Uniformity in the Bogomolov conjecture

Abstract: The Bogomolov Conjecture, proved by Ullmo and Zhang, classifies the torsion subvarieties of an abelian variety as the subvarieties of small Arakelov height.
This raises the question whether an explicit positive lower bound can be given for the heights of non-torsion subvarieties. This question has been addressed by David and Philippon. More recently, Kuehne and Yuan showed independently the existence of non-explicit, but uniform bounds using techniques from Arakelov theory. Even more, Yuan proved lower bounds that are actually better than uniform. Before this, similar bounds have been only obtained in the case of self-products of an elliptic curve through related work of David and Philippon.

After giving an overview of the results mentioned, I will present the ideas used in the proof of the latter result. I will conclude by sketching how to improve upon this proof by replacing the construction of an auxiliary function with a motivated use of techniques of Arakelov Theory. This allows to improve the uniform bounds given by David and Philippon.

Seminarraum 05.002, Spiegelgasse 5