{"id":"04e2708d-17d7-4485-8690-0919f555d60c","arxiv_id":"2411.16764","paper_version":3,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A proposed extension of Kodaira-Hodge theory to complex manifolds whose complex structure degenerates along a divisor, built from degenerate complex Monge-Ampère solutions.","lead":"This paper defines a new class of complex manifolds with singular complex structures and claims a degenerate Kodaira-Hodge theory for them. The intended payoff is new tools for mirror symmetry and for embedding certain three-dimensional CR boundaries into projective space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's central Hodge decomposition depends on inequality (33) in §9.3, which is asserted without proof and is not implied by the stated metric assumptions; until it is verified, Theorem 4 and the subsequent Kodaira results are unsupported.","rationale":"The reader's verdict identifies two fragile premises: the unverified regularity theorem from [1] and the asserted inequality (33). I agree with the reader that the paper does not currently support its central claims, but I focus on (33) because it is the immediate load-bearing step inside this manuscript: Theorem 4 uses it explicitly, and all of the Hodge and Kodaira statements depend on Theorem 4. The self-cited theorem from [1] is also necessary, but it is external to this paper and the paper itself gives no indication of how to check it. Inequality (33), by contrast, is stated and then used without proof in the middle of the paper, and it is not a standard consequence of the hypotheses in Definition 2. The concern is not internal inconsistency but an omitted proof of the key analytic estimate; that is sufficient to reject the preprint as evidence for the claimed results. I would not move to 'conditional' because the missing estimate is not a small gap: it is the main analytic difficulty of the paper. The recommendation is therefore REJECT, matching the reader's verdict, though the concern is narrower and more specific than the reader's composite list.","tokens_in":43929,"tokens_out":2728,"duration_ms":30729,"concrete_test":"Verify inequality (33) in the explicit flat model of Section 3: take Ω ⊂ C² with π(z,w) = (z,w²), the degenerate structure J'_0, and the weighted function φ_0 = −|w|^4. For a sequence of test forms f_ε supported in |w| ≤ ε, compute the three weighted norms in (33) explicitly with the metric from Definition 2. If there is a sequence with ∂̄_{J'_0} f_ε and ∂̄*_{J'_0} f_ε both tending to zero in weighted L² while ‖f_ε‖_{χ(φ_0)} does not tend to zero, then (33) is false in the model case and the proof of Theorem 4 fails. If the inequality holds in the model, independently derive it from the assumptions of Definition 2 and Lemma 6, checking all boundary terms and curvature contributions, to certify the step used in (20).","verdict_should_be":"REJECT","load_bearing_attack":"The single most load-bearing concern is the unproved weighted estimate (33) in §9.3. This inequality is the mechanism by which the paper obtains local ∂̄_{J'}-solvability in a neighborhood of the singular divisor, and it is used directly in the proof of Theorem 4: inequality (20) is quoted from (33) to bound the sequence {φ_b} in L²_{J'}, which is then the crucial step producing the L² solution of ∂̄_{J'}u = α − α_0. Without (33), the sequence need not be bounded, and the argument collapses. The paper does not derive (33); it states that 'one then can prove the inequality ... for appropriate constants' and refers vaguely to arguments from [8] and [23]. This is not a routine invocation of Hörmander's L² theory, because the complex structure J' itself degenerates along D and the background metric is only assumed to have bounded Ricci curvature in Definition 2. The hypothesis of Definition 2 does not obviously supply the curvature, connection, and coercivity controls needed for the standard weighted ∂̄-estimate on the degenerate structure. Lemma 6 only asserts boundedness of ∂̄_{J'} of certain local frames by a 'simple computation' whose cancellations are not written out. If (33) fails, then Theorem 8 fails, and with it Theorem 4, Corollary 4, Theorem 5, the ∂∂̄ lemma, and the Kodaira embedding theorem all lose their foundation. The self-cited regularity theorem from [1] is also a premise, but the immediate missing analytic step inside this paper is the derivation of (33).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a class of singular complex manifolds (Definition 2) modeled on degenerations obtained from solutions of degenerate complex Monge-Ampère equations, and claims a full L2 Hodge package for the degenerate complex structure J': Hodge decomposition for the ∂̄_{J'}-complex (Theorem 4), smoothness of harmonic forms (Theorem 5), a singular ∂∂̄ lemma (Lemma 4), vanishing of positive-degree harmonic forms for J'-positive line bundles (Theorem 6), and a Kodaira embedding theorem for X\\U_ε (Theorem 7). The local analytic engine is Theorem 8, a local ∂̄_{J'}-solvability statement whose proof is sketched in Section 9.3 and depends on the unproved weighted inequality (33). The proof of Theorem 4 uses (33) through inequality (20) to bound the sequence {φ_b} and to obtain an L2 solution. Section 2 imports from the author's previous work [1] the existence of the degenerate Kähler form ω' and the smooth solution of the DCMA equation. The paper also contains a degenerate Poincaré lemma in Appendix A.","tokens_in":44367,"tokens_out":6428,"duration_ms":63954,"significance":"If the main theorems were fully established, this would be a substantial extension of Hodge and Kodaira theory to spaces whose complex structure itself degenerates along a divisor; such a theory would be of real interest, particularly in connection with the twistor construction in Section 2. The explicit local model in Section 3, the careful formulation of Definition 2, and the attempt to prove a degenerate Poincaré lemma in Appendix A are genuine strengths. However, the central results currently rest on unproved analytic estimates and on existence assertions imported from the author's earlier papers. The contribution is therefore a promising framework rather than a completed proof, and the significance can only be assessed after the missing analytic core is supplied.","major_comments":[{"comment":"The weighted estimate (33) is asserted without proof, and it is load-bearing: in the proof of Theorem 4 it is invoked as inequality (20) to bound the sequence {φ_b}, and without that bound the L2 solution of ∂̄_{J'}u = α − α_0 is not obtained. The sentence that one can prove (33) 'by the same arguments as in [8] and ([23])' is not a derivation, especially because J' degrades along D and Definition 2 only assumes bounded Ricci curvature; the curvature, connection, and coercivity controls needed for a weighted ∂̄-estimate are not shown to follow from the stated hypotheses. A complete derivation of (33) from Definition 2, or an explicit additional assumption that yields it, is required before Theorem 4 can be accepted.","section":"§9.3, Eq. (33)"},{"comment":"The local solvability theorem is the analytic foundation for Theorem 4, but its proof is only a sketch. In Lemma 5 the computation of ∂_{J'}∂̄_{J'}ψ_0 is carried out for the flat model and the general case is compared up to O(|(x,y)|), yet the error term is not estimated with respect to the degenerate metric. Lemma 6 is dismissed as 'a simple computation' whose claimed cancellation of singularities is not written out. The density statement for D_{p,q+1}(U) is imported from [8] and [23] without verifying the necessary hypotheses in the degenerate setting. Theorem 8 needs a complete proof, including the verification that the operators T and S are well-defined closed operators with the stated domains.","section":"Theorem 8 and §9.1–9.2"},{"comment":"The proof of the Kodaira embedding theorem assumes a special covering V = {U_α} of the blown-up manifold with the properties that exactly one open set meets D and all higher intersections have trivial cohomology. The existence of such a covering is not demonstrated. The subsequent Čech-type argument for extending holomorphic sections depends on this covering, so without a construction or a reference for it, the embedding theorem is unsupported. The same issue affects the claim that the blow-up of the degenerate manifold again belongs to the class of Definition 2.","section":"Theorem 7, §8.2"},{"comment":"The asymptotic inequality 2n−k+a−2>3b is imposed as part of the definition of the class, but the paper verifies it only for the Monge-Ampère example, where a=2 and b=1. Lemma 1 and Lemma 2, and hence the L2 decomposition and Hodge theory, depend essentially on this inequality. The manuscript should either prove that the inequality holds for all examples in the class or state it as a hypothesis whose verification is supplied for each application.","section":"Definition 2(4)"},{"comment":"The existence of the degenerate Kähler form ω' with properties (i)–(iii) and the smooth solution of the degenerate complex Monge-Ampère equation are imported from the author's preprint [1]. The manuscript does not state the precise theorem from [1] that is being used, nor does it list the regularity estimates at D that the subsequent arguments require. Since every example and the entire class in Definition 2 depend on this input, the dependence should be made explicit, and the needed estimates should be stated either as theorems with proofs or as clearly identified assumptions.","section":"Section 2 and [1]"}],"minor_comments":[{"comment":"There are numerous typographical errors and misspellings ('strudy', 'whcih', 'forht', 'sequaent', 'harmonique', 'pluisubharmonic', 'dergative'); the manuscript needs a careful proofreading pass.","section":"Abstract and Introduction"},{"comment":"The relation between the real submanifold D of dimension k and the complex divisor in the main example (real codimension 2) should be stated explicitly, since k is also used elsewhere to denote the number of branched factors in Section 3.","section":"Definition 2"},{"comment":"The map Φ:S^2→Ω^2(M) in equation (3) uses E_1,E_2,E_3, but the text then writes J' without explaining how J' corresponds to a point of the twistor sphere; the notation should be made consistent.","section":"§3, Eq. (3)"},{"comment":"In the proof of Lemma 2, the phrase 'forht and fifth line' is a typo, and the constants in the Cauchy–Schwarz argument are not tracked; a cleaner presentation with named constants would improve readability.","section":"Lemma 2 proof"},{"comment":"The statement that 'the same argument holds' for d and d* is not immediate, because the decomposition d=∂+∂̄ is only available away from D and the boundary terms require separate treatment; this needs to be spelled out or referenced.","section":"Corollary 5"},{"comment":"The homotopy formula in the degenerate Poincaré lemma is classical, but the claim that the resulting forms lie in L^2 with respect to the degenerate metric is not fully justified, and the Čech correction step would benefit from explicit L^2 estimates on the intersections.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is heavily self-referential: the main existence and solvability inputs come from the author's own papers [1] and [8]. This is not disqualifying by itself, but it raises the threshold for clarity; the referee needs explicit statements of the imported results. The main reason for my recommendation of major_revision rather than reject is that the missing pieces are local analytic arguments and a covering construction that could in principle be supplied in a revision. If the weighted estimate (33) and the local solvability theorem cannot be proved at the stated level of generality, then Theorems 4–7 and the central claims of the paper are unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The idea is real, but the preprint is not ready. The genuinely new thing is the combination: a degenerate Kähler structure whose complex structure itself is singular along a divisor, built from degenerate Monge-Ampère solutions, and then a program of L² Hodge theory for that structure. That does go beyond Treves's hypoanalytic structures and Cheeger–Dai's conical metrics. The local model in Section 3—the ramified double cover—is explicit and concrete, and the twistor motivation is clear.\n\nThe soft spot is exactly where the stress test put it. Inequality (33) in §9.3 is the load-bearing step: it supplies the weighted L² estimate that bounds the approximating sequence in Theorem 4, and without it Theorem 8's local ∂̄-solvability, and then Corollary 4, Theorem 5, the ∂∂̄ lemma, and the Kodaira statements all lose their support. The paper does not derive (33). It says \"by the same arguments as in [8] and [23]\" and that is not enough, because J' is degenerating along D and Definition 2 only assumes bounded metric and Ricci curvature. Those hypotheses do not obviously imply the curvature, connection, and coercivity controls needed for a Hörmander-type estimate on a degenerate structure. Lemma 6 is also only a sketch: the cancellations are asserted, not shown.\n\nThere are smaller issues. Theorem 7 uses a covering with properties (i)–(ii) whose existence is not demonstrated. The whole paper inherits the author's earlier preprint [1] for the existence of smooth degenerate Monge-Ampère solutions; that is a self-citation and not independently verified. The text is badly OCR-mangled, which makes checking formulas harder, though that is not itself a scientific flaw.\n\nProportionate bottom line: the paper is not obviously wrong, and the architecture suggests a plausible program. But the central claim is unsupported in the current text. I would not accept it for publication and would not cite it yet. The author needs to prove (33) in detail, or state it explicitly as an assumption, and then the rest can be evaluated. As an editor, I would not send this out for full review in its present form; I would ask the author to fill the gap and resubmit.","headline":"A genuinely new program for L² Hodge theory on singular complex structures, but inequality (33) is asserted rather than proved, and the whole paper leans on it.","tokens_in":44790,"tokens_out":3002,"would_cite":false,"duration_ms":32173,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","53C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new class of singular complex manifolds is shown to carry a full degenerate Kodaira–Hodge theory, including Hodge decomposition, vanishing, and projective embedding away from the divisor.","keywords":["singular complex manifolds","degenerate Monge-Ampère equation","degenerate Kähler metric","L2 Hodge theory","Kodaira embedding","∂̄ solvability","involutive structures","hyperkähler structure"],"falsifier":"Work out the flat local model of Section 3, the pullback $J'_0$ of the quaternionic structure under $(z,w)\\mapsto(z,w^2)$, and compute the weighted estimate (33) for that model's degenerate $\\bar\\partial_{J'}$-operator: the inequality is stated without proof, and if it fails in the model, the local solvability theorem and everything it feeds collapses. A second decisive check is to compute the $L^2$ harmonic $(1,1)$-forms of the model and test whether the claimed Hodge decomposition and positive-degree vanishing hold for the trivial line bundle.","tokens_in":92,"feed_emoji":"🧮","tokens_out":13999,"duration_ms":178307,"temperature":0.7,"pith_summary":"This paper introduces a class of singular complex manifolds—smooth manifolds equipped with a complex structure that degenerates along a smooth divisor—and develops a degenerate Kodaira–Hodge theory for them. The central aim is to show that the standard Kähler package survives the singularity: an $L^2$ Hodge decomposition for the degenerate $\\bar\\partial_{J'}$-complex, smoothness of the harmonic representatives, a singular $\\partial\\bar\\partial$ lemma, vanishing of positive-degree harmonic forms for positive line bundles, and a Kodaira embedding theorem for the complement of a small neighborhood of the singular divisor. The examples come from degenerate complex Monge–Ampère equations on Kähler surfaces of general type with canonical divisor; the solutions produce a degenerate hyperkähler structure, and the degenerate complex structure $J'$ studied here is one member of the twistor sphere. If the theorems are right, projective embedding and cohomology control extend to spaces where the complex structure itself is singular, not merely the metric. The foundational existence statement is cited to the author's earlier preprint and not proved here, and the local solvability proof invokes a weighted estimate that is asserted rather than proved.","feed_headline":"New singular complex manifolds get a full Hodge–Kodaira theory","feed_subtitle":"Spaces whose complex structure degenerates along a divisor still support Hodge decomposition and projective embedding.","key_machinery":"The central object is the degenerate complex structure $J'$: inside the twistor sphere of the hyperkähler structure on $X\\setminus D$, it is the member orthogonal to the original complex structure, compatible with the degenerate Kähler form $\\Re\\Omega$ and the degenerate Ricci-flat metric $g'$, and it is singular exactly along the canonical divisor $D$. The argument is carried by the axiomatic class of Definition 2, whose asymptotic conditions—boundedness of the metric and Ricci tensor, the coordinate model $\\det g' = |r|^a + O(r^a)$, the normal decay $\\|dr\\|_{g'}= r^b + O(r^b)$, and the inequality $2n-k+a-2>3b$—are chosen so that the weighted Sobolev embedding is compact, integration by parts holds, and the fundamental inequality for $\\Delta_{g'}$ is available. The key local mechanism is Theorem 8, solvability of $\\bar\\partial_{J'}$ in a neighborhood of $D$: a plurisubharmonic exhaustion built from $-|\\sigma|^4$ and $-\\log F$ plus the weighted estimate (33) produces solutions, and this solvability converts the weak $L^2$ Hodge decomposition into the strong Hodge and Kodaira theorems.","core_discovery":"The paper's central claim is that for the singular complex manifolds of Definition 2—built from the degenerate hyperkähler structure and satisfying four asymptotic conditions on the metric, the Ricci curvature, and the rate of degeneration along the divisor $D$—the full $L^2$ Hodge package holds for the degenerate structure $J'$. Specifically, Theorem 4 gives Hodge decomposition for $\\bar\\partial_{J'}$-closed forms, Theorem 5 states that $\\Delta_{g'}$-harmonic forms are smooth on all of $X$ and are both $\\bar\\partial_{J'}$- and $\\bar\\partial_{J'}^*$-closed, Lemma 4 gives the singular $\\partial\\bar\\partial$ lemma, Theorem 6 gives vanishing of harmonic forms of bidegree $(p,q)$ with $p+q>n$ for $J'$-positive line bundles, and Theorem 7 embeds $X\\setminus U_\\varepsilon$, the complement of an arbitrarily small tubular neighborhood of $D$, into a projective space using holomorphic sections of a high power of a $J'$-positive line bundle. The proof route goes through a weak $L^2$ Hodge theory modeled on the theory for spaces with non-isolated conical singularities, a local $\\bar\\partial_{J'}$-solvability theorem near $D$ (Theorem 8), and a degenerate Poincaré lemma, with the final corollary that the $L^2$ de Rham cohomology of the singular space is isomorphic to the ordinary de Rham cohomology of $X$.","pith_inferences":["The introduction's mirror-symmetry motivation suggests that the $J'$-holomorphic sections whose zero sets are produced by the Kodaira embedding could yield Lagrangian cycles Poincaré dual to classes orthogonal to $c_1$; the paper does not construct those cycles, but the embedding theorem makes the search concrete.","The flat branched-cover model of Section 3 is the natural test case: if inequality (33) can be verified there by an explicit calculation, the local solvability theorem would stand on much firmer ground.","A quantitative version of the smoothness theorem might follow from identifying $|\\sigma|^2\\Delta_{g'}$ as a Grushin-type operator: the degeneration is subelliptic, with a controlled loss of derivatives in the normal directions, rather than elliptic in the usual sense.","The class of Definition 2 may also cover the higher-dimensional branched covers constructed from the maps $\\pi_{n,k}$ in Section 3; the paper does not check the asymptotic conditions there, so whether the full Hodge package extends to those examples remains open."],"forward_implications":["For a Kähler surface of general type with canonical divisor $D$, the complement $X\\setminus U_\\varepsilon$ embeds holomorphically into a projective space for arbitrarily small $\\varepsilon$, even though a generic CR structure on its three-dimensional boundary is not embeddable.","The degenerate $L^2$ cohomology $H^{p,q}_{(2)}(X)$ is finite-dimensional and isomorphic to the space of $L^2$ harmonic forms, and every harmonic form is smooth across the singular divisor.","A $d$-closed $(p,q)$-form orthogonal to harmonic forms is $\\partial_{J'}\\bar\\partial_{J'}$-exact, so cohomology classes on the singular manifold admit $\\partial\\bar\\partial$-type potentials.","For a $J'$-positive line bundle, all harmonic forms of degree greater than half the real dimension vanish, giving the Kodaira vanishing half of the embedding theorem."],"supporting_citations":[{"why":"Supplies the existence theorem for smooth solutions of the degenerate Monge–Ampère equation and the degenerate Kähler form with the asymptotic properties on which all examples and theorems rest.","marker":"[1]"},{"why":"Provides the definitions of involutive and hypo-analytic structures and of the characteristic set used to describe the singularity of $J'$.","marker":"[28]"},{"why":"Supplies the weak $L^2$ Hodge theory framework for spaces with non-isolated conical singularities, including the criterion that the Hodge map be an isomorphism.","marker":"[31]"},{"why":"Gives the pseudo-concavity statement for $X\\setminus U_\\varepsilon$ and the boundary function used to build the plurisubharmonic exhaustion in the $\\bar\\partial$-solvability proof.","marker":"[10]"},{"why":"Provides the density and weighted-estimate pattern for $\\bar\\partial$-solvability in hypo-analytic manifolds that Theorem 8 adapts to the degenerate setting.","marker":"[8]"},{"why":"Identifies operators of the form $|\\sigma|^2\\Delta$ as Grushin-type, which the smoothness proof for harmonic forms invokes.","marker":"[19]"},{"why":"Supplies the standard $L^2$ estimates and density arguments for $\\bar\\partial$ that the local solvability proof follows.","marker":"[23]"},{"why":"Underlies the strong Hodge theory criterion via finite-dimensional $L^2$ cohomology and Stokes's theorem on cone-like singularities.","marker":"[29]"}],"fun_headline_variants":["Singular complex manifolds admit full L2 Hodge theory","Hodge theory survives degenerate complex structures","L2 Hodge package for singular complex manifolds","Singular manifolds: Hodge decomposition still holds","New singular manifolds with complete Hodge theory"],"cache_read_input_tokens":46848,"weakest_assumption_plain":"The load-bearing premise is the author's earlier existence theorem—stated without proof here—that a degenerate complex Monge–Ampère equation on a Kähler surface with canonical divisor has a smooth solution producing the degenerate Kähler form $\\omega'$ with the stated asymptotic behavior; on top of that, the local $\\bar\\partial_{J'}$-solvability proof leans on a weighted estimate, inequality (33), that is asserted but not demonstrated.","fun_headline_variants_meta":{"raw":{"variants":["Singular complex manifolds admit full L2 Hodge theory","Hodge theory survives degenerate complex structures","L2 Hodge package for singular complex manifolds","Singular manifolds: Hodge decomposition still holds","New singular manifolds with complete Hodge theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000554,"raw_usage":{"total_tokens":2601,"prompt_tokens":867,"completion_tokens":1734,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1660}},"tokens_in":483,"tokens_out":1734,"duration_ms":10693,"temperature":1.0,"reasoning_tokens":1660,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:39:54.862317+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out the flat local model of Section 3, the pullback $J'_0$ of the quaternionic structure under $(z,w)\\mapsto(z,w^2)$, and compute the weighted estimate (33) for that model's degenerate $\\bar\\partial_{J'}$-operator: the inequality is stated without proof, and if it fails in the model, the local solvability theorem and everything it feeds collapses. A second decisive check is to compute the $L^2$ harmonic $(1,1)$-forms of the model and test whether the claimed Hodge decomposition and positive-degree vanishing hold for the trivial line bundle.","supporting_citations":[{"cited_title":"Degenerate Complex Monge Ampere Equation Part I","cited_arxiv_id":"2302.14106","evidence_quote":"Supplies the existence theorem for smooth solutions of the degenerate Monge–Ampère equation and the degenerate Kähler form with the asymptotic properties on which all examples and theorems rest."},{"cited_title":"Hypo-analytic structures, a local theory, P riceton University Press, 1992","cited_arxiv_id":null,"evidence_quote":"Provides the definitions of involutive and hypo-analytic structures and of the characteristic set used to describe the singularity of $J'$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the weak $L^2$ Hodge theory framework for spaces with non-isolated conical singularities, including the criterion that the Hodge map be an isomorphism."},{"cited_title":"Representants lagrangiens de l'homologie des surfaces projectives complexes","cited_arxiv_id":"0903.4490","evidence_quote":"Gives the pseudo-concavity statement for $X\\setminus U_\\varepsilon$ and the boundary function used to build the plurisubharmonic exhaustion in the $\\bar\\partial$-solvability proof."},{"cited_title":"A note on the solvability of ̄/u1D715in a class of hypo-anlytic manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the density and weighted-estimate pattern for $\\bar\\partial$-solvability in hypo-analytic manifolds that Theorem 8 adapts to the degenerate setting."},{"cited_title":"On a class of elliptic pseudodifferential operators degenerate on a submani- fold, Mathematics of the USSR-Sbornik, 1971, Vol 13, No","cited_arxiv_id":null,"evidence_quote":"Identifies operators of the form $|\\sigma|^2\\Delta$ as Grushin-type, which the smoothness proof for harmonic forms invokes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard $L^2$ estimates and density arguments for $\\bar\\partial$ that the local solvability proof follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underlies the strong Hodge theory criterion via finite-dimensional $L^2$ cohomology and Stokes's theorem on cone-like singularities."}],"review_version":1}