{"id":"beb69605-151a-4841-937a-e30a430bbf53","arxiv_id":"2607.14858","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If α is nef and c1(L)−α is a positive current, then H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) = 0 for p ≥ n − nd(α) + 1.","lead":"This paper proves a new vanishing theorem on projective manifolds: if a line bundle L is a nef class plus a positive current, then many cohomology groups of L twisted by forms and a multiplier ideal vanish. It extends Bogomolov's classical vanishing theorem to singular metrics and multiplier ideals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the strong-openness step in §4 is under-justified but standard and fixable.","rationale":"The reader correctly identifies the strong-openness application as the weakest spot in the proof. The paper cites Theorem 2.2, which is stated for negative psh functions and only as a union over ε>0, not as an equality for a fixed small ε. I checked whether this is a genuine obstruction. It is not: quasi-psh functions can be locally normalized to negative psh functions, and the finite generation of multiplier ideal sheaves on a compact manifold gives a uniform ε. The remaining steps of the proof—curvature computation, induction via hyperplane sections, and the exact sequences—are consistent after accounting for the torsion-freeness of I(ψ) in the tensor product with the Poincaré residue sequence. Therefore no fatal objection emerges, and the conditional verdict remains appropriate pending a short clarification.","tokens_in":8862,"tokens_out":52950,"duration_ms":412632,"concrete_test":"Supply the omitted normalization: on a coordinate chart, write ψ = u + smooth and φ = v + smooth, choose constants so u, v are negative psh, apply Theorem 2.2 to get I(u) = ∪_{ε>0} I(u+εv). Since I(u) is coherent, finite generation gives ε0 with I(u+εv) = I(u) for ε < ε0; cover X by finitely many charts and intersect the ε0's. Then verify the same δ can be chosen to satisfy both δ ≪ ε0 and the curvature lower bound δω − (1−δ)εω > 0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only potentially load-bearing step is the equality I(h_L) = I(ψ + δφ) = I(ψ) in §4, attributed to strong openness (Theorem 2.2). The cited theorem is stated for negative psh functions and gives I(φ) = ∪_{ε>0} I(φ+εφ0), not directly equality for a fixed small δ. If this equality failed, the big-case reduction to Theorem 1.5(2) would collapse. However, the gap is only expository: locally ψ and φ are quasi-psh, so one can write ψ = u + smooth, φ = v + smooth and subtract constants to make u, v negative psh. Theorem 2.2 applies, and because I(u) is coherent and finitely generated, the union stabilizes locally at some ε0; compactness yields a uniform δ. Thus the argument is mathematically sound and the concern does not land.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Bogomolov-type vanishing theorem (Theorem 1.6). For a projective manifold X of dimension n, a line bundle L, a nef class {α}, and a semipositive current β + i∂∂ψ representing c1(L) − {α}, the theorem asserts H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) = 0 for p ≥ n − nd({α}) + 1. When L is nef this gives H^n(X, Ω_X^p ⊗ L) = 0 for p ≥ n − nd(L) + 1. The proof first handles the big case nd({α}) = n by constructing a singular metric with positive curvature and invoking the authors' earlier Theorem 1.5(2), using strong openness of multiplier ideals. The general case is proved by induction on dimension, combining Xia's restriction formula, a logarithmic Serre vanishing theorem (Theorem 1.2), and slicing of numerical dimension. The paper also proves Theorem 1.2, a logarithmic-type Serre vanishing theorem for coherent sheaves.","tokens_in":9102,"tokens_out":17477,"duration_ms":136935,"significance":"If the proof is made rigorous, Theorem 1.6 is a clean and useful Bogomolov-type statement in the setting of nef classes and multiplier ideal sheaves, complementing Kawamata-Viehweg-type results. The organization is clear and the paper is honest about relying on substantial prior results: strong openness, Xia's restriction formula, and Theorem 1.5(2), the last due partly to the same group. However, the genuinely new content is largely a reduction: the big case is essentially Theorem 1.5(2), and the non-big case is a standard hyperplane induction. There are two technical gaps that need repair before the proof is acceptable.","major_comments":[{"comment":"The displayed exact sequence is false as stated. For D = D1, n = 2, p = 1 it reads 0 → Ω^1_X(log A)(−A) → Ω^1_X → Ω^1_A → 0; the first two sheaves have the same rank, so an injection would be an isomorphism and the quotient cannot be the nonzero locally free sheaf Ω^1_A. The standard residue sequence is 0 → Ω^p_X(log(D−D1))(−D1) → Ω^p_X(log D) → i_*Ω^{p−1}_{D1}(log(D−D1)|_{D1}) → 0. The proof of Theorem 1.2 in §3 uses the printed (2.c) to obtain the exact sequence displayed there, so that part of the proof is invalid. The line-bundle case still follows directly from Theorem 1.1, as the text notes, but the erroneous proposition and the proof depending on it must be corrected or removed.","section":"§2, Proposition 2.5(2.c)"},{"comment":"The equality I(h_L) = I(ψ + δφ) = I(ψ) is attributed to Theorem 2.2 without sufficient justification. Theorem 2.2 is local, requires negative plurisubharmonic weights, and gives I(ψ) = ∪_{ε>0} I(ψ + εφ), not equality for a fixed small δ. To make the argument work one must explain that ψ and φ can be made negative psh after subtracting smooth functions, that the union from strong openness is an increasing family of coherent ideal sheaves whose union is I(ψ), and that coherence/Noetherianity implies stabilization, so equality holds for all sufficiently small δ. Without this, the reduction to Theorem 1.5(2) in the big case, which is the base of the induction, is not justified.","section":"§4, big case, strong openness step"}],"minor_comments":[{"comment":"The function ψ should be explicitly declared quasi-plurisubharmonic; the condition β + i∂∂ψ ≥ 0 implies it, but the multiplier ideal I(ψ) is defined before that regularity is stated.","section":"Theorem 1.6 statement"},{"comment":"The notation 'ε ≪ δ ≪ 1' is garbled, and the formula I(h_L) = I(ψ + εφ) mixes ε and δ. The coefficient in the metric is δ, and the strong-openness parameter should be δ.","section":"§4, notation"},{"comment":"The sheaves on D1 in sequences (2.b) and (2.c) should be written consistently with the pushforward i_*, as is done in the proof. The current notation may confuse the reader about where these sheaves live.","section":"§2, Proposition 2.5"},{"comment":"Once Proposition 2.5(2.c) is corrected, the 'simple proof' for line bundles no longer works. Consider deleting it and relying on the already-cited Theorem 1.1, which gives the line-bundle case directly.","section":"§3, Theorem 1.2 proof"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is plausible and the main technical gap—the strong-openness step—is standard and fixable. The false residue sequence in Proposition 2.5 is localized to that proposition and to the optional 'simple proof' of Theorem 1.2; the theorem itself remains true because it follows from Theorem 1.1. That said, the novelty is modest: the big case is essentially the authors' earlier Theorem 1.5(2), and the non-big case is a standard slicing argument. The editors may wish to weigh the paper's contribution relative to the journal's standards. The paper should not be accepted in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a Bogomolov-type vanishing theorem: for a projective manifold X, a line bundle L, a nef class α, and a current β + i∂∂̄ψ ≥ 0 representing c1(L) − α, one gets H^n(X, Ω^p_X ⊗ L ⊗ I(ψ)) = 0 for p ≥ n − nd(α) + 1. The statement is clean and likely correct. But it is a modest extension: the case nd(α) = n is essentially the same group's earlier theorem (Theorem 1.5(2) in [LMNWZ25]) after choosing a metric, and the remaining case is a standard hyperplane induction using a logarithmic Serre vanishing theorem. The genuinely new part is the metric construction in the big case and the observation that the induction goes through.\n\nWhat is done well: the organization is clear, the arguments are standard and mostly correct, and the paper includes a self-contained proof of the logarithmic Serre vanishing theorem (Theorem 1.2) which may be handy. I checked the proof of Theorem 1.2 and the sequence used there; Proposition 2.5(c) with quotient Ω^p_{D1} is the right one, so the reader's concern about a conflicting exponent does not land.\n\nThe soft spots are minor. The application of strong openness in §4 is under-explained: Theorem 2.2 is stated for negative psh functions on a polydisk, while ψ and φ are only quasi-psh. One can fix this locally by subtracting smooth functions and then use coherence to get a uniform δ, so it is not a load-bearing problem, but it should be made explicit. The paper leans heavily on the authors' own prior work; that is not circularity, but it does mean the novelty is modest. Also, the 'in particular' corollary for nef L is already covered by known Bogomolov-type results, though the singular-metric version with multiplier ideal sheaves is a bit broader.\n\nWho is this for: specialists in complex algebraic geometry and the MMP who work with vanishing theorems and multiplier ideal sheaves. It is a useful reference, not a breakthrough. It deserves a serious referee; I would accept it for review and recommend minor revisions, mainly the normalization in the strong-openness step.","headline":"A solid, correct, but modest extension of the authors' own Bogomolov vanishing theorem; the proof is a clean reduction to prior work plus standard slicing, with a small normalization gap in the strong-openness step.","tokens_in":9580,"tokens_out":11418,"would_cite":true,"duration_ms":81903,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32L20","14F18","32L10","32C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that on a projective manifold, the cohomology groups H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) vanish for p ≥ n − nd({α}) + 1 whenever the first Chern class of L differs from a nef class {α} by a semi-positive current.","keywords":["Bogomolov vanishing theorem","nef line bundle","multiplier ideal sheaf","numerical dimension","singular Hermitian metric","projective manifold","strong openness","logarithmic differential forms"],"falsifier":"A concrete way to test the theorem is to search for a projective manifold X, a line bundle L, a nef class {α}, and a quasi-psh function ψ satisfying c1(L) − {α} = {β + i∂∂̄ψ} with β + i∂∂̄ψ ≥ 0, such that H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) is nonzero for some p ≥ n − nd({α}) + 1. At the level of the proof, one could try to construct a pair (ψ, φ) as in the big-case reduction where I(ψ + εφ) ≠ I(ψ) for all small ε > 0 despite α + i∂∂̄φ ≥ ω; this would invalidate the reduction step even if the theorem itself might still hold.","tokens_in":8784,"feed_emoji":"","tokens_out":5616,"duration_ms":43137,"temperature":0.7,"pith_summary":"The paper establishes a Bogomolov-type vanishing theorem: for a projective manifold X of dimension n, a line bundle L, and a nef class {α} such that c1(L) − {α} is represented by a semi-positive current, the cohomology group H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) vanishes for all p ≥ n − nd({α}) + 1. In the special case where L itself is nef, this gives H^n(X, Ω_X^p ⊗ L) = 0 for p ≥ n − nd(L) + 1, recovering and extending Bogomolov's classical vanishing theorem. The result matters because such vanishing statements are central tools in the classification of higher-dimensional algebraic varieties, and the theorem covers line bundles with singular Hermitian metrics and multiplier ideal sheaves, which arise naturally in minimal model theory. The proof combines a big-case reduction via the strong openness property of multiplier ideals with an induction on dimension using hyperplane sections and logarithmic differential forms.","feed_headline":"Bogomolov vanishing holds for nef line bundles with multiplier ideals","feed_subtitle":"Generalizes a classical cohomology vanishing theorem to line bundles that are nef up to a semi-positive twist.","key_machinery":"The proof relies on three tools: the strong openness property of multiplier ideal sheaves (Theorem 2.2), used to identify the multiplier ideal of a perturbed metric with I(ψ) for small perturbation parameter; the restriction formula for multiplier ideals under hyperplane sections (Theorem 2.3); and an induction on dimension that uses the Poincaré residue exact sequence for logarithmic differential forms together with a logarithmic Serre vanishing theorem (Theorem 1.2) to control the error term. The numerical dimension nd({α}) serves as the threshold, and its invariance under slicing by ample divisors makes the induction step work.","core_discovery":"Theorem 1.6 is the central claim. Let X be a projective algebraic manifold of dimension n and L a line bundle. Assume {α} is a nef class and c1(L) − {α} = {β + i∂∂̄ψ} with β + i∂∂̄ψ ≥ 0 as currents. Then H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) = 0 for p ≥ n − nd({α}) + 1, where nd({α}) is the numerical dimension of the nef class. As a direct consequence, if L is nef, then H^n(X, Ω_X^p ⊗ L) = 0 for p ≥ n − nd(L) + 1. This is the conjugate or Bogomolov-type counterpart of the Kawamata–Viehweg–Nadel vanishing theorem, and it generalizes earlier results that were limited to big line bundles or to metrics with strictly positive curvature.","pith_inferences":["The projective assumption may be relaxable: the induction uses hyperplane sections, but a similar slicing argument with general smooth divisors in a base-point-free linear system could work on compact Kähler manifolds, though the logarithmic Serre theorem might need adaptation.","Via Serre duality, the vanishing of H^n(X, Ω_X^p ⊗ L) is dual to H^0(X, Ω_X^{n−p} ⊗ L^{−1}); the theorem therefore also constrains the existence of holomorphic tensor fields twisted by negative line bundles, which is the original Bogomolov viewpoint.","The strong-openness step suggests a general principle: multiplier ideals are stable under perturbing a quasi-psh weight by a small multiple of any quasi-psh function whose curvature term is positive in the sense of currents, provided the perturbation preserves the total class. If formalized, this would simplify similar reductions in other vanishing theorems.","A natural testable extension is to replace the single nef class {α} by a nef (1,1)-class that is only relatively nef over a fibration, which could yield relative vanishing theorems with multiplier ideals."],"forward_implications":["If L is nef, the theorem yields H^n(X, Ω_X^p ⊗ L) = 0 for p ≥ n − nd(L) + 1, a direct generalization of Bogomolov's vanishing theorem to nef line bundles.","For big line bundles (nd = n), it recovers the known vanishing H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) = 0 for all p ≥ 1, now as a special case of a uniform statement.","The result applies to line bundles carrying singular Hermitian metrics with semi-positive curvature up to a nef twist, giving a vanishing statement with multiplier ideal sheaves in that generality.","Combined with the companion Kawamata–Viehweg–Nadel type theorem (H^q(K_X ⊗ L ⊗ I(ψ)) = 0), it gives a complete package of vanishing results for pseudo-effective line bundles.","The logarithmic Serre vanishing theorem (Theorem 1.2) proved in Section 3 may be of independent interest for further applications."],"fun_headline_variants":["Bogomolov-type vanishing for nef line bundles","Nef bundles get Bogomolov vanishing with multiplier ideals","Vanishing theorem generalized to nef line bundles","Multiplier ideals enable Bogomolov vanishing for nef bundles","Bogomolov vanishing extended to nef line bundles"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of the big case requires that the multiplier ideal of the perturbed weight ψ + εφ equals I(ψ) for small ε, which relies on the strong openness theorem for negative plurisubharmonic functions; the perturbation φ, however, is not explicitly normalized to be negative, so the applicability of that theorem is not fully justified as written.","fun_headline_variants_meta":{"raw":{"variants":["Bogomolov-type vanishing for nef line bundles","Nef bundles get Bogomolov vanishing with multiplier ideals","Vanishing theorem generalized to nef line bundles","Multiplier ideals enable Bogomolov vanishing for nef bundles","Bogomolov vanishing extended to nef line bundles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1195,"prompt_tokens":591,"completion_tokens":604,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":335,"completion_tokens_details":{"reasoning_tokens":520}},"tokens_in":335,"tokens_out":604,"duration_ms":4954,"temperature":1.0,"reasoning_tokens":520,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T00:53:25.711470+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the theorem is to search for a projective manifold X, a line bundle L, a nef class {α}, and a quasi-psh function ψ satisfying c1(L) − {α} = {β + i∂∂̄ψ} with β + i∂∂̄ψ ≥ 0, such that H^n(X, Ω_X^p ⊗ L ⊗ I(ψ)) is nonzero for some p ≥ n − nd({α}) + 1. At the level of the proof, one could try to construct a pair (ψ, φ) as in the big-case reduction where I(ψ + εφ) ≠ I(ψ) for all small ε > 0 despite α + i∂∂̄φ ≥ ω; this would invalidate the reduction step even if the theorem itself might still hold.","supporting_citations":[],"review_version":1}