Weak analytic hyperbolicity of generic hypersurfaces of high degree in 4
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 2, p. 369-383

In this article we prove that every entire curve in a generic hypersurface of degree d593 in 4 is algebraically degenerated i.e there exists a proper subvariety which contains the entire curve.

Dans cet article nous démontrons que toute courbe entière dans une hypersurface générique de degré d593 dans 4 est algébriquement dégénérée i.e il existe une sous-variété propre qui contient la courbe entière.

@article{AFST_2007_6_16_2_369_0,
     author = {Rousseau, Erwan},
     title = {Weak analytic hyperbolicity of generic hypersurfaces of high degree in $\mathbb{P}^{4}$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {2},
     year = {2007},
     pages = {369-383},
     doi = {10.5802/afst.1152},
     mrnumber = {2331545},
     zbl = {pre05236230},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_2_369_0}
}
Rousseau, Erwan. Weak analytic hyperbolicity of generic hypersurfaces of high degree in $\mathbb{P}^{4}$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 2, pp. 369-383. doi : 10.5802/afst.1152. http://www.numdam.org/item/AFST_2007_6_16_2_369_0/

[1] Bogomolov (F.A.).— Holomorphic tensors and vector bundles on projective varieties, Math. USSR Izvestija 13, p. 499-555 (1979). | Zbl 0439.14002

[2] Clemens (H.).— Curves on generic hypersurface, Ann. Sci. Ec. Norm. Sup., 19, p. 629-636 (1986). | Numdam | MR 875091 | Zbl 0611.14024

[3] Demailly (J.-P.).— Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proc. Sympos. Pure Math., vol.62, Amer. Math.Soc., Providence, RI, p.285-360 ( 1997). | MR 1492539 | Zbl 0919.32014

[4] Demailly (J.-P.), El Goul J..— Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math 122, p. 515-546 (2000). | MR 1759887 | Zbl 0966.32014

[5] Ein (L.).— Subvarieties of generic complete intersections, Invent. Math., 94, p. 163-169 (1988). | MR 958594 | Zbl 0701.14002

[6] Fulton (W.).— Intersection theory, Springer-Verlag, Berlin (1998). | MR 1644323 | Zbl 0541.14005

[7] Green (M.), Griffiths P..— Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979, Proc. Inter. Sympos. Berkeley, CA, 1979, Springer-Verlag, New-York, p. 41-74 (1980). | MR 609557 | Zbl 0508.32010

[8] Kobayashi (S.).— Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York (1970). | MR 277770 | Zbl 0207.37902

[9] McQuillan (M.).— Diophantine approximations and foliations, in Publ. Math. IHES (1998). | Numdam | MR 1659270 | Zbl 1006.32020

[10] Paun (M.).— Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity, preprint (2005).

[11] Rousseau (E.).— Etude des jets de Demailly-Semple en dimension 3, Ann. Inst. Fourier, 56, p. 397-421 (2006). | Numdam | MR 2226021 | Zbl 1092.58003

[12] Rousseau (E.).— Equations différentielles sur les hypersurfaces de 4 , to appear in J. Math. Pures Appl. (2006). | MR 2257847 | Zbl 05125005

[13] Siu (Y.-T.).— Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel, Springer, Berlin, p. 543-566 (2004). | MR 2077584 | Zbl 1076.32011

[14] Voisin (C.).— On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom., 44, p. 200-213 (1996). | MR 1420353 | Zbl 0883.14022