Degeneration of Calabi-Yau metrics and canonical basis

2025/05/16 by Yang Li, Li, Yang · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Basis (linear algebra) #Degeneration (medical) #Differential Geometry (math.DG) #Dimension (graph theory) #FOS: Mathematics #Field (mathematics) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Limit (mathematics) #Metric (unit) #Tensor (intrinsic definition)

paper · pdf · doi:10.48550/arxiv.2505.11087

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

For polarised degenerations of Calabi-Yau manifolds whose essential skeleton has dimension 1≤ m≤ n, we show that the C⁰ potential theoretic limit of the Calabi-Yau metrics agrees with the non-archimedean Calabi-Yau metric on the Berkovich analytification. Moreover, this limit data can be encoded into the unique minimiser of the Kontorovich functional of an optimal transport problem, under some algebro-geometric assumptions on the existence of a canonical basis of sections for tensor powers of the polarisation line bundle.

Citations

• Toric degenerations of Calabi--Yau complete intersections and metric SYZ conjecture
• Intermediate complex structure limit for Calabi-Yau metrics
• Solvability of Monge-Ampère equations and tropical affine structures on reflexive polytopes
• Metric SYZ conjecture for certain toric Fano hypersurfaces
• Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces
• Global pluripotential theory on hybrid spaces
• Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces
• Special Lagrangian fibrations, Berkovich retraction, and crystallographic groups
• Metric SYZ conjecture and non-archimedean geometry
• Complex structure degenerations and collapsing of Calabi-Yau metrics
• Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces
• Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry
• The non-archimedean SYZ fibration
• Degeneration of endomorphisms of the complex projective space in the hybrid space
• Tropical and non-Archimedean limits of degenerating families of volume forms
• Theta functions on varieties with effective anti-canonical class
• The Non-Archimedean Monge-Ampere Equation
• The essential skeleton of a degeneration of algebraic varieties
• Mirror Symmetry and the Strominger-Yau-Zaslow conjecture
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Theta functions and mirror symmetry
• Solution to a non-Archimedean Monge-Ampère equation
• Singular semipositive metrics in non-Archimedean geometry
• Large Complex Structure Limits of K3 Surfaces

Cited by

• Generic regularity of intermediate complex structure limits

Related

• Intermediate complex structure limit for Calabi-Yau metrics
• Complex structure degenerations and collapsing of Calabi-Yau metrics
• Generic regularity of intermediate complex structure limits
• Completion of the moduli space for polarized Calabi-Yau manifolds
• Note on equivalences for degenerations of Calabi-Yau manifolds