$\DeclareMathOperator\Var{Var}$Let $K\_{0}(\Var\_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of a variety, $X$, in $K\_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are varieties then we say that they are *piecewise isomorphic* if there are finite locally closed stratifications, $\{X\_{i}\}$ of $X$, and $\{Y\_{i}\}$ of $Y$, so that there exist isomorphisms $X\_{i}\rightarrow Y\_{i}$.


Note, if $X$ and $Y$ are piecewise isomorphic then we have $[\,X\,]=[\,Y\,]$ in $K\_{0}(\Var\_{\mathbb{C}})$. I expect the converse should be false, perhaps even generically so (in some sense).


**Question.** What is a simple example of a pair of varieties with equivalent classes in $K\_{0}(\Var\_{\mathbb{C}})$ which are not piecewise isomorphic?


Here is a candidate example; let $C$ be the affine cone inside $\mathbb{A}^{3}$ given by the equation $x^{2}+y^{2}+z^{2}=0$, and denote by $q\mathrel{:=}[\,\mathbb{A}^{1}\,]$ the Lefschetz motive. Then we have $[\,C\,]=q^{2}$, since removing the cone point we obtain a (Zariski locally trivial) $\mathbb{C}^{\*}$ bundle over $\mathbb{P}^{1}$. I expect that $C$ is not piecewise isomorphic to $\mathbb{A}^{2}$. Another example is given by $\operatorname{SL}(2)$, which has class $q^{3}-q$, and which I presume is not piecewise isomorphic to $\mathbb{A}^{3}$ with a line removed.


**Edit.** As noted in the [comments](https://mathoverflow.net/questions/380473/constructible-isomorphism-versus-equivalence-in-grothendieck-ring#comment966401_380473) the example $C$ above is in fact piecewise isomorphic to the affine plane.



Piecewise isomorphism versus equivalence in Grothendieck ring
There are no simple examples as yet; it's been an open question going back to at least [Larsen and Lunts - Motivic measures and stable birational geometry](https://arxiv.org/abs/math/0110255), which has been open for about 15 years, and some of us believed that it should be true.


The first counterexample for smooth non-projective varieties was constructed by Borisov as a consequence on his work on L-zero divisors: [Borisov - The class of the affine line is a zero divisor in the Grothendieck ring](https://arxiv.org/abs/1412.6194).


There are currently no counterexamples known for smooth projective varieties. Specifically it is not known if $X$, $Y$ are smooth connected projective varieties over a field of characteristic zero such that $[X] = [Y]$, whether $X$ and $Y$ must be birational; they are stably birational by the work of Larsen and Lunts above.


