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.

中文翻译：

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衍生的等价形式和Grothendieck环：以12度的K3曲面和阿贝尔变种为例

在本文中，我们讨论了是否可以消除光滑的投影变种的Grothendieck环中的Fourier–Mukai对（X，  Y）对的类别的差（[[X]-[Y] \）被仿射行的类\（\ mathbb {L} = [\ mathbb {A} ^ 1] \）的某些幂。我们对阶数为12的非常普通的K3曲面的傅立叶–穆凯对给出肯定的回答。另一方面，我们证明了在大于一的每个维度中，存在一个阿贝尔变体，使得与它的对偶的差不会被消除\（\ mathbb {L} \）的任何幂，从而对问题给出否定的答案。我们还将讨论该问题的变体。