论文标题
双曲线品种的界限
Boundedness of hyperbolic varieties
论文作者
论文摘要
令$ k $为特征零的代数封闭场，让$ x/k $成为投射品种。 demailly-green- griffiths的猜想 - 肯定，$ x $的每个积分子变量是一般类型的，并且仅当$ x $是代数上是多余的，即，对于任何足够的线条捆绑$ \ nathcal $ \ nathcal $ \ nathcal {l} $ $ \ MATHCAL {l} $，使得每条光滑的投影曲线$ c/k $ g（c）$和每个$ k $ -morphism $ f \ colon c \ to x $ to x $，$ \ text {deg} _cf} _cf} _cf^*\ Mathcal {l} 在这项工作中，我们证明，如果$ x/k $是一个投影量的变化，那么每个积分子变量都是一般类型的，那么对于$ x $上的每一个充分的线条捆绑$ \ nathcal {l} $，每个整数$ g \ geq 0 $，都有一个整数$α（x，x，\ nathcal {l}，$ ntecl { $ g $，使得每种光滑的投影曲线$ c/k $属$ g $和每个$ k $ -morphism $ f \ colon c \ colon c \ to x $，不等式$ \ text {deg} _cf^*\ Mathcal {l Mathcal {l} $ \ usewissline {\ text {hom}} _ k（c，x）$是投影。
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 $α(X,\mathcal{L})$, 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 α(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 $α(X,\mathcal{L},g)$, 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 $\text{deg}_Cf^*\mathcal{L} \leq α(X,\mathcal{L},g)$ holds, or equivalently, the Hom-scheme $\underline{\text{Hom}}_k(C,X)$ is projective.
