There is a well-defined map $$P:K(Var)\to \mathbb{Z}[t]$$ which sends smooth projective varieties to their Poincare polynomial. This is in fact enough to define $P$ on all elements of $K(Var)$.
I wonder if there are any sufficient conditions under which $P([X])$ is the Poincare polynomial of $X$, other then $X$ being smooth projective?