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Weak analytic hyperbolicity of generic hypersurfaces of high degree in \(\mathbb P^4\). (English) Zbl 1132.32010

Let \(X\) be a general hypersurface in the complex projective space \(\mathbb P^n\). A famous conjecture of Kobayashi claims that if the degree of \(X\) is at least \(2n-1\), the variety \(X\) is hyperbolic, i.e. every morphism from \(\mathbb{C} \rightarrow X\) is constant. This very difficult conjecture has attracted many mathematicians : Y.-T. Siu has shown that the conjecture holds for hypersurfaces of very high degree [in: The legacy of Niels Henrik Abel. Oslo 2002. 543–566 (2004; Zbl 1076.32011)]. J.-P. Demailly and J. El Goul have shown that a very general surface of degree at least \(21\) in projective threespace is hyperbolic [Am. J. Math. 122, No. 3, 515–546 (2000; Zbl 0966.32014)].
In the paper under review a weak version of the Kobayashi’s conjecture is established in dimension three. More precisely, we have the following theorem: Let \(X\) be a general hypersurface in \(\mathbb P^4\) of degree at least \(593\). Then every entire curve \(f:\mathbb{C} \rightarrow X\) is algebraically degenerated, i.e. its image is contained in a proper algebraic subvariety.
The proof of the theorem follows a general strategy in hyperbolicity questions to construct global sections of the bundle of jet differentials that vanish on an ample divisor. These sections give algebraic differential equations every entire curve in \(X\) has to satisfy [cf. J.-P. Demailly, in: Algebraic geometry. Santa Cruz 1995: AMS Proc. Symp. Pure Math. 62(pt.2), 285–360 (1997; Zbl 0919.32014)]. Based on his earlier paper [J. Math. Pures Appl. (9) 86, No. 4, 322–341 (2006; Zbl 1115.14009)], the author provides an effective bound for the dimension of the space of global sections. A second important ingredient is the careful construction of meromorphic vector fields on the space of vertical 3-jets of the universal hypersurface.

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14J70 Hypersurfaces and algebraic geometry
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References:

[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). · 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). · Zbl 0919.32014
[4] Demailly (J.-P.), ElGoul J..— Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math 122, p. 515-546 (2000). · Zbl 0966.32014
[5] Ein (L.).— Subvarieties of generic complete intersections, Invent. Math., 94, p. 163-169 (1988). · Zbl 0701.14002
[6] Fulton (W.).— Intersection theory, Springer-Verlag, Berlin (1998). · 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). · Zbl 0508.32010
[8] Kobayashi (S.).— Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York (1970). · Zbl 0207.37902
[9] McQuillan (M.).— Diophantine approximations and foliations, in Publ. Math. IHES (1998). · 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). · Zbl 1092.58003
[12] Rousseau (E.).— Equations différentielles sur les hypersurfaces de \({\mathbb{P}}^4\), to appear in J. Math. Pures Appl. (2006). · Zbl 1115.14009
[13] Siu (Y.-T.).— Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel, Springer, Berlin, p. 543-566 (2004). · Zbl 1076.32011
[14] Voisin (C.).— On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom., 44, p. 200-213 (1996). · Zbl 0883.14022
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