Metric SYZ Conjecture

• Metric SYZ Conjecture is a precise formulation describing how Ricci-flat Calabi–Yau metrics collapse to an affine manifold with a real Monge–Ampère structure and special Lagrangian torus fibrations.
• It integrates techniques from pluripotential theory, non-Archimedean analytic geometry, tropical methods, and convex analysis to rigorously model the large complex structure limit.
• The approach leverages local analytic models and variational frameworks to characterize the Gromov–Hausdorff limit and elucidate the nature of singularities along the discriminant locus.

The metric SYZ conjecture is a mathematically precise formulation of the large-complex-structure-limit behavior of Ricci-flat Calabi–Yau metrics, motivated by the Strominger–Yau–Zaslow conjecture in mirror symmetry. It asserts that, for a maximally degenerating family of polarized Calabi–Yau manifolds, the unique Ricci-flat metrics collapse—after appropriate rescaling—in the Gromov–Hausdorff sense to a metric space endowed with the structure of an affine manifold with a real Monge–Ampère metric, and that, away from a small singular locus, the original spaces admit special Lagrangian torus fibrations over this base. Realization of this conjecture in a broad range of algebraic-geometric settings has required the integration of pluripotential theory, non-Archimedean analytic geometry, tropical methods, and convex analysis.

1. Formulation and Geometric Structure

Let $\{X_t\}_{0<|t|<1}$ denote a maximally degenerating family of polarized $n$-dimensional Calabi–Yau manifolds over the punctured disc, equipped with ample line bundle $L$ and unique Ricci-flat Kähler metric $\omega_t \in c_1(L)$. The metric SYZ conjecture predicts that for any $\varepsilon>0$ and sufficiently small $t$, there exists an open set $U_t\subset X_t$ (with $\mathrm{vol}_{\omega_t}(U_t)\geq 1-\varepsilon$) and a special Lagrangian $T^n$-fibration

$\pi_t: U_t \to B_t,$

such that as $n$0, the Riemannian manifolds $n$1 Gromov–Hausdorff converge to the metric completion of a $n$2-dimensional real affine manifold $n$3, where $n$4 is a Hessian metric associated to a convex function $n$5 solving $n$6 on the smooth locus $n$7 (Li, 2023, Chan, 2014, Li, 2022, Li, 2020).

The semi-flat model describes locally the metric by

$n$8

where $n$9 are local affine and fiber coordinates.

2. Variational and Monge–Ampère Frameworks

A rigorous mechanism for identifying the base metric relies on convex variational problems and Monge–Ampère equations, which take different but equivalent forms in real, tropical, and non-Archimedean categories. For explicit toric or Gross–Siebert degenerations, a global convex potential $L$0 is obtained as the unique minimizer (modulo constants) of a functional,

$L$1

with $L$2 the Legendre transform, and $L$3 Lebesgue measures on dual polytopes $L$4 (Li, 2023). The corresponding Euler–Lagrange equation is

$L$5

with analogous dual equations for $L$6, and $L$7 set by normalization. Measure-preserving properties and Alexandrov solution theory are used to guarantee existence, uniqueness, and regularity up to a discriminant locus of codimension at least two.

In the non-Archimedean context (Berkovich analytification), the limit Calabi–Yau metric emerges as the solution to a Monge–Ampère equation for a continuous semipositive metric on the skeleton, typically solved via intersection theory (Li, 2020, Goto et al., 2024).

3. Applications in Toric, Tropical, and Gross–Siebert Degenerations

For Calabi–Yau hypersurfaces in smooth toric Fano manifolds, e.g., defined via a reflexive Delzant polytope $L$8, the large complex structure limit is governed by the real-analytic Monge–Ampère metric associated to a convex potential over $L$9, constructed via global variational analysis (Li, 2023). In such settings:

• The open subset $\omega_t \in c_1(L)$0 corresponds to the preimage under the moment map of the regular part of $\omega_t \in c_1(L)$1.
• The Gromov–Hausdorff limit is the metric completion of $\omega_t \in c_1(L)$2.
• The non-Archimedean Monge–Ampère potential matches the real-analytic one due to the “weak comparison property.”

The generalization to the Gross–Siebert program encompasses toric degenerations of complete intersections. Existence of convex solutions to the (piecewise-smooth) real Monge–Ampère equation on the dual intersection complex $\omega_t \in c_1(L)$3—coupled with identification with the unique semipositive toric metric solving the non-Archimedean Monge–Ampère equation—entirely determines the limiting metric geometry (Goto et al., 2024). Under these circumstances, the metric SYZ conjecture holds for all Gross–Siebert toric degenerations of Batyrev–Borisov complete intersections.

Tropical and optimal-transport-theoretic techniques have established SYZ-type metric collapse results for large classes of polytopes, particularly Weyl polytopes, with the weak metric SYZ conjecture verified in the centrally symmetric toric case (Delcroix et al., 2024).

4. Non-Archimedean and Tropical Approaches

Recent advances have unified the real, tropical, and non-Archimedean Monge–Ampère theories in the study of special Lagrangian collapse:

• The “tropical Monge–Ampère equation” admits a unique symmetric convex minimizer, up to constants, whose pushforward measure under the $\omega_t \in c_1(L)$4-subgradient map realizes the Lebesgue measure on the base skeleton (Hultgren et al., 2022).
• The associated non-Archimedean solution is obtained as the restriction of a continuous semipositive toric metric to the essential skeleton, with the Monge–Ampère measure coinciding with the tropical one.
• Collapsing results by Li show that Ricci-flat metrics converge in measured Gromov–Hausdorff sense to the real Monge–Ampère metric, and for any $\omega_t \in c_1(L)$5, large portions of the original Calabi–Yau admit special Lagrangian torus fibrations (Li, 2020, Li, 2022).

In practical terms, the reduction of the metric SYZ conjecture to properties of convex potentials and Monge–Ampère measures allows explicit identification of the limit metric structure in key examples, including Fermat-type hypersurfaces, toric complete intersections, and Gross–Siebert degenerations.

5. Combinatorial and Canonical Basis Techniques

In polarised maximal degenerations of Calabi–Yau varieties, the existence of a canonical basis of the section ring for the polarisation line bundle, satisfying valuative independence, enables direct verification of the metric SYZ conjecture (Li, 1 May 2026). Valuative independence allows for explicit control over the non-Archimedean Fubini–Study metric and identification of the limiting convex potential on the essential skeleton, ensuring the weak comparison property and the desired Gromov–Hausdorff convergence to a metric space with Monge–Ampère affine structure. This approach provides an explicit convex-geometric and combinatorial mechanism for constructing limits and verifying the existence of special Lagrangian torus fibrations.

6. Local Models, Singularities, and Analysis near the Discriminant

Local analytic models, such as Taub–NUT and Ooguri–Vafa metrics, provide the structure of the Calabi–Yau metric near singularities (codimension-two loci) of the affine base (Li, 2019). Away from the discriminant, the semiflat approximation is exponentially accurate, while degeneration to higher codimension loci is governed by matching to these local models. These singularities are responsible for the incomplete extension of the global fibration, and the topology and metric structure of the discriminant are the subject of continuing research, including explicit computations of affine monodromy and monodromy invariants.

7. Open Problems and Future Directions

Several aspects of the metric SYZ conjecture remain open:

• Full analytic construction of global special Lagrangian torus fibrations in compact settings beyond tori and K3 surfaces is unresolved (Chan, 2014, Li, 2022).
• The regularity of the real Monge–Ampère metric across codimension-one faces and the nature of the discriminant remain challenging. The discriminant is expected to have codimension at least two and govern the singular set of convergence.
• Proving the non-Archimedean to real Monge–Ampère comparison property in complete generality is a central open conjecture.
• The extension to Landau–Ginzburg and more general Fano settings, as well as applications to mirror symmetry via discrete Legendre transforms and tropical geometry, continues to be intensively developed.

The metric SYZ conjecture serves as a central unifying thread connecting Ricci-flat collapse, special Lagrangian geometry, convex and tropical analysis, and non-Archimedean algebraic geometry, and has driven significant technical advances across a broad swath of current mathematical research.