On the Grothendieck ring of varieties

Abstract

Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0 (\operatorname{Var}_k)$ are piecewise isomorphic. Gromov asked if a birational self-map of a $k$-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group. Furthermore, if $\mathfrak B$ denotes the multiplicative monoid of birational equivalence classes of irreducible $k$-varieties then we also prove that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb Z[\mathfrak B]$.