{"id":"0db592a4-d44f-45c4-962d-6f7c3ffcd0e4","arxiv_id":"2506.07336","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On compact Hermitian manifolds, degenerate Monge-Ampere equations are solvable for non-closed pseudoeffective (1,1)-forms with positive Bott-Chern volume and a bounded potential, with stability estimates.","lead":"This paper proves existence, uniqueness, and stability for degenerate complex Monge-Ampere equations on compact Hermitian manifolds, allowing the background form to be non-closed as long as a bounded subsolution and a positive volume condition hold. For specialists, it extends earlier closed-form results and gives partial answers to two conjectures in Hermitian pluripotential theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Domination principle for non-closed β is cited from [GL22] rather than proved; its compactness step is the load-bearing unverified input for Theorems 5.1–5.3.","rationale":"I read the main construction in good faith: the architecture is coherent, and most estimates are routine adaptations of [GL23, BGL24] to the non-closed class. The domination principle is the lynchpin: it is used for monotonicity of the approximating sequence, uniqueness, the L∞ oscillation bound, and the stability estimate. The reader's weakest_assumption identifies this same point. My independent check did not find a separate, more basic error in the Monge-Ampère flow; Theorem 6.1 contains an impossible constant condition (q<m with b<ε), but the intended estimate can be recovered from boundedness of the potentials, so it is a repair rather than a structural break. The real risk is the compactness step inside Proposition 2.2, which is delegated to a reference and may fail for non-closed β unless the mass upper bound is guaranteed. Since the central claims depend on this, a conditional verdict is justified: the paper should supply a self-contained proof of the domination principle for non-closed β, or a precise citation with hypotheses verified. I therefore keep the reader's CONDITIONAL verdict; the concern does not by itself force rejection, but it prevents acceptance as written.","tokens_in":24425,"tokens_out":46613,"duration_ms":482102,"concrete_test":"Verify [GL22, Proposition 2.8] in its original statement: does it assume a closed reference form, and does its proof establish L1 compactness for u_b - sup_X u_b when β is non-closed? If yes, check that all hypotheses (including any mass upper bound) are satisfied in Proposition 2.2. If not, supply a self-contained proof of the compactness step using only Vol(β)>0 and the bounded potential ρ, or exhibit a non-closed β with Vol(β)>0 and bounded β-psh u,v for which u_b - sup_X u_b has no subsequence converging in L1 to a finite β-psh function. Completion of this check would settle whether uniqueness in Theorem 5.1 and the stability estimate in Theorem 5.3 are valid as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The structural backbone of the paper is Proposition 2.2 (domination principle). Its proof is one short paragraph after \"The proof follows from [GL22, Proposition 2.8]\". The decisive step is the assertion that u_b - sup_X u_b converges in L1 and a.e. to u_∞∈PSH(X,β) for u_b=P_β(bu-(b-1)v). For non-closed β this compactness is not the classical closed-class result unless one has an upper bound on the Monge-Ampère masses of u_b; Vol(β)>0 only gives a lower bound on each total mass, and no uniform upper bound is supplied. The membership of the limit in PSH(X,β) also needs a non-closed compactness theorem. Since Corollaries 2.5–2.6, Lemma 3.1, Theorem 3.1, Lemma 4.1 and the uniqueness/stability proofs of Theorems 5.1–5.3 all invoke Proposition 2.2, a gap here would leave the L∞ estimates and uniqueness in the main theorems without proof. The paper's own Remark 5.1 admits that the known uniqueness devices do not adapt to the non-closed case, which makes this dependence on a cited result especially delicate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies degenerate complex Monge-Ampère equations of the form (β+dd^c φ)^n = e^{λφ} f ω^n on compact Hermitian manifolds, where β is a smooth (1,1)-form that is allowed to be non-closed. Under the assumptions that β admits a bounded β-plurisubharmonic function and that Vol(β)>0, the authors prove L∞ a priori estimates, existence of bounded solutions for λ>0, existence up to a constant for λ=0, and a stability estimate in the L^p norm of the densities. These results are then applied to obtain partial answers to the extended Tosatti–Weinkove and Demailly–Păun conjectures. The paper is a sequel to earlier work by the same authors and builds substantially on the framework of Boucksom–Guedj–Lu, Guedj–Lu, and Nguyen.","tokens_in":24603,"tokens_out":36064,"duration_ms":354011,"significance":"If the results are correct, they provide a natural and nontrivial extension of degenerate complex Monge-Ampère theory from closed to non-closed Bott-Chern classes on Hermitian manifolds. The main theorems are clearly stated, the proofs are detailed and follow a coherent global strategy, and the stability estimate in Theorem 5.3 is strong enough to imply uniqueness. The paper is also honest about the limitations of its methods, notably in Remark 5.1, where it acknowledges that uniqueness for the λ=0 equation remains open. The stress-test concern about the domination principle does not, on reading, invalidate the proof: the compactness step needed in Proposition 2.2 is the standard precompactness of quasi-plurisubharmonic functions with sup=0, which does not require a uniform upper bound on Monge-Ampère masses. Nevertheless, this step should be spelled out explicitly, since it is the structural backbone of the paper and the current text compresses it into a single assertion.","major_comments":[{"comment":"The proof asserts without further justification that the sequence u_b−sup_X u_b converges in L^1 and almost everywhere to a function u_∞∈PSH(X,β). This is the load-bearing compactness step for the domination principle, and for non-closed β it is not literally the classical compactness theorem for a fixed closed class. The needed fact is the standard precompactness of quasi-plurisubharmonic functions with sup=0, which does not require a uniform upper bound on the Monge-Ampère masses; the assumption Vol(β)>0 is used only to guarantee positive mass on the contact set D, not to bound the masses from above. Please expand this step, for example by adding a local potential argument and citing a precise compactness statement such as [Ngu16, Proposition 1.1]. Without this clarification, the proof of Proposition 2.2, and hence of the later comparison and uniqueness arguments, is not verifiable as written.","section":"Section 2.2, Proposition 2.2"}],"minor_comments":[{"comment":"The proof of Theorem 5.3 contains a verbatim duplicate: the paragraph beginning 'Up to rescaling we may assume without loss of generality that λ=1' is repeated almost word for word after the first 'reversing the inequality we conclude the proof'. Remove the duplicate.","section":"Section 5, Theorem 5.3"},{"comment":"The definition of ε in the stability proof is garbled as rendered. It should be ε = (c^{-1} e^{sup_X φ})^{1/n} ||f−g||_p^{1/n} (or an equivalent formula) so that the subsequent identity ε^n c h = e^{sup_X φ}( |f−g| + ||f−g||_p ) holds. Please correct the displayed formula.","section":"Section 5, Theorem 5.3"},{"comment":"In Step 2 of the proof, the identity should read 1/((1+t)^2 g'(t)) = μ(φ<ρ−t); the displayed expression '1/(1+t)^2 g'(t) = μ(φ<ρ−t)' is missing parentheses and is ambiguous.","section":"Section 3, Theorem 3.1"},{"comment":"The inequality Vol(β_j) ≥ Vol(β) is invoked from [BGL24, Proposition 3.7]. Since PSH(X,β_j) is not contained in PSH(X,β) when β_j = β+ε_jω, the monotonicity is not immediate; please state the precise result from [BGL24] being used and check that its hypotheses are satisfied in this setting.","section":"Section 4, Lemma 4.1"},{"comment":"In the statement of Lemma 7.2, 'mertic' should be 'metric', and in the proof of Theorem 7.2, 'adimit' should be 'admit'.","section":"Section 7.2, Lemma 7.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a serious contribution and the main arguments are plausible and well aligned with the cited literature. My main concern is not correctness but verifiability: the domination principle is the structural backbone, and its proof currently relies on a one-line compactness assertion and on [GL22, Proposition 2.8]. If the authors expand this step and clean up the duplicated proof and typographical errors, the manuscript would be suitable for publication. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The main results—existence, uniqueness for λ>0, and an L∞ stability estimate for the degenerate Monge-Ampere equation when β is possibly non-closed, has Vol(β)>0, and admits a bounded β-psh function—are a genuine extension of [LWZ24] (closed β) and [GL23] (semipositive β). The partial results on the extended Tosatti-Weinkove and Demailly-Paun conjectures are real applications, not window dressing. The overall architecture is coherent: domination principle, L∞ estimates, subsolution construction, approximation by β+εω. If the domination principle holds, the main theorems follow.\n\nThe soft spot is exactly where the reader pointed. Proposition 2.2 is the backbone, and its proof is a sketch deferring to [GL22, Prop 2.8]. The decisive compactness claim—that u_b − sup u_b converges to a β-psh limit—is not automatic for non-closed β. Vol(β)>0 is a lower bound on mass, not an upper bound, so you don't get the usual L1 compactness for free. This is not a proven error: [GL22] may well contain the missing bound, but the paper needs to either state the precise result being cited or give the argument. A referee should start here.\n\nTwo other things to fix. The proof of Theorem 5.3 contains a large duplicated block; the second copy looks like a paste artifact, which is sloppy. Theorem 6.1 has a confusing constant chain and a level-set estimate with f(t)=1/µ(...) that does not obviously have finite integral; that section needs rewriting. Remark 5.1 is honest about the λ=0 uniqueness remaining open, and the paper does not oversell it.\n\nBottom line: the paper deserves a serious referee. It is not in final form, but the central claims look defensible and the extension to non-closed classes is meaningful. I'd accept for review with the expectation of a substantial revision.","headline":"Strong extension of closed-case Monge-Ampere results to non-closed β, but the domination principle proof is too condensed to certify; needs referee scrutiny and a revision.","tokens_in":25202,"tokens_out":5612,"would_cite":true,"duration_ms":58187,"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":"The paper claims that positivity of the Bott–Chern volume, not closedness, is what makes degenerate complex Monge–Ampère equations solvable on compact Hermitian manifolds.","keywords":["degenerate complex Monge-Ampère equation","Bott-Chern volume","Hermitian manifolds","plurisubharmonic functions","L-infinity a priori estimates","stability","Tosatti-Weinkove conjecture","Demailly-Paun conjecture"],"falsifier":"Exhibit bounded $\\beta$-psh functions $u,v$ and $c<1$ satisfying $1_{\\{u<v\\}}(\\beta+\\mathrm{dd}^{\\mathrm{c}}u)^n\\leq c\\,1_{\\{u<v\\}}(\\beta+\\mathrm{dd}^{\\mathrm{c}}v)^n$ while $u\\not\\geq v$; or exhibit two distinct bounded solutions of the $\\lambda=0$ equation with the same $f$ and the same constant $c$. Either example would falsify the paper's central reduction to the domination principle.","tokens_in":24157,"feed_emoji":"📐","tokens_out":11562,"duration_ms":104814,"temperature":0.7,"pith_summary":"This paper proves that the degenerate complex Monge–Ampère equation $(\\beta+\\mathrm{dd}^{\\mathrm{c}}\\varphi_\\lambda)^n=e^{\\lambda\\varphi_\\lambda} f\\,\\omega^n$ has a unique bounded $\\beta$-plurisubharmonic solution on a compact Hermitian manifold whenever $\\beta$ is a smooth, possibly non-closed $(1,1)$-form admitting a bounded $\\beta$-plurisubharmonic potential and satisfying $\\mathrm{Vol}(\\beta)>0$. It proves the same for the unnormalized equation $(\\beta+\\mathrm{dd}^{\\mathrm{c}}\\varphi)^n=c f\\,\\omega^n$, with the constant $c$ uniquely determined by the data, and it proves the stability estimate $\\|\\varphi-\\psi\\|_\\infty\\leq C\\|f-g\\|_p^{1/n}$ for the $\\lambda>0$ family. If the paper is right, closedness of $\\beta$—a standing hypothesis in previous results of this type—is not needed; positive Bott–Chern volume plus a bounded potential is enough. The applications give partial positive answers to the extended Tosatti–Weinkove and Demailly–Păun conjectures for non-closed Bott–Chern classes.","feed_headline":"Non-closed Monge–Ampère equations solved via Bott–Chern volume","feed_subtitle":"Positive volume of a possibly non-closed form yields unique bounded solutions and L∞ stability.","key_machinery":"The load-bearing mechanism is the Bott–Chern lower volume $\\mathrm{Vol}(\\beta)$ together with the domination principle for bounded $\\beta$-plurisubharmonic functions (Proposition 2.2): if $0\\leq c<1$ and $1_{\\{u<v\\}}(\\beta+\\mathrm{dd}^{\\mathrm{c}}u)^n\\leq c\\,1_{\\{u<v\\}}(\\beta+\\mathrm{dd}^{\\mathrm{c}}v)^n$, then $u\\geq v$. Its proof forms rooftop envelopes $u_b=P_\\beta(bu-(b-1)v)$ and uses $\\mathrm{Vol}(\\beta)>0$ to force positive Monge–Ampère mass on the contact set, then passes $u_b-\\sup_X u_b$ to a limiting $\\beta$-psh function, so that the comparison cannot fail. From this principle the paper derives uniqueness, the lower and upper bounds in the subsolution construction, and the stability estimate. The $L^\\infty$ a priori estimates follow the quasi-plurisubharmonic envelope method: for a concave increasing weight $\\chi$, one bounds $P_\\beta(\\chi\\circ(\\varphi-\\rho)+\\rho)$ and controls its Monge–Ampère energy using the integrability of $\\mathrm{PSH}(X,\\beta)$ in $L^m(\\mu)$ and the positivity of $\\mathrm{Vol}(\\beta)$. Mixed-type inequalities for several possibly different reference forms $\\beta_1,\\dots,\\beta_n$ (Lemma 2.3) let the argument wedge $\\beta_j+\\mathrm{dd}^{\\mathrm{c}}u_j$ against a fixed Hermitian metric, which is how the non-closedness of $\\beta$ is absorbed.","core_discovery":"The central claim is that the degenerate complex Monge–Ampère equations of the title are solvable for a possibly non-closed reference form $\\beta$ in the Bott–Chern space $\\mathrm{BC}^{1,1}(X)$, under exactly two assumptions: some bounded $\\beta$-plurisubharmonic function exists and the lower volume $\\mathrm{Vol}(\\beta)=\\inf_{u\\in\\mathrm{PSH}(X,\\beta)\\cap L^\\infty}\\int_X(\\beta+\\mathrm{dd}^{\\mathrm{c}}u)^n$ is positive. For each $\\lambda>0$ the solution to $(\\beta+\\mathrm{dd}^{\\mathrm{c}}\\varphi_\\lambda)^n=e^{\\lambda\\varphi_\\lambda} f\\,\\omega^n$ exists, is unique in $\\mathrm{PSH}(X,\\beta)\\cap L^\\infty(X)$, and obeys a uniform bound depending only on $\\lambda$, $\\beta$, $p$, $\\|f\\|_p$, $X$, and $\\omega$. The $\\lambda=0$ equation is solved up to the constant $c$, the constant is uniquely fixed by the data, and the oscillation of the solution is controlled. The stability estimate $\\|\\varphi-\\psi\\|_\\infty\\leq C\\|f-g\\|_p^{1/n}$ is proved for the exponential family and implies uniqueness there. On the application side, the paper derives a logarithmic-pole $\\beta$-psh function when $\\sum_i\\tau_i^n<\\mathrm{Vol}(\\beta)$ (extended Tosatti–Weinkove) and bigness of $\\{\\beta\\}$—existence of a Hermitian current—when additionally $\\mathrm{Vol}_{n-1}(\\beta)<+\\infty$ (partial extended Demailly–Păun). The paper leaves open the uniqueness of the $\\lambda=0$ solution, noting that existing methods do not directly apply.","pith_inferences":["Editorial: the same pair of hypotheses—bounded potential and positive Bott–Chern volume—may replace closedness in other Hermitian pluripotential statements, since the proofs here use only those two inputs through the domination principle and envelope estimates.","Editorial: the stability exponent $1/n$ and the use of $L^p$ densities suggest a route to quantitative Demailly–Păun criteria: the size of the mass in the Hermitian current could be controlled by $\\mathrm{Vol}_{n-1}$ and the $L^p$ data, which the paper does not state.","Editorial: the paper's Remark 2.2 indicates the domination principle survives under the weaker condition $\\int_X(\\beta+\\mathrm{dd}^{\\mathrm{c}}u)^n>0$ for every bounded $u$; if that holds, the main theorems might extend beyond the $\\mathrm{Vol}(\\beta)>0$ hypothesis.","Editorial: uniqueness for $\\lambda=0$ could be tested by letting $\\lambda\\to0$ in the stability estimate and tracking whether the constant $C$ degenerates; the paper leaves this as open."],"forward_implications":["For every $\\lambda>0$ and every $0\\leq f\\in L^p(X,\\omega^n)$, $p>1$, with $\\|f\\|_p>0$, the equation $(\\beta+\\mathrm{dd}^{\\mathrm{c}}\\varphi_\\lambda)^n=e^{\\lambda\\varphi_\\lambda}f\\,\\omega^n$ has a unique bounded $\\beta$-psh solution, and all such solutions share one uniform $L^\\infty$ bound.","The $\\lambda=0$ equation $(\\beta+\\mathrm{dd}^{\\mathrm{c}}\\varphi)^n=c f\\,\\omega^n$ is solvable with the constant $c$ uniquely determined by $f$, $\\beta$, and the oscillation of $\\varphi$, and with the oscillation bounded by data.","Solutions for $\\lambda>0$ are stable: $\\|\\varphi_\\lambda-\\psi_\\lambda\\|_\\infty\\leq C\\|f-g\\|_p^{1/n}$, so the solution map from $L^p$ densities to bounded $\\beta$-psh functions is Hölder continuous.","Whenever $\\sum_{i=1}^N\\tau_i^n<\\mathrm{Vol}(\\beta)$, there is a $\\beta$-psh function with prescribed logarithmic poles $O(\\tau_j\\log|z|)$ at the given points, extending the Tosatti–Weinkove conclusion to non-closed classes.","If in addition $\\mathrm{Vol}_{n-1}(\\beta)<+\\infty$, then $\\{\\beta\\}$ is big: it contains a Hermitian current, giving a partial Demailly–Păun-type statement without closedness."],"supporting_citations":[{"why":"Introduces the Bott–Chern space and lower volumes of possibly non-closed $(1,1)$-forms and solves the big-case equation, the starting point of this paper.","marker":"[BGL24]"},{"why":"Settles the closed case of these equations and formulates the extended Tosatti–Weinkove and Demailly–Păun conjectures that the applications target.","marker":"[LWZ24]"},{"why":"Provides the envelope-based $L^\\infty$ estimates, subsolution construction, and general strategy for solving Monge–Ampère equations on Hermitian manifolds.","marker":"[GL23]"},{"why":"Supplies the quasi-plurisubharmonic envelope bounds used in Theorem 3.1 to control oscillation.","marker":"[GL21]"},{"why":"Contains the domination principle that Section 2.2 adapts; uniqueness and comparisons throughout rely on it.","marker":"[GL22]"},{"why":"Provides local solvability and stability for Dirichlet problems with arbitrary reference forms, used in the proof of the mixed-type inequalities.","marker":"[KN15]"},{"why":"Gives existence for non-degenerate Hermitian Monge–Ampère equations and compactness properties of psh functions used in the subsolution argument.","marker":"[Ngu16]"},{"why":"Supplies the Gauduchon-metric criterion for existence of a Hermitian current, used in the proof of Theorem 7.2.","marker":"[Lam99]"},{"why":"Provides the Bedford–Taylor convergence theorem, maximum principle, and comparison principle underlying the pluripotential arguments.","marker":"[BT82]"}],"fun_headline_variants":["Bott–Chern volume solves non-closed Monge–Ampère","Degenerate Monge–Ampère solved for non-closed forms","Positive Bott–Chern volume yields Monge–Ampère solutions","Non-closed Monge–Ampère equations: existence and stability","Bott–Chern volume unlocks degenerate Monge–Ampère"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on the domination principle for bounded $\\beta$-plurisubharmonic functions, whose proof requires $\\mathrm{Vol}(\\beta)>0$ to guarantee positive Monge–Ampère mass on the contact set and requires a compactness passage to a limiting $\\beta$-psh function; if this comparison step fails for non-closed $\\beta$, the uniqueness theorem and all $L^\\infty$ bounds collapse.","fun_headline_variants_meta":{"raw":{"variants":["Bott–Chern volume solves non-closed Monge–Ampère","Degenerate Monge–Ampère solved for non-closed forms","Positive Bott–Chern volume yields Monge–Ampère solutions","Non-closed Monge–Ampère equations: existence and stability","Bott–Chern volume unlocks degenerate Monge–Ampère"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000411,"raw_usage":{"total_tokens":2195,"prompt_tokens":1081,"completion_tokens":1114,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":1016}},"tokens_in":697,"tokens_out":1114,"duration_ms":9518,"temperature":1.0,"reasoning_tokens":1016,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:38:16.544665+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit bounded $\\beta$-psh functions $u,v$ and $c<1$ satisfying $1_{\\{u<v\\}}(\\beta+\\mathrm{dd}^{\\mathrm{c}}u)^n\\leq c\\,1_{\\{u<v\\}}(\\beta+\\mathrm{dd}^{\\mathrm{c}}v)^n$ while $u\\not\\geq v$; or exhibit two distinct bounded solutions of the $\\lambda=0$ equation with the same $f$ and the same constant $c$. Either example would falsify the paper's central reduction to the domination principle.","supporting_citations":[],"review_version":1}