{"id":"d6f69c67-ea50-4e86-9857-dbfc8a8d5dbf","arxiv_id":"2412.11547","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A weak convergence criterion for non-pluripolar Monge-Ampere measures is proved under only a bounded subsolution, yielding solvability for L1 densities and an L-infinity estimate.","lead":"This paper proves a criterion for weak convergence of complex Monge-Ampere measures on compact Hermitian manifolds when the reference class has only a bounded subsolution, and uses it to solve degenerate Monge-Ampere equations with L1 right-hand side. A reader interested in degenerate complex Monge-Ampere equations or the nef and pseudoeffective cones will find a broader setting than earlier semipositive results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.26(2)⇒(1) invokes a Kähler-only theorem ([DDL23, Thm 3.3]) without a Hermitian analogue; this step is needed to show u∈E in Lemma 3.5, so the proof of Theorem 1.1 has a load-bearing gap.","rationale":"I agree with the reader that the transport of [DDL23] results into the Hermitian setting is the main risk, but I locate the sharpest instance in Proposition 2.26 rather than in Theorem 3.2 or Theorem 3.13. At Theorem 3.2, the paper cites both [DDL23, Thm 2.6] and [LLZ24, Thm 2.6]; at Theorem 2.20, it also cites [LLZ24, Thm 2.6] parenthetically. Theorem 3.13 uses [DDL23, Lemma 2.5] without a Hermitian citation, but Theorem 3.13 is not used in the proof of the central Theorem 1.1. In contrast, Proposition 2.26 is directly in the main path: Lemma 3.5 uses it to convert the existence of the envelope P(Au) into the full-mass property of u. The (2)⇒(1) direction relies on [DDL23, Theorem 3.3], whose statement and hypotheses are Kähler-specific in the source paper. The paper gives no statement, proof, or Hermitian reference for the needed inequality at that exact spot. Because the main contribution is precisely to weaken the reference to a bounded β-psh subsolution, one cannot assume the Kähler theorem transfers verbatim. The comparison principle cited from [LWZ24a] may be sufficient to repair the gap, but the repair is not written. This is an addressable but genuine gap in the proof of the central claim; the verdict CONDITIONAL is appropriate.","tokens_in":28707,"tokens_out":18129,"duration_ms":141740,"concrete_test":"Give a complete proof of the inequality used in Proposition 2.26(2)⇒(1) for compact Hermitian (X,ω) with bounded β-psh ρ: for u∈PSH(X,β+dd^cρ) with P_{β+dd^cρ}(Au)∈PSH, prove ∫⟨(β+dd^cρ+dd^c u)^n⟩ ≥ (1-1/A)^n∫β^n without invoking [DDL23, Thm 3.3], e.g., by a direct argument using the Hermitian comparison principle [LWZ24a, Prop 3.11]. If the inequality fails, the full-mass conclusion in Lemma 3.5 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is proved as Theorem 3.1. In that proof, Lemma 3.5 concludes u∈E(X,β+dd^cρ) from the statement P_{β+dd^cρ}(Au)∈PSH(X,β+dd^cρ) for all A≥1, citing Proposition 2.26. The direction (2)⇒(1) of Proposition 2.26 is where this implication is justified. Its proof uses [DDL23, Theorem 3.3] to assert that, because A^{-1}P_{β+dd^cρ}(Au) ≤ u, one has ∫⟨(β+dd^cρ+dd^c u)^n⟩ ≥ ∫⟨(β+dd^cρ+dd^c A^{-1}P(Au))^n⟩ ≥ (1-1/A)^n∫β^n. [DDL23] is a compact Kähler manifold paper, and no Hermitian analogue is cited at this step (unlike Theorem 3.2, which also cites [LLZ24, Thm 2.6]). Since the whole point of the paper is to allow β that is not semipositive and ρ merely bounded, it is not automatic that the relative pluripotential theory of [DDL23] carries over. If this inequality is false in the Hermitian setting, Lemma 3.5 cannot prove u∈E, and the chain (Lemma 3.6 → Theorem 3.2 → Theorem 3.1) collapses. This is the single most load-bearing gap: it is a black-box import of a Kähler result in a place where the paper's weaker hypotheses matter most.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies weak continuity of non-pluripolar complex Monge-Ampere operators on compact Hermitian manifolds. The main result, Theorem 1.1 (= Theorem 3.1), states: if (X, omega) is compact Hermitian, beta is a smooth closed real (1,1)-form with positive total mass, rho is a bounded beta-psh function, u_j in E(X, beta+dd^c rho) satisfy a uniform bound by a non-pluripolar measure mu, and u_j converges to u in L^1, then u is in E(X, beta+dd^c rho), u_j converges to u in capacity, and the non-pluripolar Monge-Ampere measures converge weakly. The proof follows the strategy of Alehyane--Lu--Salouf, using a new envelope P_{beta+dd^c rho}. Applications include solving a degenerate Monge-Ampere equation with L^1 density (Corollary 1.3) and an L^infty estimate for finite Radon measures (Theorem 1.5).","tokens_in":28977,"tokens_out":2580,"duration_ms":23409,"significance":"If correct, the main theorem substantially extends known weak-continuity results for non-pluripolar Monge-Ampere operators: it removes semipositivity of beta, allowing any closed real (1,1)-form with a bounded psh weight, and it works on compact Hermitian manifolds rather than only Kahler manifolds. The paper also provides a useful envelope calculus for the relative class beta+dd^c rho, with several lemmas stated in a parameter-free way. The applications to degenerate equations with L^1 densities and to L^infty estimates are natural and potentially useful. The manuscript does not rely on fitted parameters or on assumptions that force the conclusion; its main debt is to prior work, which is cited clearly.","major_comments":[{"comment":"This is the most load-bearing gap. The proof of (2) => (1) uses [DDL23, Theorem 3.3] to assert that, from A^{-1}P_{beta+dd^c rho}(Au) <= u, one obtains the lower bound int <(beta+dd^c rho+dd^c u)^n> >= (1-1/A)^n int beta^n. However, [DDL23] is a compact Kahler paper, and the manuscript does not cite a Hermitian analogue at this step. Since the whole point of the paper is to allow beta that is not semipositive and rho merely bounded, it is not automatic that this relative pluripotential comparison theorem carries over to the Hermitian setting. Lemma 3.5 uses exactly this implication to conclude u in E(X, beta+dd^c rho), and the proof of Theorem 3.1 depends on Lemma 3.5. The authors should either provide a proof of this inequality in the Hermitian setting or cite a published Hermitian version.","section":"Proof of Theorem 3.1"},{"comment":"The proof invokes [DDL23, Lemma 2.5] to pass from v_{j,k} -> v_j in capacity to convergence of v_{j,k} (beta+dd^c rho+dd^c v_{j,k})^n to v_j (beta+dd^c rho+dd^c v_j)^n, and later applies the same lemma to the limit v. No Hermitian analogue is cited for this lemma, unlike Theorem 3.2, where [LLZ24, Theorem 2.6] is cited in parallel. Since Theorem 3.13 is used in the proof of Theorem 3.8 and Theorem 3.9, the lack of justification for the Hermitian validity of [DDL23, Lemma 2.5] leaves a gap in the auxiliary convergence-in-capacity results.","section":"Corollary 2.28"},{"comment":"There are many unresolved 'Remark ??' references, including in the proof of Proposition 2.26 (lines near equations (6) and (8)), in Corollary 2.28, and in Theorem 2.31. These placeholders make it impossible for the reader to verify the cited properties, such as the mass concentration of the envelope on the contact set and the comparison principle for the relative class. The proofs of Proposition 2.26 and Theorem 2.31 are directly affected, so these references must be resolved before the manuscript is complete.","section":"Throughout Section 2"},{"comment":"The proof of Theorem 3.1 is concluded with: 'The proof follows from Theorem 3.3 in [ALS24] using contradiction and applying Lemma 3.6.' This is not a self-contained argument. The contradiction step, in particular, is not described, and the reader cannot check how Lemma 3.6 combines with the uniform measure bound and L^1 convergence to yield the weak convergence of the Monge-Ampere measures. This is load-bearing for the main theorem and should be written out in detail.","section":"Section 3"}],"minor_comments":[{"comment":"The statement of Theorem 3.1 says '<(beta+dd^c u_j)^n>' converges weakly to '<(beta+dd^c u)^n>', but the theorem is formulated with u_j, u in PSH(X, beta+dd^c rho); the measures should be <(beta+dd^c rho+dd^c u_j)^n> and <(beta+dd^c rho+dd^c u)^n>. This appears to be a typo, since the proof uses the relative measures, but it should be corrected.","section":"Section 3"},{"comment":"In the proof of Lemma 2.11, the phrase 'h is bounded above by C outside a pluripolar set P' is slightly imprecise: since h is bounded above on X, one can simply say h <= C on X. The subsequent integral over B_0 \\ P is fine, but the wording could be clarified.","section":"Section 2.2"},{"comment":"The first sentence reads 'Let mu be a positive Radon measure vanishingand assume mu is absolutely continuous...' The word 'vanishing' is incomplete; presumably it should say 'vanishing on pluripolar sets'. This should be fixed.","section":"Section 4"},{"comment":"The authors state 'We can't make use of this fact in this paper as it is still hard to verify the uniform boundedness in order to use Theorem 3.9 when we are solving equations, but we believe that it will be useful in the future.' This is an honest limitation, but as written it suggests that the L^infty estimate is not applied to the existence theorem in this paper. If so, the relationship between Theorem 5.2 and the earlier sections should be stated more clearly in the introduction.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and the overall strategy is coherent, but the manuscript currently has a load-bearing gap: a Kahler-only theorem from [DDL23] is used to prove Proposition 2.26(2)=>1, and that proposition underpins Lemma 3.5 and hence Theorem 3.1. The unresolved 'Remark ??' placeholders and the terse final step of Theorem 3.1 make it hard for a reader to check the main proof. I would be willing to look at a revised version that either proves the Hermitian analogue or cites a published Hermitian version of the needed comparison theorem, and that completes the missing details."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe short version: this is a real weakening of the Alehyane–Lu–Salouf criterion, replacing semipositive β and a special Hermitian metric with a bounded β-psh subsolution, and the new envelope P_{β+ddcρ} is a genuine technical contribution. But the proof of Proposition 2.26(2)⇒(1) imports a Kähler-only theorem from [DDL23] without a Hermitian analogue, and that step is what gets you u ∈ E in Lemma 3.5. If it doesn't go through, Lemma 3.6 and Theorem 3.1 lose their foundation.\n\nWhat's genuinely new: Theorem 1.1 extends weak continuity and the domination machinery to the full Hermitian big-class setting under a mere bounded subsolution. The envelope P_{β+ddcρ} is built carefully for ρ that is only bounded, and the mass-concentration properties (Theorems 2.16, 2.20, Corollary 2.21) look right. Corollary 1.3, solving the e^{λu} equation with L^1 density, improves the L^p (p>1) result in [LWZ24a]. The L∞ estimate in Section 5 follows Guedj–Lu but is clean.\n\nThe main problem: Proposition 2.26(2)⇒(1) uses [DDL23, Thm 3.3] to assert\n∫⟨(β+ddcρ+ddc u)^n⟩ ≥ (1−1/A)^n∫β^n.\nThat theorem is for compact Kähler manifolds. The paper gives no Hermitian counterpart, and the boundedness of ρ is exactly the condition that makes the relative theory non-trivial. This is not a minor citation gap; it is the bridge from P_{β+ddcρ}(Au) ∈ PSH to u ∈ E. The authors cite [LLZ24] elsewhere as a Hermitian companion, but not here.\n\nMinor issues: the proof of Theorem 3.1 ends by referring to [ALS24, Thm 3.3] without written details, and several \"Remark ??\" cross-references are unresolved. Those are cosmetic.\n\nBottom line: this is serious work, the envelope construction is a real step forward, and I suspect the gap is fixable. But as submitted, the central proof has a load-bearing hole. I would send it to a good referee, asking them to check Proposition 2.26 and whether a Hermitian version of the [DDL23, Thm 3.3] argument can be supplied. If that's patched, I'd cite it.","headline":"A genuine extension of the weak-continuity criterion to bounded-subsolution Hermitian classes, but the proof leans on a Kähler-only import at the load-bearing step.","tokens_in":29636,"tokens_out":2326,"would_cite":true,"duration_ms":22516,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32W20","32U05","32U40","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bounded subsolution forces weak continuity of Monge-Ampere measures on Hermitian manifolds.","keywords":["non-pluripolar complex Monge-Ampere operator","convergence in capacity","compact Hermitian manifold","degenerate complex Monge-Ampere equation","L1-density","quasi-plurisubharmonic envelope","full mass class","weak convergence"],"falsifier":"Find a compact Hermitian manifold and a class $\\beta$ with a bounded $\\beta$-psh potential for which a sequence $u_j \\in E(X,\\beta)$ with $\\langle(\\beta + dd^c u_j)^n\\rangle \\leq \\mu$ converges in $L^1$ to $u$ but not in capacity; Theorem 1.1 says no such example exists. A more local test is to check whether the inequality used in Theorem 3.2, that the convergence-in-capacity limit of the measures has total mass at least the limsup of the total masses, fails when $d\\omega \\neq 0$; a counterexample there would pinpoint exactly which imported step breaks.","tokens_in":28388,"feed_emoji":"📐","tokens_out":9370,"duration_ms":69449,"temperature":0.7,"pith_summary":"This paper establishes a criterion for weak convergence of non-pluripolar complex Monge-Ampere measures on compact Hermitian manifolds. The main theorem says that if a sequence of $\\beta$-plurisubharmonic functions with full Monge-Ampere mass has its Monge-Ampere measures dominated by one fixed non-pluripolar Radon measure, then $L^1$ convergence forces convergence in capacity and weak convergence of the measures. The advance over earlier work is that the reference form $\\beta$ need not be semipositive; it only needs a bounded $\\beta$-plurisubharmonic function $\\rho$ to act as a subsolution. If the criterion is correct, the natural continuity theorem for the Monge-Ampere operator holds in the Hermitian setting under bounded-subsolution hypotheses, which is the input needed to solve degenerate Monge-Ampere equations with merely $L^1$ densities.","feed_headline":"Bounded subsolution forces Monge-Ampere weak continuity","feed_subtitle":"On compact Hermitian manifolds, L1 limits with controlled Monge-Ampere mass stay in the full mass class.","key_machinery":"The engine of the proof is a new envelope for the possibly non-smooth current $\\beta + dd^c\\rho$: $P_{\\beta + dd^c\\rho}(h) = \\sup\\{\\phi \\in PSH(X,\\beta + dd^c\\rho) : \\phi \\leq h \\text{ quasi-everywhere}\\}$, which the paper proves equals $P_\\beta(h+\\rho) - \\rho$. This envelope is the Hermitian, non-semipositive substitute for the classical Balayage envelope; it is monotone under decreasing limits, and its non-pluripolar Monge-Ampere measure is carried on the contact set $\\{P_{\\beta + dd^c\\rho}(h) = h\\}$. The envelope inequalities and the domination principle built from them let the authors pass from uniform control of Monge-Ampere mass to $L^1$-closeness of exponentials, then to convergence in capacity, and finally to weak convergence of measures.","core_discovery":"The central claim, stated as Theorem 1.1, is a compactness-continuity statement: on a compact Hermitian manifold $(X,\\omega)$, for a smooth closed real $(1,1)$-form $\\beta$ with $\\int_X \\beta^n > 0$ and a bounded $\\beta$-psh function $\\rho$, any sequence $u_j$ in the full mass class $E(X,\\beta)$ satisfying $\\langle(\\beta + dd^c u_j)^n\\rangle \\leq \\mu$ for a positive non-pluripolar Radon measure $\\mu$, with $u_j \\to u$ in $L^1$, must have $u \\in E(X,\\beta)$, have $u_j \\to u$ in capacity, and have $\\langle(\\beta + dd^c u_j)^n\\rangle \\to \\langle(\\beta + dd^c u)^n\\rangle$ weakly. The paper then uses this continuity to solve the degenerate equation $\\langle(\\beta + dd^c u)^n\\rangle = e^{\\lambda u}\\mu$ for $\\mu = f\\omega^n$ with $f \\in L^1$, and to prove an a priori $L^\\infty$ estimate for solutions when $\\mu$ is a finite Radon measure whose integrability class controls a compact family of normalized potentials.","pith_inferences":["An implicit consequence is that the obstruction to weak continuity is not the Hermitian character of $\\omega$ but the absence of a global bounded subsolution; searching for counterexamples would naturally focus on classes $\\beta$ with no bounded $\\beta$-psh function at all.","The $L^1$-density solvability might be iterable to build solutions for measures that are absolutely continuous with respect to the non-pluripolar Monge-Ampere measure of a reference potential, rather than with respect to $\\omega^n$.","The $L^\\infty$ bound's independence from Skoda-type uniform integrability could support a priori control in families where only compactness of normalized $\\beta$-psh potentials is available, as in degenerating or collapsing Hermitian classes.","A testable extension would be to relax the fixed domination $\\langle(\\beta + dd^c u_j)^n\\rangle \\leq \\mu$ to a sequence of measures with controlled exponential moments, since the exponential-weighting argument in the proof appears tolerant of such a change."],"forward_implications":["If Theorem 1.1 holds, then non-pluripolar Monge-Ampere measures are continuous under $L^1$ convergence whenever the measures are uniformly dominated by one non-pluripolar measure and the limit candidate is $\\beta$-psh.","The degenerate equation $\\langle(\\beta + dd^c u)^n\\rangle = e^{\\lambda u} f\\omega^n$ has a unique solution in the full mass class $E(X,\\beta)$ for any nonnegative $f \\in L^1(\\omega^n)$.","A finite positive Radon measure $\\mu$ satisfying $PSH(X,\\beta + dd^c\\rho) \\subset L^m(\\mu)$ for some $m>n$ and $\\mu(X)=\\int_X\\beta^n$ forces any bounded solution to have oscillation bounded by a constant depending only on $X$, $\\beta$, and an $L^m$-normalized constant $A_m(\\mu)$.","The same machinery yields, for any such measure $\\mu$, a unique constant $c>0$ and $u \\in E(X,\\beta)$ with $\\langle(\\beta + dd^c u)^n\\rangle = c\\mu$."],"supporting_citations":[{"why":"supplies the envelope-based strategy for weak convergence in the semipositive Hermitian case that this paper extends to arbitrary β with a bounded subsolution.","marker":"[ALS24]"},{"why":"provides the Kähler relative pluripotential facts (Theorem 2.6 and Lemma 2.5) used to justify integration against measures that converge in capacity.","marker":"[DDL23]"},{"why":"provides the Hermitian analogues of those convergence-in-capacity tools and the global pluripolarity lemma used in the envelope arguments.","marker":"[LLZ24]"},{"why":"defines the non-pluripolar operator and full mass class on Hermitian manifolds and supplies the bounded solution theorem for L^p densities used in the existence step.","marker":"[LWZ24a]"},{"why":"established the non-pluripolar product and the basic full mass class properties on which Definition 2.5 and the mass computations rest.","marker":"[BEGZ10]"},{"why":"supplies the envelope lemmas showing the Monge-Ampere measure of a β-psh envelope is carried on the contact set.","marker":"[GLZ19]"},{"why":"furnishes the envelope-based method for a priori L∞ estimates that Theorem 1.5 adapts to finite Radon measures.","marker":"[GL21]"},{"why":"provides the refined envelope estimate used in the proof of Theorem 1.5 and in removing boundedness in Theorem 2.20.","marker":"[GL22]"},{"why":"gives the prior Hermitian weak-convergence result for uniformly bounded functions used in Lemma 3.5 and Theorem 3.9.","marker":"[KN22]"},{"why":"supplies the local fine-topology comparison properties of the complex Monge-Ampere operator used throughout the preliminaries.","marker":"[BT87]"}],"fun_headline_variants":["Weak continuity from bounded psh subsolution","Monge-Ampere weak convergence via bounded subsolution","L1 limit keeps full mass class on Hermitian manifolds","Bounded psh functions control Monge-Ampere mass convergence","Degenerate MA equation solved by weak continuity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that two convergence-in-capacity facts proved for compact Kähler manifolds and quoted from the paper's reference [DDL23] remain valid on compact Hermitian manifolds when the metric form is not closed; if they need extra hypotheses there, the proof of continuity in Section 3 loses its foundation.","fun_headline_variants_meta":{"raw":{"variants":["Weak continuity from bounded psh subsolution","Monge-Ampere weak convergence via bounded subsolution","L1 limit keeps full mass class on Hermitian manifolds","Bounded psh functions control Monge-Ampere mass convergence","Degenerate MA equation solved by weak continuity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000557,"raw_usage":{"total_tokens":2661,"prompt_tokens":968,"completion_tokens":1693,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":1617}},"tokens_in":584,"tokens_out":1693,"duration_ms":10562,"temperature":1.0,"reasoning_tokens":1617,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:49:32.162120+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a compact Hermitian manifold and a class $\\beta$ with a bounded $\\beta$-psh potential for which a sequence $u_j \\in E(X,\\beta)$ with $\\langle(\\beta + dd^c u_j)^n\\rangle \\leq \\mu$ converges in $L^1$ to $u$ but not in capacity; Theorem 1.1 says no such example exists. A more local test is to check whether the inequality used in Theorem 3.2, that the convergence-in-capacity limit of the measures has total mass at least the limsup of the total masses, fails when $d\\omega \\neq 0$; a counterexample there would pinpoint exactly which imported step breaks.","supporting_citations":[],"review_version":1}