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Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces

Jakob Hultgren, Mattias Jonsson, Enrica Mazzon, Nicholas McCleerey

TL;DR

The paper develops a tropical–non-Archimedean Monge–Ampère framework for a class of Calabi–Yau hypersurfaces X_t in projective space, leveraging the essential skeleton Sk(X) identified with the boundary of a simplex. It proves existence and uniqueness (up to constants) of a tropical Monge–Ampère solution on the tropical base B via a variational problem on symmetric c-convex functions, and shows that the corresponding non-Archimedean Monge–Ampère solution is the restriction of a continuous semipositive toric metric on the Berkovich analytification of O(d+2). This leads to a weak metric SYZ result: using Li–Yang Li’s framework, the tropical solution governs the metric and fibration structure in degenerations, with a controlled Gromov–Hausdorff limit. Overall, the work bridges toric/tropical geometry, optimal transport, and non-Archimedean pluripotential theory to illuminate Calabi–Yau degenerations and special Lagrangian fibrations in this setting.

Abstract

For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.

Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces

TL;DR

The paper develops a tropical–non-Archimedean Monge–Ampère framework for a class of Calabi–Yau hypersurfaces X_t in projective space, leveraging the essential skeleton Sk(X) identified with the boundary of a simplex. It proves existence and uniqueness (up to constants) of a tropical Monge–Ampère solution on the tropical base B via a variational problem on symmetric c-convex functions, and shows that the corresponding non-Archimedean Monge–Ampère solution is the restriction of a continuous semipositive toric metric on the Berkovich analytification of O(d+2). This leads to a weak metric SYZ result: using Li–Yang Li’s framework, the tropical solution governs the metric and fibration structure in degenerations, with a controlled Gromov–Hausdorff limit. Overall, the work bridges toric/tropical geometry, optimal transport, and non-Archimedean pluripotential theory to illuminate Calabi–Yau degenerations and special Lagrangian fibrations in this setting.

Abstract

For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.
Paper Structure (37 sections, 29 theorems, 106 equations, 3 figures)

This paper contains 37 sections, 29 theorems, 106 equations, 3 figures.

Table of Contents

  1. Toric setup
  2. Lattices and tori
  3. Tropicalization
  4. Simplices and projective space
  5. Tropical manifolds
  6. Singular integral affine structure
  7. Pairing and symmetries
  8. Pairing in coordinate charts
  9. The c-transform and the class of c-convex functions
  10. General definitions and properties
  11. Extension property
  12. Convexity in coordinate charts
  13. A principal $\mathbb{R}$-bundle
  14. Symmetric c-convex functions and their Monge--Ampère measures
  15. Controlling the c-gradients
  16. ...and 22 more sections

Key Result

Lemma 2.1

Pick any $m\in A$, $n\in B$, and let $G(m,n)\subset G$ be the set of $g\in G$ such that $\langle m,g(n)\rangle$ is maximal. Then, for any $i\in\{0,1,\dots,d+1\}$ we have:

Figures (3)

  • Figure 1: Subset $\tau_1$ for $d=2$
  • Figure 2: Subset $T_1$ for $d=2$
  • Figure :

Theorems & Definitions (72)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 62 more