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Weighted-blowup correspondence of orbifold Gromov–Witten invariants and applications

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Abstract

Let \(\mathsf {X}\) be a compact symplectic orbifold groupoid with \(\mathsf {S}\) being a compact symplectic sub-orbifold groupoid, and \({\underline{\mathsf {X}}_{\mathfrak {a}}}\) be the weight-\({\mathfrak {a}}\) blowup of \(\mathsf {X}\) along \(\mathsf {S}\) with \(\mathsf {Z}\) being the exceptional divisor. We show that there is a weighted-blowup correspondence between some certain absolute orbifold Gromov–Witten invariants of \(\mathsf {X}\) relative to \(\mathsf {S}\) and some certain relative orbifold Gromov–Witten invariants of the pair \(({\underline{\mathsf {X}}_{\mathfrak {a}}}|\mathsf {Z})\). As an application, we prove that the symplectic uniruledness of symplectic orbifold groupoids is a weighted-blowup invariant.

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Notes

  1. Here \({\overline{\mathsf {N}}}\) is the projectification of the normal bundle of \(\mathsf {Z}\) in \({\underline{\mathsf {X}}_{\mathfrak {a}}}\), hence the \({\overline{\mathsf {N}}}_\mathsf {Z}\) in (6.12).

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Acknowledgements

The authors thank the anonymous referee for careful reading and helpful suggestions, especially for the suggestion to consider the more general version of symplectic uniruledness discussed in Sect. 7.3. The first author is supported by NSFC (nos. 11431001 and 11890663). The second author is supported by NSFC (no. 11501393) and by Sichuan Science and Technology Program (no. 2019YJ0509). The third author is supported by NSFC (nos. 11831017, 11890662, 11521101 and 11771460).

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Authors and Affiliations

  1. Department of Mathematics and Yangtze Center of Mathematics, Sichuan University, 610065, Chengdu, China

    Bohui Chen

  2. Department of Mathematics, Sichuan Normal University, 610068, Chengdu, China

    Cheng-Yong Du

  3. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Jianxun Hu

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  1. Bohui Chen
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  2. Cheng-Yong Du
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  3. Jianxun Hu
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Correspondence to Cheng-Yong Du.

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Chen, B., Du, CY. & Hu, J. Weighted-blowup correspondence of orbifold Gromov–Witten invariants and applications. Math. Ann. 374, 1459–1523 (2019). https://doi.org/10.1007/s00208-019-01850-3

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