Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T01:18:40.206Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  22 December 2015

Kazuhiko Yamaki*
Affiliation:
Institute for Liberal Arts and Sciences, Kyoto University, Kyoto 606-8501, Japan email [email protected]

Abstract

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.

Type
Research Article
Copyright
© The Author 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berkovich, V. G., Spectral theory and analytic geometry over nonarchimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematial Society, Providence, RI, 1990).Google Scholar
Berkovich, V. G., Étale cohomology for non-archimedean analytic spaces, Publ. Math. Inst. Hautes Études Sci. 78 (1993), 5161.Google Scholar
Berkovich, V. G., Vanishing cycles for formal schemes, Invent. Math. 115 (1994), 539571.CrossRefGoogle Scholar
Berkovich, V. G., Smooth p-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), 184.CrossRefGoogle Scholar
Bosch, S., Rigid analytische Gruppen mit guter Reduktion, Math. Ann. 233 (1976), 193205.Google Scholar
Bosch, S. and Lütkebohmert, W., Néron models from the rigid analytic viewpoint, J. Reine Angew. Math. 364 (1986), 6984.Google Scholar
Bosch, S. and Lütkebohmert, W., Degenerating abelian varieties, Topology 30 (1991), 653698.Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21 (Springer, Berlin, 1990).Google Scholar
Chambert-Loir, A., Mesure et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215235.Google Scholar
Cinkir, Z., Zhang’s conjecture and the effective Bogomolov conjecture over function fields, Invent. Math. 183 (2011), 517562.Google Scholar
de Jong, A. J., Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 5193.CrossRefGoogle Scholar
Faber, X. W. C., The geometric Bogomolov conjecture for curves of small genus, Experiment. Math. 18 (2009), 347367.CrossRefGoogle Scholar
Gubler, W., Local and canonical height of subvarieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), 711760.Google Scholar
Gubler, W., Tropical varieties for non-archimedean analytic spaces, Invent. Math. 169 (2007), 321376.Google Scholar
Gubler, W., The Bogomolov conjecture for totally degenerate abelian varieties, Invent. Math. 169 (2007), 377400.CrossRefGoogle Scholar
Gubler, W., Equidistribution over function fields, Manuscripta Math. 127 (2008), 485510.Google Scholar
Gubler, W., Non-archimedean canonical measures on abelian varieties, Compositio Math. 146 (2010), 683730.Google Scholar
Lang, S., Abelian varieties (Springer, New York, 1983).Google Scholar
Lang, S., Fundamentals of Diophantine geometry (Springer, New York, 1983).CrossRefGoogle Scholar
Milne, J., Abelian varieties, in Arithmetic geometry (Storrs, Conn., 1984) (Springer, New York, 1986), 103150.Google Scholar
Moriwaki, A., Bogomolov conjecture for curves of genus 2 over function fields, J. Math. Kyoto Univ. 36 (1996), 687695.Google Scholar
Moriwaki, A., Bogomolov conjecture over function fields for stable curves with only irreducible fibers, Compositio Math. 105 (1997), 125140.Google Scholar
Moriwaki, A., Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc. 11 (1998), 569600.CrossRefGoogle Scholar
Moriwaki, A., Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), 101142.Google Scholar
Mumford, D., Abelian varieties, TIFR Studies in Mathematics, vol. 5 (Hindustan Book Agency, New Delhi, 2008); with appendices by C. P. Ramanujam and Yuri Manin; corrected reprint of the second (1974) edition.Google Scholar
Scanlon, T., A positive characteristic Manin–Mumford theorem, Compositio Math. 141 (2005), 13511364.CrossRefGoogle Scholar
Silverman, J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106 (Springer, New York, 1986).Google Scholar
Szpiro, L., Ullmo, E. and Zhang, S., Equirépartition des petits points, Invent. Math. 127 (1997), 337347.Google Scholar
Ullmo, E., Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), 167179.CrossRefGoogle Scholar
Yamaki, K., Geometric Bogomolov’s conjecture for curves of genus 3 over function fields, J. Math. Kyoto Univ. 42 (2002), 5781.Google Scholar
Yamaki, K., Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields, J. Math. Kyoto. Univ. 48 (2008), 401443.Google Scholar
Yamaki, K., Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration, Manuscripta Math. 142 (2013), 273306; with an appendix by Walter Gubler, The minimal dimension of a canonical measure.Google Scholar
Zhang, S., Admissible pairing on a curve, Invent. Math. 112 (1993), 171193.Google Scholar
Zhang, S., Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), 159165.Google Scholar