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Contraction algebra and singularity of three-dimensional flopping contraction

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Abstract

Donovan and Wemyss (Duke Math J 165(8):1397–1474, 2016) introduced the contraction algebra of flopping curves in threefold. They conjectured that the contraction algebra determines the formal neighborhood of the underlying singularity of the contraction. In this paper, we prove that the contraction algebra together with its natural \(A_\infty \)-structure constructed in Hua and Toda (Int Math Res Not rnw333, 2017), determine the formal neighborhood of the singularity.

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Notes

  1. Here we use the standard grading on Hochschild cochain which differs with the grading of Efimov by 1. Efimov used the pro-nilpotent version of \({\mathfrak {g}}\) and \({\mathfrak {h}}\) because he needed to define a gauge group action on the MC locus.

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Acknowledgements

We are grateful to Will Donovan and Michael Wemyss for many valuable discussions. The research was supported by RGC Early Career Grant no. 27300214 and NSFC Science Fund for Young Scholars no. 11401501.

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Authors and Affiliations

  1. Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong

    Zheng Hua

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  1. Zheng Hua
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Correspondence to Zheng Hua.

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Hua, Z. Contraction algebra and singularity of three-dimensional flopping contraction. Math. Z. 290, 431–443 (2018). https://doi.org/10.1007/s00209-017-2024-7

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  • DOI: https://doi.org/10.1007/s00209-017-2024-7

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