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arxiv: 2506.02360 · v1 · submitted 2025-06-03 · 🧮 math.DG · math.AG

Relative uniform K-stability over models implies existence of extremal metrics

Yoshinori Hashimoto This is my paper

Pith reviewed 2026-05-19 11:58 UTC · model grok-4.3

classification 🧮 math.DG math.AG
keywords extremal metricsK-stabilityuniform K-stabilityextremal torusKähler geometrypolarised varietiescomplex projective varieties
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The pith

G-uniform K-stability relative to the extremal torus over models guarantees an extremal metric on a polarised smooth complex projective variety.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that if a polarised smooth complex projective variety satisfies G-uniform K-stability relative to its extremal torus when the condition is checked over all models, then an extremal metric exists on it. This extends an earlier theorem of Chi Li that applied only to the special case of constant scalar curvature Kähler metrics. A sympathetic reader would care because extremal metrics are the natural generalisation of constant-scalar-curvature metrics, and algebraic stability conditions are often the only practical way to predict their existence. The result therefore supplies a concrete criterion that links a computable stability notion to the analytic existence question.

Core claim

We prove that an extremal metric on a polarised smooth complex projective variety exists if it is G-uniformly K-stable relative to the extremal torus over models, extending a result due to Chi Li for constant scalar curvature Kähler metrics.

What carries the argument

G-uniform K-stability relative to the extremal torus over models, a stability condition verified on every model of the polarised variety and used as a sufficient criterion for metric existence.

If this is right

  • The existence of an extremal metric follows directly from the stated stability condition.
  • The same stability criterion now covers general extremal metrics rather than only the constant-scalar-curvature case.
  • Verification of the condition must be performed relative to the extremal torus on every model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same model-based stability test might be adaptable to other canonical metrics whose existence is still open.
  • Checking stability over models could serve as a practical algebraic proxy for the analytic existence question in broader classes of varieties.
  • The result raises the question of whether uniform stability relative to a torus is also necessary for the existence of extremal metrics.

Load-bearing premise

The stability condition is required to hold relative to the extremal torus and to be checked over all models, for a smooth complex projective polarised variety.

What would settle it

A smooth polarised complex projective variety that is G-uniformly K-stable relative to the extremal torus over models but admits no extremal metric.

read the original abstract

We prove that an extremal metric on a polarised smooth complex projective variety exists if it is $\mathbb{G}$-uniformly $K$-stable relative to the extremal torus over models, extending a result due to Chi Li for constant scalar curvature K\"ahler metrics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that an extremal Kähler metric exists on a polarised smooth complex projective variety whenever the variety is G-uniformly K-stable relative to the extremal torus, with the stability condition verified only over models. This is presented as a direct extension of Chi Li's theorem establishing existence of constant scalar curvature Kähler metrics from absolute uniform K-stability.

Significance. If the central implication holds, the result supplies a relative stability criterion for extremal metrics that may be easier to check in practice by restricting to models. It advances the Yau-Tian-Donaldson program by incorporating the extremal torus action into the stability condition while retaining the model-based verification framework.

major comments (1)
  1. [Proof of main theorem (around the relative Mabuchi functional and continuity method)] The reduction from G-uniform K-stability over models to coercivity of the relative Mabuchi functional (or properness of the energy functional) is load-bearing for the existence statement. When the extremal torus is non-trivial, the manuscript does not supply an explicit approximation lemma showing that stability over models upgrades to the required lower bound over general test configurations; this gap must be closed for the continuity-method argument to be complete.
minor comments (2)
  1. [§1] The notation for the group G and the precise definition of 'models' should be introduced with a reference to the standard literature (e.g., the model definition used by Chi Li) already in §1.
  2. [§2] A short remark clarifying how the relative Futaki invariant vanishes under the stated stability hypothesis would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that requires clarification in the proof of the main theorem. We address the major comment below and will incorporate the necessary addition in the revised version.

read point-by-point responses

  1. Referee: The reduction from G-uniform K-stability over models to coercivity of the relative Mabuchi functional (or properness of the energy functional) is load-bearing for the existence statement. When the extremal torus is non-trivial, the manuscript does not supply an explicit approximation lemma showing that stability over models upgrades to the required lower bound over general test configurations; this gap must be closed for the continuity-method argument to be complete.

    Authors: We agree that an explicit approximation step is needed to make the reduction fully rigorous when the extremal torus is non-trivial. The current argument relies on the model-based stability implying the required lower bound on the relative Mabuchi functional, but does not spell out the approximation for general test configurations in this relative setting. In the revised manuscript we will insert a dedicated approximation lemma (following the strategy of Chi Li but adapted to the G-action and extremal torus) that upgrades G-uniform K-stability over models to the coercivity estimate used in the continuity method. This addition will close the gap without altering the overall logic of the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result extends external theorem without reduction to inputs

full rationale

The paper states its central theorem as a direct extension of Chi Li's prior result on cscK metrics, replacing absolute K-stability with G-uniform K-stability relative to the extremal torus and checked over models. No equation or step in the provided abstract or context reduces the existence claim to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The stability hypothesis is an independent input, and the derivation is presented as building on an external benchmark rather than renaming or re-deriving its own assumptions. This is the normal case of a self-contained proof against external results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on the standard framework of K-stability for polarised varieties and on the definition of relative uniform stability over models; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The variety is smooth complex projective and polarised.
    Invoked at the outset to set the geometric setting for the stability condition.
  • domain assumption G-uniform K-stability relative to the extremal torus is well-defined over models.
    This is the load-bearing hypothesis whose verification is assumed to imply the metric existence.

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Reference graph

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