Let $k$ be an algebraically closed field of characteristic zero. Let $A$ be the $\mathbf{Z}$-subalgebra of the Grothendieck ring of $k$-varieties $K_0(\text{Var}_k)$ generated by classes of semi-abelian varieties. **Question 1.** What can we say about the ring homomorphism $A\to K_0(\text{Var}_k)$? It is obviously very far from being surjective. Mainly, is it integral/finite/flat/essentially of finite type? **Question 2.** The class of smooth projective varieties over $k$ generates $K_0(\text{Var}_k)$ in the obvious sense. Is there a more manageable generating class of $k$-varieties, possibly containing semi-abelian varieties at the very least? **Question 3.** In light of Question 2, does the class $\mathcal{P}$ of varieties over $k$, constructed below, have any hope to be a generating class for $K_0(\text{Var}_k)$, and if not, upon calling $B$ the $\mathbf{Z}$-subalgebra of $K_0(\text{Var}_k)$ it generates, same question as in Question 1, for the ring homomorphism $B\to K_0(\text{Var}_k)$. We call $\mathcal{P}_0$ the class of $k$-varieties whose objects are: - abelian varieties - semi-abelian varieties (extension of abelian varieties by a torus) - smooth hypersurfaces in large projective spaces over $k$ - smooth projective varieties of dimension $\le 3$ - smooth projective toric varieties - Calabi-Yau varieties - products of the above Let $\mathcal{P}_1$ be the class of $k$-varieties whose objects are obtained by blowing up an element of $\mathcal{P}_0$ along a closed $k$-subvariety that is again an element of $\mathcal{P}_0$. Finally, let $\mathcal{P}$ be the class of all those $k$-varieties such that their class in the Grothendieck ring of varieties $K_0(\text{Var}_k)$ belongs to the subring $B\subset K_0(\text{Var}_k)$ generated by the classes of the elements of $\mathcal{P}_1$. **Question 4.** Does there at least exist a morphism $f: Y\to X$ over $k$, with $Y$ in $\mathcal{P}$, and such that $f$ is surjective/flat/such that $f^*\omega_X\simeq\omega_Y$? (This last question is expected to have negative answer for $f$ required to be surjective, and a good strategy to see this is suggested in the comments.)