Hodge--Tate crystals on the logarithmic prismatic sites of semi-stable formal schemes

TL;DR

This paper explores the relationship between Hodge--Tate crystals on logarithmic prismatic sites of semi-stable formal schemes and enhanced log Higgs bundles, establishing an equivalence and an inverse Simpson functor.

Contribution

It introduces an equivalence between rational Hodge--Tate crystals and enhanced log Higgs bundles on semi-stable formal schemes.

Findings

Established an equivalence between Hodge--Tate crystals and log Higgs bundles.

Constructed an inverse Simpson functor from log Higgs bundles to generalized representations.

Provided new insights into the structure of prismatic cohomology in the logarithmic setting.

Abstract

Let $\calO_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with a perfect residue field. In this paper, for a semi-stable $p$-adic formal scheme $\frakX$ over $\calO_K$ with rigid generic fibre $X$ and canonical log structure $\calM_{\frakX} = \calO_{\frakX}\cap\calO_X^{\times}$, we study Hodge--Tate crystals over the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$. As an application, we give an equivalence between the category of rational Hodge--Tate crystals on the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$ and the category of enhanced log Higgs bundles over $\frakX$, which leads to an inverse Simpson functor from the latter to the category of generalised representations on $X_{\proet}$.