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.


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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