{"id":"be7734b2-9d42-443e-953c-84e4cb3615ce","arxiv_id":"2607.29153","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"On compact Kähler manifolds, holomorphic top forms on σ-lc centres of the same codimension extend to the ambient space, without L² estimates, under the curvature positivity condition (eq5.1).","lead":"An extension theorem for holomorphic top forms is proved on compact Kähler manifolds: sections living on log-canonical centres that are compatible with adjoint ideal sheaves extend to the ambient space when a curvature-positivity condition holds. The result is qualitative — the extensions carry no L² estimates — and is proved with harmonic theory and residue computations adapted from injectivity theorems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1's proof hinges on Prop 4.1, whose transfer to non-reduced ψ and extra B-divisors is only asserted as 'formally the same'; this is the load-bearing gap.","rationale":"The reader identified the transfer of the harmonic-residue/injectivity machinery as the weakest assumption; I agree. It is the most load-bearing because Theorem 5.1's proof cannot go through without the exact adjunction identity, whereas the existence of Ψ^(p) is an explicit condition of the theorem and its unavailability is openly stated in Remark 5.2, affecting applicability but not logical validity. The paper is otherwise coherent: the reduction to ker τ=0, the use of Bochner-Kodaira via Prop 3.3, and the induction over σ are structurally sound given Prop 4.1/4.2. The author's honest limitation statements and the conditional nature of the theorem are properly weighed; no fundamental flaw beyond the unverified transfer was found. Therefore the reviewer's CONDITIONAL verdict with moderate confidence and medium correctness risk is appropriate; no adjustment is needed.","tokens_in":20203,"tokens_out":10006,"duration_ms":98174,"concrete_test":"Re-derive the displayed integration-by-parts/residue computation in the proof of Prop 4.1 for a concrete non-reduced, mixed-polar example: X = C^2 with S = {z_1 z_2 = 0}, B = {z_1+z_2-1=0}, ψ = 2 log|z_1|^2 + log|z_2|^2, φ = |z_1|^2+|z_2|^2, σ=1, q=1. Compute both sides of ⟨δ^H w,u⟩ = 2⟨w,R(u)⟩ using the residue norms in (eq2.2) and compare the Lelong-number factors. If the equality fails or requires a factor different from 2, Prop 4.1 is invalid in the stated generality and Theorem 5.1's proof collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised extension (Theorem 5.1) is proved by showing ker τ^{σ'}_σ = 0. The chain is: (eq5.1) forces the harmonic residue R(u)=0; Prop 4.2 gives u∈im δ^H; Prop 4.1 then gives ||u||²_{lc^σ} = σ_+ ⟨w,R(u)⟩_{lc^{σ+1}} = 0. Both Prop 4.1 and Prop 4.2 are imported from [13]/[14], where ψ has a reduced polar divisor and no B-components. Here ψ may have non-reduced polar divisor (generic Lelong numbers ν_i>1 in (eq2.1)) and B-components outside the lc locus. The proof of Prop 4.1 is described only as 'formally the same' (Sec. 4) with 'extra care' for singularities along B and the Lelong-number factors ν_p. The delicate coefficient σ_+=max{1,σ} and the cancellation of ν_{p(k)} in the residue computation depend on the normalizations in (eq2.2). If the non-reduced or B data change these factors, the equality ⟨δ^H w,u⟩ = σ_+⟨w,R(u)⟩ would acquire extra constants, and the final R(u)=0 ⇒ ||u||²=0 step would fail. Prop 4.2's remaining equality also cites [13] with 'suitable adjustments.' No independent derivation or machine check is provided. The paper's own Remark 5.2 additionally concedes that the auxiliary functions Ψ^(p) satisfying (eq5.1) for σ≥1 are not known in general, limiting applicability, but the transfer gap is the load-bearing correctness risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a qualitative extension theorem for holomorphic top forms in terms of adjoint ideal sheaves on compact Kähler manifolds in the snc configuration. Theorem 5.1 states that if, for a fixed σ, there exists λ0 > 0 such that i∂∂̄(φ + λΨ^(p)) ≥ 0 on every σ-lc centre (eq5.1), then the natural maps H^q(R^σ) → H^q(J^{σ'}/J^{σ-1}) are injective and the relevant long exact sequences split; in particular, sections over lc^{σ+1}_X(S) lift to J^{σ'}/J^{σ-1}. The proof follows the harmonic-residue route: the positivity condition forces the harmonic residue R(u) to vanish, Proposition 4.2 places u in im δ^H, and Proposition 4.1's adjunction identity gives ∥u∥² = σ_+⟨w, R(u)⟩ = 0. The result generalizes [12] to non-reduced polar divisors of ψ and to extra B-components, and the author explicitly records (Remark 5.2) that the auxiliary functions Ψ^(p) for σ ≥ 1 are not known to exist in general.","tokens_in":20548,"tokens_out":10855,"duration_ms":103966,"significance":"If the key imported propositions are valid in the claimed generality, the paper makes a meaningful contribution: it sharpens the previous qualitative extension theorem by using equalities rather than inequalities, allows non-reduced and B-adapted polar data, and frames the extension as splittings of cohomology exact sequences. The main proof is coherent and the main limitation is honestly stated. However, the result is not yet fully supported because Propositions 4.1 and 4.2 are transferred from arXiv preprints with only sketched justifications, and the condition (eq5.1) may be void for σ ≥ 1. These issues are fixable but currently leave a load-bearing gap.","major_comments":[{"comment":"The identity ⟨δ^H w, u⟩_{lc^σ_{X°}(S)} = σ_+⟨w, R(u)⟩_{lc^{σ+1}_{X°}(S)} is the exact step that turns R(u) = 0 into ∥u∥² = 0 in the proof of Theorem 5.1. The proof is described only as 'formally the same' as [13, Prop. 2.3.3], with 'extra care' for the singularities along B and the coefficients ν_p. In the present setting ψ may be non-reduced (ν_i > 1 in (eq2.1)) and has components outside S, so the normalizations in (eq2.2) and the cancellation of ν_{p(k)} in the residue computation are not directly covered by the earlier reduced case. If these factors change, the equality acquires extra constants and the conclusion fails. A complete proof in the present generality, or a precise statement/reference covering exactly this case, is required.","section":"§4, Prop. 4.1"},{"comment":"The equality ker τ^{σ'}_σ = ker τ^{σ+1}_σ = im δ is also load-bearing: it is what lets the proof of Theorem 5.1 write u = δ^H w. The proof cites [13, proof of Thm. 3.4.1] with 'suitable adjustments' and then gives a direct argument that still depends on Prop. 4.1. Because Prop. 4.1 itself is only sketched, Prop. 4.2 does not remove the gap. The author should provide a self-contained derivation of this equality for the non-reduced/B-setting.","section":"§4, Prop. 4.2"},{"comment":"The local residue formula in Prop. 3.5 is used in the proof of Theorem 5.1 to identify the limits after (eq5.1) with sums of squared residues and thereby conclude R(u) = 0. Its proof is again delegated to [13] with 'extra care' for Lelong numbers and coefficients in C^∞_X[eω^±]. Since this formula is a prerequisite for the main argument and is adapted to the more general setting, the manuscript should contain a full proof or a precise adaptation statement for this proposition.","section":"§3, Prop. 3.5"},{"comment":"The theorem's hypothesis (eq5.1) is not known to be satisfiable for σ ≥ 1. The paper states that no general procedure is known to construct Ψ^(p) from ψ. This does not make the theorem false, but it substantially limits the advertised conclusion: no example is given where extension over lc centres of codimension ≥ 2 is obtained. The paper should either provide examples or constructions verifying (eq5.1), or rephrase the main theorem explicitly as conditional and discuss the non-vacuity of the hypothesis.","section":"§5, Remark 5.2"}],"minor_comments":[{"comment":"In the displayed chain defining m_k, the expression 'I(φ + mψ)' appears; this is likely intended to be 'I(φ_L + mψ)' since φ has not been introduced at that point.","section":"§1, jumping-number display"},{"comment":"The notation 'σV_{i_1...i_q}' is used in the formula for s_(p) but is never defined. It seems to denote the number of coordinate hyperplanes in the admissible open set; please define it.","section":"§4, proof of Prop. 4.1"},{"comment":"For σ = 0, the displayed residue norm formula contains (σ−1)! and ν_p, which are not defined for σ = 0. The convention R^0 is only stated afterwards; a short clarifying remark would avoid confusion.","section":"§2, (eq2.2)"},{"comment":"The proof relies substantially on [13] and [14], which are arXiv preprints. The manuscript should note this explicitly and, if possible, cite published or final versions.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the transfer of Propositions 4.1 and 4.2, and also Proposition 3.5, from the author's earlier preprints to the generalized non-reduced/B-setting. If the author can supply complete proofs or point to published statements that cover exactly this generality, the paper would be acceptable. I also recommend asking for at least one non-trivial class of examples where (eq5.1) holds for σ ≥ 1, since Remark 5.2 currently leaves the applicability of the main theorem open."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does what it says: it extends the qualitative extension theorem of Chan–Choi [12] to quasi-psh functions whose polar divisor need not be reduced and may have components outside the lc locus, and it reworks the proof around harmonic residues. That is a genuine new statement, not a repackaging. The proof strategy is coherent: curvature positivity forces the harmonic residue to vanish, Proposition 4.2 places the class in the image of the connecting map, and Proposition 4.1's adjunction identity then kills the norm. The paper also includes a fairly detailed sketch of the key residue computation, including the coefficient σ₊ and the cancellation of the Lelong-number factors, so the advertised generalization is not just asserted.\n\nThe soft spot is exactly where the stress-test points: Propositions 4.1 and 4.2 are imported from the author's earlier injectivity work [13], [14], with the proof of 4.1 described as \"formally the same\" plus extra care for B-singularities and Lelong numbers. The sketch addresses the main coefficient issue, but it is not a full derivation; Proposition 4.2's second equality also cites [13] with \"suitable adjustments.\" If those transfers fail, the chain R(u)=0 ⇒ u∈im δᴴ ⇒ ||u||²=0 collapses. This is a verifiability gap, not detected error, but it is load-bearing enough that a referee should demand a complete derivation or a clearly isolated lemma. The other fragility, noted in Remark 5.2, is that the auxiliary functions Ψ⁽ᵖ⁾ satisfying (eq5.1) for σ≥1 are not known to exist in general; that limits applicability but is openly stated and does not undermine the conditional theorem.\n\nCitation pattern is fine: heavy self-citation is appropriate because the machinery is the author's own and the cited results are the actual basis. The paper is honest about the snc assumption and the lack of L² estimates. It deserves a serious referee and likely publication after the residue transfer is checked. If I worked in this area I would cite it; it is a step forward in the adjoint-ideal-sheaf program.","headline":"A real generalization of the Chan–Choi extension theorem to non-reduced polar data, proved via harmonic-residue machinery; the main risk is the transfer of prior injectivity identities, and the paper is honest about its limitations.","tokens_in":21193,"tokens_out":1852,"would_cite":true,"duration_ms":21802,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32J25","32Q15","14B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every holomorphic top form on a union of log-canonical centres of fixed codimension extends to the ambient compact Kähler manifold, provided a curvature semipositivity condition holds on each centre, and the extension lies in the adjoint id","keywords":["adjoint ideal sheaf","multiplier ideal sheaf","log-canonical centre","Ohsawa–Takegoshi extension theorem","harmonic residue","residue exact sequence","injectivity theorem","compact Kähler manifold"],"falsifier":"Compute the adjunction identity of Proposition 4.1 for ψ = a log|s_S|² + b log|s_B|² with a > 1, b > 0 and B disjoint from the lc locus: if the equality gains a factor depending on a or b, the transfer from the reduced case fails and Theorem 5.1 collapses. Alternatively, find a compact Kähler example satisfying the curvature condition at some σ where H^0(J^{σ′}/J^{σ−1}) → H^0(J^{σ′}/J^σ) is not surjective; the asserted splitting would then be false.","tokens_in":19909,"feed_emoji":"📐","tokens_out":12421,"duration_ms":120904,"temperature":0.7,"pith_summary":"This paper proves a qualitative extension theorem in the spirit of Ohsawa–Takegoshi: on a compact Kähler manifold, holomorphic top forms defined on the union of log-canonical centres of a fixed codimension extend to the whole manifold, not with L² estimates but with a prescribed local integrability condition encoded by adjoint ideal sheaves. The main statement says that if a curvature condition i∂∂̄(φ+λΨ^{(p)}) ≥ 0 holds on each σ-codimensional log-canonical centre for small λ, then certain cohomology sequences split, so each section over the (σ+1)-lc centres lifts first to the σ-lc centres and, by induction, to the ambient space when the condition holds at every σ. The proof rewrites extension as an equality problem: harmonic residues and an adjunction identity show that positivity forces the relevant residue to vanish, and with it the harmonic representative of the section. This matters because existing L² extension theorems usually do not control which adjoint ideal sheaf the extension belongs to, whereas this theorem gives a sheaf-theoretic description of the non-integrable locus while working one codimension at a time.","feed_headline":"Every holomorphic top form extends from log-canonical centres","feed_subtitle":"A curvature condition on each centre makes the cohomology steps split, so extension carries a local L² condition.","key_machinery":"The engine is the filtration by adjoint ideal sheaves J^σ(φ_L;ψ) — germs f locally satisfying ∫ |f|² e^{−φ_L−ψ} |ψ|^{−σ} (log|eψ|)^{−1−ε} < ∞ for every ε > 0 — together with the residue exact sequence 0 → J^{σ−1} → J^σ → R^σ → 0, where R^σ is the residue sheaf supported on the σ-codimensional lc centres. On the level of harmonic forms, the central object is the harmonic residue R(u), the collection of residues of (∂ψ^{(p)})⌟u_p over the (σ+1)-lc centres inside each σ-lc centre. Two identities carry the argument: the adjunction formula ⟨δ^H w, u⟩_{lc^σ} = σ_+ ⟨w, R(u)⟩_{lc^{σ+1}} and the equality ker τ^{σ′}_σ = ker τ^{σ+1}_σ = im δ. Under the curvature hypothesis, the proof forces R(u) = 0; t","core_discovery":"The paper's central claim is Theorem 5.1. For a system (X, φ_L, ψ) in an snc configuration, write J^σ for the adjoint ideal sheaves and suppose that for a given σ ≤ σ_mlc there is λ0 > 0 such that i∂∂̄(φ + λΨ^{(p)}) ≥ 0 on each σ-lc centre S^σ_p for all λ ∈ [0, λ0]. Then, for every σ′ ≥ σ, the short exact sequence 0 → R^σ → J^{σ′}/J^{σ−1} → J^{σ′}/J^σ → 0 induces long exact sequences that split into short exact sequences 0 → H^q(R^σ) → H^q(J^{σ′}/J^{σ−1}) → H^q(J^{σ′}/J^σ) → 0 for all q ≥ 0. In particular, every holomorphic top form f on the union of (σ+1)-lc centres, i.e. f ∈ H^0(J^{σ′}/J^σ), admits an extension F^{σ−1} on the σ-lc centres with f ≡ F^{σ−1} mod J^σ; if the curvature conditio","pith_inferences":["Editorial inference: a general procedure for constructing Ψ^{(p)} from ψ would make Theorem 5.1 unconditional at every σ; Remark 5.2 identifies this as the key open step, so the theorem's scope currently depends on a case-by-case curvature check.","Editorial inference: if existing L² estimates for codimension-one extensions can be composed along the induction, the qualitative extension should upgrade to one with explicit norm control; the paper raises this as a natural question but leaves it open.","Editorial inference: since the proof only needs the full harmonic residue R(u) to vanish, a weaker positivity condition that kills R(u) without killing each summand — as happens in the injectivity theorem — would likely widen the class of examples; Remark 5.3 explicitly asks for such a weakening.","Editorial inference: the equality-based mechanism suggests that the same adjunction identities could be used to split higher direct image cohomology under proper Kähler morphisms, connecting the extension statement to injectivity-type vanishing on snc spaces."],"forward_implications":["For each q, the cohomology of the successive quotients decouples: H^q(J^{σ′}/J^{σ−1}) is isomorphic to H^q(R^σ) ⊕ H^q(J^{σ′}/J^σ) whenever the curvature condition holds at level σ.","Extension works in all cohomological degrees, not only for global sections.","If the curvature condition holds at every σ, sections over lc centres of any fixed codimension extend to ambient sections in H^0(J^{σ′}), so the whole union of lc centres of the same codimension can be extended at once.","The case σ = 0 recovers the earlier qualitative extension over the lc locus, and the proof supplies a one-codimension-at-a-time inductive mechanism.","The extended form automatically lies in the prescribed adjoint ideal sheaf, so it satisfies the local L² condition controlled by |ψ|^{−σ′} rather than merely being holomorphic."],"fun_headline_variants":["Curvature lets top forms extend from lc centres","Top forms extend from lc centres under curvature condition","No L² estimates: top forms extend from lc centres","Under curvature, top forms extend from log-canonical centres"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument stands or falls on whether the harmonic-residue identities and kernel equalities that were checked for simpler, reduced polar data remain true for the more general ψ treated here (the paper says the proof is 'formally the same'), together with the existence of auxiliary functions Ψ^{(p)} satisfying the curvature condition at every codimension — which Remark 5.2 explicitly leaves open in general.","fun_headline_variants_meta":{"raw":{"variants":["Curvature lets top forms extend from lc centres","Top forms extend from lc centres under curvature condition","No L² estimates: top forms extend from lc centres","Under curvature, top forms extend from log-canonical centres"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001939,"raw_usage":{"total_tokens":7441,"prompt_tokens":782,"completion_tokens":6659,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":6591}},"tokens_in":526,"tokens_out":6659,"duration_ms":49249,"temperature":1.0,"reasoning_tokens":6591,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:37:21.592944+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the adjunction identity of Proposition 4.1 for ψ = a log|s_S|² + b log|s_B|² with a > 1, b > 0 and B disjoint from the lc locus: if the equality gains a factor depending on a or b, the transfer from the reduced case fails and Theorem 5.1 collapses. Alternatively, find a compact Kähler example satisfying the curvature condition at some σ where H^0(J^{σ′}/J^{σ−1}) → H^0(J^{σ′}/J^σ) is not surjective; the asserted splitting would then be false.","supporting_citations":[],"review_version":1}