$\DeclareMathOperator\Var{Var}$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 constructibly 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 constructibly 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 constructibly 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 constructibly 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 constructibly equivalent to $\mathbb{A}^{3}$ with a line removed.

Edit. As noted in the comments the example $C$ above is in fact constructibly equivalent to the affine plane.

$\DeclareMathOperator\Var{Var}$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 the example $C$ above is in fact piecewise isomorphic to the affine plane.

Let $K_{0}(Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are varieties we say that they are constructibly 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 constructibly 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 constructibly 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 $q:=[\,\mathbb{A}^{1}\,]$ 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 constructibly isomorphic to $\mathbb{A}^{2}$. Another example is given by $SL(2)$, which has class $q^{3}-q$, and which I presume is not constructibly equivalent to $\mathbb{A}^{3}$ with a line removed.

Edit. As noted in the comments the example $C$ above is in fact constructibly equivalent to the affine plane.

$\DeclareMathOperator\Var{Var}$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 constructibly 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 constructibly 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 constructibly 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 constructibly 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 constructibly equivalent to $\mathbb{A}^{3}$ with a line removed.

Edit. As noted in the comments the example $C$ above is in fact constructibly equivalent to the affine plane.

Let $K_{0}(Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are varieties we say that they are constructibly 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 constructibly 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 constructibly 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 $q:=[\,\mathbb{A}^{1}\,]$ 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 constructibly isomorphic to $\mathbb{A}^{2}$. Another example is given by $SL(2)$, which has class $q^{3}-q$, and which I presume is not constructibly equivalent to $\mathbb{A}^{3}$ with a line removed.

Let $K_{0}(Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are varieties we say that they are constructibly 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 constructibly 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 constructibly 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 $q:=[\,\mathbb{A}^{1}\,]$ 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 constructibly isomorphic to $\mathbb{A}^{2}$. Another example is given by $SL(2)$, which has class $q^{3}-q$, and which I presume is not constructibly equivalent to $\mathbb{A}^{3}$ with a line removed.

Edit. As noted in the comments the example $C$ above is in fact constructibly equivalent to the affine plane.