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Localized Chern characters for 2-periodic complexes

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Abstract

For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. We explore some basic properties of this localized Chern character. In particular, we show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. We apply this equivalence to the comparison of virtual classes of moduli of \({\varepsilon }\)-stable quasimaps and moduli of the corresponding LG \({\varepsilon }\)-stable quasimaps, in full generality.

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Notes

  1. Their paper is appeared in [6].

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Acknowledgements

B. Kim would like to thank Yongbin Ruan for drawing his attention to the comparison question of virtual classes, Andrei Okounkov for stimulating comments, and Arkady Vaintrob for answering a question. The authors would like to thank Ionuţ Ciocan-Fontanine, Tom Graber and Taejung Kim for helpful comments in shaping the paper. This material is based upon work supported by NSF grant DMS-1440140 while the first author was in residence at MSRI in Berkeley during Spring 2018 semester. J. Oh would like to thank Sanghyeon Lee for useful discussions and University of California, Berkeley for excellent working conditions. B. Kim is partially supported by KIAS individual grant MG016404. J. Oh is partially supported by KIAS individual grant MG063002.

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Authors and Affiliations

  1. Korea Institute for Advanced Study, 85 Hoegiro, Seoul, Dongdaemun-gu, 02455, Republic of Korea

    Bumsig Kim

  2. Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom

    Jeongseok Oh

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  1. Bumsig Kim
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  2. Jeongseok Oh
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Correspondence to Bumsig Kim.

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Kim, B., Oh, J. Localized Chern characters for 2-periodic complexes. Sel. Math. New Ser. 28, 23 (2022). https://doi.org/10.1007/s00029-021-00743-1

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