Any smooth projective variety $X$ gives an object $h(X)$ in the category of pure Chow motives. If $X$ is a generalized flag variety, i.e. a quotient $G/P$ where $G$ is semisimple linear algebraic group and $P$ is a parabolic subgroup, I believe $h(X)$ is a direct sum of tensor powers of the Lefschetz motive, because $X$ can be decomposed into Schubert varieties which are copies of $\mathbb{A}^n$ for various $n$.

If this is correct, I'd like to know: **which other smooth projective varieties give pure Chow motives that are direct sums of tensor powers of the Lefschetz motive?**

My intuition is that any variety with something like a "Schubert decomposition" — roughly, a well-behaved way of expressing it as a disjoint union of copies of $\mathbb{A}^n$'s — will have this property. I feel there should be plenty. But I don't actually know any varieties with this property, apart from flag varieties!

Any variety $X$ of dimension $d$ over $\mathbb{F}_p$ having this property will have an associated polynomial $N_X$ of degree $d$ with natural number coefficients:

$$ N_X(q) = \sum_{n = 0}^d a_n q^n $$

such that $X$ has $N_X(q)$ points over $\mathbb{F}_q$ when $q$ is any power of $p$.

**Is the converse true? Is a smooth projective variety $X$ over $\mathbb{F}_p$ with a polynomial $N_X$ having this property always a direct sum of tensor powers of the Lefschetz motive?**