Two Calabi-Yau theorems for degenerations of compact Kähler manifolds

2025/10/03 by Nyström, David Witt
#32P05 #32Q15 #32Q26 #53C55 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.03044

Abstract

We discuss two closely related Calabi-Yau theorems for degenerations of compact Kähler manifolds. The first is a Calabi-Yau theorem for big test configurations, that generalizes a result in [WN24]. It follows from recent joint work with Mesquita-Piccione [MW25], but is here given a more direct proof. The second result is a Calabi-Yau theorem for a wider class of degenerations, formulated in the language of non-Archimedean Kähler geometry. It was first proved in the algebraic setting by Boucksom-Jonsson [BJ22], building on earlier work of Boucksom-Favre-Jonsson [BFJ15], while the general Kähler case was established in [MW25]. Our main focus here is on the connection between these results and the theory of big cohomology classes and their volumes.

Citations

• A YTD correspondence for constant scalar curvature metrics
• A non-Archimedean theory of complex spaces and the cscK problem
• Derivative of volumes of big cohomology classes
• Measures of finite energy in pluripotential theory: a synthetic approach
• A transcendental approach to non-Archimedean metrics of pseudoeffective classes
• Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces
• A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations
• Deformations of Kähler manifolds to normal bundles and restricted volumes of big classes
• K-stability and Fujita approximation
• Metric SYZ conjecture and non-archimedean geometry
• Differentiability of relative volumes over an arbitrary non-Archimedean field
• Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)
• K-stability for Kähler Manifolds
• Duality between the pseudoeffective and the movable cone on a projective manifold
• A variational approach to the Yau-Tian-Donaldson conjecture
• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π
• K-stability and Kähler-Einstein metrics
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Solution to a non-Archimedean Monge-Ampère equation
• A variational approach to complex Monge-Ampere equations
• Monge-Ampère equations in big cohomology classes
• Convex Bodies Associated to Linear Series
• Differentiability of volumes of divisors and a problem of Teissier
• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• Numerical characterization of the Kähler cone of a compact Kähler manifold

Related

• Convergence of Calabi-Yau manifolds
• Note on equivalences for degenerations of Calabi-Yau manifolds
• A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem
• Degenerations of Ricci-flat Calabi-Yau manifolds
• Degeneration of Ricci-flat Calabi-Yau manifolds and its applications