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On singular Lagrangian fibrations and applications to symplectic embeddings I

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Pith's one-line read The paper constructs explicit singular Lagrangian torus fibrations on disk cotangent bundles of spheres of revolution, of the 6-dimensional ellipsoids $E(1,1,c,c)$, and of the Lagrangian bidisk, and uses them to prove that the Gromov…

desk verdict A genuinely useful construction method with explicit base formulas, but the main existence theorem is built on an unproved foliation modification that must be fixed before the paper's central claims are solid. read the letter →

arxiv 2506.01556 v2 pith:WDWO26HC submitted 2025-06-02 math.SG

Santiago Achig-Andrango , Renato Vianna , Alejandro Vicente This is my paper
classification math.SG MSC 53D1253D3537J35
keywords singularLagrangianfibrationsalmosttoricdiskcotangentbundlesspheresofrevolutionGromovwidthbidiskdomainsfocus-focussingularities
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

The paper's thesis is that certain disk cotangent bundles, which are not toric, can still be given explicit singular Lagrangian torus fibrations whose base diagrams are computable. The main structural result, Proposition 2.5, turns a Hamiltonian $T^{n-1}$-action together with a foliation of reduced 2-manifolds by simple closed curves into a singular Lagrangian torus fibration, with nodal singularities appearing when a chosen curve passes through a fixed point. Applying this to a sphere of revolution with one equator yields an almost toric fibration with exactly two nodal singularities, and analogous fibrations are built for $D^*E(1,1,c,c)$ in dimension 6 and for the Lagrangian bidisk. These fibrations recover previously known toric domains and give a new proof that the Gromov width of $D^*S^3$ is $2\pi$. The reason to care is that the base diagram becomes a tool for symplectic embedding problems, converting ball-packing questions into comparisons of explicit planar or 3-dimensional regions.

What carries the argument

The engine is Proposition 2.5, a construction that converts a Hamiltonian $T^{n-1}$-action on $T^*M$ with moment map $\mu$, together with a family of simple closed curves foliating the reduced 2-dimensional image $f_\lambda(\mu^{-1}(\lambda))$, into a singular Lagrangian torus fibration on $D^*M$. For each pair $(\lambda,s)$, the lift $L_{\lambda,s}=f_\lambda^{-1}(\gamma_{\lambda,s})$ is an embedded Lagrangian torus, and the second coordinate of the fibration is the symplectic area $A$ of a disk whose boundary lies on $L_{\lambda,s}$. The mechanism's control of singularities is the freedom to choose the foliation on the singular level set $\mu=0$: arranging leaves so that no leaf meets both fixed points yields exactly two nodal (focus-focus) singularities, while other choices alter the base diagram by nodal slides. In the sphere case the load-bearing identity is equation (3), which expresses $\eta_3^2u(\xi_3)^2(1+u'(\xi_3)^2)=u(\xi_3)^2|\eta|^2-\mu^2/4\pi^2$ and identifies the leaves as closed curves in the $(\xi_3,\eta_3)$-plane.

What would settle it

Take a profile function $u$ with a single equator, choose the alternative foliation of $X^2_0$ described in the proof of Theorem 1.1, and compute the monodromy of each of the two purported nodal singularities directly from the lifted tori; if either monodromy is not conjugate to the focus-focus matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$, the claim of exactly two nodal singularities fails. A simpler concrete test is to check, via equation (3), whether a leaf of the modified foliation that passes through one fixed point has smooth part that is isotropic: if not, Proposition 2.5 does not hold for that foliation.

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Extended reading notes

Core claim

This paper establishes that a sphere of revolution $S\subset\mathbb R^3$ with exactly one equator has a disk cotangent bundle $(D^*S,\omega_{\mathrm{can}})$ admitting an almost toric fibration $f:(D^*S,\omega_{\mathrm{can}})\to B_S\subset\mathbb R\times\mathbb R_{\ge0}$ with exactly two nodal singularities, with the base region $B_S$ described explicitly inside the proof. The same construction produces a singular Lagrangian fibration on the 6-dimensional ellipsoid bundle $D^*E(1,1,c,c)$ with a computable 3-dimensional base region, and an almost toric fibration on the Lagrangian bidisk $P_L$. From these fibrations the paper recovers the toric domains obtained in earlier work on spheres of revolution and on the Lagrangian bidisk, and proves that $c_{\mathrm{Gr}}(D^*S^3)=2\pi$: the lower bound comes from a Traynor-trick ball embedding into a pyramid inside the fibration's base, and the upper bound comes from the symplectic identification of $D^*S^3$ with the Fermat quadric minus a hyperplane section, whose Gromov width is known to be $2\pi$. The authors present this as a general technique for converting integrable-system data into singular Lagrangian fibrations that can be used to study symplectic embeddings.

Load-bearing premise

The construction depends on an unproved freedom to replace the natural level-set foliation on the singular level $\mu=0$ by any foliation by simple closed curves that retract to a point, and to have the lifted fibration still be Lagrangian with exactly nodal singularities.

Editorial extensions

If this is right

  • Theorem 1.1 gives every sphere of revolution with one equator an almost toric fibration whose base has exactly two nodal points, so the disk cotangent bundle is represented by a base diagram in $\mathbb R\times\mathbb R_{\ge0}$.
  • For $D^*E(1,1,c,c)$, the singular Lagrangian fibration exists with an explicit area function, giving a 3-dimensional base region; at $c=1$ this yields the Gromov width lower bound $2\pi$ for $D^*S^3$.
  • The toric domains obtained in earlier work for spheres of revolution and for the Lagrangian bidisk are recovered from these fibrations by transferring-the-cut operations, $SL(2,\mathbb Z)$ changes of coordinates, and nodal trades.
  • The Traynor trick applies to the resulting fibrations: a pyramid contained in the base embeds a symplectic ball, giving $c_{\mathrm{Gr}}(D^*E(1,1,c,c))\ge 2\pi c$ for $c<1$ and $\ge 2\pi$ for $c\ge1$.
  • Because $D^*S^3$ symplectically embeds into scaled copies of $D^*E(1,1,c,c)$, monotonicity of symplectic capacities upgrades these lower bounds to the stated Gromov width estimates.

Reading between the lines

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

  • A direct implication the authors leave implicit is that the same construction should produce almost toric fibrations on cotangent disks of any surface of revolution with finitely many equators, with the number of nodal points tracking the number of critical circles.
  • The explicit base-region formulas are quantitative enough that one could compute other symplectic capacities or embedding obstructions for $D^*E(1,1,c)$ and $D^*E(1,1,c,c)$, not only the Gromov width.
  • The authors state a programme of extending Proposition 2.5 from free $T^{n-1}$-actions to $T^k$-actions; if this extension succeeds, the higher-dimensional bidisk fibrations would follow from the same area-of-disk coordinate rather than from special symmetries.
  • A testable extension is to check whether the foliation-choice step in Theorem 1.1 remains Lagrangian for every profile function $u$ with one equator, or whether additional conditions on $u$ are needed to guarantee exactly two nodal singularities.
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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

4 major / 4 minor

Summary. The paper develops a general method for constructing singular Lagrangian torus fibrations on disk cotangent bundles equipped with a Hamiltonian torus action, then applies it to spheres of revolution with one equator, ellipsoids of revolution E(1,1,c) and E(1,1,c,c), and the Lagrangian bidisk. The main technical tool is Proposition 2.5, which builds a fibration from a moment map μ, a T^{n-1}-invariant map f_λ to a 2-manifold, and a family of simple closed curves foliating the image. The authors compute explicit area integrals defining the base regions, recover toric domains from [FRV23] and [Ram17], and use the constructions to prove c_G(D*S^3)=2π and lower bounds for D*E(1,1,c,c).

Significance. The proposed framework is attractive and, if completed, would give explicit almost toric fibrations with computable base diagrams on several previously inaccessible spaces. Strengths are the concrete coordinate computations, the closed-form area formulas (8), (9), (21) and (27), and the direct ball-embedding argument for the lower bound in Proposition 1.8. However, the central existence theorem (Theorem 1.1) and its higher-dimensional analogues are not yet proved as stated: the crucial modification of the foliation on the singular level set μ=0 is asserted rather than constructed, and the nodal nature of the singular fibres is not verified. The paper therefore establishes a promising method and conditional statements, but the main claims require substantial additional work.

major comments (4)
  1. [§2 (Prop. 2.5); §3.1] Proposition 2.5 proves only that the smooth part of each L_{λ,s} is isotropic; it does not identify the local structure of the fibres where the torus action is not free. The proof ends by saying the singular fibres occur over a stratified subset of codimension at least two, which is not enough to conclude that the fibration is almost toric with nodal singularities. Since the theorems in Section 1 claim almost toric fibrations with exactly two nodal singularities, a local normal-form or monodromy computation at each singular value is required.
  2. [§3.1, proof of Theorem 1.1] For the level set μ=0 the proof states that one is 'free to choose a different foliation' of X^2_0 and arranges it 'so that no leaf intersects both points', asserting that this yields exactly two nodal singularities. No such foliation is constructed, and no argument shows that it fits into a smooth family γ_{λ,s} satisfying condition (iv) of Proposition 2.5 for λ in a neighbourhood of 0. The fibration is defined piecewise (|η|-foliation for μ≠0, modified foliation for μ=0), so the area coordinate A could fail to be continuous or smooth across μ=0; the base diagram in Figure 1 is therefore not rigorously obtained.
  3. [§3.2, proof of Theorem 1.4; §3.3, proof of Theorem 1.6] The same gap appears in the 6-dimensional construction: for μ1=μ2=0 the proof chooses a foliation by simple closed curves avoiding both distinguished points, but gives no explicit family and no verification that it extends smoothly to nearby λ; the statement that the base diagram contains only nodal singularities is not derived from a local model. Because Theorem 1.4 is used in the proofs of Proposition 1.8 and Theorem 1.10, those results are conditional on this missing step. The same remark applies to the proof of Theorem 1.6 in §3.3, where the chosen foliation of f(P_L∩μ^{-1}(0)) is not explicitly constructed.
  4. [§3.1, §3.2, boundary discussion] The assertion that changing the foliation 'does not affect the boundary of the base diagram' is not justified. The boundary of the standard base diagram is the image of |η|=1, but the area coordinate A of a disk bounding γ_{λ,s} depends on the choice of disk and on the foliation; replacing the foliation on μ=0 may change the moment map in a neighbourhood of that level set and hence the base region. This point needs a proof, since the shape of B_S and A_c is central to the applications.
minor comments (4)
  1. [§3.1, proof of Theorem 1.1] The factor 2π in the moment map is dropped with the promise to reinstate it later, but the convention is never explicitly fixed in the final formulas; the area integral (8) and the base curve μ↦(μ,A_S(μ,1)) should state clearly which convention is being used.
  2. [§3.1, equation (9)] In equation (9) the expression 'µ2|η2|' appears to be a typo; judging from the preceding integral it should probably be µ²|η|².
  3. [References and Remark 1.9] The reference [Sym] is incomplete (no year or volume), [LMZ13] is cited only as an arXiv preprint, and Remark 1.9's attribution to [KS18] needs a precise pointer since that paper's title does not indicate the result.
  4. [§4, Lemma 4.3] In the proof of Lemma 4.3 the sentence 'Note that P_ε is contained in the image of the moment map' is asserted without verification; please indicate how the vertices were checked against the base region A_c for c=1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fibrations are built from explicit area integrals and external tools; self-citations are definitional or comparative and do not force the results.

full rationale

The central claims (Theorems 1.1, 1.4, 1.6 and the width applications) are not derived by fitting target outcomes into inputs. The area functions A_S, A_c, and A(μ) are computed from first principles via Stokes' theorem and explicit integrals (equations (8), (9), (21), (27)), and the base diagrams are defined by these computed functions. The lower bound for c_Gr(D*S^3) is a direct Traynor ball construction (Lemma 4.3), while the upper bound uses the external theorem [LMZ13] on Hermitian symmetric spaces, so neither bound is imported from the paper's own assumptions. Self-citations to [AA24] appear only to adopt notation and definitions for base diagrams and restricted almost-toric fibrations, and [FRV23] is used as a comparison target ('recover results'), not as an input that constructs the fibrations. The main logical weakness is not circularity: the proof of Theorem 1.1 asserts a modified foliation on μ=0, saying 'we arrange the foliation on X_0^2 so that no leaf intersects both points... This setup yields exactly two nodal singularities', without proving smoothness in λ or verifying by computation that the singularity type is nodal. This is an unproved construction step, but it does not reduce the theorem to its own conclusion; the fibration is still built from explicit data, and the nodal claim is an unverified assertion about that data, not an input. Hence no circular step in the sense of the review's taxonomy.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central construction assumes Arnold-Liouville, the given T^{n-1} action with free-orbit hypotheses, the existence of the foliation γ_{λ,s}, and the informal freedom to modify the foliation on μ=0. No fitted constants are introduced; parameters such as c are inputs. The external facts [LMZ13] and the symplectic isotopy theorem are used as black boxes.

assumptions (6)
  • standard math Arnold-Liouville theorem: regular fibers of a Lagrangian fibration with compact connected fibers are tori.
    Used in Remark 2.2 and throughout to describe L_{λ,s} as T^{n-1} x S^1 tori.
  • domain assumption The Hamiltonian T^{n-1}-action on T*M preserves the fibrewise norm and is free except on a finite union of orbits in each μ-level.
    Conditions (i) and (ii) of Proposition 2.5; verified for the rotation actions in Sections 3.1 through 3.3 but not proven generally.
  • domain assumption For each λ, the set f_λ(D*M ∩ μ^{-1}(λ)) is simply connected and admits a smooth foliation by simple closed curves γ_{λ,s}.
    Condition (iv) of Proposition 2.5; established by Claims 3.1, 3.2, and 3.3 for the examples.
  • ad hoc to paper The foliation on the singular level μ=0 may be changed freely to any foliation by simple closed curves retracting to a point, without changing the Lagrangian property or the boundary of the base.
    Used in the proof of Theorem 1.1 and again in Theorem 1.4; no proof is supplied, though boundary independence is asserted.
  • domain assumption The local model of nodal or focus-focus singularities from Example 2.7 describes the singularities of the constructed fibrations.
    The paper identifies singularities as nodal because the Hessians agree with the model in Example 2.7; this is asserted rather than proved for the constructed fibration.
  • domain assumption External results: [LMZ13] gives c_G(Q^n)=2π for the Fermat quadric, and the symplectic isotopy extension theorem from [Sch17] allows transfer of embedded balls through cuts.
    Used in the proof of Proposition 1.8 and Lemma 4.3; cited without proof.

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Pith. "Pith review of On singular Lagrangian fibrations and applications to symplectic embeddings I." pith.science (2026). https://pith.science/paper/WDWO26HC

@misc{pith2026250601556,
  author       = {Pith},
  title        = {Pith review of: On singular Lagrangian fibrations and applications to symplectic embeddings I},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDWO26HC}},
  note         = {Machine review of arXiv:2506.01556}
}
read the original abstract

In this paper, we construct singular Lagrangian fibrations on some examples of disk cotangent bundles in dimensions 4 and 6. As an application, we show how this construction can be used to obtain toric domains in some cases. In particular, we recover results from Ferreira, Ramos, and Vicente on the Gromov width of the disk cotangent bundle of spheres of revolution, as well as results from Ramos on the Lagrangian bidisk. We also briefly discuss how this technique can be used to study symplectic embedding problems.

Figures

Figures reproduced from arXiv: 2506.01556 by the authors.

Figure 1
Figure 1. Base diagram for the almost toric fibration of Proposition 1.2 on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Base diagram for the almost toric fibration on [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Base diagram for the singular Lagrangian fibration of Theorem 1.4 on [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Base diagram for the almost toric fibration of Theorem 1.6 on the Lagrangian Bidisk. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Base diagram of the Lagrangian Bidisk after transferring the cut operation. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: The fibration π. notice that the subspace generated by the Hessians of µ and pf ˝ πq agree with that of the Hamiltonian system1 pH1, H2q : C 2 Ñ R given by pH1, H2qpz1, z2q “ ` i 2 Impz 2 1 ` z 2 2 q, Impz1z¯2q ˘ . Using the idea in Proposition 2.5, we can construct a …
Figure 7
Figure 7. Figure 7: Different foliations for X2 0 . As shown in the final part of the proof of Proposition 2.5, the singular fibers of the Lagrangian torus fibration correspond to points p P f ´1 λ ppq for which the fiber is not homeomorphic to S 1 . These singularities arise precisely at…
Figure 8
Figure 8. Figure 8: Base diagram of D˚Ep1, 1, cq. 3.2 Disk cotangent bundle of 3-dimensional ellipsoids Ep1, 1, c, cq Proof of Theorem 1.4. Let’s consider D˚Ep1, 1, c, cq Ă C 4 given by: D˚ Ep1, 1, c, cq :“ $ ’’’’& ’’’’% ξ ` iη P C 4 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ξ 2 1 ` ξ 2 2 ` ξ 2 3 c 2 ` ξ 2 4 c…

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Works this paper leans on

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