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Asymptotically holomorphic theory for symplectic orbifolds

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We extend Donaldson’s asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a compact symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient orbifold satisfies those properties.

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Acknowledgements

The authors are very grateful to Fran Presas for useful discussions concerning the subtleties of the perturbation near the isotropy locus. We also thank warmly to Nieves Alamo for her contribution in the initial steps of this paper. Lastly, we would also like to thank Steven Zelditch for useful feedback on the first version of the preprint, concerning Remark 1.2 and the equidistribution property for the zero sets of the sections. The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 772479) during the preparation of this work. The second author is partially supported by Project MINECO (Spain) PID2020-118452GB-I00. The third author is supported by National Natural Science Foundation of China under Grant No. 12288201 and 12231010.

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  1. Fabio Gironella

    Present address: CNRS and Laboratoire de Mathématiques Jean Leray, Nantes Université, Nantes, France

Authors and Affiliations

  1. Department of Mathematics, Humboldt University, Berlin, Germany

    Fabio Gironella

  2. Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Campus de Teatinos, s/n, 29071, Malaga, Spain

    Vicente Muñoz

  3. Morningside Center of Mathematics and Institute of Mathematics AMSS, CAS, Beijing, China

    Zhengyi Zhou

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  1. Fabio Gironella
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Gironella, F., Muñoz, V. & Zhou, Z. Asymptotically holomorphic theory for symplectic orbifolds. Math. Z. 304, 21 (2023). https://doi.org/10.1007/s00209-023-03277-8

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