On the locus of higher order jets of entire curves in complex projective varieties

TL;DR

This paper investigates how positivity properties of the cotangent bundle influence the distribution of entire curves in complex projective varieties, providing new differential equations that restrict their locus.

Contribution

It introduces new results on differential equations that constrain entire curves in foliated or directed varieties under positivity assumptions.

Findings

Existence of differential equations constraining entire curves

Results applicable to foliated or directed varieties

Strengthening of the Green-Griffiths-Lang conjecture

Abstract

For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green-Griffiths-Lang conjecture stipulates that entire curves drawn on a variety of general type should all be contained in a proper algebraic subvariety. We present here new results on the existence of differential equations that strongly restrain the locus of entire curves in the general context of foliated or directed varieties, under appropriate positivity conditions.