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Heights on toric varieties for singular metrics: Global theory

Gari Y. Peralta Alvarez

Abstract

In this paper, we develop a toric analog of the theory of adelic divisors on quasi-projective arithmetic varieties introduced by Yuan and Zhang, and extend the convex-analytic descriptions of the Arakelov geometry of projective toric arithmetic varieties given by Burgos, Philippon, and Sombra. Our main result is that the arithmetic self-intersection number of a semipositive toric adelic divisor is given by the integral of a concave function on a compact convex set. These generalized arithmetic intersection numbers coincide with the ones introduced by Burgos and Kramer in 2024, and therefore, can be used to compute heights of toric arithmetic varieties with respect to line bundles equipped with toric singular metrics.

Heights on toric varieties for singular metrics: Global theory

Abstract

In this paper, we develop a toric analog of the theory of adelic divisors on quasi-projective arithmetic varieties introduced by Yuan and Zhang, and extend the convex-analytic descriptions of the Arakelov geometry of projective toric arithmetic varieties given by Burgos, Philippon, and Sombra. Our main result is that the arithmetic self-intersection number of a semipositive toric adelic divisor is given by the integral of a concave function on a compact convex set. These generalized arithmetic intersection numbers coincide with the ones introduced by Burgos and Kramer in 2024, and therefore, can be used to compute heights of toric arithmetic varieties with respect to line bundles equipped with toric singular metrics.
Paper Structure (25 sections, 47 theorems, 191 equations)

This paper contains 25 sections, 47 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Heights and singular metrics
  3. The global Arakelov theory of projective toric varieties
  4. Main results
  5. Acknowledgments
  6. Notation and conventions
  7. Review of Yuan-Zhang's theory of adelic divisors
  8. Adelic divisors
  9. A global-local approach
  10. Nefness, semipositivity and intersection numbers
  11. Toric arithmetic varieties
  12. Toric schemes and canonical models
  13. Toric arithmetic varieties and arithmetic fans
  14. Proper toric morphisms
  15. Toric divisors
  16. ...and 10 more sections

Key Result

Theorem 1

There is a bijection between the cone of semipositive toric adelic divisors $\overline{\mathcal{D}}$ of $\mathcal{U}_S$ over $\mathbb{Z}$ and the cone of $v$-tuples $(\vartheta_v)$ of closed concave functions $\vartheta_v \colon M_{\mathbb{R}} \rightarrow \mathbb{R}_{-\infty}$ satisfying the followi

Theorems & Definitions (130)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.1.1
  • Definition 2.1.2
  • Remark 2.1.3
  • Theorem 2.1.4
  • Lemma 2.1.5
  • Definition 2.2.1
  • Proposition 2.2.2
  • ...and 120 more