Anabelian presentation of the motivic Galois group in characteristic zero

Abstract

Let k be a field endowed with a complex embedding, and let

be the associated motivic Galois group. The goal of this paper is to show that

G mot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot

admits a natural manifestation in anabelian geometry. Roughly speaking, we prove that

G mot $\mathcal {G}_{\mathrm {mot}}$ script upper G Subscript mot

coincides with the automorphism group of the functor

X ↦ π 1 geo ( X ) $X\mapsto \pi _1^{\mathrm {geo}}(X)$ upper X right arrow from bar pi 1 Superscript geo Baseline left parenthesis upper X right parenthesis

sending a k -variety X to the geometric completion of its topological fundamental group. This can be considered as a motivic version of the Ihara–Matsumoto–Oda Conjecture, greatly generalised and proven by Pop. In a sequel to this paper, we will develop the parallel story for the

ℓ $\ell $ script l

-adic realisation. For this reason, many of the intermediate results are developed at a greater level of generality than strictly necessary for the main results proven here.