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K-semistability of log Fano cone singularities

Yuchen Liu, Yueqiao Wu

TL;DR

The paper develops a non-Archimedean framework for local K-semistability of log Fano cone singularities, proving a full equivalence between NA functional positivity, delta-criterion, and stability against special test configurations. It extends global K-stability ideas to the local cone setting, establishing a continuous dependence on Reeb fields and a valuative criterion that reduces checks to special configurations. A key contribution is linking special test configurations to lc places of torus-equivariant bounded complements, yielding a torus-equivariant existence theory for complements. The results unify analytic and algebro-geometric perspectives, providing practical criteria for verifying local K-semistability and informing moduli-like questions for singularities with torus actions.

Abstract

We give a non-Archimedean characterization of K-semistability of log Fano cone singularities, and show that it agrees with the definition originally defined by Collins--Székelyhidi. As an application, we show that to test K-semistability, it suffices to test special test configurations. We also show that special test configurations give rise to lc places of torus equivariant bounded complements.

K-semistability of log Fano cone singularities

TL;DR

The paper develops a non-Archimedean framework for local K-semistability of log Fano cone singularities, proving a full equivalence between NA functional positivity, delta-criterion, and stability against special test configurations. It extends global K-stability ideas to the local cone setting, establishing a continuous dependence on Reeb fields and a valuative criterion that reduces checks to special configurations. A key contribution is linking special test configurations to lc places of torus-equivariant bounded complements, yielding a torus-equivariant existence theory for complements. The results unify analytic and algebro-geometric perspectives, providing practical criteria for verifying local K-semistability and informing moduli-like questions for singularities with torus actions.

Abstract

We give a non-Archimedean characterization of K-semistability of log Fano cone singularities, and show that it agrees with the definition originally defined by Collins--Székelyhidi. As an application, we show that to test K-semistability, it suffices to test special test configurations. We also show that special test configurations give rise to lc places of torus equivariant bounded complements.
Paper Structure (18 sections, 36 theorems, 99 equations)

This paper contains 18 sections, 36 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Valuations and norms
  4. The Berkovich analytification
  5. $\mathbb{T}$-invariant valuations
  6. Filtrations and norms
  7. Log Fano cone singularities
  8. Non-Archimedean pluripotential theory
  9. Local K-stability
  10. K-stability of log Fano pairs
  11. K-stability of log Fano cone singularities
  12. Comparison of local and global K-stability
  13. Torus equivariant complements
  14. A valuative criterion for K-semistability
  15. Non-Archimedean functionals
  16. ...and 3 more sections

Key Result

Theorem 1.1

For a fixed normal test configuration $(\mathcal{X}, \mathcal{B}; \xi, \eta)$ of a log Fano cone singularity $(X, B; \xi)$, the function is continuous on the Reeb cone for any $F\in \{M, D\}$. Moreover, when $\xi$ is rational, they recover the corresponding global NA functionals.

Theorems & Definitions (95)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 85 more