Skip to main content

Scattering Diagrams, Sheaves, and Curves


We review the recent proof of the N. Takahashi’s conjecture on genus 0 Gromov-Witten invariants of (ℙ2, E), where E is a smooth cubic curve in the complex projective plane ℙ2. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of (ℙ2, E) and the world of moduli spaces of coherent sheaves on ℙ2. Using this bridge, the N. Takahashi’s conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on ℙ2. This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing, 9–12 September 2019.

This is a preview of subscription content, access via your institution.


  1. Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne-Faltings pairs II. Asian J. Math., 18(3), 465–488 (2014)

    MathSciNet  Article  Google Scholar 

  2. Abramovich, D., Chen, Q., Gross, M., et al.: Decomposition of degenerate Gromov-Witten invariants. Compos. Math., 156(10), 2020–2075 (2020)

    MathSciNet  Article  Google Scholar 

  3. Abramovich, D., Marcus, S., Gross, M., et al.: Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations. Ann. Inst. Fourier, 64(4), 1611–1667 (2014)

    MathSciNet  Article  Google Scholar 

  4. Abramovich, D., Wise, J.: Invariance in logarithmic Gromov-Witten theory. Compos. Math., 154(3), 595–620 (2018)

    MathSciNet  Article  Google Scholar 

  5. Arcara, D., Bertram, A.: Bridgeland-stable moduli spaces for K-trivial surfaces. J. Eur. Math. Soc. (JEMS), 15(1), 1–38 (2013) With an appendix by Max Lieblich

    MathSciNet  Article  Google Scholar 

  6. Bayer, A., Macrì, E.: The space of stability conditions on the local projective plane. Duke Math. J., 160(2), 263–322 (2011)

    MathSciNet  Article  Google Scholar 

  7. Bousseau, P. A proof of N. Takahashi’s conjecture for (ℙ2, E) and a refined sheaves/Gromov-Witten correspondence, preprint, arXiv:1909.02992 (2019)

  8. Bousseau, P.: Scattering diagrams, stability conditions and coherent sheaves on ℙ2, preprint, arX-iv:1909.02985 (2019)

  9. Bousseau, P., Fan, H., Guo, S., et al.: Holomorphic anomaly equation for (ℙ2, E) and the Nekrasov-Shatashvili limit of local ℙ2. preprint, arXiv:2001.05347 (2020)

  10. Bridgeland, T.: Stability conditions on triangulated categories. Ann. of Math. (2), 166(2), 317–345 (2007)

    MathSciNet  Article  Google Scholar 

  11. Bridgeland, T.: Stability conditions on K3 surfaces. Duke Math. J., 141(2), 241–291 (2008)

    MathSciNet  Article  Google Scholar 

  12. Bridgeland, T.: Hall algebras and curve-counting invariants. J. Amer. Math. Soc., 24(4), 969–998 (2011)

    MathSciNet  Article  Google Scholar 

  13. Carl, M., Pumperla, M., Siebert, B.: A tropical view of Landau-Ginzburg models. (2010)

  14. Chen, Q.: Stable logarithmic maps to Deligne-Faltings pairs I. Ann. of Math. (2), 180(2), 455–521 (2014)

    MathSciNet  Article  Google Scholar 

  15. Choi, J., van Garrel, M., Katz, S., et al.: Local BPS invariants: Enumerative aspects and wall-crossing. Int. Math. Res. Not. IMRN, 17, 5450–5475 (2020)

    MathSciNet  Article  Google Scholar 

  16. Choi, J., van Garrel, M., Katz, S., et al.: Log BPS numbers of log Calabi-Yau surfaces. Trans. Amer. Math. Soc., 374(1), 687–732 (2021)

    MathSciNet  Article  Google Scholar 

  17. Choi, J., van Garrel, M., Katz, S., et al.: Contributions of degenerate stable log maps, preprint, arX-iv:1908.10906 (2019)

  18. Gräfnitz, T.: Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs, preprint, arX-iv:2005.14018 (2020)

  19. Gross, M., Pandharipande, R.: Quivers, curves, and the tropical vertex. Port. Math., 67(2), 211–259 (2010)

    MathSciNet  Article  Google Scholar 

  20. Gross, M., Pandharipande, R., Siebert, B.: The tropical vertex. Duke Math. J., 153(2), 297–362 (2010)

    MathSciNet  Article  Google Scholar 

  21. Gross, M., Siebert, B.: From real affine geometry to complex geometry. Ann. of Math. (2), 174(3), 1301–1428 (2011)

    MathSciNet  Article  Google Scholar 

  22. Gross, M., Siebert, B.: Logarithmic Gromov-Witten invariants. J. Amer. Math. Soc., 26(2), 451–510 (2013)

    MathSciNet  Article  Google Scholar 

  23. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, Springer, 2010

  24. Joyce, D., Song, Y.: A theory of generalized Donaldson-Thomas invariants. Mem. Amer. Math. Soc., 217(1020), iv+199, 2012

    MathSciNet  MATH  Google Scholar 

  25. Klemm, A., Maulik, D., Pandharipande, R., et al.: Noether-Lefschetz theory and the Yau-Zaslow conjecture. J. Amer. Math. Soc., 23(4), 1013–1040 (2010)

    MathSciNet  Article  Google Scholar 

  26. Konishi, Y.: Integrality of Gopakumar-Vafa invariants of toric Calabi-Yau threefolds. Publ. Res. Inst. Math. Sci., 42(2), 605–648 (2006)

    MathSciNet  Article  Google Scholar 

  27. Konishi, Y.: Pole structure of topological string free energy. Publ. Res. Inst. Math. Sci., 42(1), 173–219 (2006)

    MathSciNet  Article  Google Scholar 

  28. Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: The Unity of Mathematics, Volume 244 of Progr. Math., Birkhöuser Boston, Boston, MA, 2006, 321–385

  29. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435 (2008)

  30. Le Potier, J.: Faisceaux semi-stables de dimension 1 sur le plan projectif. Rev. Roumaine Math. Pures Appl., 38(7–8), 635–678 (1993)

    MathSciNet  MATH  Google Scholar 

  31. Li, C., Zhao, X.: Smoothness and Poisson structures of Bridgeland moduli spaces on Poisson surfaces. Math. Z., 291(1–2), 437–447 (2019)

    MathSciNet  Article  Google Scholar 

  32. Li, J.: Stable morphisms to singular schemes and relative stable morphisms. J. Differential Geom., 57(3), 509–578 (2001)

    MathSciNet  Article  Google Scholar 

  33. Maulik, D., Nekrasov, N., Okounkov, A., et al.: Gromov-Witten theory and Donaldson-Thomas theory. I. Compos. Math., 142(5), 1263–1285 (2006)

    MathSciNet  Article  Google Scholar 

  34. Maulik, D., Pandharipande, R., Thomas, R. P.: Curves on K3 surfaces and modular forms. J. Topol., 3(4), 937–996 (2010) With an appendix by Aaron Pixton

    MathSciNet  Article  Google Scholar 

  35. Mikhalkin, G.: Enumerative tropical algebraic geometry in ℝ2. J. Amer. Math. Soc., 18(2), 313–377 (2005)

    MathSciNet  Article  Google Scholar 

  36. Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J., 135(1), 1–51 (2006)

    MathSciNet  Article  Google Scholar 

  37. Pandharipande, R., Thomas, R. P.: Curve counting via stable pairs in the derived category. Invent. Math., 178(2), 407–447 (2009)

    MathSciNet  Article  Google Scholar 

  38. Takahashi, N.: Curves in the complement of a smooth plane cubic whose normalizations are \({{\mathbb{A}}^1}\), preprint, alg-geom/9605007 (1996)

  39. Takahashi, N.: Log mirror symmetry and local mirror symmetry. Comm. Math. Phys., 220(2), 293–299 (2001)

    MathSciNet  Article  Google Scholar 

  40. Toda, Y.: Stability conditions and curve counting invariants on Calabi-Yau 3-folds. Kyoto J. Math., 52(1), 1–50 (2012)

    MathSciNet  Article  Google Scholar 

  41. Woolf, M.: Nef and effective cones on the moduli space of torsion sheaves on the projective plane, preprint, arXiv:1305.1465 (2013)

Download references


I thank Xiaobo Liu, Rahul Pandharipande, Emanuel Scheidegger, and Qizheng Yin for the organization of the Beijing-Zurich moduli workshop. I thank Michel van Garrel for sharing his notes of my lectures. Finally, I thank the referee for useful comments.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pierrick Bousseau.

Additional information

Supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bousseau, P. Scattering Diagrams, Sheaves, and Curves. Acta. Math. Sin.-English Ser. 37, 1005–1022 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • moduli spaces
  • Gromov-Witten invariants
  • coherent sheaves
  • scattering diagrams

MR(2010) Subject Classification

  • 14N35