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Hyperbolicity of Projective Hypersurfaces pp 11–24Cite as

Algebraic hyperbolicity

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Part of the book series: IMPA Monographs ((IMPA,volume 5))

Abstract

This chapter deals with an algebraic analogue of the Kobayashi hyperbolicity, introduced in [17]. We shall explain how the Kobayashi hyperbolicity implies restrictions on the ratio between the genus and the degree of algebraic curves contained in a hyperbolic projective algebraic manifold, and shall take this property as a definition. Then, we will discuss some general conjecture and related results, in particular Bogomolov’s proof of the finiteness of rational and elliptic curves on surfaces whose Chern classes satisfy a certain inequality. In the second part, an account of known results about algebraic hyperbolicity of generic projective hypersurfaces of high degree will be done.

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Notes

  1. 1.

    A K3 surface is a simply connected surface X with irregularity \(q(X) = h^{1}(X,\mathcal{O}_{X}) = 0\) and trivial canonical bundle \(K_{X} \simeq \mathcal{O}_{X}\).

  2. 2.

    Let T be a two dimensional complex torus with a base point chosen. The involution ι: T → T has exactly 16 fixed points, namely the points of order 2 on T, so that the quotient \(T/\langle 1,\iota \rangle\) has sixteen ordinary double points. Resolving the double points we obtain a smooth surface X, the Kummer surface Km(T) of T. Kummer surfaces are special case of K3 surfaces.

Bibliography

  1. F. A. Bogomolov. Holomorphic tensors and vector bundles on projective manifolds. Izv. Akad. Nauk SSSR Ser. Mat., 42(6):1227–1287, 1439, 1978.

    Google Scholar 

  2. Serge Cantat. Deux exemples concernant une conjecture de Serge Lang. C. R. Acad. Sci. Paris Sér. I Math., 330(7):581–586, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  3. Herbert Clemens. Curves on generic hypersurfaces. Ann. Sci. École Norm. Sup. (4), 19(4):629–636, 1986.

    MathSciNet  MATH  Google Scholar 

  4. Herbert Clemens and Ziv Ran. Twisted genus bounds for subvarieties of generic hypersurfaces. Amer. J. Math., 126(1):89–120, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  5. Jean-Pierre Demailly. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In Algebraic geometry—Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., pages 285–360. Amer. Math. Soc., Providence, RI, 1997.

    Google Scholar 

  6. Olivier Debarre, Gianluca Pacienza, and Mihai Păun. Non-deformability of entire curves in projective hypersurfaces of high degree. Ann. Inst. Fourier (Grenoble), 56(1):247–253, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  7. Jacques Dufresnoy. Théorie nouvelle des familles complexes normales. Applications à l’étude des fonctions algébroïdes. Ann. Sci. École Norm. Sup. (3), 61:1–44, 1944.

    Google Scholar 

  8. Lawrence Ein. Subvarieties of generic complete intersections. Invent. Math., 94(1): 163–169, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  9. Mark Green and Phillip Griffiths. Two applications of algebraic geometry to entire holomorphic mappings. In The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), pages 41–74. Springer, New York, 1980.

    Google Scholar 

  10. Mark L. Green. Holomorphic maps into complex projective space omitting hyperplanes. Trans. Amer. Math. Soc., 169:89–103, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  11. Shoshichi Kobayashi. Hyperbolic manifolds and holomorphic mappings, volume 2 of Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1970.

    Google Scholar 

  12. Steven Shin-Yi Lu and Yoichi Miyaoka. Bounding codimension-one subvarieties and a general inequality between Chern numbers. Amer. J. Math., 119(3):487–502, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  13. Michael McQuillan. Diophantine approximations and foliations. Inst. Hautes Études Sci. Publ. Math., (87):121–174, 1998.

    Google Scholar 

  14. Gianluca Pacienza. Subvarieties of general type on a general projective hypersurface. Trans. Amer. Math. Soc., 356(7):2649–2661 (electronic), 2004.

    Google Scholar 

  15. Gianluca Pacienza and Erwan Rousseau. On the logarithmic Kobayashi conjecture. J. Reine Angew. Math., 611:221–235, 2007.

    MathSciNet  MATH  Google Scholar 

  16. Yum-Tong Siu. Hyperbolicity in complex geometry. In The legacy of Niels Henrik Abel, pages 543–566. Springer, Berlin, 2004.

    Google Scholar 

  17. Claire Voisin. On a conjecture of Clemens on rational curves on hypersurfaces. J. Differential Geom., 44(1):200–213, 1996.

    MathSciNet  MATH  Google Scholar 

  18. Geng Xu. Subvarieties of general hypersurfaces in projective space. J. Differential Geom., 39(1):139–172, 1994.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. CNRS, Institut de Mathématiques de Jussieu – Paris Rive Gauche, Sorbonne Universités Univ Paris Diderot, Sorbonne Paris Cité, Paris, France

    Simone Diverio

  2. Aix Marseille Université, CNRS, Centrale Marseille, Marseille, France

    Erwan Rousseau

Authors
  1. Simone Diverio
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  2. Erwan Rousseau
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Diverio, S., Rousseau, E. (2016). Algebraic hyperbolicity. In: Hyperbolicity of Projective Hypersurfaces. IMPA Monographs, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-32315-2_2

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