Boundedness in families with applications to arithmetic hyperbolicity

Abstract

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 nonzero 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.

arXiv comment39 pages. Final version

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Cited by in corpus (5)

• Demailly's notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps
• The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties
• Good reduction and cyclic covers
• Finiteness properties of pseudo-hyperbolic varieties
• Hyperbolicity and model-complete fields