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Scattering Diagrams, Sheaves, and Curves

Abstract

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.

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References

  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. http://www.math.uni-hamburg.de/home/siebert/preprints/LGtrop.pdf (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)

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Acknowledgements

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.

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Correspondence to Pierrick Bousseau.

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

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Bousseau, P. Scattering Diagrams, Sheaves, and Curves. Acta. Math. Sin.-English Ser. 37, 1005–1022 (2021). https://doi.org/10.1007/s10114-021-0060-z

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  • DOI: https://doi.org/10.1007/s10114-021-0060-z

Keywords

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

MR(2010) Subject Classification

  • 14N35