{"id":"088a2774-73c2-4f2e-9768-2723841953a9","arxiv_id":"2412.01586","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves the Kolodziej-Tosatti conjecture on Morse-type integrals for nef classes on compact Hermitian manifolds with bounded mass property.","lead":"This paper proves a conjecture of Kolodziej and Tosatti about Morse-type integrals in nef classes, assuming the manifold has a bounded mass property. It also derives two related bigness conjectures in that setting, though one was already covered by earlier work.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof rests on an unproved uniform bound for mixed Monge-Ampere integrals; Eq. (2.13) visibly drops a first-order i∂∂bar term, so the O(ε)-convergence arguments in Theorems 1.1 and 1.2 are not established.","rationale":"I agree with the reader that the decisive gap is the uniform control of mixed Monge-Ampere integrals. The paper's stated bounded mass property only controls top-degree volumes, while the proofs of Theorem 1.1 and Theorem 1.2 repeatedly need uniform bounds on mixed integrals of the form ∫(2ω+i∂∂bar φ)^{n-k}∧ω^k. On a non-Kähler Hermitian manifold, the exact terms i∂∂bar φ do not integrate to zero against ω, and the text supplies no Stokes or comparison argument establishing the required estimates. Equation (2.13) makes the gap concrete: a first-order i∂∂bar(uε+ψε) term is omitted, and only an O(ε) estimate is visible, not the claimed O(ε^2). This is a genuine incompleteness in the central argument, not merely a failure of exposition. I do not see reason to reject the theorem itself: the overall strategy is plausible, the envelope regularity and Lamari's lemma are standard tools, and known results on bounded mass property may supply the missing estimates if properly cited and proved. The reader's CONDITIONAL verdict is therefore appropriate, and my read does not move it.","tokens_in":9232,"tokens_out":23831,"duration_ms":223823,"concrete_test":"Recompute Eq. (2.13) with v=uε+ψε: expand (α+i∂∂bar v+εω)^{n-1} and isolate the term n(n-1)ε∫_X ω∧α^{n-2}∧i∂∂bar v. Use Stokes to rewrite it as n(n-1)ε∫_X v ∂∂bar(ω∧α^{n-2}) up to terms involving ∂ω; on a Hermitian non-Kähler manifold this integral has no reason to vanish. The decisive check is to exhibit one compact Hermitian non-Kähler example (e.g. an Iwasawa or Hopf manifold), a closed form α, and smooth functions uε,ψε satisfying (2.9), for which the sequence ∫_X ω∧α^{n-2}∧i∂∂bar v is unbounded while ∫_X (2ω+i∂∂bar v)^n stays bounded. If such a sequence exists, the uniform mixed-bound premise behind (2.1), (2.7) and (2.13) fails; if it cannot exist, the proof must supply the missing estimate explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing gap is the unproved assertion that the bounded mass property (vol(ω)<∞) yields uniform-in-ε bounds for the mixed Monge-Ampere integrals used in the O(ε) expansions. As defined in the paper, vol(ω) only bounds ∫(ω+i∂∂bar f)^n for ω+i∂∂bar f>0; it does not by itself bound ∫(2ω+i∂∂bar φ)^{n-k}∧ω^k when ω is not Kähler. The exact term i∂∂bar φ is not cohomologically trivial against ω, and pointwise a bound on the top elementary symmetric function does not control the lower ones (e.g. eigenvalues (R,1/R,1,...,1) give bounded determinant but unbounded mixed terms). Equations (2.1) and (2.7) in the proof of Theorem 1.1 both depend on this unstated premise, so the claimed O(ε) convergence and the limsup argument are unsupported. In Theorem 1.2 the same problem is sharper: Eq. (2.13) expands ~α_ε^{n-1} but drops the first-order term in i∂∂bar(uε+ψε). Keeping it gives n(n-1)ε∫_X ω∧α^{n-2}∧i∂∂bar(uε+ψε), which is generally O(ε), not O(ε^2), because ∂ω≠0 prevents Stokes from killing it. The claimed O(ε^2) in (2.13) is therefore not established, and inequality (2.11), on which (2.8) rests, lacks a proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":45,"tokens_out":14750,"duration_ms":235706,"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":[{"comment":"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":"Section 2.1, Eq. (2.1)"},{"comment":"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":"Section 2.1, Eqs. (2.5)–(2.7)"},{"comment":"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":"Section 2.2, Eq. (2.13)"},{"comment":"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.","section":"Section 3, Eq. (3.6) and Step 3"}],"minor_comments":[{"comment":"The title, abstract, and body contain the corrupted string ‘Ko/suppress lodziej’; this should be ‘Kołodziej’.","section":"Throughout"},{"comment":"The display is typographically corrupted: ‘ω_X ∧’ is unexplained, the wedge powers are misprinted, and binomial coefficients are missing from the summation.","section":"Section 2.2, Eq. (2.13)"},{"comment":"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).","section":"Section 2.2"},{"comment":"‘A classes [α] ∈ H^{1,1}_{BC}(X;R)’ should be ‘A class [α] …’; the notation psh(X, α+εω) is used without definition.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The reader’s report accurately identifies the main issue: the unproved uniform control of mixed Monge–Ampère integrals. I agree with the conditional assessment in the stress-test note. The error in Eq. (2.13) is concrete and visible, but I do not think it is necessarily fatal: with Gårding/mixed-discriminant bounds and a careful treatment of uniformity in ω_G, the strategy could in principle be repaired. The manuscript is not ready in its current form, and the amount of missing detail is substantial; a major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThis 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.\n\nWho 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.","headline":"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.","tokens_in":10140,"tokens_out":11063,"would_cite":false,"duration_ms":88308,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32W20","53C55","32U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["bounded mass property","nef class","Kołodziej-Tosatti conjecture","Morse-type integrals","complex Monge-Ampère equation","big class","Kähler current","Hermitian manifold"],"falsifier":"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.","tokens_in":9040,"feed_emoji":"📐","tokens_out":6183,"duration_ms":49431,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bounded mass property settles Kołodziej-Tosatti conjecture","feed_subtitle":"On Hermitian manifolds with finite Monge-Ampère volume, nef classes obey the Morse-type volume identity.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Kołodziej-Tosatti state Conjecture 1 and prove special cases; the present proof follows their strategy.","marker":"[18]"},{"why":"Guedj-Lu introduce and study the bounded mass property and prove it is invariant under bimeromorphic maps.","marker":"[15]"},{"why":"Tosatti-Weinkove solve the complex Monge-Ampère equation on compact Hermitian manifolds, used in Step 1 of Theorem 1.2.","marker":"[27]"},{"why":"Lamari's Gauduchon-metric criterion (Lemma 2.1) is used to prove that the nef class is big.","marker":"[19]"},{"why":"Tosatti's Theorem 1.3 is applied to deduce Conjecture 3 from bigness, and his observation motivates applications to Monge-Ampère equations.","marker":"[23]"},{"why":"Berman's Proposition 3.1(iii) gives the identity used for the Monge-Ampère mass of the envelope h_ε in (2.5).","marker":"[1]"},{"why":"Demailly-Păun's mass concentration method and their Conjecture 2 frame the alternate proof in Section 3.","marker":"[14]"},{"why":"Tosatti-Weinkove's conjecture and their partial results are the target of Theorem 1.2.","marker":"[28]"}],"fun_headline_variants":["Bounded mass property clinches Morse-type equality","Mass bound yields equality of Morse integrals","One mass condition, three conjecture proofs","Hermitian mass bound forces volume identity","Morse integral identity under bounded mass"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bounded mass property clinches Morse-type equality","Mass bound yields equality of Morse integrals","One mass condition, three conjecture proofs","Hermitian mass bound forces volume identity","Morse integral identity under bounded mass"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1656,"prompt_tokens":815,"completion_tokens":841,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":778}},"tokens_in":431,"tokens_out":841,"duration_ms":8591,"temperature":1.0,"reasoning_tokens":778,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:28:56.137967+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Morse-type integrals on non-K¨ ahler manifolds.Pure Appl","cited_arxiv_id":null,"evidence_quote":"Kołodziej-Tosatti state Conjecture 1 and prove special cases; the present proof follows their strategy."},{"cited_title":"Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Amp` ere volumes","cited_arxiv_id":null,"evidence_quote":"Guedj-Lu introduce and study the bounded mass property and prove it is invariant under bimeromorphic maps."},{"cited_title":"The complex Monge-Amp` ere equation on compact Hermitian ma ni- folds","cited_arxiv_id":null,"evidence_quote":"Tosatti-Weinkove solve the complex Monge-Ampère equation on compact Hermitian manifolds, used in Step 1 of Theorem 1.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lamari's Gauduchon-metric criterion (Lemma 2.1) is used to prove that the nef class is big."},{"cited_title":"The Calabi-Yau theorem and K¨ ahler currents","cited_arxiv_id":null,"evidence_quote":"Tosatti's Theorem 1.3 is applied to deduce Conjecture 3 from bigness, and his observation motivates applications to Monge-Ampère equations."},{"cited_title":"Bergman kernels and equilibrium measures for line bundles o ver projective manifolds","cited_arxiv_id":null,"evidence_quote":"Berman's Proposition 3.1(iii) gives the identity used for the Monge-Ampère mass of the envelope h_ε in (2.5)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demailly-Păun's mass concentration method and their Conjecture 2 frame the alternate proof in Section 3."},{"cited_title":"Plurisubharmonic functions and nef classes on complex mani folds","cited_arxiv_id":null,"evidence_quote":"Tosatti-Weinkove's conjecture and their partial results are the target of Theorem 1.2."}],"review_version":1}