Boundedness of hyperbolic varieties

TL;DR

This paper proves that for projective varieties with all subvarieties of general type, the degrees of morphisms from curves of fixed genus are uniformly bounded, confirming a key aspect of hyperbolicity conjectures.

Contribution

It establishes uniform bounds on degrees of morphisms from curves of fixed genus to hyperbolic varieties, extending previous results to all subvarieties of general type.

Findings

Bounded degrees for morphisms from curves of fixed genus

Hom-scheme $ ext{Hom}_k(C,X)$ is projective

Supports conjectures relating hyperbolicity and subvariety types

Abstract

Let $k$ be an algebraically closed field of characteristic zero, and let $X/k$ be a projective variety. The conjectures of Demailly--Green--Griffiths--Lang posit that every integral subvariety of $X$ is of general type if and only if $X$ is algebraically hyperbolic i.e., for any ample line bundle $\mathcal{L}$ on $X$ there is a real number $\alpha(X,\mathcal{L})$, depending only on $X$ and $\mathcal{L}$, such that for every smooth projective curve $C/k$ of genus $g(C)$ and every $k$-morphism $f\colon C\to X$, $\text{deg}_Cf^*\mathcal{L} \leq \alpha(X,\mathcal{L})\cdot g(C) $ holds. In this work, we prove that if $X/k$ is a projective variety such that every integral subvariety is of general type, then for every ample line bundle $\mathcal{L}$ on $X$ and every integer $g\geq 0$, there is an integer $\alpha(X,\mathcal{L},g)$, depending only on $X,\mathcal{L},$ and $g$, such that for every smooth projective curve $C/k$ of genus $g$ and every $k$-morphism $f\colon C\to X$, the inequality $\text{deg}_Cf^*\mathcal{L} \leq \alpha(X,\mathcal{L},g)$ holds, or equivalently, the Hom-scheme $\underline{\text{Hom}}_k(C,X)$ is projective.