Semipositive Torus-Invariant Singular Metric

• Semipositive torus-invariant singular metrics are Hermitian metrics defined on toric varieties that combine torus symmetry with semipositivity, ensuring nef curvature.
• They enable computation of local heights and arithmetic invariants by linking roof functions over moment polytopes with integrability criteria in both archimedean and non-archimedean settings.
• Their explicit construction aids in cases without Zariski decompositions, bridging complex geometry, convex analysis, and arithmetic applications.

A semipositive torus-invariant singular metric is a Hermitian metric with specific invariance and curvature properties, defined on toric (and often torus or abelian fibered) geometric structures. It plays a central role in arithmetic and complex geometry, particularly in the study of toric line bundles and their associated heights and invariants. Such metrics encode singularities (often tailored to the toric symmetries), and their semipositivity ensures nefness in the appropriate sense (such as positivity of curvature currents or approximability by nef models). The interplay between convex-analytic data and algebraic geometry is characteristic, providing explicit classification and computation frameworks.

1. Torus-Invariant Singular Metrics: Definition and Semipositivity

Let $X_\Sigma$ be a (quasi-)projective toric variety of dimension $d$ over a local field $K$, with dense torus $U \simeq \mathbb{G}_m^d$, and let $D$ be a toric Cartier divisor, with associated line bundle $L = \mathcal{O}_{X_\Sigma}(D)$. A singular Hermitian metric $\|\cdot\|$ on $L^{\mathrm{an}}$ is torus-invariant if, for a toric trivialization and a global section $s$, the function

$$g(p) = -\log \|s(p)\|^2,\quad p \in U^{\mathrm{an}}$$

is invariant under the compact subtorus $\mathbb{S} \subset U^{\mathrm{an}}$, and extends continuously off $|D|$.

Semipositivity, or nefness, is characterized as follows:

• In the archimedean case, $g$ is required to be locally integrable and plurisubharmonic ($\ddc g \geq 0$ as a current).
• In the non-archimedean case, semipositivity requires that $\|\cdot\|$ is the uniform limit of nef model metrics, equivalently that $\overline{D} = (D, g)$ lies in the closure of toric model divisors from nef $\mathbb{Q}$-models (Alvarez, 20 Jan 2026).

Koike’s construction gives explicit torus-invariant, semipositive singular metrics with minimal singularities for big toric line bundles and demonstrates their utility even where Zariski decompositions do not exist (Koike, 2013).

2. Convex-Analytic Classification via Moment Polytopes and Roof Functions

Toric semipositive singular metrics are classified through a convex-analytic correspondence:

• The support function $\Psi_D : N_\mathbb{R} \to \mathbb{R}$ is piecewise-linear and concave for nef divisors.
• The moment polytope $\Delta_D$ is

$$\Delta_D = \{ m \in M_\mathbb{R} : \langle m, v_\rho \rangle \geq \Psi_D(v_\rho) \;\forall \rho \}$$

When $D$ is nef, one establishes a bijection between nef toric compactified divisors and closed concave functions (roof functions) $$\vartheta : \Delta_D \to \mathbb{R} \cup \{-\infty\}$$. The metric is described by the roof function

$$\vartheta_{\overline{D}}(m) = \inf_{u \in N_\mathbb{R}} (\langle m, u \rangle - g(u))$$

which is concave and possibly $-\infty$ on the boundary if the metric is singular (Alvarez, 20 Jan 2026).

In the projective case, the equivalence between the metric data $g$, a continuous concave function $\gamma : N_\mathbb{R} \to \mathbb{R}$ (with $\Psi_D - \gamma$ bounded), and a continuous concave roof function $\vartheta : \Delta_D \to \mathbb{R}$ is established.

3. Local Heights and Their Convex-Analytic Expression

For a nef toric arithmetic divisor $\overline D = (D, g)$, the local toric height is given by

$$\hbar^{\mathrm{tor}}(X_\Sigma; \overline D) = (d+1)! \int_{\Delta_D} \vartheta_{\overline D}(m)\, d\mathrm{Vol}_M(m)$$

where $\mathrm{Vol}_M$ is the Haar volume on $M_\mathbb{R}$, normalized by the lattice. In the quasi-projective or singular setup, $\overline D$ can be approached as the limit of nef model divisors, and the same integral formula holds, provided the roof function is integrable ($\vartheta \in L^1(\Delta_D)$) (Alvarez, 20 Jan 2026).

If the integrability fails (i.e., if $\vartheta \notin L^1(\Delta_D)$), the local height can be $-\infty$, illustrating new phenomena compared to the continuous case.

4. Explicit Constructions, Minimality, and Nonexistence of Zariski Decompositions

In the setting of toric bundles over complex tori, minimal singular metrics are constructed explicitly by maximizing a family of plurisubharmonic (psh) weights indexed by a “nef polytope” $\Box_{\mathrm{Nef}}(L_0, h)$. The resulting metric is torus-invariant, semipositive ($c_1(L, h_{\min}) \geq 0$ in currents), and has minimal singularities among all semipositive singular metrics.

In Nakayama's examples, where the nef set is non-polyhedral or irrational, Zariski decompositions fail to exist, but the minimal semipositive metric remains well-defined and explicit (Koike, 2013).

5. Pathologies, Counterexamples, and Integrability Criteria

Allowing singularities in the metric extends the possible behavior of roof functions. For instance, $\vartheta$ may tend to $-\infty$ on parts of $\partial \Delta_D$, leading to local heights that can be $-\infty$.

Explicit examples (e.g., functions on the standard simplex or degenerate polytopal cases) exhibit both finite and infinite heights. The precise criterion for finiteness is the $L^1$-integrability of the roof function over the polytope (Alvarez, 20 Jan 2026). Sturm-type approximations are employed to handle general closed concave functions for convergence questions.

Conversely, in the context of relative Ricci-flat metrics on families of Calabi–Yau manifolds (such as elliptic fibrations), it has been shown that semipositivity does not generally hold: torus-invariant (translation-invariant) singular metrics fail to be semipositive in total spaces whenever the complex structure varies nontrivially. This disproves previous folklore conjectures and isolates specific geometric obstructions, ultimately tied to the collapse of the potential to a function depending only on base variables and the negative sign of the base curvature when the moduli vary (Cao et al., 2019).

6. Applications and Algebraic Consequences

Semipositive torus-invariant singular metrics serve as tools for resolving questions on the structure and invariants of toric and toric bundle varieties:

• Non-nef loci associated to these metrics are Zariski closed (by inspecting the set of points where the minimal psh weight's Lelong numbers are positive), and are unions of toric divisors, hence algebraic (Koike, 2013).
• The openness of multiplier ideals associated to such metrics (related to the Demailly–Kollár conjecture) is affirmed; explicit combinatorial/monomial descriptions can be given in toric charts.

The convex-analytic framework, together with basic properties of the roof function and moment polytope, enables efficient computation and comparison of local and arithmetic invariants in both the geometric and arithmetic context (Alvarez, 20 Jan 2026).

7. Broader Context, Generalizations, and Limitations

The concept of semipositive torus-invariant singular metrics unifies complex differential geometry, convex analysis, and arithmetic geometry. It provides a bridge between plurisubharmonic singular metrics, toric combinatorics, and analytic/arithmetic invariants. The flexibility in singularity structure allows for powerful generalizations—handling line bundles lacking Zariski decompositions and singular base/fiber settings—while the precise geometric obstructions (as in the breakdown of semipositivity in families with torus-invariant Ricci–flat metrics and nontrivial moduli variation) delineate the domain of validity for associated positivity conjectures (Cao et al., 2019).

Further research directions focus on density theorems for convex-analytic data, extensions to nonpolyhedral settings, and interplay with direct image flatness and additional bundle-theoretic positivity properties. The analytic and combinatorial techniques established for toric varieties have influenced approaches to more general settings, including abelian and Calabi–Yau fibered varieties.