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Degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds and applications II

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.07336 v1 pith:HCL37Q37 submitted 2025-06-09 math.CV

Haoyuan Sun , Zhiwei Wang This is my paper
classification math.CV MSC 32W2032U0532U4053C55
keywords degeneratecomplexMonge-AmpèreequationBott-ChernvolumeHermitianmanifoldsplurisubharmonicfunctionsL-infinityaprioriestimatesstabilityTosatti-WeinkoveconjectureDemailly-Paun
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

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.

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 (1)
  1. [Section 2.2, Proposition 2.2] 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.
minor comments (5)
  1. [Section 5, Theorem 5.3] 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.
  2. [Section 5, Theorem 5.3] 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.
  3. [Section 3, Theorem 3.1] 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.
  4. [Section 4, Lemma 4.1] 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.
  5. [Section 7.2, Lemma 7.2] In the statement of Lemma 7.2, 'mertic' should be 'metric', and in the proof of Theorem 7.2, 'adimit' should be 'admit'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central theorems rest on external tools and do not reduce to their assumptions; self-citations are motivational, not load-bearing.

full rationale

Walking the derivation chain, the paper's central theorems (5.1–5.3) are obtained from the domination principle (Proposition 2.2), the L∞ a priori estimates (Theorem 3.1), the subsolution construction (Lemma 4.1), and mixed-type inequalities (Lemmas 2.2–2.3). Proposition 2.2 is quoted from [GL22, Proposition 2.8], whose authors (Guedj–Lu) do not overlap with Sun–Wang, and the remaining structural tools come from [BGL24], [GL21], [GL23], [KN15], and [Lam99]. The assumptions that a bounded β-psh function ρ exists and that Vol(β)>0 are inputs to the mass and comparison estimates; they are not restatements of the existence, uniqueness, or stability conclusions. No equation in Section 5 is equivalent by construction to these assumptions, and no fitted parameter is renamed as a prediction. The applications in Section 7 use the newly proved Theorems 5.1 and 5.2 and copy proof schemes from [LWZ24] with the statement that the closedness of β is not required, which is an extension rather than a renaming of [LWZ24, Theorems 1.10 and 1.11]. The only self-referential element is that the extended Tosatti–Weinkove and Demailly–Păun conjectures were posed in [LWZ24], which includes the second author, but these citations motivate the applications rather than supply the load-bearing argument. The compressed compactness step inside Proposition 2.2 is a potential correctness concern about the cited external result, not a circularity of the present paper. Therefore no circular step is identified.

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

All hypotheses are standard geometric or analytic assumptions drawn from prior literature. No numerical parameters are fitted to data, and no new physical or geometric entities are introduced. The main unstated cost is the assumption that the Bedford-Taylor and Guedj-Lu machinery extends to non-closed reference forms under Vol(beta)>0, which the paper argues but does not fully axiomatize.

assumptions (6)
  • domain assumption Vol(beta)>0 and the existence of a bounded beta-psh function rho
    Main hypothesis of all theorems. It replaces bigness or closedness and is used to guarantee positive Monge-Ampere mass in the domination principle and a uniform lower volume for approximating forms beta+epsilon_j omega.
  • standard math Bedford-Taylor calculus for bounded psh functions applies to non-closed beta
    The paper defines (beta+dd^c u)^n for bounded beta-psh u and invokes the maximum principle, convergence theorem, and comparison principle; it cites [BT82, BT87] and follows [BGL24].
  • domain assumption The Guedj-Lu L^infinity a priori estimate framework transfers to Vol(beta)>0 and non-closed beta
    Theorem 3.1 is adapted from [GL23, Theorem 2.2]; the proof is sketched and is the main source of uniform bounds used in the existence theorems.
  • standard math Known solvability of Hermitian Monge-Ampere equations for positive reference forms
    Used in the approximation step with beta_j=beta+epsilon_j omega; cited to [KN15, Theorem 4.2], [Ngu16, Theorem 0.1], and [BGL24, Theorem 4.5].
  • standard math Lamari's criterion for Hermitian currents via Gauduchon metrics
    Used in Theorem 7.2 to detect bigness of the Bott-Chern class; cited to [Lam99].
  • standard math Skoda's uniform integrability theorem
    Used in Theorem 5.2 to find epsilon>0 such that e^{-epsilon v_j} f is uniformly in L^q for some 1<q<p; cited to [GZ17, Theorem 8.11].

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Pith. "Pith review of Degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds and applications II." pith.science (2026). https://pith.science/paper/HCL37Q37

@misc{pith2026250607336,
  author       = {Pith},
  title        = {Pith review of: Degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds and applications II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCL37Q37}},
  note         = {Machine review of arXiv:2506.07336}
}
abstract

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$, equipped with a Hermitian metric $\omega$. Let $\beta$ be a possibly non-closed smooth $(1,1)$-form on $X$ such that $\int_X\beta^n>0$. Assume that there is a bounded $\beta$-plurisubharmonic function $\rho$ on $X$ and $\underline{\mathrm{Vol}}(\beta) > 0$. In this paper, we establish solutions to the degenerate complex Monge-Amp\`ere equations on $X$ within the Bott-Chern space of $\beta$ (as introduced by Boucksom-Guedj-Lu) and derive stability results for these solutions. As applications, we provide partial resolutions to the extended Tosatti-Weinkove conjecture and Demailly-P\u aun conjecture.

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