Modulus of continuity of Monge--Amp\`ere potentials in big cohomology classes

Hoang Hiep Pham; Hoang-Son Do; Quang-Tuan Dang

REVIEW 4 major objections 4 minor 35 references

Modulus of continuity of Monge--Amp\`ere potentials in big cohomology classes

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper proves that solutions to degenerate complex Monge–Ampère equations in big cohomology classes admit a uniform modulus of continuity away from the singularities of the weight, even when the right-hand side is not integrable.

desk verdict A genuine improvement over Di Nezza–Lu and Dang, with a mostly standard proof and one internal gap worth a referee's attention before publication. read the letter →

arxiv 2504.15763 v2 pith:IRVVNRXY submitted 2025-04-22 math.CV

Quang-Tuan Dang , Hoang-Son Do , Hoang Hiep Pham This is my paper · ORCID

classification math.CV MSC 32U1532W2032Q15

keywords complexMonge-AmpèreequationsbigcohomologyclassesmodulusofcontinuityHöldercontinuousmeasuresquasi-plurisubharmonicfunctionsKiselman-Legendretransformnon-pluripolarproductcapacityestimates

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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 a uniform modulus of continuity for solutions of the degenerate complex Monge-Ampère equation $(\theta+dd^c u)^n=e^{-\psi}\mu$ on a compact Kähler manifold, where $\mu$ is Hölder continuous and $\psi$ is quasi-psh, and the cohomology class $\{\theta\}$ is big. The right-hand side is allowed to be non-integrable, so solutions may be unbounded and need not be Hölder continuous. The main theorem shows that for each relatively compact set $U$ inside the ample locus $\operatorname{Amp}(\theta)$ with $U\cap\{\psi=-\infty\}=\varnothing$, there is a continuous function $F_U$ with $F_U(0)=0$ such that $|u(z_1)-u(z_2)|\le F_U(\operatorname{dist}(z_1,z_2))$ for all $z_1,z_2\in U$. The modulus depends only on the fixed geometric data and on an auxiliary bound for $\int_X e^{2(V_\theta-u)/a}d\mu$, not on the individual solution. This matters because uniform control of the modulus is what feeds into geometric consequences such as diameter bounds and Gromov–Hausdorff convergence for singular Kähler metrics.

What carries the argument

The load-bearing object is the Kiselman–Legendre transform $\Phi_{c,\delta}(z)=\inf_{0<t\le\delta}\big[\rho_t u(z)+K(t^2-\delta^2)+K(t-\delta)-c\log(t/\delta)\big]$ of an unbounded $\theta$-psh function, where $\rho_t$ is the $\delta$-regularization along geodesics of the exponential map. The cited Lemma 2.1 from [KN19] asserts that for unbounded $u$ the transform still provides the Hessian lower bound $\theta+dd^c\Phi_{c,\delta}\ge-(Ac+2K\delta)\omega_X$ and that $\rho_t u+Kt^2$ is increasing in $t$. The proof then forms the comparison subsolution $u_{c,\delta}=B_0\Psi_0+(1-B_0)\Phi_{c,\delta}$ with $\Psi_0$ a negative $\theta$-psh function of analytic singularities, and uses a bootstrap iteration with capacity estimates for Hölder continuous measures to convert $L^1$ smallness of $(\rho_\delta u-u)_+$ into pointwise control of $u$. The whole argument reduces the modulus question to quantitative control of this $L^1$ difference.

What would settle it

Exhibit an unbounded $\theta$-psh function $u$ with full non-pluripolar Monge–Ampère mass for which the Kiselman–Legendre transform violates the Hessian bound of Lemma 2.1, or find a Hölder continuous measure $\mu$ and quasi-psh $\psi$ satisfying the hypotheses of Theorem 1.1 whose solution $u$ is discontinuous at some point of $\operatorname{Amp}(\theta)\setminus\{\psi=-\infty\}$. The radial singularity examples of the paper's example section are a natural testing ground for the second option.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: given a compact Kähler manifold $(X,\omega_X)$ of dimension $n$, a smooth closed $(1,1)$-form $\theta$ with big cohomology class, a Hölder continuous measure $\mu$ with constants $B>0$ and $0<\beta\le 1$, and a quasi-psh weight $\psi$ with $\omega_X+a_0dd^c\psi\ge 0$ and $\int_X e^{-\psi}d\mu=\operatorname{Vol}(\theta)$, every solution $u\in E(X,\theta)$ of $(\theta+dd^c u)^n=e^{-\psi}\mu$ with $\sup_X u=0$ is uniformly continuous on each $U\Subset\operatorname{Amp}(\theta)\setminus\{\psi=-\infty\}$, with a modulus $F_U$ that is continuous at $0$ and depends only on $X$, $U$, $\omega_X$, $n$, $\theta$, $a_0$, $B$, $\beta$, $\sup_U(-\psi)$, and an upper bound for $H(a)=\int_X e^{2(V_\theta-u)/a}d\mu$. In the Kähler case $\theta=\omega_X$, the corollary upgrades the previously known continuity result to equicontinuity of the whole family of solutions satisfying $\int_X h(-\psi)e^{-\psi}d\mu\le C_0$ with fixed increasing concave $h$ and constant $C_0$.

Load-bearing premise

Everything rests on Lemma 2.1, taken from [KN19], which states that the Kiselman–Legendre transform of an unbounded $\theta$-psh function obeys $\theta+dd^c\Phi_{c,\delta}\ge-(Ac+2K\delta)\omega_X$ and that $\rho_t u+Kt^2$ is increasing in $t$; the proof of that lemma is not reproduced in this paper, and if it fails without boundedness of $u$, the comparison subsolution and the proof of Theorem 1.1 collapse.

Editorial extensions

If this is right

For every compact $U$ inside $\operatorname{Amp}(\theta)\setminus\{\psi=-\infty\}$, the solution set of (1.2) with a fixed upper bound on $H(a)=\int_X e^{2(V_\theta-u)/a}d\mu$ is equicontinuous on $U$.
In the Kähler case $\theta=\omega_X$, the family of solutions with $\int_X h(-\psi)e^{-\psi}d\mu\le C_0$ for fixed increasing concave $h$ and constant $C_0$ is equicontinuous on each $U\Subset X\setminus\{\psi=-\infty\}$ (Corollary 1.2).
The previous continuity results for such equations are recovered as special cases, now with a modulus independent of the particular solution.
Because the modulus depends only on the listed data, families of solutions satisfying uniform bounds are precompact in the $C^0$ topology on compact subsets of $\operatorname{Amp}(\theta)\setminus\{\psi=-\infty\}$, which is the kind of control needed for diameter and convergence statements for singular Kähler metrics.

Reading between the lines

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

A natural test would be to compute the abstract modulus $F_U$ explicitly in the radial singularity examples of the paper; wherever $\psi$ is bounded above, one may expect the modulus to simplify to a power function, though that is not asserted here.
The same proof scheme should transfer to other geometric settings—Hermitian manifolds, or big classes with prescribed singularity type—once an analogue of Lemma 2.1 is available.
The explicit dependence of the scale $\kappa(\delta)$ on the curvature constants suggests that quantitative versions of the modulus could be extracted from the proof, yielding concrete estimates for families of Kähler currents.
One could also read Theorem 1.1 as a stability statement: small $L^1$ deviation of a potential from its regularization forces small sup-norm deviation away from singularities, so the modulus is ultimately a quantitative form of the comparison principle.

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Editorial analysis

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

Desk Editor's Note

read the letter

The paper proves a uniform modulus of continuity for solutions to degenerate complex Monge–Ampère equations in big cohomology classes, improving the pointwise continuity results of Di Nezza–Lu and Dang. That is a real, modest step forward. The equicontinuity statement in the Kähler case (Corollary 1.2) is also new. The main theorem is stated cleanly: for a Hölder continuous measure and a quasi-psh weight, the solution admits a continuous modulus on each compact subset of the ample locus away from the poles of the weight.

The proof follows the established toolbox — Demailly regularization, Kiselman–Legendre transform, De Giorgi iteration, capacity estimates — and the authors are honest about borrowing key lemmas. Lemma 2.1 is quoted from Kołodziej–Nguyen with a one-line justification, and Proposition 2.12 is sketched with reference to Darvas–Di Nezza–Lu. That is acceptable for the field, though a referee should check the unbounded version of Lemma 2.1 carefully.

The real soft spot is the final quantitative step in Theorem 1.1. The authors invert h1, h2, and κ without proving monotonicity or range. M5 is only assumed decreasing; h1(a) = aε0mU + 2M5(a) need not be monotone, and h2 is defined using h1^{-1} in a way that is not justified. This is not cosmetic — the existence of the modulus F_U depends on choosing c and δ so that the scale κ(δ) works. The stress-test note points out that for some admissible upper-bound functions the required c(δ)/δ → ∞ may fail. I read that as a genuine gap, not a manufactured one. It is likely fixable with a more careful choice of parameters, but as written the proof of Theorem 1.1 is incomplete.

There are also minor issues: the definition of dc changes between the introduction (i/2) and the preliminaries (i/2π), a normalization inconsistency that should be reconciled. The self-citation of Dang for Lemma 2.11 is fine — it is an external theorem, not circular.

Who this is for: specialists in pluripotential theory and degenerate Monge–Ampère equations. The result is plausible and the strategy is sound, but the paper needs a referee pass on the inversion steps before it is publishable. I would send it to a serious referee rather than desk-reject. Whether it survives depends on whether the parameter choice can be repaired.

Referee Report

4 major / 4 minor

Summary. The paper claims a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge–Ampère equations in big cohomology classes on compact Kähler manifolds. The main theorem (Theorem 1.1) states that if the right-hand side is e^{-ψ} μ where μ is Hölder continuous and ψ is quasi-psh with a fixed lower bound on its Hessian, then any normalized solution u has a modulus of continuity that depends only on the fixed data and on an upper bound for the integrals ∫ e^{2(Vθ−u)/a} dμ, away from the pole set of ψ. A corollary treats the Kähler case under an additional integrability condition on ψ. The proof combines Demailly regularization, a Kiselman–Legendre transform, capacity estimates for Hölder measures, and De Giorgi-type iteration lemmas.

Significance. If the proof is correct, the result is a meaningful improvement over earlier work of Di Nezza–Lu and of the first author, providing a quantitative modulus of continuity without assuming the right-hand side lies in L^p. The dependence of the modulus on an upper bound for the exponential integrals is a parameter-free, falsifiable statement. The paper also supplies useful auxiliary estimates (e.g., Proposition 2.10 and Lemma 3.3) that may be of independent interest. The overall strategy is coherent and the reliance on prior results (KN19, DDL18, DN14, Dan22) is explicit, though some steps are only sketched.

major comments (4)

[Section 3, proof of Theorem 1.1, after Eq. (3.8)] The proof defines h1(a)=aε0m_U+2M5(a) and chooses a=h1^{-1}(1/√B0), then defines h2(t)=M7(h1^{-1}(1/√t)t)/t^{3/2} and chooses c so that h2(B0)=δ^{-β/(2n+1)}. However, M5 is only assumed decreasing, so h1 need not be monotone; no branch of h1^{-1} is specified. Moreover, no argument is given that h2 is continuous or that the equation h2(B0)=δ^{-β/(2n+1)} has a solution with B0 in the admissible range (0,1/2). Since B0=(Ac+2Kδ)/ε0, this is an implicit equation for c(δ), and the required condition c(δ)/δ→∞ from Lemma 2.4 is not verified. Without such a verification, the existence of the modulus F_U is not established as written.
[Section 3, final step of the proof of Theorem 1.1] After obtaining the bound on ρ_{κ(δ)}u−u, the proof replaces δ with κ(δ) and uses κ^{-1}(δ) without proving that the function κ(δ)=δ exp(−4A(C_U B0+2√B0+2Kδ)/(ε0 B0)) is monotone or has the required range. Additionally, the final inference from the regularization bound to |u(z1)−u(z2)|≤F_U(dist(z1,z2)) is not written out; the usual triangle inequality involving the Lipschitz bound of ρ_δ u is missing. These are internal, checkable steps, and as they stand they leave the conclusion unsupported.
[Section 2, Lemma 2.1 and definition of u_{c,δ}] Lemma 2.1 is asserted to hold for unbounded θ-psh functions with the proof described as 'identical' to [KN19, Lemma 4.1]. Since the unbounded case is essential for the construction of the comparison subsolution u_{c,δ} in (2.5) and for Lemma 2.4, the reader needs either a self-contained proof or an explicit statement in the cited reference that covers unbounded functions. The manuscript does not provide either.
[Section 2, Eq. (2.4), and Section 3 proof of Theorem 1.1] There is an inconsistency in the definition of u_{c,δ}: in (2.5) it is B0Ψ0+(1−B0)Φ_{c,δ}, while in the proof of Theorem 1.1 it is written as 2B0Ψ0+(1−2B0)Φ_{c,δ}. Lemma 2.4 is stated for the (2.5) version. If the 2B0 version is intended, the condition 1−2B0≥1/2 (which is used implicitly in the estimates) requires B0≤1/4, yet the manuscript only guarantees B0<1/2. This discrepancy needs to be resolved, and the hypothesis of Lemma 2.4 must be stated accordingly.

minor comments (4)

[Section 1, Theorem 1.1 statement] In the statement of Theorem 1.1, the modulus function is called F_U but the sentence 'F(0)=0' uses F without subscript; please unify the notation.
[Section 2, Proposition 2.12] The word 'Höler' in the proof of Proposition 2.12 should be 'Hölder'.
[Section 2, Example 2.16] In items (3) and (4) of Example 2.16, the displayed integrals appear to have inconsistent exponents (e.g., '(log(−t))^{nα−n}' versus the preceding line), and the final integral in (4) is written without a convergence condition; please correct the formulas.
[Section 3, Lemma 3.6] The proof of Lemma 3.6 relies on [DH12, Propositions 2.10 and 2.11]; it would be helpful to state the precise property used, since the conclusion that u∈E(X,ωX) is not immediate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the modulus bound is derived by a priori estimates, and the only self-citation is an independent prior theorem rather than a re-importation of the target claim.

full rationale

The paper's derivation chain does not reduce to its own inputs. Theorem 1.1 concludes a uniform modulus bound for a solution u, while the hypotheses (Holder continuity of mu with constants B, beta; quasi-psh weight psi with omega_X + a0 dd^c psi >= 0; compatibility integral e^{-psi} dmu = Vol(theta)) impose no bound on the pointwise oscillation of u. The proof is an a priori estimate: Lemma 3.3 converts mass and energy bounds into a lower bound for u, and the chain after (3.8) converts that into a bound on (rho_delta u - u)_+ on U via Lemma 2.3 and the De Giorgi-type Lemmas 3.1-3.5. The final modulus F_U is constructed from the stated constants and from an assumed upper-bound function for H(a) = integral e^{2(V_theta - u)/a} dmu; H(a) is a standard a priori datum for such equations, not a hidden copy of the modulus of u. The only self-citation by the authors is [Dan22], used in Lemma 2.11 to obtain integral e^{m(V_theta - u)} dmu < infinity for u in E(X, theta). That cited result is a previously published, parameter-free theorem whose assumptions do not include the modulus estimate, so it is independent support and does not raise the circularity score. Lemma 2.1 is quoted from [KN19], not from the authors' own work, and its omission of proof is a standard reliance on an external result. The skeptical concern about the well-definedness of h1^{-1}, h2, and kappa^{-1} is a correctness or gap-checking issue about the proof as written, not a circularity: even if those inverses are not justified, no hypothesis equals the conclusion and no fitted parameter is renamed as a prediction. Hence no circular step is present.

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

The paper introduces no free parameters or new entities. It uses standard pluripotential theory results and stated structural assumptions on mu and psi. Several external lemmas from prior papers are imported; this is normal practice in the field.

assumptions (6)

standard math Existence of Demailly regularization and ample locus for big classes (Psi_0 with analytic singularities, theta + dd^c Psi_0 >= epsilon_0 omega_X).
Section 2.1; used throughout to construct the subsolution u_{c,delta}.
standard math Lemma 2.1 (Kiselman-Legendre transform estimate), borrowed from [KN19, Lemma 4.1].
Used in Lemma 2.4 and the main proof; proof not reproduced in this paper.
domain assumption Holder continuity of mu with respect to L1 on C, yielding Proposition 2.12's exponential capacity bound.
Assumed in Theorem 1.1; imported from DN14 Proposition 4.4 and DDL18 Proposition 4.30.
domain assumption psi quasi-psh with omega_X + a0 dd^c psi >= 0 and integral e^{-psi} dmu = Vol(theta).
Standing hypothesis of Theorem 1.1.
domain assumption Condition (1.3) in Corollary 1.2: integral h(-psi) e^{-psi} dmu <= C0 for increasing concave h with h(infinity)=infinity.
Needed for uniform control of H(a) and equicontinuity of the family of solutions.
standard math Comparison principle for non-pluripolar Monge-Ampere measures [DDL21, Lemma 2.3].
Used in Lemma 3.1 and subsequent estimates.

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Pith. "Pith review of Modulus of continuity of Monge--Amp\`ere potentials in big cohomology classes." pith.science (2026). https://pith.science/paper/IRVVNRXY

@misc{pith2026250415763,
  author       = {Pith},
  title        = {Pith review of: Modulus of continuity of Monge--Amp\`ere potentials in big cohomology classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRVVNRXY}},
  note         = {Machine review of arXiv:2504.15763}
}

read the original abstract

In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Amp\`ere equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

a hler- E instein metrics and the K \

R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi. K\" a hler- E instein metrics and the K \" a hler- R icci flow on log F ano varieties. J. Reine Angew. Math. , 751:27--89, 2019

work page 2019

[2] [2]

Berman and J.-P

R. Berman and J.-P. Demailly. Regularity of plurisubharmonic upper envelopes in big cohomology classes. In Perspectives in analysis, geometry, and topology , volume 296 of Progr. Math. , pages 39--66. Birkh\" a user/Springer, New York, 2012

work page 2012

[3] [3]

Boucksom, P

S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi. Monge- A mp\`ere equations in big cohomology classes. Acta Math. , 205(2):199--262, 2010

2010

[4] [4]

Boucksom

S. Boucksom. Divisorial Z ariski decompositions on compact complex manifolds. Ann. Sci. \' E cole Norm. Sup. (4) , 37(1):45--76, 2004

work page 2004

[5] [5]

Bedford and B

E. Bedford and B. A. Taylor. Fine topology, S ilov boundary, and (dd^c)^n . J. Funct. Anal. , 72(2):225--251, 1987

work page 1987

[6] [6]

Q.-T. Dang. Continuity of M onge- A mp\`ere potentials in big cohomology classes. Int. Math. Res. Not. IMRN , (14):11180--11201, 2022

work page 2022

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