Non-Archimedean Pluripotential Theory

• Non-Archimedean pluripotential theory is a framework studying psh functions on Berkovich spaces using semipositive model metrics and the Monge–Ampère operator.
• It employs variational principles and energy functionals to analyze measures and metrics, providing insights into stability and degenerations in algebraic geometry.
• The theory connects analytic techniques with algebraic K-stability, tropical geometry, and synthetic formulations to address both local and global geometric problems.

Non-Archimedean pluripotential theory is a framework for potential theory on Berkovich analytic spaces, modelled in close analogy with complex pluripotential theory but adapted to the non-Archimedean context. It centers on the paper of plurisubharmonic (psh) functions, semipositive metrics, and the Monge–Ampère operator on analytic spaces over non-Archimedean fields, with connections to K-stability, degenerations of algebraic varieties, and arithmetic geometry.

1. Berkovich Spaces, Model Metrics, and Plurisubharmonic Functions

Let $k$ be a complete non-Archimedean valued field, possibly with trivial valuation. For a $k$-scheme $X$ of finite type, the Berkovich analytification $X^{\mathrm{an}}$ consists of multiplicative seminorms on the coordinate ring, extending the valuation on $k$. This space is compact and Hausdorff when $X$ is proper.

A line bundle $L$ on $X$ admits the analytification $L^{\mathrm{an}}$ over $X^{\mathrm{an}}$. A continuous metric $\|\cdot\|$ on $L^{\mathrm{an}}$ assigns to each section $s$ a continuous function $x\mapsto \|s(x)\|$. Model metrics arise from line bundles on integral models of $X$, and a model metric is called semipositive if the associated model is nef on the special fiber.

A non-Archimedean plurisubharmonic function (abbreviated NA psh function) $\phi$ is a continuous function such that the metric $\|\cdot\|_0 e^{-\phi}$ is semipositive, i.e., a uniform limit of semipositive model metrics. Such psh functions are continuous and, for toric or retraction-invariant settings, restrict to convex functions on skeleta or dual complexes (Gil et al., 2016, Wu, 19 Jun 2024, Gil et al., 2021).

2. The Non-Archimedean Monge–Ampère Operator

For model metrics and their continuous limits, the first Chern form $c_1(L,\phi)$ is defined as a closed $(1,1)$-form in the sense of Chambert–Loir–Ducros. For projective varieties (or local cones), the Monge–Ampère operator assigns to an $n$-tuple of continuous psh metrics a Radon probability measure:

$$\mathrm{MA}(\phi_1,\ldots,\phi_n) = \frac{1}{V} \left(c_1(\phi_1)\wedge \cdots \wedge c_1(\phi_n)\right)$$

where $V = \deg_L X$. In the case of model metrics, the measure is atomic, supported at Berkovich points corresponding to irreducible components of the special fiber, with weights given by intersection theory (Gil et al., 2016, Boucksom, 11 Oct 2025).

Extension of this operator to broader classes, including upper semicontinuous psh functions and finite energy classes $\mathcal{E}^1$, relies on monotone convergence, decreasing nets of model metrics, and Hölder regularity estimates (Wu, 19 Jun 2024, Boucksom et al., 2023).

3. Energy Functionals and Variational Principles

The energy (or volume) functional plays a central role:

$$E(\phi) = \frac{1}{n+1} \sum_{j=0}^n \int_{X^{\mathrm{an}}} \phi \; c_1(\phi)^j \wedge c_1(\phi_0)^{n-j}$$

where $\phi_0$ is a reference metric. The Monge–Ampère measure is the first variation of this energy functional:

$$\frac{d}{dt}\Bigl|_{t=0} E(\phi + t f) = \int_{X^{\mathrm{an}}} f\, \mathrm{MA}(\phi)$$

This variational structure underlies differentiability of the non-Archimedean volume (asymptotic growth of small sections) and is essential for formulating the non-Archimedean Calabi–Yau and Kähler–Einstein problems (Gil et al., 2016, Boucksom et al., 2023).

Functionals $I$, $J$, and $M$ (Mabuchi functional), and their Legendre duals on spaces of finite energy measures, provide the bridge to stability questions in algebraic geometry. A functional is coercive if $F(\mu) \geq \varepsilon J(\mu) + A$, and coercivity thresholds correspond to (uniform) K-stability (Boucksom et al., 2023, Boucksom, 2018).

4. The Non-Archimedean Monge–Ampère Equation and K-Stability

The central equation is

$\mathrm{MA}(\phi) = \mu$

for a prescribed Radon probability measure $\mu$ of appropriate total mass. Existence and uniqueness theory relies on variational methods using the energy functional and psh envelopes, paralleling Yau's theorem in the Archimedean setting.

Boucksom–Favre–Jonsson's variational solution and the differentiability of the non-Archimedean volume yield orthogonality properties and regularity of psh envelopes. When $X$ is smooth, $L$ is ample, and the residue characteristic is zero, solutions exist and are unique up to additive constants for measures supported on skeleta (Gil et al., 2016).

Non-Archimedean pluripotential theory gives a variational and slope-theoretic interpretation of K-stability, connecting the positivity and coercivity of the non-Archimedean Mabuchi functional to the algebro-geometric stability of $(X, L)$ (Boucksom, 2018, Boucksom et al., 2023). The "slope formula" relates geodesic rays in the space of Kähler potentials to their non-Archimedean avatars, providing a precise link between energy asymptotics and stability invariants.

5. Local, Global, and Hybrid Extensions

The theory extends beyond projective settings. In the local (affine cone) setting with a polarized affine cone $X = \operatorname{Spec} R$ over a trivially valued field and Reeb field $\xi$, the framework adapts to compact links $X_0$ in the analytification, with $\xi$-equivariant Fubini–Study functions and associated energy, $I$, $J$, and Monge–Ampère structures. All core functional-analytic, comparison, and duality properties persist, with additional care for rational approximations and fixed polarization (Wu, 19 Jun 2024).

Global Pluripotential Theory for adelic line bundles connects families of Monge–Ampère measures on analytifications of projective arithmetic varieties to the quasi-projective case, characterizing non-degenerate subvarieties and constructing the global Monge–Ampère measure for adelic line bundles over Berkovich spaces (Morrow, 14 Jul 2025).

Hybrid spaces bridge the complex and non-Archimedean theories, furnishing tools to understand degenerations; psh metrics and Monge–Ampère measures converge in the hybrid analytification, linking complex and non-Archimedean models through tropical and skeleton maps (Boucksom, 11 Oct 2025).

6. Toric and Tropical Pluripotential Theory

The case of toric varieties admits a refined description. Tropical toric varieties provide a stratified real space $N_\Sigma$ equipped with tropical psh functions and Lagerberg calculus. There is a canonical correspondence between $S$-invariant psh functions on complex, tropical, and non-Archimedean toric varieties, with isomorphisms preserved by the Bedford–Taylor Monge–Ampère product. For invariant Monge–Ampère equations on toric and abelian varieties, explicit convex analytic methods solve the second boundary-value problem on $N_\Sigma$, and lift the solution to a semipositive metric on $X_\Sigma^{\mathrm{an}}$ (Gil et al., 2021).

7. Transcendental and Synthetic Formulations

Transcendental non-Archimedean metrics are constructed as projective limits over all ample classes, admitting operations including envelopes, Legendre–Fenchel transforms, and infimal convolution. The envelope conjecture, proven in this framework, states that the regularized supremum of bounded families of non-Archimedean psh metrics is again psh, paralleling the complex theory (Xia, 2023).

Synthetic pluripotential theory unifies Archimedean and non-Archimedean pluripotential concepts via a formalism built on test functions and multilinear intersection pairings, leading to complete metric structures on spaces of finite energy measures and providing natural dualities between energies and measures, as well as openness results for stability thresholds (Boucksom et al., 2023).

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The theory thus provides a comprehensive machinery for potential theory on Berkovich spaces, including the construction and analysis of Monge–Ampère operators, the structure of energy functionals, variational methods, existence and regularity of solutions to the Monge–Ampère equation, and deep connections to K-stability, tropical geometry, and arithmetic intersections. This framework is essential for the analysis of degenerations, uniform and divisorial stability, and extensions of complex geometric techniques to non-Archimedean analytic and arithmetic geometry.