie in the Bourbaki seminar). As the usual slope filtration (or filtration by decreasing slopes) is also opposed to the filtration by increasing slopes, both are opposed to the same filtration. They are usually distinct (but I don't know the answer to your question). $\endgroup$ – Xarles Aug 8 at 13:15

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