Do birationally equivalent Calabi-Yau manifolds have the same classes in the Grothendieck ring of varieties?


Here a *Calabi-Yau manifold* is a smooth complex projective variety with trivial canonical bundle.


This is true for Calabi-Yau threefolds or holomorphic symplectic fourfolds.


Related conjectures:


1. Birationally equivalent Calabi-Yau manifolds have equivalent derived categories. This is known for dimension three.
2. Consider two derived equivalent Calabi-Yau manifolds which are simply connected. Then the difference of their classes in the Grothendieck ring of varieties is annihilated by some power of the class of the affine line. This is open even for K3 surfaces.
3. Two derived equivalent smooth projective varieties have isomorphic Chow motives. This is known for dimension two.



Classes of birationally equivalent Calabi-Yau manifolds in the Grothendieck ring
This is not known. Motivic integration provides equality of classes of K-equivalent varieties (in particular, for birational with trivial canonical class) in the appropriate localization of the Grothendieck ring. This implies that birational Calabi-Yau varieties have equal Hodge numbers.


I believe the sharpest known result in this direction is Corollary 7.3 is in Kawamata's paper: <https://arxiv.org/pdf/1710.07370.pdf>.


