Integral $p$-adic Hodge filtrations in low dimension and ramification

Abstract

Given an integral $p$-adic variety, we observe that if the integral Hodge–de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral $p$-adic Hodge theory, we show that such a nice behavior is guaranteed if the $p$-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne–Illusie, Fontaine–Messing, and Kato. Lastly, we discuss an example illustrating the necessity of the aforementioned lifting condition, which is of independent interest.