Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Families of singular Kähler-Einstein metrics

Eleonora Di Nezza, Vincent Guedj, Henri Guenancia

TL;DR

The paper develops uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Kahler manifolds and extends these to families of possibly singular Kahler varieties. By refining Yau/Kolodziej techniques and introducing capacity methods, uniform integrability, and a pluripotential framework in families, it obtains C^0 controls for KE potentials and analyzes their behavior across general type, Calabi–Yau, and log Calabi–Yau degenerations. It also constructs and analyzes semi-stable models, canonical densities, and Green-type inequalities to handle normalization, irreducibility, and singularities in a unified way. The results illuminate moduli of stable varieties, provide stability results for singular KE metrics, and establish convergence patterns in both collapsing and non-collapsing regimes, with broad implications for moduli theory and geometric analysis of degenerating families.

Abstract

Refining Yau's and Kolodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler-Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman-Guenancia and Song, as well as on the behavior of singular Ricci flat metrics on (log) Calabi-Yau varieties, generalizing works by Rong-Ruan-Zhang, Gross-Tosatti-Zhang, Collins-Tosatti and Tosatti-Weinkove-Yang.

Families of singular Kähler-Einstein metrics

TL;DR

The paper develops uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Kahler manifolds and extends these to families of possibly singular Kahler varieties. By refining Yau/Kolodziej techniques and introducing capacity methods, uniform integrability, and a pluripotential framework in families, it obtains C^0 controls for KE potentials and analyzes their behavior across general type, Calabi–Yau, and log Calabi–Yau degenerations. It also constructs and analyzes semi-stable models, canonical densities, and Green-type inequalities to handle normalization, irreducibility, and singularities in a unified way. The results illuminate moduli of stable varieties, provide stability results for singular KE metrics, and establish convergence patterns in both collapsing and non-collapsing regimes, with broad implications for moduli theory and geometric analysis of degenerating families.

Abstract

Refining Yau's and Kolodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler-Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman-Guenancia and Song, as well as on the behavior of singular Ricci flat metrics on (log) Calabi-Yau varieties, generalizing works by Rong-Ruan-Zhang, Gross-Tosatti-Zhang, Collins-Tosatti and Tosatti-Weinkove-Yang.

Paper Structure

This paper contains 33 sections, 41 theorems, 305 equations.

Table of Contents

  1. Chasing the constants
  2. Preliminaries on capacities
  3. Proof of Theorem \ref{['thm:uniform1']}
  4. Domination by capacity
  5. Bounding the solution from below
  6. More general densities
  7. Big cohomology classes
  8. Uniform integrability
  9. Notations
  10. Uniform integrability index
  11. Skoda's integrability theorem in families: the projective case
  12. Skoda's integrability theorem in families: the general case
  13. Normalization in families
  14. Irreducibility of the fibers
  15. The locally trivial case
  16. ...and 18 more sections

Key Result

Theorem A

Let $X$ be a compact Kähler manifold of complex dimension $n \in {\mathbb{N}}^*$ and let $\omega$ be a semi-positive form such that $V:= \int_X \omega^n >0.$ Let $\nu$ and $\mu=f \, \nu$ be probability measures, with $0 \leqslant f \in L^p(\nu)$ for some $p>1$. Assume the following assumptions are s Let $\varphi$ be the unique $\omega$-psh solution $\varphi$ to the complex Monge-Ampère equation n

Theorems & Definitions (97)

  • Theorem A
  • Remark 1
  • Proposition B
  • Theorem C
  • Proposition D
  • Theorem E
  • Theorem F
  • Theorem G
  • Theorem 1.1
  • Lemma 1.2
  • ...and 87 more