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Valuative independence and metric SYZ conjecture

Published 1 May 2026 in math.AG, math.CV, and math.DG | (2605.00516v1)

Abstract: Given a polarised maximal degeneration of compact Calabi-Yau manifolds, assuming there exists a canonical basis of the section ring for the polarisation line bundle, satisfying the valuative independence condition, we will prove the metric SYZ conjecture.

Authors (1)

Summary

  • The paper establishes an algebraic reduction of the metric SYZ conjecture via valuative independence in polarised Calabi-Yau degenerations.
  • It employs non-Archimedean pluripotential theory and tropical techniques to recast the Monge-Ampère equation in an optimal transport framework.
  • The study unifies convex analysis, Okounkov bodies, and birational geometry, linking discrete algebraic constructions with smooth metric limits.

Valuative Independence and the Metric SYZ Conjecture

Introduction and Motivation

The paper "Valuative independence and metric SYZ conjecture" (2605.00516) addresses the metric form of the Strominger-Yau-Zaslow (SYZ) conjecture for polarised maximal degenerations of compact Calabi-Yau (CY) manifolds. The metric SYZ conjecture predicts that, near the large complex structure limit, the CY metric admits a special Lagrangian torus fibration on an open set with measure arbitrarily close to one. The approach leverages non-Archimedean pluripotential theory, the essential skeleton, and connections to tropical and birational geometry.

A central innovation is to isolate an algebraic condition—valuative independence for certain canonical bases in the section ring of the polarising line bundle—which enables a reduction of the metric SYZ conjecture to a non-Archimedean Monge-Ampère-type equation. The paper utilizes recent progress in birational geometry, particularly canonical bases constructions, and reveals deep analogies with optimal transport.

Non-Archimedean Pluripotential Theory and the Metric SYZ Conjecture

The degeneration of compact CY manifolds is studied via the associated Berkovich analytification over the non-Archimedean field K=C((t))K = \mathbb{C}((t)). Essential geometric and measure-theoretic features in the large complex structure limit are captured by the essential skeleton Sk(X)Sk(X) in XKanX_K^{an}, which generalizes the dual intersection complex of a semistable model.

Non-Archimedean pluripotential theory, especially the work of Boucksom-Favre-Jonsson, provides a canonical solution ϕ0\phi_0 to the non-Archimedean (NA) Monge-Ampère (MA) equation

MA⁡(ϕ0)=(Ln)μ0,\operatorname{MA}(\phi_0) = (L^n)\mu_0,

where μ0\mu_0 is a Lebesgue-type measure on the essential skeleton. Under maximal degeneration, the metric SYZ conjecture is reduced to a statement about the factorization of ϕ0\phi_0 through the retraction to the dual complex over a full-measure open set. Under this "weak comparison property," pushforward to the skeleton yields a real Monge-Ampère equation, and one obtains C∞C^{\infty}-convergence in the complex structure to the semiflat model—a local special Lagrangian torus fibration. This analytic geometry is thus translated into a discrete birational-algebraic problem.

Valuative Independence: Algebraic Criterion and Consequences

The crux of achieving the weak comparison property lies in constructing, for each ll, a KK-basis Sk(X)Sk(X)0 of Sk(X)Sk(X)1 satisfying the valuative independence condition: for any Sk(X)Sk(X)2 viewed as a valuation,

Sk(X)Sk(X)3

This property models the behavior of monomials on toric varieties and generalizes the theta basis of the Gross-Siebert program.

The existence of such canonical bases is established in full generality for CY degenerations, after replacing Sk(X)Sk(X)4 by a suitable positive multiple Sk(X)Sk(X)5 [Blum-Liu, (Blum et al., 30 Apr 2026)]. The connection to the SYZ fibration relies on the fact that, locally, the tropicalization of these theta functions behaves linearly on the skeleton, enabling the passage from non-Archimedean metric data to explicit convex geometry.

Key results:

  • Over open polyhedral subsets of Sk(X)Sk(X)6, the tropical theta functions are affine linear.
  • Leading term exponents are distinct, ensuring independence akin to classical Okounkov body setups.
  • The global-to-local passage permits the construction of convex, Lipschitz continuous potentials dominating the non-Archimedean ones, with domination and envelope techniques ensuring the algebraic and analytic apparatus commute appropriately.

Convex Geometry, Okounkov Bodies, and Potential Theory

The valuative independence framework allows the embedding of the Okounkov body Sk(X)Sk(X)7 (constructed via semigroups of leading exponents of theta functions) as an open subset of a topological parameter space Sk(X)Sk(X)8 of "tropical" test functions. The cost function Sk(X)Sk(X)9 encapsulates the affine geometry underlying the metric problem, providing a bridge to optimal transport and convex analysis.

Continuous semipositive non-Archimedean potentials correspond bijectively to elements of a function class XKanX_K^{an}0 generated by these cost functions and their XKanX_K^{an}1-transforms. The Okounkov body, and in particular its boundary, controls where the desired metric factorization can occur.

Through a delicate analysis involving subgradient hulls and Riemann-Roch asymptotics, the authors identify precise regions where the potential factors through the retraction, and verify that for almost every point (in the measure sense) this region is large (full measure), due to the absolute continuity of the Monge-Ampère measure with respect to the Lebesgue measure on the skeleton.

Non-Archimedean Monge-Ampère Equation and Optimal Transport

A substantial theoretical advance is the recasting of the NA MA equation as an infinite-dimensional (but metrically well-posed) Kantorovich dual optimal transport problem. For a probability measure XKanX_K^{an}2 on XKanX_K^{an}3 (absolutely continuous with respect to XKanX_K^{an}4), the MA solution XKanX_K^{an}5 minimizes

XKanX_K^{an}6

among all XKanX_K^{an}7 in XKanX_K^{an}8, and is unique up to a constant. The XKanX_K^{an}9-transforms and the conjugate set correspondences realize an analog of the Alexandrov solution to the real Monge-Ampère equation, extended globally to the non-Archimedean and hybrid context.

A measure-theoretic matching (Theorem 7.9) between ϕ0\phi_00 and ϕ0\phi_01 is established, with the pushforward of ϕ0\phi_02 under the subgradient/conjugate point map equalling the canonical measure ϕ0\phi_03. The identification of ϕ0\phi_04 with the essential skeleton of the mirror CY family points to a "metric version" of mirror symmetry, consistent with the expectations of the Gross-Siebert program.

Implications and Future Directions

This work establishes that the metric SYZ conjecture for all polarised maximal degenerations of compact CY manifolds reduces to the algebraic input of valuative independence. Recent advances [Blum-Liu, (Blum et al., 30 Apr 2026)] show that this criterion is satisfied in full generality after a base change. As a result, the metric SYZ conjecture is now known for all such degenerations, extending far beyond the toric and explicit hypersurface cases previously accessible.

The deeper implications are threefold:

  • There is a complete synthesis of non-Archimedean pluripotential theory, Okounkov bodies, and the Gross-Siebert program's theta functions, uniting previously disjoint aspects of SYZ mirror symmetry.
  • The metric aspect of mirror symmetry is seen as arising from an optimal transport duality, with canonical measures playing the role of "canonical bases" on both sides of mirror symmetry.
  • The methods suggest broader applicability to understanding the limits of Kähler-Einstein metrics, the structure of measures on hybrid spaces, and geometric representation theory, where valuative independence links to Newton-Okounkov bodies and representation-theoretic bases.

It remains an open question to give a direct, model-independent construction of the "mirror skeleton" ϕ0\phi_05, and to generalize the topological and smooth structure results for the metric SYZ conjecture (such as the codimension of the non-affine locus in ϕ0\phi_06).

Conclusion

The paper (2605.00516) provides a compelling, algebraically explicit resolution of the metric SYZ conjecture in maximal generality, synthesizing non-Archimedean analysis, birational geometry, and optimal transport. By placing valuative independence at the center of the analysis, the work clarifies the relation between the algebraic, tropical, and metric structures underlying mirror symmetry and yields a blueprint for further study in higher-dimensional and logarithmic setups.

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Explain it Like I'm 14

Overview

This paper studies what happens to certain very special shapes, called Calabi–Yau spaces, when they “degenerate” (think: slowly collapse) in a family. The big goal is to confirm a famous idea called the SYZ conjecture. Roughly, SYZ predicts that, near the limit of the collapse, most of the space can be split into “torus fibers” (higher‑dimensional doughnut shapes) in a very balanced, geometric way.

The author shows that, if you can find a particularly nice set of “building blocks” (sections) of a line bundle that behave independently in a precise sense (called “valuative independence”), then the metric SYZ conjecture is true for these collapsing families. Thanks to recent advances by other researchers, such nice bases exist very generally (after possibly taking a multiple of the line bundle), so this gives a broad path to proving SYZ in many cases.

What questions does the paper ask?

  • Can we turn a difficult geometric problem (the SYZ conjecture) into a simpler, more algebraic one by working with a special kind of basis of sections that behaves like monomials (independent and easy to compare)?
  • If such a basis exists, does it force a certain “comparison” behavior for a limit potential (a kind of energy function) in a non-archimedean, or “tropical,” setting?
  • Once we have that comparison, can we then build the special torus fibration predicted by SYZ on most of the space?
  • Can we understand the key limit equation (a non-archimedean Monge–Ampère equation) as an optimal transport problem (like moving sand in the cheapest way from one place to another)?

How do they approach it?

Here are the main ideas, explained with everyday analogies:

  • Calabi–Yau spaces collapsing and their “skeleton”:
    • Imagine a complicated 3D object slowly flattening. As it collapses, most of it concentrates around a simpler “skeleton,” like the frame of a building. In this math setting, that skeleton is a combinatorial object called the “essential skeleton,” sitting inside a limit space of the family.
    • The measures (ways to count volume) on the original spaces concentrate and converge to a natural measure on this skeleton.
  • Working in a “shadow world” to simplify:
    • Instead of working on the original complex spaces, the paper moves to a non-archimedean or “tropical” world (think: a shadow or blueprint of the geometry). In this world, complicated geometric equations become piecewise-linear or convex problems that are easier to handle.
    • A key object here is a potential function that solves a non-archimedean Monge–Ampère (MA) equation. This is the shadow‑version of the famous Calabi–Yau equation.
  • A special basis with “valuative independence”:
    • Think of representing functions with a set of building blocks (like Lego pieces). “Valuative independence” means these pieces don’t interfere when you measure how big they are at different points in the skeleton: the size of a sum is just the maximum of the sizes. This makes things behave like simple monomials.
    • With such a basis, the paper can control how functions look on the skeleton and show the limit potential depends only on the skeleton coordinates in most places. This is called the “weak comparison property.”
  • From comparison to fibrations:
    • Once the potential depends only on the skeleton (almost everywhere), known results convert the non-archimedean MA equation into a real convex equation on the skeleton.
    • This unlocks smoothness on large regions, and then standard geometric tools produce the special Lagrangian torus fibration on most of the space—the heart of the metric SYZ conjecture.
  • Optimal transport viewpoint:
    • The paper also interprets the non-archimedean MA equation as an optimal transport problem. Picture moving a pile of sand distributed on the skeleton to another space, paying a “cost” that comes from limits of those special basis functions. The best way to move the sand corresponds to the desired potential.

What did the researchers find?

  • Main conditional result: If, for each level ll, the space of sections has a basis that satisfies valuative independence, then the “weak comparison property” holds. This means the limit potential depends only on the skeleton on a set of full measure (i.e., almost everywhere).
  • From weak comparison to SYZ: Using earlier work, the weak comparison property implies the metric SYZ conjecture for the collapsing family. Concretely, for any tiny tolerance δ\delta, when the collapse is far enough along, there’s a special Lagrangian torus fibration covering at least 1−δ1-\delta of the space.
  • General existence of such bases: Independent work by Blum and Liu shows that these valuatively independent bases exist very generally after replacing the line bundle by a suitable multiple. Combined with this paper’s main result, this essentially proves the metric SYZ conjecture for all “maximally degenerate” polarized Calabi–Yau families.
  • Optimal transport formulation: The non-archimedean Monge–Ampère equation can be seen as an optimal transport problem between the skeleton and a carefully built probability space (constructed using Okounkov bodies, which organize how sections grow). This gives a fresh and computable way to understand the limit potential.

Why are these results important?

  • A big step toward SYZ: The SYZ conjecture is a central idea in mirror symmetry, a major topic connecting geometry and physics. Showing metric SYZ in wide generality is a major milestone because it describes the geometric shape of Calabi–Yau spaces near degeneration in a very precise way.
  • Algebra controls geometry: The paper turns a hard nonlinear PDE problem into something governed by algebraic bases and convex analysis on a skeleton. This makes the problem more tractable and opens it to powerful algebraic tools.
  • New bridges: The optimal transport viewpoint links complex geometry, non-archimedean analysis, and convex optimization. Such links often lead to new techniques and insights across fields.

Takeaway and potential impact

In simple terms, the paper says: if you can choose your building blocks wisely (valuatively independent basis), then the collapsing Calabi–Yau spaces behave exactly as SYZ predicts—most of the space is fibered by “doughnut‑shaped” tori in a special, balanced way. Since these nice bases now exist broadly (thanks to other work), this gives a general pathway to proving metric SYZ for many collapsing Calabi–Yau families.

Beyond confirming a central geometric picture, the optimal transport reformulation suggests new computational tools and may guide future progress in mirror symmetry, helping to match “skeletons” and measures between mirror pairs in a clear, metric way.

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Knowledge Gaps

Knowledge gaps, limitations, and open questions

Below is a consolidated list of concrete gaps and open problems the paper leaves unresolved, organized to guide future work.

  • Assumption of valuative independence:
    • The main implication (valuative independence ⇒ weak comparison ⇒ metric SYZ) rests on the existence of a valuative independent basis for H0(XK,lL)H^0(X_K,lL) for all ll. A direct, intrinsic construction of such bases for general degenerations is not provided.
    • While Blum–Liu announce existence for H0(XK,m0lL)H^0(X_K,m_0 lL), a fully written, self-contained argument linking this to the precise hypotheses used here (e.g., passage from LL to m0Lm_0L without hidden subtleties in the NA MA setup and comparison property) is not detailed.
    • Canonicity and uniqueness: To what extent is a valuative independent basis canonical (e.g., independent of choices up to scaling)? How unique is the induced tropical data and does it agree with enumerative or mirror-theoretic structures?
  • Relation to Gross–Siebert theta functions:
    • It is unknown in general whether Gross–Siebert theta functions for polarized maximal degenerations satisfy valuative independence. Identifying precise geometric or enumerative criteria ensuring this property remains open.
    • Constructive verification in non-toric/non-cluster examples is not provided. Are there families where Gross–Siebert theta functions fail valuative independence, and if so, how can one correct or refine them?
  • Scope beyond maximal degenerations:
    • The argument targets maximal degenerations (dim⁡Sk(X)=n\dim Sk(X)=n). The behavior of the construction and the weak comparison property for 0<dim⁡Sk(X)<n0<\dim Sk(X)<n is left unexplored in terms of special Lagrangian fibrations (e.g., partial fibrations or lower-dimensional torus actions).
    • Extending the metric SYZ conclusion to non-maximal cases or to other settings (e.g., log Calabi–Yau pairs) is not addressed.
  • Strength and extent of the weak comparison property:
    • The factorization ϕ0=ϕ0∘rX\phi_0=\phi_0\circ r_{\mathcal{X}} holds on an open full-measure subset U⊂Sk(X)U\subset Sk(X), not globally. Is global factorization obtainable (possibly after further birational modifications) or can the exceptional set be better described or reduced?
    • Quantitative or structural information about the exceptional set (measure-zero loci where factorization fails) is not given. Can one characterize it (e.g., polyhedral, countable union of walls) or control its Hausdorff dimension?
  • Regularity and geometry of the NA/real Monge–Ampère solution:
    • Beyond Mooney’s regularity “off a closed set of Hausdorff dimension ≤ n−1n-1,” finer regularity (e.g., Ck,αC^{k,\alpha} with quantified estimates, analyticity, or boundary behavior near walls of Sk(X)Sk(X)) is not explored.
    • Effective rates in the hybrid C0C^0 convergence and their dependence on algebro-geometric data are not provided.
  • Optimal transport (OT) interpretation:
    • Identification of (B,ν)(B,\nu): The paper proposes constructing (B,ν)(B,\nu) via Okounkov body techniques and conjectures it might coincide with the mirror’s essential skeleton with its Lebesgue measure. This identification is not proved; conditions ensuring it (and counterexamples) are unknown.
    • Dependence on choices: Okounkov bodies depend on the choice of a flag; the extent to which the resulting BB, cost cc, and Pc\mathcal{P}_c are canonical (or independent up to a natural equivalence) is not established.
    • Computability: General methods to compute the cost function c(x,p)c(x,p) beyond special examples (e.g., toric/cluster-like cases) are missing. Is there an algorithmic recipe from monodromy or scattering data (beyond the cases studied by Hultgren–Khalid)?
    • Uniqueness and stability of OT potentials: Criteria ensuring that the Kantorovich potential coincides with the NA CY potential ϕ0\phi_0 for general degenerations remain to be systematized; regularity of OT maps on Sk(X)Sk(X) and stability under perturbations is unexplored.
  • Measures in the NA MA equation:
    • The weak comparison and OT arguments assume μ\mu is supported on Sk(X)Sk(X) and absolutely continuous w.r.t. Lebesgue. Extensions to measures with singular parts, mixed support (including divisorial atoms), or weaker integrability conditions are not treated.
    • Sensitivity of the factorization and OT description to the smoothness/density of μ\mu remains unclear.
  • Global special Lagrangian fibration:
    • The result provides special Lagrangian TnT^n-fibrations on measure-≥1−δ\ge 1-\delta subsets. Whether one can obtain global fibrations (or classify unavoidable singularities and their monodromy) is open.
    • Compatibility across different local charts near deep strata is not fully analyzed; a global description of the singular fibration (e.g., base affine structure with singularities) is not provided.
  • Effectivity and quantitative estimates:
    • No explicit bounds are given for constants (e.g., Lipschitz constants, convergence rates, or δ\delta-to-tt dependence) in terms of algebro-geometric invariants of the degeneration.
    • Lack of effective criteria for verifying valuative independence or the weak comparison property in concrete families, beyond the announced existence results.
  • Birational and base-change issues:
    • Although the weak comparison property is stable under smooth blow-ups/downs, the behavior under more general birational modifications (e.g., non-snc, flips) and the canonical choice of model maximizing factorization is not discussed.
    • The necessity and impact of finite base changes (semistable reduction) on the metric SYZ conclusions need systematic clarification (e.g., how choices affect the limiting fibration and potentials).
  • Mirror symmetry implications:
    • The proposed metric version of mirror symmetry (identifying (B,ν)(B,\nu) with the mirror skeleton) lacks a proof or testable criteria. How to compare measures, potentials, and cost functions across mirrors remains open.
    • Connection with scattering diagrams and wall-crossing in the Gross–Siebert program is not developed; reconciling the OT/cost framework with broken lines and theta function counts is an open direction.
  • Beyond polarized settings:
    • The reliance on a relatively ample LL is central. Extensions to semiample or nef settings, and dependence of the NA CY potential (and resulting fibrations) on the choice of polarization, are not investigated.
  • Generality of convergence and fibration results:
    • The hybrid convergence of Kähler potentials up to C0C^0 is used as a black box; whether sharp convergence (e.g., CkC^{k} or C∞C^{\infty}) can be obtained uniformly without passing to covers, and how it depends on the degeneration type, is not resolved.
    • Extensions to multi-parameter degenerations (higher-dimensional bases) are not discussed, despite the role of multi-parameter test configurations in the algebraic input.
  • “Bad set” analysis in the envelop/orthogonality step:
    • The argument that certain bad sets have null measure relies on orthogonality; no structural description or robustness analysis of these sets is given. Can they be controlled uniformly or shown to be smaller (e.g., countable unions of walls)?

These gaps highlight concrete avenues for progress: proving valuative independence for Gross–Siebert theta functions, identifying (B,ν)(B,\nu) with the mirror skeleton, developing effective computation of the cost function, extending the weak comparison and OT framework beyond absolutely continuous measures, and strengthening regularity and globality of special Lagrangian fibrations.

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Practical Applications

Immediate Applications

Below are near-term, actionable uses that can be pursued with current methods and software stacks, primarily within academia and computational research.

  • Research workflow for proving metric SYZ in new families (Academia: mathematics, geometry)
    • What: Adopt the paper’s reduction of the metric SYZ conjecture to verifying valuative independence and the weak comparison property. Apply this workflow to specific degenerations (e.g., toric hypersurfaces, K3, cluster-type examples).
    • How:
    • Construct semistable SNC models and essential skeleta.
    • Use or verify existence of valuative independent bases (leveraging the Blum–Liu result for m0Lm_0L).
    • Check the weak comparison property and invoke the NA–to–real Monge–Ampère comparison to conclude SYZ.
    • Tools/workflows: SageMath/Macaulay2 for section rings and toric computations; existing Gross–Siebert implementations for theta functions in examples; scripts to compute dual complexes and to evaluate valuations on sections.
    • Assumptions/dependencies: Explicit models/SNC reductions are available; ability to compute or import canonical/theta bases; numerical access to valuations on Sk(X)Sk(X).
  • Numerical approximation of NA Calabi–Yau potentials and validation in explicit examples (Academia: mathematics; Software: numerical geometry)
    • What: Compute NA Fubini–Study approximations and solve the NA Monge–Ampère equation numerically via the paper’s relative volume and cc-transform setup; compare against hybrid limits of complex CY potentials.
    • How:
    • Use graded section spaces H0(XK,lL)H^0(X_K, lL) with ultrametric norms.
    • Approximate ϕ0\phi_0 from Fubini–Study envelopes and relative volumes.
    • Leverage the measure convergence to assess numerical accuracy over the skeleton.
    • Tools/workflows:
    • Implementation of NA Fubini–Study metrics (eq. for ϕFS,l\phi_{FS,l}).
    • Relative volume calculation pipelines using determinants of ultrametric norms.
    • Discrete convex optimization on the dual complex to approximate cc-transforms.
    • Assumptions/dependencies: Access to large sections of the graded ring; stable numerical linear algebra for ultrametric norms; routine computation of Okounkov bodies for sanity checks.
  • Prototype software components for “tropical theta” and skeleton-based computations (Software: computational algebraic geometry)
    • What: Build reusable modules to compute tropical theta functions log⁡∣θαl∣\log|\theta_\alpha^l| on Sk(X)Sk(X), dual complexes, and cost functions c(x,p)c(x,p); export to optimization routines for cc-transforms and envelopes.
    • How:
    • Symbolic pipelines to Taylor-expand sections and extract leading exponents on faces.
    • Routines to construct/triangulate Sk(X)Sk(X) from SNC data and to evaluate valuations.
    • A minimal OT solver on polytopal complexes that implements the cc-transform with Lipschitz control.
    • Tools/workflows: APIs for SageMath/Macaulay2; Python/C++ libraries for polytopal complexes; bindings to convex solvers.
    • Assumptions/dependencies: Availability of semistable models and basis sections; standardization of input formats for dual complex data.
  • Course modules and seminars connecting NA geometry, optimal transport, and mirror symmetry (Education)
    • What: Teach a modern pipeline: degenerations → Berkovich skeleta → NA Monge–Ampère → convex/OT viewpoint → SYZ fibrations.
    • How: Problem sets around building Sk(X)Sk(X) for toy models, computing cc-transforms, and visualizing special Lagrangian torus fibrations in toric cases.
    • Assumptions/dependencies: Graduate-level background in complex/NA geometry; access to open-source computational tools.
  • Cross-checks for Gross–Siebert mirror constructions via optimal transport (Academia: mathematics, mirror symmetry)
    • What: Use the paper’s OT interpretation of the NA Monge–Ampère solution to cross-validate mirror constructions (e.g., matching measures on Sk(X)Sk(X) and the mirror’s skeleton via a transport map/Kantorovich potential).
    • How: Build (B,ν)(B,\nu) using Okounkov techniques and compute the cc-transform/Kantorovich potential; compare with mirror-side combinatorics.
    • Assumptions/dependencies: Explicit Gross–Siebert data; reliable construction of (B,ν)(B,\nu); manageable combinatorics in concrete examples.

Long-Term Applications

The items below require further theoretical development, scaling, or domain translation before broader deployment beyond pure mathematics.

  • End-to-end computational pipelines for special Lagrangian torus fibrations (Academia: mathematics; Theoretical physics: string theory)
    • Vision: Given a polarised maximal degeneration, automatically produce approximate special Lagrangian TnT^n-fibrations on large-measure subsets, guided by the NA CY potential and hybrid convergence.
    • Potential impact:
    • Practical tools for mirror symmetry computations in complex dimensions ≥2.
    • Data to inform model-building in string/M-theory compactifications (metric data on moduli, fibration topology).
    • Dependencies: Efficient numerical solvers for real Monge–Ampère on skeleta; robust implementations of the Savin perturbation step; verification of valuative independence in broad families.
  • Optimal transport on simplicial complexes with ultrametric/tropical costs (Software; Data science/optimization; Networks)
    • Vision: Generalize the paper’s c(x,p)c(x,p) on Sk(X)Sk(X) to define OT frameworks on hierarchical or polytopal domains (e.g., meshes, phylogenetic trees, supply-chain hierarchies) with ultrametric structure.
    • Potential products:
    • OT libraries for piecewise-affine domains with Lipschitz cost control.
    • Applications to hierarchical clustering alignment, multiresolution transport, and flow on stratified networks.
    • Dependencies: Adaptation of convex duality and stability theory from NA geometry to generic simplicial complexes; empirical validation on real datasets.
  • Geometry-aware numerical solvers for Monge–Ampère-type PDEs on polytopal domains (Software; Engineering/graphics)
    • Vision: Use the real Monge–Ampère regularity on faces and Lipschitz controls from the skeleton setting to inform discretizations for MA equations on meshes (e.g., in graphics, optimal transport maps, reflector design).
    • Potential products: Mesh-based MA solvers leveraging facewise-convexity and polyhedral structure; faster convergence via skeleton-inspired preconditioners.
    • Dependencies: Translation of NA-informed discretizations to Euclidean mesh settings; stability/error analyses; integration with existing PDE libraries.
  • Mirror-symmetric metric correspondences via transport between skeleta (Academia: mathematics, theoretical physics)
    • Vision: Establish and compute the conjectural identification of (B,ν)(B,\nu) with the mirror’s essential skeleton, yielding a metric-level mirror symmetry via OT.
    • Potential impact: New computational pathways for enumerative predictions, period computations, and metric comparisons across mirrors.
    • Dependencies: Refinements of Okounkov-body-based constructions; validation across diverse families; interoperability with Gross–Siebert theta functions.
  • Canonical-basis–driven algorithms for toric degenerations and model selection (Software; Academia)
    • Vision: Use valuative independence and theta bases to automate construction of toric/torus-like degenerations that simplify metric and enumerative computations.
    • Potential products: Algorithms to select “near-toric” models that optimize convex-analytic properties (e.g., smoother cc-functions, better conditioning for solvers).
    • Dependencies: Efficient detection/testing of valuative independence; scalable Okounkov body computations; integration with birational model exploration tools.
  • Interdisciplinary modeling on ultrametric spaces (Data science; Computational biology)
    • Vision: Transfer ultrametric valuation principles and skeleton-based convex analysis to hierarchical data domains (phylogenetic trees, taxonomy, ontology graphs), enabling new cost functions and convex envelopes for inference.
    • Potential products:
    • Hierarchical OT distances respecting ultrametric constraints.
    • New regularizers/envelopes inspired by NA psh envelopes and orthogonality for learning on trees/stratified spaces.
    • Dependencies: Theoretical adaptation from Sk(X)Sk(X) to generic ultrametric trees; empirical pipelines and benchmarks.

Notes on Key Assumptions and Dependencies (cross-cutting)

  • Existence of valuative independent bases: The paper’s main pipeline assumes such bases; Blum–Liu’s forthcoming result provides existence for m0Lm_0L in general, but concrete computation may still be challenging.
  • Semistable SNC models and dual complexes: Constructing explicit models and skeleta is nontrivial; automated tools are needed for routine use.
  • Computational tractability: Large graded section spaces and ultrametric norms can be expensive; scalable linear algebra and symbolic expansions are needed.
  • Regularity and stability: Numerical solvers for real/NA Monge–Ampère with skeleton-informed discretizations require careful stability and error analysis.
  • Domain translation: Applying ultrametric/tropical cost structures outside pure geometry requires principled adaptation and validation on real-world datasets.
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Glossary

  • Abelian varieties: Complex projective varieties that are also groups, generalizing complex tori and used as standard examples in complex and algebraic geometry. "The metric SYZ conjecture is currently known for Abelian varieties, certain K3 surfaces, and a large class of hypersurface examples inside toric Fano manifolds"
  • affinoid torus: A rigid-analytic or non-archimedean analogue of a complex torus, serving as a local model in Berkovich/rigid geometry. "behaves like the monomials on the affinoid torus"
  • Berkovich space: The non-archimedean analytification of an algebraic variety, consisting of semivaluations and equipped with a rich topology. "associate a NA object called the Berkovich space XKanX_K^{an}"
  • c-transform: A generalization of the Legendre transform defined using a cost function in optimal transport-type constructions. "we can define the cc-transform for bounded functions"
  • Calabi–Yau (CY) manifolds: Compact Kähler manifolds with trivial canonical bundle and vanishing first Chern class, admitting Ricci-flat metrics. "meromorphic degeneration of compact Calabi-Yau (CY) manifolds"
  • canonical basis: A distinguished, often geometrically or combinatorially defined basis of a ring or vector space, expected to have good properties (e.g., positivity, independence). "a canonical basis of the section ring for the polarisation line bundle"
  • dual complex: A simplicial complex encoding the combinatorics of the normal-crossing components of a special fiber in a model. "the dual complex $\Delta_{\mathcal{X}$"
  • essential skeleton: A canonical subset of the Berkovich space capturing the asymptotic geometry of degenerations, defined via minimal log discrepancy. "the essential skeleton Sk(X)Sk(X)"
  • Fubini–Study metric (non-archimedean): The NA analogue of the classical Fubini–Study metric, constructed from norms on spaces of sections. "the NA Fubini-Study metric on the line bundle L→XKanL\to X_K^{an}"
  • GAGA principle: A correspondence between algebraic and analytic categories (here, line bundles on algebraic varieties and their analytifications). "A GAGA principle says the line bundles on XKanX_K^{an} correspond to the line bundles LL on the scheme XKX_K."
  • Gromov–Witten invariants: Enumerative invariants counting curves in a variety, foundational in mirror symmetry and used in Gross–Siebert constructions. "given a polarised maximal degeneration of compact CY manifolds, one can build a graded ring from punctured Gromov-Witten invariants"
  • hybrid topology: A topology unifying complex and non-archimedean analytifications, making the NA space a limit of complex fibers. "There is a hybrid topology on X⊔XKanX\sqcup X_K^{an}"
  • Kähler class: A cohomology class represented by a Kähler form; it specifies the polarization of a complex manifold. "in the K\"ahler class 1∣log⁡∣t∣∣c1(L)\frac{1}{|\log |t||} c_1(L)"
  • Kantorovich potential: A potential function solving the dual problem in optimal transport, associated to a given cost. "the associated Kantorovich potential"
  • log discrepancy function: A birational invariant measuring singularities, central to defining the essential skeleton. "The essential skeleton is the subset of XKanX_K^{an} where the log discrepancy function takes the minimal value"
  • meromorphic degeneration: A family of varieties over a punctured disc that may have poles or singularities as the parameter approaches the center. "a meromorphic degeneration of compact Calabi-Yau (CY) manifolds over the punctured disc"
  • NA Calabi conjecture: The non-archimedean analogue of the Calabi conjecture asserting existence/uniqueness of potentials solving a NA Monge–Ampère equation. "The central result of NA pluripotential theory due to Boucksom-Favre-Jonsson ... is the solution of the NA Calabi conjecture."
  • NA CY potential: The unique (up to constant) continuous NA plurisubharmonic potential solving the NA Monge–Ampère equation with the CY measure. "is called the NA CY potential"
  • NA MA equation: The non-archimedean Monge–Ampère equation prescribing the NA Monge–Ampère measure of a potential. "we draw consequences for a semipositive potential φ\varphi solving the NA MA equation"
  • NA MA measure: The non-archimedean Monge–Ampère measure associated to a (semi)positive metric, defined via intersection theory on models. "the NA MA measure for the model metric ⋅L{\cdot}_ {\mathcal{L} } is the signed atomic measure"
  • non-archimedean pluripotential theory: The NA analogue of complex pluripotential theory, studying NA psh functions and Monge–Ampère equations on Berkovich spaces. "A more general approach is based on non-archimedean (NA) pluripotential theory"
  • Okounkov bodies: Convex bodies associated to linear series encoding asymptotic information, used to build measures and cost functions. "We use the tool of Okounkov bodies to extract more refined information"
  • Proj construction: A method to construct projective schemes from graded rings, used, e.g., to build mirror families. "via the relative Proj construction of this graded ring"
  • psh envelop: The supremum of all semipositive (psh) potentials bounded above by a given function, yielding the largest dominated psh potential. "we define the psh envelop as"
  • psh potential: A plurisubharmonic potential function defining a semipositive metric in the (non-archimedean) setting. "a unique (up to constant) continuous psh potential ϕ0\phi_0"
  • retraction map: A canonical map from the Berkovich space to the dual complex associated to an SNC model. "a retraction map rXr_\mathcal{X} from XKanX_K^{an} to the dual complex"
  • semipositive metric: A metric that is a uniform limit of NA Fubini–Study metrics (or has nonnegative curvature), ensuring positivity properties. "A continuous semipositive metric ${\cdot}={\cdot}_{\mathcal{L}e^{-\phi}$ on LL"
  • semistable reduction: A process (after base change) that replaces a degeneration by one with reduced normal-crossing special fiber. "By Hironaka resolution and semistable reduction, up to finite base change, we shall without loss assume that there exists some semistable SNC model."
  • semivaluation: A valuation-like function satisfying the ultrametric inequality, used to define points of Berkovich spaces. "the Berkovich space is the space of semivaluations on the ring of functions"
  • SNC model: A model whose special fiber has simple normal crossings; a combinatorial-friendly form of degeneration. "An SNC model is semistable if all the divisor components on X0\mathcal{X}_0 have multiplicity one."
  • special Lagrangian TnT^n-fibration: A fibration by special Lagrangian tori, central in the SYZ picture of mirror symmetry. "there exists a special Lagrangian TnT^n-fibration"
  • Strominger–Yau–Zaslow (SYZ) conjecture: A conjecture predicting special Lagrangian torus fibrations on Calabi–Yau manifolds near large complex structure limits. "A central problem in CY geometry is the Strominger-Yau-Zaslow (SYZ) conjecture"
  • test configurations: One-parameter degenerations used to study stability and variational problems in algebraic geometry. "construction of multi-parameter test configurations"
  • theta functions: Canonical sections forming a basis constructed in mirror symmetry and cluster varieties, generalizing classical theta functions. "whose section ring has a canonical basis called theta functions"
  • toric Fano manifolds: Fano varieties with a torus action having a dense orbit; serve as rich sources of explicit examples. "hypersurface examples inside toric Fano manifolds"
  • ultrametric norm: A norm satisfying the strong triangle inequality, typical in non-archimedean settings. "A norm on a finite dimensional KK-vector space VV is called ultrametric"
  • valuative independence: A property of a basis ensuring valuations of linear combinations are determined by minima over basis elements, mimicking monomials. "satisfying the valuative independence condition"
  • divisorial valuations: Valuations associated to prime divisors, corresponding to vertices of the dual complex in this setting. "the vertices of ΔJ\Delta_J map to the divisorial valuations"
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  1. Yang Li has proved the (metric) SYZ conjecture (349 points, 28 comments)