Local constancy of pro-unipotent Kummer maps

TL;DR

This paper generalizes Kim-Tamagawa's theorem on the local constancy of pro-unipotent Kummer maps from curves to higher-dimensional varieties and extends it to the case where helle equals p, using advanced comparison theorems.

Contribution

It extends the Kim-Tamagawa theorem to higher-dimensional varieties and to the case helle=p, including a new proof of an étale-de Rham comparison theorem for pro-unipotent fundamental groupoids.

Findings

Kim-Tamagawa theorem extended to higher dimensions

Established helle=p case for arbitrary dimension

Proved étale-de Rham comparison theorem for pro-unipotent groupoids

Abstract

It is a theorem of Kim-Tamagawa that the $\mathbb Q_\ell$-pro-unipotent Kummer map associated to a smooth projective curve $Y$ over a finite extension of $\mathbb Q_p$ is locally constant when $\ell\neq p$. The present paper establishes two generalisations of this result. Firstly, we extend the Kim-Tamagawa Theorem to the case that $Y$ is a smooth variety of any dimension. Secondly, we formulate and prove the analogue of the Kim-Tamagawa Theorem in the case $\ell = p$, again in arbitrary dimension. In the course of proving the latter, we give a proof of an \'etale-de Rham comparison theorem for pro-unipotent fundamental groupoids using methods of Scholze and Diao-Lan-Liu-Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.