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Motivated by conjectures of Demailly, Green-Griffiths, Lang, and Vojta, weshow that several notions related to hyperbolicity behave similarly infamilies. We apply our results to show the persistence of arithmetichyperbolicity along field extensions for projective normal surfaces withnonzero irregularity. These results rely on the mild boundedness ofsemi-abelian varieties. We also introduce and study the notion ofpseudo-algebraic hyperbolicity which extends Demailly's notion of algebraichyperbolicity for projective schemes.

Boundedness in families with applications to arithmetic hyperbolicity