Local constancy of pro-unipotent Kummer maps

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 $\ell \ne p$. 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 étale-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.

arXiv comment31 pages, comments welcome