Ko{\l}odziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property

Lei Zhang

REVIEW 4 major objections 4 minor 30 references

Ko{\l}odziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Kołodziej-Tosatti conjecture is true on every compact Hermitian manifold whose Hermitian metric has bounded mass: the Morse-type integral identity holds for every nef class.

desk verdict Plausible conjecture, but the proof repeatedly uses bounded mass property to control mixed Monge-Ampere integrals that do not follow from the definition; as written, the main results are not proven. read the letter →

arxiv 2412.01586 v1 pith:B5KIV3W3 submitted 2024-12-02 math.DG

Lei Zhang This is my paper · ORCID

classification math.DG MSC 32W2053C5532U05

keywords boundedmasspropertynefclassKołodziej-TosatticonjectureMorse-typeintegralscomplexMonge-AmpèreequationbigKählercurrentHermitianmanifold

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 proves the Kołodziej-Tosatti conjecture under the bounded mass property. The conjecture claims that for a nef class [α] on a compact Hermitian manifold, the total mass ∫X αⁿ equals the infimum of integrals of (α+i∂∂̄u)ⁿ over the set where the smooth representative is nonnegative; this identity is the non-Kähler analogue of the volume of a nef class. The author shows that whenever the Hermitian metric has finite Monge-Ampère volume (vol(ω) < ∞), the identity holds. As a consequence, the Demailly-Păun conjecture (nef plus positive self-intersection implies big) and the Tosatti-Weinkove conjecture (existence of α-psh functions with prescribed logarithmic poles) also hold under the same hypothesis. The proof follows the Kołodziej-Tosatti strategy, using plurisubharmonic envelopes and the bounded mass property to control mixed Monge-Ampère integrals uniformly in ε.

What carries the argument

The bounded mass property vol(ω) = sup{∫_X (ω+i∂∂̄f)ⁿ : f∈C∞(X,R), ω+i∂∂̄f>0} < ∞ is the central object; it supplies the uniform O(ε) bounds on the mixed Monge-Ampère integrals that appear when expanding (α+εω+i∂∂̄u_ε)ⁿ. The second ingredient is the plurisubharmonic envelope h_ε = sup{φ∈psh(X,β+εω) : φ≤0}, which is $C^{{1,1}}$ and has its Monge-Ampère mass supported on the contact set {h_ε=0}; this transfers the mass estimate to the positivity set X(β+εω,0). For the big-class consequence, the proof uses Tosatti-Weinkove's solution of the complex Monge-Ampère equation and Lamari's characterization of classes admitting positive currents via Gauduchon metrics.

What would settle it

A concrete check: compute the mixed integral nε∫_X ω∧(α+εω+i∂∂̄(u_ε+ψ_ε))^{n−1} on a compact Hermitian manifold with vol(ω)<∞ and verify whether it is in fact O(ε²) as equation (2.13) claims; if an O(ε) term proportional to ∫_X ω∧$α^{{n−1}}$ persists for every choice of u_ε and ψ_ε, the domination step in Case 2 of Section 2.2 fails and the proof of Theorem 1.2 as written does not go through. Alternatively, exhibit a compact Hermitian manifold with bounded mass property and a nef class [α] with ∫_X αⁿ>0 but no Kähler current in [α]; that would directly refute Theorem 1.2.

Watch

Extended reading notes

Core claim

Theorem 1.1 establishes that for every nef Bott-Chern class [α] on a compact Hermitian manifold with bounded mass property, ∫_X αⁿ = inf_{u∈C∞(X,R)} ∫_{X(α+i∂∂̄u,0)} (α+i∂∂̄u)ⁿ. The inequality in one direction is a standard result of Demailly, so the content is the reverse inequality, obtained by constructing envelopes h_ε whose Monge-Ampère masses converge to ∫_X αⁿ from below while each stays below the integral over the positivity set. The bounded mass property is used to control the mixed terms (α+εω+i∂∂̄u_ε)^{n−k}∧ω^k by O(ε), uniformly in the regularizing parameter. Theorem 1.2 then derives that a nef class with ∫_X αⁿ > 0 is big, using Lamari's Gauduchon-metric criterion, and hence that Tosatti-Weinkove's logarithmic-pole potentials exist.

Load-bearing premise

The argument assumes that the bounded mass property, which controls integrals of (ω+i∂∂̄f)ⁿ, also gives uniform-in-ε bounds on all mixed Monge-Ampère integrals that appear when the nef-regularizing term εω is expanded against powers of the fixed representative; without such uniform mixed-product control, the convergence steps in equations (2.1), (2.7), (2.12), and (2.13) do not go through.

Editorial extensions

If this is right

Conjecture 1 holds on all compact Hermitian manifolds with vol(ω) < ∞, which includes every Fujiki-class manifold.
A nef class with ∫_X αⁿ > 0 is big under the bounded mass property, so it contains a Kähler current.
The Tosatti-Weinkove conjecture holds: for any prescribed points xᵢ and positive weights τᵢ with ∑τᵢⁿ < ∫_X αⁿ, there exists an α-psh function with logarithmic poles of weights τᵢ at xᵢ.
The results give a route to solving complex Monge-Ampère equations on non-Kähler Hermitian manifolds with bounded mass, extending the Calabi-Yau theorem beyond the Kähler and Fujiki settings.
Because the bounded mass property is invariant under bimeromorphic changes, the theorems transfer across bimeromorphic models of the manifold.

Reading between the lines

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

The proof's crucial unstated premise is that bounded mass of the Hermitian metric gives uniform-in-ε control of every mixed Monge-Ampère term in the expansion; if that control fails, the O(ε) arguments in equations (2.1), (2.7), (2.12), and (2.13) collapse, so the uniform mixed-product bounds are effectively an additional hypothesis.
Equation (2.13) claims an O(ε²) estimate while the displayed computation only shows O(ε); the subsequent domination step only needs the leading term to be controlled for small ε, so the gap is probably fixable, but it marks the place where the argument relies on an estimate not fully written.
A natural strengthening would replace the full bounded mass property by boundedness of the finitely many mixed products αᵖ∧ω^{n−p} or of the p-th Monge-Ampère masses; the present proof requires all of them.
If the Kołodziej-Tosatti identity can be established without bounded mass, the same steps would prove the Demailly-Păun and Tosatti-Weinkove conjectures unconditionally; conversely, a nef class satisfying the identity on a manifold with unbounded mass would show the condition is not necessary.

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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

Here's my take on Lei Zhang's note. The paper tries to prove the Kolodziej-Tosatti conjecture on compact Hermitian manifolds with bounded mass property, and pulls in Demailly-Paun and Tosatti-Weinkove as consequences. The bounded mass property is the right hypothesis—it's the condition Guedj-Lu and Boucksom-Guedj-Lu have been using to solve related Monge-Ampere problems. And the author is candid that the Demailly-Paun part may already be in Boucksom-Guedj-Lu [4]. So the only genuinely new claim is Theorem 1.1.

The problem is that the proof for Theorem 1.1 has a systematic gap. In equations (2.1) and (2.7), the bounded mass property is invoked to control mixed integrals like ∫(2ω+i∂∂bar u)^{n-k}∧ω^k. The definition of bounded mass only bounds top-degree integrals ∫(ω+i∂∂bar f)^n. When ω is not Kähler, Stokes' theorem doesn't reduce mixed integrals to the top one: i∂∂bar f is not cohomologically trivial against ω, and a pointwise bound on the product of eigenvalues doesn't control the lower elementary symmetric functions. So the O(ε) expansions are unsupported.

The same issue appears in Theorem 1.2. Equation (2.13) drops a first-order term n(n-1)ε∫ω∧α^{n-2}∧i∂∂bar(uε+ψε), which is generally O(ε), not O(ε^2). The claimed O(ε^2) requires a uniform bound on that mixed integral that is exactly what is missing. The alternative proof in Section 3 makes the same leap.

This is load-bearing, not a minor gap. The note is short and clearly written, and the author knows the literature, but the main theorem is not proven as written. If the mixed-integral bounds can be established by other means—possibly using the full strength of Boucksom-Guedj-Lu—the conjecture likely goes through, but this paper doesn't supply them.

Who is this for: people working on non-Kähler geometry and Monge-Ampere equations. I wouldn't cite it right now, and I wouldn't put it in a reading group without heavy caveats. But the question matters and the strategy is not crazy, so I'd send it to a careful referee with a request to focus on (2.1), (2.7), and (2.13). Major revision, not desk reject.

Referee Report

4 major / 4 minor

Summary. The manuscript claims to prove the Kołodziej–Tosatti conjecture on Morse-type integrals of nef (1,1)-classes on a compact Hermitian manifold satisfying the bounded mass property, and to derive the Demailly–Păun and Tosatti–Weinkove conjectures under the same hypothesis. The proof follows the Kołodziej–Tosatti strategy: perturb a nef class [α] to α+εω, use envelope and Monge–Ampère equation arguments, and expand integrals in powers of ε; the key repeated step is the assertion that the bounded mass property makes all mixed Monge–Ampère integrals O(ε) uniformly. Section 3 gives an alternative proof of Theorem 1.2 via Demailly–Păun mass concentration. The central estimates are asserted without proof, and one displayed expansion is visibly wrong, so the paper is currently a sketch rather than a complete proof.

Significance. If the main estimates were correct, Theorem 1.1 would resolve a conjecture of Kołodziej and Tosatti in the bounded-mass setting, and Theorems 1.2 would subsume or complement a line of partial results by Chiose, Popovici, Nguyen, Li–Wang–Zhou, and Guedj–Lu. The paper’s strategy is attractive and its use of the recent bounded-mass theory of Boucksom–Guedj–Lu is timely. However, the load-bearing convergence arguments are not established: the bounded mass property is used to justify uniform bounds on mixed Monge–Ampère integrals that it does not directly provide, and Eq. (2.13) drops a first-order term that is generically O(ε), not O(ε²). The significance is therefore conditional on a substantial repair.

major comments (4)

[Section 2.1, Eq. (2.1)] The step ‘by the bounded mass property and Stokes’ theorem’ is not justified. The bounded mass property controls sup_f ∫_X (ω+i∂∂f)^n for f with ω+i∂∂f>0; it does not by itself bound ∫_X (α+i∂∂u_ε)^{n−k}∧ω^k, and α+i∂∂u_ε is not a positive form (it satisfies only α+i∂∂u_ε ≥ −εω). Moreover, the pointwise inequality used to pass from α+i∂∂u_ε to 2ω+i∂∂u_ε is not a valid wedge inequality for forms with mixed signature: writing η=α+i∂∂u_ε and γ=2ω+i∂∂u_ε, the relation η≤γ does not imply η^{n−k}∧ω^k ≤ γ^{n−k}∧ω^k pointwise. A repair would require introducing P=α+εω+i∂∂u_ε>0, writing η=P−εω, and applying mixed-discriminant/Gårding inequalities; this argument is absent. Since (2.2) and (2.3) rest on (2.1), Step 1 of Theorem 1.1 is not established.
[Section 2.1, Eqs. (2.5)–(2.7)] Equation (2.7) is justified by the phrase ‘similar calculation of (2.1)’, but this repeats the same unproved mixed-integral bound, now for the C^{1,1} envelope h_ε. The applicability of Berman’s Proposition 3.1(iii) in the non-Kähler Hermitian setting is also not explained; even granting that comparison, the limsup lower bound (2.7) is exactly where the O(ε) expansion is needed. As written, the proof of the converse inequality (2.4) is incomplete.
[Section 2.2, Eq. (2.13)] The expansion of nε∫_X ω∧~α_ε^{n−1} omits the first-order term involving i∂∂(u_ε+ψ_ε). Expanding (α+εω+i∂∂v)^{n−1} with v=u_ε+ψ_ε gives a term n(n−1)ε∫_X ω∧α^{n−2}∧i∂∂v. Because ∂ω≠0, Stokes’ theorem does not make this integral vanish, and it is generically O(1); hence this contribution is O(ε), not O(ε²). The displayed formula also has typographical corruption (ω_X, missing binomial coefficients, garbled summation ranges), but the substantive issue is the dropped term. Consequently inequality (2.11) is unsupported as written. Even if one replaces O(ε²) by O(ε), the proof must show that the O(ε) constants are independent of the Gauduchon metric ω_G, since the ε in (2.11) must be chosen before ω_G is known; this uniformity is not established.
[Section 3, Eq. (3.6) and Step 3] The alternative proof of Proposition 3.2 relies on the assertion that the bounded mass property implies ∫_X ~α_ε^{n−p}∧ω_ε^p = ∫_X α^{n−p}∧ω^p + O(ε) ≤ M. This is the same mixed-integral identity, now with a family of metrics ω_ε depending on ε. The bounded mass property for ω gives no direct control of mixed integrals of ~α_ε against ω_ε, and the displayed equality is not a cohomological identity when ω and ω_ε are not closed. This also undermines the uniform mass bound for the weak limits T used in Step 3. The section therefore does not provide a valid alternative proof without substantial additional estimates.

minor comments (4)

[Throughout] The title, abstract, and body contain the corrupted string ‘Ko/suppress lodziej’; this should be ‘Kołodziej’.
[Section 2.2, Eq. (2.13)] The display is typographically corrupted: ‘ω_X ∧’ is unexplained, the wedge powers are misprinted, and binomial coefficients are missing from the summation.
[Section 2.2] The nef condition gives α+i∂∂u_ε ≥ −εω, so α_ε=α+εω+i∂∂u_ε is only nonnegative, not strictly positive; the proof silently perturbs to achieve the strict positivity used in Lemma 2.1 and the Monge–Ampère equation (2.9).
[Introduction] ‘A classes [α] ∈ H^{1,1}_{BC}(X;R)’ should be ‘A class [α] …’; the notation psh(X, α+εω) is used without definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bounded-mass property is an external hypothesis on ω, the target identity about nef α is never an input, and all load-bearing citations are independent external work; the flagged defects (uniform mixed-integral bounds in Eqs. (2.1)/(2.7)/(2.12)/(2.13), O(ε²) in (2.13)) are correctness gaps, not self-referential reductions.

full rationale

The derivation chain is not circular. The bounded mass property vol(ω)=sup_f∫_X(ω+√-1∂∂̄f)^n<∞ is an external hypothesis on the Hermitian metric ω, while the target results concern arbitrary nef classes [α], and the conjectured identity ∫_X α^n = inf_u ∫_{X(α+√-1∂∂̄u,0)}(α+√-1∂∂̄u)^n never appears as an input to any step. Theorem 1.1 splits into Demailly's inequality (2.4), re-derived via Berman's envelope result [1, Prop. 3.1(iii)], and the perturbation estimate (2.1)-(2.3) based on the bounded-mass hypothesis; no parameter is fitted to a subset and renamed a prediction, and no ansatz or uniqueness theorem is imported from the author's own work (the bibliography has no self-citations). The load-bearing results [1, 2, 4, 11, 18, 19, 23, 27] are independent external theorems, and the one case where a cited theorem equals a stated result (Theorem 1.2 is the nef-class case of Boucksom's conjecture, already credited to BGL [4, Theorem C] in the introduction) is acknowledged in the text, making it a novelty/attribution matter rather than a circular reduction. The reviewer-flagged gaps — the unproved premise that vol(ω)<∞ implies uniform-in-ε bounds on the mixed integrals in (2.1), (2.7), (2.12), (2.13) and §3 Step 1 (a premise not equivalent to vol(ω)<∞ by construction, since 2ω+√-1∂∂̄u_ε is not of the form ω+√-1∂∂̄f, and a bounded top symmetric function does not bound the lower ones), and the O(ε²) claim in (2.13) that drops the first-order term n(n-1)ε∫_X ω∧α^{n-2}∧√-1∂∂̄(u_ε+ψ_ε), which is only O(ε) because ∂ω≠0 blocks Stokes — are missing estimates or unjustified order-of-magnitude assertions, i.e. correctness risk, not steps that presuppose their own conclusions. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The central assumption is bounded mass property; the paper also implicitly assumes mixed Monge-Ampere integral bounds under that property, which are not explicitly stated as an axiom.

assumptions (6)

domain assumption The manifold (X,omega) satisfies the bounded mass property vol(omega) < infinity.
Theorems 1.1 and 1.2 are stated under this hypothesis in the introduction and used throughout the proofs.
ad hoc to paper Bounded mass property implies uniform bounds on mixed Monge-Ampere integrals such as ∫(omega+i∂∂bar f)^j ∧ omega^(n-j).
Used implicitly in Eqs. (2.1), (2.7), (2.12) and (2.13) to claim O(epsilon) terms; not stated or proved in the paper.
standard math C^{1,1} regularity of plurisubharmonic envelopes and the support property of their Monge-Ampere measures.
Invoked in Step 2 of Theorem 1.1 via [9, 24] and Berman's Proposition 3.1(iii).
standard math Solvability of complex Monge-Ampere equations on compact Hermitian manifolds (Tosatti-Weinkove).
Used in Eq. (2.9) to construct alpha_tilde_epsilon as a solution of a Monge-Ampere equation.
standard math Lamari's lemma on existence of positive currents via the Gauduchon cone.
Used as Lemma 2.1 for the bigness criterion in the proof of Theorem 1.2.
standard math Demailly-Paun mass concentration lemma.
Used as Lemma 3.1 in the alternative proof of Theorem 1.2 in Section 3.

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Cite this review

Pith. "Pith review of Ko{\l}odziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property." pith.science (2026). https://pith.science/paper/B5KIV3W3

@misc{pith2026241201586,
  author       = {Pith},
  title        = {Pith review of: Ko\lodziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B5KIV3W3}},
  note         = {Machine review of arXiv:2412.01586}
}

abstract

In this note, we show a conjecture of Ko{\l}odziej-Tosatti about Morse-type integrals in nef $(1,1)$ classes on compact Hermitian manifold with bounded mass property. As a consequence, we give positive answers to Demailly-P\u{a}un's conjecture and Tosatti-Weinkove's conjecture when compact Hermitian manifold with bounded mass property.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

Lu Volumes of Bott-Chern classes

S` ebastien Boucksom, Vincent Guedj, Chinh H. Lu Volumes of Bott-Chern classes. priprint, arXiv:2406.01090

arXiv

[2] [1]

Bergman kernels and equilibrium measures for line bundles o ver projective manifolds

Robert J Berman. Bergman kernels and equilibrium measures for line bundles o ver projective manifolds. Amer. J. Math. 131 (2009), no. 5, 1485-1524

work page 2009

[3] [2]

Internat

S` ebastien Boucksom.On the volume of a line bundle. Internat. J. Math. 13 (2002), no. 10, 1043-1063

work page 2002

[4] [3]

The pseudo-eﬀective cone of a compact K¨ ahler manifold and varieties of negative Kodair a dimension

S` ebastien Boucksom, Jean-Pierre Demailly, Mihai P˘ aun, Thomas Peternell. The pseudo-eﬀective cone of a compact K¨ ahler manifold and varieties of negative Kodair a dimension. J. Algebraic Geom. 22 (2013), no. 2, 201-248

work page 2013

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