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

Debarre, Olivier; Pacienza, Gianluca; Păun, Mihai

doi:10.5802/aif.2178

Non-deformability of entire curves in projective hypersurfaces of high degree
[Non-deformabilité des courbes entières dans les hypersurfaces projectives de grand degré]

Debarre, Olivier ¹ ; Pacienza, Gianluca ² ; Păun, Mihai 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, Tome 56 (2006) no. 1, pp. 247-253.

Résumé
Abstract

Dans cet article, nous démontrons qu’il n’existe pas de famille de rang maximal de courbes entières dans la famille universelle des hypersurfaces de degré $d \geq 2 n$ dans l’espace projectif complexe $ℙ^{n}$ . Cela peut se voir comme une version faible de la conjecture de Kobayashi affirmant qu’une hypersurface projective générale de haut degré est hyperbolique au sens de Kobayashi.

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 $d \geq 2 n$ 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.

MR Zbl

DOI : 10.5802/aif.2178

Classification : 14J70, 32Q45
Keywords: projective hypersurfaces, entire curves, Kobayashi hyperbolicity
Mot clés : hypersurfaces projectives, courbes entières, hyperbolicité au sens de Kobayashi

Affiliations des auteurs :

Debarre, Olivier ¹ ; Pacienza, Gianluca ² ; Păun, Mihai ³

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     title = {Non-deformability of entire curves in projective hypersurfaces of high degree},
     journal = {Annales de l'Institut Fourier},
     pages = {247--253},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
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     year = {2006},
     doi = {10.5802/aif.2178},
     zbl = {1096.32010},
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     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2178/}
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Debarre, Olivier; Pacienza, Gianluca; Păun, Mihai. Non-deformability of entire curves in projective hypersurfaces of high degree. Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 247-253. doi : 10.5802/aif.2178. http://www.numdam.org/articles/10.5802/aif.2178/

Bibliographie
Cité par

[1] Brody, R. Compact manifolds in hyperbolicity, Trans. Amer. Math. Soc., Volume 235 (1978), pp. 213-219 | MR | Zbl

[2] Brunella, M. On entire curves tangent to a foliation (preprint 2004)

[3] Brunella, M. Courbes entières dans les surfaces algébriques complexes (d’après McQuillan, Demailly-El Goul, $\dots$ ), Astérisque (2002) no. 282, pp. 39-61 (Séminaire Bourbaki, Vol. 2000/2001, Exp. No. 881, vii) | Numdam | MR | Zbl

[4] Clemens, H. Lower bounds on genera of subvarieties of generic hypersurfaces, Comm. Algebra (2003) no. 31, pp. 3673-3711 (Special issue in honor of Steven L. Kleiman) | DOI | MR | Zbl

[5] Demailly, Jean-Pierre Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 285-360 | MR | Zbl

[6] J.-P. Demailly, J. El Goul Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math., Volume 122 (2000), pp. 515-546 | DOI | MR | Zbl

[7] Kobayashi, S. Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1998 no. 318 | MR | Zbl

[8] McQuillan, M. Holomorphic curves on hyperplane sections of $3$ -folds, Geom. Funct. Anal. (1999) no. 9, pp. 370-392 | DOI | MR | Zbl

[9] Rousseau, E. Weak analytic hyperbolicity of generic hypersurfaces of high degree in the complex projective space of dimension 4 (preprint math.AG/0510285)

[10] S. Kobayashi, T. Ochiai Meromorphic mappings onto compact complex spaces of general type, Invent. Math. (1975) no. 31, pp. 7-16 | DOI | MR | Zbl

[11] Siu, Y.T. Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 543-566 | MR | Zbl

[12] Voisin, C. On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom. (1996) no. 44, pp. 200-214 | MR | Zbl

[13] Voisin, C. A correction: “On a conjecture of Clemens on rational curves on hypersurfaces”, J. Diff. Geom. (1998) no. 49, pp. 601-611 | MR | Zbl

Cité par Sources :