Non-Archimedean Calabi–Yau Theorem

• Non-Archimedean Calabi–Yau theorem is a framework establishing a homeomorphic correspondence between finite-energy A-psh potentials and finite energy probability measures on Kähler spaces.
• It employs variational methods, continuity of envelopes, and orthogonality to ensure existence, uniqueness, and continuous dependence of solutions to the non-Archimedean Monge–Ampère equation.
• The theorem extends classical Calabi–Yau insights to non-Archimedean settings, offering applications in mirror symmetry, tropical geometry, and uniform K-stability.

The non-Archimedean Calabi–Yau theorem is the non-Archimedean analogue of the finite-energy Calabi–Yau theorem for the Monge–Ampère equation. In its general compact Kähler form, it asserts that on the non-Archimedean space $X^{NA}$ associated to a compact Kähler manifold $(X,\alpha)$, the non-Archimedean Monge–Ampère operator

$MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$

is a homeomorphism, where $A=\pi_X^*\alpha$, $\mathcal{E}^1_{A,0}$ is the space of sup-normalized finite energy $A$-psh potentials, and $\mathcal{M}^1_A$ is the space of finite energy Radon probability measures. Equivalently, every finite energy measure admits a unique finite energy non-Archimedean potential solving $MA_A(\phi)=\mu$, and the solution depends continuously on the measure in the strong topology (Nyström, 3 Oct 2025).

1. Core statement and principal variants

In the transcendental compact Kähler setting, the theorem is formulated on the tropical analytification $X^{NA}$ and states that

$\MA_A:\mathcal E^1_{A,\sup}(X^{NA})\longrightarrow \mathcal M^1_A(X^{NA})$

is a homeomorphism. This is the general Kähler extension of the algebraic non-Archimedean Calabi–Yau theorem and gives existence, uniqueness, and bicontinuity for solutions of the non-Archimedean Monge–Ampère equation in finite energy classes (Mesquita-Piccione et al., 11 Sep 2025).

The phrase “non-Archimedean Calabi–Yau theorem” is also used in more specialized settings. For certain affine log Calabi–Yau varieties whose dual complex is a standard simplex, one seeks a continuous non-Archimedean plurisubharmonic potential $(X,\alpha)$0 on the Berkovich analytification with

$(X,\alpha)$1

where $(X,\alpha)$2 is the Lebesgue measure on the essential skeleton. In that setting, existence is proved by an explicit generalized Calabi ansatz (Wang, 24 Oct 2025). For totally degenerate abelian varieties, a semipositive toric metric is constructed so that its Chambert–Loir measure equals a prescribed smooth positive measure on the skeleton, giving an early explicit existence theorem in a highly structured case (Liu, 2010).

Setting	Equation	Result
Compact Kähler $(X,\alpha)$3	$(X,\alpha)$4	Homeomorphism $(X,\alpha)$5
Certain affine log Calabi–Yau varieties	$(X,\alpha)$6	Continuous psh solution on $(X,\alpha)$7
Totally degenerate abelian varieties	$(X,\alpha)$8	Unique semipositive toric metric up to scalar

A recurrent source of ambiguity is that the term may denote either a full existence–uniqueness theorem for a non-Archimedean Monge–Ampère equation or a concrete realization of such a theorem in a special geometric class. The compact Kähler theorem is a genuine global bijectivity statement; the affine and toric papers often provide explicit solutions in special ansätze or reductions to real Monge–Ampère equations rather than a universal existence theory.

2. Non-Archimedean spaces, potentials, and measures

In the compact Kähler theory, $(X,\alpha)$9 is the tropical analytification of $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$0: the set of semivaluations on the coherent ideal sheaves of $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$1, endowed with the topology of pointwise convergence. It is a compact Hausdorff space. Divisorial points arise from irreducible components of central fibers of SNC models, and the set of divisorial points is dense. Geometrically, $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$2 is the projective limit of the dual complexes of all models, so degenerations of $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$3 produce combinatorial approximations to the ambient non-Archimedean space (Nyström, 3 Oct 2025).

Fixing a Kähler class $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$4 and its pullback $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$5, one first defines piecewise linear functions from vertical divisors on models. A PL function $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$6 is a non-Archimedean Kähler potential if $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$7 is relatively Kähler on some model. An $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$8-psh function is then a decreasing limit of such Kähler potentials. The finite-energy space $MA_A:\mathcal{E}^1_{A,0}\to \mathcal{M}^1_A$9 is defined by the Monge–Ampère energy functional, and the dual energy defines the space $A=\pi_X^*\alpha$0 of finite energy probability measures. The strong topology on potentials is the coarsest topology finer than pointwise convergence on divisorial points for which the energy is continuous; the strong topology on measures is the coarsest topology finer than weak convergence for which the dual energy is continuous (Mesquita-Piccione et al., 11 Sep 2025).

For a PL Kähler potential $A=\pi_X^*\alpha$1 on an SNC model with central fiber $A=\pi_X^*\alpha$2, the non-Archimedean Monge–Ampère measure is the discrete probability measure

$A=\pi_X^*\alpha$3

and the operator extends by continuity to all finite energy potentials. In the “big” setting, restricted volumes replace intersection numbers, and the resulting measure is still a probability measure. This precise link with big classes and restricted volumes is one of the distinctive features of the Kähler extension (Nyström, 3 Oct 2025).

3. Relation to the classical Calabi–Yau theorem

The formal analogy with the classical theorem is exact at the level of finite-energy pluripotential theory. Classically, for a compact Kähler manifold $A=\pi_X^*\alpha$4 with volume $A=\pi_X^*\alpha$5, the Calabi–Yau theorem and its finite-energy extension state that every finite energy probability measure is the Monge–Ampère measure of a unique normalized finite-energy $A=\pi_X^*\alpha$6-psh potential. The non-Archimedean theorem replaces the complex space $A=\pi_X^*\alpha$7 by $A=\pi_X^*\alpha$8, smooth or $A=\pi_X^*\alpha$9-psh potentials by $\mathcal{E}^1_{A,0}$0-psh functions, and $\mathcal{E}^1_{A,0}$1 by the non-Archimedean Monge–Ampère measure $\mathcal{E}^1_{A,0}$2 (Nyström, 3 Oct 2025).

From the degeneration viewpoint, the two theories solve analogous equations on different parameter spaces. In the complex setting one solves the Monge–Ampère equation on each fiber. In the non-Archimedean setting one solves it on a space encoding all degenerations simultaneously. This viewpoint is sharpened by hybrid convergence results: for polarized degenerations of projective Calabi–Yau manifolds, the rescaled complex Calabi–Yau potentials converge in the $\mathcal{E}^1_{A,0}$3 hybrid topology to the non-Archimedean Calabi–Yau potential, and the complex Monge–Ampère measures converge to the non-Archimedean Monge–Ampère measure supported on the essential skeleton (Li, 16 May 2025).

For log Calabi–Yau degenerations, canonical complex volume forms converge, after explicit scaling, to a canonical non-Archimedean measure on the Berkovich analytification, supported on the essential skeleton and described by normalized Lebesgue measures on faces weighted by residue integrals. This gives a measure-theoretic non-Archimedean Calabi–Yau limit even when the complex measures may have infinite total mass in the sub log canonical case (Shivaprasad, 2019).

4. Variational method, envelopes, and orthogonality

The proof of the compact Kähler theorem follows the variational method. Given $\mathcal{E}^1_{A,0}$4, one maximizes

$\mathcal{E}^1_{A,0}$5

over normalized finite-energy potentials. A maximizing sequence has Monge–Ampère measures converging strongly to $\mathcal{E}^1_{A,0}$6, and compactness of normalized $\mathcal{E}^1_{A,0}$7-psh functions yields a limit $\mathcal{E}^1_{A,0}$8 with $\mathcal{E}^1_{A,0}$9. Injectivity follows from the quasi-metric geometry of the energy functional, and continuity of the inverse is encoded by a quasi-isometry between the strong geometry of potentials and the strong geometry of measures (Mesquita-Piccione et al., 11 Sep 2025).

Two technical pillars are continuity of envelopes and orthogonality. For a continuous function $A$0 on $A$1, the $A$2-psh envelope

$A$3

is continuous, and it satisfies the orthogonality identity

$A$4

In the transcendental theory this is proved by combining non-Archimedean pluripotential methods with the correspondence between non-Archimedean psh functions and Archimedean subgeodesic rays (Mesquita-Piccione et al., 11 Sep 2025).

A distinctive Kähler-analytic ingredient is the role of big cohomology classes and restricted volumes. For a fixed model $A$5, any probability measure on the set of irreducible components of the central fiber can be realized as the Monge–Ampère measure of a big test configuration $A$6. The weights are controlled by restricted volumes

$A$7

and the identity

$A$8

is the volume-theoretic mechanism behind the variational formalism and the orthogonality property on $A$9 (Nyström, 3 Oct 2025).

5. Skeletons, hybrid limits, and SYZ geometry

A major geometric theme is that the right-hand side of the non-Archimedean equation is frequently a canonical measure on the essential skeleton. For polarized Calabi–Yau degenerations, the normalized volume forms $\mathcal{M}^1_A$0 converge weakly in the hybrid topology to a probability measure $\mathcal{M}^1_A$1 supported on $\mathcal{M}^1_A$2, absolutely continuous with respect to Lebesgue measure on the top-dimensional faces. The non-Archimedean Calabi–Yau theorem then produces the unique semipositive metric $\mathcal{M}^1_A$3 with

$\mathcal{M}^1_A$4

and the complex Calabi–Yau metrics converge to this non-Archimedean metric at the potential level (Li, 16 May 2025).

The essential skeleton is also the base of the non-Archimedean SYZ fibration in maximally degenerate situations. For projective Calabi–Yau varieties over $\mathcal{M}^1_A$5, the non-Archimedean SYZ map

$\mathcal{M}^1_A$6

is an affinoid torus fibration away from a codimension-two subset of the base, and the induced integral affine structure agrees with the canonical piecewise integral affine structure on the skeleton (Nicaise et al., 2018). In toric hypersurface settings, the potential of the non-Archimedean Calabi–Yau metric can be constant along the fibers of such a retraction outside the discriminant, which is the non-Archimedean semi-flat behavior predicted by the SYZ picture (Pille-Schneider, 2022).

For toric degenerations of Batyrev–Borisov Calabi–Yau complete intersections, the non-Archimedean Monge–Ampère equation

$\mathcal{M}^1_A$7

is reduced, for toric semipositive metrics, to a real Monge–Ampère equation on the tropical sphere $\mathcal{M}^1_A$8. The paper proves equivalence between the toric non-Archimedean equation and the real Alexandrov equation on each facet, and shows that if such a toric solution exists then the NA MA–real MA comparison property holds, implying the metric SYZ conjecture for the degeneration (Goto et al., 2024). This does not prove existence in full generality; it identifies the analytic problem with a tropical one.

6. Explicit realizations, applications, and adjacent directions

The theorem has several explicit realizations. For totally degenerate abelian varieties $\mathcal{M}^1_A$9, the Chambert–Loir measure of a toric metric is the pushforward of the real Hessian measure of a convex Green function on the real torus $MA_A(\phi)=\mu$0. Solving the corresponding real Monge–Ampère equation yields a unique semipositive toric metric, up to scalar, with prescribed smooth positive density on the skeleton (Liu, 2010). In certain affine log Calabi–Yau varieties with simplex dual complex, a continuous psh non-Archimedean potential is written explicitly as

$MA_A(\phi)=\mu$1

where $MA_A(\phi)=\mu$2 is the Collins–Tong–Yau solution of a real Monge–Ampère free boundary problem, and this solves $MA_A(\phi)=\mu$3 on the Berkovich analytification (Wang, 24 Oct 2025).

The theorem also feeds directly into stability theory. In the transcendental Kähler setting, uniform K-stability for models is translated into coercivity of the non-Archimedean Mabuchi functional, and the non-Archimedean Calabi–Yau theorem is a key input in proving that uniform K-stability for models implies existence and uniqueness of a cscK metric in the Kähler class $MA_A(\phi)=\mu$4 (Mesquita-Piccione et al., 11 Sep 2025). In Li’s degeneration framework, the same theorem identifies the non-Archimedean Calabi–Yau potential as the $MA_A(\phi)=\mu$5 potential-theoretic limit of degenerating complex Calabi–Yau metrics and, under canonical basis assumptions, as the unique minimizer of a Kontorovich dual functional on the essential skeleton (Li, 16 May 2025).

A final clarification is methodological. Some papers use “non-Archimedean Calabi–Yau theorem” to describe canonical non-Archimedean objects attached to log Calabi–Yau varieties, such as skeleton measures, period maps, or mirror algebras. These constructions are adjacent to, but not identical with, the Monge–Ampère homeomorphism theorem. The theorem proper concerns solvability of a non-Archimedean Monge–Ampère equation; the broader program uses that theorem together with essential skeletons, hybrid spaces, and SYZ fibrations to study metric collapse, mirror symmetry, and period geometry.