Advertisement

Inventiones mathematicae

, Volume 180, Issue 1, pp 161–223 | Cite as

Effective algebraic degeneracy

  • Simone Diverio
  • Joël MerkerEmail author
  • Erwan Rousseau
Article

Abstract

We show that for every smooth projective hypersurface X⊂ℙn+1 of degree d and of arbitrary dimension n 2, if X is generic, then there exists a proper algebraic subvariety Y X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound \(d\geqslant 2^{n^{5}}\) .

Keywords

Line Bundle Chern Class Ample Line Bundle Ample Divisor Holomorphic Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bloch, A.: Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension. J. Math. 5, 19–66 (1926) Google Scholar
  2. 2.
    Brückmann, P.: The Hilbert polynomial of the sheaf ΩT of germs of T-symmetrical tensor differential forms on complete intersections. Math. Ann. 307, 461–472 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Clemens, H.: Curves on generic hypersurface. Ann. Sci. Éc. Norm. Super. 19, 629–636 (1986) MathSciNetGoogle Scholar
  4. 4.
    Demailly, J.-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In: Proc. Sympos. Pure Math., vol. 62, pp. 285–360. Am. Math.Soc., Providence (1997) Google Scholar
  5. 5.
    Demailly, J.-P., El Goul, J.: Hyperbolicity of generic surfaces of high degree in projective 3-space. Am. J. Math. 122, 515–546 (2000) zbMATHCrossRefGoogle Scholar
  6. 6.
    Diverio, S.: Differential equations on complex projective hypersurfaces of low dimension. Compos. Math. 144(4), 920–932 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Diverio, S.: Existence of global invariant jet differentials on projective hypersurfaces of high degree. Math. Ann. 344(2), 293–315 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Green, M., Griffiths, P.: Two applications of algebraic geometry to entire holomorphic mappings. In: The Chern Symposium 1979, Proc. Internat. Sympos., Berkeley, Calif., 1979, pp. 41–74. Springer, New York-Berlin (1980) Google Scholar
  9. 9.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York-Heidelberg (1977), xvi+496 pp. zbMATHGoogle Scholar
  10. 10.
    Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehren der Mathematischen Wissenschaften, vol. 318. Springer, Berlin (1998), xiv+471 pp. zbMATHGoogle Scholar
  11. 11.
    McQuillan, M.: Diophantine approximation and foliations. Inst. Hautes Études Sci. Publ. Math. 87, 121–174 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    McQuillan, M.: Holomorphic curves on hyperplane sections of 3-folds. Geom. Funct. Anal. 9, 370–392 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Merker, J.: Low pole order frames on vertical jets of the universal hypersurface. Ann. Inst. Fourier (Grenoble) 59(3), 1077–1104 (2009) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Merker, J.: An algorithm to generate all polynomials in the k-jet of a holomorphic disc D→ℂn that are invariant under source reparametrization, arxiv.org/abs/0808.3547/
  15. 15.
    Păun, M.: Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity. Math. Ann. 340, 875–892 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rousseau, E.: Étude des jets de Demailly-Semple en dimension 3. Ann. Inst. Fourier (Grenoble) 56(2), 397–421 (2006) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Rousseau, E.: Équations différentielles sur les hypersurfaces de ℙ4. J. Math. Pures Appl. (9) 86(4), 322–341 (2006) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Rousseau, E.: Weak analytic hyperbolicity of generic hypersurfaces of high degree in ℙ4. Ann. Fac. Sci. Toulouse 16(2), 369–383 (2007) MathSciNetGoogle Scholar
  19. 19.
    Siu, Y.-T.: Hyperbolicity problems in function theory. In: Chan, K.-Y., Liu, M.-C. (eds.) Five Decades as a Mathematician and Educator—on the 80th Birthday of Professor Yung-Chow Wong, pp. 409–514. World Scientific, Singapore (1995) Google Scholar
  20. 20.
    Siu, Y.-T.: Some recent transcendental techniques in algebraic and complex geometry. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. I, pp. 439–448. Higher Ed. Press, Beijing (2002) Google Scholar
  21. 21.
    Siu, Y.-T.: Hyperbolicity in complex geometry. In: The Legacy of Niels Henrik Abel, pp. 543–566. Springer, Berlin (2004) Google Scholar
  22. 22.
    Siu, Y.-T., Yeung, S.-K.: Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane. Invent. Math. 124, 573–618 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Trapani, S.: Numerical criteria for the positivity of the difference of ample divisors. Math. Z. 219(3), 387–401 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Voisin, C.: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 44(1), 200–213 (1998). A correction: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 49(3), 601–611 (1998) MathSciNetGoogle Scholar
  25. 25.
    Voisin, C.: On some problems of Kobayashi and Lang; algebraic approaches. In: Current Developments in Mathematics, vol. 2003, pp. 53–125. Int. Press, Somerville (2003) Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Istituto “Guido Castelnuovo”Sapienza Università di RomaRomeItaly
  2. 2.Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParisFrance
  3. 3.Département de Mathématiques et ApplicationsÉcole Normale SupérieureParisFrance
  4. 4.Département de MathématiquesUniversité Louis PasteurStrasbourgFrance

Personalised recommendations