Elementary anabelian varieties are anabelian

TL;DR

This paper proves that isomorphisms of fundamental groups of elementary anabelian varieties over certain fields correspond exactly to isomorphisms of the varieties themselves, confirming Grothendieck's conjectures.

Contribution

It establishes a bijective correspondence between fundamental group isomorphisms and variety isomorphisms for elementary anabelian varieties, verifying longstanding conjectures.

Findings

Isomorphisms of fundamental groups correspond to isomorphisms of varieties.

Dominant maps are characterized by cohomological injectivity conditions.

Results extend to étale homotopical generalizations.

Abstract

We show that isomorphisms of fundamental groups of elementary anabelian varieties -- varieties obtained as iterated fibrations of hyperbolic curves -- over sub-$p$-adic fields correspond bijectively to isomorphisms of varieties. Moreover, dominant maps between proper elementary anabelian varieties are in bijection with ``stably cohomologically injective'' maps of fundamental groups: open maps whose pullbacks to all open subgroups induce injections on cohomology rings with $\ell$-adic coefficients, for any prime $\ell$. This verifies conjectures of Grothendieck from his letter to Faltings. Finally, we establish \'etale homotopical generalizations of these results.