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Heights on toric varieties for singular metrics: Local theory

Gari Y. Peralta Alvarez

TL;DR

The paper advances the local Arakelov theory for toric varieties by incorporating singular torus-invariant metrics through toric compactified divisors. It builds a convex-analytic dictionary between nef toric compactified arithmetic divisors and concave roof functions on associated polytopes, enabling a precise integral formula for the local toric height: $h^{tor}(U;\overline{D})=(d+1)\int_{\Delta_{D}}\vartheta_{\overline{D}}$. Finite height is characterized by $\vartheta_{\overline{D}}\in L^{1}(\Delta_{D})$, and the framework recovers and extends the mixed-energy perspective of Burgos–Kramer–Kühn. The results hold over local fields in archimedean and non-archimedean settings and align toric geometry with Yuan–Zhang’s adelic-divisor theory, offering concrete tools for computations in singular Arakelov geometry on toric varieties. This lays groundwork for a global theory in Part II and has potential applications to explicit height calculations in arithmetic geometry of toric spaces.

Abstract

We show that the (toric) local height of a toric variety with respect to a semipositive torus-invariant singular metric is given by the integral of a concave function over a compact convex set. This generalizes a result of Burgos, Philippon, and Sombra for the case of continuous metrics and answers a question raised by Burgos, Kramer, and Kühn in 2016.

Heights on toric varieties for singular metrics: Local theory

TL;DR

The paper advances the local Arakelov theory for toric varieties by incorporating singular torus-invariant metrics through toric compactified divisors. It builds a convex-analytic dictionary between nef toric compactified arithmetic divisors and concave roof functions on associated polytopes, enabling a precise integral formula for the local toric height: . Finite height is characterized by , and the framework recovers and extends the mixed-energy perspective of Burgos–Kramer–Kühn. The results hold over local fields in archimedean and non-archimedean settings and align toric geometry with Yuan–Zhang’s adelic-divisor theory, offering concrete tools for computations in singular Arakelov geometry on toric varieties. This lays groundwork for a global theory in Part II and has potential applications to explicit height calculations in arithmetic geometry of toric spaces.

Abstract

We show that the (toric) local height of a toric variety with respect to a semipositive torus-invariant singular metric is given by the integral of a concave function over a compact convex set. This generalizes a result of Burgos, Philippon, and Sombra for the case of continuous metrics and answers a question raised by Burgos, Kramer, and Kühn in 2016.
Paper Structure (23 sections, 94 theorems, 232 equations)

This paper contains 23 sections, 94 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. The local Arakelov theory of proper toric varieties
  3. Singular metrics
  4. Main results
  5. Acknowledgments
  6. Notation and conventions
  7. Review of Yuan-Zhang's theory of compactified divisors
  8. The geometric case
  9. The local arithmetic case
  10. Nefness, measures and local heights.
  11. A toric geometric analog of Yuan-Zhang's theory
  12. Toric compactified divisors
  13. The case over a field
  14. The case over a DVR
  15. A toric local arithmetic analog of Yuan-Zhang's theory
  16. ...and 8 more sections

Key Result

Theorem 1

The assignment $\overline{D} \mapsto \vartheta_{\overline{D}}$ induces a bijective correspondence The singularities of the associated metric are determined by the growth of $\vartheta_{\overline{D}}$ along the boundary of the compact convex set $\Delta_{D}$ associated to the toric compactified divisor $D$.

Theorems & Definitions (255)

  • Theorem 1
  • Theorem 2
  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Remark 2.1.6
  • Definition 2.1.7
  • Definition 2.1.9
  • ...and 245 more