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Non-Archimedean Calabi-Yau Potentials on Certain Affine Varieties

Ying Wang

TL;DR

This work constructs and analyzes non-Archimedean Calabi–Yau potentials on the Berkovich analytification of affine Calabi–Yau varieties obtained from log Calabi–Yau pairs with simplex dual complexes. It reduces the NA Monge–Ampère equation to a real Monge–Ampère problem via tropicalization, constructs a NA Calabi–Yau potential from a convex solution to a CTY-type PDE, and proves that MA$^{\mathrm{na}}(\psi^{\mathrm{na}})=\mu^{\mathrm{na}}$ with $\mu^{\mathrm{na}}$ the Lebesgue measure on the essential skeleton. The paper then develops hybrid spaces to connect NA and Archimedean geometries, showing continuity of hybrid potentials built from TY, CL, and generalized Calabi ansatz data, and demonstrating measure convergence on hybrid spaces. As a consequence, the NA framework recovers the expected homogeneous degree of the generalized Calabi ansatz and provides a bridge to Odaka’s conjecture relating volume growth to the essential skeleton, thereby strengthening the Archimedean–non-Archimedean dictionary in affine Calabi–Yau geometry.

Abstract

We solve a non-Archimedean Monge-Ampère equation on the Berkovich analytification of a complex log Calabi-Yau pair whose dual complex is a standard simplex, answering a question of Collins-Li and offering a non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties. This work builds on the solution to a complex Monge-Ampère equation obtained by Collins-Li and Collins-Tong-Yau. We also show the suitably rescaled limits of the complex potentials coincide with their non-Archimedean counterparts in some situations, strengthening their connections.

Non-Archimedean Calabi-Yau Potentials on Certain Affine Varieties

TL;DR

This work constructs and analyzes non-Archimedean Calabi–Yau potentials on the Berkovich analytification of affine Calabi–Yau varieties obtained from log Calabi–Yau pairs with simplex dual complexes. It reduces the NA Monge–Ampère equation to a real Monge–Ampère problem via tropicalization, constructs a NA Calabi–Yau potential from a convex solution to a CTY-type PDE, and proves that MA with the Lebesgue measure on the essential skeleton. The paper then develops hybrid spaces to connect NA and Archimedean geometries, showing continuity of hybrid potentials built from TY, CL, and generalized Calabi ansatz data, and demonstrating measure convergence on hybrid spaces. As a consequence, the NA framework recovers the expected homogeneous degree of the generalized Calabi ansatz and provides a bridge to Odaka’s conjecture relating volume growth to the essential skeleton, thereby strengthening the Archimedean–non-Archimedean dictionary in affine Calabi–Yau geometry.

Abstract

We solve a non-Archimedean Monge-Ampère equation on the Berkovich analytification of a complex log Calabi-Yau pair whose dual complex is a standard simplex, answering a question of Collins-Li and offering a non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties. This work builds on the solution to a complex Monge-Ampère equation obtained by Collins-Li and Collins-Tong-Yau. We also show the suitably rescaled limits of the complex potentials coincide with their non-Archimedean counterparts in some situations, strengthening their connections.
Paper Structure (26 sections, 23 theorems, 143 equations, 1 figure)

This paper contains 26 sections, 23 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The generalized Calabi ansatz
  3. General Setup
  4. Calabi model space
  5. Interlude I: Berkovich spaces
  6. Tropical spectrum over a trivially valued field
  7. Quasimonomial valuations
  8. Essential skeletons
  9. Interlude II: The Monge--Ampère operator on Berkovich spaces
  10. Moment maps and tropical charts
  11. Functions and metrics
  12. Characteristic polyhedron
  13. Non-Archimedean Monge--Ampère operator
  14. Real Monge--Ampère operator and comparison
  15. Non-Archimedean Calabi--Yau potentials
  16. ...and 11 more sections

Key Result

Theorem A

Let $n, d$ be integers with $n > d \geq 1$, and let $\bar{X}$ be a smooth projective Fano variety of dimension $n$, with a reduced simple normal crossing anticanonical divisor $D$ whose dual complex is the standard $(d-1)$-simplex. Set $X = \bar{X} \backslash D$, which is an affine Calabi--Yau varie The solution $\psi^\mathrm{na}$ is built on the Calabi ansatz from Cal79, TY90, CL24, and CTY24.

Figures (1)

  • Figure 1: The left illustrates $\mathbf{L}_{\sigma}$ that does not contain $\Delta$. From the case analysis, we have $\mathrm{MA}_{\mathbf{R}}(v|_{\mathbf{L}_{\sigma}}) = 0$. The right illustrates $\mathbf{L}_{\sigma}$ that contains $\Delta$, where we have shown $\mathrm{MA}_{\mathbf{R}}(v|_{\mathbf{L}_{\sigma}}) = c \cdot (n-d)!^{-1}\cdot \mathrm{Leb}_\Delta$.

Theorems & Definitions (70)

  • Theorem A
  • Theorem B
  • Example 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 3.1
  • Remark 3.2: Center
  • Remark 3.3: Pullback
  • Remark 3.4: Topologies and the structure sheaf
  • Definition 3.5
  • ...and 60 more