Question: Consider algebraic manyfold assume it is number of points is q^n ( [n+1]_q ) does it apply any geometric relation to A^n (P^n) ? In particular is there equivalence in Grothendieck ring of varieties ? Or may be birational equivalence ?

If it is not true in general, may be some additional reasonable requirments on a manyfold will force that to be true ?

Motivation: one can see that some examples of identities on the level of F_q points enumeration can be lifted to geometric relations:

Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?

Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$