Inventiones mathematicae

, Volume 213, Issue 3, pp 1017–1097 | Cite as

Gopakumar–Vafa invariants via vanishing cycles

  • Davesh Maulik
  • Yukinobu Toda


In this paper, we propose an ansatz for defining Gopakumar–Vafa invariants of Calabi–Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem–Li, which is itself based on earlier ideas of Hosono–Saito–Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov–Witten theory and Pandharipande–Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem–Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.



We are grateful to Tomoyuki Abe, Jim Bryan, Duiliu-Emanuel Diaconescu, Igor Dolgachev, Jesse Kass, Young-Hoon Kiem, Jun Li, Georg Oberdieck, Rahul Pandharipande, Christian Schnell and Richard Thomas for many discussions and valuable comments. We are also grateful to the referees for many suggestions and comments. This article was written while Y. T.  was visiting Massachusetts Institute of Technology from 2015 October to 2016 September by the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers. D. M. is supported by NSF Grants DMS-1645082 and DMS-1564458. Y. T. is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research Grant (No. 26287002) from MEXT, Japan.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

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