Equidistribution for semiabelian varieties over number fields

Technical Summary: Equidistribution for Semiabelian Varieties over Number Fields

Problem Statement
This paper addresses the gap between the equidistribution of small points on semiabelian varieties and the classical Bogomolov conjecture, specifically in settings where the defining metrics are not "quasi-canonical" in the sense required by the Yuan–Zhang equidistribution theorem. While Yuan and Zhang established equidistribution for quasi-canonical adelic line bundles on quasi-projective varieties, and Kühne proved equidistribution for canonical semiabelian metrics, a significant class of metrics—specifically those on semiabelian compactifications governed by toric metrics that are semipositive, monocritical, and arithmetically T-effective—remained outside the direct scope of these theorems.

The central question is whether generic equidistribution and the Bogomolov property (that a non-special subvariety cannot contain a Zariski-dense set of points of arbitrarily small height) hold for these more general toric metrics. The difficulty lies in the fact that for such metrics, the global height of the variety may be strictly less than its essential minimum ($h_L(G) < \mu^{\text{ess}}_L(G)$), preventing the direct application of Yuan–Zhang's framework.

Methodology
The author, Wei Xue, develops a proof strategy that isolates the asymptotic estimates in Kühne's argument and reinterprets them through "explicit compression paths." The methodology proceeds in four distinct stages:

1. Local Trivialization and Adelic Construction: The paper extends Kühne's archimedean local trivialization to all places of a number field $K$. Using Raynaud–Bosch–Lütkebohmert uniformization for abelian varieties and $p$-adic theta functions, the author constructs an adelic line bundle $\mathcal{G}(D)$ on the semiabelian compactification $\overline{G}$ associated with a toric metrized divisor $D$. This construction ensures compatibility with the local trivializations at every place, allowing for a precise comparison between the height on the semiabelian variety and the absolute minimum on the toric part.

2. Canonical Metric Case: The paper first establishes equidistribution for the canonical metric case. By constructing an auxiliary $n$-division tower and extending Kühne's asymptotic estimates to arbitrary ample toric divisors, the author proves that generic small sequences equidistribute with respect to the canonical measure. This involves bounding Picard-zero translation terms and establishing a "canonical lower bound" for heights.

3. Quasi-Canonical Compression: For metrics that are quasi-canonical but not necessarily canonical, the paper introduces "explicit compression paths." The metric is compressed toward a quasi-canonical model singled out by its critical tropical center. The author constructs explicit compression parameters $m(n)$ and proves that the lower bounds, measure estimates, and arithmetic volume comparisons hold uniformly along these paths. This step effectively transports Kühne's Haar-slice and difference-morphism arguments from the standard $(\mathbb{P}^1)^t$ compactification to an arbitrary normal proper toric compactification $X_\Sigma$.

4. Monocritical Reduction: The final step handles the general monocritical case. Using the toric measure theory of Burgos Gil, Philippon, Rivera-Letelier, and Sombra (BGPS), the author shows that monocritical metrics can be reduced to quasi-canonical metrics via a "quasi-canonical replacement." This relies on the arithmetic T-effectivity of the metric, which fixes the origin in the stability set of the roof function. The limiting measure for the monocritical metric is shown to coincide with that of the associated quasi-canonical metric.

Key Contributions and Results

• Generic Equidistribution (Theorem 1.3): The paper proves that for a semiabelian variety $G$ over a number field, equipped with a line bundle $L = \mathcal{G}(D) \otimes \pi^*N$ where $D$ is a big, T-effective, semipositive, monocritical, and arithmetically T-effective toric metrized divisor, every generic $L$-small sequence satisfies $v$-adic equidistribution at every place $v$.
• Bogomolov Theorem (Theorem 1.4): The paper establishes the Bogomolov theorem for this specific metric class. It proves that if a closed subvariety $X \subseteq G$ satisfies $\mu^{\text{ess}}_L(X) = \mu^{\text{ess}}_L(G)$, then $X$ is a special subvariety (a translate of an algebraic subgroup by a torsion point). Conversely, if $G(K)$ contains $L$-special points, then $X$ is $L$-special if and only if $\mu^{\text{ess}}_L(X) = \mu^{\text{ess}}_L(G)$.
• Kühne Transport Assembly: A significant technical contribution is the "Kühne Transport Assembly" (Proposition 96). This package rigorously adapts Kühne's local-trivialization analysis, difference morphisms, and small-box contradiction arguments to arbitrary toric compactifications and quasi-canonical metrics. It demonstrates that the pushforward identities and Haar-slice descriptions used in the canonical case remain valid under these generalizations, provided the theta-factor conventions are correctly managed.
• Strict Equidistribution: The paper derives a strict equidistribution theorem (Theorem 102) for quasi-canonical metrics, showing that strict small sequences equidistribute toward the canonical probability measure. This is achieved by combining the Bogomolov property with the generic equidistribution result, avoiding circularity by using the "restricted quasi-canonical model" on minimal special envelopes.

Significance and Claims
The paper claims to extend the scope of equidistribution and Bogomolov theorems beyond the quasi-canonical range of Yuan–Zhang and the canonical range of Kühne. The contribution is not a new proof of the canonical theorem nor a result for all semipositive toric metrics. Instead, it targets the specific class of monocritical and arithmetically T-effective toric metrics.

The significance lies in the "comparison mechanism" established between canonical, quasi-canonical, and general toric metrics. By isolating the asymptotic estimates and reinterpreting them via compression paths, the author provides a bridge that allows the powerful machinery of Kühne's local-trivialization arguments to be applied to metrics that are not strictly quasi-canonical. The paper explicitly notes that the "general metric" referred to in the context of the Bogomolov application is restricted to this controlled class, and extending the argument to arbitrary semipositive metrics would require new replacement or deformation theorems beyond the current scope.

The work relies heavily on the interplay between tropical geometry (critical vectors, roof functions), arithmetic intersection theory (adelic line bundles, arithmetic volumes), and analytic uniformization (Raynaud extensions, theta functions). It successfully integrates the BGPS monocritical criteria with the semiabelian height decomposition to resolve the equidistribution problem for this specific, non-trivial class of metrics.