Crystalline comparison for rigid-analytic varieties

Question

Let $k$ be a finite extension of $\mathbb{Q}_p$. In this paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says:

...it should be possible to deduce (log-)crystalline comparison theorems...

1. What should be the correct statements of the more refined Hodge theoretic comparison theorems in the context of rigid-analytic varieties?
2. Are there any mathematical difficulties with proving these statements, or did Scholze just not feel like proving them?
3. Is there any subsequent work aiming to prove these statements?

EDIT: in general, I have the following question (which may be trivial, and I only have it become of my ignorance). What is "the reduction $\mathrm{mod} \, p$" of a non-algebraic smooth proper rigid-analytic variety over $\mathbb{Q}_p$? Does it see the non-algebraicity?

Answer 1

The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.

Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the \'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.

One reference is the paper by Tan-Tong (https://arxiv.org/abs/1510.05543), settling the above result over unramified bases. They follow the Faltings-Scholze's method, build on previous work of Andreatta-Iovita (https://arxiv.org/abs/1212.3813), and also prove a result for non-trivial coefficient systems.

A reference for the general statement is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.

The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.