Singularity of non-pluripolar cohomology classes

Duc-Bao Nguyen; Duc-Viet Vu; Shuang Su

REVIEW 2 major objections 3 minor 1 cited by

Singularity of non-pluripolar cohomology classes

T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Full-mass divisors force vanishing Lelong numbers

desk verdict The conditional theorem is plausible, but the global vanishing claim is likely false for big non-nef classes. read the letter →

arxiv 2508.14669 v2 pith:MDNOF5BS submitted 2025-08-20 math.CV math.AG

Duc-Bao Nguyen , Shuang Su , Duc-Viet Vu This is my paper · ORCID

classification math.CVmath.AG MSC 32U0532U2532J25

keywords Lelongnumbersnon-pluripolarproductsbigclassesrestrictedvolumefullmasspositivecurrentspluripotentialtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The 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.

The 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.

There 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[Abstract, 'In particular' claim]
[Abstract, definition of 'Lelong numbers of the non-pluripolar class']

minor comments (3)

[Abstract] 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.
[Abstract] 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.
[Abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected in the abstract-level derivation chain.

full rationale

The review is based only on the abstract, since full text is not available. The paper's claimed chain is: (1) establish a relation between Lelong numbers and the full mass property of relative non-pluripolar products; (2) use this relation to show that full restricted volume along an effective divisor implies vanishing of Lelong numbers of the non-pluripolar class on the divisor support; (3) infer the projective-manifold statement. None of these steps, as stated, reduces by definition to its own inputs. The hypothesis (full mass of restricted volume) does not contain the conclusion (zero Lelong numbers), and there is no fitted parameter, renamed quantity, or self-citation invoked in the abstract. The mathematical content may be challenging or potentially open to skeptical counterexamples, but skepticism about correctness is not circularity. No quote or equation is available to exhibit a specific reduction, and the hard rules require quoting the paper and exhibiting the reduction to flag circularity. Therefore the appropriate finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The result uses established machinery from pluripotential theory, especially non-pluripolar products and restricted volumes. The full-mass hypothesis is a theorem assumption, not an extra axiom.

assumptions (3)

domain assumption Existence and well-definedness of non-pluripolar products for closed positive currents in big cohomology classes.
The paper uses non-pluripolar products as an established construction; this is standard background in pluripotential theory.
domain assumption The theory of relative non-pluripolar products and restricted volumes, including their behavior along divisors.
The central relation in the abstract is stated for relative non-pluripolar products; the paper assumes this theory.
domain assumption Lelong numbers for non-pluripolar classes are well-defined and satisfy standard properties.
Lelong numbers for such classes are assumed to exist and behave as in the standard literature; this is background for the statement.

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Cite this review

Pith. "Pith review of Singularity of non-pluripolar cohomology classes." pith.science (2026). https://pith.science/paper/MDNOF5BS

@misc{pith2026250814669,
  author       = {Pith},
  title        = {Pith review of: Singularity of non-pluripolar cohomology classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDNOF5BS}},
  note         = {Machine review of arXiv:2508.14669}
}

abstract

We establish a relation between Lelong numbers and the full mass property of relative non-pluripolar products. We use this relation to prove that if the restricted volume of a big class $\alpha$ along an effective divisor $D$ is of full mass, then the Lelong numbers of the non-pluripolar class $\langle \alpha^{n-1}\rangle$ at every point in the support of $D$ are zero. In particular, we obtain that on projective manifolds, the Lelong numbers of the non-pluripolar class $\langle \alpha^{n-1}\rangle$ of a big class $\alpha$ are zero.

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