A sharp Bogomolov-type bound

TL;DR

This paper establishes a precise lower bound for the essential minimum of certain subvarieties in abelian varieties, advancing the understanding of their arithmetic complexity and extending previous results.

Contribution

It generalizes Galateau's result to a broader class of varieties, contributing to the abelian analogue of a conjecture related to toric varieties and applications in anomalous intersections.

Findings

Proves a sharp lower bound for the essential minimum.

Extends previous results of Galateau to new settings.

Provides tools for further research in arithmetic geometry.

Abstract

We prove a sharp lower bound for the essential minimum of a non-translate variety in certain abelian varieties. This uses and generalises a result of Galateau. Our bound is a new step in direction of an abelian analogue by David and Philippon of a toric conjecture of Amoroso and David and has applications in the framework of anomalous intersections.