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Scattering Diagrams, Sheaves, and Curves

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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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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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  1. Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, Building CLV, 8092, Zurich, Switzerland

    Pierrick Bousseau

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  1. Pierrick Bousseau
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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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