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Connections between K-stability and Vojta's conjecture

Jackson S. Morrow, Yueqiao Wu

Abstract

In this note, we use recent advances concerning the K-stability of $\mathbb{Q}$-Fano varieties to provide settings for which Vojta's conjecture holds.

Connections between K-stability and Vojta's conjecture

Abstract

In this note, we use recent advances concerning the K-stability of -Fano varieties to provide settings for which Vojta's conjecture holds.
Paper Structure (21 sections, 11 theorems, 14 equations)

This paper contains 21 sections, 11 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Ideas and relation to other works
  4. Organization
  5. Acknowledgements
  6. Preliminaries
  7. Fields
  8. Varieties and divisors
  9. Toric geometry
  10. $\mathop{\mathrm{b}}\nolimits$-divisors
  11. Local Weil functions and height functions
  12. Birational local Weil functions
  13. Vojta's conjecture
  14. Volume of a filtration
  15. General definitions and arithmetic importance
  16. ...and 6 more sections

Key Result

Theorem 1

Let $F$ be a number field and let $X$ be a $\mathbb{Q}$-Fano $F$-variety such that $X_{\overline{F}}$ has canonical singularities and infinite automorphism group. Then there exists a finite extension $F'/F$ and a $\mathop{\mathrm{b}}\nolimits$-divisor $\mathbb{E}$ on $X_{F'}$ such that inequalities

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.9
  • Definition 2.11
  • Proposition 2.12: RuVojta:BirationalNevanlinna
  • Example 2.13
  • ...and 24 more