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Boundedness in families with applications to arithmetic hyperbolicity

Raymond van Bommel, Ariyan Javanpeykar, Ljudmila Kamenova

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.

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.

Paper Structure

This paper contains 25 sections, 76 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Stable under generisation
  3. Countable-openness of the hyperbolic locus
  4. Mildly bounded varieties
  5. Persistence of arithmetic hyperbolicity
  6. Diophantine applications
  7. Predictions made by Vojta's geometric conjecture
  8. Outline of paper
  9. Characterising boundedness and algebraic hyperbolicity
  10. Testing algebraic hyperbolicity
  11. Testing boundedness
  12. Testing mild boundedness
  13. Extending maps over Dedekind schemes
  14. Zariski-countable openness of the hyperbolic locus
  15. Varieties of general type
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $X\to S$ be a proper morphism of schemes. Then the set of $s$ in $S$ such that $X_s$ is of general type is an open subscheme of $S$.

Theorems & Definitions (179)

  • Conjecture 1.1: Demailly, Green--Griffiths, Lang, Vojta
  • Theorem 1.2: Nakayama
  • Theorem 1.3: Generisation
  • Theorem 1.4: Demailly
  • Theorem 1.5: Countable-openness of boundedness
  • Theorem 1.6: Countable-openness of $(n,m)$-boundedness
  • Theorem 1.7: Countable-openness of every subvariety being of general type
  • Definition 1.8: Mildly bounded varieties
  • Proposition 1.9
  • Theorem 1.10
  • ...and 169 more