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Kähler-Einstein metrics on families of Fano varieties

Chung-Ming Pan, Antonio Trusiani

TL;DR

This work analyzes how Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties behave in one-parameter degenerations by developing a relative pluripotential theory in families. The authors introduce convergence in families for quasi-psh functions, prove upper semi-continuity of Monge–Ampère energies, and establish a Demailly–Kollár type stability result along families using adapted measures. They derive uniform a priori estimates for Kähler–Einstein potentials, prove openness of the KE condition in degenerating families, and obtain higher-order and Moser–Trudinger-type controls, including a log Fano extension. The results provide analytic tools for moduli problems of Fano varieties, yielding an intrinsic, variational approach to the openness and stability of KE metrics in singular and degenerate settings, independent of existing stability criteria. Overall, the paper advances the understanding of KE metrics in degenerations and offers a robust framework for studying moduli of Fano varieties via pluripotential methods.

Abstract

Given a one-parameter family of $\mathbb{Q}$-Fano varieties such that the central fibre admits a unique Kähler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique Kähler-Einstein metric. Our results go beyond by establishing uniform a priori estimates on the Kähler-Einstein potentials along fully degenerate families of $\mathbb{Q}$-Fano varieties. In addition, we show the continuous variation of these Kähler-Einstein currents, and establish uniform Moser-Trudinger inequalities and uniform coercivity of the Ding functionals. Central to our article is introducing and studying a notion of convergence for quasi-plurisubharmonic functions within families of normal Kähler varieties. We show that the Monge-Ampère energy is upper semi-continuous with respect to this topology, and we establish a Demailly-Kollár result for functions with full Monge-Ampère mass.

Kähler-Einstein metrics on families of Fano varieties

TL;DR

This work analyzes how Kähler–Einstein metrics on -Fano varieties behave in one-parameter degenerations by developing a relative pluripotential theory in families. The authors introduce convergence in families for quasi-psh functions, prove upper semi-continuity of Monge–Ampère energies, and establish a Demailly–Kollár type stability result along families using adapted measures. They derive uniform a priori estimates for Kähler–Einstein potentials, prove openness of the KE condition in degenerating families, and obtain higher-order and Moser–Trudinger-type controls, including a log Fano extension. The results provide analytic tools for moduli problems of Fano varieties, yielding an intrinsic, variational approach to the openness and stability of KE metrics in singular and degenerate settings, independent of existing stability criteria. Overall, the paper advances the understanding of KE metrics in degenerations and offers a robust framework for studying moduli of Fano varieties via pluripotential methods.

Abstract

Given a one-parameter family of -Fano varieties such that the central fibre admits a unique Kähler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique Kähler-Einstein metric. Our results go beyond by establishing uniform a priori estimates on the Kähler-Einstein potentials along fully degenerate families of -Fano varieties. In addition, we show the continuous variation of these Kähler-Einstein currents, and establish uniform Moser-Trudinger inequalities and uniform coercivity of the Ding functionals. Central to our article is introducing and studying a notion of convergence for quasi-plurisubharmonic functions within families of normal Kähler varieties. We show that the Monge-Ampère energy is upper semi-continuous with respect to this topology, and we establish a Demailly-Kollár result for functions with full Monge-Ampère mass.
Paper Structure (23 sections, 27 theorems, 205 equations)

This paper contains 23 sections, 27 theorems, 205 equations.

Table of Contents

  1. Preliminaries
  2. Monge--Ampère energy
  3. $L^1$ metric geometry
  4. Variational approach to Kähler--Einstein metrics
  5. Convergence of quasi-plurisubharmonic functions in families
  6. Setting and known facts
  7. Definition of the convergence in families
  8. Hartogs type results
  9. Upper semi-continuity of Monge--Ampère energies in families
  10. Preparations
  11. Proof of Proposition \ref{['prop:usc_energy_family']}
  12. A Demailly--Kollár result in families
  13. Lelong number of functions in the finite energy class
  14. A Skoda--Zeriahi type estimate
  15. A Demailly--Kollár type result for functions with full Monge--Ampère mass
  16. ...and 8 more sections

Key Result

Theorem A

Let $\mathcal{X}$ be an $(n+1)$-dimensional $\mathbb{Q}$-Gorenstein variety and let $\pi: \mathcal{X} \to \mathbb{D}$ be a proper holomorphic surjective map with connected fibres. Assume that $-K_{\mathcal{X}/\mathbb{D}}$ is relatively ample, $X_0$ is klt, and $\mathop{\mathrm{Aut}}\nolimits(X_0)$ i

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Proposition C
  • Proposition D
  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3: Darvas_Rubinstein_2017Dinezza_Guedj_2018
  • Theorem 1.4: Berndtsson_2015, BBEGZ_2019
  • Remark 2.1
  • Remark 2.2
  • ...and 44 more