Hyperbolicity might nevertheless be algebraic

Abstract

A compact complex manifold X is hyperbolic if every holomorphic map from the complex plane to X is constant. Obstructions to hyperbolicity are given by holomorphic maps from CP^1 and holomorphic maps from complex tori. S. Lang (1986) conjectured that for projective manifolds, these are the only obstructions to hyperbolicity. In this talk, I will present some recent progress on this conjecture that is joint work with Frédéric Campana.