Motivated by conjectures of Demailly, Green–Griffiths, Lang and Vojta, we show that several notions related to hyperbolicity behave similarly in families. We apply our results to show the persistence of arithmetic hyperbolicity along field extensions for projective normal surfaces with non-zero irregularity. These results rely on the mild boundedness of semi-abelian varieties. We also introduce and study the notion of pseudo-algebraic hyperbolicity which extends Demailly's notion of algebraic hyperbolicity for projective schemes.

中文翻译：

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族有界性及其在算术双曲性中的应用

受 Demailly、Green-Griffiths、Lang 和 Vojta 猜想的启发，我们证明了与双曲性相关的几个概念在家庭中表现相似。我们应用我们的结果来显示算术双曲性沿着具有非零不规则性的射影法线表面的场延伸的持久性。这些结果依赖于半阿贝尔簇的温和有界性。我们还介绍并研究了伪代数双曲性的概念，它扩展了德迈利射影格式的代数双曲性概念。