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.

Non-density of small points on divisors on Abelian varieties and the Bogomolov conjecture

“…This completes the proof. ✷ Let A be an abelian variety K. By [30,Lemma 7.9], there exists a unique abelian subvariety of A that is maximal among the nowhere degenerate abelian subvarieties. This abelian subvariety is called the maximal nowhere degenerate abelian subvariety of A ([30, Definition 7.10]).…”

“…Since A is nowhere degenerate, A 1 is nowhere degenerate and has trivial K/k-trace (cf. [30,Lemma 7.8 (2)] and [31,Remark 5.4]). Then by the Poincaré complete reducibility theorem, A is isogenous to t × A 1 .…”

Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension $5$ with trivial trace

“…In [1], Gubler has proved the conjecture holds under the assumption that A is totally degenerate at some place. In [7], we have generalized Gubler's work and have shown that the conjecture for A is reduced to the conjecture for its maximal nowhere degenerate abelian subvariety m, where "being nowhere degenerate" means "having everywhere good reduction" and "maximal" means "maximal with respect to the inclusion" (cf. [7,Definition 7.10]).…”

“…In [7], we have generalized Gubler's work and have shown that the conjecture for A is reduced to the conjecture for its maximal nowhere degenerate abelian subvariety m, where "being nowhere degenerate" means "having everywhere good reduction" and "maximal" means "maximal with respect to the inclusion" (cf. [7,Definition 7.10]). It is known that there exists such an m uniquely.…”

“…The case of dim(X) = 5 is trivial, and the case of dim(X) = 0 is classically known (cf. [7,Remark 7.4]). Further by Lemma 2.6, we have dim(X) = 1 and dim(X) = 4.…”