{"id":"7e815fe7-9166-4a77-8dc2-69edbaf2c70a","arxiv_id":"1908.09518","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Uniform relative Ding stability of a Fano manifold implies the necessary numerical condition ϑ(M) < 1 for the existence of Mabuchi solitons.","lead":"A Fano manifold that is stable in a specific algebraic sense must satisfy the numerical condition ϑ(M) < 1, which is the known obstruction to the existence of a Mabuchi soliton metric. The paper develops new geometric tools for relative stability, including an integral formula and a convex-body description.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uniform-stability half of Theorem 1 hinges on an omitted convex-geometric claim: the infimum defining JNA_T(X,Lc) is asserted to be attained at trivial twisting ρ=0, but no proof is supplied.","rationale":"The reader's weakest-assumption analysis pinpoints exactly the place where the argument for the uniform-stability half of Theorem 1 is least secure: the unproved convex-geometric statement that the infimum defining the reduced non-Archimedean J-functional for the deformation-to-normal-cone family is attained at the trivial twisting ρ=0. My independent reading confirms this is the single most load-bearing concern. The semistable implication ϑ(M)≤1 is supported by a detailed local computation and an explicit expansion (6.4); the uniform implication, however, depends on an omitted estimate that is asserted to be 'clear' from the convex-geometry description. Without that estimate, the inequality DNA_Z≥δ·JNA_T does not yield the strict bound ϑ(M)<1. I do not see a more serious defect: the intersection-theoretic inner product, the HRR-based limit formula, and the Okounkov-body description are coherent and substantially developed. The implicit assumption that the extremal vector field Z lies in the chosen torus T is a scope condition inherited from Definitions 34 and 45; it should be stated explicitly in Theorem 1, but it is not a flaw in the main argument. Because the concern is exactly the one already identified by the reader and the appropriate response is to require a proof of the missing convex-geometric inequality, the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":31647,"tokens_out":8917,"duration_ms":88101,"concrete_test":"Prove or disprove the omitted estimate directly: for the concave function G_c(x)=min{x1−c,0} on the Okounkov body Δ(L), and for each ρ∈R^m with affine part L_ρ coming from (7.12), show that J(ρ):=sup_Δ(G_c+L_ρ)−∫_Δ(G_c+L_ρ) ≥ J(0)=c^{n+1}/((n+1)vol(Δ)) for all 0<c<c0. Since adding a constant to L_ρ does not change J(ρ), it suffices to check mean-zero linear perturbations: verify sup_Δ(G_c+L) ≥ sup_Δ G_c for every mean-zero L in the span of the torus weights. As a numerical sanity check, take a concrete Fano manifold with a torus action, e.g. CP^n with a standard subtorus, compute Δ(L), the weights ν*_i, and evaluate J(ρ) on a grid of ρ for small c; if any ρ gives J(ρ)<J(0), the claim in Theorem 52 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 8, proof of Theorem 52. The uniform-stability half of Theorem 1 reduces to evaluating the reduced non-Archimedean J-functional on the deformation-to-normal-cone family (X,Lc). The proof asserts, without argument, that for c≪1 the infimum over ρ∈R^m in Definition 44 is attained at ρ=0, so that JNA_T(X,Lc)=JNA(X,Lc)=c^{n+1}/((n+1)L^n). The only justification offered is that this is 'clear from the convex-geometry description' together with facts (1)-(2) about the infinitesimal Okounkov body Δ(L). This claim is load-bearing: if some nonzero twisting ρ gives a strictly smaller reduced J-functional, then the uniform-stability inequality DNA_Z ≥ δ·JNA_T yields only a weaker bound than ϑ(M)≤1−δ, and the strict conclusion ϑ(M)<1 does not follow from the expansion (6.4). Concretely, the missing step is an inequality of the form sup_Δ(G_c+ℓ_ρ) − ∫_Δ(G_c+ℓ_ρ) ≥ sup_Δ G_c − ∫_Δ G_c, where G_c=min{x1−c,0} is the concave function of the filtration F_c and ℓ_ρ are the affine functions from (7.12). The qualitative facts (1)-(2) describe the shape of Δ but do not by themselves imply this variational inequality; the convex-geometry description gives a picture, not a proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":31909,"tokens_out":15476,"duration_ms":169985,"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":[{"comment":"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.","section":"§8, proof of Theorem 52"}],"minor_comments":[{"comment":"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.","section":"Theorem 1 and §5.4"},{"comment":"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.","section":"§7.6"},{"comment":"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":"Abstract and Figure 1.1"},{"comment":"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":"Section 4, Theorem 3"},{"comment":"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.","section":"Section 3, Proposition 10"}],"recommendation":"major_revision","confidential_remarks":"The main gap is the omitted convex-geometric proof in Theorem 52. This is a local claim that could plausibly be repaired, and the rest of the paper contains substantial original material, so I do not recommend rejection if the author can supply a complete proof or a precise reference for the minimizer claim. The semistable half alone is strong, but the uniform-stability half is the advertised main theorem, so the manuscript should not be accepted in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nAll 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.\n\nTwo 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.\n\nWho 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.","headline":"Solid technical core, genuinely useful tools, but the uniform-stability half of the main theorem currently rests on an unproved convex-geometric claim.","tokens_in":32485,"tokens_out":4081,"would_cite":true,"duration_ms":41549,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","14L24","32Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Mabuchi solitons","relative Ding stability","extremal vector field","test configurations","non-Archimedean J-functional","Okounkov body","Fano manifolds","Duistermaat-Heckman measure"],"falsifier":"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 $\\vartheta(M)<1$.","tokens_in":31395,"feed_emoji":"📐","tokens_out":11093,"duration_ms":94786,"temperature":0.7,"pith_summary":"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.","feed_headline":"Relative Ding stability forces Mabuchi-soliton obstruction below 1","feed_subtitle":"A Fano manifold that is uniformly Ding-stable must satisfy ϑ(M)<1, the precise necessary condition for a Mabuchi soliton.","key_machinery":"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 $J^{NA}(X,L_c)=\\frac{c^{n+1}}{(n+1)L^n}$.","core_discovery":"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 $\\vartheta(M)<1$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the original inner product of actions and the relative-stability framework that the paper's intersection-theoretic definition extends.","marker":"[Sz]"},{"why":"provides DH measures, admissible filtrations, and the non-Archimedean functionals used to define relative Ding stability and compute $J^{NA}(X,L_c)$.","marker":"[BHJ1]"},{"why":"defines the extremal vector field through the Futaki-Mabuchi bilinear form, the field whose normalized Hamiltonian maximum is $\\vartheta(M)$.","marker":"[FM]"},{"why":"introduces Mabuchi solitons and the necessary condition $\\max\\theta_Z<1$ encoded in the invariant now written $\\vartheta(M)$.","marker":"[Ma1]"},{"why":"establishes the toric case where relative D-stability is equivalent to the $\\vartheta(M)$ bound, the model this paper extends to general Fano manifolds.","marker":"[Ya]"},{"why":"shows admissible filtrations induce concave functions on Okounkov bodies and identifies their pushforward with the limit measure, the basis of the convex-geometry description.","marker":"[BC]"},{"why":"associates an admissible filtration to a test configuration and computes the deformation-to-normal-cone filtration used in the proof of Theorem 52.","marker":"[WN]"},{"why":"introduces the reduced non-Archimedean $J$-functional by twisting with one-parameter subgroups, the norm used to define uniform relative D-stability.","marker":"[Hi3]"},{"why":"identifies the non-Archimedean Ding functional as the limit slope of the classical Ding functional, making the relative Berman-Ding invariant the stability quantity.","marker":"[Ber]"}],"fun_headline_variants":["Uniform relative Ding stability forces theta(M)<1 for Mabuchi solitons","Stable Fano manifolds satisfy the sharp Mabuchi-soliton obstruction","Ding-stable Fano manifolds imply theta(M)<1, the Mabuchi-soliton bound","Uniform D-stability gives the Mabuchi soliton necessary condition","Relative Ding stability upgrades to the precise theta(M)<1 bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Uniform relative Ding stability forces theta(M)<1 for Mabuchi solitons","Stable Fano manifolds satisfy the sharp Mabuchi-soliton obstruction","Ding-stable Fano manifolds imply theta(M)<1, the Mabuchi-soliton bound","Uniform D-stability gives the Mabuchi soliton necessary condition","Relative Ding stability upgrades to the precise theta(M)<1 bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00098,"raw_usage":{"total_tokens":4177,"prompt_tokens":979,"completion_tokens":3198,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":3097}},"tokens_in":595,"tokens_out":3198,"duration_ms":20832,"temperature":1.0,"reasoning_tokens":3097,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:09:40.086144+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\vartheta(M)<1$.","supporting_citations":[],"review_version":1}