{"id":"45f5d02d-312d-489f-b2f6-ea0509118cb2","arxiv_id":"2412.02564","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If a smooth Fano manifold admits a Kähler-Ricci soliton, then for all sufficiently large k the canonical cone of X times complex projective k-space has a Calabi-Yau cone structure.","lead":"A theorem in Kähler geometry shows that if a smooth Fano manifold carries a Kähler-Ricci soliton, then multiplying it by a sufficiently large complex projective space yields a Calabi-Yau cone on its canonical bundle. This is the first asymptotic confirmation of the Mabuchi-Nakagawa conjecture, obtained through a new openness result for weighted soliton metrics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the main theorem follows correctly from Han-Li's weighted YTD criterion and [3, Prop.2], applied within their stated hypotheses.","rationale":"The reader's weakest assumption is the dependence on Han-Li's weighted YTD theorem. I agree this is the most external and least re-derived ingredient, but I do not see a load-bearing defect in relying on it: the weights v_N are smooth positive functions with vanishing v-Futaki on a smooth Fano manifold, which is precisely the class covered by [37, Theorem 1.7]. The approximation argument is sound: the family W(a,ξ) is smooth with a unique critical point, the implicit function theorem gives a path τ_a with τ_0 equal to the Tian-Zhu vector field, and hence ξ_N = τ_{-1/N} -> τ. The convergence v_N -> e^{⟨τ,x⟩} is C^0, matching the openness theorem. The final cone conversion is also correct: on Z = X × P^{N-n-2}, the weight is ℓ_N^{-(dim Z + 2)} with ℓ_N positive affine, exactly the input of [3, Prop.2]. The non-quantitative nature of k0 is acknowledged in the paper. The proof is a chain of substantial cited results, which justifies moderate rather than high confidence, but no internal gap or misplaced hypothesis surfaced in this stress-test.","tokens_in":29201,"tokens_out":37034,"duration_ms":382848,"concrete_test":"As a verification step, recompute the v_N-Futaki condition directly: for ξ_N the unique minimizer of V_N, check that ∫_X ⟨ζ,μ⟩ (1 - ⟨μ,ξ_N⟩/N)^{-N} ω^n = 0 for every ζ ∈ t, and verify that on Z = X × P^{N-n-2} the product weight is exactly ℓ_N^{-(dim Z + 2)} with ℓ_N(z) = 1 - ⟨x, ξ_N⟩/N > 0 on P_Z. Then confirm that both Han-Li's [37, Theorem 1.7] and [3, Prop.2] are cited with all their hypotheses satisfied for this family of weights.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. Theorem 1.2 is a clean application of two external results: Han-Li's weighted Yau-Tian-Donaldson theorem (Theorem 2.1, i.e. [37, Theorem 1.7]), which identifies S(X) with the coercivity locus D(X), and the authors' earlier correspondence [3, Prop.2] between ℓ^{-(m+2)}-solitons and Calabi-Yau cone metrics. Both are used exactly in the setting for which they were proved. The weights v_N(x) = (1 - ⟨x, ξ_N⟩/N)^{-N} are smooth and positive on the canonical polytope, their v-Futaki vanishes by construction of ξ_N as the minimizer of V_N, and they converge uniformly to the KRS weight e^{⟨τ,x⟩}. The openness Theorem 1.1 — or equivalently the LeBrun-Simanca style Corollary A.1 for the smooth family v_a — then yields v_N-solitons for N large. On Z = X × P^{N-n-2} the product metric is a w_N-soliton with w_N(z) = ℓ_N(x)^{-(dim Z + 2)} for the positive affine function ℓ_N(z) = 1 - ⟨x, ξ_N⟩/N, so the hypothesis of [3, Prop.2] is met and the canonical cone of Z inherits a Ricci-flat Kähler cone metric. The absence of a quantitative k0 is a stated limitation, not a mathematical gap. The internal arguments, including the implicit function theorem step giving ξ_N -> τ, are coherent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the space S(X) of weight functions v on the momentum polytope of a smooth Fano manifold for which a v-soliton exists. The main structural result, Theorem 1.1, states that the coercivity locus D(X) of the v-weighted Ding functional is an open convex cone inside the vanishing-v-Futaki space F(X); combined with Han-Li's criterion S(X)=D(X), this yields quantitative openness of S(X). The main application, Theorem 1.2, shows that if X admits a Kähler-Ricci soliton, then for all sufficiently large k the product Z=X×P^k admits an ℓ^{-(dim Z+2)}-soliton, hence the canonical cone of Z is a Calabi-Yau cone, giving an asymptotic version of the Mabuchi-Nakagawa conjecture. The paper also proves relative openness of transversal KRS Sasaki structures (Theorem 1.3), an upper bound on the soliton potential (Theorem 1.4), and a weighted Fujita volume bound (Theorem 1.5), with an appendix giving a LeBrun-Simanca style deformation argument.","tokens_in":29471,"tokens_out":10397,"duration_ms":111925,"significance":"If the results hold, the paper gives a clean new route to constructing Calabi-Yau cones by perturbing KRS weights, and establishes openness of the v-soliton locus in C^0 topology as a corollary of Han-Li's analytic criterion. The proof of Theorem 1.2 is a genuine consequence of external results applied in their valid regime: Han-Li's weighted YTD theorem (Theorem 2.1) identifies S(X)=D(X), and the authors' earlier correspondence [3, Prop.2] converts ℓ^{-(m+2)}-solitons into Ricci-flat cone metrics. The approximation of the KRS weight by v_N via the implicit function theorem is explicit and coherent, and the absence of a quantitative k0 is honestly stated as a limitation. The secondary results give useful complements: a new Lichnerowicz-type obstruction and a weighted Fujita bound.","major_comments":[],"minor_comments":[{"comment":"The organization paragraph says 'In Sect. 5, we prove Theorem 1.4' and then 'The final Sect. 5, we prove Theorem 1.5', but Theorem 1.5 is proved in Section 6; the numbering should be corrected.","section":"Introduction, organization paragraph"},{"comment":"There are several typos that should be fixed: 'caries' should be 'carries', 'Nikagawa' should be 'Nakagawa', 'Licherowicz' should be 'Lichnerowicz', 'Fujita'a' should be 'Fujita's', and 'Ricci-flal' should be 'Ricci-flat'.","section":"Abstract and throughout"},{"comment":"The Duistermaat-Heckman measure is denoted dµDH in the text but the displayed equation (2.5) writes dµDM; the notation should be made uniform.","section":"Equation (2.5)"},{"comment":"The proof of convexity is too terse: the sentence 'The convexity of D(X) follows from the fact that the subspace of normalized weight functions is linearly convex' does not by itself explain why D(X) is a cone, since a cone also requires dilation invariance; a one-sentence clarification would be helpful.","section":"Proof of Theorem 1.1"},{"comment":"In the comparison of normalized weights it is assumed that inf(˚v - ˚v0) = -λ0 with λ0 > 0; since both weights are normalized the case λ0 = 0 forces ˚v = ˚v0, but this reasoning is implicit and could be stated explicitly.","section":"Corollary 3.1 and proof of Theorem 1.1"},{"comment":"The maximum principle argument for positivity of the transversal scalar curvature is very compressed: at a minimum of Scal the displayed identity (B.2) gives ΔScal ≥ 0 and |Ric|² - Scal ≥ 0, which does not by itself show Scal > 0; the sign conventions for the Laplacian and the precise maximum principle argument should be expanded.","section":"Appendix B, Corollary B.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a competent and mostly sound application of the Han–Li weighted Yau–Tian–Donaldson criterion and the authors' own cone correspondence; the main novelty is the openness observation rather than new analytic technology. The self-citations are frequent but appear as legitimate framework citations rather than circular support. I recommend acceptance after minor polishing; the referee does not need the external black-box theorems to be reproved, but the terse maximum principle in Appendix B and a few organizational issues should be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper proves an asymptotic version of Mabuchi–Nakagawa that actually holds up. If X is a smooth Fano with a Kähler–Ricci soliton, then for all sufficiently large k the canonical cone of X × P^k carries a Ricci-flat Kähler cone metric. The k0 is not effective, but the existence statement is solid.\n\nWhat's genuinely new: Theorem 1.1, the set D(X) of weights whose v-weighted Ding functional is T^C-coercive is an open convex cone in F(X). Since Han–Li proved S(X)=D(X), this is effective openness of the soliton locus. The approximation lemma, writing the KRS weight e^{<τ,x>} as a limit of powers of affine-linear functions, is the clever bit that makes the product construction work. The paper also recovers Petrecca's theorem on deformations of transversal KRS and adds a weighted Fujita volume bound and a Lichnerowicz-type obstruction. These are useful side results, honestly derived.\n\nWhere the soft spots are: the main theorem leans entirely on Han–Li's weighted YTD theorem (Theorem 2.1) as a black box. That's not a flaw in itself — the theorem is published and independent — but it means the paper's contribution is conditional on that criterion being as general as stated. The non-quantitative k0 is admitted and is a true limitation. The maximum principle argument in Appendix B is terse; Corollary B.1 (positive transversal scalar curvature) is sketched rather than proved in detail, though it looks correct. The self-citation count is high, but the cited results are used as framework, not as the target, so I don't read it as circular.\n\nI checked the stress-test note's chain: the weights v_N are smooth and positive, their v-Futaki vanishes by construction, they converge uniformly to e^{<τ,x>}, and then either Theorem 1.1 or Corollary A.1 gives v_N-solitons. The product with P^k and the conversion to a Calabi–Yau cone uses their earlier [3, Prop.2] exactly in the stated setting. That all lines up.\n\nWho this is for: people working in Kähler–Ricci solitons, weighted K-stability, and Sasaki geometry. It's a clean, well-written paper that deserves a serious referee. I'd send it to review and expect acceptance after the usual checks.","headline":"Asymptotic Mabuchi–Nakagawa is proved cleanly via Han–Li's weighted YTD criterion; the paper deserves a serious referee.","tokens_in":30055,"tokens_out":2248,"would_cite":true,"duration_ms":21987,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","53C55","32Q20","14J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Fano manifold with a Kähler–Ricci soliton gives Calabi–Yau cones after adding a large projective-space factor.","keywords":["Kähler–Ricci soliton","Calabi–Yau cone","Sasaki–Einstein metric","v-soliton","weighted Ding functional","weighted K-stability","Fano manifold","coercivity"],"falsifier":"Take a smooth Fano $X$ with a Kähler–Ricci soliton and compute the weighted $\\beta$-invariant of a $T$-equivariant prime divisor for the approximating weights $v_N$; finding $\\beta_{v_N} < 0$ for arbitrarily large $N$ would contradict the openness of $S(X)$ and invalidate Theorem 1.2. Alternatively, producing weights $v_N \\in F(X)$ that converge in $C^0$ to an existing soliton weight but admit no $v_N$-soliton for infinitely many $N$ would falsify the paper's central mechanism.","tokens_in":28961,"feed_emoji":"","tokens_out":12254,"duration_ms":124578,"temperature":0.7,"pith_summary":"The paper aims to show that soliton-type canonical metrics on a Fano manifold produce Ricci-flat cone metrics on a related space. More precisely, it claims that if the smooth Fano manifold $X$ admits a Kähler–Ricci soliton, then for every sufficiently large $k$ the canonical cone $K^\\times_Z$ of $Z = X \\times \\mathbb{P}^k_{\\mathbb{C}}$ carries a Ricci-flat Kähler cone metric, also called a Calabi–Yau cone. This is an asymptotic version of the conjecture that $k=0$ already works. The engine is Theorem 1.1: the set of weights $v$ for which the $v$-weighted Ding functional is coercive relative to the complexified torus is an open convex cone in the space of weights with vanishing $v$-Futaki invariant. The argument approximates the soliton weight $e^{\\langle\\tau,x\\rangle}$ by explicit rational weights, obtains $v$-solitons on $X$ for all sufficiently large $N$, and converts the product soliton on $Z$ into a Calabi–Yau cone metric.","feed_headline":"Kähler–Ricci solitons force Calabi–Yau cones on large products","feed_subtitle":"For any Fano manifold with a Kähler–Ricci soliton, large projective-space products carry Ricci-flat canonical cones.","key_machinery":"The carrying object is the $v$-weighted Ding functional $D_v$ on the space of $T$-invariant Kähler potentials: its critical points are exactly the $v$-solitons, and its coercivity relative to the complex torus $T^{\\mathbb{C}}$ is the analytic criterion for existence. Theorem 1.1 proves that the coercive locus $D(X)$ is open and convex in $F(X)$ by a comparison estimate: if normalized weights $\\bar v_1,\\bar v_2$ satisfy $\\inf_{P_X}(\\bar v_1-\\bar v_2)=-\\lambda_0$, then $D_{v_1}-D_{v_2} \\geq -\\lambda_0 J(\\omega) + \\text{constant}$, so coercivity of slope $\\Lambda_0$ at one weight transfers to nearby weights with slope $\\Lambda_0-\\lambda_0$. Convexity comes from linearity of the weighted Aubin–Mabuchi functional in the weight. The approximation sequence $v_N$ arises from convex volume functionals and an implicit-function-theorem deformation of the Kähler–Ricci soliton weight.","core_discovery":"Let $X$ be a smooth Fano manifold and $T$ a maximal compact torus of automorphisms, with canonically normalized momentum polytope $P_X$. Let $F(X)$ be the cone of positive weight functions $v$ on $P_X$ whose $v$-Futaki invariant vanishes. Theorem 1.1 asserts that the subset $D(X)$ where the $v$-weighted Ding functional $D_v$ is $T^{\\mathbb{C}}$-coercive is open and convex in $F(X)$. By the criterion of [37], a $v$-soliton exists exactly when $D_v$ is $T^{\\mathbb{C}}$-coercive, so $S(X)=D(X)$. The main application, Theorem 1.2, constructs for a manifold $X$ admitting a Kähler–Ricci soliton a sequence of weights $v_N(x) = (1 - \\langle x, \\xi_N\\rangle/N)^{-N}$ in $F(X)$ that converges uniformly to $e^{\\langle\\tau,x\\rangle}$; for $N$ large these weights are in $S(X)$, the product of the resulting $v_N$-soliton with the Fubini–Study metric is a $v_N$-soliton on $Z = X \\times \\mathbb{P}^{N-n-2}_{\\mathbb{C}}$ with weight of the form $\\ell^{-(\\dim Z+2)}$, and the correspondence of [3] turns this into a Ricci-flat Kähler cone metric on the canonical cone $K^\\times_Z$.","pith_inferences":["The openness theorem implies that the soliton locus is convex in weight space, so convex combinations of soliton weights in $F(X)$ should again admit solitons; the paper does not isolate this as a separate statement.","If the coercivity slope could be estimated explicitly from the Kähler–Ricci soliton weight, the proof would yield a quantitative $k_0$; the paper only establishes existence of some such $k_0$.","The weighted Fujita bound can be read as an obstruction on the size of admissible weights: a normalized $v$-soliton weight cannot have its minimum too small relative to the anti-canonical volume.","The same deformation mechanism, run through the Appendix A argument, should give openness of transversal Kähler–Ricci solitons for Fano orbifolds, an extension the paper gestures at in Remark 4.5 but does not state as a theorem."],"forward_implications":["If $X$ admits a Kähler–Ricci soliton, then for every $k \\geq k_0$ the canonical cone of $X \\times \\mathbb{P}^k_{\\mathbb{C}}$ admits a Ricci-flat Kähler cone metric; the proof gives no explicit bound on $k_0$.","The projective-space factor can be replaced by any $k$-dimensional Kähler–Einstein Fano manifold.","A transversal Kähler–Ricci soliton on a Sasaki structure persists under nearby Sasaki–Reeb polarizations, recovering a known deformation theorem as Theorem 1.3.","A Kähler–Ricci soliton on $X$ forces $\\langle\\tau,x\\rangle < n$ on the canonical polytope $P_X$, with the analogous bound $\\langle\\tau_\\xi,x\\rangle < n(\\langle\\xi,x\\rangle+1)$ for transversal Kähler–Ricci solitons.","For a normalized $v$-soliton weight, the first Chern number satisfies $c_1(X)^n \\leq \\left(\\frac{n+1}{\\inf_{P_X} v}\\right)^n$, a weighted version of Fujita's volume bound."],"supporting_citations":[{"why":"Supplies the criterion that a $v$-soliton exists exactly when the $v$-weighted Ding functional is $T^{\\mathbb{C}}$-coercive, and the weighted $\\beta$-invariant machinery used in Theorem 1.5.","marker":"[37]"},{"why":"Establishes the correspondence carrying a $v$-soliton of weight $\\ell^{-(n+2)}$ on a Fano manifold to a Ricci-flat Kähler cone metric on its canonical cone.","marker":"[3]"},{"why":"Provides the Kähler–Ricci soliton theory, the normalized soliton vector field, and the Futaki-type vanishing used to define the approximated weight.","marker":"[59]"},{"why":"States the conjecture that a Fano manifold with a Kähler–Ricci soliton has a Calabi–Yau canonical cone; Theorem 1.2 is its asymptotic version.","marker":"[50]"},{"why":"Develops the Sasaki–Einstein and Calabi–Yau cone correspondence together with volume minimization fixing the Reeb vector field normalization.","marker":"[51]"},{"why":"Supplies the openness-under-perturbation method that Appendix A adapts to $v$-solitons as an alternative route to Theorems 1.2 and 1.3.","marker":"[43]"},{"why":"Provides the weighted scalar curvature and weighted Futaki formalism, and the computations on which the deformation lemma in Appendix A relies.","marker":"[41]"},{"why":"Proves the Fujita volume bound for Kähler–Einstein Fano manifolds whose weighted version is Theorem 1.5.","marker":"[29]"}],"fun_headline_variants":["Large CP products turn KRS into CY cones","Solitons on Fanos yield Ricci-flat cones on CP products","Open v-soliton weights force Calabi–Yau cones on products","KRS on Fano give CY cones on large products"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a $v$-soliton exists precisely when the $v$-weighted Ding functional is coercive; if that equivalence fails for some smooth positive weights, the openness theorem and the Calabi–Yau cone conclusion do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Large CP products turn KRS into CY cones","Solitons on Fanos yield Ricci-flat cones on CP products","Open v-soliton weights force Calabi–Yau cones on products","KRS on Fano give CY cones on large products"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0016,"raw_usage":{"total_tokens":6410,"prompt_tokens":1016,"completion_tokens":5394,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":5322}},"tokens_in":632,"tokens_out":5394,"duration_ms":44405,"temperature":1.0,"reasoning_tokens":5322,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:21:12.644155+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth Fano $X$ with a Kähler–Ricci soliton and compute the weighted $\\beta$-invariant of a $T$-equivariant prime divisor for the approximating weights $v_N$; finding $\\beta_{v_N} < 0$ for arbitrarily large $N$ would contradict the openness of $S(X)$ and invalidate Theorem 1.2. Alternatively, producing weights $v_N \\in F(X)$ that converge in $C^0$ to an existing soliton weight but admit no $v_N$-soliton for infinitely many $N$ would falsify the paper's central mechanism.","supporting_citations":[{"cited_title":"Han and C","cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that a $v$-soliton exists exactly when the $v$-weighted Ding functional is $T^{\\mathbb{C}}$-coercive, and the weighted $\\beta$-invariant machinery used in Theorem 1.5."},{"cited_title":"Apostolov, S","cited_arxiv_id":null,"evidence_quote":"Establishes the correspondence carrying a $v$-soliton of weight $\\ell^{-(n+2)}$ on a Fano manifold to a Ricci-flat Kähler cone metric on its canonical cone."},{"cited_title":"Tian, K-stability and K¨ ahler–Einstein metrics","cited_arxiv_id":null,"evidence_quote":"Provides the Kähler–Ricci soliton theory, the normalized soliton vector field, and the Futaki-type vanishing used to define the approximated weight."},{"cited_title":"Mabuchi, Multiplier Hermitian structures on K¨ ahler manifolds, Nagoya J","cited_arxiv_id":null,"evidence_quote":"States the conjecture that a Fano manifold with a Kähler–Ricci soliton has a Calabi–Yau canonical cone; Theorem 1.2 is its asymptotic version."},{"cited_title":"Mabuchi and Y","cited_arxiv_id":null,"evidence_quote":"Develops the Sasaki–Einstein and Calabi–Yau cone correspondence together with volume minimization fixing the Reeb vector field normalization."},{"cited_title":"LeBrun and S","cited_arxiv_id":null,"evidence_quote":"Supplies the openness-under-perturbation method that Appendix A adapts to $v$-solitons as an alternative route to Theorems 1.2 and 1.3."},{"cited_title":"Lahdili, K¨ ahler metrics with constant weighted scalar curvature and weighted K-stability , Proc","cited_arxiv_id":null,"evidence_quote":"Provides the weighted scalar curvature and weighted Futaki formalism, and the computations on which the deformation lemma in Appendix A relies."},{"cited_title":"Fujita, Optimal bounds for the volumes of K¨ ahler-Einstein Fano manifolds , American J","cited_arxiv_id":null,"evidence_quote":"Proves the Fujita volume bound for Kähler–Einstein Fano manifolds whose weighted version is Theorem 1.5."}],"review_version":1}