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On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

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Pith's one-line read Two counterexamples refute Gonzales' fixed-data classification, and a corrected version with rational-surface reduced spaces is proved and shown to preserve Cho's Fano classification.

arxiv 2505.14000 v1 pith:OIOPREMM submitted 2025-05-20 math.SG

Liat Kessler , Nikolas Wardenski This is my paper · ORCID
classification math.SG
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The reading

Symplectic geometry studies spaces that carry a way of measuring area. This paper looks at six-dimensional such spaces with a circle action that has a momentum map, a function measuring how the circle rotates. A classical question is whether the space is completely determined by the fixed point data: the critical levels of the momentum map, the fixed pieces sitting on them, and how they index. Gonzales had claimed that this data determines the whole space. The authors give two counterexamples. The first is made by gluing two identical pieces of S2 x S2 x S2 with the two S2 factors swapped along an overlap. The resulting space has the same fixed point data and same local charts as the original, but the two fixed spheres at different critical levels represent the same homology class in one space and different classes in the other, so no equivariant diffeomorphism can exist. The second example is built from two open toric polytopes that agree below a level and differ above it; the spaces agree below the critical level and have the same small fixed point data there, but the classes that the fixed sphere produces just below the level have different symplectic sizes. The positive part of the paper proves that if every four-dimensional reduced space is a symplectic rational surface, if the families of reduced forms are rigid, and if interior fixed surfaces lie on at most one critical level, then the fixed point data do determine the isomorphism type. The proof uses J-holomorphic curve methods to control exceptional spheres in rational surfaces, allowing an isomorphism below a critical level to be glued across it. This repaired theorem is enough for Cho's classification of positive monotone six-manifolds with semi-free circle actions.
Extended reading notes

Core claim

Theorems 1.9 and 1.10 assert that, under Setting 1.8 (four-dimensional reduced spaces below each critical value are symplectic rational surfaces, and the reduced form families are rigid) plus the stated restrictions on non-extremal fixed components, two compact simply-connected semi-free Hamiltonian S^1-manifolds of dimension six with the same *-small fixed point data are isomorphic. If correct, this restores a controlled version of Gonzales' classification and implies Theorem 1.14, that Cho's positive monotone Fano classification still holds.

Load-bearing premise

The rigidity assumption (Definition 1.4) for every interval of regular values below the relevant critical level, applied in Proposition 4.13, Lemma 4.16, and the proof of Theorem 1.9 to extend an isomorphism to arbitrarily close to a critical level. It is known to hold only for a finite family of rational surfaces (Theorem 1.13); for a general four-dimensional reduced space there is no known mechanism, so the theorem's scope depends entirely on this property. The companion assumption that all four-dimensional reduced spaces below each critical value are symplectic rational surfaces is also load-bearing, since it provides the J-holomorphic exceptional-class control in Lemma 6.3 and Lemma 5.22.

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

No numerical free parameters or invented physical entities appear; the ledger consists of explicit hypotheses and external theorems. The key added content is the rational-surface and rigidity hypotheses plus the restriction on interior fixed surfaces.

assumptions (6)
  • domain assumption Rigidity assumption (Definition 1.4): for every interval of regular values below each critical level, the family of reduced symplectic forms is rigid.
    Explicit hypothesis in Setting 1.8 and Theorem 1.10; used in Proposition 4.13 to extend isomorphisms to arbitrarily close to a critical level. It is not derived from fixed point data.
  • domain assumption Four-dimensional reduced spaces below each critical value are symplectic rational surfaces.
    Explicit in Setting 1.8; enables the J-holomorphic exceptional-class characterization in Lemma 6.3 and the uniqueness and isotopy results for symplectomorphisms of rational surfaces used in Lemma 5.22 and Theorem 5.2.
  • standard math The classification results for symplectomorphism groups of S2 x S2 and CP2#kCP2 with k at most 4 (Theorem 1.13, citing [Gr85, AM00, LP04, Pin08, Ev11, LLW15]) are correct.
    Citable external theorems, not reproved in this paper; they supply rigidity for the examples and for Cho's application.
  • standard math [KK17, Lemma 2.12 and Theorem 3.12]: an exceptional class represented in one blowup form is represented in every blowup form, and minimal exceptional classes in reduced blowups have the listed structure.
    Used in Lemma 6.3 and Proposition 6.1 to identify the classes D^i with E'^i. One author is a coauthor of [KK17], but it is a published theorem with a proof, not derived in this paper.
  • standard math Local normal form for semi-free Hamiltonian S^1-actions in dimension six (Sections 3.3 and 3.4): fixed components have dimension 0, 2, or 4 with weights of the listed types.
    Background from equivariant Darboux and Atiyah-Bott theory; used throughout to describe fixed point sets and the Morse flow.
  • ad hoc to paper Non-extremal fixed surfaces are restricted to at most one critical level in the main classification.
    This is a new extra hypothesis, not a consequence of fixed point data; it is exactly the gap exposed by Example 2.1, where two non-extremal fixed surfaces sit on two different levels.

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Pith. "Pith review of On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data." pith.science (2026). https://pith.science/paper/OIOPREMM

@misc{pith2026250514000,
  author       = {Pith},
  title        = {Pith review of: On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIOPREMM}},
  note         = {Machine review of arXiv:2505.14000}
}
abstract

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian $S^1$-manifold of dimension six is determined by certain data on the critical levels. We first give counter examples showing that Gonzales' assumptions are not sufficient for a positive answer. Then we prove that it is enough to further assume that the reduced spaces of dimension four are symplectic rational surfaces and the interior fixed surfaces are restricted to at most one level. The additional assumptions allow us to use results proven by $J$-holomorphic methods. Gonzales' answer was applied by Cho in proving that if the underlying symplectic manifold is positive monotone then the space is isomorphic to a Fano manifold with a holomorphic $S^1$-action. We show that our variation is enough for Cho's application.

Figures

Figures reproduced from arXiv: 2505.14000 by the authors.

Figure 1
Figure 1. On the left: the images of the fixed points of N under µN . On the right: the toric momentum image of N, where the red lines represent the level sets λ and λ ′ of µN . We will construct a new semi-free Hamiltonian S 1 -manifold M′ by gluing µ −1 ((−∞, λ′ )) and µ −1 ((λ, ∞) along µ −1 ((λ, λ′ )) by a gluing map that is an equivariant symplectomorphism on µ −1 ((λ, λ′ )) and is not the identity. To define the gluing … view at source ↗
Figure 2
Figure 2. On the left, the momentum polytope of the con-compact sym￾plectic toric manifold we started with, on the right the momentum polytope of the same manifold after twisting the action. By the classification of non-compact symplectic toric manifolds in [KL15, Theorem 1.3.2], ρ2 on N′ is T 2 -equivariantly symplectomorphic to ((λ, λ′ ) × S 1 ) × S 2 endowed with the standard T 2 -action, and with the standard symplectic f… view at source ↗
Figure 3
Figure 3. We will show that (S 1 ⟲ M, µ) and (S 1 ⟲ M′ , µ′ ) are not µ−S 1 -diffeomorphic, meaning that there is no equivariant diffeomorphism between the spaces that respects the momentum maps. In particular, (S 1 ⟲ M, ωM, µ) and (S 1 ⟲ M′ , ωM′, µ′ ) can not be isomorphic as Hamiltonian S 1 -manifolds. The main point is that the two fixed spheres at level λ and λ ′ in M represent the same homology in H2(M), whereas they do… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The T 2 -momentum images of the reduced spaces at y = 0 for both open symplectic toric manifolds. The red lines correspond to the fixed spheres in the respective symplectic toric manifolds and are of the same length. The arrows indicate how the cross section changes as…
Figure 5
Figure 5. Figure 5: The preimage of the red region under the momentum map from the toric symmetric S 2 × S 2 to R 2 is the embedded closed ball B. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: The T 2 -momentum images of the reduced space at y = −0.4 for both open symplectic toric manifolds. The red line corresponds to the class S1, on which the symplectic form evaluates to 1 + 0.4 = 1.4, and the blue line corresponds to the class S2, on which the form evalu…
Figure 7
Figure 7. Figure 7: The momentum image of a T 2 -action on a Hirzebruch surface Et . The red line corresponds to s(S 2 ), and this sphere has self-intersection −k if the blue line is going in direction (k, −1). The sphere corresponding to the green line has self-intersection k, where the …
Figure 8
Figure 8. Figure 8: Toric moment polytopes of blowups in two ways. where jS2 is the almost complex structure induced from a complex atlas on S 2 . If u is an embedding, we call its image u(S 2 ) an embedded J-holomorphic sphere. Sketch of proof. By the positivity of intersections of J-hol…

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