{"id":"84a1bcf1-313e-4661-8892-b532b66bc108","arxiv_id":"2602.14341","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symplectic forms on hypersurface algebroids yield generically symplectic Poisson structures whose variation along the degeneracy locus is controlled by the obstruction to lifting truncated-polynomial representations.","lead":"This paper develops a new way to build generically symplectic Poisson structures that become singular along a hypersurface but keep a nontrivial twist in the singular foliation. The twist is governed by groups of truncated polynomials, leading to large families of new examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.3's map Φ depends on an unproven global-extension choice; Lemma 7.4 covers only odd k, and the proof does not show independence of the chosen extension.","rationale":"The reader identified the global-extension assumption as a load-bearing weakness. I agree that this is the weakest point, and I sharpen it: the issue is not only existence but also well-definedness of Φ under different extensions. The paper's own Lemma 7.4 supplies extensions only for odd k and through choices of singularizing functions, so the unconditional phrasing in the abstract overstates the scope. This does not invalidate the local Proposition 5.4 or the explicit odd-k examples, so the existing CONDITIONAL verdict remains appropriate rather than moving to REJECT.","tokens_in":53425,"tokens_out":35391,"duration_ms":323132,"concrete_test":"Take W = T³ with cosymplectic data a = dx¹, B = dx²∧dx³, and let M = T⁴ = T³ × S¹ with the symplectic form B + a∧du. For k = 2, attempt to construct two different global extensions of the local A(a)-symplectic form by choosing two closed 2-forms on M\\W that represent distinct classes in H²(M) and agree with the local model near W. Then run the deformation construction of Theorem 7.3 for a fixed nontrivial σ ∈ M₂(M,W,a) and compare the resulting Poisson structures modulo isotopy preserving W and F = ker(a). If the results are not isotopic, Φ is not well-defined. Also check directly whether any monotone interpolating function f₂ can be chosen globally on both sides of W; if not, the even-k hypothesis of Theorem 7.3 is vacuous for this standard example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction of Φ in Theorem 7.3 begins with a local HS symplectic form on a tubular neighbourhood of W (Eq. 7.1) and assumes it extends to a global algebroid symplectic form on M. The algebroid A(a) itself extends as a subsheaf of TM once ν_W is trivializable, but the closed nondegenerate section of ∧²A(a)* does not extend automatically. Lemma 7.4 provides an extension only for odd k via the singularization trick, and even then it uses a choice of interpolating function f_k. For even k no monotone f_k exists on a full two-sided neighbourhood, so the theorem's hypothesis is not known to be satisfiable in that case. Moreover, the proof never shows that the resulting class in Pois(M,W,F) is independent of the chosen extension or of the choice of f_k. Different choices of the smooth closed 2-form on M\\W can yield non-isotopic Poisson structures, so Φ may depend on input data not contained in (M,W,a). Since the abstract claims a map from the G_k-character variety to the moduli space of Poisson structures, this well-definedness gap is load-bearing. If the global extension is meant to be part of the input data, the theorem should say so explicitly, and the claimed even-k version remains unsupported by examples.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops the symplectic geometry of hypersurface (HS) algebroids, a class of Lie algebroids generalizing b^k-tangent bundles. Its main results are: (i) a cohomology decomposition for HS algebroids (Theorem 4.19), with a universal cdga S_k governing the singular parts; (ii) a normal form and structure theory for HS symplectic forms, culminating in Proposition 5.4, which identifies the symplectic variation of the induced foliation with the restriction of the algebroid's extension class; (iii) deformation theorems (Theorems 6.6 and 6.7) that allow symplectic forms to be deformed along deformations of the algebroid; (iv) a construction, in Theorem 7.3, of maps from a G_k-character variety into the moduli space of Poisson structures, with the variation detecting non-triviality of the family. Explicit examples include genus-g surfaces, the Heisenberg group, Arnold's cat map, and the universal algebroids E_{2n+1}, connecting to the Babenko--Taimanov symplectic forms.","tokens_in":53714,"tokens_out":8421,"duration_ms":80142,"significance":"If the main claims hold, the paper gives a genuinely new invariant: the symplectic variation of an HS Poisson structure is controlled by the obstruction to lifting a G_k-representation of π_1(W) to G_{k+1}. This distinguishes HS Poisson structures from b^k-symplectic structures and provides a computable way to detect non-triviality of deformations. The explicit computations of cohomology, the universal algebroids, and the detailed examples (particularly the mapping-torus families of Corollary 7.6) are substantial contributions. The paper also ships a large amount of concrete, reproducible algebraic data, including the explicit formulas for the universal cdga and the cohomology computations, which is a strength. However, the central construction of the map Φ in Theorem 7.3 is conditional on a global extension assumption whose independence is not established, and the even-order case is not supported by any construction or example. These issues need to be resolved before the main theorem can be accepted as stated.","major_comments":[{"comment":"The construction of Φ begins with the local algebroid symplectic form ω = B - k a ∧ dt/t^{k+1} on a tubular neighbourhood of W and assumes this form extends to a global algebroid symplectic form on M. The proof never shows that the resulting class in Pois(M,W,F) is independent of the chosen extension, nor of the choice of interpolating function f_k in Lemma 7.4. Different smooth closed extensions on M\\W can lead to non-isotopic Poisson structures, so the map from M_k(M,W,a) alone is not well-defined unless the extension is declared to be part of the input data and the target is adjusted accordingly. This is load-bearing for the paper's central claim that there is a map from a character variety to a moduli space of Poisson structures.","section":"§7.1.4, Theorem 7.3"},{"comment":"Lemma 7.4 supplies the required global extension only for odd positive integers k. For even k, no construction or example is given; the monotonicity argument for f_k fails on a two-sided neighbourhood, and Remark 7.5 only mentions a doubling construction without details. Since Theorem 7.3 and the abstract claim maps from G_k-character varieties for general k, the even-k case remains unsupported. The statement should either be restricted to odd k or an even-k extension construction must be provided.","section":"Lemma 7.4 and Remark 7.5"},{"comment":"The deformation theorems depend essentially on the 'nice' condition (Definition 6.5), used in Lemma 6.11 and Corollary 6.13 to ensure H^0(W,S_k(L)) = R and smooth primitives. This excludes the monodromy-contained-in-{±1} case, which includes the classical b^k-symplectic setting. The condition is stated explicitly, but the paper's presentation in the introduction and abstract suggests a fully general deformation method; the limitation should be made prominent, and the consequences for the b^k comparison should be spelled out.","section":"§6, Theorems 6.6 and 6.7"}],"minor_comments":[{"comment":"Proposition 6.3 is stated without proof. If it is not needed for the main results, it should be deleted or explicitly marked as a remark; if it is intended to justify Assumption 6.4, a proof or reference is required.","section":"Proposition 6.3"},{"comment":"The abstract states that the paper constructs maps from a G_k-character variety into the moduli space of Poisson structures without mentioning the global extension assumption. This should be qualified, since Theorem 7.3's hypothesis 'Assume that ω extends' is essential and is not satisfied by any known construction for even k.","section":"Abstract and Introduction"},{"comment":"The splitting S defined in Equation (4.6) depends on the tubular neighbourhood, bump function, metric, and splitting data. The proof of Theorem 4.19 and Lemma 6.9 use this dependence; it would help the reader if the smooth dependence on σ were stated explicitly, since it is used in the deformation arguments.","section":"Section 4.5, Eq. (4.6)–(4.7)"},{"comment":"In the displayed formula for ω after the definition of the canonical form, the term 'γe∧(3at+ 2bt^2 − ct^3)' appears to contain a typographical error; the factor '3at' should likely be '3a' or '3a t' with clearer notation.","section":"Example 5.9"},{"comment":"The uniqueness statement compares cohomology classes [ω_1(t)] = [ω_2(t)] ∈ H^2(A(t)) for different t, but no canonical identification of these cohomology groups across t is specified. This should be clarified.","section":"Theorem 6.6, uniqueness statement"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution with substantial new content, and the main invariant (Proposition 5.4) is likely correct. However, the central map Φ in Theorem 7.3 is not yet well-defined without proving independence of the global extension, and the even-k case is unsupported. I recommend major revision. The heavy reliance on the previous preprint [BdPW25] for classification and extension-class results is acceptable but puts pressure on the referee process; the editor may wish to ensure that [BdPW25] is publicly available in a stable form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper is worth taking seriously. Proposition 5.4 — symplectic variation equals minus the extension class — is a clean, genuinely new invariant, and the universal algebroids E_k and U_k plus the deformation-to-character-variety construction are useful new machinery. The examples (Heisenberg, Arnold cat map, genus surfaces) are explicit enough to check, and the paper is honest about its overlap with [BdPW23], so the reader can separate new material from background. The cohomology computations in Section 4 are detailed and plausible, and the citation practice looks fair.\n\nThe main soft spot is exactly the one the stress-test flags. Theorem 7.3 assumes the local HS symplectic form extends to a global algebroid symplectic form on M. That is stated as a hypothesis, so the theorem is not false, but the paper never shows that the resulting class in Pois(M,W,F) is independent of the chosen extension, nor of the choice of interpolating function in Lemma 7.4. Different extensions could plausibly give non-isotopic Poisson structures, and the proof does not address this. Lemma 7.4 only constructs extensions for odd k; the even case is left to a remark about doubling the hypersurface. So the advertised map from the whole G_k-character variety is really a map from the locus where a global extension is known to exist, and even there the well-definedness is open. I would want that fixed, or the theorem restated so that the extension is explicitly part of the input data.\n\nSmaller issues: Proposition 6.3 is stated without proof, but the paper says it is not needed, so that is minor. The classification of HS algebroids is imported from the authors' earlier work, which limits self-containedness but is not a defect. The omitted proofs are few, and the central formula in Proposition 5.4 is derived directly rather than fitted to examples.\n\nWho this is for: Poisson geometers and people working on singular symplectic structures, especially those who care about b^k-symplectic geometry and its generalizations. It deserves a serious referee; the referee should focus on the well-definedness of Φ and the status of even k. I would recommend sending it to review, with the expectation of a revision.","headline":"New symplectic-variation formula and a deformation machine from character varieties, but the map Φ has a real well-definedness gap and the even-k case is only conditional.","tokens_in":54205,"tokens_out":2788,"would_cite":true,"duration_ms":27421,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D17","53D05","53C12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Poisson structure that degenerates along a hypersurface gets its symplectic variation from the obstruction to lifting a truncated-polynomial representation of the fundamental group to the next degree.","keywords":["hypersurface algebroids","singular Poisson structures","b^k-symplectic structures","symplectic foliation","symplectic variation","truncated polynomial groups","extension class","character variety"],"falsifier":"In the explicit order-5 example on the compact nilmanifold (with η1=a, η2=b, η3=-c), the extension class is -2 a∧c; Proposition 5.4 predicts var(ω_F)=2a∧c. Directly computing the foliated symplectic forms on the level sets of γ and computing their variation would settle whether the formula holds; if the variation vanishes or differs from the predicted class, the central claim fails.","tokens_in":53277,"feed_emoji":"🍃","tokens_out":11202,"duration_ms":92711,"temperature":0.7,"pith_summary":"This paper develops a formalism for geometric structures that become singular to polynomial order along a hypersurface, and applies it to Poisson geometry. Its central claim is that a generically symplectic Poisson structure which drops rank along a hypersurface W can carry genuinely non-zero 'symplectic variation' of its leaf symplectic forms, and that this variation is exactly the obstruction to lifting a representation of the fundamental group of W from the group of degree-k truncated polynomials to degree k+1. Earlier b^k-symplectic structures always had vanishing variation; the paper shows that vanishing is not forced, and it constructs families of Poisson structures on compact manifolds where the variation is non-zero and computable. A sympathetic reader should care because this connects an invariant of singular foliations to a Lie-algebroid and group-cohomology invariant, and it turns the G_k-character variety of W into a source of new Poisson structures whose non-triviality can be detected through the variation.","feed_headline":"A lifting obstruction sets the variation of singular symplectic leaves","feed_subtitle":"Generalizing b^k-symplectic structures, these Poisson structures have leaf symplectic forms that genuinely vary, with computable variation.","key_machinery":"The central object is a hypersurface (HS) algebroid, a Lie algebroid whose anchor map is an isomorphism away from W and drops to corank 1 along W, with order equal to the vanishing order of the anchor determinant. Such an algebroid is equivalent to a splitting σ of the k-th order jet algebroid of the normal bundle, encoded by a flat connection plus twisted 1-forms satisfying a Maurer-Cartan equation; the classification data is a representation φ: π_1(W) → G_k, where G_k is the group of degree-k truncated polynomials under composition. The argument is carried by the extension class e(φ) ∈ H^2_W(ν_W^{-k}), the obstruction to lifting φ to G_{k+1}. Proposition 5.4 identifies minus the restrictio","core_discovery":"On the paper's own terms: let A be an order k+1 hypersurface algebroid for a hypersurface W, classified by a representation φ: π_1(W) → G_k, and let ω be an algebroid symplectic form with associated Poisson structure Q = ρ(ω^{-1}). Then Q induces a corank-1 symplectic foliation (F, ω_F) on W, and Proposition 5.4 states var(ω_F) = -e(φ)|_F, where e(φ) ∈ H^2_W(ν_W^{-k}) is the extension class obstructing the lift of φ through the extension 0 → R → G_{k+1} → G_k → 0. Because e(φ) can be non-zero, the variation of the symplectic leaves is generally non-trivial, in contrast to b^k-symplectic structures. The paper also proves a cohomology decomposition H^•(A) ≅ H^•(M) ⊕ H^{•-1}(W, S_k(ν_W)) and co","pith_inferences":["Editorial inference: the variation formula gives a practical invariant for distinguishing singular Poisson structures up to isotopy: if two HS Poisson structures on the same foliation have different symplectic variations, no isotopy preserving W and F can relate them, even if their underlying algebroids are isotopic. This could serve as a coarse Torelli-type invariant for the class.","Editorial inference: the deformation machinery is quite general and suggests that the same 'drag along a deformation path' procedure works whenever the principal part of a closed form can be kept closed in a fixed cochain complex. One testable extension is to construct HS symplectic forms on hypersurfaces that are not mapping tori (the paper's Question 1.2) by solving the twisted Maurer-Cartan equ","Editorial inference: the restriction to odd k in the global-extension step points to a double-cover mechanism for even k; a natural prediction is that every even-order HS symplectic structure produced by this method is the quotient of an odd-order one on a Z/2 cover, with the symplectic variation anti-invariant under the deck transformation."],"forward_implications":["If the central formula is right, the symplectic variation of the foliation on W is a group-cohomology invariant: var(ω_F) = -e(φ)|_F, so it vanishes exactly when the G_k-representation lifts to G_{k+1}.","Because the extension class can be non-zero, HS Poisson structures form a distinct class from b^k-symplectic structures, whose symplectic variation always vanishes; the paper constructs explicit examples exhibiting this difference.","The cohomology decomposition H^•(A) ≅ H^•(M) ⊕ H^{•-1}(W, S_k(ν_W)) gives a practical tool for computing Lie algebroid cohomology, generalizing the known decompositions for logarithmic and b^k-tangent bundles.","The character-variety map produces many new compact examples: in the order-4 mapping-torus case, the resulting families are parameterized by a sphere of dimension dim H^1(N)_{e^λ} + dim H^1(N)_{e^{2λ}} - 1.","The universal algebroids and their quotients yield Poisson structures whose symplectic leaves include the compact symplectic forms previously used to build non-formal simply connected symplectic manifolds; those forms now appear as leaves of a global Poisson structure."],"fun_headline_variants":["Non-trivial symplectic variation from lifting obstructions in Poisson geometry","Truncated polynomials yield Poisson structures with varying symplectic leaves","Symplectic variation beyond b^k: the role of a lifting obstruction","Corank-1 Poisson with non-zero leaf variation from G_k lifts"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper's main construction (Theorem 7.3) assumes that the local singular symplectic form extends globally to a hypersurface algebroid on M with trivializable normal bundle, and that the induced flat connection on the normal bundle has monodromy not contained in {±1}; the paper proves such global extensions only for odd k, so if these fail, the character-variety map is only local or may not exist.","fun_headline_variants_meta":{"raw":{"variants":["Non-trivial symplectic variation from lifting obstructions in Poisson geometry","Truncated polynomials yield Poisson structures with varying symplectic leaves","Symplectic variation beyond b^k: the role of a lifting obstruction","Corank-1 Poisson with non-zero leaf variation from G_k lifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1220,"prompt_tokens":804,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":548,"tokens_out":416,"duration_ms":3997,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T23:13:23.387570+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the explicit order-5 example on the compact nilmanifold (with η1=a, η2=b, η3=-c), the extension class is -2 a∧c; Proposition 5.4 predicts var(ω_F)=2a∧c. Directly computing the foliated symplectic forms on the level sets of γ and computing their variation would settle whether the formula holds; if the variation vanishes or differs from the predicted class, the central claim fails.","supporting_citations":[],"review_version":1}