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Stability thresholds for big classes

Chenzi Jin, Yanir A. Rubinstein, Gang Tian

TL;DR

The paper extends the Tian–Odaka–Sano program from ample to big classes by introducing volume quantiles $\bm{\delta}_\tau$ and a new invariant $\tilde{\alpha}$, along with discrete counterparts, to characterize K-stability and the existence of twisted Kähler–Einstein metrics on the big cone. It develops a general theory of sub-barycenter estimates via a generalized Neumann–Hammer theorem and generalized Fujita inequalities, providing a framework that interpolates between classical alpha/delta criteria and modern Fujita–Odaka results. A key outcome is a generalized Tian–Odaka–Sano criterion for big classes, yielding Ding–stability and K–stability conclusions for a wide range of $\tau\in[0,1]$, with discrete variants. The paper also includes an explicit cubic-surface Eckardt-point example, illustrating how sub-barycenter and quantile techniques give transparent, conceptual proofs of stability criteria in concrete settings.

Abstract

In 1987, the $α$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $δ$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $τ\in[0,1]$. The special degenerate (collapsing) case $τ=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.

Stability thresholds for big classes

TL;DR

The paper extends the Tian–Odaka–Sano program from ample to big classes by introducing volume quantiles and a new invariant , along with discrete counterparts, to characterize K-stability and the existence of twisted Kähler–Einstein metrics on the big cone. It develops a general theory of sub-barycenter estimates via a generalized Neumann–Hammer theorem and generalized Fujita inequalities, providing a framework that interpolates between classical alpha/delta criteria and modern Fujita–Odaka results. A key outcome is a generalized Tian–Odaka–Sano criterion for big classes, yielding Ding–stability and K–stability conclusions for a wide range of , with discrete variants. The paper also includes an explicit cubic-surface Eckardt-point example, illustrating how sub-barycenter and quantile techniques give transparent, conceptual proofs of stability criteria in concrete settings.

Abstract

In 1987, the -invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to -Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the -invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles . The special degenerate (collapsing) case on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.

Paper Structure

This paper contains 11 sections, 11 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Valuative coercivity criteria on the ample cone
  3. A new alpha-invariant on the big cone
  4. Coercivity criteria on the big cone for all quantiles
  5. Generalized Neumann--Hammer theorem: sub-barycenter estimates
  6. A slice-projection generalization of the Neumann--Hammer barycenter estimate
  7. Generalized functional Neumann--Hammer inequality
  8. A generalized Fujita inequality of the first kind
  9. A generalized Fujita inequality of the second kind
  10. Proof of Theorem \ref{['GenTianThm']}
  11. K-stability of cubic surfaces with an Eckardt point

Key Result

Theorem 1.1

(Tian 1987) Let $X$ be a smooth $n$-dimensional Fano manifold, i.e., $-K_X$ is ample. Let $G$ be a compact subgroup of $\mathop{\mathrm{Aut}}\nolimits(X)$. Then $\alpha_G {>} \frac{n}{n+1}$ implies the existence of a Kähler--Einstein metric.

Figures (2)

  • Figure 1: An illustration of Proposition \ref{['concave comparison']}.
  • Figure 2: The graph of $\tau\mapsto\frac{A(v)}{{\mathcal{S}}_\tau(v)}$. The dashed line is the lower bound for Theorem \ref{['GenTianThm']} to apply.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • Definition 1.9
  • Remark 1.10
  • Theorem 1.11
  • ...and 15 more