{"id":"31e8b7ab-2bf0-438b-90e8-20318453b059","arxiv_id":"2411.17574","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A carefully built 10-dimensional toric Fano manifold satisfies a relative K-instability criterion, so it cannot carry an extremal Kahler metric in c1.","lead":"This paper constructs a 10-dimensional toric Fano manifold that admits no extremal Kahler metric in its first Chern class, the first such example known. It resolves a folklore conjecture and Mabuchi's question in the negative, and yields examples in every dimension at least 10 via products.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim rests entirely on unshipped computer algebra; an arithmetic or transcription error in the exact values of Vol(P^-) or the integral would invalidate (2.7).","rationale":"The reader's weakest_assumption correctly identifies the unshipped computer algebra as the critical point. My stress-test concurs: the proof of Theorem 1.4 is a direct verification of inequality (2.7) using printed exact rationals, and without independently reproducible code the argument cannot be fully checked. The typo in Example 3.1 strengthens this concern because it demonstrates that the manuscript is not free of transcription errors in the polytope data. I considered whether there is a theoretical gap in the use of Proposition 2.5, but the reduction is standard and the criterion is a known theorem; the only place where the argument could fail is the numerical data. Therefore the verdict should remain CONDITIONAL as the reader recommended, pending independent verification of the exact arithmetic.","tokens_in":79687,"tokens_out":9678,"duration_ms":75079,"concrete_test":"Using the vertices of P listed in Section 5.2, independently recompute the integrals b_i = integral over P of x_i dv and c_ij = integral over P of x_i x_j dv in exact rational arithmetic (e.g., with LattE, Normaliz, or a different CAS), solve the linear system (2.8)-(2.9) to obtain theta_P, and compute the polytope P^- = {x in P : 1 - theta_P(x) >= 0}. Then recompute Vol(P^-) and the integral over P^- of (1 - theta_P)^2 dv exactly and compare the rational numbers to the printed a/b and c/d in Section 5.1. If the recomputed values match and inequality (2.7) holds with the same sign, the central claim is confirmed; any mismatch identifies a concrete error that invalidates Theorem 1.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem is proven by a finite computation that is printed but not reproducible from the paper alone. Specifically, Theorem 1.4 follows only if (i) the potential function theta_P printed in Section 3.2 is the unique solution of equations (2.8)-(2.9) for the moment polytope P whose vertices are listed in Section 5.2; (ii) the polytope P^- = {x in P : 1 - theta_P(x) >= 0} is correctly computed and its 346 vertices are correctly listed in Section 5.3; and (iii) the exact values Vol(P^-) = a/b and integral over P^- of (1 - theta_P)^2 dv = c/d printed in Section 5.1 are correct. These values involve integers with hundreds of digits, and no code or verification script is provided. A single transcription error in any of these data, for example a typo in one vertex of P^- or one digit of a/b, would change the sign of the difference in (2.7) and destroy the conclusion. The presence of a genuine typo in Example 3.1 (a duplicated vertex and a missing vertex e3), although repaired by Example 4.1, shows that transcription errors do occur in this manuscript. Since the existence theorem is established entirely by this computation, the printed exact arithmetic is the load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":79926,"tokens_out":6387,"duration_ms":59610,"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":[{"comment":"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.","section":"§3.2"},{"comment":"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.","section":"§3.4 and §5.1"},{"comment":"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.","section":"§5.3"},{"comment":"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.","section":"Example 3.1"}],"minor_comments":[{"comment":"The inequality in Definition 2.3 is typeset with a nonstandard symbol `greaterorequalslant`; it should be the usual ≥.","section":"§2.2"},{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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.","section":"Section 5.1"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The result, if correct, is of high interest and the construction is natural. The main obstruction to acceptance is the lack of a verifiable computational record: the central theorem is proved by one large exact computation, and the printed data contain visible typos, including an incomplete potential function. I would advise the editor to require the authors to deposit the SageMath/LattE code and output, or at minimum a machine-readable file with the full polytope data and exact integrals, and to correct the incompleteness of θ_P before this can be considered for publication. The mathematical framework around the computation appears sound, and the concern is about verification rather than about the validity of the instability criterion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this looks like the first genuine counterexample to the folklore conjecture that every toric Fano manifold admits an extremal Kähler metric in c1, and it answers Mabuchi's Question 2 negatively. If the computation is right, that is a real result, not a marginal one. The construction of X_2 as a double blow-up of P^4×P^4×P^1×P^1, generalizing the five-dimensional ID:788, is sensible, and the reduction to inequality (2.7) via YZ19 is clean. The paper also gives a second, independent route to non-existence of Mabuchi solitons through the Mabuchi constant, and product arguments cover all dimensions n≥10. The exact rationals in Section 5.1 are useful and let a referee check the sign of (1−c) − integral/volume without redoing the entire polytope computation.\n\nThe soft spot is exactly what the stress-test note says: the theorem rests on unshipped computer algebra. The potential function, the 346 vertices of P^-, and the hundred-digit numerators/denominators are printed but not reproducible without Sage/LattE scripts. The margins are not razor-thin—the difference is about -1.36, relative to values around 27.98 and 73.70—so one small transcription error in a vertex list is unlikely to change the sign by accident. Still, the typo in Example 3.1 (duplicated vertex, missing e3) shows transcription errors do occur, and the listed vertices of P^- contain at least one obvious formatting artifact (a ']' in one line) that makes the printed data not fully self-contained. This is a checkable issue, not a conceptual one. I do not see a circularity problem: applying YZ19's criterion and computing is legitimate, and choosing X_2 because it was expected to satisfy the criterion is legitimate search. The self-citations are to the published criterion and to prior low-dimensional work; that does not bother me.\n\nBottom line: the paper is for Kähler geometers working on YTD-type stability, and for anyone interested in the limits of computational proof in geometry. It deserves a serious referee. The right demand is simple: ship the Sage/LattE scripts and the input data, or have an independent computation confirm the three key numbers. Without that, acceptance should not happen; with it, I expect the result will hold.","headline":"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.","tokens_in":80466,"tokens_out":2292,"would_cite":false,"duration_ms":25968,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","14L24","14M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["toric Fano manifold","extremal Kähler metric","relative K-stability","moment polytope","potential function","Mabuchi constant","reflexive lattice polytope","Donaldson-Futaki invariant"],"falsifier":"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.","tokens_in":79474,"feed_emoji":"🔷","tokens_out":13383,"duration_ms":105951,"temperature":0.7,"pith_summary":"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 $n\\ge 10$.","feed_headline":"10D toric Fano manifold has no extremal Kähler metric","feed_subtitle":"A reflexive-polytope computation disproves the folklore conjecture for toric Fano manifolds.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $r\\ge 3$."],"supporting_citations":[{"why":"Supplies the instability criterion (Proposition 2.5): inequality (2.7) yields a piecewise-linear function with negative $L_P$, hence relative K-instability.","marker":"[YZ19]"},{"why":"Defines the functional $L_P$ and the linear equations determining the potential function $\\theta_P$, and proves the boundary-volume identities used in the computation.","marker":"[ZZ]"},{"why":"Provides Theorem 1.2, which turns relative K-instability into non-existence of an extremal Kähler metric.","marker":"[SS]"},{"why":"The computer algebra system used to compute the potential function coefficients and the vertices of $P^{-}$.","marker":"[Sage]"},{"why":"The software used to integrate over the rational polytope $P^{-}$ and produce $\\operatorname{Vol}(P^{-})$ and $\\int_{P^{-}}(1-\\theta_P)^2\\,dv$.","marker":"[LattE]"},{"why":"Provides the database of smooth reflexive lattice polytopes from which the 5-dimensional seed with ID 788 is taken.","marker":"[P]"},{"why":"Poses Problem 1.3, the question answered in the negative by the paper.","marker":"[Ma11]"}],"fun_headline_variants":["10D toric Fano with no extremal Kähler metric","Toric Fano counterexample: no extremal metric in dim 10","First 10D toric Fano without extremal Kähler metric","Dimension 10 toric Fano lacks extremal Kähler metric"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["10D toric Fano with no extremal Kähler metric","Toric Fano counterexample: no extremal metric in dim 10","First 10D toric Fano without extremal Kähler metric","Dimension 10 toric Fano lacks extremal Kähler metric"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000507,"raw_usage":{"total_tokens":2452,"prompt_tokens":907,"completion_tokens":1545,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":1464}},"tokens_in":523,"tokens_out":1545,"duration_ms":11802,"temperature":1.0,"reasoning_tokens":1464,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:57:37.072470+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}