Anabelian geometry with étale homotopy types

Abstract

Anabelian geometry with étale homotopy types generalizes in a natural way classical anabelian geometry with étale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field $k$ that is finitely generated over $\mathbb{Q}$ has a fundamental system of (affine) anabelian Zariski-neighborhoods. This was predicted by Grothendieck in his letter to Faltings.