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Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties

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Abstract

In this paper, we discuss the problem of whether the difference \([X]-[Y]\) of the classes of a Fourier–Mukai pair (X, Y) of smooth projective varieties in the Grothendieck ring of varieties is annihilated by some power of the class \(\mathbb {L} = [ \mathbb {A}^1 ]\) of the affine line. We give an affirmative answer for Fourier–Mukai pairs of very general K3 surfaces of degree 12. On the other hand, we prove that in each dimension greater than one, there exists an abelian variety such that the difference with its dual is not annihilated by any power of \(\mathbb {L}\), thereby giving a negative answer to the problem. We also discuss variations of the problem.

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Acknowledgements

We thank Kenji Hashimoto and Daisuke Inoue for collaboration at an early stage of this work; this note is originally conceived as a joint project with them. We also thank Genki Ouchi for Remark 5.1 and the reference [13], and Kota Yoshioka for the reference [33]. We also thank Yujiro Kawamata, Keiji Oguiso, Evgeny Shinder, Hokuto Uehara, and Takehiko Yasuda for useful discussions. We also thank the anonymous referees for reading the manuscript carefully and suggesting a number of improvements. A. I. was supported by Grants-in-Aid for Scientific Research (14J01881,17K14162). M. M. was supported by Korea Institute for Advanced Study. S. O. was partially supported by Grants-in-Aid for Scientific Research (16H05994, 16K13746, 16H02141, 16K13743, 16K13755, 16H06337, 18H01120) and the Inamori Foundation. K. U. was partially supported by Grants-in-Aid for Scientific Research (24740043, 15KT0105, 16K13743, 16H03930).

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  1. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Atsushi Ito

  2. Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea

    Makoto Miura

  3. Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan

    Shinnosuke Okawa

  4. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Kazushi Ueda

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  1. Atsushi Ito
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Ito, A., Miura, M., Okawa, S. et al. Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. Sel. Math. New Ser. 26, 38 (2020). https://doi.org/10.1007/s00029-020-00561-x

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