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Shifted Poisson geometry and meromorphic matrix algebras over an elliptic curve

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Abstract

In this paper we classify symplectic leaves of the regular part of the projectivization of the space of meromorphic endomorphisms of a stable vector bundle on an elliptic curve, using the study of shifted Poisson structures on the moduli of complexes from our previous work (Hua and Polishchuk in Adv Math 338:991–1037, 2018). This Poisson ind-scheme is closely related to the ind Poisson–Lie group associated to Belavin’s elliptic r-matrix, studied by Sklyanin, Cherednik and Reyman and Semenov-Tian-Shansky. Our result leads to a classification of symplectic leaves on the regular part of meromorphic matrix algebras over an elliptic curve, which can be viewed as the Lie algebra of the above-mentioned ind Poisson–Lie group. We also describe the decomposition of the product of leaves under the multiplication morphism and show the invariance of Poisson structures under autoequivalences of the derived category of coherent sheaves on an elliptic curve.

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Notes

  1. For an algebraic stack, there always exists an obstruction theory satisfying the conditions in Artin’s representability theorem. We refer to Section 5 [1] for details.

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Acknowledgements

We are grateful to Jiang-Hua Lu and Yongchang Zhu for many valuable discussions. The second author thanks Institut Mathematique Jussieu and Institut des Hautes Etudes Scientifiques for hospitality and excellent working conditions during preparation of this paper. The research of Z.H. is supported by RGC General Research Fund No. 17330316, No. 17308818 and NSFC Science Fund for Young Scholars No. 11401501. The research of A.P. is supported in part by the NSF Grant DMS-1700642 and by the Russian Academic Excellence Project ‘5-100’.

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Authors and Affiliations

  1. Department of Mathematics, The University of Hong Kong, Hong Kong, SAR, China

    Zheng Hua

  2. University of Oregon, Eugene, USA

    Alexander Polishchuk

  3. National Research University Higher School of Economics, Moscow, Russia

    Alexander Polishchuk

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  1. Zheng Hua
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  2. Alexander Polishchuk
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Correspondence to Zheng Hua.

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Hua, Z., Polishchuk, A. Shifted Poisson geometry and meromorphic matrix algebras over an elliptic curve. Sel. Math. New Ser. 25, 42 (2019). https://doi.org/10.1007/s00029-019-0489-4

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