# Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?

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?

• Search for "exceptional Enriques surfaces". – Jason Starr 2 days ago
• 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. – Piotr Achinger 2 days ago
• @PiotrAchinger You are right. I misread that paragraph. I assume you mean $\dim > p$, in any case. – Kim 2 days ago