{"id":"46e4af8b-52c8-4025-b62f-1d5c2c08b02d","arxiv_id":"2608.03815","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of solutions to complex Hessian-type equations on projective manifolds is equivalent to uniform positivity of certain subvariety integrals when the associated polynomial is strictly right-Noetherian.","lead":"This paper proves a numerical test for when certain nonlinear complex equations have smooth solutions on projective manifolds: checking positive integrals over subvarieties is enough. The test covers complex Hessian and Hessian quotient equations, extending earlier results to a general family of polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 9.3's 'We only need to assume that r=1' is unsupported: a single blowup need not resolve a singular subvariety, and the closedness of the continuity set depends on Lemma 9.3 through Theorem 8.1.","rationale":"The reader already identified Lemma 9.3's reduction 'We only need to assume that r=1' as a load-bearing gap and assigned CONDITIONAL. My stress test finds the same gap and no new reason to move to REJECT, because the gap is local and might be repairable by supplying the missing induction over the resolution sequence. However, the assertion is genuinely load-bearing: Lemma 9.3 is the only input to Theorem 8.1 that provides cone metrics near arbitrary subvarieties, and Theorem 8.1 is the only mechanism for closedness in §8. Without Lemma 9.3, the continuity argument stops at t0 = 1. The paper has substantial independent structure—Section 3 lemmas are flagged as AI-revised, but most of the proof is conventional—yet a single unsupported reduction in the extension theorem blocks the main theorem. Therefore the appropriate verdict remains CONDITIONAL: accept only if the r=1 reduction (or a full multi-blowup induction) is supplied, and Remark 8.2's transfer is either proven or replaced by a citation with matching hypotheses.","tokens_in":23485,"tokens_out":9282,"duration_ms":108623,"concrete_test":"Work Lemma 9.3 for a projective surface Z whose embedded resolution has length at least two, e.g., a compactification of the Whitney umbrella {x^2 = y z^2} ⊂ C^3 (singular locus the z-axis; the strict transform after blowing up the z-axis still has a singular point). Attempt to construct the neighborhood U and the cone metric ω_U = ω_0 + √−1∂∂̄ψ_U using only the first blowup, following the proof verbatim. If the construction fails to give a smooth Kähler form satisfying the cone condition on a full neighborhood of Z, then Lemma 9.3 is not justified as written. Alternatively, if a rigorous induction over the resolution sequence can be supplied, check that the numerical estimates (19) and Lemma 9.4 hold with constants independent of the stage.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 9, Lemma 9.3 is the only source of the neighborhoods U with cone metrics ω_U demanded by Theorem 8.1, and Theorem 8.1 is what closes the continuity set I in §8. In the singular case the proof invokes a canonical resolution π: M_r → … → M_0 = M and then asserts 'We only need to assume that r=1.' No argument is given. A resolution of a singular subvariety is generally a composition of several blowups along smooth centers; one blowup does not in general make the proper transform smooth (e.g., a Whitney umbrella surface needs more than one blowup). The induction in Lemma 9.3 is on the dimension m of Z, so after one blowup the new proper transform Z~ still has dimension m, and the induction hypothesis does not apply to it. The subsequent estimates (19) and Lemma 9.4 are written only for a single exceptional divisor; for a longer resolution sequence the current T~_s and the gluing argument at the end of Lemma 9.3 would have to be re-run at each stage, with no indication that the uniform positivity or cone-extension estimates survive. Since this extension lemma is a hypothesis of Theorem 8.1, the implication (3)⇒(2) of Theorem 1.1 is not proved unless the reduction to r=1 is justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves numerical (Nakai–Moishezon type) criteria for complex Hessian-type equations ω^d∧χ^{n−d} = Σ a_k ω^k∧χ^{n−k} on compact connected smooth projective manifolds. Theorem 1.1 treats degree-n polynomials that are strongly strictly right-Noetherian, with variable coefficient c_0, and establishes equivalence among existence of a smooth solution in the cone, existence of a C-subsolution, and a numerical positivity condition on all subvarieties of dimension p0 in Lemma 9.4 and hence to obtain the smooth solutions ω̃_{s,t} on Z̃. The claim is plausible—the leading term Φ_m((1+s)π*ω_0,π*χ) has positive integral by the original numerical condition and the error terms are O(s)—but no uniform-in-s estimate is written out. Because Lemma 9.4 is then used to produce the current Θ̃_s and complete the extension argument, this omission is load-bearing.","section":"§9, Lemma 9.3 and Lemma 9.4"},{"comment":"Theorem 8.1 is presented as Proposition 4.1 of [11], but Remark 8.2 transfers it to the present general cone by asserting that the proof “goes through without any changes because the only condition used there is the convexity of the cone condition, and that the cone condition implies the Kähler condition.” The original proposition is not quoted, and no verification is given that the gluing/regularized-maximum argument depends only on those two properties in the exact form needed here, particularly for currents with positive Lelong numbers along Y and strict cone condition off Y. Since Theorem 8.1 is the step that converts the mass concentration of §7 into closedness of I, the transfer needs to be documented explicitly or the original theorem stated with all hypotheses checked.","section":"§8, Remark 8.2"}],"minor_comments":[{"comment":"Notation: “ω_ϵ ⊂ [ω]∩Υ_{F_ϵ}” and similar expressions should be “ω_ϵ ∈ [ω]∩Υ_{F_ϵ}.”","section":"§4, proof of Corollary 1.3"},{"comment":"The proof says “By Lemma 2.5 of [18],” but Lemma 2.5 is not stated. Please include its statement or give a precise reference so the reduction is self-contained.","section":"§2, Lemma 2.15"},{"comment":"The proposition includes p=n, but the proof uses condition (3) for that value, which is not part of Theorem 1.2 and is not implied by integrability for a general g. The later uses in Theorem 1.2 only need p0) would fix this.","section":"§5, Proposition 5.1"},{"comment":"The reduction from variable c_0 to constant c′_0 says the cone condition and numerical condition are “the same” for the two equations. This is true for Υ^1 because that cone depends only on derivatives of the polarization, hence not on c_0, but the point deserves one sentence of explanation.","section":"§6"},{"comment":"In the estimate near the end of the lemma, “s_2 is a fixed number” is introduced without definition; please specify the choice of s_2 and why the integral with χ̃_{s_2} controls the corresponding integral with χ̃_s.","section":"§9, Lemma 9.4"}],"recommendation":"major_revision","confidential_remarks":"The r=1 reduction in Lemma 9.3 is the main risk: if it cannot be repaired, the closedness argument for Theorem 1.1 fails. The transfer of Datar–Pingali in Remark 8.2 is also under-documented. The rest of the architecture is credible and the result, if completed, would be significant. The AI-use declaration is transparent and does not affect my assessment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the take: the paper proves a genuinely new uniform Nakai-Moishezon criterion for strictly right-Noetherian polynomials of arbitrary degree on projective manifolds, with uniform Szekelyhidi and Murakami corollaries. That is a real advance if the proof holds. The broad architecture is standard—continuity path, mass concentration, local extension—and large parts are carefully written. The cone-inclusion lemmas in Section 3 are useful, and the flag about AI assistance is honest; they are not the problem.\n\nThe problem is the singular extension step. Section 9, Lemma 9.3 introduces a resolution π: M_r → ... → M_0 and then says 'We only need to assume that r=1.' No argument is given, and the statement is not generally true: one blowup along smooth centers does not resolve an arbitrary singular subvariety. The induction in that lemma is on dimension m of Z, and after one blowup the proper transform still has dimension m, so the induction hypothesis does not apply. The estimates (19) and Lemma 9.4 are written only for a single exceptional divisor. For a longer resolution sequence you would have to re-run the argument at each stage, and nothing in the text shows the uniform positivity or cone-extension estimates survive. Because Lemma 9.3 is what supplies the neighborhoods U demanded by Theorem 8.1, and Theorem 8.1 is what closes the continuity set, the implication (3)⇒(2) of Theorem 1.1 is not proved as written. This is not a cosmetic gap; it is load-bearing.\n\nSecond, Remark 8.2 transfers the Datar-Pingali gluing theorem to the present cone by assertion. The proof in [11] may well go through, but the cone here is not identical, and the paper should show it.\n\nIf those two gaps are fixed, the paper is a substantial contribution. The corollaries are clean, the uniform statement is genuinely new, and the mass concentration and blow-up expansion are intricate and mostly convincing. As it stands, the central claim is plausible but the proof is incomplete.\n\nFor a colleague: this is worth a serious referee, but the referee should insist on a complete proof of Lemma 9.3 and Remark 8.2. I would bring it to the reading group and would cite the uniform statement if the revision closes the gaps.","headline":"A genuinely new uniform Nakai-Moishezon criterion for strictly right-Noetherian equations, but the singular extension step in Section 9 is asserted rather than proved, so the paper is conditional pending a complete proof of Lemma 9.3 and Remark 8.2.","tokens_in":24281,"tokens_out":2747,"would_cite":true,"duration_ms":29672,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32Q25","32W20","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, on projective manifolds, solvability of complex Hessian-type equations is equivalent to a uniform numerical positivity condition on all analytic subvarieties.","keywords":["complex Hessian equations","Hessian quotient equations","k-Hessian equations","Nakai-Moishezon criterion","right-Noetherian polynomials","Fang-Ma-Gårding polynomials","cone condition","projective manifolds"],"falsifier":"Check Lemma 9.3 on a singular subvariety whose canonical resolution needs at least two blowups, and see whether inequality (19) can be established by iterating the one-step argument. If the 'r=1' reduction fails, the closedness argument fails. Alternatively, run the gluing proof of Theorem 8.1 with the Fang-Ma-Gårding cone of this paper in place of the cone used in [11]; if any step uses a property that the cone lacks, the transfer in Remark 8.2 is the point of failure.","tokens_in":23384,"feed_emoji":"📐","tokens_out":7544,"duration_ms":75206,"temperature":0.7,"pith_summary":"This paper tries to establish that on projective manifolds, a large family of fully nonlinear equations of complex Hessian type is governed by a numerical criterion: a smooth solution exists exactly when certain intersection numbers with every analytic subvariety stay strictly positive, uniformly. The family is picked out by a condition on the roots of the associated degree-d polynomial and its derivatives (strictly right-Noetherian). For degree-n polynomials with the stronger 'strongly strictly' condition, the paper proves the equivalence without needing to assume a prior subsolution. For arbitrary degrees it proves a uniform version and derives new Nakai-Moishezon criteria for complex Hessian quotient and k-Hessian equations. A sympathetic reader would care because the result converts an analytic existence question into finitely many checkable algebro-geometric inequalities.","feed_headline":"Uniform subvariety inequalities settle Hessian-type equations","feed_subtitle":"For projective manifolds, smooth solutions exist exactly when these intersection numbers are uniformly positive.","key_machinery":"The load-bearing object is the right-Noetherian property of the univariate polynomial f: the largest real roots of f, f′, …, f^{(d-1)} exist and form a nonincreasing (strict, for the main results) sequence. This root condition makes f a Fang-Ma-Gårding polynomial, and its polarization F is a multi-affine polynomial whose positivity cone Υ_F (defined by eigenvalue conditions on χ^{-1}ω) is convex, permutation-invariant, and Kähler-implying. The proof also uses two cone-inclusion lemmas: multiplying f by x+T keeps the class and shrinks the cone, and subtracting ε f′ makes the polynomial strongly strictly right-Noetherian while enlarging the cone; these allow perturbations along the continuity","core_discovery":"The central claim is Theorem 1.2: for a compact connected smooth projective manifold of complex dimension n, if f(x)=x^d − Σ a_k x^k is strictly right-Noetherian and the integral of the equation matches, then the following are equivalent: (1) there is ε₀>0 such that every p-dimensional irreducible subvariety V satisfies the intersection inequality (3) with the uniform lower bound ε₀∫_V χ^p; (2) for every sufficiently small ε>0 there is a form ω_ε in the same class satisfying the cone condition for the perturbed polynomial f+ε. This is then specialized to Hessian quotient equations (Corollary 1.3) and complex k-Hessian equations (Corollary 1.5), giving uniform Nakai-Moishezon criteria. The pr","pith_inferences":["If the proof closes, the same strategy should extend to other strictly right-Noetherian equations, since the cone-inclusion lemmas are purely polynomial and the analytic steps (mass concentration, gluing) appear equation-independent.","The known counterexample shows the projective assumption is essential; the uniform ε₀ condition may fail exactly in the non-projective regime where a path-condition is needed, so testing (3) on those examples would delimit the criterion's boundary.","The reduction of c₀(z) to a constant by a point value suggests a dimension-counting heuristic: for strongly right-Noetherian equations, variable coefficients may matter only through a single mean in a way that could extend to lower-order coefficients."],"forward_implications":["For complex k-Hessian equations on projective manifolds, a smooth solution exists iff there is an ε₀-uniform lower bound ∫_V ω^{k−n+p}∧χ^{n−k} ≥ ε₀∫_V χ^p for every subvariety V of dimension n−k ≤ p < n.","For complex Hessian quotient equations, the uniform positivity condition (with two families of inequalities) is equivalent to smooth solvability, and this direction does not require a priori existence of a C-subsolution.","For degree-n strongly strictly right-Noetherian equations, the three conditions — existence of a unique smooth cone solution, existence of a cone form, and the integral inequality (3) for all p