Consider an algebraic manifold whose number of points is $q^n ([n+1]_q)$. Is there a geometric relation to $A^n (P^n)$? In particular, is there an equivalence in the Grothendieck ring of varieties or could there be a birational equivalence?
If there is no such equivalence in general, might some additional reasonable requirements on a manifold force there to be such an equivalence?
Motivation: one can see that some examples of identities on the level of enumerating $F_q$ points can be lifted to geometric relations:
