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Logarithmic jets and hyperbolicity. (English) Zbl 1048.32016

S. Kobayashi [Hyperbolic manifolds and holomorphic mappings (1970; Zbl 0207.37902)] conjectured that the complement of a generic plane curve of degree \(d \geq 5\) is hyperbolic. In the author’s previous work with J. P. Demailly [Am. J. Math. 122, No. 3, 515–546 (2000; Zbl 0966.32014)] it was shown that the conjecture holds for very generic curves of degree \(d \geq 21\). In the present paper the bound is improved to \(15\).
The basic strategy of proof is similar to that of the paper with Demailly, but using the logarithmic generalization by G. Dethloff and S. Lu [Osaka J. Math. 38, No. 1, 185–237 (2001; Zbl 0982.32022)] of the projectivized jet bundles introduced by J. P. Demailly [Proc. Symp. Pure Math. 62, 285–360 (1997; Zbl 0919.32014)].
The author also generalizes a result of M. McQuillan [Publ. Math., Inst. Hautes √Čtud. Sci. 87, 121–174 (1998; Zbl 1006.32020)] to the logarithmic context, proving that for an algebraic surface \((\bar X,D)\) of log-general type, any entire curve contained in a leaf of a holomorphic foliation on \(\bar X\) and omitting the divisor \(D\) is algebraically degenerate. This result will certainly be of independant interest.

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14J29 Surfaces of general type
32S65 Singularities of holomorphic vector fields and foliations
53C12 Foliations (differential geometric aspects)
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