{"id":"f4f1ecec-aad0-4590-83c7-a3cb69e10d86","arxiv_id":"2411.18580","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On irrational ruled symplectic 4-manifolds, homologically trivial cyclic actions of order k>2 extend to Hamiltonian circle actions, but explicit involutions with connected fixed point sets do not.","lead":"Finite-order symplectic symmetries of irrational ruled surfaces behave differently depending on order: cyclic actions of order greater than two extend to continuous circle actions, while involutions with connected fixed point sets cannot. The paper constructs the first such non-extendable involutions on minimal examples and uses them to answer an open question about symplectic versus holomorphic actions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-extendability of the exotic involutions relies on an unproved fixed-point assertion for symplectic circle actions; Proposition 3.7 only covers finite Z_k actions with k>2.","rationale":"The reader correctly identified Theorem 2.7 from [44] as the key external input. My concern sharpens that dependency: the specific load-bearing step in Theorem 1.2 is the fixed-point shape of symplectic circle actions, which the paper cites to Proposition 3.7 even though that proposition only treats finite Z_k-actions with k>2. The circle-action version is very likely correct and derivable from the same pseudoholomorphic machinery, so this is a repairable gap rather than a fatal flaw. Other issues in the written proofs, such as the false assertion in Theorem 4.1 that H_1(S~) surjects onto H_1(S) for a double cover, and the weight normalization gap in Theorem 5.1, are also real but easily patched. Since the central claims appear sound but the written proofs contain genuine gaps, the reader's CONDITIONAL verdict remains appropriate.","tokens_in":30350,"tokens_out":23340,"duration_ms":230393,"concrete_test":"Prove the circle-action analogue of Proposition 3.7: for any connected symplectic S^1-action on an irrational ruled 4-manifold with base genus at least two and any S^1-invariant ω-tamed J, use Theorem 2.7 to show that the action preserves every fiber and that each smooth fiber carries an effective S^1-action with exactly two fixed points, yielding two disjoint sections; then check the singular-fiber and blow-up cases for b2>2 separately. If a counterexample with a connected fixed component or with only one section exists, the obstruction used in Corollary 4.2 fails; if the derivation succeeds, the non-extendability proof is complete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central obstruction in Theorem 1.2 and Corollary 4.2 is that a homologically trivial symplectic involution with connected fixed bisection cannot extend to a symplectic circle action. The proof asserts that it is 'known (see Proposition 3.7)' that any homologically trivial symplectic circle action on an irrational ruled 4-manifold with base genus at least two has two disjoint sections. But Proposition 3.7 is stated and proved only for finite cyclic Z_k-actions with k>2, and its weight argument uses k>2 essentially (a is not congruent to -a mod k). For an S^1-action the analogous statement is plausible and can probably be proved from Theorem 2.7 by choosing an S^1-invariant J: connectedness forces the action to preserve the fiber class and hence the fibers, and an effective S^1-action on CP^1 has exactly two fixed points per fiber. However, the paper never supplies this argument, and this fixed-point fact is exactly what rules out extension for every possible symplectic form. If a circle action on some irrational ruled surface with a non-product or non-minimal symplectic form had only one section, or a connected fixed component, the non-extendability argument would collapse. The b2>2 blow-up case inherits the same gap, since singular fibers require additional justification that the two sections persist.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite-order symplectic and Hamiltonian diffeomorphisms on irrational ruled symplectic 4-manifolds, motivated by Kedra's question whether Hamiltonian cyclic actions extend to Hamiltonian circle actions. It constructs homologically trivial symplectic involutions with connected fixed bisection on Sigma x S^2 that do not extend to symplectic circle actions even after changing the symplectic form, and it extends this to b2 >= 2 via equivariant blow-ups. It also proves that homologically trivial Z_k-actions with k > 2 on minimal irrational ruled surfaces with base genus at least two extend to Hamiltonian S^1-actions after a possible change of symplectic form, and it classifies finite groups of symplectomorphisms acting on such manifolds.","tokens_in":30635,"tokens_out":44376,"duration_ms":405103,"significance":"If the technical gaps identified below are repaired, the paper makes a substantial contribution to the cyclic-to-circle extension problem in dimension four. The main new idea is to use the fixed point set as an obstruction: the authors produce explicit homologically trivial symplectic involutions whose connected fixed bisection cannot occur for a circle action, and contrast this with the k > 2 case where two fixed sections force an extension. The paper also gives a fairly complete classification of finite symplectic group actions on irrational ruled 4-manifolds and supplies algebraic examples via Maruyama's classification, including an answer to a question of Weimin Chen. The reliance on the second author's published moduli fibration theorem [44] is explicit and not circular; the main arguments are geometric and the proofs are plausible, but several load-bearing steps are missing or misstated.","major_comments":[{"comment":"The non-extendability proof in Corollary 4.2 relies on the assertion that any homologically trivial symplectic S^1-action on an irrational ruled 4-manifold with base genus at least two has fixed point set consisting of two disjoint sections, and it cites Proposition 3.7 for this. Proposition 3.7, however, is stated and proved only for finite cyclic Z_k-actions with k > 2, and its weight argument uses k > 2 essentially. Since this fixed-point fact is exactly what rules out extension for every possible symplectic form, the proof needs a separate argument for circle actions. One can restrict an S^1-invariant almost complex structure (Lemma 2.11) to a finite subgroup Z_k with k > 2 and apply Proposition 3.7, but this is not written in the paper; the analogue for the blow-up case b2 > 2 also requires additional justification concerning singular fibers. Please supply this argument or give a precise reference.","section":"Corollary 4.2 / Theorem 1.2"},{"comment":"The extension constructed in Theorem 5.1 is defined by requiring the tangent action at the fixed section Sigma_1 to be multiplication by e^{2*pi*i*theta}. The original generator f, however, acts on a smooth fiber with weights a and -a mod k for some a coprime to k, and Proposition 3.7 does not guarantee a = 1. Consequently the equality phi((Sigma_1(p), Sigma_2(p)), 2*pi/k, p) = f claimed in the proof is generally false. The construction can be repaired by using the constant weight a along Sigma_1, as guaranteed by the weight-constancy statement in Section 2.3, and by requiring the tangent action at Sigma_1 to be e^{2*pi*i*a*theta}; the theorem statement should then clarify what 'standard embedding' means for this parametrization.","section":"Theorem 5.1 / Key fact in Section 5"},{"comment":"The blow-down step asserts that if S is a -1-sphere and f(S) has the same class, then positivity of intersections of J-holomorphic curves forces f(S) = S. This requires S to be J-holomorphic for the f-invariant almost complex structure J, which is not established for an arbitrary embedded symplectic -1-sphere. The standard fact that every exceptional class admits a unique J-holomorphic representative for any tamed J should be invoked, or the argument should be replaced by a direct equivariant blow-down argument. As written, the reduction from b2 > 2 to b2 = 2 is incomplete.","section":"Proposition 5.2"}],"minor_comments":[{"comment":"The text says 'the Z2-action generated by h' but the involution is denoted q; please fix the notation.","section":"Theorem 4.1"},{"comment":"The statement that 'every non-trivial element of S3 induces a non-trivial automorphism of C, so H1 is the trivial group' is inconsistent with the subsequent exact sequence 1 -> Z3 -> G -> Z2 -> 1; the notation H1 should be clarified.","section":"Section 6.2"},{"comment":"The 'Key fact' would benefit from an explicit statement of how the identification of each fiber with CP^1 is chosen to depend smoothly on the base point, since the global smoothness of the resulting S^1-action depends on this choice.","section":"Section 5, 'Key fact'"},{"comment":"In the proof, the step 'since f^k = id we have f'^n = id' contains a typo (n should be k), and the decomposition of H_t into base and fiber parts should be justified more carefully.","section":"Theorem 5.3"},{"comment":"The proof cites [21] for the self-intersection of the fixed point set; a brief statement of the exact formula used would improve readability.","section":"Proposition 3.9"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are plausible and the gaps I identified appear repairable, so the paper merits revision rather than rejection. The proof of Theorem 5.1 has a more serious local error (the weight mismatch), but the intended argument is clear and can be fixed by carrying an integer weight a. The non-extendability theorem also needs a short but explicit circle-action fixed-point argument. I would encourage the editor to send the revision back to the same referee pool, since the corrections are technical and require checking the details of the weight bookkeeping and the J-holomorphic exceptional sphere facts."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main story is real: homologically trivial symplectic Z_k-actions with k>2 on minimal irrational ruled surfaces extend to Hamiltonian circle actions, while involutions do not, even after changing the symplectic form. The minimal examples with connected bisection fixed sets are new, as are the Klein four-group actions, the finite group classification, and the answer to Weimin Chen's question. The constructions are clever and the paper is honest about what is imported, especially the moduli fibration from [44], which is a published, parameter-free theorem and not a self-citation problem.\n\nThere are two gaps that need fixing before this is referee-ready. First, Corollary 4.2 relies on the assertion that a homologically trivial symplectic circle action on an irrational ruled 4-manifold with base genus at least two has two disjoint sections, citing Proposition 3.7. But Proposition 3.7 is proved only for finite cyclic Z_k with k>2, and its weight argument uses k>2 essentially. The S^1 statement is probably true and can be proved by taking an S^1-invariant almost complex structure, using the moduli fibration, and noting that the induced action on the base is trivial because the base has genus at least two and S^1 is connected; then each fiber has an S^1 action with exactly two fixed points. The paper just says \"known\" and points to a proposition that does not cover it. This is load-bearing for non-extendability, so the proof must be added or a precise reference given.\n\nSecond, in Theorem 5.1 the constructed circle action is defined by requiring the tangent weight at one fixed section to be e^{2πiθ}. But the original Z_k action has some unit weight a mod k at that section, not necessarily 1. To agree with f under the standard embedding Z_k⊂S^1, the parameterization needs the weight aθ, or one must show a=1 can be assumed. This is a concrete fix, but as written the proof only works when a=1.\n\nA smaller issue: Corollary 1.5 misuses Li's theorem [25], claiming each fixed component of a Hamiltonian circle action has π1 equal to π1(M). Li's theorem gives a surjection from π1 of some fixed component to π1(M), not equality, and the argument in the h=0 case assumes the components are surfaces of genus exactly g, which need not hold. This is a side corollary, but as written the proof is not correct.\n\nOne concern the reader raised is not real: the claim in Theorem 4.1 that [S0] and [F] span H2(S×S2,R) is fine, since H2(S×S2) has rank two with basis [S] and [F].\n\nThis paper is for symplectic topologists working on group actions and Hamiltonian circle actions. The main theorems are likely correct and the gaps are in written proofs, not in the overall vision. Send it to a serious referee; expect a revision before acceptance.","headline":"A significant, mostly sound paper with two repairable gaps: the S^1 fixed-point fact behind non-extendability is quoted rather than proved, and the circle-extension weight in Theorem 5.1 needs a unit correction.","tokens_in":31127,"tokens_out":13323,"would_cite":true,"duration_ms":124686,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D35","57S17","57R17","53D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that on minimal irrational ruled 4-manifolds, symplectic cyclic actions of order $k>2$ extend to Hamiltonian circle actions, while certain symplectic involutions do not, even after changing the symplectic form.","keywords":["symplectic 4-manifolds","irrational ruled surfaces","Hamiltonian circle actions","finite-order symplectomorphisms","homologically trivial actions","fixed point sets","pseudoholomorphic curves"],"falsifier":"Take the involution $q$ on $\\Sigma_g\\times S^2$ from Theorem 4.1 for $g\\ge2$ and search for any cohomologous symplectic form and any smooth $S^1$-action extending $q$; the paper predicts that the fixed set of any such circle action must contain two disjoint sections, whereas $q$ has a connected bisection, so producing one such extension would refute Theorem 1.2.","tokens_in":30159,"feed_emoji":"🔄","tokens_out":14296,"duration_ms":115790,"temperature":0.7,"pith_summary":"The paper asks when finite-order symplectic symmetries of irrational ruled 4-manifolds—the symplectic blowups of $S^2$-bundles over Riemann surfaces of positive genus—extend to continuous Hamiltonian circle actions. Its central claim is that the answer depends sharply on the order of the symmetry: on minimal such manifolds with base genus at least two, homologically trivial (trivial on integral homology) symplectic $\\mathbb{Z}_k$-actions with $k>2$ always extend to Hamiltonian $S^1$-actions, possibly after deforming the symplectic form, whereas homologically trivial symplectic involutions exist that cannot be extended even after any such modification. The mechanism is a fixed-point dichotomy: the higher-order actions force a fixed set made of two disjoint sections, while the constructed involutions have a single connected bisection as fixed set. If the paper is right, discrete-to-continuous symmetry extension fails precisely at order two in the minimal cyclic case, and it produces symplectic actions on Kähler surfaces that are not smoothly equivalent to any holomorphic action.","feed_headline":"Non-extendable symplectic involutions exist on minimal ruled surfaces","feed_subtitle":"Higher-order cyclic symplectic actions extend to Hamiltonian circles; involutions do not.","key_machinery":"The load-bearing object is the moduli map $f:(M,\\omega)\\to\\Sigma$ produced by the existence theorem for pseudoholomorphic subvarieties: for every $\\omega$-tamed almost complex structure $J$ on an irrational ruled symplectic 4-manifold, the fibers of $f$ are connected pseudoholomorphic curves in the fiber class, with smooth sphere fibers over all but finitely many base points and a continuous base of genus $g$; when $b_2(M)=2$ the fibration is a smooth $S^2$-bundle. Homologically trivial finite symplectomorphisms preserve this map and, for $g\\ge2$, act fiberwise by Möbius transformations. The argument then turns on fixed-point weights: a $\\mathbb{Z}_k$-action with $k>2$ has two fixed points per fiber with weights $a$ and $-a$ satisfying $a\\not\\equiv -a \\pmod{k}$, forcing two disjoint sections; an involution has equal weights, permitting the connected bisection that blocks circle extension.","core_discovery":"The central discovery is a structural dichotomy for homologically trivial finite symplectic group actions on irrational ruled symplectic 4-manifolds with base genus $g\\ge 2$. For any $\\omega$-tamed almost complex structure, a moduli map fibers the manifold by pseudoholomorphic spheres in the fiber class; a homologically trivial finite symplectic group preserves this fibration, and when $g\\ge 2$ it acts fiberwise. For a cyclic action of order $k>2$, each fiber carries a finite-order Möbius transformation with two fixed points of distinct weights, so the fixed point set splits into two disjoint sections; the authors extend the action to a circle by rotating every fiber through elliptic Möbius transformations with those fixed points, then average to produce an $S^1$-invariant symplectic form for which the action is Hamiltonian. For $k=2$ the distinct-weight argument collapses, and the paper constructs homologically trivial symplectic involutions on $\\Sigma_g\\times S^2$ whose fixed set is a connected bisection of genus $2g-1$; no symplectic circle action can have such a fixed set, so the involutions are non-extendable, and equivariant blowups spread the obstruction to every $b_2\\ge 2$.","pith_inferences":["The paper leaves implicit that order two is the only genuinely discrete cyclic case on these manifolds: every homologically trivial cyclic action of order $k>2$ is continuous in disguise, so exotic discrete symmetries must be searched among involutions or non-cyclic groups.","The construction in Proposition 4.3 gives a template likely to produce non-extendable involutions on other symplectic manifolds admitting a double cover with an equivariant retraction to a circle, not only on ruled surfaces.","The paper's open questions about $\\mathbb{Z}_{2k}$-actions with $4\\le b_2\\le 2k$ suggest a natural next test: decide whether the non-extendability of involutions propagates to even-order cyclic actions of composite order in non-minimal cases."],"forward_implications":["Every homologically trivial symplectic $\\mathbb{Z}_k$-action with $k>2$ on a minimal irrational ruled 4-manifold with base genus $\\ge2$ extends to a Hamiltonian $S^1$-action for some symplectic form (Theorem 1.6).","On $\\Sigma_g\\times S^2$ with $g\\ge1$, any Hamiltonian diffeomorphism of finite order $k>2$ generates a Hamiltonian circle action under the standard embedding (Theorem 1.7).","Homologically trivial symplectic involutions exist on irrational ruled 4-manifolds with base genus $\\ge2$ and every $b_2\\ge2$ that do not extend to any symplectic circle action, whatever symplectic form is chosen (Theorem 1.2).","The same construction yields a symplectic involution on the Kähler surface $C\\times\\mathbb{P}^1$, for a projective curve $C$ of genus $\\ge1$, that is not smoothly equivalent to any holomorphic action (Corollary 1.4).","Finite symplectic symmetry groups of these 4-manifolds are classified up to short exact sequences whose outer terms are finite subgroups of $\\mathrm{SO}(3)$ and of $\\mathrm{SL}_{2g}(\\mathbb{Z})$ (or $\\mathbb{Z}_p\\times\\mathbb{Z}_q$ in genus one) (Theorem 1.9)."],"supporting_citations":[{"why":"Supplies the moduli-map theorem: any tamed almost complex structure on an irrational ruled 4-manifold admits a fibration by pseudoholomorphic spheres in the fiber class, the backbone of the whole fixed-point analysis.","marker":"[44]"},{"why":"Provides the earlier homologically trivial symplectic involution on a non-minimal irrational ruled surface and the method of proving non-extendability by comparing fixed-point sets.","marker":"[10]"},{"why":"Gives the affirmative extension result for rational ruled surfaces that the paper contrasts with the irrational ruled non-extension examples.","marker":"[9]"},{"why":"Underlies the theorem that a periodic homeomorphism of a genus-at-least-two surface acting trivially on $H_1$ is trivial, which makes the group action act fiberwise.","marker":"[41]"},{"why":"Supplies the symplectic-topology machinery: compatible almost complex structures, evaluation maps, and the criterion that an $S^1$-action with fixed points is Hamiltonian.","marker":"[34]"},{"why":"Provides the classification of automorphism groups of ruled surfaces used in the algebraic examples and in the proof that the constructed involutions are not holomorphically realizable.","marker":"[30]"},{"why":"Gives the signature formula used to determine the self-intersection of the fixed point set of an involution, completing the dichotomy between two sections and a bisection.","marker":"[21]"},{"why":"Classifies homologically trivial periodic maps on the torus, used in the genus-one structural results.","marker":"[3]"}],"fun_headline_variants":["Finite symplectic actions split: cycles extend, involutions don't","Involutions on ruled surfaces break Hamiltonian circle extension","Symplectic order k>2 lifts to circles; k=2 resists","Ruled-surface involutions permanently disrupt circular symmetry","Why higher-order symplectic cycles become circles, but involutions don't"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported theorem that every $\\omega$-tamed almost complex structure on an irrational ruled symplectic 4-manifold admits a continuous fibration by pseudoholomorphic spheres in the fiber class, together with the consequence that a homologically trivial finite symplectic group preserves that fibration; if either gave way, the fixed-point analysis and both the extension theorem for $k>2$ and the non-extension theorem for involutions would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Finite symplectic actions split: cycles extend, involutions don't","Involutions on ruled surfaces break Hamiltonian circle extension","Symplectic order k>2 lifts to circles; k=2 resists","Ruled-surface involutions permanently disrupt circular symmetry","Why higher-order symplectic cycles become circles, but involutions don't"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1429,"prompt_tokens":959,"completion_tokens":470,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":388}},"tokens_in":575,"tokens_out":470,"duration_ms":4605,"temperature":1.0,"reasoning_tokens":388,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:06:52.006581+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the involution $q$ on $\\Sigma_g\\times S^2$ from Theorem 4.1 for $g\\ge2$ and search for any cohomologous symplectic form and any smooth $S^1$-action extending $q$; the paper predicts that the fixed set of any such circle action must contain two disjoint sections, whereas $q$ has a connected bisection, so producing one such extension would refute Theorem 1.2.","supporting_citations":[{"cited_title":"Zhang,Moduli space ofJ-holomorphic subvarieties, Selecta Mathematica","cited_arxiv_id":null,"evidence_quote":"Supplies the moduli-map theorem: any tamed almost complex structure on an irrational ruled 4-manifold admits a fibration by pseudoholomorphic spheres in the fiber class, the backbone of the whole fixed-point analysis."},{"cited_title":"Chiang and L","cited_arxiv_id":null,"evidence_quote":"Provides the earlier homologically trivial symplectic involution on a non-minimal irrational ruled surface and the method of proving non-extendability by comparing fixed-point sets."},{"cited_title":"Chiang and L","cited_arxiv_id":null,"evidence_quote":"Gives the affirmative extension result for rational ruled surfaces that the paper contrasts with the irrational ruled non-extension examples."},{"cited_title":"Thurston,On the geometry and dynamics of diffeomorphisms of surfaces, Amer- ican Mathematical Society","cited_arxiv_id":null,"evidence_quote":"Underlies the theorem that a periodic homeomorphism of a genus-at-least-two surface acting trivially on $H_1$ is trivial, which makes the group action act fiberwise."},{"cited_title":"McDuff, D","cited_arxiv_id":null,"evidence_quote":"Supplies the symplectic-topology machinery: compatible almost complex structures, evaluation maps, and the criterion that an $S^1$-action with fixed points is Hamiltonian."},{"cited_title":"Maruyama,On automorphism groups of ruled surfaces, J","cited_arxiv_id":null,"evidence_quote":"Provides the classification of automorphism groups of ruled surfaces used in the algebraic examples and in the proof that the constructed involutions are not holomorphically realizable."},{"cited_title":"J¨ annich, E","cited_arxiv_id":null,"evidence_quote":"Gives the signature formula used to determine the self-intersection of the fixed point set of an involution, completing the dichotomy between two sections and a bisection."},{"cited_title":"Russian Journal of Nonlinear Dynamics, 2023, vol","cited_arxiv_id":null,"evidence_quote":"Classifies homologically trivial periodic maps on the torus, used in the genus-one structural results."}],"review_version":1}