Towards the green-griffiths-lang conjecture

Abstract

The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over ℂ, there exists a proper algebraic subvariety of X containing all non constant entire curves f : ℂ → X. Using the formalism of directed varieties, we prove here that this assertion holds true in case X satisfies a strong general type condition that is related to a certain jet semistability property of the tangent bundle TX. We then give a sufficient criterion for the Kobayashi hyperbolicity of an arbitrary directed variety (X, V).