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REVIEW 4 major objections 4 minor 40 references

Toric Fano manifolds that do not admit extremal K\"ahler metrics

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Pith's one-line read This paper constructs a 10-dimensional toric Fano manifold that is relatively K-unstable, hence admits no extremal Kähler metric in the first Chern class.

desk verdict A genuinely important construction whose proof is a large unshipped computation; likely true, but the paper should not be accepted until the arithmetic is independently checkable. read the letter →

arxiv 2411.17574 v2 pith:PWVK5SLI submitted 2024-11-26 math.AG math.DG

DongSeon Hwang , Hiroshi Sato , Naoto Yotsutani This is my paper · ORCID
classification math.AGmath.DG MSC 53C5514L2414M25
keywords toricFanomanifoldextremalKählermetricrelativeK-stabilitymomentpolytopepotentialfunctionMabuchiconstantreflexivelatticeDonaldson-Futakiinvariant
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

This paper constructs a 10-dimensional toric Fano manifold and proves that it is relatively K-unstable, which by Theorem 1.2 of the paper implies that it admits no extremal Kähler metric in its first Chern class. This answers Problem 1.3 in the negative and refutes the folklore conjecture that every toric Fano manifold carries an extremal Kähler metric in its first Chern class. The argument is combinatorial: a specific reflexive Delzant polytope $P$ is produced, the potential function $\theta_P$ of the extremal vector field is computed, and an instability inequality involving the region $P^{-}=\{1-\theta_P\le 0\}$ is verified. Taking products with arbitrary toric Fano manifolds gives such examples in every dimension $n\ge 10$.

What carries the argument

The load-bearing objects are the moment polytope $P$ of the toric Fano manifold, its potential function $\theta_P$ — the unique affine-linear function with zero average satisfying $L_P(1)=L_P(x_i)=0$ — and the sublevel polytope $P^{-}=\{x\in P:1-\theta_P(x)\le 0\}$. The mechanism is the instability criterion from [YZ19]: if $\operatorname{Vol}(P^{-})\ne 0$ and $1-c < \int_{P^{-}}(1-\theta_P)^2\,dv/\operatorname{Vol}(P^{-})$, then some simple piecewise-linear convex function makes $L_P$ negative, so the manifold is relatively K-unstable. The verification is a large rational computation: first the volume and first and second moments of $P$ are computed exactly, then the linear system for $\theta_P$ is solved, then the 346 vertices of $P^{-}$ are listed, and finally the two integrals over the rational polytope $P^{-}$ are evaluated to confirm the inequality.

What would settle it

Recompute $\operatorname{Vol}(P^{-})$ and $\int_{P^{-}}(1-\theta_P)^2\,dv$ for the 10-dimensional polytope of Example 3.1 with independent exact rational arithmetic. If the difference $1-c-\int_{P^{-}}(1-\theta_P)^2\,dv/\operatorname{Vol}(P^{-})$ is not negative, inequality (2.7) fails and the proof of relative K-instability collapses.

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Extended reading notes

Core claim

The paper's central claim is that there exists a relatively K-unstable toric Fano manifold of dimension 10. The manifold $X$ is the toric variety of a 10-dimensional reflexive Delzant polytope $P$ with 500 vertices, obtained from $\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1 \times \mathbb{P}^1$ by two rounds of star subdivisions along torus-invariant curves, generalizing the 5-dimensional toric Fano manifold with ID 788. On $P$ one defines the potential function $\theta_P$, the unique affine-linear function with zero average satisfying $L_P(1)=L_P(x_i)=0$, where $L_P$ is the modified Futaki functional. The paper's key computation concerns the subpolytope $P^{-}=\{x\in P: 1-\theta_P(x)\le 0\}$ and verifies the inequality $1-c < \int_{P^{-}}(1-\theta_P)^2\,dv/\operatorname{Vol}(P^{-})$, where $c$ is the constant term of $\theta_P$. By the instability criterion of [YZ19], this inequality yields a simple piecewise-linear convex function $f$ with $L_P(f)<0$, proving relative K-instability. Theorem 1.2 of the paper then gives the non-existence of an extremal Kähler metric in the first Chern class.

Load-bearing premise

The proof rests on the correctness of the computer evaluations — $\operatorname{Vol}(P^{-})\approx 27.9812$ and $\int_{P^{-}}(1-\theta_P)^2\,dv\approx 73.7005$, printed as enormous exact rationals — and on the assertion that the 18-vertex polytope in Example 3.1 is a smooth reflexive Fano polytope; an arithmetic slip would invalidate inequality (2.7) and with it the theorem.

Editorial extensions

If this is right

  • The folklore conjecture that every toric Fano manifold admits an extremal Kähler metric in the first Chern class is false.
  • Problem 1.3, asking whether every smooth polarized toric Fano manifold is relatively K-polystable, is answered negatively in dimension 10.
  • By taking products with any toric Fano manifold, there are toric Fano manifolds of every dimension $n\ge 10$ with no extremal Kähler metric in the first Chern class.
  • Because a Mabuchi soliton would induce an extremal metric, these examples also admit no Mabuchi soliton; the paper proves this directly from the Mabuchi constant $M_X\approx 2.45>1$.
  • The paper leaves open whether the members $X_r$ of its constructed family fail to admit extremal metrics for all $r\ge 3$.

Reading between the lines

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

  • A natural test is to run the same computation on $X_3$, the dimension-15 member of the family, to see whether the instability inequality persists; this would indicate the construction is structural rather than an isolated example.
  • Because the paper prints exact rational values but ships no code, an independent re-evaluation of $\operatorname{Vol}(P^{-})$ and $\int_{P^{-}}(1-\theta_P)^2\,dv$ with exact rational arithmetic would remove the residual doubt about the computer-assisted step.
  • The same search strategy — checking the inequality against databases of smooth reflexive polytopes — could determine whether dimension 10 is minimal, settling the paper's Question 1.11 for dimensions 6 through 9.
  • The construction via repeated blow-ups suggests that extremal-metric obstructions cluster near the boundary of the Fano property, since the manifold is exactly on the edge of ceasing to be Fano.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper constructs an explicit 10-dimensional toric Fano manifold X and proves, using the relative K-instability criterion of Yotsutani--Zhou (Proposition 2.5), that it is relatively K-unstable. Since relative K-polystability is necessary for the existence of an extremal Kähler metric in the first Chern class (Theorem 1.2), this gives a toric Fano manifold of dimension 10 with no extremal metric in c1, answering Problema 1.3 of Mabuchi in the negative and disproving the folklore conjecture in dimension 10. Taking products with arbitrary toric Fano manifolds extends the conclusion to dimensions n ≥ 10. The proof is computational: the potential function θ_P is obtained from explicit integrals via SageMath, the polytope P^- is computed, and the key inequality (2.7) is verified using LattE with exact rational values printed in Section 5.1.

Significance. If the computation is correct, this is a substantial result: it provides the first known Fano manifold (indeed a toric Fano manifold) without an extremal Kähler metric in the first Chern class, resolves a folklore conjecture in the negative, and gives explicit higher-dimensional examples. The paper is honest in printing exact rational values for the decisive computation and in making the geometric construction (Example 4.1) relatively transparent. However, the verification is not reproducible from the manuscript alone: no code or scripts are provided, and the printed data contain observable transcription errors, including an incomplete definition of θ_P. Because the conclusion depends on a finite but enormous exact arithmetic computation, the absence of a machine-checkable verification path is a serious gap that prevents full confidence in the stated theorem.

major comments (4)
  1. [§3.2] The displayed formula for the potential function θ_P is incomplete: it lists coefficients for x1 through x7 and for x9 and x10, but no coefficient for x8. Since θ_P is supposed to be an affine function on R^10 and is used in §3.4 to define P^- = { x ∈ P : 1 - θ_P(x) ≤ 0 }, the missing term makes the printed potential function ill-defined and prevents any independent check of the central inequality (2.7). This must be corrected, either by supplying the missing coefficient or by explicitly stating that it is zero.
  2. [§3.4 and §5.1] The proof of Theorem 1.4 rests entirely on the equality (1-c) - (∫_{P^-}(1-θ_P)^2 dv)/Vol(P^-) < 0, where the two integrals are produced by LattE and SageMath runs. The paper does not provide the code, scripts, or even the precise versions of the software used, and the printed exact values involve integers with hundreds of digits. The numerical margin is only about -1.36, so even a small arithmetic or transcription error could change the sign and invalidate the conclusion. Since the existence theorem is established by this computation alone, a machine-readable supplement or an independently repeatable verification protocol is needed.
  3. [§5.3] The list of 346 vertices of P^- does not serve as a reliable certified record: it contains duplicated entries (e.g., the two identical lines beginning (-1,1,-1,-1,99514132805180591354737040230560552119486341818763655374550342285975576726358200069906545235135/... )), malformed entries with missing parentheses, and at least one line ending with a square bracket instead of a parenthesis (the entry beginning (4,-1,-1,-1,3,0,-1,-1,-18720596285543647624721728228191765292829918396410865837951549025248659390659160140472531192861/... )). These problems are not merely cosmetic: they mean the printed data cannot be used to re-verify the volume and integral values in Section 5.1.
  4. [Example 3.1] The list of 18 putative vertices of Δ contains the vector (0,1,0,0,0,0,0,0,0,0) twice and omits the basis vector e3 = (0,0,1,0,0,0,0,0,0,0). The intended polytope is clarified only by the separate construction in Example 4.1. This concrete typo in the central combinatorial definition illustrates that transcription errors do occur in the manuscript's large printed data, reinforcing the need for a corrected and machine-verifiable version of all computational input.
minor comments (4)
  1. [§2.2] The inequality in Definition 2.3 is typeset with a nonstandard symbol `greaterorequalslant`; it should be the usual ≥.
  2. [Abstract and Introduction] There are several typographical errors in the opening text, such as 'ans wering' in the abstract and 'Mabuchi solitions' in Section 1; these should be corrected in a final revision.
  3. [Section 5.1] The exact rational values are printed with line breaks inside the integers, which is acceptable for the print version, but a supplementary text or data file with the values as single machine-readable tokens would greatly improve verifiability.
  4. [References] The paper cites SageMath and LattE by name but does not record the precise versions used; given the computational nature of the proof, version numbers should be included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorem is obtained by applying an external instability criterion to an explicitly constructed polytope and reporting direct computations.

full rationale

The derivation chain is not circular. The authors construct a specific toric Fano polytope (Example 4.1), take its dual moment polytope P, compute the affine potential function θ_P by solving the defining linear equations (2.8)-(2.9), and then compute the subpolytope P^-, its volume, and the integral of (1-θ_P)^2. The inequality (2.7) is verified numerically and reproduced as exact rational data in Section 5.1. The instability criterion used, Proposition 2.5, is quoted from [YZ19] as a general theorem: it states that if Vol(P^-)≠0 and inequality (2.7) holds, then the toric Fano manifold is relatively K-unstable. This criterion does not assume the target result, does not contain the present example, and is not fitted to the data; it is an independent, parameter-free theorem. The use of [SS, Theorem 1.4] to pass from relative K-instability to non-existence of an extremal metric is likewise an external theorem. The computations are not circular: θ_P is not chosen to make (2.7) hold; it is determined by the fixed normalization equations. The self-citations to [YZ19], [Yo], and [YZ23] concern previously published results and do not smuggle in the conclusion. The genuine caveats are correctness/reproducibility risks: the exact arithmetic values are printed but no code is shipped, and Example 3.1 contains a duplicated/missing vertex that is repaired by the precise construction in Example 4.1. Those are verification concerns, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted. The argument uses published stability criteria and the assertion that the constructed polytope is a smooth Fano polytope.

assumptions (3)
  • domain assumption The instability criterion of Yotsutani-Zhou (Proposition 2.5): if Vol(P^-) is nonzero and 1 - c < integral over P^- of (1-theta_P)^2 dv divided by Vol(P^-), then XP is relatively K-unstable.
    Invoked from [YZ19, Theorem 1.4(2)] in Section 2.3 and used as the sufficient condition for Theorem 1.4.
  • domain assumption A manifold admitting an extremal Kahler metric is relatively K-polystable (Stoppa-Szekelyhidi, Theorem 1.2).
    Bridges the relative K-instability of X to the non-existence of extremal metrics in Corollary 1.5.
  • domain assumption The polytope Delta_r constructed in Example 4.1 is a smooth reflexive Fano polytope, hence the associated toric variety is a Fano manifold.
    Asserted in Example 4.1 ('One can check that X_r is a toric Fano manifold'); necessary for the problem to apply. The printed list of vertices in Example 3.1 has a typo, but the construction in Example 4.1 defines the intended polytope.

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Pith. "Pith review of Toric Fano manifolds that do not admit extremal K\"ahler metrics." pith.science (2026). https://pith.science/paper/PWVK5SLI

@misc{pith2026241117574,
  author       = {Pith},
  title        = {Pith review of: Toric Fano manifolds that do not admit extremal K\"ahler metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWVK5SLI}},
  note         = {Machine review of arXiv:2411.17574}
}
abstract

We show that there exists a toric Fano manifold of dimension $10$ that does not admit an extremal K\"ahler metric in the first Chern class, answering a question of Mabuchi. By taking a product with a suitable toric Fano manifold, one can also produce a toric Fano manifold of dimension $n$ admitting no extremal K\"ahler metric in the first Chern class for each $n \geq 11$.

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