Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: The minimal dimension of a canonical measure)

TL;DR

This paper formulates the geometric Bogomolov conjecture for abelian varieties, providing partial results under degeneracy conditions and analyzing canonical measures via tropical geometry to advance understanding of small points distribution.

Contribution

It introduces a partial proof of the conjecture under degeneracy conditions and relates the minimal dimension of canonical measure components to the abelian part's dimension.

Findings

Non-special subvarieties lack dense small points under degeneracy conditions

The minimal dimension of canonical measure components equals the abelian part's dimension

Application of tropical equidistribution theory to study small points

Abstract

In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety does not have a dense subset of small points if it is a non-special subvariety. The key of the proof is the study of the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety. Then we can apply the tropical version of equidistribution theory due to Gubler. This article includes an appendix by Walter Gubler. He shows that the minimal dimension of the components of a canonical measure is equal to the dimension of the abelian part of the subvariety. We can apply this result to make a further contribution to the geometric Bogomolov conjecture.