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H\"older estimates for degenerate complex Monge-Amp\`ere equations

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Pith's one-line read Uniform Hölder estimates hold for degenerate complex Monge-Ampère equations on singular Kähler varieties.

desk verdict First Holder estimates on smoothable singular KE varieties, with a real gap in the curve-comparison lemma and fixable exponent typos. read the letter →

arxiv 2508.20933 v1 pith:OSBCIYHF submitted 2025-08-28 math.CV math.DG

Bin Guo , Slawomir Kolodziej , Jian Song , Jacob Sturm This is my paper · ORCID
classification math.CVmath.DG MSC 32W2032Q2032U05
keywords complexMonge-AmpèreequationsHölderestimatessingularKählervarietiesKähler-EinsteinmetricspartialC^0estimateKodairaembeddingssmoothablepluripotentialtheory
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 solutions to degenerate complex Monge-Ampère equations on singular Kähler varieties are Hölder continuous, provided the singular variety is smoothable and the surrounding family satisfies a uniform Ricci lower bound and a diameter bound. The core result is a quantitative comparison, uniform over a whole family of polarized manifolds, between the Kähler potential and the extrinsic Fubini-Study distance coming from a Kodaira embedding. The paper then passes this estimate to singular limits and uses it for two targets: Kähler-Einstein currents on smoothable Kähler-Einstein varieties, and degenerate Monge-Ampère equations on smoothable mildly singular (klt) varieties with an L^p density and a quasi-plurisubharmonic bound on the right-hand side. This gives the first uniform Hölder estimates on singular Kähler varieties in the smoothable case and confirms the expected Hölder regularity of Kähler-Einstein potentials there.

What carries the argument

The main engine is the quantitative Kodaira embedding supplied by the partial C^0 estimate: for a family of compact Kähler manifolds with Ricci curvature bounded below and diameter bounded above, some fixed power of the polarization embeds every member uniformly, with Bergman kernels bounded above and below by constants. Around this, the paper combines Bergman-potential approximation of φ_X, with error proportional to log A over m; an effective finite-generation theorem and a Skoda-type division theorem used to estimate covariant derivatives of sections, giving |∇φ_m| ≤ C m^{κ−1}; and a weak effective finite-generation statement obtained by contradiction from Gromov-Hausdorff compactness of

What would settle it

Compute, for a degenerating family of smooth Kähler surfaces in F(2,D) with points x_j,y_j approaching a singular point, the ratio |φ_j(x_j)−φ_j(y_j)| / d_FS(x_j,y_j)^α; if for every α>0 this ratio is unbounded as j grows, Theorem 1 fails. For the singular statement, explicitly lift a short arc in the regular part of a singular limit to nearby smooth fibers and check whether the curve lengths converge as claimed; a sequence where the intrinsic distances do not converge to a Hölder power of the limit distance would break Theorem 2.

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Extended reading notes

Core claim

The central discovery is a uniform Hölder estimate for the potentials φ_X that compare a polarized manifold's Kähler metric ω_X with the Fubini-Study metric ω_FS pulled back from a projective embedding. For every dimension n and diameter bound D, the paper finds k, C, α depending only on n and D such that every member of the bounded family F(n,D) satisfies |φ_X(x) − φ_X(y)| ≤ C d_FS(x,y)^α, with φ_X normalized to have supremum zero. The proof approximates φ_X by Bergman potentials at level m, uses the partial C^0 estimate to control the approximation error, and uses effective finite generation together with a Skoda division theorem to bound the derivatives of sections by a power of m. Choosi

Load-bearing premise

The passage from smooth approximating manifolds to the singular limit rests on the assumption that any short arc in the regular part of the limit variety can be replaced by arcs in the nearby smooth fibers with essentially the same length, a step asserted via [15] and a partition of unity without a detailed proof.

Editorial extensions

If this is right

  • Kähler-Einstein currents on smoothable projective Kähler-Einstein varieties are Hölder continuous with respect to the ambient Fubini-Study distance of the smoothing family, confirming a conjecture of Guedj, Guenancia, and Zeriahi in this case.
  • For any smoothable mildly singular projective variety, the degenerate Monge-Ampère equation with e^F in L^p and −F quasi-plurisubharmonic has a unique Hölder continuous solution with respect to the ambient-projective distance.
  • If F is smooth on the variety, the solution to the degenerate equation is automatically Hölder continuous with respect to the ambient distance.
  • The constants in the main estimate depend only on dimension, diameter, and the Ricci lower bound, so the estimate survives passage to Gromov-Hausdorff limits.
  • The result establishes a quantitative bridge between intrinsic Kähler geometry and extrinsic projective geometry on singular Kähler varieties.

Reading between the lines

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

  • Editorial inference: the same strategy—approximate by smooth fibers, prove uniform estimates, pass to the limit—could yield Hölder bounds for other canonical currents, such as twisted or weighted Monge-Ampère equations, whenever uniform partial C^0 and finite-generation estimates are available.
  • Editorial inference: the one-sided comparison in the main theorem suggests a natural testable question: whether the reverse comparison also holds, giving a bi-Hölder equivalence between intrinsic and extrinsic distances on smoothable Kähler-Einstein varieties.
  • Editorial inference: the weakest step in the limit passage is the curve-approximation lemma; a counterexample or a completed proof of that lemma would directly determine whether the smoothable assumption in the applications can be relaxed.
  • Editorial inference: one could test the stability of the Hölder exponent numerically by approximating a known singular Kähler-Einstein variety by smooth fibers and computing the ratio of potential differences to powers of the ambient distance near the singular set.
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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

3 major / 4 minor

Summary. The paper develops a geometric regularization method, based on quantitative Kodaira embeddings and effective finite generation, to prove uniform Hölder estimates for complex Monge-Ampère equations on certain families of Kähler manifolds, and then applies this to singular Kähler varieties. Theorem 1 establishes a uniform extrinsic Hölder estimate for the potentials φ_X defined by ω_X = ω_FS + i∂∂̄φ_X over the family F(n,d). Theorem 2 uses this to prove Hölder continuity of Kähler-Einstein potentials on smoothable projective Kähler-Einstein varieties. Theorem 3 extends the method to degenerate Monge-Ampère equations on smoothable klt varieties. The overall strategy is plausible and rests on deep external results: the partial C0 estimate, effective finite generation, Gromov-Hausdorff/algebraic compactness, Lojasiewicz inequalities, and stability of complex Monge-Ampère equations.

Significance. If the technical gaps are repaired, this would be a substantial contribution: it would provide the first uniform Hölder estimates on singular Kähler varieties in the smoothable case, confirm a conjecture of Guedj-Guenancia-Zeriahi for smoothable Kähler-Einstein varieties, and give a framework for quantitative comparisons between intrinsic and extrinsic metric structures. The proof strategy is largely non-circular: no desired Hölder bound is assumed as an input, and the estimates are derived from independent external theorems. The applications in Theorems 2 and 3 are significant. However, several load-bearing points in the written proof are incomplete or internally inconsistent, so the current manuscript is not yet conclusive.

major comments (3)
  1. [Section 4, final paragraph of Proof of Theorem 1] The exponent computation is internally inconsistent. Lemma 3 is stated for m = (n+2)^r, but the final paragraph chooses r so that (n+1)^r ≤ d^{-1/κ} < (n+1)^{r+1}. The displayed inference '1/(n+1)^{r+1} ≤ d^{1/κ} =? ...' has the wrong inequality direction and does not yield the claimed bound. Moreover, α is first set to 1/κ and later to log(n+1)/log D, with D never defined. Since Theorem 1 is the engine for the rest of the paper, this gap must be repaired.
  2. [Section 5, proof of Proposition 2] The passage from the single-graph case to a finite cover is asserted without proof. 'We choose a finite cover and construct γ_j using a partition of unity' on the parameter interval produces convex combinations in P^N, which do not generally lie in X_j. To prove (5.2) one needs a chart-by-chart construction of γ_j with short connecting arcs in the overlaps and uniform control of transition functions and overlap sizes. No such construction or estimate is supplied. This is load-bearing: (5.2) is exactly how Theorem 1 is transferred to the singular limit in Theorem 2, and a similar comparison is needed in Lemma 13/Proposition 3.
  3. [Section 6, Lemma 13] Theorem 1 is applied to (X_t, ω_{t,j}) after Corollary 4 gives only Ric(ω_{t,j}) ≥ -Λ ω_{t,j} and diam ≤ D, not the normalized conditions Ric ≥ -1 and diam ≤ d required by the definition of F(n,d). The application is valid only after an explicit rescaling of ω_{t,j} by a factor depending on Λ and a corresponding change of polarization; this rescaling is not stated. Since Lemma 13 provides the uniform Hölder bound needed for Theorem 3, this step must be made explicit.
minor comments (4)
  1. [Section 4, Lemma 3 proof] The claim before (4.19) states 'for all u ∈ H^0(X, (n+1)^r L)', but the surrounding argument and the application to u_p ∈ H^0((n+2)^{r-1}L) require (n+2)^r; the variable U/u is also mixed. This appears to be a typo but should be corrected as part of the exponent tangle.
  2. [Section 5, proof of Proposition 2] There are duplicated 'Proof of Proposition 2' headings and a typo 'defnitions'. In the final part of the proof, the line 'd_FS(x_j,y_j) ≤ ℓ(γ_j)' should refer to the intrinsic distance d_{X_j}; as written it is not the quantity needed for (5.2).
  3. [Section 6, Lemma 8] The sentence 'for fixed j > 0, limsup_{j→∞} lim_{t→0} c_{t,j} = 0' has the limits in the wrong order relative to the intended statement; the normalization constants c_{t,j} should be controlled uniformly in t and j as stated in the proof. Please clarify.
  4. [Throughout] There are several small repetitions and typos, e.g., 'with klt singularities with klt singularities' in Corollary 1, and inconsistent use of C, C0, C1, C2 in the final display of Theorem 1. These should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and driven by external partial-C0, finite-generation, GH-limit, Lojasiewicz, and stability theorems.

full rationale

I walked the claimed derivation chain and found no circular step that can be exhibited with a reduction. Theorem 1 proves uniform Holder estimates for the family F(n,d) by comparing phi_X with Bergman potentials phi_m. The approximation error |phi_X - phi_m| <= log A/m comes from the partial C0 estimate (Corollary 2, from Zhang [59] and Croke [11]), not from any Holder bound. The gradient estimate for phi_m is obtained from effective finite generation (Theorem 5, proved in the paper via Siu's division theorem) and Proposition 1, whose proof uses GH limits from Donaldson-Sun [15]. No parameter in the conclusion (C, alpha) is fitted to the functions phi_X, and no equation is defined in terms of the target Holder estimate. Theorem 2 applies Theorem 1 to smooth approximants X_j and then uses Proposition 2 to compare extrinsic/intrinsic distances. Proposition 2 rests on Lojasiewicz's semianalytic arc theorem and algebraic convergence from [15], not on an imported uniqueness theorem or on the Holder estimate itself. The proof does contain a terse step: 'we choose a finite cover and construct gamma_j using a partition of unity' (Section 5). This is a potential omitted proof or correctness gap, because a naive partition of unity on the parameter interval need not keep the curve inside X_j; however, that is not circularity. It does not define the conclusion in terms of its input or rename a known result. Theorem 3 similarly uses the stability theorem of Dinew-Zhang [12] and Theorem 1 on smoothings; its auxiliary estimates come from Guedj-Guenancia-Zeriahi, Song-Tian, and the diameter estimates [22] (Guo-Phong-Song-Sturm). Those self-citations are to external theorems with stated assumptions and do not assume the target Holder conclusion. I therefore find no circularity and score 0.

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

The central results rest on a network of deep external theorems in Kaehler geometry, pluripotential theory, and algebraic geometry, all explicitly cited. The paper introduces no free parameters in the fitting sense; constants such as A(n,d), D(n,d), kappa(n,d) are outputs of the external results. No new geometric or physical entities are postulated.

assumptions (6)
  • domain assumption Uniform partial C0 estimate for the family F_r(n,d) (Zhang's Theorem 4, extending Tian, Donaldson-Sun, Liu-Szekelyhidi).
    Used to bound the Bergman kernel rho uniformly, giving the estimate |phi_X - phi_m| <= log A / m. This is an external deep theorem, not proved here. See Section 2, Theorem 4 and Corollary 2.
  • domain assumption Effective finite generation of the section ring by H^0(X,L) (Li's Theorem 5, adapted from Ric>0 to Ric >= -omega/2).
    Used to decompose sections into products with uniform bounds, leading to Lemma 2 and Lemma 3. The authors state the Ric>0 to Ric >= -omega/2 modification requires 'minor modifications' and provide a proof. Section 2, Remarks after Corollary 3.
  • domain assumption Gromov-Hausdorff compactness theory for Kaehler manifolds with Ricci lower bound (Donaldson-Sun, Liu-Szekelyhidi).
    Used in the contradiction proof of Proposition 1 and in Lemma 12 to obtain GH limits, algebraic convergence, and partial C0 estimates. Section 3.1.
  • standard math Lojasiewicz inequalities for semianalytic arcs and the resolution structure of normal varieties.
    Used in Proposition 2 to control intrinsic arc length by a power of extrinsic Fubini-Study distance. Section 5, proof of Proposition 2.
  • domain assumption Stability of degenerate complex Monge-Ampere equations (Dinew-Zhang).
    Used in Theorem 3 to pass from the approximating potentials phi_{0,j} to the solution phi. Section 6, Proposition 3.
  • domain assumption Uniform bound on Tian's alpha invariant for the smoothing family (Guedj-Guenancia-Zeriahi, Li).
    Used in Lemma 10 to obtain uniform L-infinity bounds for the approximating solutions phi_{t,j}. Section 6.

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Pith. "Pith review of H\"older estimates for degenerate complex Monge-Amp\`ere equations." pith.science (2026). https://pith.science/paper/OSBCIYHF

@misc{pith2026250820933,
  author       = {Pith},
  title        = {Pith review of: H\"older estimates for degenerate complex Monge-Amp\`ere equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OSBCIYHF}},
  note         = {Machine review of arXiv:2508.20933}
}
abstract

Uniform $L^\infty$ and H\"older estimates were proved by the Kolodziej for complex Monge-Amp\`ere equations on compact K\"ahler manifolds with $L^p$ volume measure with $p>1$. On the other hand, establishing H\"older estimates on singular K\"ahler varieties has remained open. In this paper, we establish uniform H\"older continuity for a family of complex Monge-Amp\`ere equations on K\"ahler varieties, by developing a geometric regularization based on the partial $C^0$ estimate, i.e., quantitive Kodaira embeddings. As an application, we prove that local potentials of smoothable K\"ahler-Einstein varieties are H\"older continuous.

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