## 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 $$n=3$$.

### MSC:

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

### Citations:

Zbl 0966.32014; Zbl 1076.32011
Full Text:

### References:

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