Non-Archimedean Calabi-Yau Potentials on Certain Affine Varieties

2025/10/24 by Wang, Ying
#14J32 #14T90 #32P05 #32Q25 #53C23 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.21677

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.

Citations

• A non-Archimedean theory of complex spaces and the cscK problem
• Calabi-Yau metrics of Calabi type with polynomial rate of convergence
• A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics
• Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces
• Global pluripotential theory on hybrid spaces
• Special Lagrangian fibrations, Berkovich retraction, and crystallographic groups
• A non-Archimedean approach to K-stability, II: divisorial stability and openness
• Pluripotential theory for tropical toric varieties and non-archimedean Monge-Ampére equations
• Polystable log Calabi-Yau varieties and Gravitational instantons
• Moduli of hybrid curves I: Variations of canonical measures
• Convergence of volume forms on a family of log-Calabi-Yau varieties to a non-Archimedean measure
• Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces
• The non-archimedean SYZ fibration
• Global pluripotential theory over a trivially valued field
• Degeneration of endomorphisms of the complex projective space in the\n hybrid space
• Tropical and non-Archimedean limits of degenerating families of volume forms
• Whitney Extension Theorems for convex functions of the classes C¹ and C^1,ω
• Metrization of differential pluriforms on Berkovich analytic spaces
• Degenerations of amoebae and Berkovich spaces
• Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros)
• Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Espaces de Berkovich, polytopes, squelettes et théorie des modèles
• Solution to a non-Archimedean Monge-Ampère equation
• Valuations and asymptotic invariants for sequences of ideals
• On asymptotics of complete Ricci-flat Kähler metrics on open manifolds

Related

• The Non-Archimedean Monge-Ampere Equation
• Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces
• Singular Calabi-Yau metrics
• A non-archimedean analogue of Calabi-Yau theorem for totally degenerate abelian varieties
• Non-Archimedean Green's functions and Zariski decompositions