Mathematische Zeitschrift

, Volume 279, Issue 1–2, pp 99–137 | Cite as

Majorations explicites des fonctions de Hilbert–Samuel géométrique et arithmétique

Article
First Online:
Received:
Accepted:
  • 189 Downloads

Résumé

En utilisant l’approche de \(\mathbb R\)-filtration en géométrie d’Arakelov, on établit des majorations explicites des fonctions de Hilbert–Samuel géométrique et arithmétique pour les fibrés inversibles sur une variété projective et les fibrés inversibles hermitiens sur une variété projective arithmétique.

This is a preview of subscription content, log in to check access.

Notes

Remerciements

Je voudrais remercier Éric Gaudron pour des remarques qui m’ont beaucoup aidé à améliorer la rédaction de l’article. Pendant la préparation et la rédaction de l’article, j’ai bénéficié des discussions avec Sebastien Boucksom, je tiens à lui exprimer mes gratitudes. Je suis aussi reconnaissant à Xinyi Yuan et Tong Zhang pour m’avoir communiqué leur article et pour des discussions très intéressantes. Enfin, je voudrais remercier le rapporteur anonyme pour sa lecture soigneuse de l’article et pour ses suggestions précieuses. Ce travail a été soutenu par le fond de recherche NSFC11271021.

Références

  1. 1.
    Abbes, A., Bouche, T.: Théorème de Hilbert–Samuel “arithmétique”. Université de Grenoble. Annales de l’Institut Fourier 45(2), 375–401 (1995)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Banaszczyk, W.: Inequalities for convex bodies and polar reciprocal lattics in \({ R}^n\). Discrete Comput. Geom. 13(2), 217–231 (1995)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Betke, U., Böröczky Jr, K.: Asymptotic formulae for the lattice point enumerator. Can. J. Math. 51(2), 225–249 (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Blichfeld, H.F.: Notes on geometry of numbers. Announcement to the October meeting of the San Francisco Section. Bull. Am. Math. Soc. 27(4), 152–153 (1921)Google Scholar
  5. 5.
    Bombieri, E., Vaaler, J.: On Siegel’s lemma. Invent. Math. 73(1), 11–32 (1983)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bost, J.-B., Chen, H.: Concerning the semistability of tensor products in Arakelov geometry. Journal de Mathématiques Pures et Appliquées. Neuvième Série 99(4), 436–488 (2013)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bost, J.-B., Künnemann, K.: Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers. Adv. Math. 223(3), 987–1106 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series. Compos. Math. 147(4), 1205–1229 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chardin, M.: Une majoration de la fonction de Hilbert et ses conséquences pour l’interpolation algébrique. Bulletin de la Société Mathématique de France 117(3), 305–318 (1989)MATHMathSciNetGoogle Scholar
  10. 10.
    Chen, H.: Positive degree and arithmetic bigness (2008). arXiv:0803.2583
  11. 11.
    Chen, H.: Maximal slope of tensor product of Hermitian vector bundles. J. Algebraic Geom. 18(3), 575–603 (2009)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Chen, H.: Arithmetic Fujita approximation. Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 43(4), 555–578 (2010)MATHGoogle Scholar
  13. 13.
    Chen, H.: Convergence des polygones de Harder–Narasimhan. Mémoires de la Société Mathématique de France 120, 1–120 (2010)Google Scholar
  14. 14.
    Faltings, G.: Calculus on arithmetic surfaces. Ann. Math. Second Ser. 119(2), 387–424 (1984)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Fujita, T.: Approximating Zariski decomposition of big line bundles. Kodai Math. J. 17(1), 1–3 (1994)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gaudron, É.: Pentes de fibrés vectoriels adéliques sur un corps globale. Rendiconti del Seminario Matematico della Università di Padova 119, 21–95 (2008)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gaudron, É.: Géométrie des nombres adélique et lemmes de Siegel généralisés. Manuscr. Math. 130(2), 159–182 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Gaudron, É., Rémond, G.: Corps de siegel, à paraître dans Journal für die reine und angewandte Mathematik. Prépublication (2013)Google Scholar
  19. 19.
    Gaudron, É., Rémond, G.: Minima, pentes et algèbre tensorielle. Isr. J. Math. 195(2), 565–591 (2013)CrossRefMATHGoogle Scholar
  20. 20.
    Gieseker, D.: Stable vector bundles and the Frobenius morphism. Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 6, 95–101 (1973)MATHMathSciNetGoogle Scholar
  21. 21.
    Gillet, H., Soulé, C.: On the number of lattice points in convex symmetric bodies and their duals. Isr. J. Math. 74(2–3), 347–357 (1991)CrossRefMATHGoogle Scholar
  22. 22.
    Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Institut des Hautes Études Scientifiques. Publications Mathématiques (11), 167 (1961)Google Scholar
  23. 23.
    Henk, M.: Successive minima and lattice points, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento, (2002). no. 70, part I, pp. 377–384. IV International Conference in Stochastic Geometry, Convex Bodies, Empirical Measures and Applications to Engineering Science, vol. I (Tropea, 2001)Google Scholar
  24. 24.
    Henze, M.: A Blichfeldt-type inequality for centrally symmetric convex bodies. Monatshefte für Mathematik 170(3–4), 371–379 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig (1997)Google Scholar
  26. 26.
    Kaveh, K., Khovanskii, A.: Algebraic equations and convex bodies. In: Itenberg, I., Jöricke, B., Passare, M. (eds.) Perspectives in Analysis, Geometry, and Topology, Progress in Mathematics, vol. 296, pp. 263–282. Birkhäuser/Springer, New York (2012)Google Scholar
  27. 27.
    Kollár, J., Matsusaka, T.: Riemann–Roch type inequalities. Am. J. Math. 105(1), 229–252 (1983)CrossRefMATHGoogle Scholar
  28. 28.
    Lazarsfeld, R.: Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 48. Springer, Berlin (2004). Classical setting: line bundles and linear seriesGoogle Scholar
  29. 29.
    Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Annales Scientifiques de l’École Normale Supérieure, Quatrième Série 42(5), 783–835 (2009)MATHGoogle Scholar
  30. 30.
    Liu, Q.: Algebraic geometry and arithmetic curves. Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford (2002). Translated from the French by Reinie Erné, Oxford Science PublicationsGoogle Scholar
  31. 31.
    Moriwaki, A.: Continuity of volumes on arithmetic varieties. J. Algebraic Geom. 18(3), 407–457 (2009)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. Second Ser. 82, 540–567 (1965)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Nesterenko, Y., Philippon, P. (eds.): Introduction to Algebraic Independence Theory. Lecture Notes in Mathematics, vol. 1752. Springer, Berlin (2001)Google Scholar
  34. 34.
    Nesterenko, Y.V.: Estimates for the characteristic function of a prime ideal. Math. USSR-Sbornik 51(1), 9–32 (1985)CrossRefMATHGoogle Scholar
  35. 35.
    Roy, D., Thunder, J.L.: An absolute Siegel’s lemma. Journal für die Reine und Angewandte Mathematik 476, 1–26 (1996)MATHMathSciNetGoogle Scholar
  36. 36.
    Sombra, M.: Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz. J. Pure Appl. Algebra 117/118, 565–599 (1997) Algorithms for algebra (Eindhoven, 1996)Google Scholar
  37. 37.
    Soulé, C., Abramovich, D., Burnol, J.-F., Kramer, J.: Lectures on Arakelov Geometry, Cambridge Studies in Advanced Mathematics, vol. 33. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  38. 38.
    Takagi, S.: Fujita’s approximation theorem in positive characteristics. J. Math. Kyoto Univ. 47(1), 179–202 (2007)MATHMathSciNetGoogle Scholar
  39. 39.
    Thunder, J.L.: An adelic Minkowski-Hlawka theorem and an application to Siegel’s lemma. Journal für die Reine und Angewandte Mathematik 475, 167–185 (1996)MATHMathSciNetGoogle Scholar
  40. 40.
    Yuan, X.: On volumes of arithmetic line bundles. Compos. Math. 145(6), 1447–1464 (2009)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Yuan, X., Zhang, T.: Effective bound of linear series on arithmetic surfaces. Duke Math. J. 162(10), 1723–1770 (2013)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Yuan, X., Zhang, T.: Relative Noether inequality on fibered surfaces. Prépublication (2013)Google Scholar
  43. 43.
    Yuan, X., Zhang, T.: Effective bounds of linear series on algebraic varieties and arithmetic varieties. Prépublication (2014)Google Scholar
  44. 44.
    Zhang, S.: Positive line bundles on arithmetic varieties. J. Am. Math. Soc. 8(1), 187–221 (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut Fourier (UMR 5582)Université Grenoble AlpesSaint Martin d’HèresFrance

Personalised recommendations