Strict supports of canonical measures and applications to the geometric Bogomolov conjecture

TL;DR

This paper advances the understanding of the geometric Bogomolov conjecture by analyzing canonical measures and supports, showing the conjecture holds for abelian varieties with anywhere good reduction, thus extending previous partial results.

Contribution

It demonstrates that the geometric Bogomolov conjecture is true for all abelian varieties with anywhere good reduction, building on the analysis of canonical measures and their supports.

Findings

The conjecture holds for abelian varieties with anywhere good reduction.

Support of canonical measures is crucial in proving the conjecture.

Partial results are extended to full generality under certain conditions.

Abstract

The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvarieties. In the arithmetic setting over number fields, the Bogomolov conjecture for abelian varieties has already been established as a theorem of Ullmo and Zhang, but in the geometric setting over function field, it has not yet been solved completely. There are only some partial results known such as the totally degenerate case due to Gubler and our recent work generalizing Gubler's result. The key in establishing the previous results on the Bogomolov conjecture is the equidistribution method due to Szpiro--Ullmo--Zhang with respect to the canonical measures. In this paper, we exhibit the limit of this method, making an important contribution to the geometric version of the conjecture. In fact, by the crucial investigation of the support of the canonical measure on a subvariety, we show that the conjecture in full generality holds if the conjecture holds for abelian varieties which have anywhere good reduction. As a consequence, we establish a partial answer that generalizes our previous result.