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A hyperbolic sextic surface in \(\mathbb P^3(\mathbb C)\). (Une sextique hyperbolique dans \(\mathbb P^3(\mathbb C)\).) (French) Zbl 1071.14045

The aim of this paper is to construct a hyperbolic sextic in \(\mathbb{P}^3(\mathbb{C})\), therefore showing the existence of such a surface. A subset of \(\mathbb{P}^3(\mathbb{C})\) is called hyperbolic if it does not contain an entire curve, this is a non-constant holomorphic image of \(\mathbb{C}\). The interest in the subject comes from Kobayashi’s conjecture that states that a generic surface of degree greater than 4 in \(\mathbb{P}^3(\mathbb{C})\) is hyperbolic. The result has been proved first for degree \(\geq 36\) by M. McQuillan [Geom. Funct. Anal. 9, No. 2, 370–392 (1999; Zbl 0951.14014)], and then for degree \(\geq 21\) by J.-P. Demailly and J. El Goul [Am. J. Math. 122, No. 3, 515–546 (2000; Zbl 0966.32014)]. Parallel to this, an effort to construct examples of hyperbolic surfaces of lowest degree possible has been performed: prior to the article under review, the best known result corresponded to degree 8.

MSC:

14J70 Hypersurfaces and algebraic geometry
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14N05 Projective techniques in algebraic geometry
14J29 Surfaces of general type
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References:

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