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On product identities and the Chow rings of holomorphic symplectic varieties

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Abstract

For a moduli space \({\mathsf M}\) of stable sheaves over a K3 surface X, we propose a series of conjectural identities in the Chow rings \(CH_\star ({\mathsf M}\times X^\ell ),\, \ell \ge 1,\) generalizing the classic Beauville–Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring \(R_\star ({\mathsf M}) \subset CH_\star ({\mathsf M}).\) The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on \(CH_\star ({\mathsf M})\), which we also discuss. We prove the proposed identities when \({\mathsf M}\) is the Hilbert scheme of points on a K3 surface.

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Acknowledgements

We thank Mark de Cataldo, Lie Fu, Daniel Huybrechts, Eyal Markman, Davesh Maulik, Andrei Negut, Dragos, Oprea, Junliang Shen, Charles Vial, Qizheng Yin, and Ruxuan Zhang for helpful discussions and correspondence. Conversations with Andrei Neguț regarding the Chow induction on number of points in the Hilbert scheme context played a particularly important role. I.B. is supported by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK. L.F. was supported by the NSF through Grant DMS 1803082. A.M. was supported by the NSF through Grants DMS 1601605 and 1902310, as well as by the Radcliffe Institute for Advanced Study at Harvard, through a 2019–2020 Radcliffe Fellowship. R.S. was supported by the NSF through Grant DMS 1645877. This project was initiated at Northeastern University under the auspices of the NSF-funded RTG grant DMS 1645877.

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Authors and Affiliations

  1. LMO, Université Paris-Saclay, Rue Michel Magat, Bât. 307, 91405, Orsay, France

    Ignacio Barros

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139, USA

    Laure Flapan

  3. Department of Mathematics, Northeastern University, 360 Huntington Ave., Boston, MA, 02115, USA

    Alina Marian & Rob Silversmith

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  1. Ignacio Barros
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  2. Laure Flapan
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Correspondence to Alina Marian.

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Barros, I., Flapan, L., Marian, A. et al. On product identities and the Chow rings of holomorphic symplectic varieties. Sel. Math. New Ser. 28, 46 (2022). https://doi.org/10.1007/s00029-021-00729-z

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