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

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  22 December 2015

Kazuhiko Yamaki
Kazuhiko Yamaki*
Affiliation:
Institute for Liberal Arts and Sciences, Kyoto University, Kyoto 606-8501, Japan email yamaki.kazuhiko.6r@kyoto-u.ac.jp
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Abstract

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The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvariety. 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 fields, 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 and Zhang with respect to the canonical measures. In this paper we exhibit the limits 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.

Keywords

geometric Bogomolov conjecturenon-archimedean geometrycanonical measures

MSC classification

Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Secondary: 11G50: Heights 14G22: Rigid analytic geometry
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 997 - 1040
Copyright
© The Author 2015 

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