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Open problems in K-stability of Fano varieties

Chenyang Xu, Ziquan Zhuang

TL;DR

This note surveys open problems in K-stability of Fano varieties, linking the analytic existence of Kähler-Einstein metrics to algebraic criteria such as the delta-invariant and the stability threshold. It outlines the construction and properties of K-moduli, discusses explicit families (hypersurfaces, bundles) and low-dimensional cases, and emphasizes local stability through normalized volumes and stable degenerations. The manuscript also surveys Gibbs stability, positivity of tangent bundles, and symplectic/contact geometry connections, while articulating broader questions on higher-rank valuations, graded ideal sequences, and positive characteristics. Together, these topics illuminate how K-stability informs moduli, birational geometry, and singularity theory, with implications for both theory and future computations.

Abstract

In this note, we discuss a number of open problems in K-stability theory.

Open problems in K-stability of Fano varieties

TL;DR

This note surveys open problems in K-stability of Fano varieties, linking the analytic existence of Kähler-Einstein metrics to algebraic criteria such as the delta-invariant and the stability threshold. It outlines the construction and properties of K-moduli, discusses explicit families (hypersurfaces, bundles) and low-dimensional cases, and emphasizes local stability through normalized volumes and stable degenerations. The manuscript also surveys Gibbs stability, positivity of tangent bundles, and symplectic/contact geometry connections, while articulating broader questions on higher-rank valuations, graded ideal sequences, and positive characteristics. Together, these topics illuminate how K-stability informs moduli, birational geometry, and singularity theory, with implications for both theory and future computations.

Abstract

In this note, we discuss a number of open problems in K-stability theory.
Paper Structure (19 sections, 5 theorems, 44 equations)

This paper contains 19 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Kähler-Einstein problem
  3. K-stability of Fano varieties in algebraic geometry
  4. K-stability of projective varieties
  5. K-stability of explicit Fano varieties
  6. Fano hypersurfaces
  7. Moduli of vector bundles
  8. Low dimensional cases
  9. Positivity of tangent bundle
  10. Gibbs stability
  11. Local K-stability
  12. Symplectic singularities and contact Fano varieties
  13. Local volumes
  14. Miscellaneous Questions
  15. MMP and local volume
  16. ...and 4 more sections

Key Result

Theorem 2

We have the following: A log Fano pair $(X,\Delta)$ is

Theorems & Definitions (7)

  • Theorem 2
  • Theorem 4: K-moduli
  • Conjecture 15: Berman
  • Theorem 17: Stable degeneration
  • Claim
  • Theorem 26: XZ-localbound
  • Theorem 33: LXZ-HRFGXZ-HRFGChen-HRFG