Towards the Green-Griffiths-Lang conjecture

TL;DR

This paper advances the understanding of the Green-Griffiths-Lang conjecture by establishing conditions under which entire curves are contained in proper subvarieties of projective varieties of general type, linking jet-semistability and hyperbolicity.

Contribution

It proves the conjecture for varieties satisfying a strong general type condition related to jet-semistability and provides a criterion for Kobayashi hyperbolicity of directed varieties.

Findings

Proves the Green-Griffiths-Lang conjecture under a jet-semistability condition.

Provides a sufficient criterion for Kobayashi hyperbolicity.

Links geometric stability properties to hyperbolic behavior.

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 $\rightarrow$ 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). This work is dedicated to the memory of Professor Salah Baouendi.