Géométrie, points entiers et courbes entières  [ Geometry, integral points and integral curves ]
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, p. 221-239

Let X be a projective variety over a number field K (resp. over ). Let H be the sum of “sufficiently many positive divisors” on X. We show that any set of quasi-integral points (resp. any integral curve) in X-H is not Zariski dense.

Soit X une variété projective sur un corps de nombres K (resp. sur ). Soit H la somme de « suffisamment de diviseurs positifs » sur X. On montre que tout ensemble de points quasi-entiers (resp. toute courbe entière) dans X-H est non Zariski-dense.

DOI : https://doi.org/10.24033/asens.2094
Classification:  14G25,  11J97,  11G35
Keywords: arithmetic geometry, height, integral points, diophantine approximation, hyperbolicity
@article{ASENS_2009_4_42_2_221_0,
     author = {Autissier, Pascal},
     title = {G\'eom\'etrie, points entiers et courbes enti\`eres},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {4e s{\'e}rie, 42},
     number = {2},
     year = {2009},
     pages = {221-239},
     doi = {10.24033/asens.2094},
     zbl = {1173.14016},
     mrnumber = {2518077},
     language = {fr},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_2_221_0}
}
Autissier, Pascal. Géométrie, points entiers et courbes entières. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, pp. 221-239. doi : 10.24033/asens.2094. http://www.numdam.org/item/ASENS_2009_4_42_2_221_0/

[1] F. Angelini, An algebraic version of Demailly's asymptotic Morse inequalities, Proc. Amer. Math. Soc. 124 (1996), 3265-3269. | MR 1389502 | Zbl 0860.14019

[2] P. Corvaja & U. Zannier, On a general Thue's equation, Amer. J. Math. 126 (2004), 1033-1055. | MR 2089081 | Zbl 1125.11022

[3] P. Corvaja & U. Zannier, On integral points on surfaces, Ann. of Math. 160 (2004), 705-726. | MR 2123936 | Zbl 1146.11035

[4] J.-P. Demailly, L 2 vanishing theorems for positive line bundles and adjunction theory, in Transcendental methods in algebraic geometry (Cetraro, 1994), Lecture Notes in Math. 1646, Springer, 1996, 1-97. | MR 1603616 | Zbl 0883.14005

[5] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366. | MR 718935 | Zbl 0588.14026

[6] G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. 133 (1991), 549-576. | MR 1109353 | Zbl 0734.14007

[7] W. Fulton, Intersection theory, 2e éd., Ergebnisse der Mathematik und ihrer Grenzgebiete 2, Springer, 1998. | MR 1644323 | Zbl 0885.14002

[8] J.-P. Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics 42, Birkhäuser, 1983. | MR 725671 | Zbl 0519.14002

[9] S. Lang, Number theory. III, Encyclopaedia of Mathematical Sciences 60, Springer, 1991. | MR 1112552 | Zbl 0744.14012

[10] R. Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 48, Springer, 2004. | MR 2095471 | Zbl 1093.14500

[11] A. Levin, Generalizations of Siegel's and Picard's theorems, à paraître dans Annals of Math. arXiv :math.NT/0503699. | MR 2552103 | Zbl 1250.11067

[12] H. P. Schlickewei, The 𝔭-adic Thue-Siegel-Roth-Schmidt theorem, Arch. Math. (Basel) 29 (1977), 267-270. | MR 491529 | Zbl 0365.10026

[13] W. M. Schmidt, Diophantine approximation, Lecture Notes in Math. 785, Springer, 1980. | MR 568710 | Zbl 0421.10019

[14] P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Math. 1239, Springer, 1987. | MR 883451 | Zbl 0609.14011

[15] P. Vojta, A refinement of Schmidt's subspace theorem, Amer. J. Math. 111 (1989), 489-518. | MR 1002010 | Zbl 0662.14002

[16] P. Vojta, Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), 133-181. | MR 1408559 | Zbl 1011.11040

[17] P. Vojta, On Cartan's theorem and Cartan's conjecture, Amer. J. Math. 119 (1997), 1-17. | MR 1428056 | Zbl 0877.11040

[18] S. Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), 281-300. | MR 1311351 | Zbl 0861.14019