A log prismatic-crystalline comparison theorem

TL;DR

This paper establishes a comparison theorem between log prismatic and log crystalline cohomology for semistable $p$-adic formal schemes, leading to progress on the $C_{st}$-conjecture with semistable local system coefficients.

Contribution

It proves a new comparison theorem linking log prismatic and log crystalline cohomology in the semistable case, advancing the understanding of $p$-adic Hodge theory.

Findings

Proves the log prismatic-crystalline comparison theorem for semistable schemes.

Combines with Tian's prismatic-étale comparison to verify the $C_{st}$-conjecture.

Extends the scope of prismatic cohomology applications in $p$-adic Hodge theory.

Abstract

We show a comparison theorem between log prismatic cohomology and log crystalline cohomology for a $p$-adic formal scheme with semistable reduction. Combined with the prismatic-\'etale comparison theorem recently proved by Tian, this implies the $C_{\mathrm{st}}$-conjecture in the semistable case with coefficients given by semistable local systems.