  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.
