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?  

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

https://mathoverflow.net/questions/299581/is-there-a-lift-of-the-q-vandermonde-identity-to-some-geometric-motivic-identi

https://mathoverflow.net/questions/299748/can-one-divide-algebraic-manifolds-make-sense-gr2-n-gr2-nm-pn-1-p?noredirect=1&lq=1