[Paper Review] Arithmetic geometry of toric varieties. Metrics, measures and heights

José Ignacio Burgos Gil, Patrice Philippon+1 more|arXiv (Cornell University)|May 27, 2011

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes a formula expressing the height of a toric variety with respect to a toric metrized line bundle as an integral over a polytope of an adelic family of concave functions. By leveraging Arakelov geometry and convex analysis, the authors link metrics, measures, and heights on toric varieties to objects like Legendre-Fenchel duality, real Monge-Ampère measures, and polyhedral complexes, providing a closed-form computation for heights in terms of piecewise affine functions and linear forms.

ABSTRACT

We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Ampère measures, and Legendre-Fenchel duality. We also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height of toric projective curves with respect to the Fubini-Study metric, and of some toric bundles.

Motivation & Objective

• To express the height of a toric variety with respect to a toric metrized line bundle as an integral over a polytope.
• To establish a bridge between Arakelov geometry of toric varieties and convex analysis.
• To develop tools for computing heights using adelic families of concave functions and their duals.
• To provide explicit formulas for heights in geometrically meaningful cases, such as Fubini-Study metrics and toric curves.
• To generalize height theory to toric schemes over discrete valuation rings and study positivity properties of T-Cartier divisors.

Proposed method

• Utilize metrized line bundles on toric varieties over adelic fields, distinguishing Archimedean and non-Archimedean cases.
• Apply Berkovich analytic spaces to model non-Archimedean metrics and define associated measures.
• Employ Legendre-Fenchel duality to translate geometric data into convex analytic objects like concave functions and their superdifferentials.
• Introduce a closed-form integration formula for compositions of univariate functions with linear forms over polytopes.
• Use piecewise affine concave functions and their V-representations to compute mixed Monge-Ampère measures.
• Relate the height of a toric variety to the integral of a family of concave functions over a polytope, leveraging the real Monge-Ampère measure.

Experimental results

Research questions

• RQ1 How can the height of a toric variety be expressed in terms of convex geometry and adelic functions?
• RQ2 What is the precise relationship between metrized line bundles on toric varieties and concave functions on polytopes?
• RQ3 How do Legendre-Fenchel duality and Monge-Ampère measures arise naturally in the context of toric Arakelov geometry?
• RQ4 Can explicit formulas be derived for the height of toric varieties under specific metrics, such as Fubini-Study?
• RQ5 What are the positivity and integrability conditions on toric schemes and T-Cartier divisors that ensure well-defined height invariants?

Key findings

• The height of a toric variety with respect to a toric metrized line bundle is given by the integral over a polytope of an adelic family of concave functions.
• A closed-form formula is derived for the integral of a univariate function composed with a linear form over a polytope, enabling explicit height computations.
• The height of a toric projective curve with respect to the Fubini-Study metric is computed explicitly using the integration formula.
• The height of toric bundles and polarized toric varieties is expressed via mixed Monge-Ampère measures of concave functions on polyhedral complexes.
• The authors establish a correspondence between semipositive metrics on toric varieties and concave functions via Legendre-Fenchel duality.
• The real Monge-Ampère measure of a concave function on a polytope is shown to coincide with the measure associated to a semipositive metrized line bundle.