Inventiones mathematicae

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

Gopakumar–Vafa invariants via vanishing cycles

  • Davesh Maulik
  • Yukinobu TodaEmail author


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.


  1. 1.
    Ben-Bassat, O., Brav, C., Bussi, V., Joyce, D.: A ‘Darboux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications. Geom. Topol. 19, 1287–1359 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brav, C., Bussi, V., Dupont, D., Joyce, D., Szendrői, B.: Symmetries and stabilization for sheaves of vanishing cycles. With an appendix by Jörg Schürmann. J. Singul. 11, 85–151 (2005)zbMATHGoogle Scholar
  3. 3.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces I. Asterisque 100, 5–171 (1982)Google Scholar
  4. 4.
    Brav, C., Dyckerhoff, T.: Relative Calabi–Yau Structures, preprint. arXiv:1606.00619
  5. 5.
    Behrend, K.: Donaldson–Thomas invariants via microlocal geometry. Ann. Math. 170, 1307–1338 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bussi, V., Joyce, D., Meinhardt, S.: On Motivic Vanishing Cycles of Critical Loci, preprint. arXiv:1305.6428
  7. 7.
    Bryan, J., Pandharipande, R.: The local Gromov–Witten theory of curves. J. Am. Math. Soc. 21, 101–136 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bridgeland, T.: Flops and derived categories. Invent. Math. 147, 613–632 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bussi, V.: Generalized Donaldson–Thomas Theory Over Fields \({K}\ne {\mathbb{C}}\), preprint. arXiv:1403.2403
  10. 10.
    Calabrese, J.: Donaldson–Thomas invariants and flops. J. Reine Angew. Math. 716, 103–145 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cossec, F., Dolgachev, I.: Enriques Surfaces I. Progress in Mathematics, vol. 76. Birkhäuser, Boston (1989)CrossRefzbMATHGoogle Scholar
  12. 12.
    Chuang, W.Y., Diaconescu, D.E., Pan, G.: BPS States and the \(P=W\) Conjecture, Moduli Spaces, London Mathematical Society Lecture Note Series, vol. 411, pp. 132–150. Cambridge University Press, Cambridge (2014)Google Scholar
  13. 13.
    de Cataldo, M., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties. Ann. Math. 175, 1329–1407 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dimca, A.: Sheaves in Topology. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  15. 15.
    Efimov, A.: Quantum Cluster Variables Via Vanishing Cycles, preprint. arXiv:1112.3601
  16. 16.
    Gillet, H.: K-theory and intersection theory, Handbook of K-theory. Springer, Berlin, Heidelberg, pp 253–293 (2005)Google Scholar
  17. 17.
    Gopakumar, R., Vafa, C.: M-Theory and Topological Strings II. arXiv:hep-th/9812127
  18. 18.
    Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174, 555–624 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hosono, S., Saito, M., Takahashi, A.: Relative Lefschetz actions and BPS state counting. Int. Math. Res. Not. 15, 783–816 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ionel, E.N., Parker, T.H.: The Gopakumar–Vafa Formula for Symplectic Manifolds, preprint. arXiv:1306.1516
  21. 21.
    Joyce, D.: A classical model for derived critical loci. J. Differ. Geom. 101, 289–367 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Joyce, D., Song, Y.: A theory of generalized Donaldson-Thomas invariants. Mem. Am. Math. Soc. 217, 1–216 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jiang, Y., Thomas, R.: Virtual Signed Euler Characteristics, preprint. arXiv:1408.2541
  24. 24.
    Katz, S.: Genus zero Gopakumar–Vafa invariants of contractible curves. J. Differ. Geom. 79, 185–195 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kiem, Y.H., Li, J.: Categorification of Donaldson–Thomas Invariants Via Perverse Sheaves, preprint. arXiv:1212.6444
  26. 26.
    Kollár, J.: Rational Curves on Algebraic Varieties, Ergebnisse Math. Grenzgeb.(3), vol. 32. Springer, Berlin (1996)Google Scholar
  27. 27.
    Kool, M., Thomas, R.P.: Reduced classes and curve counting on surfaces I: theory. Algebra. Geom. 3, 334–383 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 142, 1263–1285 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maulik, D., Pandharipande, R.: New calculations in Gromov–Witten theory. Pure Appl. Math. Q. 4, 469–500 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Melo, M., Rapagnetta, A., Viviani, F.: Fine Compactified Jacobians of Reduced Curves, preprint. arXiv:1406.2299
  31. 31.
    Migliorini, L., Shende, V.: A support theorem for Hilbert schemes of planar curves. J. Eur. Math. Soc. 15, 2353–2367 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Maruyama, M., Yokogawa, K.: Moduli of parabolic stable sheaves. Math. Ann. 293, 77–99 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Maulik, D., Yun, Z.: Macdonald formula for curves with planar singularities. J. Reine Angew. Math. 694, 27–48 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Nekrasov, N., Okounkov, A.: Membranes and Sheaves, preprint. arXiv:1404.2323
  35. 35.
    Pandharipande, R.: Hodge integrals and degenerate contributions. Commun. Math. Phys. 208, 489–506 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pandharipande, R., Pixton, A.: Gromov–Witten/Pairs Correspondence for the Quintic 3-fold, preprint. arXiv:1206.5490
  37. 37.
    Preygel, A.: Some Remarks on Shifted Symplectic Structures on Non-compact Mapping Spaces, preprintGoogle Scholar
  38. 38.
    Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178, 407–447 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pandharipande, R., Thomas, R.P.: Stable pairs and BPS invariants. J. Am. Math. Soc. 23, 267–297 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Pandharipande, R., Thomas, R.P.: 13/2 Ways of Counting Curves, Moduli Spaces, London Mathematical Society Lecture Note Series, vol. 411, pp. 282–333. Cambridge University Press, Cambridge (2014)Google Scholar
  41. 41.
    Pantev, T., Toën, B., Vaquie, M., Vezzosi, G.: Shifted symplectic structures. Publ. Math. IHES 117, 271–328 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
  43. 43.
    Saito, M.: A Young Person’s Guide to Mixed Hodge Modules, preprint. arXiv:1605.00435
  44. 44.
    Schnell, C.: An Overview of Morihiko Saito’s Theory of Mixed Hodge Modules, preprint. arXiv:1405.3096
  45. 45.
    Thomas, R.P.: A holomorphic Casson invariant for Calabi–Yau 3-folds and bundles on \({K3}\)-fibrations. J. Differ. Geom. 54, 367–438 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Toda, Y.: Multiple Cover Formula of Generalized DT Invariants II: Jacobian Localizations, preprint. arXiv:1108.4993
  47. 47.
    Toda, Y.: Birational Calabi–Yau 3-folds and BPS state counting. Commun. Number Theory Phys. 2, 63–112 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Toda, Y.: Stability conditions and curve counting invariants on Calabi–Yau 3-folds. Kyoto J. Math. 52, 1–50 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Toda, Y.: Curve counting theories via stable objects II. DT/ncDT flop formula. J. Reine Angew. Math. 675, 1–51 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Toda, Y.: Multiple cover formula of generalized DT invariants I: parabolic stable pairs. Adv. Math. 257, 476–526 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Toda, Y.: Flops and the \(S\)-duality conjecture. Duke Math. J. 164, 2293–2339 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations