Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
 With cedram.org version française
 Home 1975-2014 archives All online articles Advanced Search Latest articles Search for an article Table of contents for this volume | Previous article Kazuhiko YamakiSurvey on the geometric Bogomolov conjecturePublications mathématiques de Besançon (2017), p. 137-193, doi: 10.5802/pmb.19 Article PDF Class. Math.: 14G40, 11G50Keywords: Geometric Bogomolov conjecture, Bogomolov conjecture, canonical heights, canonical measures, small points AbstractThis is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory over function fields and a quick review on basic notions on nonarchimedean analytic geometry. Bibliography[1] Ahmed Abbes, Hauteurs et discrétude (d’après Szpiro, L., Ullmo, E., Zhang, S.), Séminaire Bourbaki 1996/97, Astérisque 245, Société Mathématique de France, 1997, p. 141–166 [2] Vladimir G. Berkovich, Spectral theory and analytic geometry over non-archimedean fields, Mathematical Surveys and Monographs 33, American Mathematical Society, 1990 [3] Vladimir G. Berkovich, “Etale cohomology for non-archimedean analytic spaces”, Publ. Math., Inst. Hautes Étud. Sci. 78 (1993), p. 5-161 [4] Vladimir G. Berkovich, “Vanishing cycles for formal schemes”, Invent. Math. 115 (1994) no. 3, p. 539-571 [5] Vladimir G. Berkovich, “Smooth p-adic analytic spaces are locally contractible”, Invent. Math. 137 (1999) no. 1, p. 1-84 [6] Fedor Alekseivich Bogomolov, “Points of finite order on Abelian Variety”, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980) no. 4, p. 782-804 [7] Enrico Bombieri & Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs 4, Cambridge University Press, 2006 [8] Antoine Chambert-Loir, “Mesures et équidistribution sur les espaces de Berkovich”, J. Reine Angew. Math. 595 (2006), p. 215-235 [9] Zubeyir Cinkir, “Zhang’s conjecture and the effective Bogomolov conjecture over function fields”, Invent. Math. 183 (2011) no. 3, p. 517-562 [10] X. W. C. Faber, “The geometric Bogomolov conjecture for curves of small genus”, Exp. Math. 18 (2009) no. 3, p. 347-367 [11] Walter Gubler, “Local heights of subvarieties over non-archimedean fields”, J. Reine Angew. Math. 498 (1998), p. 61-113 [12] Walter Gubler, “Local and canonical heights of subvarieties”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 2 (2003) no. 4, p. 711-760 [13] Walter Gubler, “The Bogomolov conjecture for totally degenerate abelian varieties”, Invent. Math. 169 (2007) no. 2, p. 377-400 [14] Walter Gubler, “Tropical varieties for non-archimedean analytic spaces”, Invent. Math. 169 (2007) no. 2, p. 321-376 [15] Walter Gubler, “Equidistribution over function fields”, Manuscr. Math. 127 (2008) no. 4, p. 485-510 [16] Walter Gubler, “Non-archimedean canonical measures on abelian varieties”, Compos. Math. 146 (2010) no. 3, p. 683-730 [17] A.J. de Jong, “Smoothness, semi-stability and alterations”, Publ. Math., Inst. Hautes Étud. Sci. 83 (1996), p. 51-93 [18] Shu Kawaguchi, Atsushi Moriwaki & Kazuhiko Yamaki, Introduction to Arakelov geometry, in Algebraic geometry in East Asia (Kyoto, 2001), World Scientific, 2002, p. 1-74 [19] Serge Lang, “Division points on curves”, Ann. Mat. Pura Appl. 70 (1965), p. 229-234 [20] Serge Lang, Abelian varieties, Springer, 1983 [21] Serge Lang, Fundamentals of Diophantine Geometry, Springer, 1983 [22] Laurent Moret-Bailly, Métriques permises, Séminaire sur les pinceaux arithmétiques: La Conjecture de Mordell, Astérisque 127, Société Mathématique de France, 1985, p. 29–87 [23] Atsushi Moriwaki, “Bogomolov conjecture for curves of genus $2$ over function fields”, J. Math. Kyoto Univ. 36 (1996) no. 4, p. 687-695 [24] Atsushi Moriwaki, “A sharp slope inequality for general stable fibrations of curves”, J. Reine Angew. Math. 480 (1996), p. 177-195 [25] Atsushi Moriwaki, “Bogomolov conjecture over function fields for stable curves with only irreducible fibers”, Compos. Math. 105 (1997) no. 2, p. 125-140 [26] Atsushi Moriwaki, “Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves”, J. Am. Math. Soc. 11 (1998) no. 3, p. 569-600 [27] Atsushi Moriwaki, “Arithmetic height functions over finitely generated fields”, Invent. Math. 140 (2000) no. 1, p. 101-142 [28] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press, 1970 [29] Johannes Nicaise, “Berkovich skeleta and birational geometry”, https://arxiv.org/abs/1409.5229, 2014 [30] Richard Pink & Damian Roessler, “On $\psi$-invariant subvarieties of semiabelian varieties and the Manin–Mumford conjecture”, J. Algebr. Geom. 13 (2004) no. 4, p. 771-798 [31] Michel Raynaud, “Courbes sur une variété abélienne et points de torsion”, Invent. Math. 71 (1983), p. 207-233 [32] Michel Raynaud, Sous-variétés dúne variété abélienne et points de torsion, Arithmetic and geometry, vol. I: Arithmetic, Progress in Mathematics 35, Birkhäuser, 1983, p. 327–352 [33] Thomas Scanlon, “Diophantine geometry from model theory”, Bull. Symb. Log. 7 (2001) no. 1, p. 37-57 [34] Thomas Scanlon, “A positive characteristic Manin–Mumford theorem”, Compos. Math. 141 (2005) no. 6, p. 1351-1364 [35] Lucien Szpiro, Emmanuel Ullmo & Shou-Wu Zhang, “Équirépartition des petits points”, Invent. Math. 127 (1997) no. 2, p. 337-347 [36] Emmanuel Ullmo, “Positivité et discrétion des points algébriques des courbes”, Ann. Math. 147 (1998) no. 1, p. 167-179 [37] Kazuhiko Yamaki, “Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension 5 with trivial trace”, to appear in Math. Res. Lett. [38] Kazuhiko Yamaki, “Geometric Bogomolov’s conjecture for curves of genus $3$ over function fields”, J. Math. Kyoto Univ. 42 (2002) no. 1, p. 57-81 [39] Kazuhiko Yamaki, “Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields”, J. Math. Kyoto Univ. 48 (2008) no. 2, p. 401-443 [40] Kazuhiko Yamaki, “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)”, Manuscr. Math. 142 (2013) no. 3-4, p. 273-306 [41] Kazuhiko Yamaki, “Strict supports of canonical measures and applications to the geometric Bogomolov conjecture”, Compos. Math. 152 (2016) no. 5, p. 997-1040 Article[42] Kazuhiko Yamaki, “Trace of abelian varieties over function fields and the geometric Bogomolov conjecture”, J. Reine Angew. Math. (2016), https://doi.org/10.1515/crelle-2015-0086 Article[43] Kazuhiko Yamaki, “Non-density of small points on divisors on abelian varieties and the Bogomolov conjecture”, J. Am. Math. Soc. 30 (2017) no. 4, p. 1133-1163 [44] Xinyi Yuan, “Big line bundles over arithmetic varieties”, Invent. Math. 173 (2008) no. 3, p. 603-649 [45] Shou-Wu Zhang, “Admissible pairing on a curve”, Invent. Math. 112 (1993) no. 1, p. 171-193 [46] Shou-Wu Zhang, “Small points and adelic metrics”, J. Algebr. Geom. 4 (1995) no. 2, p. 281-300 [47] Shou-Wu Zhang, “Equidistribution of small points on abelian varieties”, Ann. Math. 147 (1998) no. 1, p. 159-165 [48] Shou-Wu Zhang, “Gross-Schoen cycles and dualising sheaves”, Invent. Math. 179 (2010) no. 1, p. 1-73