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By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?

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    $\begingroup$ Search for "exceptional Enriques surfaces". $\endgroup$
    – Jason Starr
    Oct 17, 2020 at 16:29
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    $\begingroup$ Deligne-Illusie do not claim that; in fact they cite Mumford's paper "Pathologies of modular surfaces" for a counterexample. What they say they don't know is whether the assumption $p>\dim X$ in their theorem is necessary; that is, whether there exists a variety liftable to $W_2(k)$, necessarily of dimension $\leq p$, for which the Hodge-de Rham spectral sequence does not degenerate. This is an open problem, to the best of my knowledge. $\endgroup$
    – Piotr Achinger
    Oct 17, 2020 at 17:31
  • $\begingroup$ @PiotrAchinger You are right. I misread that paragraph. I assume you mean $\dim > p$, in any case. $\endgroup$
    – Kim
    Oct 17, 2020 at 18:51

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