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.
Birationally equivalent Calabi-Yau manifolds have equivalent derived categories. This is known for dimension three.
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.
Two derived equivalent smooth projective varieties have isomorphic Chow motives. This is known for dimension two.