Boundedness of hyperbolic varieties

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.