Logarithmic jets and hyperbolicity. (English) Zbl 1048.32016

El Goul, Jawher

Osaka J. Math. 40, No. 2, 469-491 (2003).

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 Étud. 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.

Reviewer: Gerd Dethloff (Brest)

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)

Keywords:

Kobayashi hyperbolicity; surfaces of logarithmic general type; logarithmic jet bundles; holomorphic foliations

Citations:

Zbl 0207.37902; Zbl 0966.32014; Zbl 0982.32022; Zbl 0919.32014; Zbl 1006.32020