{"id":"87fcfa5b-e700-4fce-bd05-746b36da1dab","arxiv_id":"1908.05842","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any holomorphic family of bounded strongly pseudoconvex domains in a Kähler manifold, the variation of complete Kähler-Einstein metrics is positive definite on the total space when the total domain is strongly pseudoconvex, and extends as a positive current under a scalar-curvature bound.","lead":"This paper proves that, in a holomorphic family of strongly pseudoconvex domains in a Kähler manifold, the fiberwise Kähler-Einstein metrics combine into a positive form when the total space is also strongly pseudoconvex, and that this form extends across singular fibers under a uniform scalar-curvature condition. It extends earlier positivity results from Euclidean spaces and compact fibers to general Kähler manifolds and arbitrary holomorphic maps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ratio c(ρ)/c(τ_r)→1 in Proposition 4.5 depends on mixed-derivative estimates |φ_{αs}|, |φ_{sβ}| = O(|r|^{-1/2-ε}) that are cited from the Euclidean case [5] and not proven for a general Kähler base; if they fail, Theorem 1.1 collapses.","rationale":"The reader's weakest_assumption correctly identifies the Achilles heel. I re-read the proof of Proposition 4.5. The only step that converts the fiberwise asymptotic (4.5) into control of the base-fiber mixed derivatives is the sentence 'Applying the Schauder estimates to φ_s and φ_s... see Section 3.3 in [5]'. This is the sole bridge between the known fiberwise Cheng-Yau estimates and the boundary blow-up of c(ρ). The rest of Proposition 4.5 (the algebraic decomposition of c(ρ) and boundedness of R3, R4) is local and depends only on the defining function r, not on the ambient metric; it should transfer from [4] unchanged. The genuine new ingredient is the dependence of the Monge-Ampère data on the base coordinate through Ric(ω_y), which is absent in [5]. I considered two other potential problems. (1) The use of real-analyticity in Proposition 4.2 is unnecessary: the strong minimum principle applied to Δc(ρ)-(n+1)c(ρ)≤0 with c(ρ)≥0 gives the same conclusion without real-analyticity, so this is only a presentational gap. (2) In Theorem 1.2, the uniform volume bound for ~ω_y^n follows from Diederich-Pinchuk after comparing ~ωD with the Euclidean metric on the relatively compact coordinate chart; this is standard and not load-bearing. Therefore the central concern is exactly the unverified mixed-derivative estimate. Since the authors cite a prior paper in a different setup and do not indicate how the Kähler-base case is reduced to it, the paper should be accepted only after this estimate is checked. This matches the reader's CONDITIONAL verdict.","tokens_in":11640,"tokens_out":39545,"duration_ms":383828,"concrete_test":"Re-derive the estimate for φ_s in the present setting: differentiate the fiberwise Monge-Ampère equation (3.1) with respect to the base coordinate s, obtaining Δ_g φ_s -(n+1)φ_s = -F_s + terms involving ∂_s(Ric(ω_y)|_{D_y}) and ∂_s r; compute the leading singularity of the source near r=0 for a model Kähler metric with nontrivial base dependence, e.g., ω_X = i∂∂bar(|z|^2+|s|^2+λ|z|^2|s|^2) on a neighborhood of r=|z|^2+|s|^2-1<0, and verify by explicit barrier functions whether |φ_{αs}| ≤ C|r|^{-1/2-ε} holds. If the barrier yields only |r|^{-1}, the proof of Proposition 4.5 is invalid as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the boundary estimate for parameter-derivatives of the Monge-Ampère potentials. Equation (4.5) bounds only the fiber-fiber components φ_{αβ̄}. The mixed components φ_{αs} and φ_{sβ̄}, which appear in R1 and control R1/c(τ_r)→0, are obtained by 'applying the Schauder estimates to φ_s and φ_s' with the citation 'for detailed proof, see Section 3.3 in [5]'. But [5] treats holomorphic families of domains in C^n with the coordinate projection, where the reference metric ω0_r is built from the flat Kähler form and the defining function has the special form r(z,s). In the present setting, ω0_r = -(1/(n+1))Ric(ω_y)+i∂∂bar(-log(-r_y)) acquires additional s-dependence through the ambient Kähler metric, and p is an arbitrary holomorphic map. Differentiating (3.1) in s yields an elliptic equation for φ_s whose coefficients and source contain ∂_s of the ambient Ricci form; these terms were absent in [5]. If the resulting growth of φ_{αs} is O(|r|^{-1}) rather than O(|r|^{-1/2-ε}), then R1/c(τ_r) need not vanish, the ratio in Proposition 4.5 would not tend to 1, and the boundary blow-up c(ρ)→∞ (Proposition 4.3) would not follow. Without that, the almost-maximum-principle argument has no lower bound for c(ρ) and the strict positivity in Theorem 1.1 fails. This is not a disagreement with consensus; it is an unverified transfer of a load-bearing analytic estimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a holomorphic family p:D→S of bounded strongly pseudoconvex domains in a Kähler manifold X, with the fibers assumed to admit complete Kähler-Einstein metrics. The family of fiberwise Kähler-Einstein metrics induces a smooth (1,1)-form ρ on D′, and the main theorem (Theorem 1.1) asserts that ρ is positive-definite on D′ whenever the total space D is strongly pseudoconvex. The proof follows Schumacher's framework: ρ satisfies an elliptic equation for its geodesic curvature c(ρ), and positivity is obtained by combining Yau's almost maximum principle with a boundary blow-up of c(ρ), proved by comparing c(ρ) with the geodesic curvature of the reference form τ_r=i∂∂(−log(−r)) on each fiber. A second theorem (Theorem 1.2) claims that under an additional uniform scalar-curvature lower bound, ρ extends across the singular fibers as a positive current, following Păun's method with Demailly approximation, Ohsawa-Takegoshi extension, and a Schwarz lemma volume estimate.","tokens_in":12002,"tokens_out":12075,"duration_ms":126850,"significance":"If correct, the result would extend the known positivity of variations of Kähler-Einstein metrics from compact fibers (Schumacher) and from Euclidean-coordinate families (Choi) to families of strongly pseudoconvex domains in an arbitrary Kähler manifold with arbitrary holomorphic base map. The use of Cheng-Yau theory and Schumacher's PDE is appropriate, and the overall strategy—boundary comparison of geodesic curvatures plus an almost maximum principle—is natural and promising. The paper also proposes an extension across singular fibers, which is a useful contribution. However, the analytic core of the main proof is not self-contained: several load-bearing estimates are delegated to earlier papers by the first author in the Euclidean setting, and the necessary modifications for a general Kähler base are not proved here. I see no circularity: the conclusion is not assumed, and the cited results are independent. The contribution would be significant if the missing estimates are supplied.","major_comments":[{"comment":"The proof of Proposition 4.5 requires the boundary estimates |φ_{αs}|=O(|r|^{-1/2-ε}) and |φ_{sβ}|=O(|r|^{-1/2-ε}), but these are not proved in the present setting. The paper states that they follow by applying Schauder estimates to φ_s and φ_{\\bar s}, with the citation \"for detailed proof, see Section 3.3 in [5]\". Reference [5] treats holomorphic families of domains in complex Euclidean space under the coordinate projection, where the reference metric has no additional s-dependence. In the present situation, ω0_{r_y}=ω0_y−i∂∂log(−r_y) and F_y in equation (3.1) depend on y through the ambient Kähler form Ric(ω_y) and through the defining function r_y. Differentiating (3.1) in s introduces terms involving ∂_s of the ambient Ricci form and of r, terms that are absent in the Euclidean case. These mixed-derivative estimates control the terms R1 and R2 in the expression for c(ρ)/c(τ_r), and their decay is essential for the claimed limit c(ρ)/c(τ_r)→1. Without a proof valid in the general Kähler setting, Proposition 4.5 is not established, and therefore the boundary blow-up c(ρ)→∞ in Proposition 4.3 and the strict positivity in Theorem 1.1 are unsupported.","section":"§4.2, Proposition 4.5"},{"comment":"The comparison in Proposition 4.5 also relies on two matrix identities, h^{βα}−(g0_r)^{βα}=(g0_r)^{βγ}N_{γδ}(g0_r)^{δα} and (g0_r)^{βα}−(gr)^{βα}=(gr)^{βγ}M_{γδ}(gr)^{δα}, together with Lemmas 4.6 and 4.7. These are cited from [4] (equations (5.3) and Lemmas 5.3–5.4), whose proofs are described as \"essentially the same\". However, [4] works in a Euclidean/coordinate-projection setting, and the general Kähler case introduces additional s-dependence through g0=(1/(n+1))Ψ_U and through ω0_y. In particular, Lemma 4.6 asserts (g0_r)^{βα}=O(|r|) in U∩D_V, and Lemma 4.7 gives the growth of N_y; both are needed to show that R2/c(τ_r)→0. Since the verification is entirely delegated and the transfer is not automatic, this is a second load-bearing gap in the proof of Proposition 4.5.","section":"§4.2, equations before Lemmas 4.6 and 4.7"},{"comment":"The proof of the extension theorem contains a gap in the volume estimate. After applying the Schwarz lemma (Theorem 5.4), the paper obtains (ωKE_y)^n≤C(~ω_y)^n on D_y and then states that it is enough to show ∫_{U_y}(~ω_y)^n<C, citing Theorem 5.6 (Diederich–Pinchuk) for the uniform boundedness of Euclidean volumes of analytic slices. This does not control the volume with respect to an arbitrary complete Kähler metric ~ω_D on D. In Remark 5.5 the metric is taken to be the Poincaré metric on a neighborhood U biholomorphic to the unit ball, but the restriction of the Poincaré metric to a slice U_y has infinite volume in general, so the claimed reduction cannot work as stated. Thus the uniform bound on VolKE(U_y), which is essential for Demailly's approximation argument, is not proved.","section":"§5, Theorem 1.2"}],"minor_comments":[{"comment":"The manuscript contains many typographical errors, including \"extensioin\", \"Holomophic\", \"F amily\", \"str ongly\", and inconsistent spacing in the title and abstract; a careful proofreading pass is needed.","section":"Throughout"},{"comment":"The reduction to the case of a one-dimensional base is stated without justification. The authors should explain why positivity of the geodesic curvature on every holomorphic disk pullback implies positive-definiteness of the (1,1)-form ρ on D′.","section":"§4.1"},{"comment":"The estimate |(ϕ_y)_{αβ}|=O(|r_y|^{n−3/2−ε}) is quoted for each fiber, but it is not stated whether the constants are uniform in y∈V; the subsequent Schauder argument would require such uniformity near the boundary.","section":"§4.2, equation (4.5)"},{"comment":"In the statement of the Schwarz lemma, the inequality (i∂∂log V)^n≥K2V needs clarification: for V=(ωKE_y)^n the Ricci form is negative definite, so the sign of the left-hand side depends on the complex dimension n. The authors should state the theorem with absolute values or with the correct sign convention.","section":"§5, Theorem 5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper extends the first author's earlier work [4,5], and the main proof relies heavily on those papers for the boundary estimates of the Monge-Ampère potentials. If the authors can supply a complete proof of the mixed-derivative estimates and the matrix estimates in the general Kähler setting, the main theorem would be convincing. As it stands, the central analytic step is a transfer of results from the Euclidean case with no verification of the additional terms, so I cannot recommend acceptance. The gap in the volume estimate for Theorem 1.2 also needs to be addressed. I would encourage the editor to request a revision rather than reject, since the strategy appears sound and the missing estimates may be provable by adapting [4,5]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper proves the expected positivity of the variation of Kähler–Einstein metrics for holomorphic families of strongly pseudoconvex domains in a general Kähler manifold, extending the first author's earlier Euclidean/projection result. It is a real extension and deserves referee time, but the proof leans on a boundary estimate that is imported, not derived, and the transfer from the Euclidean case is not automatic.\n\nWhat is genuinely new: the setup allows an arbitrary surjective holomorphic map between Kähler manifolds, not just a coordinate projection, and the fibers are noncompact strongly pseudoconvex domains. The main theorem (positivity of the fiberwise Kähler–Einstein metric on the total space) is plausible and the overall strategy is standard: use Schumacher's PDE, Cheng–Yau theory, and careful boundary behavior of the geodesic curvature. The extension across singular fibers in Theorem 1.2 also follows a known route via Păun's method, with a reasonable extra assumption to control fiber volumes.\n\nThe soft spot is exactly where the stress-test note lands. In Proposition 4.5, the estimates |φ_{αs}| and |φ_{sβ}| = O(|r|^{-1/2-ε}) are load-bearing: they make the remainder R1 vanish relative to c(τ_r). The paper cites Section 3.3 of [5] for the proof, but [5] treats domains in C^n with a flat reference metric and coordinate projection. Here ω0_r = -Ric(ω_y)/(n+1) + i∂∂̄(-log(-r_y)) carries s-dependence through the ambient Kähler metric and the arbitrary map p. Differentiating the Monge–Ampère equation in s introduces terms with ∂_s of the ambient Ricci form that simply are not in [5]. If those extra terms degrade the estimate to O(|r|^{-1}), the ratio R1/c(τ_r) need not tend to 0, and the boundary blow-up of c(ρ) does not follow. The authors may well have a valid argument in mind—maybe the same Schauder machinery still works with the extra smooth coefficients—but they have not written it down. A referee should ask for the calculation.\n\nOther minor concerns: Proposition 4.4 is also dispatched by a reference to [4], and the proof that φ is smooth on D′ is sketched via an implicit function theorem in a Banach space that is not fully detailed. These are less worrying because the references exist, but they add to the sense that the paper is an assembly of prior technology as much as a new proof.\n\nThe citation pattern is first-author-heavy, but that is not inherently a flaw because the cited results are real and published. The paper is honestly written and the theorems are clearly stated.\n\nBottom line: this is a conditional accept. It deserves a serious referee, and the referee's main task is to verify the mixed-derivative boundary estimates in the general Kähler setting, or to find a counterexample. If that check passes, the paper is a solid contribution. Send it out.\n\nBest,\n[Name]","headline":"A genuine generalization of Schumacher-style positivity to noncompact fibers, but the load-bearing mixed-derivative boundary estimate is cited from the Euclidean setting rather than proved here, so the verdict should wait on that check.","tokens_in":12518,"tokens_out":2373,"would_cite":true,"duration_ms":23972,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32T15","32Q20","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"A holomorphic family of strongly pseudoconvex domains in a Kähler manifold has positive variation of Kähler-Einstein metrics whenever the total space is strongly pseudoconvex.","keywords":["Kähler-Einstein metrics","strongly pseudoconvex domains","holomorphic families","variation of Kähler-Einstein metrics","geodesic curvature","positive currents","Monge-Ampère equation","relative canonical bundle"],"falsifier":"Compute the boundary asymptotics of the mixed derivatives $\\phi_{\\alpha s}$ and $\\phi_{s\\beta}$ for a strongly pseudoconvex domain family in a non-flat Kähler manifold, for instance with a Riemann-surface base carrying a nonzero curvature; if any such derivative fails to be $O(|r|^{-1/2-\\varepsilon})$, then the comparison $c(\\rho)/c(\\tau_r)\\to 1$ fails and Theorem 1.1 collapses.","tokens_in":11418,"feed_emoji":"📐","tokens_out":10272,"duration_ms":84913,"temperature":0.7,"pith_summary":"This paper proves that the family of Kähler-Einstein metrics on the fibers of a holomorphic family of strongly pseudoconvex domains in a Kähler manifold assembles into one smooth form $\\rho$ on the total space, and that $\\rho$ is positive-definite whenever the total space itself is strongly pseudoconvex. Positivity in the base direction is the new content, since positivity along each fiber is already built in. This matters because it gives a metric version of the positivity of the relative canonical bundle for noncompact fiber families, extending results known for Euclidean domains to arbitrary Kähler bases and arbitrary surjective holomorphic maps. The paper also shows that, under a completeness and scalar-curvature condition, $\\rho$ extends as a positive current across singular fibers.","feed_headline":"Kähler-Einstein fiber metrics positive when total space pseudoconvex","feed_subtitle":"Positivity of the relative canonical bundle holds for noncompact fiber families.","key_machinery":"The load-bearing object is the variation form $\\rho$, a smooth d-closed $(1,1)$-form that restricts to the complete Kähler-Einstein metric on every generic fiber. Its positivity is controlled by the geodesic curvature $c(\\rho)$, defined by comparing $\\rho^{n+1}$ with $\\rho^n \\wedge i\\,ds\\wedge d\\bar{s}$; this is a smooth function on each fiber, and $\\rho$ is positive-definite exactly when $c(\\rho)>0$ on every fiber. The argument uses two identities: the elliptic equation $-\\Delta c(\\rho) + (n+1)c(\\rho) = \\|\\bar{\\partial} v_\\rho\\|^2$ on each fiber, and the boundary comparison $c(\\rho)/c(\\tau_r)\\to 1$ as $x\\to\\partial D_y$, where $\\tau_r = i\\partial\\bar{\\partial}\\log(-r)$ is the complete model metric built from the defining function $r$. The comparison turns the known blow-up of $c(\\tau_r)$ at the boundary into the same blow-up for $c(\\rho)$, which excludes the vanishing alternative supplied by the real-analytic maximum principle.","core_discovery":"The central discovery is that the variation of Kähler-Einstein metrics $\\rho$, defined on the smooth part $D'$ of the family by $\\rho = \\frac{1}{n+1}\\Theta_{h_{X'/Y'}} + i\\partial\\bar{\\partial}(-\\log(-r)+\\phi)$, is positive-definite on $D'$ whenever the total domain $D$ is strongly pseudoconvex in $X$. Because $\\rho\\vert_{D_y}$ equals the fiberwise Kähler-Einstein metric, the theorem is really about the base direction: the geodesic curvature $c(\\rho)$, defined by $\\rho^{n+1}=c(\\rho)\\,\\rho^n \\wedge i\\,ds\\wedge d\\bar{s}$, is shown to satisfy an elliptic equation and to blow up to $+\\infty$ at the boundary of every generic fiber. This forces $c(\\rho)>0$ by the maximum principle and real analyticity, and hence forces $\\rho>0$ on $D'$. A second result extends $\\rho$ as a positive current across the singular fibers when the total space admits a complete Kähler metric whose restriction to the fibers has scalar curvature bounded below.","pith_inferences":["One consequence the authors leave implicit is that the positivity of $\\rho$ can be read as a form of relative negative curvature in the base direction; this suggests a link to hyperbolicity of the family that is not developed here.","A testable extension would be to prove the mixed-derivative estimate $|\\phi_{\\alpha s}|=O(|r|^{-1/2-\\varepsilon})$ directly from bounded geometry of the Kähler base, removing the present reliance on the Euclidean argument.","The scalar-curvature bound in the extension theorem is likely stronger than necessary; a local uniform bound on the fiberwise Kähler-Einstein volumes might suffice by itself, and searching for an example without the bound would clarify whether the hypothesis is sharp."],"forward_implications":["The relative canonical bundle $K_{D'/S'}$ acquires a smooth hermitian metric whose curvature form is $(n+1)\\rho$, so it is positive in a strong sense.","The boundary blow-up $c(\\rho)\\to\\infty$ along each fiber gives a quantitative lower bound for the fiberwise Kähler-Einstein metric near the boundary.","Under the completeness and scalar-curvature hypothesis, $\\rho$ extends as a positive current across singular fibers, preserving positivity in a weak sense on the degenerate fibers.","The result applies to arbitrary surjective holomorphic maps between Kähler manifolds, so curved bases are allowed and not only coordinate projections in $\\mathbb{C}^n$."],"supporting_citations":[{"why":"Proves existence and uniqueness of the complete Kähler-Einstein metric via the complex Monge-Ampère equation.","marker":"[3]"},{"why":"Supplies the refined defining function and the boundary asymptotics for the solution of the Monge-Ampère equation.","marker":"[7]"},{"why":"Derives the elliptic equation for the geodesic curvature and the maximum-principle argument used to get positivity.","marker":"[13]"},{"why":"Sets up the variation form and proves the boundary blow-up of $c(\\tau_r)$ in the Euclidean setting.","marker":"[4]"},{"why":"Contains the boundary estimates for the mixed derivatives that are cited to justify the same decay in the Kähler-manifold case.","marker":"[5]"},{"why":"Provides the extension strategy for the curvature current across singular fibers.","marker":"[11]"},{"why":"Supplies the $L^{2/m}$ extension theorem used to produce holomorphic approximants in the extension proof.","marker":"[1]"},{"why":"Gives the volume-form comparison used to bound fiberwise Kähler-Einstein volumes.","marker":"[10]"}],"fun_headline_variants":["KE fiber metrics turn positive in pseudoconvex families","Pseudoconvex total space gives positive fiberwise KE metrics","Fiber KE metrics are positively curved when total space is pseudoconvex","Positive fiberwise KE metrics from pseudoconvex total domain","Total space pseudoconvex forces positive KE fiber metrics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the boundary decay rate for the mixed second derivatives of the Monge-Ampère potentials, $|\\phi_{\\alpha s}| = O(|r|^{-1/2-\\varepsilon})$; if that rate is wrong for general Kähler bases, the ratio $c(\\rho)/c(\\tau_r)\\to 1$ used to force positivity need not hold.","fun_headline_variants_meta":{"raw":{"variants":["KE fiber metrics turn positive in pseudoconvex families","Pseudoconvex total space gives positive fiberwise KE metrics","Fiber KE metrics are positively curved when total space is pseudoconvex","Positive fiberwise KE metrics from pseudoconvex total domain","Total space pseudoconvex forces positive KE fiber metrics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000544,"raw_usage":{"total_tokens":2591,"prompt_tokens":922,"completion_tokens":1669,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":1587}},"tokens_in":538,"tokens_out":1669,"duration_ms":12261,"temperature":1.0,"reasoning_tokens":1587,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:03:43.956697+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the boundary asymptotics of the mixed derivatives $\\phi_{\\alpha s}$ and $\\phi_{s\\beta}$ for a strongly pseudoconvex domain family in a non-flat Kähler manifold, for instance with a Riemann-surface base carrying a nonzero curvature; if any such derivative fails to be $O(|r|^{-1/2-\\varepsilon})$, then the comparison $c(\\rho)/c(\\tau_r)\\to 1$ fails and Theorem 1.1 collapses.","supporting_citations":[{"cited_title":"Cheng, S.-T","cited_arxiv_id":null,"evidence_quote":"Proves existence and uniqueness of the complete Kähler-Einstein metric via the complex Monge-Ampère equation."},{"cited_title":"van Coevering, K¨ ahler-Einstein metrics on strictly pseudoconvex domain s, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the refined defining function and the boundary asymptotics for the solution of the Monge-Ampère equation."},{"cited_title":"Schumacher, Positivity of relative canonical bundles and applications , Invent","cited_arxiv_id":null,"evidence_quote":"Derives the elliptic equation for the geodesic curvature and the maximum-principle argument used to get positivity."},{"cited_title":"Choi, Variations of K¨ ahler-Einstein metrics on stronlgy pseudoconvex domains, Math","cited_arxiv_id":null,"evidence_quote":"Sets up the variation form and proves the boundary blow-up of $c(\\tau_r)$ in the Euclidean setting."},{"cited_title":"Choi, A study of variations of pseudoconvex domains via K¨ ahler-E instein metrics , Math","cited_arxiv_id":null,"evidence_quote":"Contains the boundary estimates for the mixed derivatives that are cited to justify the same decay in the Kähler-manifold case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the extension strategy for the curvature current across singular fibers."},{"cited_title":"Berndtsson, M","cited_arxiv_id":null,"evidence_quote":"Supplies the $L^{2/m}$ extension theorem used to produce holomorphic approximants in the extension proof."},{"cited_title":"Mok, S.-T","cited_arxiv_id":null,"evidence_quote":"Gives the volume-form comparison used to bound fiberwise Kähler-Einstein volumes."}],"review_version":1}