TY - JOUR
T1 - Derived equivalence and Grothendieck ring of varieties
T2 - the case of K3 surfaces of degree 12 and abelian varieties
AU - Ito, Atsushi
AU - Miura, Makoto
AU - Okawa, Shinnosuke
AU - Ueda, Kazushi
N1 - Funding Information:
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).
Funding Information:
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 and the reference [], and Kota Yoshioka for the reference []. 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).
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
PY - 2020/7/1
Y1 - 2020/7/1
N2 - 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 L= [A1] 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 L, thereby giving a negative answer to the problem. We also discuss variations of the problem.
AB - 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 L= [A1] 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 L, thereby giving a negative answer to the problem. We also discuss variations of the problem.
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U2 - 10.1007/s00029-020-00561-x
DO - 10.1007/s00029-020-00561-x
M3 - Article
AN - SCOPUS:85086387831
SN - 1022-1824
VL - 26
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 3
M1 - 38
ER -