Local constancy of pro‐unipotent Kummer maps

It is a theorem of Kim–Tamagawa that the Qℓ${\mathbb {Q}}_\ell$ ‐pro‐unipotent Kummer map associated to a smooth projective curve Y$Y$ over a finite extension of Qp${\mathbb {Q}}_p$ is locally constant when ℓ≠p$\ell \ne p$ . This paper establishes two generalisations of this result. First, we extend the Kim–Tamagawa theorem to the case that Y$Y$ is a smooth variety of any dimension. Second, we formulate and prove the analogue of the Kim–Tamagawa theorem in the case ℓ=p$\ell =p$ , 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.