{"id":"5a219807-8913-4989-89fc-6e156e78032b","arxiv_id":"2412.04451","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The natural map from complex bordism to derived orbifold bordism splits, giving complex-cobordism-valued Gromov-Witten invariants for arbitrary closed symplectic manifolds.","lead":"The paper proves a splitting theorem for complex bordism and derived orbifold bordism, yielding complex-cobordism-valued Gromov-Witten invariants for all closed symplectic manifolds. A specialist would read it because it supplies integral stable-complex invariants of holomorphic-curve moduli spaces, not merely rational homology classes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.18 depends on Theorem 6.6, a complex analytic functorial embedded resolution theorem only sketched and attributed to an external reference; if this theorem lacks the stated pullback functoriality, the Cartesian diagrams in Proposition 6.15 break.","rationale":"The reader's verdict correctly identifies Theorem 6.6 as the weakest load-bearing assumption. The central claim Theorem 1.18 is a splitting of homology theories, and its proof is a delicate local-to-global resolution algorithm. The paper provides substantial original work—Section 3's abelianization, Section 4's destackification, the FOP regularity framework, and the universal zero locus analysis—but the step that bridges local algebraic resolutions to global analytic orbifold charts is explicitly outsourced to a sketch and a possibly garbled citation. I found no circular reasoning or data fabrication; the internal logic is otherwise coherent. However, the manuscript's own Remark 6.14 concedes that an algebraic proof of the crucial Cartesian compatibility is unknown, reinforcing that Theorem 6.6 is not a decorative assumption but the keystone. The proposed test—verifying the cited theorem or, failing that, testing the analytic maximal-contact step on a concrete non-algebraic singularity—would settle whether the gap is real. Since the reader already conditioned the verdict on this exact point, my assessment does not change the verdict.","tokens_in":62668,"tokens_out":8337,"duration_ms":79366,"concrete_test":"Verify the pivotal Step (2) of the sketch of Theorem 6.6 on a non-algebraic analytic pair: take Y=C^2 and X={z1^2+exp(z2)=0}, compute the maximum locus of the invariant inv_p(I_X) (as defined in [ATW19, §5.1]) and check that it is a closed complex analytic subspace and that the associated weighted blow-up is independent of the choice of local coordinates. If the maximum locus is not analytic, or if the blow-up depends on the coordinate chart, then Theorem 6.6 does not hold for arbitrary complex analytic pairs, and Proposition 6.15 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central splitting map Z in Theorem 1.18 is constructed by resolving the zero loci of regular FOP sections. The passage from local resolutions to a global orbifold relies on Proposition 6.15, which asserts a Γ2-equivariant Cartesian diagram (6.31) for the resolutions of the universal zero loci after restricting to V^{Γ2}_+. This proposition is proved using the complex analytic functorial embedded resolution theorem, Theorem 6.6, which is only given a three-step sketch and attributed to a mangled citation (\"[W/suppress lo24]\"). Remark 6.14 explicitly says the algebraic category proof is not known, so the analytic theorem is load-bearing. If Theorem 6.6 does not hold exactly as stated—for instance, if the functoriality under holomorphic submersions fails, or if the uniqueness of maximal contact hypersurfaces and the coefficient ideal arguments only work in the algebraic setting—then the Cartesian diagram (6.31) does not follow. Without (6.31), the colimit construction in Section 7.2 cannot be patched, and the resulting map is not a well-defined natural transformation of homology theories. The paper's own sketch does not supply the missing analytic details, and the external reference is not verifiable from the text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper adapts functorial algorithms for resolution of singularities from algebraic geometry to construct a natural transformation Z : dΩU → ΩU of homology theories on orbispaces, splitting the inclusion of orbifold bordism into derived orbifold bordism (Theorem 1.18). The construction proceeds in three stages: abelianization of normally complex orbifolds via iterated blow-ups (Section 3), desingularization of coarse spaces via blow-ups and root stacks (Section 4), and resolution of the zero loci of regular FOP sections using functorial embedded resolution of singularities (Sections 6 and 7). The splitting is then applied to prove splitting statements for equivariant geometric vs. homotopical bordism (Theorems 1.3, 1.7, 1.10, 1.12), to construct complex cobordism-valued Gromov–Witten invariants for arbitrary closed symplectic manifolds (Theorem 1.14), and to derive a constraint on Hamiltonian fibrations over S^2 (Theorem 1.16). The central technical novelty is the replacement of the canonical Whitney stratification of [BX22b] by resolutions of universal zero loci, which requires a complex analytic functorial embedded resolution theorem that is only sketched.","tokens_in":62916,"tokens_out":6165,"duration_ms":70217,"significance":"If the main theorem is established, the paper provides a substantial structural result: a natural, module-linear splitting of derived orbifold bordism into ordinary orbifold bordism, unifying several previously separate phenomena in equivariant stable homotopy and symplectic topology. The applications to integral complex cobordism-valued Gromov–Witten invariants and to the topology of Hamiltonian fibration spaces are significant and would likely trigger further work. The paper is also praiseworthy for its concrete algorithmic perspective, its careful treatment of normally complex orbifolds as the natural category, and its explicit identification of the ways in which the algebraic resolution machinery must be adapted to the smooth/analytic setting. However, the central result currently rests on two under-supported inputs: the complex analytic functorial embedded resolution theorem (Theorem 6.6) and the generic regularity of FOP sections (Lemma 7.6). Because the paper itself notes in Remark 6.14 that the algebraic category proof is not known, these gaps are load-bearing and must be closed before the claims can be accepted.","major_comments":[{"comment":"Theorem 6.6, the complex analytic functorial embedded resolution theorem, is load-bearing for the paper's local-to-global construction: it is used in Proposition 6.15 to establish the Γ2-equivariant Cartesian diagram (6.31), which in turn guarantees that the local resolutions of universal zero loci can be patched into a global resolution in Section 7.2 (the colimit (7.45)). The proof of Theorem 6.6 occupies only a three-step sketch, and the citation given is the unverifiable string \"[W/suppress lo24]\". The sketch does not establish the crucial functoriality property for holomorphic submersions—namely that the resolution of a pullback pair is the pullback of the resolution—nor does it provide the analytic counterparts of the maximal-contact hypersurface uniqueness and coefficient ideal arguments in [ATW19]. Since Remark 6.14 explicitly states that the algebraic version of Proposition 6.15 is not known, Theorem 6.6 cannot be replaced by the existing algebraic Theorem 6.2. Unless a complete proof or a precise, verifiable reference with the exact statement is supplied, the Cartesian diagram (6.31), and hence the naturality of the splitting map in Theorem 1.18, is not rigorously established.","section":"§6.1, Theorem 6.6"},{"comment":"Lemma 7.6 asserts that a generic FOP section is regular, but the proof is a single paragraph that invokes Sard–Smale on a Fréchet space of smooth Γ-equivariant bundle maps. The paper does not verify the hypotheses of the infinite-dimensional transversality theorem it appeals to: it does not specify the Banach manifold setup, the nonlinear Fredholm operator whose linearization must be surjective, nor the sense in which the universal zero loci in Definition 7.3 are smooth Banach submanifolds of the relevant mapping spaces. Moreover, Definition 7.3 requires transversality simultaneously for all conjugacy classes γ'' and all stabilizer subgroups Γ', and Proposition 7.4 requires compatibility of these conditions under the partial order of stabilizers; Lemma 7.6 must show that a single generic section satisfies all these constraints at once, uniformly over the compact base. As written, the proof of Lemma 7.6 is an assertion rather than a demonstration, and the resolution procedure of Section 7.2 depends essentially on this regularity statement.","section":"§7.1, Lemma 7.6"},{"comment":"Even assuming Theorem 6.6, the proof of Proposition 6.15 has a gap in the passage from the local statement of Lemma 6.16 to the global Cartesian diagram (6.31). Lemma 6.16 constructs a complex analytic left inverse θd only locally near points of ZΓ1d(V,W)◦ ∩ VΓ2+ × PolyΓ1d(V,W), and the proof of Lemma 6.17 asserts that the resulting local diagrams \"can be patched together\" using functoriality. The manuscript does not provide a descent argument explaining why the local left inverses, which depend on choices of a subspace lv0 and Lagrange interpolation data, can be chosen compatibly on overlaps of the open neighborhood, nor why the resulting local submersions induce a single well-defined morphism of pairs to which the resolution functor can be applied globally. Without such an argument, the Cartesian-ness of (6.31) is not fully established, and the patching of resolutions from local charts to the global orbifold is not justified.","section":"§6.2, Proposition 6.15"}],"minor_comments":[{"comment":"The name \"Włodarczyk\" is consistently garbled as \"W/suppress lo\" in the text and in the citation for Theorem 6.6; the reference should be corrected to a proper, verifiable source.","section":"Throughout"},{"comment":"The text claims that the only auxiliary choice in the destackification algorithm is the choice of flattened Riemannian metric, but the subsequent FOP perturbation step (Section 7) involves additional choices, including the degree d and the approximating section; the paper should clarify which choices are shown to be irrelevant or contractible for the bordism class.","section":"§5.2, Algorithm I"},{"comment":"The commutative diagram in Lemma 7.5 has a typo in its display (the arrow labels f1 and f2 appear to be missing targets), and the proof would benefit from explicitly naming the point in M1 used in the pullback.","section":"§7.1, Lemma 7.5"},{"comment":"The definition of a straightened connection is notationally dense and does not explicitly state that the parallelism ρΓ' is used to identify the connection on the disc bundle with the pulled-back connection; a short explanatory sentence would improve readability.","section":"§2.3, Definition 2.21"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a high-profile problem and contains many original ideas, but the central splitting theorem is not yet proven because two key analytic inputs—Theorem 6.6 and Lemma 7.6—are only sketched and the citation for Theorem 6.6 is not verifiable. I recommend requesting a major revision in which the authors either give complete proofs of these two statements or cite precise, published theorems with the exact functoriality used in Section 6.2 and Section 7.1. I would also ask them to address the descent issue in the proof of Proposition 6.15. The applications and the overall framework are promising, and I do not see a fundamental obstruction, but the current draft does not meet the standard of proof required for publication in a leading journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Abouzaid and Bai prove something genuinely new: the natural map from complex bordism to derived orbifold bordism splits, and similarly geometric equivariant bordism splits off homotopical equivariant bordism. Combined with recent Kuranishi work, this gives complex-cobordism-valued Gromov–Witten invariants for arbitrary closed symplectic manifolds. The main theorems are not in the prior literature, and they do not reduce to earlier results. The architecture is clear and the paper is unusually detailed—long sections actually prove the abelianization and destackification algorithms from Bergh–Rydh and Bergh in the normally complex orbifold category.\n\nThe soft spots are where the reader says they are. The splitting map is constructed by resolving zero loci of regular FOP sections, and the local-to-global compatibility (Proposition 6.15) depends on a complex analytic functorial embedded resolution theorem (Theorem 6.6) that is presented as a three-step sketch and attributed to external work. Remark 6.14 explicitly says the algebraic proof is not known, so the analytic theorem is load-bearing. If the pullback functoriality under holomorphic submersions fails, the Cartesian diagram (6.31) breaks and the colimit construction in Section 7.2 does not give a natural transformation. I think the theorem is very likely true—Włodarczyk's methods should work analytically—but the paper should not leave it as a sketch. The same holds for Lemma 7.6 (generic regularity of FOP sections), which is argued via Sard–Smale in a few lines; the compactness and covering argument is plausible but will need more detail to be a proof.\n\nThese are gaps in presentation, not evidence that the central argument is wrong. I found no circularity, no fitted parameters, and the authors are honest about limitations (e.g., failure of multiplicativity, module structure over the larger ring). The citation to \"[W/suppress lo24]\" is mangled—presumably Włodarczyk—and needs fixing.\n\nWho is this for? Stable homotopy theorists and symplectic topologists working on virtual fundamental chains and equivariant bordism. It deserves a serious referee: an editor should send it out. The referee report should ask for a complete proof or precise delegation of Theorem 6.6 and a fuller proof of Lemma 7.6 before final acceptance.","headline":"Substantial, carefully built splitting theorems for derived orbifold bordism and equivariant bordism; the main caveat is a load-bearing analytic resolution theorem that is only sketched.","tokens_in":63456,"tokens_out":2039,"would_cite":true,"duration_ms":87361,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N22","57R90","14E15","53D45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every compact stably complex derived orbifold is complex bordant to an honest stably complex manifold, via functorial resolution of singularities.","keywords":["complex bordism","derived orbifolds","orbifold bordism","resolution of singularities","Gromov-Witten invariants","equivariant bordism","normally complex orbifolds","Fukaya-Ono-Parker sections"],"falsifier":"Apply the construction to a compact stably complex derived orbifold whose local model is a non-abelian quotient, such as a generic equivariant polynomial zero locus for $\\Gamma = S_3$ acting faithfully, and check that the Cartesian square of Proposition 6.15 commutes after restricting stabilizers; any failure would make the splitting map fail to be a natural transformation. Equivalently, exhibit a complex-analytic pair for which the resolution functor of Theorem 6.6 disagrees with the algebraic resolution on a common algebraic pair, which would break the local-to-global gluing and hence Theorem 1.18.","tokens_in":62457,"feed_emoji":"","tokens_out":8197,"duration_ms":80598,"temperature":0.7,"pith_summary":"The paper establishes a structural splitting in complex bordism theory: the natural map from orbifold bordism to derived orbifold bordism has a left inverse. Concretely, every compact stably complex derived orbifold, meaning the zero locus of a section of a complex vector bundle over an orbifold, is complex bordant to an honest stably complex orbifold, and the correspondence is natural for maps of orbispaces. The proof works by resolving the singularities of zero loci of carefully perturbed sections, using functorial embedded resolution of singularities together with abelianization and destackification algorithms. If correct, the result yields complex cobordism-valued Gromov-Witten invariants for arbitrary closed symplectic manifolds and a splitting of equivariant complex bordism for every finite group.","feed_headline":"Every derived orbifold is bordant to a genuine manifold","feed_subtitle":"Resolution of singularities splits complex bordism and yields Gromov-Witten classes for all closed symplectic manifolds.","key_machinery":"The carrying mechanism is a three-stage resolution algorithm applied to a normally complex orbifold. First, Fukaya-Ono-Parker sections, equivariant polynomial perturbations of a section, make zero loci regular in a sense modeled on holomorphic transversality. Their universal zero loci, built from $\\Gamma$-equivariant polynomial maps between representations, are then resolved using the Abramovich-Temkin-Wlodarczyk functorial embedded resolution of singularities, with a complex-analytic version used to glue local resolutions into global Cartesian diagrams. Finally, the resulting orbifold is made geometrically abelian by iterated blow-ups following Bergh-Rydh, then destackified by blow-ups and root stacks following Bergh so that its coarse space is a smooth manifold. Compatibility of the resolutions under restriction, stabilization, and products is what makes the construction a natural transformation of homology theories.","core_discovery":"On the paper's own terms, the central claim is Theorem 1.18: there is a natural transformation $Z: d\\Omega^U \\to \\Omega^U$ of homology theories on orbispaces, respecting the $\\Omega^U$-module structures, which splits the inclusion of orbifold bordism in its derived version. In geometric terms, every compact stably complex derived orbifold, presented as the zero locus of a section of a complex vector bundle over an orbifold, is complex bordant to an honest stably complex orbifold, and the correspondence is natural for maps of orbispaces. Passing to topological spaces through the functor $R$ gives the splitting of manifold bordism into its orbifold version, and specializing to finite groups gives the equivariant splittings of Theorems 1.3 and 1.7. The same construction, combined with global Kuranishi charts, defines integral complex cobordism-valued Gromov-Witten classes for arbitrary closed symplectic manifolds.","pith_inferences":["If the splitting is compatible with external but not internal equivariant products, the natural home for the invariants may be a global equivariant bordism spectrum rather than a single $\\Gamma$-equivariant module; the paper does not settle that refinement.","The algebraic analogue of complex-bordism Gromov-Witten invariants would require a different general-position argument, since Lemma 7.6 uses smooth Sard-Smale transversality; the paper explicitly leaves the purely algebraic route open.","A testable extension is to compute the new Gromov-Witten classes in genus zero for simple targets such as $\\mathbb{CP}^n$ and check the expected splitting axioms, which would provide independent evidence for the naturality of the resolution construction.","If the methods can be refined to respect fiber products, they could support a Hamiltonian Floer homotopy type as an $MU$-module; the paper notes its current construction is not multiplicative enough for that step."],"forward_implications":["The inclusion of complex orbifold bordism in derived orbifold bordism splits naturally, so stably complex derived orbifolds introduce no new bordism classes beyond ordinary stably complex orbifolds and manifolds.","For each finite group $\\Gamma$, the map from complex bordism to $\\Gamma$-equivariant bordism and the map from geometric to homotopical equivariant bordism split as modules over $\\Omega^U_*$.","Closed symplectic manifolds admit complex cobordism-valued Gromov-Witten invariants that are integral and independent of the almost complex structure, agreeing with the classical class whenever the moduli space is a transverse manifold.","The null-homotopy statement for $\\Omega\\mathrm{Ham}(X,\\omega) \\wedge S^1$ constrains the topology of Hamiltonian fibrations over $S^2$, giving a family version of the homological splitting theorem.","Because the algorithm applies to normally complex orbifolds, the same resolution machinery is available for moduli spaces with Lagrangian boundary conditions, not only for closed holomorphic curves."],"supporting_citations":[{"why":"Supplies the functorial embedded resolution of singularities algorithm used to resolve the universal zero loci in Section 6.","marker":"[ATW19]"},{"why":"Supplies the abelianization algorithm of iterated blow-ups along non-abelian loci used in Section 3 to make stabilizers abelian.","marker":"[BR19]"},{"why":"Supplies the destackification procedure of blow-ups and root stacks used in Section 4 to make the coarse space of an abelian orbifold smooth.","marker":"[Ber17]"},{"why":"Provides the FOP perturbation scheme, normal complex structures, and the universal zero loci that the resolution algorithm is applied to.","marker":"[BX22b]"},{"why":"Provides the RepOrbTop model for orbispaces in which the homology theories $d\\Omega^U$ and $\\Omega^U$ are formulated.","marker":"[Par21]"},{"why":"Shows every derived orbifold arises as a global quotient, used to pass to effective ambient orbifolds in the constructions.","marker":"[Par22]"},{"why":"Supplies the equivariant transversality results used to prove that generic regular FOP sections exist in Lemma 7.6.","marker":"[Was69]"},{"why":"Provides the global Kuranishi chart presentations of moduli spaces of stable maps used in the Gromov-Witten application.","marker":"[AMS21]"}],"fun_headline_variants":["Splitting complex bordism via singularity resolution","Derived orbifolds break bordism barrier","Gromov-Witten for all closed symplectic manifolds","Resolution yields equivariant splittings and GW classes","Bordism split by orbifold resolution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the complex-analytic functorial embedded resolution theorem stated as Theorem 6.6 in Section 6.1, which the paper presents only as a sketch and attributes to external work; all local resolutions of universal zero loci are patched into global Cartesian diagrams through that theorem, so if it fails for complex-analytic pairs exactly as stated, the splitting map would not be a natural transformation.","fun_headline_variants_meta":{"raw":{"variants":["Splitting complex bordism via singularity resolution","Derived orbifolds break bordism barrier","Gromov-Witten for all closed symplectic manifolds","Resolution yields equivariant splittings and GW classes","Bordism split by orbifold resolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000533,"raw_usage":{"total_tokens":2554,"prompt_tokens":922,"completion_tokens":1632,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":1557}},"tokens_in":538,"tokens_out":1632,"duration_ms":11688,"temperature":1.0,"reasoning_tokens":1557,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:23:22.574751+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Apply the construction to a compact stably complex derived orbifold whose local model is a non-abelian quotient, such as a generic equivariant polynomial zero locus for $\\Gamma = S_3$ acting faithfully, and check that the Cartesian square of Proposition 6.15 commutes after restricting stabilizers; any failure would make the splitting map fail to be a natural transformation. Equivalently, exhibit a complex-analytic pair for which the resolution functor of Theorem 6.6 disagrees with the algebraic resolution on a common algebraic pair, which would break the local-to-global gluing and hence Theorem 1.18.","supporting_citations":[],"review_version":1}