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 requirments on a manifold will force there to be such an equivalence?

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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

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:

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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

Question: Consider an algebraic manifold; assume its number of points is $q^n ( [n+1]_q )$, is there a geometric relation to $A^n (P^n)$ applicable? In particular, is there an equivalence in the Grothendieck ring of varieties? Or could there be a birational equivalence?

If it is not true in general, maybe some additional reasonable requirments on a manifold 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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

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 requirments on a manifold will force there to be such an equivalence?

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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

Question: Consider algebraic manyfold assume its 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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$

Question: Consider an algebraic manifold; assume its number of points is $q^n ( [n+1]_q )$, is there a geometric relation to $A^n (P^n)$ applicable? In particular, is there an equivalence in the Grothendieck ring of varieties? Or could there be a birational equivalence?

If it is not true in general, maybe some additional reasonable requirments on a manifold 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: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$