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

TL;DR

This paper provides a simple criterion to determine when the Green-Griffiths locus equals the entire manifold, and applies it to manifolds uniformized by higher-rank bounded symmetric domains, clarifying an old example.

Contribution

It introduces a straightforward criterion for the Green-Griffiths locus to be the whole manifold and applies it to specific uniformized manifolds, clarifying previous ambiguities.

Findings

The Green-Griffiths locus equals the whole manifold under certain conditions.

Manifolds uniformized by bounded symmetric domains of rank > 1 have the entire Green-Griffiths locus.

Clarification of an old example by Green and Lang.

Abstract

We give a very simple criterion in order to ensure that the Green-Griffiths locus of a projective manifold is the whole manifold. Next, 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 example given by M. Green to S. Lang.