Algebro-geometric criteria for Kobayashi hyperbolicity of

complex projective manifolds

Simone Diverio (Paris VI)

Let X be a complex projective manifold. We shall explain how the abundance of sections of a certain holomorphic vector bundle of algebraic differentials equations with values in an anti-ample divisor gives informations about the geometry of the entire curves in the manifold itself.

We will then apply these techniques to describe recent results about the algebraic degeneracy of entire curves in generic projective hypersurfaces of high degree.

In particular, we will give a confirmation of the Kobayashi conjecture for three-folds in projective 4-space.