--- title: Non-Archimedean Monge-Ampère Equation url: https://www.emergentmind.com/topics/non-archimedean-monge-ampere-equation type: topic --- # Non-Archimedean Monge-Ampère Equation The non-Archimedean Monge-Ampère equation is the non-Archimedean analogue of the complex Monge-Ampère equation, posed on Berkovich analytic spaces over non-Archimedean fields. It prescribes the Monge-Ampère measure associated to a continuous semipositive metric on an ample line bundle, capturing both potential-theoretic and intersection-theoretic content in non-Archimedean pluripotential theory. The equation and its variational formulation occupy a central role in non-Archimedean geometry, particularly in the study of Calabi-Yau type problems, degenerations, and tropical methods. ## 1. Berkovich Spaces, Metrics, and the Chambert-Loir Monge-Ampère Measure Let $K$ be a complete discretely valued non-Archimedean field of residue characteristic zero, $R$ its valuation ring, $X$ a smooth projective variety of dimension $n$ over~$K$, and $L$ an ample line bundle on~$X$. The analytification $X^\mathrm{an}$ in the sense of Berkovich is a compact Hausdorff space whose points are real valuations on the coordinate ring of~$X$ extending the given norm. For a model $(\mathcal{X}, \mathcal{L})$ of $(X, L)$, with $\mathcal{L}$ a line bundle on the projective $R$-scheme $\mathcal{X}$ extending $L$, one defines a "model metric" $\varphi_{\mathcal{L}}$ on $L^\mathrm{an}$ where $|s|_{\varphi_{\mathcal{L}}}=1$ on the locus where $s$ is regular for local sections~$s$ of $m\mathcal{L}$. A model metric is **semipositive** if $\mathcal{L}|_C$ has nonnegative degree on every irreducible curve $C$ of the special fiber $\mathcal{X}_k$; equivalently, $\mathcal{L}$ can be chosen nef on $\mathcal{X}_k$ (Zhang's criterion). Chambert-Loir's Monge-Ampère measure for a semipositive model metric on $L^\mathrm{an}$ is defined using intersection theory on models. If $\varphi$ is determined by $(\mathcal{X}, \mathcal{L})$ and the special fiber $\mathcal{X}_k = \sum_i b_i E_i$, one sets \[ \mathrm{MA}(\varphi) = \sum_i b_i \left( c_1(\mathcal{L}|_{E_i})^n \right) \delta_{x_{E_i}}, \] where $x_{E_i}$ is the divisorial point associated to $E_i$. For general continuous semipositive metrics, $\mathrm{MA}(\varphi)$ is defined as the limit of Monge-Ampère measures of approximating model metrics in the uniform topology [1502.05724]. The total mass is always $(L^n)$. ## 2. The Non-Archimedean Monge-Ampère Equation: Statement and Solutions The non-Archimedean Monge-Ampère equation seeks $\varphi \in \mathrm{PSH}^0(L^\mathrm{an})$ (the space of continuous semipositive metrics, i.e., uniform limits of semipositive model metrics) such that \[ \mathrm{MA}(\varphi) = \mu, \] for a given positive Radon measure $\mu$ on $X^\mathrm{an}$ with $\mu(X^\mathrm{an}) = (L^n)$. This is the non-Archimedean Calabi-Yau problem. Existence and uniqueness results are given by Boucksom–Favre–Jonsson and rely on the variational approach [1502.05724]. - **Existence:** If $X$ is defined over the fraction field of the completion of a smooth $k$-curve and $\mu$ is supported on the dual complex of some SNC model, a continuous semipositive solution exists. - **Uniqueness:** Any two solutions differ by an additive constant. The solution space exhibits compactness properties: the quotient $\mathrm{PSH}(L^\mathrm{an})/\mathbb{R}$ is compact in the topology of uniform convergence on dual complexes, and regularization by decreasing nets of semipositive model metrics is always possible. ## 3. Variational Method and Pluripotential Theory The variational strategy extends the Aubin–Mabuchi functional to the non-Archimedean setting. Fix a reference semipositive model metric $\varphi_0$; the energy functional is \[ E(\varphi) = \frac{1}{n+1} \sum_{j=0}^n \int_{X^\mathrm{an}} (\varphi - \varphi_0)\, (dd^c\varphi)^j \wedge (dd^c\varphi_0)^{n-j}. \] The Monge-Ampère operator on general metrics is defined by approximation, using the Chambert-Loir measure. Given a measure $\mu$ of the correct total mass, consider \[ F_\mu(\varphi) = E(\varphi) - \int_{X^\mathrm{an}} (\varphi - \varphi_0)\,d\mu, \] which, as a functional on the (compact) space $\mathrm{PSH}(L^\mathrm{an})/\mathbb{R}$, attains its maximum. Differentiability of the energy along "psh envelopes" is a key ingredient: for a continuous metric $\psi$, the envelope $P(\psi)$ is the supremum of all semipositive metrics $\leq \psi$, and $t \mapsto E(P(\psi + t f))$ is differentiable at $t=0$ with derivative $\int f\,\mathrm{MA}(P(\psi))$. The critical point condition implies $\mathrm{MA}(\varphi^*)=\mu$. For mixed characteristic, regularization is performed using perturbation-friendly test ideals [2203.12282], extending regularization arguments beyond equicharacteristic~$0$. Continuity of the plurisubharmonic envelope follows under suitable resolution hypotheses. ## 4. Comparison with the Complex Monge-Ampère Theory Both the complex and non-Archimedean theories seek to prescribe the Monge-Ampère measure of a semipositive continuous metric, using variational and potential-theoretic techniques. In the complex case, the Monge-Ampère operator is defined for (smooth) plurisubharmonic potentials with $(dd^c\varphi)^n$ associated to non-pluripolar positive measures; regularization is achieved via convolution. In the non-Archimedean case, the operator is algebraic and measures are supported on divisorial or monomial points, with regularization tied to passage to suitable algebraic or formal models and use of multiplier or test ideals [1502.05724, 2203.12282]. Both contexts employ energy and capacity estimates, enabling the elevation of weak to continuous solutions [1201.0188, 1608.01919]. A pivotal result is the comparison theorem: for a metric arising from a convex function on an open face of a skeleton, the non-Archimedean Monge-Ampère measure equals, up to a scalar, the real Monge-Ampère measure of that function [1912.06220]. This establishes a dictionary between non-Archimedean pluripotential theory and real convex analysis on skeleta. ## 5. Skeletal, Tropical, and Polyhedral Perspectives In settings with toric degenerations or semistable models, the Berkovich space $X^\mathrm{an}$ retracts onto a skeleton modeled on a polyhedral or polytopal complex. The Chambert-Loir Monge-Ampère measure, when restricted to skeleta, coincides with the pushforward of the real Monge-Ampère measure of the associated convex potential, scaled by a residue degree and a factorial factor [1912.06220, 2208.13697]. The polyhedral approach generalizes this to balanced polyhedral spaces: polyhedrally plurisubharmonic (psh) functions are defined via pointwise decreasing limits of piecewise affine convex "model" functions. Their Monge-Ampère measures, defined via tropical intersection theory, extend uniquely to continuous polyhedral psh functions by monotone approximation [2603.08664]. A table summarizing the operator comparison: | Setting | Measure support | MA computation | |------------------------------|--------------------------|---------------------------| | Complex | Full variety, smooth | $dd^c \varphi$ (forms) | | Non-Archimedean, general | Divisorial/monomial pts | Chambert-Loir intersection| | Skeletal (toric/model) | Skeleton faces/vertices | Real MA & scaling | This correspondence underpins explicit formulas and variational solutions for the non-Archimedean Monge-Ampère equation in toric, tropical, or maximally degenerate settings [2510.21677, 2208.13697]. ## 6. Existence, Uniqueness, and Regularity The non-Archimedean Monge-Ampère equation admits a unique (up to constants) continuous semipositive solution for any positive measure of the correct mass supported on a skeleton, under suitable regularity and resolution assumptions [1502.05724, 1608.01919, 2203.12282]. The continuity of the solution is established via Kołodziej-type capacity estimates and compactness properties of the space of semipositive metrics. Regularity results for curves demonstrate that, on skeleton edges, the solution gains regularity in accordance with the source measure: if $\mu|_\tau = f(x)\,dx$ with $f \in C^k$, then $\varphi|_\tau \in C^{k+2}$ [1912.06220]. Piecewise linearity arises in the case where $\mu$ is a sum of Dirac masses at skeleton vertices. ## 7. Applications and Connections Solving the non-Archimedean Monge-Ampère equation has significant implications for non-Archimedean Kähler geometry, metric SYZ conjectures, degenerations of Calabi-Yau, and the study of hybrid spaces connecting Archimedean and non-Archimedean limits [2510.21677, 2208.13697]. The polyhedral and tropical frameworks yield explicit solutions for Ricci-flat metrics on maximally degenerate Calabi-Yau hypersurfaces, and link to the existence of special Lagrangian torus fibrations in the context of the SYZ conjecture [2603.08664]. In all cases, the interface between algebraic models, convex geometry, and pluripotential theory is central, with the non-Archimedean Monge-Ampère measure serving as the unifying bridge. Source: https://www.emergentmind.com/topics/non-archimedean-monge-ampere-equation