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Hyperbolicité du complémentaire d’une courbe dans \(\mathbb P^2\): Le cas de deux composantes (Hyperbolicity of the complements of plane algebraic curves: The two component case). (French. Abridged English version) Zbl 1034.32017
Summary: We prove that the complement of a very generic complex curve with two components in \(\mathbb P^2\) of degrees \(d_1 \leqslant d_2\) is hyperbolic in the sense of Kobayashi in the following cases: \(d_1 \geqslant 5\); \(d_1=4\) and \(d_2 \geqslant 7\); \(d_1=d_2=4\); \(d_1=3\) and \(d_2 \geqslant 9\); \(d_1=2\) and \(d_2 \geqslant 12\). We consider logarithmic jets developed by G. Dethloff and S. Lu [Osaka J. Math. 38, 185–237 (2001; Zbl 0982.32022)], who generalized to the logarithmic situation J.-P. Demailly’s jet bundles [Proc. Symp. Pure Math. 62, 285–360 (1997; Zbl 0919.32014)], and used by El Goul to obtain results about the hyperbolicity of the complement of a very generic curve in \(\mathbb P^2\) in the case of a single component.

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text: DOI
[1] Demailly, J.-P., Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, (), 285-360 · Zbl 0919.32014
[2] Dethloff, G.; Lu, S., Logarithmic jet bundles and applications, Osaka J. math., 38, 185-237, (2001) · Zbl 0982.32022
[3] J. El Goul, Logarithmic jets and hyperbolicity, Prépublication, 2000
[4] Hirzebruch, F., Topological methods in algebraic geometry, Grundlehren math. wiss., 131, (1966), Springer Heidelberg · Zbl 0138.42001
[5] McQuillan, M., Diophantine approximations and foliations, Publ. math. IHES, 87, 121-174, (1998) · Zbl 1006.32020
[6] Nadel, A.M., Hyperbolic surfaces in \(P\^{}\{3\}\), Duke math. J., 58, 749-771, (1989) · Zbl 0686.32015
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