Heights on toric varieties for singular metrics: Global theory

This paper establishes a toric analog of Yuan and Zhang's theory of adelic divisors to show that the arithmetic self-intersection number of a semipositive toric adelic divisor equals the integral of a concave function over a compact convex set, thereby enabling the computation of heights for toric arithmetic varieties with singular metrics.

The Gist

Imagine you are an architect trying to measure the "complexity" or "size" of a building. In the world of pure math, these buildings are called varieties, and the tools we use to measure them are called heights.

For a long time, mathematicians could only measure these buildings if they were perfectly smooth and well-behaved (like a pristine, modern skyscraper). But in the real world of number theory, many of these "buildings" are rough, jagged, or have holes in them (singularities). Trying to measure a crumbling, ancient ruin with a ruler designed for a glass tower doesn't work.

This paper, by Gari Y. Peralta Alvarez, is like inventing a new, ultra-flexible measuring tape that can handle any kind of building, even the most broken-down ones, provided the building has a specific, symmetrical shape called a Toric Variety.

Here is the breakdown of the paper's journey, using simple analogies:

1. The Problem: Measuring the Unmeasurable

Think of a Toric Variety as a building constructed entirely from a grid of identical, repeating blocks (like a giant, symmetrical Lego castle). Because of this symmetry, mathematicians have a special trick: they can translate the shape of the building into a simple geometric shape on a piece of paper, called a polytope (a multi-sided polygon).

• The Old Way: Previously, if the "roof" of the building (the metric) was smooth, they could calculate the building's height by integrating a simple curve over that polygon.
• The New Problem: Sometimes, the roof is jagged, has spikes, or even holes. The old math breaks down. It's like trying to calculate the volume of a pile of sand using a formula for a solid brick.

2. The Solution: The "Adelic" Toolkit

The author uses a powerful new toolkit developed by Yuan and Zhang called Adelic Divisors.

• The Analogy: Imagine you want to describe a mountain. You can't just look at it from one spot. You need to look at it from every angle, at every time of day, and from every distance (local views) and then stitch all those views together into one global picture.
• Adelic Divisors are exactly that: a way to describe a mathematical object by looking at it through every possible "lens" (every prime number and every real number) simultaneously.

The author's big breakthrough is taking this complex, global toolkit and applying it specifically to our symmetrical Lego castles (Toric Varieties).

3. The Magic Trick: Turning Math into Geometry

The paper's main result is a beautiful translation. It says:

"To measure the complexity of this jagged, singular building, you don't need to do hard calculus. You just need to draw a specific concave function (a shape that curves downward like a frown or a dome) on a convex set (a shape where you can draw a straight line between any two points inside it)."

• The Roof Function: Think of the building's "roof" not as a physical roof, but as a landscape map. If the building is smooth, the map is a gentle hill. If the building is broken, the map might have cliffs or deep valleys.
• The Formula: The paper proves that the "height" (the complexity) of the building is simply the volume under this landscape map.
  • Old Math: "Sum up all the local measurements."
  • New Math: "Calculate the area under this curve."

4. Why This Matters: The "Singular" Cases

The most exciting part of the paper is that it works even when the "roof" is singular (broken).

• The Analogy: Imagine a roof made of glass. If it's perfect, you can measure it easily. But what if it's shattered?
  • Previous theories said, "We can't measure this; the glass is too broken."
  • Burgos and Kramer (2024) said, "We can measure it if the glass is only slightly broken (logarithmic singularities)."
  • This Paper says, "We can measure it even if the glass is shattered into dust!"

The author constructs examples of "buildings" where the roof is so broken that previous methods failed completely, yet their new formula still gives a finite, meaningful number.

5. The "Global" vs. "Local" Dance

The paper connects two worlds:

1. The Local World: Looking at the building through the lens of a single prime number (like looking at a brick through a magnifying glass).
2. The Global World: Looking at the whole building at once.

The author shows that the "Global Height" is just the sum of all these "Local Heights," but calculated in a way that respects the jagged nature of the roof. It's like saying the total weight of a pile of rocks is the sum of the weights of individual rocks, even if some rocks are crumbling into dust.

Summary: The Big Picture

This paper is a bridge.

• On one side: The messy, complex world of Singular Metrics (broken, jagged mathematical objects).
• On the other side: The clean, simple world of Convex Geometry (shapes, areas, and integrals).

The author builds a bridge between them by showing that for symmetrical objects (Toric Varieties), you can ignore the messy details of the "breaks" and simply calculate the volume under a specific curve. This allows mathematicians to finally compute the "heights" of objects that were previously considered too broken to measure, opening the door to solving deep problems in number theory and geometry.

In a nutshell: The author figured out how to measure the complexity of a broken, symmetrical mathematical structure by turning the problem into a simple geometry problem: "How much space is under this curved roof?"

Technical Summary

Here is a detailed technical summary of the paper "Heights on Toric Varieties for Singular Metrics: Global Theory" by Gari Y. Peralta Alvarez.

1. Problem Statement and Motivation

Context:
The paper addresses the computation of arithmetic heights and intersection numbers for quasi-projective toric arithmetic varieties equipped with singular metrics.

• Background: Classical Arakelov geometry (Gillet-Soulé) defines heights via smooth hermitian metrics on line bundles over projective models. However, many natural geometric objects (e.g., Hodge bundles, Siegel-Jacobi forms) carry metrics with singularities (logarithmic or worse) along the boundary.
• Existing Frameworks:
  • Yuan-Zhang [YZ26]: Developed a theory of adelic divisors on quasi-projective varieties using a completion of the direct limit of model divisors. This allows for singular metrics but results in a complex, abstract group structure where explicit computations are difficult.
  • Burgos-Kramer [BK24]: Extended Yuan-Zhang's theory using "mixed relative energy" to handle weaker positivity conditions, but their framework relies heavily on analytic tools (Berkovich spaces, relative energy) and is less amenable to explicit combinatorial calculations.
  • Burgos-Philippon-Sombra [BPS11, BMPS16]: Provided a powerful convex-analytic description for projective toric varieties, where arithmetic intersection numbers are computed as integrals of concave "roof functions" over convex polytopes.

The Gap:
While the convex-analytic approach works beautifully for projective toric varieties with smooth metrics, and the adelic approach handles quasi-projective varieties with singular metrics, there was no unified framework that combined the explicit convex-analytic formulas of the toric setting with the generality of Yuan-Zhang's adelic divisors for singular metrics on quasi-projective varieties.

Objective:
To develop a toric analog of Yuan-Zhang's theory of adelic divisors. The goal is to extend the convex-analytic descriptions of Arakelov geometry to the quasi-projective setting, allowing for the explicit computation of heights and intersection numbers for line bundles with toric singular metrics.

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2. Methodology

The author constructs a bridge between the abstract theory of adelic divisors and the combinatorial geometry of toric varieties.

A. Toric Adelic Divisors

The paper defines the group of toric adelic divisors, denoted $\text{Div}_T(U_S/\mathcal{O}_K)$, for a split torus $U_S$ over a ring of $S$-integers.

• Construction: It is defined as the completion of the direct limit of toric model arithmetic divisors over all toric projective models of $U_S$, with respect to a boundary norm.
• Arithmetic Fans: A key innovation is the use of arithmetic fans (a collection of fans in $N_\mathbb{R} \times \mathbb{R}_{\geq 0}$ indexed by primes) to describe projective toric arithmetic varieties. This allows the author to translate geometric properties of models into combinatorial data.

B. Tropicalization and Green's Functions

The core methodology relies on the tropicalization functor ($\text{Trop}_v$).

• Tropical Green's Functions: For a toric divisor, the Green's function (metric) is $S$-invariant. The author shows that such functions descend to continuous functions on the tropical toric variety (a polyhedral complex).
• Tropical Green's Functions as Limits: A toric adelic divisor is represented by a Cauchy sequence of model divisors. The author proves that the tropical Green's functions of these models converge in a specific Banach space ($G(N_\mathbb{R})$) to a limit function $\gamma_D = (\gamma_{D,v})_v$.
• Legendre-Fenchel Duality: The tropical Green's functions are related to roof functions ($\vartheta$) via the Legendre-Fenchel transform. This duality is the engine that converts analytic data (metrics) into convex geometry (concave functions on polytopes).

C. Approximation and Convergence

To handle singularities and unbounded functions, the paper employs:

• Decreasing Sequences: Approximating general adelic divisors by sequences of "model" divisors.
• Monotone Convergence: Using measure-theoretic tools to interchange limits and integrals, ensuring that the integral formulas remain valid even when the roof functions are unbounded or the number of non-zero terms in the sum grows.

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3. Key Contributions and Results

1. Characterization of Semipositive Toric Adelic Divisors (Theorem 1 / Corollary 5.1.3)

The paper establishes a bijection between the cone of semipositive toric adelic divisors and a specific cone of $v$-tuples of closed concave functions (roof functions) $\vartheta_v: M_\mathbb{R} \to \mathbb{R} \cup \{-\infty\}$.

• Conditions: These functions must satisfy specific rationality conditions at finite places, share a common compact convex domain $\Delta$, and exhibit asymptotic behavior where they vanish outside a finite set of places (up to a small error).
• Significance: This provides a concrete, combinatorial classification of adelic divisors, replacing the abstract Cauchy sequence definition with explicit functions.

2. Characterization of Nefness (Theorem 2 / Theorem 5.2.10)

A semipositive toric adelic divisor $D$ is nef if and only if its global roof function $\vartheta_D = \sum \delta_v \vartheta_{D,v}$ is non-negative everywhere on its domain $\Delta_D$.

• This extends the classical result for projective varieties to the singular, quasi-projective setting.

3. The Integral Formula for Arithmetic Self-Intersection (Theorem 3 / Theorem 6.1.3)

This is the main result. For a semipositive toric adelic divisor $D$ with a well-defined global roof function $\vartheta_D$ (which exists if $D$ is generically big), the arithmetic self-intersection number is given by:
$$\widehat{\text{deg}}(D^{d+1}) = (d+1)! \int_{\Delta_D} \vartheta_D(y) \, d\text{Vol}_M(y)$$
Equivalently, as a sum over places:
$$\widehat{\text{deg}}(D^{d+1}) = (d+1)! \sum_{v \in M(K)} \delta_v \int_{\Delta_D} \vartheta_{D,v}(y) \, d\text{Vol}_M(y)$$

• Finiteness: The intersection number is finite if and only if $\vartheta_D \in L^1(\Delta_D)$.
• Generality: This formula holds even if the roof functions are unbounded (highly singular metrics) and even if the divisor is not nef (though the result may be $-\infty$).

4. Compatibility and Improvements over Existing Theories

• vs. Yuan-Zhang: The paper proves that for nef divisors, the toric intersection numbers coincide with Yuan-Zhang's. However, the toric theory is more general: it allows for unbounded roof functions (singular metrics) where Yuan-Zhang's original formulation might require boundedness or specific regularity.
• vs. Burgos-Kramer: The paper shows that the Burgos-Kramer generalized intersection product (using mixed relative energy) coincides with the toric integral formula in the toric case.
• Crucial Improvement: The author constructs examples (Section 6.2) of toric adelic divisors where:
  1. The divisor is not nef and not integrable (in the sense of Burgos-Kramer).
  2. All local roof functions are unbounded (singularities worse than logarithmic).
  3. The arithmetic self-intersection number is finite.
  • Significance: These examples demonstrate that the Burgos-Kramer theory (which relies on relative energy and often assumes nefness or specific energy bounds) cannot compute these numbers, whereas the toric convex-analytic approach succeeds.

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4. Significance and Impact

1. Explicit Computability: The paper transforms the abstract theory of adelic divisors on quasi-projective varieties into a computable convex-analytic framework. Researchers can now calculate heights for singular metrics on toric varieties by simply integrating concave functions over polytopes.
2. Handling Strong Singularities: It successfully handles metrics with singularities that are "worse than logarithmic" (e.g., those appearing in Siegel-Jacobi forms) and where the divisor is not nef, a regime previously inaccessible to standard intersection theories.
3. Unification: It unifies the combinatorial elegance of toric geometry (Burgos-Philippon-Sombra) with the global flexibility of Yuan-Zhang's adelic divisors.
4. New Examples: The construction of divisors with finite intersection numbers despite having unbounded local roof functions and non-nef properties provides new test cases for conjectures in arithmetic geometry (e.g., Bogomolov, Mordell-Lang) and highlights the limitations of previous theories.
5. Foundation for Future Work: By establishing the "Global Theory" for toric varieties, this paper sets the stage for applying these convex-analytic tools to broader classes of varieties and potentially to non-toric settings via toric degenerations.

In summary, Peralta Alvarez provides a robust, explicit, and computationally powerful extension of Arakelov geometry to singular metrics on quasi-projective toric varieties, resolving the tension between abstract adelic definitions and concrete convex-analytic formulas.