Demailly's notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Abstract

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.

27 pages. Included new result (Theorem 1.14). No other significant changes

References in corpus (5)

• Hypersurfaces in projective schemes and a moving lemma
• Boundedness in families with applications to arithmetic hyperbolicity
• Ampleness of canonical divisors of hyperbolic normal projective varieties
• Good reduction and cyclic covers
• Finiteness properties of pseudo-hyperbolic varieties

Cited by in corpus (7)

• Boundedness in families with applications to arithmetic hyperbolicity
• The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties
• Non-archimedean hyperbolicity and applications
• Finiteness properties of pseudo-hyperbolic varieties
• Non-archimedean hyperbolicity of the moduli space of curves
• Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3
• Hyperbolicity and model-complete fields