{"id":"3b372579-3c38-40f5-ab52-3b560bdf05ca","arxiv_id":"2504.15763","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new theorem gives a uniform modulus of continuity for solutions to degenerate complex Monge-Ampere equations in big cohomology classes, improving earlier continuity and Holder results.","lead":"This mathematics paper proves that solutions to a family of degenerate complex Monge-Ampere equations have a uniform modulus of continuity, meaning nearby points have nearby solution values with a controlled rate. It gives the first such uniform estimate in big cohomology classes when the right-hand side measure is not integrable, improving earlier continuity results of Di Nezza-Lu and of the first author.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 relies on implicit inversions of h1, h2, and κ without establishing monotonicity or range, so the existence of the modulus F_U is not fully proved as written.","rationale":"The reader's conditional verdict is appropriate, but I do not think Lemma 2.1 is the main risk: the authors explicitly cite KN19 and the assertion that the proof remains valid without boundedness is plausible for Demailly's regularization. The real gap is the chain of implicit function inversions in the proof of Theorem 1.1. Constructing F_U requires selecting a(δ), c(δ), and then inverting κ(δ); none of these selections is shown to exist with the required properties. In particular h1 need not be monotone because M5 is decreasing while aε0m_U is increasing, and the proof silently chooses a small branch without defining it. h2 inherits this issue, and its range is not established; since B0 is bounded below by a multiple of δ, the condition c(δ)/δ→∞ required by Lemma 2.4 is not automatic. The algebra may be repairable by taking generalized inverses and a limiting argument, and the model case suggests the theorem is true, so this is a completeness issue rather than a counterexample. The rest of the argument is a reasonable adaptation of DDG+14 and EGZ09, with no circularity or data-fitting concerns. The verdict should remain conditional pending a careful referee pass on this inversion passage.","tokens_in":16872,"tokens_out":20017,"duration_ms":186492,"concrete_test":"In the model case u=Vθ, ψ=0, μ=ω_X^n (so H(a)=Vol(θ) and one may take M_j(a)≡C), compute h1(a)=aε0m_U+2C, solve a from h1(a)=1/√B0 with B0=(Ac+2Kδ)/ε0, substitute into h2(B0), and solve h2(B0)=δ^{-2β/(2n+1)} for c(δ). Verify that c(δ)/δ→∞ and c(δ)<c1 for all small δ. Repeat with M_j(a)=C a^{-N} for N=1,2,3; if for some N the equation forces c(δ)=O(δ), or if h1 has no continuous branch from 0 to a0, then the inversion argument fails and Theorem 1.1 needs a different quantitative argument. Also check monotonicity of κ on (0,δ1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing hole is in the final quantitative step of the proof of Theorem 1.1, not in the cited Lemma 2.1. After (3.8) the authors set h1(a)=aε0m_U+2M5(a), choose a=h1^{-1}(1/√B0), then define h2(t)=M7(h1^{-1}(1/√t)t)/t^{3/2} and choose c so that h2(B0)=δ^{-2β/(2n+1)}. Since M5 is only assumed decreasing, h1 need not be monotone (e.g. M5(a)=C/a gives a minimum), so h1^{-1} is not well-defined unless a small branch is specified; no proof is given that h2 is continuous, monotone, or has the required range. This is not cosmetic: B0=(Ac+2Kδ)/ε0 is bounded below by 2Kδ/ε0, and Lemma 2.4 requires c(δ)/δ→∞. The h2 equation may force c=O(δ) for some admissible upper-bound functions, in which case Lemma 2.4 cannot be applied. Finally, κ(δ)=δ exp(-4A(CUB0+2√B0+2Kδ)/(ε0B0)) is replaced by κ^{-1}(δ) without proving that κ is invertible. These are internal, checkable steps; as written they leave the existence of F_U unproved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17150,"tokens_out":8414,"duration_ms":70066,"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":[{"comment":"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":"Section 3, proof of Theorem 1.1, after Eq. (3.8)"},{"comment":"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":"Section 3, final step of the proof of Theorem 1.1"},{"comment":"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":"Section 2, Lemma 2.1 and definition of u_{c,δ}"},{"comment":"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.","section":"Section 2, Eq. (2.4), and Section 3 proof of Theorem 1.1"}],"minor_comments":[{"comment":"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":"Section 1, Theorem 1.1 statement"},{"comment":"The word 'Höler' in the proof of Proposition 2.12 should be 'Hölder'.","section":"Section 2, Proposition 2.12"},{"comment":"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":"Section 2, Example 2.16"},{"comment":"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.","section":"Section 3, Lemma 3.6"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant problem and the announced result is plausible, but the proof of Theorem 1.1 contains several gaps in the final quantitative step, mainly the unjustified inversions of h1, h2, and κ. These are likely fixable by adding monotonicity/range arguments or by altering the choice of auxiliary functions, but as written the central claim is not proven. I recommend major revision rather than rejection because the issues are local and the overall framework appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThere 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.\n\nWho 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.","headline":"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.","tokens_in":17732,"tokens_out":634,"would_cite":true,"duration_ms":7390,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32U15","32W20","32Q15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["complex Monge-Ampère equations","big cohomology classes","modulus of continuity","Hölder continuous measures","quasi-plurisubharmonic functions","Kiselman-Legendre transform","non-pluripolar Monge-Ampère product","capacity estimates"],"falsifier":"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.","tokens_in":16650,"feed_emoji":"📐","tokens_out":12234,"duration_ms":91848,"temperature":0.7,"pith_summary":"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.","feed_headline":"Uniform continuity for Monge–Ampère potentials in big classes","feed_subtitle":"Even with non-integrable right-hand sides, solutions stay uniformly continuous away from the weight's singular set.","key_machinery":"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.","core_discovery":"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 $C_0$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Lemma 2.1, the Kiselman–Legendre regularization estimate for unbounded $\\theta$-psh functions on which the comparison subsolution is built.","marker":"[KN19]"},{"why":"Provides the overall proof template and the $L^1$ bound for the $\\delta$-regularization (Lemma 2.3) reused here.","marker":"[DDG+14]"},{"why":"Defines Hölder continuous measures and supplies the capacity-decay and weak-moderation estimates used to control measure on sublevel sets.","marker":"[DN14]"},{"why":"Establishes the non-pluripolar Monge–Ampère product and the big cohomology class framework in which the equation lives.","marker":"[BEGZ10]"},{"why":"Proved continuity of solutions in the Kähler case that Corollary 1.2 upgrades to equicontinuity.","marker":"[DL17]"},{"why":"Earlier continuity result for big classes by the first author that Theorem 1.1 strengthens into a uniform modulus.","marker":"[Dan22]"},{"why":"Source of the scalar bootstrap lemma (Lemma 3.2) used to turn capacity decay into sup-norm control.","marker":"[EGZ09]"},{"why":"Gives the $\\delta$-regularization construction and its Hessian estimates, the analytic foundation of the transform.","marker":"[Dem94]"}],"fun_headline_variants":["Monge-Ampère potentials in big classes get uniform continuity","Uniform continuity for degenerate Monge-Ampère equations in big classes","Potentials in big cohomology stay uniformly continuous","Equicontinuity for Monge-Ampère solutions in the Kähler case","Uniform continuity for Monge-Ampère potentials with Hölder measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Monge-Ampère potentials in big classes get uniform continuity","Uniform continuity for degenerate Monge-Ampère equations in big classes","Potentials in big cohomology stay uniformly continuous","Equicontinuity for Monge-Ampère solutions in the Kähler case","Uniform continuity for Monge-Ampère potentials with Hölder measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000643,"raw_usage":{"total_tokens":2926,"prompt_tokens":886,"completion_tokens":2040,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":1962}},"tokens_in":502,"tokens_out":2040,"duration_ms":14221,"temperature":1.0,"reasoning_tokens":1962,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:17:59.598522+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}