The Grothendieck ring of varieties and of the theory of algebraically closed fields

In each characteristic, there is a canonical homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of sets definable in the theory of algebraically closed fields. We prove that this homomorphism is an isomorphism in characteristic zero. In positive characteristics, we exhibit specific elements in the kernel of the corresponding homomorphism of Grothendieck semirings. The comparison of these two Grothendieck rings in positive characteristics seems to be an open question, related to the difficult problem of cancellativity of the Grothendieck semigroup of varieties.