A Conjecture of Bhatt--Lurie and weakly $p$-nilpotent Hodge--Tate stacks

TL;DR

This paper constructs a new weakly p-nilpotent Hodge--Tate stack for smooth varieties over perfect fields of characteristic p, linking Hodge--Tate crystals with p-nilpotent Higgs bundles and addressing a conjecture of Bhatt and Lurie.

Contribution

It introduces the weakly p-nilpotent Hodge--Tate stack, establishing its structure as a gerbe and connecting its obstruction class to Frobenius liftings.

Findings

The weakly p-nilpotent Hodge--Tate stack is a gerbe banded by T_{X/k}⊗α_p.

The obstruction class of the stack matches the obstruction to Frobenius lifting.

The stack provides a local description of certain Hodge--Tate crystals via weakly p-nilpotent Higgs bundles.

Abstract

Let $k$ be a perfect field of characteristic $p$, and let $X/k$ be a smooth variety. It is known that given a Frobenius lifting of $X$, we can identify prismatic crystals and nilpotent Higgs bundles, known as a positive characteristic version of the Simpson correspondence of $X$. However, Ogus--Vologodsky point out in their original paper of non-abelian Hodge theory in characteristic $p$ that, if we are just given a smooth lifting over $\W_2(k)$, there is a non-abelian Hodge theory on $p$-nilpotent Higgs bundles. Hence, it is natural to ask that whether there exists a subcategory of Hodge--Tate crystals on $X$, which can be described as $p$-nilpotent Higgs bundles. In this paper, we construct an analogue of the Hodge--Tate stack, so called the weakly $p$-nilpotent Hodge--Tate stack, on which the vector bundles are identified with certain Hodge--Tate crystals on $X$ that can be locally described by weakly $p$-nilpotent Higgs bundles. Furthermore, we prove that the weakly $p$-nilpotent Hodge--Tate stack is indeed a gerbe banded by $T_{X/k}\otimes{\alpha_p}$, and the obstruction class coincides with the obstruction of the existence of a Frobenius lifting of $X$, which is a conjecture of Bhatt and Lurie.