Bogomolov Conjecture in Semiabelian Varieties

• Bogomolov Conjecture for semiabelian varieties is a central diophantine statement characterizing subvarieties that admit dense sets of points with arbitrarily small canonical heights.
• The approach employs advanced analytic techniques including equidistribution of small points and birational mappings to reduce the complex geometry to manageable abelian cases.
• Recent progress extends the conjecture’s validity to function fields and globally valued fields, offering new insights into Manin–Mumford and Mordell–Lang applications.

A semiabelian variety is an algebraic group that generalizes abelian varieties by allowing extensions of abelian varieties by algebraic tori. The Bogomolov Conjecture for semiabelian varieties is a central statement characterizing subvarieties that contain Zariski-dense sets of points of small canonical height. Its importance lies in diophantine geometry, especially in the study of unlikely intersections, equidistribution of small points, and arithmetic dynamics. Recent advances establish the conjecture over function fields and globally valued fields (GVFs), revealing new phenomena unique to the semiabelian setting.

1. Definitions and Algebraic Framework

Let $k$ be an algebraically closed field, $\mathcal{B}$ a normal projective $k$-variety with function field $K = k(\mathcal{B})$, and $G/K$ a semiabelian variety. $G$ is defined by an exact sequence

$$0 \to \mathbb{G}_m^t \to G \xrightarrow{\pi} A \to 0,$$

where $A/K$ is abelian of dimension $g$ and $t \geq 0$. Extensions are classified by $\eta \in (A^\vee(K))^t$. Choosing an ample symmetric line bundle $N$ on $A$, the theory produces a canonical (Néron–Tate) height $\hat h_L$ on $G$. For subvarieties $X \subset G$, the essential minimum

$$\operatorname{ess}_L(X) = \sup_{U \subset X \text{ open}} \inf_{x \in U(K)} \hat h_L(x)$$

is a key invariant. The stabilizer $\operatorname{Stab}(X)$ consists of $g \in G$ with $g + X = X$, and $X$ is constant if pulled back from a $k$-model.

In the more general context of globally valued fields (GVFs), a height function $h$ is fixed on $K$, satisfying axioms analogous to the number field or function field case, and admitting a notion of canonical height attached to $G$ via an ample symmetric line bundle on $A$ and toric divisors coming from the torus part (Hultberg, 8 Jan 2026).

2. Statement of the Bogomolov Conjecture for Semiabelian Varieties

The conjecture can be stated as follows (Luo et al., 12 May 2025, Kühne, 2018, Hultberg, 8 Jan 2026): Let $G/K$ be a semiabelian variety, $L$ its canonical line bundle, and $X \subset G$ a closed subvariety. Then the following are equivalent:

• For every $\epsilon > 0$, the set $$X(\epsilon) = \{x \in X(K) \mid \hat h_L(x) \leq \epsilon\}$$ is Zariski dense in $X$.
• Modulo its stabilizer, $X$ is a torsion translate of a constant subvariety; precisely, writing $\widetilde G = G/\operatorname{Stab}_0(X)$, there exist a torsion point $\xi \in \widetilde G(K)$, a semiabelian $G_0$ over $k$, a finite kernel homomorphism $h: (G_0)_k \otimes K \to \widetilde G$, and a $k$-subvariety $X_0 \subset G_0$ such that

$$X/\operatorname{Stab}_0(X) = h(X_0 \otimes_k K) + \xi.$$

In this context, $X$ is called special. Thus, outside the special locus, all points have a uniform positive lower bound for $\hat h_L$.

3. Proof Strategy and Key Techniques

The proof of the conjecture consists of several structural reductions and analytic arguments:

1. Reduction to Trivial Stabilizer: Passing to the quotient by $\operatorname{Stab}_0(X)$, both the abundance of small points and the special property are invariant (Lemma 3.2, Prop 3.1 in (Luo et al., 12 May 2025)).
2. Quasi-Split Case: If $G \cong G_0 \times_k A_1$ with $A_1$ of trivial $K/k$-trace, Yamaki's relative height techniques and prior results for abelian varieties (Gubler, Xie–Yuan) imply that $X$ is a product of a constant variety with a torsion translate (Prop 3.7, Cor 3.8 in (Luo et al., 12 May 2025)).
3. Birational Faltings–Zhang Maps: In the general case, maps

$$\alpha_n, \beta_n: G^n_{/A} \to (\mathbb{G}_m^t)^{n-1} \times A, \; (\mathbb{G}_m^t)^{n-1} \times G$$

encode differences and project to the abelian quotient. For $n \gg 0$, these become birational on $X^n_{/S}$ (Sec 4.3 in (Luo et al., 12 May 2025)).

1. Vanishing Lemmas and Intersection Theory: Applying Lemma 4.1, intersections involving boundary divisors and pullbacks of $N$ vanish when $\operatorname{ess}_L(X) = 0$, constraining the geometry of $X$ via intersection numbers.
2. Constant–Torsion Test: Theorem 4.3 enforces that, under the vanishing conditions, $G$ is quasi-split over $k$ and $X$ is special.

An essential innovation is that, unlike the abelian case, special subvarieties can fail to have Zariski-dense sets of height-0 points: the relevant criterion is the abundance of arbitrarily small points (Sec 1.2, Ex 1.3 in (Luo et al., 12 May 2025)).

4. Equidistribution and the Gap Principle

For number fields and more generally for GVFs, equidistribution of small points is a fundamental analytic tool (Kühne, 2018, Hultberg, 8 Jan 2026). For a sequence of small points $(x_i)$ with $\hat h_L(x_i) \to 0$ that is $X$-generic, analytic measures on the Berkovich (resp. complex) analytifications equidistribute to canonical measures constructed from $L$. The proof over arbitrary semiabelian varieties is nontrivial because canonical heights of non-split $G$ can be negative, circumvented by an asymptotic isogeny scaling and a twisting argument.

The new gap principle (Hultberg, 8 Jan 2026) states: for $X$ with a finite stabilizer generating $G$, the locus of points

$$\{ P \in X(K) \mid \widehat h(P) \leq c_1 \max\{1, h(G)\} \}$$

for a constant $c_1$ is contained in a proper Zariski-closed subset of $X$. This gap principle is shown to be logically equivalent to the Bogomolov conjecture via model-theoretic ultraproduct arguments and height continuity, unifying the arithmetic height lower bounds and the geometric structure theorem for small points.

5. Reductions and Extensions: Positive Characteristic, Abelian Case

The proof in characteristic zero reduces the Bogomolov conjecture for semiabelian varieties to the already established abelian case: through isogeny invariance, passage to quasi-split forms, and product decompositions, the general case is composed from abelian, toric, and constant components (Hultberg, 8 Jan 2026). In positive characteristic, the reduction also applies, but the abelian Bogomolov conjecture beyond elliptic curves remains open, so unconditional results are currently restricted.

The following table summarizes logical dependencies for the conjecture in various settings:

Field Type	Semiabelian BC Proven?	Reduction
Char 0 GVF	Yes	Reduced to abelian BC
Number field	Yes	Szpiro-Ullmo-Zhang, (Kühne, 2018)
Function field	Yes	(Luo et al., 12 May 2025) (general semiabelian)
Char $p>0$ GVF	Partial (elliptic quotients)	Reduced to abelian BC

6. Consequences, Applications, and Open Questions

The establishment of the Bogomolov conjecture for semiabelian varieties has several consequences:

• Manin–Mumford and Mordell–Lang: The strong equidistribution of small points implies the Manin–Mumford conjecture for semiabelian varieties whose nowhere-degenerate part is constant, and facilitates diophantine applications via the Mordell–Lang framework (Luo et al., 12 May 2025, Kühne, 2018).
• Quantitative and Effective Results: Further progress hinges on effectivity in height bounds and quantitative versions of the gap principle, connecting to Zhang inequalities and Arakelov geometry (Luo et al., 12 May 2025, Hultberg, 8 Jan 2026).
• Positive Characteristic Obstructions: A central open conjecture remains the full Bogomolov statement for abelian varieties of dimension $\ge2$ in positive characteristic; unrestricted results for semiabelian varieties await its resolution (Hultberg, 8 Jan 2026).
• Model-Theoretic Questions: The possible existence of a model companion for GVFs may clarify or simplify ultraproduct-based arguments and compactness methods.

A new geometric phenomenon in the semiabelian setting—the potential sparsity of height-0 points even on special subvarieties—reflects genuine toric effects absent from the pure abelian case, signaling departures in the arithmetic and geometric theory of small points.

7. References and Further Reading

• Wenbin Luo, Jiawei Yu, "Geometric Bogomolov conjecture for semiabelian varieties" (Luo et al., 12 May 2025)
• Lars Kühne, "Points of Small Height on Semiabelian Varieties" (Kühne, 2018)
• Will Sawin, "New gap principle for semiabelian varieties using globally valued fields" (Hultberg, 8 Jan 2026)