Anabelian geometry with étale homotopy types

ALEXANDER SCHMIDT AND JAKOB STIX

Anabelian geometry with etale homotopy types
generalizes in a natural way classical anabelian geometry with etalefundamental 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.

https://arxiv.org/abs/1504.01068