Hyperbolicity of generic surfaces of high degree in projective 3-space.

*(English)*Zbl 0966.32014The main result of this paper is to prove that a very generic surface \(X\) in \({\mathbb P}^3\) of degree \(d \geq 21\) is Kobayashi hyperbolic, that is there is no nonconstant holomorphic map from \({\mathbb C} \rightarrow X\). As a consequence of the proof, they also prove that the complement of a very generic curve in \({\mathbb P}^2\) is hyperbolic and hyperbolically imbedded for all degrees \(d \geq 21\). We note that previously, Siu-Yeung proved the hyperbolicity of the complement of a generic smooth curve of high degree in \({\mathbb P}^2\). The approach roughly is divided into the following steps: First use the Riemann-Roch calculations to prove the existence of suitable jet differentials which vanish on an ample divisor; then use Ahlfors-Schwarz lemma to conclude that the image of \(f\) sits in the base locus of the global sections of jet differentials; finally, it is hoped to show, by analysing the base locus carefully, that the base locus actually is a proper subvariety of \(X\).

Reviewer: Min Ru (Houston)