{"id":"72420f5f-0322-4ede-955d-878564fd79ea","arxiv_id":"2508.20933","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hölder estimates for degenerate complex Monge-Ampère equations are established on smoothable singular Kähler varieties, confirming a conjecture for Kähler-Einstein potentials.","lead":"This paper proves that certain singular spaces called smoothable Kähler-Einstein varieties have Hölder continuous potentials, meaning the functions describing their metrics cannot oscillate wildly. It introduces a geometric regularization method based on projective embeddings and effective finite generation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2's partition-of-unity construction of approximating curves is unjustified, and Theorem 2 depends on it.","rationale":"The reader's verdict is CONDITIONAL, and the weakest assumption identified is exactly the curve approximation in Proposition 2. My independent reading reaches the same conclusion: the proof's finite-cover step is asserted in one sentence ('construct γ_j using a partition of unity') but a partition of unity does not naturally produce a curve lying in the variety X_j. Since Proposition 2 is the only mechanism that converts the uniform Hölder estimates of Theorem 1 into the intrinsic-to-extrinsic distance comparison needed for Theorem 2, this is the most load-bearing gap. Other issues I noticed—the duplicated definition of α, the (n+1) vs (n+2) exponents in the final line of Theorem 1's proof, and the off-by-one in the induction range of Lemma 3—appear to be fixable typos or routine repairs; the Proposition 2 issue is structural and needs a real argument. I therefore do not propose a different verdict from the reader's CONDITIONAL; the paper should not be fully accepted until the partition-of-unity construction is either justified rigorously or replaced by a valid gluing argument.","tokens_in":23778,"tokens_out":26265,"duration_ms":278968,"concrete_test":"Write out the finite-cover step of Proposition 2 explicitly: choose a partition of unity subordinate to the intervals I_a where γ∞⊂V^a∞, define the lift in X_j via the local graphs ψ^a_j:U^a→V^a_j, and at overlap/transition boundaries join the two endpoints by a curve in X_j of length O(ε), using that the endpoints are O(ε)-close in PN and that X_j is a smooth submanifold with charts converging to those of X∞. Then check the total added length is O(ε). As a numerical/nontrivial model case, test the assertion on the degeneration of smooth quadrics in CP^3 to a quadric cone, with points on different rulings approaching the vertex: estimate d_{X_j}(x_j,y_j) and compare to the Lojasiewicz bound C d_FS(x∞,y∞)^μ. If the explicit construction cannot be completed, or the model case violates the bound, Proposition 2 and hence Theorem 2 are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The bridge from Theorem 1 to Theorem 2 is Proposition 2 (Section 5). It must show that, for algebraic approximations X_j→X∞ and points x_j→x∞, y_j→y∞, limsup_j d_{X_j}(x_j,y_j) ≤ C d_{X∞}(x∞,y∞)^μ. The proof first uses Lojasiewicz to find a regular arc γ∞ in X∞^{reg} of length ≤ C d_FS(x∞,y∞)^μ, then needs curves γ_j in X_j with ℓ(γ_j) ≤ ℓ(γ∞)+ε. When γ∞ is not contained in a single local graph, the paper says: 'we choose a finite cover and construct γ_j using a partition of unity.' A partition of unity on the parameter interval produces convex combinations in the ambient PN, which do not generally lie in the subvariety X_j. To make this work one must construct the curve chart-by-chart and connect the chart-wise lifts in overlaps by short arcs in X_j, with uniform control on transition functions and overlap sizes. No such construction or estimate is supplied. This is not a cosmetic gap: without it, the limiting inequality (5.2) is unproved, so Theorem 2 does not follow from Theorem 1. The same type of comparison is needed implicitly in Lemma 13/Proposition 3, so the concern is load-bearing for the paper's main applications.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24079,"tokens_out":11206,"duration_ms":108242,"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":[{"comment":"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.","section":"Section 4, final paragraph of Proof of Theorem 1"},{"comment":"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.","section":"Section 5, proof of Proposition 2"},{"comment":"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.","section":"Section 6, Lemma 13"}],"minor_comments":[{"comment":"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.","section":"Section 4, Lemma 3 proof"},{"comment":"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).","section":"Section 5, proof of Proposition 2"},{"comment":"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.","section":"Section 6, Lemma 8"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the first genuine Holder estimate on singular Kahler varieties for degenerate Monge-Ampere, in the smoothable case. If it holds up, it confirms the Guedj-Guenancia-Zeriahi conjecture and gives a new quantitative bridge between intrinsic and extrinsic metrics. Second, the paper is not fully verified as written: the bridge from the smooth family to the singular limit (Proposition 2) contains a partition-of-unity argument that does not work as stated, and the proof of Theorem 1 has several exponent typos that block line-by-line checking. Both look fixable, but they are not cosmetic.\n\nWhat is actually new: the main mechanism is a geometric regularization. The authors combine the partial C0 estimate with Li's effective finite generation to get uniform Bergman approximations, then use their Proposition 1 to control gradients of sections in sufficiently divisible degree. That yields Theorem 1, a uniform Holder estimate for the whole family F(n,d) — that is the engine. The application to Kahler-Einstein potentials on smoothable varieties is the payoff. The strategy is natural, the external tools (Donaldson-Sun, Liu-Szekelyhidi, Zhang, Li) are used appropriately, and the citation pattern is honest.\n\nThe soft spots are real. Proposition 2 claims that when the Lojasiewicz arc does not lie in a single graph, one can “construct γ_j using a partition of unity.” That does not work as stated: a partition of unity on the parameter interval produces convex combinations in affine space, which need not lie on X_j. One needs a chart-by-chart construction with uniform control on overlaps; nothing like that is supplied. This is load-bearing for Theorem 2 and also implicitly for Lemma 13/Proposition 3. The stress-test concern is correct. Separately, in the proof of Theorem 1 the definition of alpha switches from 1/κ to log(n+1)/log D, and the exponent has (n+1)^r in one place while Lemma 3 uses (n+2)^r. The induction statement in Lemma 3 is also stated for (n+1)^r but proved for (n+2)^r. These inconsistencies are probably typos — the estimates look oriented in the right direction — but as written they prevent verification.\n\nNone of this is circular: the Holder bounds are derived from external compactness, stability, and finite-generation results, not assumed as inputs. The paper does not hide its limitations; it explicitly notes that more regularity than Holder should not be expected.\n\nWho this is for: anyone working on Monge-Ampere regularity, singular Kahler-Einstein metrics, or compactness of moduli. It deserves a serious referee despite the gaps. My recommendation: send it out. A careful revision that fixes the exponent typos and, with more work, patches the curve approximation in Proposition 2 would make this a clear accept.","headline":"First Holder estimates on smoothable singular KE varieties, with a real gap in the curve-comparison lemma and fixable exponent typos.","tokens_in":24557,"tokens_out":3016,"would_cite":true,"duration_ms":30482,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32W20","32Q20","32U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Uniform Hölder estimates hold for degenerate complex Monge-Ampère equations on singular Kähler varieties.","keywords":["complex Monge-Ampère equations","Hölder estimates","singular Kähler varieties","Kähler-Einstein metrics","partial C^0 estimate","Kodaira embeddings","smoothable varieties","pluripotential theory"],"falsifier":"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.","tokens_in":23669,"feed_emoji":"📐","tokens_out":10523,"duration_ms":94885,"temperature":0.7,"pith_summary":"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.","feed_headline":"Uniform Hölder bounds proven on singular Kähler varieties","feed_subtitle":"New estimates for degenerate Monge-Ampère equations and Kähler-Einstein potentials on smoothable singular varieties.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the refined partial C^0 estimate giving uniform two-sided bounds on Bergman kernels for the family.","marker":"[59]"},{"why":"Supplies Gromov-Hausdorff compactness, algebraic convergence, and the curve approximations used to pass to singular limits.","marker":"[15]"},{"why":"Extends the partial C^0 and Gromov-Hausdorff framework to families satisfying only a Ricci lower bound.","marker":"[36]"},{"why":"Supplies effective finite generation of sections with uniform coefficient bounds, used to bound derivatives of sections.","marker":"[30]"},{"why":"Provides the Skoda-type division theorem used in proving the effective finite-generation estimate.","marker":"[40]"},{"why":"Croke's Sobolev inequality gives the uniform Sobolev constants needed to invoke the partial C^0 estimate.","marker":"[11]"},{"why":"Lojasiewicz's semianalytic arc theorem bounds the length of arcs in the singular limit by a power of the ambient distance.","marker":"[37]"},{"why":"Stability theorem for bounded solutions of degenerate complex Monge-Ampère equations, used to pass from approximations to the limit in Theorem 3.","marker":"[12]"},{"why":"Provides uniform L^∞ estimates on smooth fibers via the α-invariant, needed in the proof of Theorem 3.","marker":"[19]"}],"fun_headline_variants":["Hölder bounds for Monge-Ampère on singular Kähler varieties","Partial C^0 estimate yields Hölder regularity on singular Kähler varieties","Hölder estimates for degenerate Monge-Ampère on singular Kähler","Hölder continuity for Kähler-Einstein potentials on smoothable varieties"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hölder bounds for Monge-Ampère on singular Kähler varieties","Partial C^0 estimate yields Hölder regularity on singular Kähler varieties","Hölder estimates for degenerate Monge-Ampère on singular Kähler","Hölder continuity for Kähler-Einstein potentials on smoothable varieties"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001309,"raw_usage":{"total_tokens":5138,"prompt_tokens":677,"completion_tokens":4461,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":421,"completion_tokens_details":{"reasoning_tokens":4387}},"tokens_in":421,"tokens_out":4461,"duration_ms":30477,"temperature":1.0,"reasoning_tokens":4387,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:42:39.832427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"PDE 14 (2021), no","cited_arxiv_id":null,"evidence_quote":"Supplies the refined partial C^0 estimate giving uniform two-sided bounds on Bergman kernels for the family."},{"cited_title":"213 (2014), no","cited_arxiv_id":null,"evidence_quote":"Supplies Gromov-Hausdorff compactness, algebraic convergence, and the curve approximations used to pass to singular limits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the partial C^0 and Gromov-Hausdorff framework to families satisfying only a Ricci lower bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies effective finite generation of sections with uniform coefficient bounds, used to bound derivatives of sections."},{"cited_title":"Techniques for the Analytic Proof of the Finite Generation of the Canonical Ring","cited_arxiv_id":"0811.1211","evidence_quote":"Provides the Skoda-type division theorem used in proving the effective finite-generation estimate."},{"cited_title":"Some isoperimetric inequalities and eigenvalue estimates , Ann","cited_arxiv_id":null,"evidence_quote":"Croke's Sobolev inequality gives the uniform Sobolev constants needed to invoke the partial C^0 estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lojasiewicz's semianalytic arc theorem bounds the length of arcs in the singular limit by a power of the ambient distance."},{"cited_title":"and Zhang, Z","cited_arxiv_id":null,"evidence_quote":"Stability theorem for bounded solutions of degenerate complex Monge-Ampère equations, used to pass from approximations to the limit in Theorem 3."},{"cited_title":"and Zeriahi, A","cited_arxiv_id":null,"evidence_quote":"Provides uniform L^∞ estimates on smooth fibers via the α-invariant, needed in the proof of Theorem 3."}],"review_version":1}