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**Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity.**
*(English)*
Zbl 1137.32010

This paper in complex algebraic geometry makes a contribution to the literature on the Kobayashi conjecture that states that a generic complex hypersurface \(X\) of degree at least \(2n-1\) in the projective space \({\mathbb P}^n\) is hyperbolic in the sense that any entire holomorphic function \(\varphi:{\mathbb C}\to X\) reduces to a constant. The author builds upon the work of Siu and Demailly and their coauthors to prove the following theorem.

Theorem 1. Let \(X\subset{\mathbb P}^3\) be a generic hypersurface of degree at least \(18\). Then there is a divisor \(Y\subset{\mathbb P}(T_X)\) in the projectivization of the tangent bundle of \(X\) such that the projectivized derivative of any holomorphic curve \(\varphi:{\mathbb C}\to X\) lies in \(Y\). In particular, \(X\) is hyperbolic.

The main references for the proof are [Am. J. Math. 122, No. 3, 515–546 (2000; Zbl 0966.32014)] by J.-P. Demailly and J. El Goul, and [Hyperbolicity in complex geometry. The legacy of Niels Henrik Abel, 543–566 (2004; Zbl 1076.32011)] by Y. Siu. In §1 the author constructs some explicit meromorphic vector fields on the manifold of vertical jets of order two of the incidence manifold of degree-\(d\) forms on \({\mathbb P}^3\). Then he applies those vector fields to derive a contradiction from the assumption that an entire holomorphic curve has Zariski dense first derivative \(\varphi_{[1]}:{\mathbb C}\to X_{[1]}={\mathbb P}(T_X)\).

The author announces his hope to use further refinements of his methods presented in the article to prove in a sequel the above mentioned conjecture of Kobayashi for \(n=3\).

Theorem 1. Let \(X\subset{\mathbb P}^3\) be a generic hypersurface of degree at least \(18\). Then there is a divisor \(Y\subset{\mathbb P}(T_X)\) in the projectivization of the tangent bundle of \(X\) such that the projectivized derivative of any holomorphic curve \(\varphi:{\mathbb C}\to X\) lies in \(Y\). In particular, \(X\) is hyperbolic.

The main references for the proof are [Am. J. Math. 122, No. 3, 515–546 (2000; Zbl 0966.32014)] by J.-P. Demailly and J. El Goul, and [Hyperbolicity in complex geometry. The legacy of Niels Henrik Abel, 543–566 (2004; Zbl 1076.32011)] by Y. Siu. In §1 the author constructs some explicit meromorphic vector fields on the manifold of vertical jets of order two of the incidence manifold of degree-\(d\) forms on \({\mathbb P}^3\). Then he applies those vector fields to derive a contradiction from the assumption that an entire holomorphic curve has Zariski dense first derivative \(\varphi_{[1]}:{\mathbb C}\to X_{[1]}={\mathbb P}(T_X)\).

The author announces his hope to use further refinements of his methods presented in the article to prove in a sequel the above mentioned conjecture of Kobayashi for \(n=3\).

Reviewer: Imre Patyi (Atlanta)

### Keywords:

Kobayashi conjecture for surfaces in projective three-space; Kobayashi hyperbolic surfaces; meromorphic vector fields on projective manifolds
Full Text:
DOI

### References:

[1] | Bogomolov F.A. (1979). Holomorphic tensors and vector bundles on projective varieties. Math. USSR Izv. 13: 499–555 · Zbl 0439.14002 |

[2] | Brody R. (1978). Compact manifolds and hyperbolicity. Trans. Am. Math. Soc. 235: 213–219 · Zbl 0416.32013 |

[3] | Clemens H. (1986). Curves on generic hypersurface. Ann. Sci. Éc. Norm. Sup. 19: 629–636 · Zbl 0611.14024 |

[4] | Demailly, J.-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In: Algebraic Geometry–Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Part 2, pp. 285–360. Am. Math. Soc., Providence (1997) · Zbl 0919.32014 |

[5] | Demailly J.-P. and El Goul J. (2000). Hyperbolicity of generic surfaces of high degree in projective 3-space. Am. J. Math. 122: 515–546 · Zbl 0966.32014 |

[6] | Ein L. (1988). Subvarieties of generic complete intersections. Invent. Math. 94: 163–169 · Zbl 0701.14002 |

[7] | Green, M., Griffiths, P.: Two applications of algebraic geometry to entire holomorphic mappings. In: The Chen Symposium 1979, Proc. Inter. Sympos. Berkeley, CA, 1979, pp. 41–74. Springer, New York (1980) |

[8] | Kobayashi S. (1970). Hyperbolic Manifolds and Holomorphic Mappings. Marcel Dekker, New York · Zbl 0207.37902 |

[9] | Lu S.S.-Y. and Yau S.T. (1990). Holomorphic curves in surfaces of general type. Proc. Nat. Acad. Sci. USA 87: 80–82 · Zbl 0702.32015 |

[10] | McQuillan M. (1998). Diophantine approximations and foliations. Publ. Math. IHES 87: 121–174 · Zbl 1006.32020 |

[11] | Miyaoka Y. (1983). Algebraic surfaces with positive indices Katata Symp. Proc. 1982. Prog. Math. 39: 281–301 |

[12] | Rousseau, E.: Weak analytic hyperbolicity of generic hypersurfaces of high degree in the complex projective space of dimension 4, math.AG/0510285 |

[13] | Sakai, F.: Symmetric powers of the cotangent bundle and classification of algebraic varieties. In: Proc. Copenhagen Meeting in Alg. Geom. (1978) |

[14] | Schneider M. and Tancredi A. (1988). Almost-positive vector bundles on projective surfaces. Math. Ann. 280: 537–547 · Zbl 0685.14015 |

[15] | Siu Y.-T. (1987). Defect relations for holomorphic maps between spaces of different dimensions. Duke Math. J. 55: 213–251 · Zbl 0623.32018 |

[16] | Siu Y.-T. and Yeung S.K. (1996). Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane. Invent. Math. 124: 576–618 · Zbl 0856.32017 |

[17] | Siu Y.-T. and Yeung S.K. (1997). Defects for ample divisors of abelian varieties, Schwarz lemma and hyperbolic hypersurfaces of low degrees. Am. J. Math. 119: 1139–1172 · Zbl 0947.32012 |

[18] | Siu, Y.-T.: Hyperbolicity in Complex Geometry. The Legacy of Niels Henrik Abel, pp. 543–566. Springer, Berlin (2004) |

[19] | Voisin C. (1996). On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 44: 200–213 · Zbl 0883.14022 |

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