  Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang
conjecture on varieties of general type, states that an algebraically
hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to
provide evidence for Demailly's conjecture by verifying several predictions it
makes. We first define what an algebraically hyperbolic projective variety is,
extending Demailly's definition to (not necessarily smooth) projective
varieties over an arbitrary algebraically closed field of characteristic zero,
and we prove that this property is stable under extensions of algebraically
closed fields. Furthermore, we show that the set of (not necessarily
surjective) morphisms from a projective variety $Y$ to a projective
algebraically hyperbolic variety $X$ that map a fixed closed subvariety of $Y$
onto a fixed closed subvariety of $X$ is finite. As an application, we obtain
that Aut$(X)$ is finite and that every surjective endomorphism of $X$ is an
automorphism. Finally, we explore "weaker" notions of hyperbolicity related to
boundedness of moduli spaces of maps, and verify similar predictions made by
the Green-Griffiths-Lang conjecture on hyperbolic projective varieties.
