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

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.

Reference [28] is added. To appear in Compositio Mathematica. Different from the published version

References in corpus (3)

• The Bogomolov conjecture for totally degenerate abelian varieties
• Tropical varieties for non-archimedean analytic spaces
• 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)

Cited by in corpus (6)

• Trace of abelian varieties over function fields and the geometric Bogomolov conjecture
• Geometric Bogomolov conjecture in arbitrary characteristics
• Effective faithful tropicalizations associated to linear systems on curves
• Non-density of small points on divisors on abelian varieties and the Bogomolov conjecture
• Survey on the geometric Bogomolov conjecture
• Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension $5$ with trivial trace