Derived equivalence and Grothendieck ring of varieties

Abstract: 

Bounded derived category of coherent sheaves of an algebraic variety is a relatively new but very important invariant. There are examples of non-singular projective varieties XX and YY which have the equivalent derived category but are even not birational to each other (meaning, their geometric relationship is not so obvious). Though it is expected that derived equivalent XX and YY should share some invariants, we have only partial knowledge in this direction.

On the other hand, the Grothendieck ring of varieties is defined as the free abelian group generated by the set of isomorphism classes of algebraic varieties (over a fixed field kk) modulo the relations [X]=[Z]+[X−Z][X]=[Z]+[X−Z], where ZZ is a closed subscheme of XX. It is a commutative ring with the product [X][Y]=[X×Y][X][Y]=[X×Y]. Some of the important invariants of a variety can be recovered from its class in the Grothendieck ring, such as the number of rational points (when kk is a finite field), topological Euler number (when k=Ck=C), and the Hodge numbers (again when k=Ck=C).

In this talk I would like to discuss a question, with a couple of examples, which asks if a pair of derived equivalent algebraic varieties share the same class in "some" modification of the Grothendieck ring of varieties.