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

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.

23 pages; v4: Added Section 7, Note, and references. Several changes here and there

References in corpus (7)

• Semiorthogonal decomposition for algebraic varieties
• Derived categories of coherent sheaves on abelian varieties and equivalences between them
• Homological projective duality for Grassmannians of lines
• Derived categories of coherent sheaves and motives
• The Grothendieck group of algebraic stacks
• Cremona transformations and derived equivalences of K3 surfaces
• Intersections of two Grassmannians in ${\mathbf{P}}^9$

Cited by in corpus (6)

• Cremona transformations and derived equivalences of K3 surfaces
• Intersections of two Grassmannians in ${\mathbf{P}}^9$
• L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces
• An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent
• $G_2$-Grassmannians and derived equivalences
• Integration of Voevodsky motives