[Paper Review] Equidistribution of small subvarieties of an abelian variety

Matthew Baker, Su-Ion Ih|ArXiv.org|Apr 21, 2004

Algebraic Geometry and Number Theory5 references5 citations

TL;DR

This paper establishes a general equidistribution theorem for small, strict subvarieties in an abelian variety over a number field, extending the Szpiro-Ullmo-Zhang equidistribution theorem from points to higher-dimensional subvarieties. Using canonical heights and Arakelov geometry, it proves that such subvarieties become equidistributed with respect to the normalized Haar measure on the complex points of the abelian variety as their canonical height tends to zero.

ABSTRACT

We prove an equidistribution result for small subvarieties of an abelian variety which generalizes the Szpiro-Ullmo-Zhang theorem on equidistribution of small points.

Motivation & Objective

• To extend the Szpiro-Ullmo-Zhang equidistribution theorem from small points to small subvarieties of higher dimension in abelian varieties.
• To establish a canonical height theory for subvarieties using arithmetic intersection theory and hermitian line bundles.
• To prove that small, strict sequences of subvarieties equidistribute with respect to the normalized Haar measure on the complex points of the abelian variety.
• To resolve a higher-dimensional analog of the Bogomolov conjecture for subvarieties via equidistribution.
• To show that the equidistribution property holds under the strictness condition, which prevents concentration on proper torsion subvarieties.

Proposed method

• Define the height of a subvariety using arithmetic intersection theory on a model of the abelian variety over the ring of integers of a number field.
• Construct a sequence of models and hermitian line bundles with cubical metrics to define the canonical height $\hat{h}_L(Y)$ as a limit of arithmetic intersection numbers.
• Introduce the notion of a 'strict' sequence of subvarieties, meaning no subsequence lies in a proper torsion subvariety.
• Use the generalized Bogomolov conjecture (Zhang) to show that small, generic subvarieties cannot be Zariski dense in a non-torsion subvariety unless it is itself torsion.
• Apply weak convergence of measures via approximation by smooth functions and use the equidistribution of small points to deduce convergence of the measures $\mu_n$ to the Haar measure $\mu$.
• Prove equidistribution by contradiction and limiting arguments, using the fact that the limit of the measures $\mu_n$ must equal the Haar measure $\mu$.

Experimental results

Research questions

• RQ1 Can the equidistribution of small points on abelian varieties be generalized to higher-dimensional subvarieties?
• RQ2 What conditions ensure that a sequence of small subvarieties becomes equidistributed with respect to the Haar measure?
• RQ3 How does the canonical height of a subvariety relate to its equidistribution behavior?
• RQ4 Is there a higher-dimensional analog of the Bogomolov conjecture that prevents small subvarieties from being Zariski dense in non-torsion subvarieties?
• RQ5 Under what conditions does a small sequence of subvarieties fail to equidistribute, and how can such obstructions be ruled out?

Key findings

• The canonical height $\hat{h}_L(Y)$ of a subvariety $Y$ is defined as the limit of arithmetic intersection numbers involving hermitian line bundles with cubical metrics.
• A sequence of subvarieties is equidistributed with respect to the normalized Haar measure $\mu$ on $A(\mathbb{C})$ if it is small and strict.
• The equidistribution result holds uniformly for all continuous functions $f$ on $A(\mathbb{C})$, with $\int f \, d\mu_n \to \int f \, d\mu$.
• The strictness condition ensures that no subsequence lies in a proper torsion subvariety, which is essential to avoid concentration on lower-dimensional tori.
• The proof relies on the generalized Bogomolov conjecture (Zhang) to show that small, generic subvarieties in a non-torsion subvariety cannot be Zariski dense unless the subvariety is torsion.
• The equidistribution of small points (Szpiro-Ullmo-Zhang) is used as a key ingredient in the proof, extended via approximation and limiting arguments to subvarieties.