## Hyperbolicity of generic surfaces of high degree in projective 3-space.(English)Zbl 0966.32014

The 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)

### MSC:

 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32H30 Value distribution theory in higher dimensions
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