Skip to main content
SpringerLink
Log in

On the Quantum Cohomology of Blow-ups of Four-dimensional Quadrics

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We propose a conjecture relevant to Galkin’s lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture \({\cal O}\) holds in these two cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bayer, A., Manin, Y. I.: (Semi)simple exercises in quantum cohomology. In: The Fano Conference, Univ. Torino, Turin, 2004, 143–173

    Google Scholar 

  2. Cheong, D., Han, M.: Galkin’s lower bound conjecture for Lagrangian and orthogonal Grassmannians. Bull. Korean Math. Soc., 57(4), 933–943 (2020)

    MathSciNet  Google Scholar 

  3. Cox, D. A., Katz, S.: Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, Vol. 68. American Mathematical Society, Providence, RI, 1999

    Google Scholar 

  4. Evans, L., Schneider, L., Shifler, R. M., et al.: Galkin’s lower bound conjecture holds for the Grassmannian. Comm. Algebra, 48(2), 857–865 (2020)

    Article  MathSciNet  Google Scholar 

  5. Frobenius, G.: Über Matrizen aus Nicht Negativen Elementen, S.-B. Presuss. Akad. Wiss., Berlin, 1912, 456–477

    Google Scholar 

  6. Galkin, S.: The conifold point. arXiv:1404.7388 (2014)

  7. Galkin, S., Golyshev, V., Iritani, H.: Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures. Duke Math. J., 165(11), 2005–2077 (2016)

    Article  MathSciNet  Google Scholar 

  8. Galkin, S., Iritani, H.: Gamma conjecture via mirror symmetry. In: Primitive Forms and Related Subjects, Advanced Studies in Pure Mathematics Vol. 83, Math. Soc. Japan, Tokyo, 2019, 55–115

    Google Scholar 

  9. Gathmann, A.: Gromov-Witten invariants of blow-ups. J. Algebraic Geom., 10(3), 399–432 (2001)

    MathSciNet  Google Scholar 

  10. Griffiths, P., Harris, J.: Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994

    Book  Google Scholar 

  11. Harris, J.: Algebraic Geometry: A First Course. Graduate Texts in Mathematics, Vol. 133, Springer-Verlag, New York, 1995

    Google Scholar 

  12. Hu, J.: Gromov–Witten invariants of blow-ups along points and curves. Math. Z., 233, 709–739 (2000)

    Article  MathSciNet  Google Scholar 

  13. Hu, J.: Gromov–Witten invariants of blow-ups along surfaces. Compositio Math., 125(3), 345–352 (2001)

    Article  MathSciNet  Google Scholar 

  14. Hu, J., Ke, H.-Z., Li, C., et al.: Gamma conjecture I for del Pezzo surfaces. Adv. Math., 386, 107797 (2021)

    Article  MathSciNet  Google Scholar 

  15. Hu, J., Li, T.-J., Ruan, Y.: Birational cobordism invariance of uniruled symplectic manifolds. Invent. Math., 172, 231–275 (2008)

    Article  MathSciNet  Google Scholar 

  16. Ke, H.-Z.: On Conjecture \({\cal O}\) for projective complete intersections. Int. Math. Res. Not., DOI:https://doi.org/10.1093/imrn/rnad141 (2023)

  17. Lai, H.-H.: Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles. Geom. Topology, 13, 1–48 (2009)

    Article  MathSciNet  Google Scholar 

  18. Li, C., Shifler, R. M., Yang, M., et al.: On Frobenius-Perron dimension. Proc. Amer. Math. Soc., 150(12), 5035–5045 (2022)

    MathSciNet  Google Scholar 

  19. Maulik, D., Pandharipande, R.: A topological view of Gromov-Witten theory. Topology, 45, 887–918 (2006)

    Article  MathSciNet  Google Scholar 

  20. Mihalcea, L. C.: Equivariant quantum Schubert calculus. Adv. Math., 203(1), 1–33 (2006)

    Article  MathSciNet  Google Scholar 

  21. Minc, H.: Nonnegative Matrices, John Willy & Sons, New York, 1988

    Google Scholar 

  22. Perron, O.: Zur Theorie der Matrices. Math. Ann., 64(2), 248–263 (1907)

    Article  MathSciNet  Google Scholar 

  23. Rietsch, K.: Quantum cohomology rings of Grassmannians and total positivity. Duke Math. J., 110(3), 523–553 (2001)

    Article  MathSciNet  Google Scholar 

  24. Szurek, M., Wisniewski, J. A.: On Fano manifolds, which are Pk-bundles over P2. Nagoya Math. J., 120, 89–101 (1990)

    Article  MathSciNet  Google Scholar 

  25. Yang, Z.: Gamma conjecture I for blowing up ℙn along ℙr. arXiv:2022.04234 (2020)

Download references

Acknowledgements

The authors would like to thank Kwokwai Chan and Heng Xie for useful discussions.

Author information

Authors and Affiliations

  1. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Jian Xun Hu, Hua Zhong Ke, Chang Zheng Li & Lei Song

Authors
  1. Jian Xun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hua Zhong Ke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang Zheng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lei Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang Zheng Li.

Ethics declarations

The authors declare no conflict of interest.

Additional information

Hu is supported by NSFC Grant (Grant Nos. 11890662 and 11831017), Ke is supported by NSFC Grant (Grant Nos. 12271532 and 11831017), Li is supported by NSFC Grant (Grant No. 11831017) and Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No. 2017ZT07X355), Song is supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2020A1515010876)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, J.X., Ke, H.Z., Li, C.Z. et al. On the Quantum Cohomology of Blow-ups of Four-dimensional Quadrics. Acta. Math. Sin.-English Ser. 40, 313–328 (2024). https://doi.org/10.1007/s10114-024-2236-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-024-2236-9

Keywords

MR(2010) Subject Classification

Search

Navigation