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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 \(n=3\).

MSC:

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