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In this paper, we formulate the geometric Bogomolov conjecture for abelianvarieties, and give some partial answers to it. In fact, we insist in a maintheorem that under some degeneracy condition, a closed subvariety of an abelianvariety does not have a dense subset of small points if it is a non-specialsubvariety. The key of the proof is the study of the minimal dimension of thecomponents of a canonical measure on the tropicalization of the closedsubvariety. Then we can apply the tropical version of equidistribution theorydue to Gubler. This article includes an appendix by Walter Gubler. He showsthat the minimal dimension of the components of a canonical measure is equal tothe dimension of the abelian part of the subvariety. We can apply this resultto make a further contribution to the geometric Bogomolov conjecture.

manuscripta mathematica

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)