# Poincaré polynomial and blow-up

Consider $X,Z$ smooth projectives varieties and $Y = Bl_Z(X)$ the blow-up of $X$ with center $Z$. Finally let $E$ be the exceptional divisor of the blow-up. Is is true that $p(Y) - p(E) = p(X) - p(Z)$ where $p$ is the Poincaré polynomial ? Thanks in advance !

Yes, both sides compute the Poicare polynomial of cohomology with compact supports of $Y \setminus E = X \setminus Z$.