Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Calabi-Yau theorems for degenerations of compact Kähler manifolds

David Witt Nyström

TL;DR

The paper addresses degenerations of compact Kähler manifolds and develops two Calabi–Yau theorems: (i) a Calabi–Yau result for big test configurations, extending the classical CY equation to a broader setting, and (ii) a non-Archimedean Calabi–Yau theorem for finite-energy potentials, formulated in non-Archimedean Kähler geometry. The main approach combines a direct proof for big test configurations with a variational/non-Archimedean framework, leveraging the theory of big cohomology classes, restricted volumes, and orthogonality to establish surjectivity and continuity of the Monge–Ampère operator in these contexts. Key contributions include a direct proof of Theorem A showing the Monge–Ampère measure of big test configurations is a probability measure and a non-Archimedean CY theorem that intertwines complex-analytic degenerations with non-Archimedean potential theory, underpinning variational approaches to the Yau–Tian–Donaldson program in the Kähler setting. Together, these results illuminate how volumes, positivity cones, and restricted volumes govern degenerations and canonical metrics, bridging complex and non-Archimedean perspectives and strengthening the foundation for stability criteria in Kähler geometry.

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.

Two Calabi-Yau theorems for degenerations of compact Kähler manifolds

TL;DR

The paper addresses degenerations of compact Kähler manifolds and develops two Calabi–Yau theorems: (i) a Calabi–Yau result for big test configurations, extending the classical CY equation to a broader setting, and (ii) a non-Archimedean Calabi–Yau theorem for finite-energy potentials, formulated in non-Archimedean Kähler geometry. The main approach combines a direct proof for big test configurations with a variational/non-Archimedean framework, leveraging the theory of big cohomology classes, restricted volumes, and orthogonality to establish surjectivity and continuity of the Monge–Ampère operator in these contexts. Key contributions include a direct proof of Theorem A showing the Monge–Ampère measure of big test configurations is a probability measure and a non-Archimedean CY theorem that intertwines complex-analytic degenerations with non-Archimedean potential theory, underpinning variational approaches to the Yau–Tian–Donaldson program in the Kähler setting. Together, these results illuminate how volumes, positivity cones, and restricted volumes govern degenerations and canonical metrics, bridging complex and non-Archimedean perspectives and strengthening the foundation for stability criteria in Kähler geometry.

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.

Paper Structure

This paper contains 30 sections, 12 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. The classical Calabi-Yau theorem
  3. A Calabi-Yau theorem for big test configurations
  4. Test configurations and models
  5. Kähler test configurations
  6. Big test configurations
  7. A non-Archimedean Calabi-Yau theorem
  8. Tropical analytification
  9. Divisorial points
  10. Dual complexes
  11. Vertical divisors and PL functions
  12. Kähler potentials and $A$-psh functions
  13. Finite energy potentials
  14. The non-Archimdedean Monge-Ampère operator
  15. Finite energy measures
  16. ...and 15 more sections

Key Result

Theorem 2.1

For any volume form $dV$ on $X$ such that $\int_XdV=V$, there is a unique Kähler form $\omega'\in\alpha$ such that $(\omega')^n=dV$.

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • ...and 15 more