Non-deformability of entire curves in projective hypersurfaces of high degree

Olivier Debarre[1]; Gianluca Pacienza[2]; Mihai Păun[3]

  • [1] Université L. Pasteur et CNRS Institut de Recherche Mathématique Avancée 7, rue René Descartes 67084 Strasbourg Cedex (France)
  • [2] Institut de Recherche Mathématique Avancée Université L. Pasteur et CNRS 7, rue René Descartes 67084 Strasbourg Cédex (France)
  • [3] Université Henri Poincaré Nancy 1 Institut Élie Cartan B.P. 239 54506 Vandœuvre-lès-Nancy Cedex (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 1, page 247-253
  • ISSN: 0373-0956

Abstract

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In this article, we prove that there does not exist a family of maximal rank of entire curves in the universal family of hypersurfaces of degree d ≥ 2 n in the complex projective space ℙ n . This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.

How to cite

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Debarre, Olivier, Pacienza, Gianluca, and Păun, Mihai. "Non-deformability of entire curves in projective hypersurfaces of high degree." Annales de l’institut Fourier 56.1 (2006): 247-253. <http://eudml.org/doc/10140>.

@article{Debarre2006,
abstract = {In this article, we prove that there does not exist a family of maximal rank of entire curves in the universal family of hypersurfaces of degree $d\ge 2n$ in the complex projective space $\{\mathbb\{P\}\}^n$. This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.},
affiliation = {Université L. Pasteur et CNRS Institut de Recherche Mathématique Avancée 7, rue René Descartes 67084 Strasbourg Cedex (France); Institut de Recherche Mathématique Avancée Université L. Pasteur et CNRS 7, rue René Descartes 67084 Strasbourg Cédex (France); Université Henri Poincaré Nancy 1 Institut Élie Cartan B.P. 239 54506 Vandœuvre-lès-Nancy Cedex (France)},
author = {Debarre, Olivier, Pacienza, Gianluca, Păun, Mihai},
journal = {Annales de l’institut Fourier},
keywords = {projective hypersurfaces; entire curves; Kobayashi hyperbolicity; hyperbolic; Kobayashi conjecture; entire curve},
language = {eng},
number = {1},
pages = {247-253},
publisher = {Association des Annales de l’institut Fourier},
title = {Non-deformability of entire curves in projective hypersurfaces of high degree},
url = {http://eudml.org/doc/10140},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Debarre, Olivier
AU - Pacienza, Gianluca
AU - Păun, Mihai
TI - Non-deformability of entire curves in projective hypersurfaces of high degree
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 1
SP - 247
EP - 253
AB - In this article, we prove that there does not exist a family of maximal rank of entire curves in the universal family of hypersurfaces of degree $d\ge 2n$ in the complex projective space ${\mathbb{P}}^n$. This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.
LA - eng
KW - projective hypersurfaces; entire curves; Kobayashi hyperbolicity; hyperbolic; Kobayashi conjecture; entire curve
UR - http://eudml.org/doc/10140
ER -

References

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  8. M. McQuillan, Holomorphic curves on hyperplane sections of 3 -folds, Geom. Funct. Anal. (1999), 370-392 Zbl0951.14014MR1692470
  9. E. Rousseau, Weak analytic hyperbolicity of generic hypersurfaces of high degree in the complex projective space of dimension 4 Zbl1132.32010
  10. T. Ochiai S. Kobayashi, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. (1975), 7-16 Zbl0331.32020MR402127
  11. Y.T. Siu, Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel (2004), 543-566, Springer, Berlin Zbl1076.32011MR2077584
  12. C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom. (1996), 200-214 Zbl0883.14022MR1420353
  13. C. Voisin, A correction: “On a conjecture of Clemens on rational curves on hypersurfaces”, J. Diff. Geom. (1998), 601-611 Zbl0994.14026MR1669712

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