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Blow-up formulae of high genus Gromov–Witten invariants for threefolds

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Abstract

Using the degeneration formula and absolute/relative correspondence, we study the change of Gromov–Witten invariants under blow-ups for smooth projective threefolds, and obtain several closed blow-up formulae for high genus Gromov–Witten invariants. Our formulae also imply some simple relations among generalized BPS numbers.

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Acknowledgements

The authors would like to thank Prof. Pandharipande for pointing out his early results about the degenerate contributions to them, and Prof. Yongbin Ruan and Pedro Acosta for their useful comments on earlier drafts. We would also like to thank the referee for many very useful suggestions. Hua-Zhong would like to thank Prof. Jian Zhou for sharing his ideas on Gromov–Witten theory generously, and Xiaowen Hu and Hanxiong Zhang for helpful discussions. Weiqiang and Hua-Zhong would like to thank Department of Mathematics of University of Michigan for its hospitality during their visiting. Weiqaing He is partially supported by China Scholarship Council and Tsinghua Postdoc Grant 100410019, Jianxun Hu is partially supported by NSFC Grants 11771460, 11228101, 11371381 and 11521101, and Hua-Zhong Ke is partially supported by NSFC Grants 11601534 and 11771461.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Weiqiang He

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

    Weiqiang He, Jianxun Hu & Hua-Zhong Ke

  3. Sino-French Institute of Nuclear and Technology, Sun Yat-sen University, Tang Jia Wan, Zhuhai, 519082, China

    Xiaoxia Qi

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  1. Weiqiang He
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  2. Jianxun Hu
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  3. Hua-Zhong Ke
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  4. Xiaoxia Qi
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Correspondence to Hua-Zhong Ke.

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He, W., Hu, J., Ke, HZ. et al. Blow-up formulae of high genus Gromov–Witten invariants for threefolds. Math. Z. 290, 857–872 (2018). https://doi.org/10.1007/s00209-018-2043-z

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  • DOI: https://doi.org/10.1007/s00209-018-2043-z

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