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$\mathbb{G}$-uniform weighted K-stability for models on klt varieties

Jiyuan Han, Yaxiong Liu

TL;DR

The paper extends the Yau–Tian–Donaldson paradigm to polarized klt varieties with weights by proving that $oldsymbol{G}$-uniform weighted K-stability for models implies the $oldsymbol{G}$-coercivity of the weighted Mabuchi functional ${f M}_{v,w}$ under the envelope property. It develops a comprehensive weighted non-Archimedean pluripotential theory, including a weighted Calabi–Yau theorem, and establishes coercivity via a Birational Chi Li-type argument, with a toric openness result that yields existence of weighted extremal metrics on resolutions when $v$ is log-concave. The framework uses polynomial and smooth weight approximations, multiplier-ideal techniques, and a robust NA–Archimedean dictionary to connect algebraic stability with analytic energy functionals in singular settings. These results extend prior smooth-case work and integrate recent advances in the weighted/non-Archimedean theory, opening paths toward singular weighted cscK and extremal metrics.

Abstract

In this paper, we make a generalization of the results in \cite{Li22a} to the singular and weighted setting. In particular, we show that on a polarized projective klt variety, the $\mathbb{G}$-uniform weighted K-stability for models implies the $\mathbb{G}$-coercivity of the weighted Mabuchi functional. In the toric case, we further show that the $(\mathbb{C}^{\times})^n$-uniform $(\mathrm{v},\mathrm{w}\cdot\ell_{\mathrm{ext}})$-weighted K-stability is preserved when perturbing the polarization on the resolution, which implies the existence of the weighted extremal metric(s) on the resolution if the weight function $\mathrm{v}$ is log-concave.

$\mathbb{G}$-uniform weighted K-stability for models on klt varieties

TL;DR

The paper extends the Yau–Tian–Donaldson paradigm to polarized klt varieties with weights by proving that -uniform weighted K-stability for models implies the -coercivity of the weighted Mabuchi functional under the envelope property. It develops a comprehensive weighted non-Archimedean pluripotential theory, including a weighted Calabi–Yau theorem, and establishes coercivity via a Birational Chi Li-type argument, with a toric openness result that yields existence of weighted extremal metrics on resolutions when is log-concave. The framework uses polynomial and smooth weight approximations, multiplier-ideal techniques, and a robust NA–Archimedean dictionary to connect algebraic stability with analytic energy functionals in singular settings. These results extend prior smooth-case work and integrate recent advances in the weighted/non-Archimedean theory, opening paths toward singular weighted cscK and extremal metrics.

Abstract

In this paper, we make a generalization of the results in \cite{Li22a} to the singular and weighted setting. In particular, we show that on a polarized projective klt variety, the -uniform weighted K-stability for models implies the -coercivity of the weighted Mabuchi functional. In the toric case, we further show that the -uniform -weighted K-stability is preserved when perturbing the polarization on the resolution, which implies the existence of the weighted extremal metric(s) on the resolution if the weight function is log-concave.

Paper Structure

This paper contains 18 sections, 42 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Background on non-Archimedean pluripotential theory
  5. Berkovich analytification and psh functions
  6. Norms and Fubini--Study operator
  7. Weighted Archimedean functionals and geodesic
  8. Weighted non-Archimedean Monge-Ampère measure, weighted non-Archimedean functionals and geodesic rays
  9. Weighted non-Archimedean Monge-Ampère measure and functionals for the polynomial weight
  10. Weighted Okounkov body and weighted volume
  11. Weighted non-Archimedean functional for smooth weight
  12. Weighted non-Archimedean Monge--Ampère equations
  13. Multiplier ideal approximations
  14. Maximal geodesic rays
  15. Weighted K-stability for models
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $(X,L)$ be a polarized projective variety with at most klt singularities. Assume $(X,L)$ satisfies the envelope property $($see Definition def:envelope_property$)$. Let $\mathrm v\in C^\infty(P,\mathbb{R}_{>0})$ and $\mathrm w\in C^\infty(P,\mathbb{R})$. If $(X,L)$ is $\mathbb{G}$-uniformly weig

Theorems & Definitions (92)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1: Li22a, BHJ17
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: BJ22
  • Definition 2.5
  • Definition 2.6
  • ...and 82 more