{"id":"e45e8ba8-2cfd-4e6e-b817-957c20f5178d","arxiv_id":"2502.02041","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Singular Kähler metrics with Ricci curvature bounded below and rational cohomology class induce non-collapsed RCD spaces homeomorphic to the projective variety, under a resolution condition on the anti-canonical bundle.","lead":"This paper proves uniform Sobolev estimates for Laplace solutions on Kähler manifolds with bounded Nash entropy and Calabi energy, then uses them to show that certain singular Kähler metrics induce non-collapsed RCD spaces homeomorphic to the original projective variety. It connects singular Kähler geometry to synthetic Ricci curvature theory and produces many new RCD examples from algebraic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Circular use of Proposition 4.1: the proofs of Theorems 7.1 and 1.3 invoke ω≥cθ_X before the regularization that Proposition 4.1 requires has been constructed; the asserted uniform Nash entropy for the approximating ω_i is also unsupported.","rationale":"The paper contains substantial original analytic content: Theorem 1.1’s uniform W^{2,2} and W^{1,4} estimates from a Calabi-energy bound are new, and the Bochner computations in Section 2 appear coherent. Proposition 3.1’s eigenfunction Lipschitz argument is elaborate and plausible, and the connection to RCD spaces is an interesting direction. However, the proof of the central theorem has a structural, not merely typographical, problem: Proposition 4.1 is used before the regularization it requires has been constructed. This is a genuine circular dependency affecting both Theorem 7.1 and the general case in Section 8. The second asserted step — that the approximate solutions ω_i automatically lie in V(X,θ_X,n,A,p,K′) — is also unsupported, since a uniform upper bound on φ_i alone does not imply the entropy bound (1.3). I do not see these as fatal to the truth of the statement; they may be repairable, for instance by constructing the regularization first via Proposition 5.1 without using ω≥cθ_X, and then deriving the lower bound. But as written, the proof of Theorem 1.3 is incomplete at exactly the point where the reduction to the approximating sequence is made. The reader’s CONDITIONAL verdict is therefore appropriate: the theorem is plausible but should not be accepted as fully proved until the circularity is removed and the uniform Nash-entropy bound is either proved or replaced by a weaker sufficient condition. I agree with the reader that Section 8 is the weak point, but I identify the primary problem as the circular appeal to Proposition 4.1 rather than the PSH-class mismatch, which is probably a harmless typo since η=η0+i∂∂̄ψ with η≥0 would normally give ψ∈PSH(X,η0).","tokens_in":25647,"tokens_out":11151,"duration_ms":115914,"concrete_test":"Analytical dependency check: write down, for each invocation of Proposition 4.1 in Sections 7 and 8, the list of hypotheses (3.1)–(3.3) that the proposition requires, and verify whether those hypotheses are established before the invocation. Then re-derive Proposition 5.1 and check whether its construction of the twisted cscK metrics ω_ε uses the lower bound ω≥cθ_X or the bound −ω≤Ric(ω)≤Cω anywhere. If Proposition 5.1 does not need these bounds, the circularity is removable by reordering the proof; if it does, Theorem 1.3 currently lacks a non-circular proof of the lower bound. In the same re-derivation, test the claimed uniform Nash entropy of the sequence ω_i defined by (8.2) by checking whether the L^p norm in (1.3) can be bounded even when ψ is unbounded below on the singular set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is a circularity in the reduction to twisted cscK approximations. In the proof of Theorem 7.1 (Section 7), after setting η=Ric(ω)+ω, the text says: “By Proposition 4.1, ω dominates θX, i.e., there exists C>0 such that −ω≤Ric(ω)≤Cω.” But Proposition 4.1 is stated and proved under the hypothesis that ω already admits a regularization (Y,{ω_j}) satisfying (3.1)–(3.3). At that point in Theorem 7.1 no regularization has been constructed; constructing one is precisely the task of the proof, via Proposition 5.1. The same circular step reappears at the start of Section 8: to reduce to the case where η is a Kähler current dominating θX, the text sets η′=ω+η and says “η′ is indeed a Kähler current by Proposition 4.1.” Again, Proposition 4.1 cannot be invoked before the regularization is established. This is not a cosmetic issue: the lower bound ω≥cθX is itself one of the conclusions of Theorem 1.3, and it is being used as an input to the approximation scheme. The rest of Section 8 — choosing η0, solving Ric(ω_i)=−ω_i+η_i, and applying Theorem 7.1 — depends on η being a Kähler current. A second serious gap is the assertion after (8.2) that ω_i=ω0+i∂∂̄φ_i lies in V(X,θ_X,n,A,p,K′) with uniform Nash entropy for all i. The preceding argument only gives a uniform upper bound for φ_i; it does not establish the L^p bound on |log(V^{-1}ω_i^n/θ_X^n)| required by (1.3), and the text gives no control of ψ_i from below. Without this bound, Theorem 7.1 cannot be applied to the sequence. The PSH-class mismatch in Lemma 8.1 (ψ∈PSH(X,(1+ε)η0) versus the needed ψ∈PSH(X,η0)) and the KY/−KY sign typo are real but appear secondary; the circularity and the missing entropy bound are the load-bearing problems.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops uniform Sobolev and gradient estimates for solutions of the Laplace equation on families of Kähler metrics with bounded Nash entropy and Calabi energy, and applies them to show that certain singular Kähler metrics on normal projective varieties induce non-collapsed RCD spaces that are homeomorphic to the underlying variety. The main global result, Theorem 1.3, asserts that under a resolution condition on the anti-canonical bundle and a lower Ricci bound as currents, every admissible singular Kähler metric with rational cohomology class yields an RCD(-1,2n) space whose regular set is the smooth locus and whose singular set has Hausdorff dimension at most 2n-3.","tokens_in":26017,"tokens_out":5127,"duration_ms":53152,"significance":"If the main theorem is correct, it provides a substantial bridge between pluripotential theory on singular varieties and the RCD theory of metric measure spaces, producing abundant examples of RCD spaces that are topologically and holomorphically equivalent to algebraic varieties. The Bochner-based estimates in Section 2 are coherent and are a genuine contribution: they upgrade the existing W^{1,2} theory to uniform W^{2,2} and W^{1,4} bounds under a Calabi energy bound, and the Green function estimates are a natural corollary. The paper is also careful in many places to cite the prior groundwork in [24,25,26] and [42]. However, several load-bearing steps in Sections 7 and 8 are asserted rather than proved, and the logical ordering and hypotheses there need correction before the main theorem can be accepted.","major_comments":[{"comment":"The proof invokes Proposition 4.1 to conclude that ω dominates θ_X, writing \"By Proposition 4.1, ω dominates θ_X\", before the regularization hypotheses of Proposition 4.1 have been established. Proposition 4.1 is stated and proved under the assumption that ω admits a regularization (Y,{ω_j}) satisfying (3.1)-(3.3); at this point in Theorem 7.1 no such regularization has yet been constructed, since its construction is precisely the content of Proposition 5.1. The proof can likely be repaired by first invoking Proposition 5.1 to obtain the regularization and then applying Proposition 4.1, but as written the argument is circular and the assertion used in the proof of (7.2) is not justified.","section":"Section 7, proof of Theorem 7.1"},{"comment":"The reduction to the smooth twisted case writes η = η_0 + i∂∂̄ψ and states that ψ ∈ PSH(X,(1+ε)η_0) ∩ C^∞(X^∘). Lemma 8.1 and the extension theorem [13] that supports it require ψ ∈ PSH(X,η_0), not merely (1+ε)η_0-plurisubharmonicity. If the stated class is correct, the regularized maximum construction in Lemma 8.1 does not apply as written, because the approximants ψ_j are required to be η_0-PSH. If instead ψ is meant to be η_0-PSH, the text should state that hypothesis and justify why η_0 + i∂∂̄ψ is a Kähler current with the required domination property.","section":"Section 8, before Lemma 8.1"},{"comment":"The assertion that the twisted Kähler-Einstein metrics ω_i = ω_0 + i∂∂̄φ_i lie in V(X,θ_X,n,A,p,K′) with a uniform K′ is not derived. The argument gives only a uniform upper bound for φ_i from the Monge-Ampère equation and the pointwise convergence of ψ_i; it does not provide the L^p bound on |log(V^{-1}ω_i^n/θ_X^n)| required by (1.3), and no lower bound on ψ_i or φ_i is established. This uniform Nash entropy control is needed before Theorem 7.1 can be applied to the approximating sequence, so the reduction is incomplete.","section":"Section 8, after equation (8.2)"},{"comment":"The statement of Theorem 7.1 says \"KY is π-nef\", but the surrounding results and the proof require −K_Y to be π-nef. Proposition 5.1, Theorem 1.3, and the discussion in the introduction all concern the anti-canonical bundle −K_Y being π-nef or π-effective. As stated, the sign in Theorem 7.1 is inconsistent with its own proof, which invokes Proposition 5.1, and the statement must be corrected to −K_Y. This is not merely a typographical issue, since the nefness of K_Y and of −K_Y are very different hypotheses.","section":"Theorem 7.1, assumption (1)"},{"comment":"Theorem 1.3 is stated for the case where −K_Y is π-nef or π-effective, but the proof in Sections 7 and 8 only addresses the π-nef case: Theorem 7.1 assumes π-nefness, and Section 8 reduces the general case to Theorem 7.1. No separate argument is supplied for the π-effective branch beyond the remark that Theorem 1.3 has been proved in [22] in that case. If the paper intends to rely on [22] for the π-effective case, that dependence should be stated explicitly in the proof of Theorem 1.3; otherwise the theorem is not proved as stated.","section":"Theorem 1.3, π-effective alternative"}],"minor_comments":[{"comment":"There are several typos and small errors: \"equppied\" in the introduction, \"no great than\" for \"no greater than\" in Theorem 1.3, \"Ccontradiction\" in the proof of Lemma 8.5, and \"small δ >> ǫ\" in Lemma 8.1 where the intended order of δ and ǫ is unclear.","section":"Throughout"},{"comment":"The notation Y is used for the Gromov-Hausdorff limit space in Lemma 8.4, which conflicts with the resolution Y used throughout the paper; a different symbol such as Z would avoid ambiguity.","section":"Lemma 8.4"},{"comment":"The sentence \"ω_{ǫ_j} converges smoothly to ω on any compact subset of Y^∘\" should presumably read \"on any compact subset of π^{-1}(X^∘)\", since ω is defined on X and π^{-1}(X^∘) is the relevant smooth region of the resolution.","section":"Section 5, Proposition 5.1"},{"comment":"The displayed condition \"0 ≤ ρ ≤ 1\" should be \"0 ≤ ρ_k ≤ 1\", and the indexing of the cut-off functions should be made consistent with the domains Ω_k.","section":"Lemma 3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution that builds on a substantial body of prior work by the authors and by Székelyhidi. The main concern is not the novelty or the plausibility of the results but the incomplete logical chain in Sections 7 and 8: the circular use of Proposition 4.1, the mismatched plurisubharmonicity class in the reduction, and the missing uniform Nash entropy bound for the approximating twisted Kähler-Einstein metrics. These issues are likely repairable by reorganizing the proof and adding the missing estimates, but they are load-bearing for Theorem 1.3. I would recommend against acceptance until the authors provide a complete proof of the reduction in Section 8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"At its core this is two papers. The first is a clean analytic result: uniform W^{2,2} and W^{1,4} estimates for Laplace solutions on Kähler manifolds with bounded Nash entropy and Calabi energy. The Bochner argument is coherent, the Green's function estimates follow from the cited work, and I think Theorem 1.1 is right. The second is an ambitious geometric conclusion: every admissible singular Kähler metric with Ricci bounded below and rational class on a variety with π-nef/effective anti-canonical resolution produces a non-collapsed RCD(-1,2n) space homeomorphic to the variety. That is a real generalization of the Kähler-Einstein cases in [42] and [11], and if it goes through it gives a wealth of new RCD examples from algebraic geometry.\n\nThe soft spots are in the proof of the general case, not in the analytic core. In the proof of Theorem 7.1, Proposition 4.1 is invoked to get ω ≥ cθX before any regularization has been constructed, even though Proposition 4.1 is stated for metrics that already admit a regularization. The same circular step appears in Section 8 where η' is declared a Kähler current via Proposition 4.1. This might be fixable by reordering—Proposition 5.1 does not obviously need the lower bound—but it is a genuine logical gap as written. The more serious issue is the assertion after (8.2) that ω_i lies in V with uniform Nash entropy. The text only establishes a uniform upper bound for φ_i at that point; the L^p control on log(ω_i^n/θ_X^n) required by the definition of V is not derived, and Lemma 8.2 later uses the V membership to prove φ_i is bounded, which is circular. There are also two smaller blemishes: Theorem 7.1 says 'KY is π-nef' where it should be '−KY', and Lemma 8.1 needs ψ ∈ PSH(X,η0) while the text only gives ψ ∈ PSH(X,(1+ε)η0). Both are easy patches.\n\nThis paper deserves a serious referee. The main analytic estimates are likely correct and useful on their own, and the geometric theorem is important enough that a careful referee should sort out whether the reduction can be repaired. I would send it to peer review with the expectation of a substantial revision.","headline":"A solid analytic core with an ambitious RCD theorem whose proof needs real repair before the general case is ready.","tokens_in":26660,"tokens_out":7646,"would_cite":true,"duration_ms":72518,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","32Q20","53C23","32W20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on normal projective varieties with log terminal singularities and a resolution with relative nef or effective anti-canonical bundle, any admissible singular Kähler metric with Ricci curvature bounded below induces…","keywords":["Kähler metrics","Nash entropy","Calabi energy","RCD spaces","singular Kähler metrics","Ricci curvature bounded below","Sobolev estimates","geometric regularization"],"falsifier":"A direct way to test the central claim: construct a normal projective variety $X$ satisfying the resolution hypothesis and a singular Kähler metric $\\omega$ with $\\operatorname{Ric}(\\omega) \\ge -\\omega$ whose metric completion $(\\hat X,d_\\omega)$ has a singular point with tangent cone splitting off $\\mathbb{R}^{2n-1}$ or $\\mathbb{R}^{2n-2}$; Theorem 1.3 predicts this cannot happen. On the proof level, one could examine a concrete example where $\\eta = \\operatorname{Ric}(\\omega)+\\omega$ has a potential $\\psi$ only in $\\operatorname{PSH}(X,(1+\\varepsilon)\\eta_0)$ and show that no decreasing sequence of smooth $\\eta_0$-PSH approximations exists, which would invalidate Lemma 8.1 as stated.","tokens_in":2357,"feed_emoji":"📐","tokens_out":3775,"duration_ms":110883,"temperature":0.7,"pith_summary":"This paper proves new uniform Sobolev estimates for solutions of the Laplace equation and for Green's functions on Kähler manifolds with bounded Nash entropy and bounded Calabi energy, and uses them to analyze singular Kähler spaces. The main theorem states that on a normal projective variety with log terminal singularities that admits a resolution with $\\pi$-nef or $\\pi$-effective anti-canonical bundle, every admissible singular Kähler metric with $\\operatorname{Ric}(\\omega) \\ge -\\omega$ induces a non-collapsed compact $\\mathrm{RCD}(-1,2n)$ space homeomorphic to the variety. The regular set of this RCD space is exactly the smooth locus, and the singular set has Hausdorff dimension at most $2n-3$. If true, the result supplies a large family of RCD spaces arising from algebraic geometry and confirms that a pluripotential-theoretic Ricci lower bound produces a synthetic Ricci lower bound in the RCD sense.","feed_headline":"Singular Kähler metrics become RCD spaces homeomorphic to the variety","feed_subtitle":"New Sobolev estimates connect singular Kähler geometry to synthetic Ricci curvature bounds.","key_machinery":"The load-bearing mechanism is a Bochner-formula estimate with a power trick: for $\\varphi = u^\\beta$ with $\\beta = 8/9$, the paper derives an integrated lower bound for $\\Delta|\\nabla\\varphi|^2$ that controls Hessian and gradient terms, with the Calabi energy bound absorbing the negative Ricci contribution. Iterating this estimate gives uniform $W^{2,2}$ and $W^{1,4}$ bounds for Laplace solutions and Green's functions (Theorem 1.1). These bounds make eigenfunctions Lipschitz via a Green's-function representation and Riesz–Thorin interpolation (Proposition 3.1), yield a Schwarz lemma $\\omega \\ge c\\theta_X$ (Proposition 4.1), and allow approximation by twisted cscK metrics (Proposition 5.1). The homeomorphism statement in the general case is carried by a regularization of the Kähler current $\\eta = \\operatorname{Ric}(\\omega)+\\omega$ by smooth forms, followed by convergence of the associated RCD spaces and partial $C^0$ holomorphic sections that separate points (Lemma 8.7 and Corollary 8.2).","core_discovery":"On the paper's own terms, the central discovery is that bounded Nash entropy together with bounded Calabi energy (equivalently an $L^2$ bound on the negative part of the Ricci form) yields higher-order linear estimates: for solutions $u$ of $\\Delta_\\omega u = f$, the quantities $|\\nabla\\nabla u|^2$, $|\\nabla\\bar\\nabla u|^2$, and $|\\nabla u|^4$ are uniformly integrable, and the same holds off small balls for the Green's function. These estimates upgrade the earlier $W^{1,2}$ theory to the regularity needed for RCD analysis. The paper then shows that, under the resolution hypothesis $-K_Y$ being $\\pi$-nef or $\\pi$-effective, any singular Kähler metric $\\omega$ in the admissible class with $\\operatorname{Ric}(\\omega) \\ge -\\omega$ admits a regularization by smooth twisted cscK metrics with uniform Nash entropy and Calabi energy, and that the metric completion $(\\hat X, d_\\omega, \\omega^n)$ is a non-collapsed $\\mathrm{RCD}(-1,2n)$ space homeomorphic to $X$, with $\\omega \\ge c\\theta_X$.","pith_inferences":["If the current-theoretic Ricci lower bound is equivalent to the synthetic RCD condition for this class, Theorem 1.3 would extend to all singular Kähler metrics with synthetic Ricci bounds, without the resolution hypothesis.","The remark that only an $L^{2-\\varepsilon}$ bound on the negative Ricci part suffices suggests the Calabi-energy hypothesis could be relaxed, potentially covering collapsing families.","A concrete test of the proof's core step: check whether the potential $\\psi$ of $\\eta = \\operatorname{Ric}(\\omega)+\\omega$ can always be chosen $\\eta_0$-PSH. If not, Lemma 8.1 and the twisted cscK approximation would need a modified construction for the general case.","The theorem implies that singular Kähler-Einstein currents on varieties of Fano type are RCD spaces, matching the recent independent result for Kähler-Einstein currents; comparing the two approximation schemes might simplify both proofs."],"forward_implications":["Every singular Kähler metric satisfying the hypotheses of Theorem 1.3 determines a non-collapsed $\\mathrm{RCD}(-1,2n)$ space whose underlying topological space is $X$ itself; the analytic metric completion recovers the original variety.","The singular set of this RCD space has Hausdorff dimension at most $2n-3$, and at most $2n-4$ when the Ricci curvature of $\\omega$ is bounded from both sides.","The uniform estimate $\\omega \\ge c\\theta_X$ makes the identity map from $(\\hat X,d_\\omega)$ to $(X,\\theta_X)$ Lipschitz, so the RCD structure is quantitatively compatible with the ambient embedding.","Theorem 1.1 supplies a general linear-analysis tool: uniform Laplace and Green's-function estimates on any family of Kähler manifolds with bounded Nash entropy and Calabi energy, independent of the metric's Ricci lower bound.","For any projectively embeddable variety with a resolution with $-K_Y$ $\\pi$-nef or $\\pi$-effective, there are infinitely many RCD spaces topologically and holomorphically equivalent to $X$, coming from Kähler currents with Ricci curvature bounded below."],"supporting_citations":[{"why":"Supplies the spectral theory, Green's functions, and $W^{1,2}$ bounds for singular Kähler spaces that the paper upgrades to higher-order estimates.","marker":"[25]"},{"why":"Provides the uniform $L^q$ Green's function estimates used in the $L^\\infty$ bound for Laplace solutions and in the Green's function derivative estimates.","marker":"[24]"},{"why":"Establishes the singular Kähler-Einstein RCD case whose approximation and partial $C^0$ methods Theorem 1.3 generalizes; its Theorem 3 and Lemma 36 are invoked directly.","marker":"[42]"},{"why":"Gives the dimension-three and $\\pi$-effective cases and the approximation strategy that Section 8 follows for the general case.","marker":"[22]"},{"why":"Used to extend $\\eta_0$-plurisubharmonic potentials to the ambient projective space, the key input in the regularization Lemma 8.1.","marker":"[13]"},{"why":"Provides partial $C^0$ estimates and holomorphic-section separation for Gromov-Hausdorff limits of Kähler manifolds with Ricci lower bounds, used in Lemma 8.7.","marker":"[30]"},{"why":"Gives the criterion that almost smooth metric measure spaces with Lipschitz eigenfunctions are RCD spaces, applied in Corollary 3.2.","marker":"[28]"},{"why":"Yields the dimension estimates for singular sets of non-collapsed RCD spaces used to bound the singular set of the limit.","marker":"[17]"}],"fun_headline_variants":["Singular Kähler metrics yield RCD spaces homeomorphic to projective varieties","Sobolev estimates connect singular Kähler geometry to RCD spaces with same topology","Bounded Nash entropy and Calabi energy regularize singular Kähler metrics into RCD","RCD spaces from singular Kähler metrics: homeomorphic to the original variety"],"cache_read_input_tokens":28544,"weakest_assumption_plain":"The argument's load-bearing step is the assertion that the singular current $\\eta = \\operatorname{Ric}(\\omega)+\\omega$ can be approximated by smooth Kähler forms via potentials that are genuinely $\\eta_0$-plurisubharmonic, and that the resulting twisted Kähler-Einstein metrics automatically lie in the admissible class with uniform Nash entropy and Calabi energy bounds; this is stated rather than fully derived, so the general proof would fail if the stated class $\\operatorname{PSH}(X,(1+\\varepsilon)\\eta_0)$ is the correct one.","fun_headline_variants_meta":{"raw":{"variants":["Singular Kähler metrics yield RCD spaces homeomorphic to projective varieties","Sobolev estimates connect singular Kähler geometry to RCD spaces with same topology","Bounded Nash entropy and Calabi energy regularize singular Kähler metrics into RCD","RCD spaces from singular Kähler metrics: homeomorphic to the original variety"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001515,"raw_usage":{"total_tokens":6049,"prompt_tokens":899,"completion_tokens":5150,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":5069}},"tokens_in":515,"tokens_out":5150,"duration_ms":36836,"temperature":1.0,"reasoning_tokens":5069,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:35:03.362846+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct way to test the central claim: construct a normal projective variety $X$ satisfying the resolution hypothesis and a singular Kähler metric $\\omega$ with $\\operatorname{Ric}(\\omega) \\ge -\\omega$ whose metric completion $(\\hat X,d_\\omega)$ has a singular point with tangent cone splitting off $\\mathbb{R}^{2n-1}$ or $\\mathbb{R}^{2n-2}$; Theorem 1.3 predicts this cannot happen. On the proof level, one could examine a concrete example where $\\eta = \\operatorname{Ric}(\\omega)+\\omega$ has a potential $\\psi$ only in $\\operatorname{PSH}(X,(1+\\varepsilon)\\eta_0)$ and show that no decreasing sequence of smooth $\\eta_0$-PSH approximations exists, which would invalidate Lemma 8.1 as stated.","supporting_citations":[{"cited_title":"and Sturm, J","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral theory, Green's functions, and $W^{1,2}$ bounds for singular Kähler spaces that the paper upgrades to higher-order estimates."},{"cited_title":"and Song, J","cited_arxiv_id":null,"evidence_quote":"Gives the dimension-three and $\\pi$-effective cases and the approximation strategy that Section 8 follows for the general case."},{"cited_title":"On the extension of quasiplurisubharmonic functions , Anal","cited_arxiv_id":null,"evidence_quote":"Used to extend $\\eta_0$-plurisubharmonic potentials to the ambient projective space, the key input in the regularization Lemma 8.1."},{"cited_title":"and Sz´ ekelyhidi, G.Gromov-Hausdorff limits of K¨ ahler manifolds with Ricci cur vature bounded below, Geom","cited_arxiv_id":null,"evidence_quote":"Provides partial $C^0$ estimates and holomorphic-section separation for Gromov-Hausdorff limits of Kähler manifolds with Ricci lower bounds, used in Lemma 8.7."},{"cited_title":"Bakry- `Emery conditions on almost smooth metric measure spaces , Anal","cited_arxiv_id":null,"evidence_quote":"Gives the criterion that almost smooth metric measure spaces with Lipschitz eigenfunctions are RCD spaces, applied in Corollary 3.2."},{"cited_title":"Non-collapsed spaces with Ricci curvature bounded from below, J","cited_arxiv_id":null,"evidence_quote":"Yields the dimension estimates for singular sets of non-collapsed RCD spaces used to bound the singular set of the limit."}],"review_version":1}