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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.

##### MSC:
 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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##### References:
 [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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