Hyperbolicity of Generic High-Degree Hypersurfaces in Complex Projective   Space
  We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets. 
Yum-Tong Siu
math.CV