{"id":"2d82d77a-0a02-48d8-911b-15b81cda3863","arxiv_id":"1908.00651","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A strictification theorem makes Lagrangian distributions on -2-shifted symplectic derived schemes strict, and gluing gives a global distribution under Hausdorff and second countability assumptions.","lead":"This paper proves that any derived scheme over C with a -2-shifted symplectic structure and Hausdorff classical points admits a globally defined Lagrangian distribution when viewed as a dg C^infinity-manifold. This is a technical bridge toward constructing shifted potentials on moduli spaces of sheaves on Calabi-Yau four-folds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 9's flat-localization claim ('can always be achieved by localization') is unproved and likely false over arbitrary algebraic dg algebras; whether it genuinely undermines Theorem 3 depends on a C^∞-local repair.","rationale":"We read the paper as attempting to prove that on any derived scheme with a −2-shifted symplectic form and Hausdorff second countable classical locus, the sheaf of Lagrangian distributions is soft, yielding a global section that will feed into the shifted-potential construction. The strategy—local Darboux models, semi-strictification via isotropic structures, interpolation/gluing via positive-definiteness of the real part—is coherent and a genuine contribution, and the comparison with [2] and [6] is honest. The proof does not rely on machine-checked formalization, but it has a clear logical spine. The single most insecure point is the flat-localization assertion before Prop. 9. That assertion is not a mere technicality: without a generating subspace with zero bracket, the homotopy kernel C^• cannot be given the Lie–Rinehart structure needed to make the semi-strictification. The paper states it as 'This can always be achieved by localization' with no argument. For algebraic dg algebras, the degree-0 piece alone would require a commuting frame of the tangent bundle of the ambient smooth algebra, which is a rigid condition not guaranteed by Zariski localization; the K3 example shows the claim is at least non-obvious and likely false. However, the central Theorem 3 is formulated for the underlying dg C^∞-manifold, and in that category the analogous local property is immediate (coordinate vector fields on a smooth chart commute and generate the tangent bundle). Thus the gap may be repairable by rewriting Section 4 to work only locally over C^∞ charts and by stating Prop. 9 for dg C^∞-rings locally. The paper does not currently make this explicit, and Prop. 16 still assumes a global frame. We therefore regard the reader's verdict of CONDITIONAL as appropriate: the central claim is plausible, but the proof as written is incomplete, and a specific repair should be provided.","tokens_in":27921,"tokens_out":28223,"duration_ms":325090,"concrete_test":"Re-derive the proof of Prop. 9 for the C^∞-case only: on a local coordinate chart of M, take V^* to be the span of the coordinate vector fields and the duals of the trivialized generating bundles, and check that the construction of C^• and the semi-strictification goes through verbatim. Simultaneously, test the algebraic flatness claim on a concrete A0 with no flat translation structure (e.g., a suitable affine open of a K3 surface): if no such V^* exists after any Zariski localization, the statement 'This can always be achieved by localization' is false as written. If the C^∞-repair succeeds, Theorem 3 stands with a revised proof; if it does not, the strictification step collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the assertion in Section 3, immediately before Prop. 9, that every well-presented dg algebra can be made flat: after localization the tangent complex TA must have a graded vector subspace V^* that generates TA over A and on which the Lie bracket vanishes. This is used in Prop. 9 to build the surjective homotopy kernel C^• and endow it with a Lie–Rinehart structure, and Prop. 9 is in turn the basis for the semi-strictification that Prop. 16 and Theorem 2 rely on. No proof or reference is given. The degree-0 part of the claim requires a commuting frame of the tangent bundle of the smooth algebra A0; Zariski localization can trivialize a vector bundle but cannot in general produce a basis of commuting vector fields. For example, an affine open of a K3 surface can have a free tangent bundle only after localization, yet need not admit any flat translation structure, so the bracket-vanishing condition imposes a strong global restriction not guaranteed by 'localization.' The paper's main theorem, Theorem 3, is about dg C^∞-manifolds, and there the analogous flat-localization holds locally by choosing coordinates on the manifold and trivializing the generating bundles, so the algebraic gap may be repairable. But the text as written claims the property for all well-presented dg algebras over C, uses it in Prop. 9, and does not state the needed C^∞-local reformulation. Consequently the strictification argument is incomplete as written, and the central claim is conditional on a proof or a careful restriction of the statement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a framework of integrable distributions, derived foliations, isotropic and Lagrangian structures on derived schemes and on dg C∞-manifolds, with the aim of proving that a derived scheme over C equipped with a −2-shifted symplectic structure and having a Hausdorff, second-countable space of classical points admits a globally defined Lagrangian distribution. Section 1 recalls and extends the language of graded mixed algebras and Lie–Rinehart algebras, and introduces derived foliations. Section 2 organizes shifted symplectic forms and isotropic structures into sheaves on the space of classical points. Section 3 proves strictification results for Lagrangian distributions, culminating in a semi-strictification statement and the claim that Lagrangianness is independent of the isotropic structure for purely derived foliations. Section 4 passes to the underlying dg C∞-manifold, uses negative definiteness with respect to the real part of the symplectic form to interpolate and glue local distributions, and concludes that the sheaf of purely derived foliations that are Lagrangian with respect to ω_im and negative definite with respect to ω_re is soft. Theorem 3 is the main global existence statement.","tokens_in":28292,"tokens_out":6537,"duration_ms":71206,"significance":"If the main theorem is correct, it is a significant step toward the authors' program: it would supply the global Lagrangian distributions needed for the shifted-potential construction for moduli spaces of sheaves on Calabi–Yau four-folds. The paper has real conceptual strengths: it gives a clean sheaf-theoretic packaging of local data, it identifies the isotropic structure as the key device that makes strictification and gluing compatible, and it isolates the role of negative definiteness with respect to the real part of the symplectic form. The dependence on prior work by Borisov–Joyce and by Brav–Bussi–Joyce is explicit and appropriate. However, the central strictification argument rests on an unproved and questionable local flatness assertion, so the significance of the paper is conditional on a repair of that step.","major_comments":[{"comment":"The definition of 'Spec(A) can be made flat' asserts that after localization the tangent complex TA has a graded vector subspace V^* that generates TA over A and on which the Lie bracket vanishes, and the text states 'This can always be achieved by localization' without proof. This assertion is load-bearing: Proposition 9 uses it to construct the surjective homotopy kernel C^• and to endow it with a Lie–Rinehart structure, and Remark 8, Theorem 1, and the strictification results in Section 4 depend on Proposition 9. The degree-zero part of the claim would require a commuting frame of the tangent sheaf of the smooth algebra A0; Zariski localization can trivialize a vector bundle, but it cannot in general produce commuting vector fields, for example on an affine open of a K3 surface. Thus the algebraic strictification argument is incomplete as written. If the intended statement is only for dg C∞-rings in local coordinates, that restricted statement should be made and proved, since Theorem 3 is about dg C∞-manifolds.","section":"Section 3, before Proposition 9"},{"comment":"The proof assumes that, after passing to a semi-strict representative, one can choose a basis {e_j} of L modulo δ(A^{-1}) and then extend it to a basis of E over all of M. For a nontrivial vector bundle E on a general well-presented dg C∞-ring, such a global basis need not exist. This matters because the displayed formula defining φ and the construction of the strict Lagrangian distribution are made relative to that global basis. The argument is local in spirit, but the proposition is stated globally; either the statement should be restricted to a coordinate chart in which E and L are trivialized, or an argument for the needed triviality should be supplied.","section":"Proposition 16, proof"},{"comment":"The proof of Theorem 2 says that a compatible system of local distributions can be converted into one global distribution 'using a good coordinate system (e.g. [2])' and by applying Proposition 17. No precise statement of what 'good coordinate system' provides is included in this paper, and Proposition 17 itself uses the explicit fibrant replacement from the proof of Proposition 9, which depends on the disputed flat-localization assertion. Consequently the sheaf-isomorphism theorem that feeds directly into Theorem 3 is not established independently of the gap in Section 3.","section":"Theorem 2, proof"}],"minor_comments":[{"comment":"The statement of Proposition 12 says 'positive definite' where the surrounding definitions and Proposition 11 require 'negative definite' with respect to ω_re; this sign inconsistency should be corrected.","section":"Proposition 12"},{"comment":"These propositions are phrased for a well-presented dg C-algebra but then use (TM, ω_im) with R-valued forms, bases over M, and classical points valued in R; the passage from the algebraic setting to the underlying dg C∞-manifold should be set out explicitly before these statements.","section":"Propositions 15 and 16"},{"comment":"The claim that local existence of semi-strict realizations follows from Propositions 8 and 9 is only valid if the flat-localization assumption is available; the statement should explicitly flag that it is conditional on that assumption.","section":"Remark 8"},{"comment":"The proof says that one can choose sections of E^1_t around p that are mapped injectively by δ and that this remains true for neighbouring values of t, but it does not justify the continuity of these choices over the parameter interval; a short argument would improve the exposition.","section":"Proposition 14, proof"},{"comment":"The verification of the descent identities for the extension of distributions and isotropic structures to the prism atlas is summarized as 'tedious but straightforward'; since the sheaf property is central to the paper, a fuller indication of the argument would be helpful.","section":"Proposition 6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a genuine technical gap in the flat-localization assertion preceding Proposition 9, and the proof of Proposition 16 has a globality issue. The main theorem concerns dg C∞-manifolds, so a C∞-local coordinate argument may well repair the algebraic gap; this seems feasible within a major revision rather than requiring a completely different approach. The reliance on [17], a conference talk, for several central notions also deserves scrutiny during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about 1908.00651. First, the core idea — using isotropic structures to strictify Lagrangian distributions, replacing Borisov–Joyce's gluing-up-to-cohomology with genuine gluing — is a real advance, and Theorem 3 gives exactly the global object the shifted-potential programme needs: any derived scheme over C with a −2-shifted symplectic structure, Hausdorff and second-countable classical locus, admits a global Lagrangian distribution as a dg C^∞-manifold. Second, as written the proof is conditional: it leans on an unproved flat-localization claim before Prop. 9 and a hidden global-framing assumption inside the proof of Prop. 16.\n\nThe paper deserves credit for where it sits. It is honest about the difference from [2] (which only glued up to cohomology because distributions were required to be strict from the start) and from [6], and the new content is precisely the strictification and the sheaf-softness argument. Theorem 1 is a genuinely useful simplification: for a purely derived foliation of half rank, the Lagrangian property is a property of the distribution, not of the chosen isotropic structure.\n\nThe soft spots are real but narrow. The claim that Spec(A) can be made flat — after localization, the tangent complex has a generating graded vector subspace with vanishing Lie bracket — is asserted with \"this can always be achieved by localization\" and no proof. Over arbitrary algebraic dg algebras that statement looks unlikely as written: Zariski localization can trivialize a bundle, but it cannot in general produce a commuting frame. The saving grace is that Theorem 3 is a statement about dg C^∞-manifolds, where the analogous flat-localization is plausible by choosing local coordinates. So the gap may be repairable, but the text currently claims the algebraic version and builds Prop. 9 on it. The second issue is smaller but real: Prop. 16 extends a local basis to a basis of E over all of M without justification.\n\nThe citation pattern is fine; [2] and [6] are published external results, and the new claims are not fitted to any parameter. I take the reader's conditional verdict as fair.\n\nBottom line: this is a serious technical paper with an important theorem, and it deserves a serious referee. I would send it out, flagging the flat-localization issue as the main thing the authors need to prove or carefully restrict.","headline":"The isotropic-structure strictification is a real advance, and Theorem 3 is the right input for the CY4 programme, but the proof currently leans on an unproved flat-localization claim (Prop. 9) and a hidden framing assumption (Prop. 16).","tokens_in":28775,"tokens_out":5543,"would_cite":true,"duration_ms":50838,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A20","14N35","14J35","14F05","55N22","53D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every derived scheme over $\\mathbb{C}$ with a $-2$-shifted symplectic structure and Hausdorff second-countable classical points admits a globally defined Lagrangian distribution.","keywords":["shifted symplectic structures","Lagrangian distributions","derived schemes","Calabi-Yau four-folds","Spin(7)-instantons","moduli spaces of sheaves","strictification","dg C-infinity rings"],"falsifier":"Find a well presented dg algebra with a strict $-2$-shifted symplectic form and a purely derived foliation satisfying the conditions of Proposition 8, for which no localization admits a graded vector subspace of the tangent complex with vanishing Lie bracket that generates it; such an example would break Proposition 9 and Theorem 3.","tokens_in":27734,"feed_emoji":"🧩","tokens_out":7234,"duration_ms":68823,"temperature":0.7,"pith_summary":"The paper establishes a global existence theorem for Lagrangian distributions on derived schemes equipped with a $-2$-shifted symplectic form, the structure that appears on moduli spaces of sheaves on Calabi-Yau four-folds. If the theorem is right, every such derived scheme, provided its underlying space of classical points is Hausdorff and second-countable, carries a globally defined distribution that is maximally isotropic with respect to the imaginary part of the symplectic form and negative definite with respect to the real part. The significance is that dividing by such a distribution produces a new space with a globally defined shifted potential, whose critical locus recovers the original derived scheme. This is the key input needed to write moduli spaces of sheaves on Calabi-Yau four-folds as derived critical loci of Spin(7)-type instanton moduli.","feed_headline":"Global Lagrangian distributions exist on -2-shifted symplectic schemes","feed_subtitle":"Every such derived scheme divides by a Lagrangian distribution to yield a global shifted potential.","key_machinery":"The load-bearing object is the purely derived foliation: a derived foliation, locally a quotient of the de Rham complex by weight-1 generators, whose distribution has no cohomology in non-positive degrees. Around it the paper builds semi-strict Lagrangian distributions, where the symplectic form vanishes only modulo $\\delta(A^{-1})$, and then uses the isotropic structure $\\lambda$ to perform the missing strictification. The gluing mechanism is the family of maximally isotropic sub-bundles interpolating between two negative-definite distributions (Proposition 11), which preserves derived-foliation and equivalence properties, letting the sheaf be shown soft.","core_discovery":"The central claim, Theorem 3, is that if $(S,\\omega)$ is a derived scheme over $\\mathbb{C}$ with a $-2$-shifted symplectic structure, and its underlying dg manifold $(M,\\omega_{\\mathrm{im}},\\omega_{\\mathrm{re}})$ has Hausdorff and second-countable space of classical points, then the sheaf on $M$ of purely derived foliations that are Lagrangian distributions for $\\omega_{\\mathrm{im}}$ and negative definite for $\\omega_{\\mathrm{re}}$ is soft. Hence it has global sections: a globally defined Lagrangian distribution exists. The proof reduces the problem to sheaves of strict Lagrangian distributions, using the isotropic structure to strictify non-strict distributions and negative definiteness to interpolate and glue via partitions of unity.","pith_inferences":["The method should carry over to derived Artin stacks that admit a principal bundle from a derived Quot-scheme, as used in Part II; the group action would distribute the chosen distribution along orbits.","Different choices of negative-definite Lagrangian distributions are connected by the interpolation family, so the resulting global potential may be unique up to homotopy; checking this would make the construction intrinsic to the moduli problem.","Because the argument uses dg $C^\\infty$-rings and partitions of unity, the global distribution can be chosen smoothly on the underlying complex manifold, which may make analytic counterparts more accessible.","Theorem 1 suggests a purely cohomological criterion for a distribution to be Lagrangian, so future constructions might verify only rank and vanishing of the symplectic form without solving for the isotropic structure."],"forward_implications":["Every derived scheme over $\\mathbb{C}$ with a $-2$-shifted symplectic structure and Hausdorff second-countable classical points has at least one global Lagrangian distribution.","The quotient of the derived scheme by such a distribution yields a new scheme or stack with a globally defined shifted potential whose critical locus recovers the original scheme.","For moduli of sheaves on Calabi-Yau four-folds, the theorem supplies the input needed to construct an algebraic-geometric version of Spin(7)-instantons as the quotient.","Lagrangian distributions can be re-encoded as strict ones, so local strict charts glue to global strict distributions rather than only up to cohomology.","The softness of the sheaf means existence can be checked locally and extended to compact subsets, allowing inductive gluing."],"supporting_citations":[{"why":"Supplies the definition of shifted symplectic structures and the theorem that moduli of sheaves on Calabi-Yau manifolds carry them.","marker":"[18]"},{"why":"Proves the local Darboux strictification result for $-2$-shifted symplectic forms used to make the symplectic form strict on charts.","marker":"[6]"},{"why":"Introduces integrable distributions, quotients by distributions, and the shifted-potential construction that motivates the main theorem.","marker":"[17]"},{"why":"Develops the gluing of local Lagrangian distributions via partitions of unity and negative definiteness that the paper adapts.","marker":"[2]"},{"why":"Provides shifted symplectic structures on derived Quot-schemes, the intended application for moduli of sheaves.","marker":"[3]"},{"why":"Part II of the sequence, uses global Lagrangian distributions on stable loci of derived Quot-stacks.","marker":"[4]"}],"fun_headline_variants":["Strictification yields global Lagrangian distributions on derived schemes","Every -2-shifted symplectic scheme has a global Lagrangian distribution","Global Lagrangians exist via strictification on derived schemes","Soft sheaves give global Lagrangians on -2-shifted symplectic schemes","Gluing strict Lagrangians: global distributions on derived schemes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on the unproved assertion in Proposition 9 that, after localization, Spec(A) can be made flat: the tangent complex has a graded vector subspace with vanishing Lie bracket that generates it; if this fails, semistrictification, and with it strictification, collapses.","fun_headline_variants_meta":{"raw":{"variants":["Strictification yields global Lagrangian distributions on derived schemes","Every -2-shifted symplectic scheme has a global Lagrangian distribution","Global Lagrangians exist via strictification on derived schemes","Soft sheaves give global Lagrangians on -2-shifted symplectic schemes","Gluing strict Lagrangians: global distributions on derived schemes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000998,"raw_usage":{"total_tokens":4130,"prompt_tokens":755,"completion_tokens":3375,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":371,"completion_tokens_details":{"reasoning_tokens":3284}},"tokens_in":371,"tokens_out":3375,"duration_ms":26982,"temperature":1.0,"reasoning_tokens":3284,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:41:00.585386+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a well presented dg algebra with a strict $-2$-shifted symplectic form and a purely derived foliation satisfying the conditions of Proposition 8, for which no localization admits a graded vector subspace of the tangent complex with vanishing Lie bracket that generates it; such an example would break Proposition 9 and Theorem 3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definition of shifted symplectic structures and the theorem that moduli of sheaves on Calabi-Yau manifolds carry them."},{"cited_title":"A Darboux theorem for derived schemes with shifted symplectic structure","cited_arxiv_id":null,"evidence_quote":"Proves the local Darboux strictification result for $-2$-shifted symplectic forms used to make the symplectic form strict on charts."},{"cited_title":"Algebra, ge- ometry and physics: a conference in honour of Maxim Kontsevich","cited_arxiv_id":null,"evidence_quote":"Introduces integrable distributions, quotients by distributions, and the shifted-potential construction that motivates the main theorem."},{"cited_title":"Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds","cited_arxiv_id":null,"evidence_quote":"Develops the gluing of local Lagrangian distributions via partitions of unity and negative definiteness that the paper adapts."},{"cited_title":"Shifted symplectic structures on de- rivedQuot-schemes","cited_arxiv_id":null,"evidence_quote":"Provides shifted symplectic structures on derived Quot-schemes, the intended application for moduli of sheaves."}],"review_version":1}