Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform K-stability of $G$-varieties of complexity 1

Yan Li, Zhenye Li

TL;DR

We address uniform K-stability for projective G-varieties of complexity 1 by giving a complete combinatorial classification of G-equivariant normal test configurations with integral central fibres, expressed through the Luna–Vust framework and associated polytopes. A central technical achievement is a closed-form formula for B-stable anti-canonical divisors in both one-parameter and quasihomogeneous settings, which simplifies Futaki-invariant computations and leads to a concrete stability criterion in terms of barycenters of polytopes $ riangle_x^O(K_X^{-1})$. Under the klt assumption, the stability criterion translates into a tractable, finite combinatorial condition, enabling explicit checks of G-equvariant uniform K-stability. The paper validates the framework on the Mukai–Umemura threefold and a SL3-based ordered-triangle family, demonstrating uniform K-stability in these symmetric examples and illustrating the practical reach of the method for constructing canonical metrics via the Yau–Tian–Donaldson perspective. Overall, the work provides a finite, combinatorial pathway to uniform K-stability for a broad class of highly symmetric Fano G-varieties of complexity 1, linking representation-theoretic data with geometric stability and metric existence results.

Abstract

Let ${\rm k}$ be an algebraically closed field of characteristic 0 and $G$ a connect, reductive group over it. Let $X$ be a projective $G$-variety of complexity 1. We classify $G$-equivariant normal test configurations of $X$ with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on $X$. Based on this formula, when $X$ is $\mathbb Q$-Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.

Uniform K-stability of $G$-varieties of complexity 1

TL;DR

We address uniform K-stability for projective G-varieties of complexity 1 by giving a complete combinatorial classification of G-equivariant normal test configurations with integral central fibres, expressed through the Luna–Vust framework and associated polytopes. A central technical achievement is a closed-form formula for B-stable anti-canonical divisors in both one-parameter and quasihomogeneous settings, which simplifies Futaki-invariant computations and leads to a concrete stability criterion in terms of barycenters of polytopes . Under the klt assumption, the stability criterion translates into a tractable, finite combinatorial condition, enabling explicit checks of G-equvariant uniform K-stability. The paper validates the framework on the Mukai–Umemura threefold and a SL3-based ordered-triangle family, demonstrating uniform K-stability in these symmetric examples and illustrating the practical reach of the method for constructing canonical metrics via the Yau–Tian–Donaldson perspective. Overall, the work provides a finite, combinatorial pathway to uniform K-stability for a broad class of highly symmetric Fano G-varieties of complexity 1, linking representation-theoretic data with geometric stability and metric existence results.

Abstract

Let be an algebraically closed field of characteristic 0 and a connect, reductive group over it. Let be a projective -variety of complexity 1. We classify -equivariant normal test configurations of with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on . Based on this formula, when is -Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.
Paper Structure (44 sections, 46 theorems, 344 equations, 4 figures, 5 tables)

This paper contains 44 sections, 46 theorems, 344 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Test configurations and K-stability
  4. $G$-varieties of complexity 1
  5. General Luna-Vust theory
  6. The case of complexity 1
  7. The central automorphism group
  8. Cartier divisors on $G$-varieties
  9. The ample divisors
  10. Associated parabolic subgroup and properties of the function $\pi$
  11. Relation with the divisorial polytope construction
  12. The affine cone over a projectively normal embedding
  13. Combinatorial data of the affine cone
  14. Admissibility
  15. Relationship with the coloured fan of $X$
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $X$ be a projective, $\mathbb Q$-Fano $G$-varieties of complexity 1 with klt singularities, which satisfies ${\rm k}(X)^B\cong{\rm k}(\mathbb P^1)$. Then $X$ is $G$-equivariantly K-semistable if and only if where $(\mathscr V_x)^\vee$ is the dual cone of $\mathscr V_x$. Moreover, the following conditions are equivalent:

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (90)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 80 more