  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.
