Remarks on polynomial count varieties

In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties. The two questions we address are the following. 1) If $X$ is smooth, polynomial count with $\#X(q)=q^n$ for some $n$, is $X$ isomorphic to $n$-dimensional affine space? 2) If $X$ is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~$h^{p,q}=0$ unless $p=q$? We show that in both cases the answer is no.