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A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants

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arxiv 2201.02688 v3 pith:SFUZQAOJ submitted 2022-01-07 math.SG math.AGmath.AT

Shaoyun Bai , Guangbo Xu This is my paper · ORCID
classification math.SGmath.AGmath.AT
keywords mathcalconditiontransversalityabouzaid-mclean-smithgromov-witteninteger-valuedinvariantssymplectic
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abstract

Following a proposal of Fukaya-Ono and the exploration by B. Parker, we introduce a new transversality condition, the FOP transversality condition, for sections of orbifold vector bundles $\mathcal{E} \rightarrow \mathcal{U}$ when both $\mathcal{E}$ and $\mathcal{U}$ have "normal complex structures." This notion allows one to define various integral virtual cycles on moduli spaces of pseudoholomorphic curves. Two immediate applications in symplectic topology are the definition of integer-valued Gromov-Witten type invariants in all genera for general compact symplectic manifolds using the global Kuranishi chart constructed by Abouzaid-McLean-Smith and Hirschi-Swaminathan, and an alternative proof of the cohomological splitting theorem for Hamiltonian fibrations over $S^2$ with integer coefficients by Abouzaid-McLean-Smith.

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Cited by 4 Pith papers

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    The natural map from complex bordism to derived orbifold bordism splits, giving complex-cobordism-valued Gromov-Witten invariants for arbitrary closed symplectic manifolds.

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  4. Reduced Gromov-Witten invariants without ghost bubble censorship

    math.SG 2026-04 unverdicted novelty 7.0 of 10

    Defines all-genus reduced Gromov-Witten invariants of symplectic manifolds via effectively supported multivalued perturbations on derived orbifold/Kuranishi charts, bypassing ghost bubble censorship.

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