Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties
Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties
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衍生等价和簇的格洛腾迪克环：12 阶 K3 曲面和阿贝尔簇的情况
DOI：
10.1007/s00029-020-00561-x
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Ueda Kazushi
中科院分区：
文献类型：
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作者：
Ito Atsushi;Miura Makoto;Okawa Shinnosuke;Ueda Kazushi
本文讨论了Grothendieck环上光滑射影簇(X，Y)的Fourier-Mukai对(X，Y)的类的差是否被仿射直线类的某些幂所消灭的问题。我们给出了非常一般的12次K3曲面的Fourier-Mukai对的一个肯定的回答。另一方面，我们证明了在每个大于一维的空间中，存在一个阿贝尔簇，使得它与其对偶的差不被的任何幂所消灭，从而给出了一个否定的答案。我们还讨论了问题的变种。
In this paper, we discuss the problem of whether the differenceof 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 classof 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, thereby giving a negative answer to the problem. We also discuss variations of the problem.