{"id":"df3e4515-73c8-40fa-b212-1923c6922c9c","arxiv_id":"2508.14669","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"If the restricted volume of a big class along an effective divisor has full mass, the Lelong numbers of the non-pluripolar product class vanish on the divisor.","lead":"This paper proves that under a 'full mass' condition, the Lelong numbers of a non-pluripolar product vanish. The result gives a clean geometric criterion and resolves a natural question about singularities of big classes on projective manifolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's 'In particular' step from restricted-volume full mass to vanishing everywhere may fail for big non-nef classes with a divisorial negative part.","rationale":"The reader's weakest assumption is the unproved key relation between Lelong numbers and full mass of relative non-pluripolar products. I agree that this relation is load-bearing and unverifiable from the abstract. However, I identify a more specific and possibly fatal gap: the 'In particular' conclusion appears to follow only if for every point of X one can find an effective divisor D with full-mass restricted volume containing that point. The abstract does not state or justify this. The blowup example α=2H+E is a projective surface with a big class whose Zariski decomposition has a negative exceptional component E; any positive current representing α must have Lelong number at least 1 along E. If the paper's definition of Lelong numbers of the non-pluripolar class coincides with the usual minimal Lelong numbers, the 'In particular' claims a false statement. If the definition is different, the abstract is ambiguous and the theorem's meaning shifts. Therefore the verdict should remain conditional pending clarification of definitions and verification on this example. I do not reject outright because the full text may contain a nonstandard notion of Lelong number for cohomology classes or an argument for the 'In particular' that I cannot see from the abstract.","tokens_in":567,"tokens_out":31103,"duration_ms":385167,"concrete_test":"Compute the paper's quantities for X=Bl_p(P^2), α=2H+E. Step 1: Using the paper's definition, evaluate ν(⟨α⟩, x) for a point x∈E. Step 2: Take D the strict transform of a conic through p (class 2H−E) and compute the restricted volume vol_{X|D}(α) and test the full-mass property of the relative non-pluripolar product. If the theorem predicted ν(⟨α⟩,x)=0 while the direct computation yields ≥1, the 'In particular' fails. If full mass fails, ask explicitly which divisor the authors use to derive the 'In particular'; if no such D exists for this α, the global statement is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is the unrestricted vanishing statement in the 'In particular.' The restricted-volume theorem only gives vanishing on supp D under a full-mass hypothesis. To reach 'all points on projective manifolds' one must know that every point is contained in some divisor D for which vol_{X|D}(α) is full mass. That is not stated and seems false for big classes with a nontrivial Zariski decomposition. Let X=Bl_p(P^2), H=π^*O(1), E the exceptional divisor, and α=2H+E. This class is big (a smooth positive 2H-form plus [E] gives a Kähler current). Its Zariski decomposition is P=2H, N=E, so α·E=-1. Any closed positive current T∈α satisfies α·E = ν_E(T) E^2 + nonnegative = -ν_E(T) + nonnegative, so ν_E(T) ≥ 1; hence every representative has Lelong number at least 1 at every point of E. If 'Lelong numbers of the class' means the usual minimal Lelong numbers (or those of the least singular representative), the 'In particular' statement is false for this projective surface. If the paper's notion of Lelong number for the non-pluripolar class discards such divisorial contributions, that needs to be stated and changes the content of the theorem. Either way, the bridge from the full-mass hypothesis to the global vanishing is the least secure step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The abstract announces a theorem relating Lelong numbers to the full mass property of relative non-pluripolar products. It states that if the restricted volume of a big class α along an effective divisor D is of full mass, then the Lelong numbers of the non-pluripolar class ⟨α^{n-1}⟩ vanish at every point of the support of D. As an 'In particular', it claims that on any projective manifold, the Lelong numbers of ⟨α^{n-1}⟩ vanish for every big class α.","tokens_in":909,"tokens_out":28072,"duration_ms":322477,"significance":"If the local restricted-volume-to-Lelong-number statement is correct, it gives a new bridge between algebraic restricted volumes and pluripotential singularity theory, and the global corollary would be a striking simplification of the Lelong-number theory of big classes. The paper's main strength is the clean formulation of this bridge. However, the global corollary is not a direct consequence of the stated local theorem and, under standard interpretations, is open to concrete counterexamples. The significance therefore depends on the precise definitions used in the full paper, which the abstract does not provide.","major_comments":[{"comment":"","section":"Abstract, 'In particular' claim"},{"comment":"","section":"Abstract, definition of 'Lelong numbers of the non-pluripolar class'"}],"minor_comments":[{"comment":"The abstract should define 'full mass' for restricted volumes and the 'full mass property of relative non-pluripolar products'. The current wording leaves the hypothesis ambiguous, especially for divisors with zero or negative intersection with α.","section":"Abstract"},{"comment":"The phrase 'the Lelong numbers of the non-pluripolar class ⟨α^{n-1}⟩' should specify whether these are the Lelong numbers of a canonical current (least singular, or positive part) and whether they are taken pointwise or generically.","section":"Abstract"},{"comment":"The example of a big class with a nontrivial Zariski decomposition (e.g., 2H+E on Bl_p(P^2)) would be a useful test case for the stated theorem; the authors should address such cases explicitly in the introduction.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is legitimate and should be sent to the authors. The 'In particular' step is not justified by the local theorem and appears false under standard interpretations of 'Lelong numbers of ⟨α^{n-1}⟩'. The paper may be fixable by clarifying the definition (e.g., using the positive part of the Zariski decomposition) and by removing or qualifying the global corollary, but as written the headline claim is not established. The full proof must be checked for this gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe abstract states a stronger claim than the theorem supports. The conditional result—full mass restricted volume along an effective divisor forces zero Lelong numbers on that divisor—is plausible and could be a useful tool. The genuinely new piece is the proposed relation between Lelong numbers and the full mass property of relative non-pluripolar products. If the proof of that lemma holds, the paper makes a clean contribution to the singularity theory of big classes.\n\nThe soft spot is the 'In particular.' Even if the conditional theorem is true, the global statement that all big classes on projective manifolds have zero Lelong numbers does not follow. You would need, for every point, a divisor D of full restricted mass. That is not established, and I suspect it is false. A concrete example: on X=Bl_p(P^2), take α=2H+E. This class is big, but α·E=-1, so any closed positive current in α has Lelong number at least 1 on the exceptional divisor. That contradicts the global claim under the usual definition of Lelong numbers of a class. If the authors intend a different definition—one that discards divisorial contributions—they need to say so explicitly, because then the statement is much weaker than it appears.\n\nThere is no way to check the load-bearing lemma from the abstract alone, so the positive verdict is conditional. The counterexample above is enough to make me think the abstract overreaches. The paper deserves a careful referee: the conditional theorem may be correct and worth publishing, but the global claim needs either proof or repair. I'd bring it to a reading group only to discuss the failure of the 'In particular' step. I wouldn't cite it yet.","headline":"The conditional theorem is plausible, but the global vanishing claim is likely false for big non-nef classes.","tokens_in":1269,"tokens_out":3428,"would_cite":false,"duration_ms":40697,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32U05","32U25","32J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Full-mass divisors force vanishing Lelong numbers","keywords":["Lelong numbers","non-pluripolar products","big classes","restricted volume","full mass","positive currents","pluripotential theory"],"falsifier":"Exhibit a projective manifold X, a big class α, and a point x in the support of an effective divisor D such that the restricted volume of α along D has full mass while the Lelong number of ⟨α^{n-1}⟩ at x is nonzero. The theorem asserts no such example exists; for n=2, one could search among big line bundle classes with a representative carrying a point singularity and compare the restricted volume with the divisor's full mass.","tokens_in":549,"feed_emoji":"🧮","tokens_out":3801,"duration_ms":37555,"temperature":0.7,"pith_summary":"This paper establishes a new bridge between two measurements attached to positive currents: Lelong numbers, which gauge the size of singularities at individual points, and the 'full mass' property of relative non-pluripolar products, a global condition on how much mass a class leaves on a divisor. The authors prove that if the restricted volume of a big cohomology class along an effective divisor has full mass, then the Lelong numbers of the non-pluripolar class ⟨α^{n-1}⟩ vanish at every point of that divisor. A direct corollary on projective manifolds is a clean global statement: the Lelong numbers of the non-pluripolar class of any big class are zero. This matters because Lelong numbers often obstruct good geometric behavior; the result removes those obstructions under a natural full-mass hypothesis.","feed_headline":"Full-mass divisor forces zero Lelong numbers","feed_subtitle":"A new relation between Lelong numbers and non-pluripolar mass kills singularities of big classes on projective manifolds.","key_machinery":"The machinery is the relative non-pluripolar product of cohomology classes: a way of multiplying currents that ignores pluripolar singularities and produces a well-defined class on a divisor, together with its full mass property. The paper's new relation ties the full mass of the relative product to the vanishing of Lelong numbers. This relation is the load-bearing bridge: once established, the vanishing theorem follows by applying it to the restricted volume of a big class along an effective divisor.","core_discovery":"The central claim is the theorem stated in the abstract: for a big class α on a compact Kähler manifold and an effective divisor D, if the restricted volume of α along D has full mass, then every point in the support of D has zero Lelong number for the non-pluripolar class ⟨α^{n-1}⟩. The proof goes through a newly proved relation between Lelong numbers and the full mass property of relative non-pluripolar products. In the projective setting, the authors derive the unconditional statement that the Lelong numbers of ⟨α^{n-1}⟩ vanish for every big class α.","pith_inferences":["The relation likely extends to higher-order non-pluripolar products ⟨α^k⟩ for k<n, yielding analogous vanishing for Lelong numbers attached to analytic subsets of higher codimension.","One could ask whether the full-mass condition is also necessary, not merely sufficient, for vanishing of Lelong numbers; the paper does not address this converse direction.","On projective manifolds, the vanishing statement means that, although individual positive representatives may carry singularities, the non-pluripolar class itself is pointwise regular; testing this on explicit examples of big line bundles would clarify the geometric content."],"forward_implications":["If the theorem is correct, big classes on projective manifolds have non-pluripolar (n−1)-products with no point singularities in the Lelong sense, simplifying the pluripotential theory of these classes.","The relation gives a new tool to detect whether a class concentrates mass on a divisor, with potential applications in Kähler geometry and transcendental algebraic geometry.","Because restricted volumes appear in birational geometry, the result links a volume-type invariant to local singularity data, suggesting further connections between the two.","On projective manifolds, the vanishing holds for every big class, so any future counterexample would have to live outside the projective realm or require a different notion of singularity."],"supporting_citations":[],"fun_headline_variants":["Zero Lelong numbers from full-mass divisor volumes","Big classes on projective manifolds have zero Lelong numbers","Full mass along a divisor forces zero Lelong numbers","Lelong numbers vanish when restricted volume is full mass","Projective big classes: Lelong numbers all zero"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof depends on a newly established relation between Lelong numbers and the full mass property of relative non-pluripolar products; if that relation is incorrectly proved, the vanishing conclusions do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Zero Lelong numbers from full-mass divisor volumes","Big classes on projective manifolds have zero Lelong numbers","Full mass along a divisor forces zero Lelong numbers","Lelong numbers vanish when restricted volume is full mass","Projective big classes: Lelong numbers all zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000164,"raw_usage":{"total_tokens":1017,"prompt_tokens":611,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":355,"completion_tokens_details":{"reasoning_tokens":330}},"tokens_in":355,"tokens_out":406,"duration_ms":4612,"temperature":1.0,"reasoning_tokens":330,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T18:20:31.881513+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a projective manifold X, a big class α, and a point x in the support of an effective divisor D such that the restricted volume of α along D has full mass while the Lelong number of ⟨α^{n-1}⟩ at x is nonzero. The theorem asserts no such example exists; for n=2, one could search among big line bundle classes with a representative carrying a point singularity and compare the restricted volume with the divisor's full mass.","supporting_citations":[],"review_version":1}