Some implications between Grothendieck's anabelian conjectures

TL;DR

This paper explores the relationships between Grothendieck's main anabelian conjectures, introduces a stronger conjecture form, and proves their validity for DM stacks over complex fields, linking orbicurves to hyperbolic curves.

Contribution

It proposes a stronger form of Grothendieck's conjecture, proves their equivalence in certain cases, and extends the conjectures' validity to DM stacks over complex fields.

Findings

Grothendieck's conjectures hold for the stronger form proposed.

The validity of the conjecture depends only on the complex points of the DM stack.

The section conjecture for hyperbolic orbicurves follows from that for hyperbolic curves.

Abstract

Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If $X$ is a DM stack over a field $k\subseteq\mathbb{C}$ finitely generated over $\mathbb{Q}$, we prove that whether $X$ satisfies the conjecture or not depends only on $X_{\mathbb{C}}$. We prove that the section conjecture for hyperbolic orbicurves stated by Borne and Emsalem follows from the conjecture for hyperbolic curves.