Title

Some implications between Grothendieck's anabelian conjectures

Speaker

Dr. Giulio Bresciani

Abstract

In his famous letter to Faltings, Grothendieck explained a series of ideas and
conjectures collected under the name of anabelian geometry. Grothendieck's
picture describes how for certain varieties, called anabelian, defined over
fields finitely generated over Q the étale fundamental group recovers all the
geometric information about the variety. In dimension one anabelian varieties
are curves with negative Euler characteristic, but in higher dimension the
picture is not so clear. Some of the anabelian conjectures have been proved,
most notably by Mochizuki. Among the anabelian conjectures, the so called
section conjecture remains largely open. We show how the section conjecture
implies a much stronger anabelian statement, from which every other anabelian
statement follows immediately. We do this in the generality of Deligne-Mumford
stacks, rather than varieties, and show that if the section conjecture holds for
curves then it holds for some other classes of DM stacks.