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Weinstein neighbourhood theorems for stratified subspaces

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Pith's one-line read The paper proves that a symplectic neighbourhood of a strongly coisotropic stratified subspace is determined by its stratified diffeomorphism type and the restricted Zariski form.

desk verdict Real result with a small fixable gap in one proposition; worth refereeing and probably citing. read the letter →

arxiv 2507.04897 v3 pith:44MMXK55 submitted 2025-07-07 math.SG math.DG

Yael Karshon , Sara B. Tukachinsky , Yoav Zimhony This is my paper · ORCID
classification math.SGmath.DG MSC 53D0558A3553D1258A40
keywords WeinsteinneighbourhoodtheoremstratifiedsubspacescoisotropicZariskitangentspaceMoser'stricksymplecticnormalformsingulargeometryLagrangiansingularities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a singular analogue of Weinstein's neighbourhood theorem. It establishes that a symplectic neighbourhood of a strongly coisotropic stratified subspace—a closed union of smooth strata with conical, smoothly locally trivial singularities—is determined, up to symplectomorphism, by the stratified diffeomorphism type of the subspace together with the pullback of the symplectic form to its Zariski tangent spaces, the tangent data encoded by derivations at points. If this is right, local models for singular subspaces such as Lagrangian pinwheels, mutation configurations, and momentum-map zero sets are unique, which is the kind of input used to construct and distinguish exotic Lagrangians. The paper also proves a strong version of Moser's trick and a tubular neighbourhood theorem for these stratified subspaces.

What carries the argument

The machinery is the class of stratified subspaces that are smoothly locally trivial with conical fibres, studied through their Zariski tangent spaces and Zariski forms. Its load-bearing components are: the relative Poincaré lemma by fiber integration along a smooth weak deformation retraction; the resulting strong Moser trick (Theorem D); Euler-like vector fields and their induced tubular neighbourhoods, chosen to be tangent to higher strata; and the concept of local extendability of a tangent-bundle isomorphism, which is the precise way a derivative-like condition is enforced at singularities. The strongly coisotropic condition (the symplectic annihilator of the Zariski tangent is contained in the tangent space of the stratum) forces the Moser vector field to point along strata, so the flow preserves the stratification.

What would settle it

A concrete falsifier would be a pair of strongly coisotropic stratified subspaces satisfying all hypotheses of Theorem A—same stratified diffeomorphism type and same Zariski form—yet having no symplectomorphic neighbourhoods. A sharper test: find a smoothly locally trivial conical stratified subspace with no smooth weak deformation retraction of any neighbourhood; then the key step in the proof of Theorem B (Section 4.4) has no mechanism to run Moser's argument, so the stated proof cannot establish the normal form.

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Extended reading notes

Core claim

The central discovery is Theorem A: for two strongly coisotropic stratified subspaces that are smoothly locally trivial with conical fibres, any stratified diffeomorphism preserving the symplectic form as a Zariski form extends, possibly after an isotopy through such diffeomorphisms, to a symplectomorphism of neighbourhoods. The same statement holds without strong coisotropicity when a symplectic tangent bundle isomorphism over the diffeomorphism is locally extendable (Theorem B), a hypothesis that reduces to the classical derivative condition when the strata are smooth submanifolds. The proof works stratum by stratum, producing a symplectomorphism on lower strata, then extending to the next stratum using a tubular neighbourhood whose radial scaling preserves the stratification and a Moser trick that adjusts the symplectic form; the strong coisotropic condition is what keeps the Moser vector field tangent to the strata.

Load-bearing premise

The load-bearing premise is that every smoothly locally trivial conical stratified subspace admits a smooth weak deformation retraction of a neighbourhood onto itself; the paper imports this from the cited reference [29] rather than proving it here, and the Moser trick of Theorem D depends on it.

Editorial extensions

If this is right

  • Any Lagrangian stratified subspace that is strongly coisotropic and embeds in a symplectic manifold has a neighbourhood determined up to symplectomorphism by its stratified diffeomorphism type and the induced Zariski form, extending the classical Lagrangian neighbourhood theorem to the singular setting.
  • Zero level sets of momentum maps for compact group actions and components of the critical set of the norm-squared momentum map gain local symplectic models, since their stratifications satisfy the conical local-triviality condition.
  • The paper's model-space application goes through under the weaker hypothesis that the skeleton embeds as a Lagrangian stratified subspace, not the full model space, because the normal form supplies the missing symplectic neighbourhood.
  • Two symplectic forms that agree on a stratified subspace are symplectomorphic on a neighbourhood whenever that subspace admits a smooth weak deformation retraction (Theorem D).
  • Near an isotropic stratified subspace, the symplectic form is exact with an explicit fiber-integration primitive (Theorem 6.1), so the normal-form neighbourhoods are candidates for Liouville and Weinstein structures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if Theorem A is correct, the uniqueness of symplectic neighbourhoods should make 'almost toric' local manipulations (node slides, Lagrangian mutations) valid in any ambient symplectic manifold once the singular subspace is symplectically embedded, not only inside the original model spaces.
  • Editorial inference: the isotopy clause in Theorem A means the classifying data are not just the stratified diffeomorphism type and the Zariski form but also a path-component of the space of stratified diffeomorphisms preserving that form; counting such components could be relevant to distinguishing exotic Lagrangians.
  • Editorial inference: a natural testable extension is to check strong coisotropicity for arboreal c-buildings; the paper leaves that open, and it would decide whether the normal form applies to that family directly.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper proves a stratified analogue of Weinstein's neighbourhood theorem. For smoothly locally trivial conical stratified subspaces A_i of symplectic manifolds, Theorem A states that if the A_i are strongly coisotropic and a stratified diffeomorphism g preserves the restricted Zariski 2-form, then symplectic neighbourhoods are symplectomorphic, with the restriction to A_0 isotopic to g through form-preserving stratified diffeomorphisms. Theorem B is a more general version assuming a locally extendable symplectic bundle isomorphism, and Theorems C and D are a non-symplectic tubular neighbourhood theorem and a Moser-type result with weak deformation retractions. The paper also proves exactness of the symplectic form near isotropic stratified subspaces and discusses applications to Lagrangian pinwheels, mutation configurations, and exotic tori.

Significance. If correct, this result unifies and generalizes several ad hoc normal form theorems in symplectic topology and gives a practical criterion: a stratified diffeomorphism preserving the Zariski form determines the symplectic neighbourhood. The paper is careful and unusually self-contained in places: the recursion in Section 5 is explicit, Theorem 3.7 on extending local bundle isomorphisms is a useful tool, and the appendices on Euler-like vector fields and on the strong deformation retraction case are valuable. The main caveats are the unproved closed-stratum case in Proposition 5.18 and the unstated dependence of Theorem B on a retraction result imported from [29]. Neither appears fatal, but both are load-bearing and need to be fixed before the proofs are complete.

major comments (2)
  1. [Section 5.2.3, Proposition 5.18] The step 'By the strongly coisotropic assumption on (A_0,S_0), X_t(p0)∈(T^Z_{p0}A_0)^{ω0}⊂T_{p0}Y' applies Eq. (2.1) of Definition 2.30, but that inclusion is only asserted for non-closed strata. In the inductive setting, U^{X_0}_0 is chosen inside M^{≥d}, so the only closed stratum of A_0 that can meet U^{X_0}_0∩A_0 is X_0 itself. For p_0∈X_0, Proposition 5.13(2) gives β_{p_0}=0, and since ω_t is nondegenerate, X_t(p_0)=0, so the desired tangency holds trivially. This is a genuine but local gap: the proof as written is incomplete for closed strata, and the missing case must be added explicitly. The same correction is needed in the proof of Proposition 5.19, which relies on Proposition 5.18 to show that the intermediate flows preserve strata.
  2. [Section 4.4, proof of Theorem B] The statement 'by the results of [29], there exists a neighbourhood V_0 of A_0 in U_0 and a smooth weak deformation retraction of V_0 to A_0' is load-bearing: it is exactly what allows Theorem D to be applied, and hence what produces the symplectomorphism in Theorem B. The paper does not state which result in [29] is being used, nor does it verify its hypotheses here. Please state the precise theorem that supplies this retraction and confirm that every smoothly locally trivial conical stratified subspace satisfies it; alternatively, prove the needed retraction statement in the present paper, for example using the Euler-like vector field machinery of Appendix A. Without this, the proof of Theorem B is not self-contained at a central point.
minor comments (4)
  1. [Throughout] Several cross-references call definitions and notations 'theorems', for example 'stratified subspaces (Theorem 2.14)', 'Theorem 2.22', and 'Theorem 2.29' in Section 1.2, and 'Recall Theorem 2.24' in Section 5.1. The final version should correct these to Definition or Notation as appropriate.
  2. [Proposition 5.2(3)] The display in case (3) contains V^{X^d_0}_i but should be V^{X^d_i}_i; as written the notation is undefined for i=1.
  3. [Proposition 5.19(2)] The set denoted W^{X_0}_0 is almost certainly meant to be W^{≤(d-1)}_0∩U^{X_0}_0∩A_0: the proof cites Theorem 5.17, which concerns W^{≤(d-1)}_0, and the subsequent use in Definition 5.1(4)(b) is about W^{≤(d-1)}_0.
  4. [Typos and notation] There are several typographical issues: 'manfiold' in Definitions 2.28 and 2.30; 'qucik into' in Section 5.2.2; 'Theorem 4.1' in Section 6 should be Definition 4.1; and Theorem D says the forms agree on 'T M|_A' where the tangent bundle of V is intended. These are presentation issues but should be cleaned up.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity; the only flagged item is a minor self-citation for the deformation-retraction input to Moser's trick.

full rationale

Theorems A and B are not circular by construction: their hypotheses (a stratified diffeomorphism preserving the restricted Zariski form, strong coisotropy, or local extendability of a symplectic tangent-bundle isomorphism) do not contain the conclusions (existence of a symplectomorphism of neighbourhoods with prescribed restriction to the stratified subspace). The proof is a genuine inductive Moser/tubular-neighbourhood argument: the 1-form beta in Theorem D is produced by fiber integration from the given forms and the retraction, the Moser vector field is then determined by beta, and no fitted parameter is renamed as a prediction. The single author-overlap citation that supplies a non-trivial input is [29], used in Section 4.4: 'By assumption, A_0 is smoothly locally trivial with conical fibers, so by the results of [29], there exists a neighbourhood V_0 of A_0 in U_0 and a smooth weak deformation retraction of V_0 to A_0.' This retraction is an input to Moser's trick rather than an output of the theorem, and [29] is prior published work whose hypotheses are exactly the conical-stratification regularity, with no assumption of the symplectic normal form being proved; under the rubric it is independent support, so the self-citation is minor rather than definitionally circular. A separate correctness issue, not a circularity, is that Proposition 5.18 invokes the strong-coisotropic inclusion without explicitly treating closed strata; at a closed stratum the Definition 2.30 inclusion is not guaranteed, although the missing case is patchable because beta, and hence the Moser vector field, vanishes on that stratum.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's results rest on standard differential-space formalism and on a class of stratified subspaces with conical fibers. No free parameters are fitted and no new particles, forces, or geometric entities are introduced. The main externally supplied input is the prior theorem [29] that such subspaces admit smooth weak deformation retractions; since one of the authors is a co-author of [29], this is the main self-citation burden.

assumptions (4)
  • standard math Sikorski differential space formalism and Zariski tangent spaces behave as stated (Definitions 2.1 through 2.13, cited to [13,18,25]).
    The central claims use pullbacks of Zariski forms and embeddings of Zariski tangents; these are standard background results assumed without proof.
  • domain assumption A stratified subspace that is smoothly locally trivial with conical fibers admits a smooth weak deformation retraction from a neighbourhood.
    Invoked as 'by the results of [29]' in Section 4.4 and Section 5.2. This is a prior theorem by one of the authors, not reproved here.
  • domain assumption Strong coisotropy as in Definition 2.30 is the right condition for the normal form; in particular the inclusion (T^Z_p A)^omega subset T_p X holds for non-closed strata.
    Definition 2.30 is imposed; the proof of Proposition 5.18 applies it and would need a small added case for closed strata.
  • domain assumption For Theorem 6.1, the stratified subspace is Whitney (B) regular, which implies the closedness of the pieces A^{<=d} and the density argument used in the proof.
    The exactness theorem relaxes the conical-fiber condition to Whitney (B) regularity and uses that regularity essentially in the proof.

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Pith. "Pith review of Weinstein neighbourhood theorems for stratified subspaces." pith.science (2026). https://pith.science/paper/44MMXK55

@misc{pith2026250704897,
  author       = {Pith},
  title        = {Pith review of: Weinstein neighbourhood theorems for stratified subspaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44MMXK55}},
  note         = {Machine review of arXiv:2507.04897}
}
read the original abstract

By analogy with Weinstein's neighbourhood theorem, we prove a uniqueness result for symplectic neighbourhoods of a large family of stratified subspaces. This result generalizes existing constructions, e.g., in the search for exotic Lagrangians. Along the way, we prove a strong version of Moser's trick and a (non-symplectic) tubular neighbourhood theorem for these stratified subspaces.

Figures

Figures reproduced from arXiv: 2507.04897 by the authors.

Figure 5.1
Figure 5.1. A schematic drawing of neighbourhoods: the stratified subspace A≤d is a union of the three lines, which represent strata of dimension d, and the vertex in the middle, which represents A≤(d−1) . (1) G Xd 0 restricts on W Xd 0 0 to a symplectomorphism W Xd 0 0 → W Xd 1 1 ; (2) G Xd 0 agrees with G ≤(d−1) on W ≤(d−1) 0 ∩ U Xd 0 0 ; (3) g Xd 0 agrees with g ≤(d−1) on W ≤(d−1) 0 ∩ U Xd 0 0 ∩ A0 and on  U Xd 0 0 \ V Xd 0… view at source ↗

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