Anabelian geometry with etale homotopy types

TL;DR

This paper extends classical anabelian geometry to include etale homotopy types, demonstrating that points on certain varieties have fundamental systems of anabelian neighborhoods, confirming a prediction by Grothendieck.

Contribution

It generalizes anabelian geometry to etale homotopy types and proves the existence of fundamental systems of anabelian neighborhoods for points on varieties over finitely generated fields.

Findings

Points on smooth varieties over finitely generated fields have fundamental systems of anabelian neighborhoods.

The generalization aligns with Grothendieck's predictions.

The work bridges classical and modern anabelian geometry.

Abstract

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