On the hyperbolicity of certain complements. (Sur l’hyperbolicité de certains complémentaires.) (French) Zbl 1009.32015

The authors give a new proof of the hyperbolicity of two complements of hypersurfaces in complex projective spaces. There has been a conjecture due to S. Kobayashi as follows [cf. G. Dethloff, G. Schumacher, and P. M. Wong, Am. J. Math. 117, No. 3, 573-599 (1995; Zbl 0842.32021)]. Let \(S\) be a hypersurface with \(p\) components in \(\mathbb{P}_n (\mathbb{C})\). If \(S\) is generic and the degree of \(S\) is not less than \(2n+1\), then \(\mathbb{P}_n (\mathbb{C}) \setminus S\) is complete hyperbolic. For this conjecture, the following classical result due to Bloch-Green-Fujimoto is well-known [cf. M. Green, Proc. Am. Math. Soc. 66, 109-113 (1977; Zbl 0366.32013)]: The complement of \(2n+1\) hyperplanes in general position in \(\mathbb{P}_n (\mathbb{C})\) is hyperbolic. On the other hand, H. Grauert [Math. Z. 200, No. 2, 149-168 (1989; Zbl 0664.32020)] and G. Dethloff, G. Schumacher and P. M. Wong [loc. cit. and Duke Math. J. 78, No. 1, 193-212 (1995; Zbl 0847.32028)] deal with the non hyperplane case. They prove that the complement of three generic curves in \(\mathbb{P}_2(\mathbb{C})\) is complete hyperbolic. The methods in their proofs are not elementary.
In this paper the authors give proofs of the above theorems by making use of the idea of Ros [cf. L. Zalcman, Bull. Am. Math. Soc., New. Ser. 35, No. 3, 215-230 (1998; Zbl 1037.30021)]. The proofs given in this paper are direct and elementary.


32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32F45 Invariant metrics and pseudodistances in several complex variables