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

TL;DR

This paper investigates Demailly's conjecture linking algebraic hyperbolicity and Kobayashi hyperbolicity, extending definitions, proving stability under field extensions, and analyzing morphism finiteness and moduli space boundedness.

Contribution

It extends the definition of algebraic hyperbolicity to broader varieties, proves stability under field extensions, and explores implications for morphism finiteness and moduli space boundedness.

Findings

Algebraic hyperbolicity is stable under field extensions.

Finiteness of morphisms from a fixed variety to a hyperbolic variety.

Aut(X) is finite and surjective endomorphisms are automorphisms.

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.