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

Atsushi Ito ^[1] ; Makoto Miura ^[2] ; Shinnosuke Okawa ^[3] ; Kazushi Ueda ^[4]
1. [1] Nagoya University
  
  Nagoya University
  
  Naka-ku, Japón
2. [2] Korea Institute for Advanced Study
  
  Korea Institute for Advanced Study
  
  Corea del Sur
3. [3] Osaka University
  
  Osaka University
  
  Kita Ku, Japón
4. [4] University of Tokyo
  
  University of Tokyo
  
  Japón
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Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 3, 2020
Idioma: inglés
DOI: 10.1007/s00029-020-00561-x
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Resumen
- 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.