Anabelian geometry with étale homotopy
types

ALEXANDER SCHMIDT AND
JAKOB STIX

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 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/pdf/1504.01068.pdf