Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

Yi Yao

REVIEW 1 major objections 5 minor 13 references

Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that uniform relative Ding stability of a Fano manifold forces $\vartheta(M)<1$, the maximum of the normalized Hamiltonian function of the extremal vector field, which is a necessary condition for the existence of…

desk verdict Solid technical core, genuinely useful tools, but the uniform-stability half of the main theorem currently rests on an unproved convex-geometric claim. read the letter →

arxiv 1908.09518 v2 pith:JQDZYBGA submitted 2019-08-26 math.DG

Yi Yao This is my paper · ORCID

classification math.DG MSC 53C5514L2432Q20

keywords MabuchisolitonsrelativeDingstabilityextremalvectorfieldtestconfigurationsnon-ArchimedeanJ-functionalOkounkovbodyFanomanifoldsDuistermaat-Heckmanmeasure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Mabuchi solitons are canonical Kähler metrics that generalize Kähler-Einstein metrics to Fano manifolds whose Futaki invariant does not vanish. The paper targets the numerical obstruction $\vartheta(M)$, the maximum of the normalized Hamiltonian function of the extremal vector field; any Mabuchi soliton requires $\vartheta(M)<1$. The paper proves that algebraic stability already enforces this obstruction: relative D-semistability forces $\vartheta(M)\le 1$, and uniform relative D-stability forces $\vartheta(M)<1$, for a Fano manifold with a torus symmetry. To reach this, it develops a pullback-invariant intersection-theoretic inner product of $\mathbb{C}^*$-actions on equivariant test configurations, with an integral formula and a limit-slope formula, and adapts Okounkov-body convex geometry to compute the reduced non-Archimedean $J$-functional. If the theorem is right, no extra ad hoc assumption is needed in the existence theory: uniform stability itself supplies the required bound.

What carries the argument

The argument runs on three linked mechanisms. First, a pullback-invariant inner product of actions $\langle\alpha,\beta\rangle$ on a $\mathbb{C}^*$-equivariant test configuration, defined by intersection numbers and computed by the integral formula $\langle\alpha,\beta\rangle=\frac{1}{(n+1)L^n}\int_X \widetilde{\Theta}_X(\Omega/2\pi)^{n+1}$; it converts limit slopes of modified energy functionals into global integrals. Second, the reduced non-Archimedean $J$-functional $J_T^{NA}(X,L)=\inf_{\rho\in\mathbb{R}^m}J^{NA}(\mathcal{F}(X,L)_\rho)$, in which twisting the filtration by a one-parameter subgroup $\rho$ shifts weight spaces; on an infinitesimal Okounkov body at a torus-fixed point, twisting by $\rho$ adds an affine function to the concave function associated with the test configuration, so $J_T^{NA}$ becomes the minimal area between a concave function and its support functions. Third, the deformation-to-normal-cone configuration $(X,L_c)$ has associated concave function $\min\{x_1-c,0\}$, so for $c\ll1$ the reduced $J$-functional is attained at the trivial twist $\rho=0$ and equals $J^{NA}(X,L_c)=\frac{c^{n+1}}{(n+1)L^n}$.

What would settle it

Compute the reduced non-Archimedean $J$-functional for the deformation-to-normal-cone configuration $(X,L_c)$ from the concave function $G[F_c]=\min\{x_1-c,0\}$ on the infinitesimal Okounkov body: if for arbitrarily small $c$ some nonzero twist $\rho$ gives a smaller value than $\rho=0$, then the proof of Theorem 52 collapses and uniform D-stability would not be known to imply $\vartheta(M)<1$.

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Extended reading notes

Core claim

Theorem 1 states that for a Fano manifold $M$ and a torus $T\subset \operatorname{Aut}_0(M)$, D-semistability relative to $T$ implies $\vartheta(M)\le 1$, while uniform D-stability relative to $T$ implies $\vartheta(M)<1$. Here $\vartheta(M)$ is the maximum of the normalized Hamiltonian function of the extremal vector field $Z$, an invariant of $M$; the Mabuchi-soliton equation has the form $(1-\theta_Z(u))\omega_u^n=e^{h_\omega-u}\omega^n$, so $\max\theta_Z(u)<1$ is a necessary condition. The proof constructs a $T$-equivariant test configuration $(X,L_c)$ by deforming to the normal cone of a $T$-fixed point where $\theta_Z$ attains its maximum, and expands the relative Berman-Ding invariant as $\operatorname{D}^{NA}_Z(X,L_c)=\frac{1-\vartheta(M)}{(n+1)c_1(M)^n}c^{n+1}+Ac^{n+2}$ for $0<c\ll 1$. Semistability is nonnegativity of this invariant, giving $\vartheta(M)\le 1$; uniform stability uses the convex-geometric expansion $J_T^{NA}(X,L_c)=\frac{c^{n+1}}{(n+1)L^n}$ to upgrade the inequality to $\vartheta(M)<1$.

Load-bearing premise

The uniform-stability half rests on the convex-geometric claim that for sufficiently small blowup parameter $c$, the infimum defining $J_T^{NA}(X,L_c)$ is attained at the trivial twist $\rho=0$; the paper states this is clear from the Okounkov-body picture but omits the details, and if the infimum were attained elsewhere the expansion $J_T^{NA}=\frac{c^{n+1}}{(n+1)L^n}$ would fail.

Editorial extensions

If this is right

If a Fano manifold admits a Mabuchi soliton, it is D-semistable relative to the relevant torus; the theorem shows the converse bound $\vartheta(M)\le1$ is a consequence of stability, not a separate hypothesis.
Uniform relative D-stability implies the strict inequality $\vartheta(M)<1$, which is precisely the prerequisite needed for continuation and variational existence arguments for Mabuchi solitons.
The new intersection-theoretic inner product is invariant under pullback, so relative stability notions are well defined on equivalence classes of test configurations and extend beyond relatively ample line bundles.
The limit-slope formula holds for general $\alpha(S^1)\times\beta(S^1)$-invariant rays, not only Phong-Sturm geodesic rays, because it uses equivariant Hirzebruch-Riemann-Roch instead of Bergman-geodesic approximation.
The extension of the Duistermaat-Heckman convergence theorem shows the DH measure of a test configuration is the weak limit of pushforward measures along any admissible invariant ray, not just geodesic rays.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the omitted minimization details are supplied, the same Okounkov-body description could yield an effective lower bound on the uniform-stability constant $\delta$, turning the existence criterion into a quantitative threshold.
The paper leaves open whether ordinary relative K-stability also forces $\vartheta(M)\le1$; testing toric Fano orbifolds with $\vartheta=1$ would separate the power of D-stability from K-stability as an obstruction.
The pullback-invariant inner product may transplant to singular Fano varieties or transcendental Kähler classes, where the original Hilbert-space definition is not available.
Since $J_T^{NA}$ is a minimal area between a concave function and its support functions, it can be computed explicitly in toric or low-dimensional examples, giving a direct numerical check of the uniform-stability inequality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the semistable half of Theorem 1 holds up, but the uniformly stable half is not proved as written. The load-bearing point is in Section 8, proof of Theorem 52: the author asserts the infimum defining J^NA_T(X,L_c) is attained at ρ=0 when c≪1, so that J^NA_T = J^NA = c^{n+1}/((n+1)L^n), and calls it clear from the convex-geometry description and two facts about Δ(L). The stress-test note is correct that the two facts describe the shape of Δ(L) but do not imply the needed variational inequality sup_Δ(G_c+ℓ_ρ) − avg(G_c+ℓ_ρ) ≥ sup_Δ G_c − avg G_c. Without that inequality, uniform D-stability gives only a weaker bound, not ϑ(M)<1. This is a genuine gap in the paper's central application.

All that said, the paper does real work. The new intersection-theoretic inner product ⟨α,β⟩ is naturally pullback-invariant, which the original H^0-based definition is not; the integral formula over the total space is clean; and the extension of Hisamoto's theorem on geodesic rays and DH measures to more general rays is a useful standalone result. The convex-geometry description of reduced NA J-functionals via infinitesimal Okounkov bodies is a good framework and likely to be reused, even if the present use is incomplete. The semistable-half computation with the deformation to normal cone is detailed and looks correct. Citation pattern is fine: the self-citation [Ya] is the toric case, used as motivation, not as a crutch.

Two smaller issues. The theorem statements in the introduction omit the standing assumption, made in Section 6, that the extremal vector field Z lies in Lie(T). That assumption is necessary: without it, the relative D-stability notion is undefined because the inner product with β_Z makes no sense. State it in the main theorem. And the omitted convex-geometric argument is not a one-line formality; if true it needs an explicit proof, and if false the theorem might still be salvageable but the current argument fails.

Who should read this: people working on Mabuchi solitons and relative stability. The tools justify referee time. I would send it to a serious referee rather than desk-reject, with instructions to focus on the convex-geometry claim and the statement of hypotheses. Fill that gap and this could be a solid paper; as it stands, the main theorem is conditional.

Referee Report

1 major / 5 minor

Summary. The paper develops tools for relative Ding stability for Fano manifolds with a torus action and applies them to Mabuchi solitons. It defines a pullback-invariant intersection-theoretic inner product of the structure action and a fiberwise action, proves a limit-slope formula for modified energy functionals and an integral formula over the total space, and extends Hisamoto's convergence of Duistermaat-Heckman measures to more general rays. It then adapts Okounkov body theory to torus actions in order to describe the reduced non-Archimedean J-functional as an infimum over twists by R^m. The main application is Theorem 1: relative D-semistability implies ϑ(M)≤1, and uniform relative D-stability implies ϑ(M)<1, where ϑ(M) is the maximum of the normalized Hamiltonian function of the extremal vector field. The first implication is proved via a deformation-to-normal-cone test configuration; the second additionally relies on an unproved convex-geometric claim in Theorem 52.

Significance. The semistable half of the paper is largely self-contained and technically substantial: the intersection-theoretic inner product, the integral formula, the equivariant HRR computations, and the localization argument for the deformation-to-normal-cone family are explicit and give a reproducible proof that D-semistability implies ϑ(M)≤1. The Okounkov-body framework for the reduced J-functional is also a useful contribution that may be of independent interest. If the missing convex-geometric inequality in Theorem 52 can be supplied, the uniform-stability half would establish a genuine Yau-Tian-Donaldson type obstruction for Mabuchi solitons in the general Fano setting, going beyond the toric case. At present the uniform-stability result is conditional on that omitted proof, so the paper's central claim is not yet fully established.

major comments (1)

[§8, proof of Theorem 52] The identity JNA_T(X,Lc)=JNA(X,Lc)=c^{n+1}/((n+1)L^n) is asserted but not proved. The text states that this is 'clear from the convex-geometry description' and then lists two facts about Δ(L), namely that Δ(L) is contained in {x1≥x2+...+xn} and that inf_Δ x1=0. These facts only describe the position of Δ(L) near the origin; they do not imply the required variational inequality sup_Δ(G_c+ℓ_ρ)−∫_Δ(G_c+ℓ_ρ) ≥ sup_Δ G_c − ∫_Δ G_c for every affine function ℓ_ρ arising from (7.12). This assertion is load-bearing: if the infimum were attained at some nonzero ρ, the uniform-stability inequality DNA_Z ≥ δ·JNA_T would give a weaker bound, and the strict conclusion ϑ(M)<1 would not follow from the expansion (6.4). A complete proof or a precise reference for the minimizer claim is needed. A one-dimensional toy computation on Δ=[0,1] with G_c=min{x−c,0} gives J(G_c+ax)=c^2/2+a(c−1/2)+O(a^2), showing that the zero-twist minimizer depends on the shape of the body and is not a formality.

minor comments (5)

[Theorem 1 and §5.4] The theorem statement says 'T a torus' without qualification, but Definition 34(1) and the proof of Theorem 36 require that for the compact torus S underlying T there exist a maximal compact subgroup K containing S such that the associated extremal vector field Z lies in Lie(S). Please state this hypothesis explicitly in Theorem 1 and at the start of Section 6.
[§7.6] The letter c is used in the support function S(ρ)=c−ℓ(ρ) and then again for the deformation-to-normal-cone parameter c in Section 8. Please use a different symbol in one of these places to avoid confusion.
[Abstract and Figure 1.1] The abstract and several displayed formulas contain formatting artifacts, and Figure 1.1 is referenced but not included in this version. A careful proofread and the addition of the figure are needed.
[Section 4, Theorem 3] The proof of Theorem 3 uses the polynomial-in-c comparison after showing convergence for sufficiently large c. This is valid, but the boundedness of the supports of the relevant measures should be stated explicitly so that the moment convergence is justified.
[Section 3, Proposition 10] In the condition (B) case, the justification of Stokes' theorem and fiber integration for C^{1,1} metrics with non-pluripolar products is compressed into a few sentences. Since this case is used later, a reference or a slightly longer argument would improve readability.

Circularity Check

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No load-bearing circularity: the ϑ(M) bounds are derived from an explicit deformation-to-normal-cone computation. The only self-citation [Ya] is background, and the uniform-stability half contains a genuine but non-circular omitted convex-geometry verification.

full rationale

The derivation chain is self-contained. Theorem 1's two implications are proved by constructing the T-equivariant deformation-to-normal-cone family (X,L_c) and directly computing both the relative Berman-Ding invariant (6.4) and the reduced non-Archimedean J-functional. No parameter is fitted to the target invariant: ⟨α,β_Z⟩ is defined by intersection numbers, then evaluated by the integral formula (3.17) and localization, yielding the coefficient (1 − max θ)/((n+1)c_1(M)^n). The expansion is a computation rather than an assumed input. The only self-cited item [Ya] concerns the toric case and the observation that critical points of Ding energy are Mabuchi solitons; it is used as motivation and background, not as the justification for the general-Fano stability implication. The proof relies on external results ([Sz], [FM], [Ber], [BHJ1], [BC], [WN], [LM], [A]) for foundational statements. The uniform-stability half does contain an asserted but unproved step in the proof of Theorem 52: the claim that the infimum defining JNA_T(X,L_c) is attained at ρ=0 for c≪1 is said to be 'clear from the convex-geometry description' with details omitted. This is a correctness gap, not a circularity: the claim is not built into the definition of JNA_T or into the uniform-stability assumption, and if it failed the proof would weaken rather than presuppose the conclusion. Accordingly no step in the paper's derivation reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard tools from Kähler geometry, algebraic geometry, and convex geometry: equivariant resolutions, equivariant HRR, Atiyah's convexity theorem, and Okounkov body theory. No new entities are postulated, and no parameters are fitted to data. The only nontrivial domain assumption is that the extremal vector field lies in the torus, which is part of the definition of the relative stability notion used.

assumptions (6)

standard math Existence of equivariant resolutions for singular test configurations preserving the inner product.
Invoked in Proposition 10, Theorem 12, and Proposition 19 to apply equivariant HRR on a smooth resolution and then pull back; the invariance under pullback is proven in Proposition 7 and 18.
standard math Equivariant Hirzebruch-Riemann-Roch formula for line bundles.
Used in Section 2.5 and Theorem 12 to relate integrals of Hamiltonian functions to equivariant Euler numbers.
standard math Atiyah's convexity theorem: the moment map image is the convex hull of images of torus-fixed points.
Used in Proposition 37 to find a T-fixed point z* at which the Hamiltonian function θ attains its maximum ϑ(M).
standard math Boucksom-Chen theorem: admissible filtrations of section rings induce concave functions on Okounkov bodies.
Used in Theorem 49 and Section 7.5 to express the limit measure and JNA as integrals over the Okounkov body.
domain assumption The description of the filtration associated to deformation to the normal cone (Lemma 5.17 in BHJ1).
Used in the proof of Theorem 52 to identify the filtration and DH measure of the test configuration (X,Lc).
domain assumption The extremal vector field Z lies in the Lie algebra of the torus T for relative D-stability to be defined.
Definition 34 requires Z ∈ Lie(K) for the relative Berman-Ding invariant; the proof of Theorem 1 assumes this condition.

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Cite this review

Pith. "Pith review of Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons." pith.science (2026). https://pith.science/paper/JQDZYBGA

@misc{pith2026190809518,
  author       = {Pith},
  title        = {Pith review of: Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQDZYBGA}},
  note         = {Machine review of arXiv:1908.09518}
}

abstract

Mabuchi solitons generalize K\"{a}hler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with K\"{a}hler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative Ding stability. For this we study the inner product of $\mathbb{C}^{*}$-actions on equivariant test-configurations and obtain an integration formula over the total space. To analyze the uniform stability, by adapting Okounkov body construction to the setting of torus action, we give a convex-geometry description for the reduced non-Archimedean J-functionals.

Figures

Figures reproduced from arXiv: 1908.09518 by the authors.

**Figure 1.1.** the shadow region gives J NA on X , we relate the integral to equivariant Euler number χ β 1 (X , kL) (Def 11). Finally, via spectral sequences, it can be further related to hα, βi. One problem of this method is that we need smooth X to apply equivariant HRR, however thanks to the pullback-invariance of hα, βi, we can apply HRR on a resolution of X and then back. As a byproduct, we obtain an integral formula for in… view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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[A] Atiyah, M.: Convexity and commuting Hamiltonians. Bulletin of the London Mathematical Society, 1982, 14(1): 1-15. [Ber] Berman, R.: K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. Inven- tiones mathematicae, 2016, 203(3): 973-1025. [BBJ] Berman, R., Boucksom, S., Jonsson, M.: A variational approach to the Yau-Tian- Donaldson con...

arXiv 1982

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work page Pith review arXiv 1905

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[LM] Lazarsfeld, R., Mustat,ă, M.: Convex bodies associated to linear series. Annales scientiﬁques de l’École normale supérieure. 2009, 42(5): 783-835. [Li] Li, C.: On equivariantly uniform stability and Yau-Tian-Donaldson conjecture for singular Fano varieties. arXiv:1907.09399,

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work page 2007

[7] [2007]

Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties

[WN] Witt Nyström, D.: Test conﬁgurations and Okounkov bodies. Compositio Mathematica, 2012, 148(6): 1736-1756. [Ya] Yao, Y.: Mabuchi Metrics and Relative Ding Stability of Toric Fano Varieties. arXiv:1701.04016,

work page Pith review arXiv 2012

[8] [2013]

Orthogonal projection of a test configuration to vector fields

[Hi1] Hisamoto, T.: On the limit of spectral measures associated to a test conﬁguration of a po- larized Kähler manifold. Journal für die reine und angewandte Mathematik (Crelles Journal), 2016, 2016(713): 129-148. [Hi2] : Orthogonal projection of a test conﬁguration to vector ﬁelds. arXiv:1610.07158,

work page Pith review arXiv 2016