The exceptional set and the Green–Griffiths locus do not always coincide

Abstract

We give a very simple criterion for the Green–Griffiths locus of a projective manifold to be the whole manifold. Then, we use it to show that the Green–Griffiths locus of any projective manifold uniformized by a bounded symmetric domain of rank greater than one is the whole manifold. In particular, this clarifies an old undetailed example given by M. Green to S. Lang.