Integral p-adic Hodge filtrations in low dimension and ramification

TL;DR

This paper explores how the behavior of the integral Hodge--de Rham spectral sequence relates the special fiber to the generic fiber's Hodge numbers in low-dimensional ramified p-adic varieties, using recent p-adic Hodge theory advancements.

Contribution

It establishes conditions under which the spectral sequence behaves well, linking the special fiber's Hodge numbers to the generic fiber in ramified mixed characteristic settings.

Findings

Spectral sequence behavior determines Hodge numbers in special fibers.

Lifting to second Witt vectors ensures nice spectral sequence behavior.

An example shows the necessity of the lifting condition.

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 and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of the aforementioned lifting condition, which is of independent interest.