I am currently reading a paper from Sankaran and Vanchinathan where they compute certain Kazhdan-Lusztig polynomials.

Sankaran, P.; Vanchinathan, P.: Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials. Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 465-480.

Let $G$ be a complex semisimple algebraic group with Weyl group $W$ and $\tau \leq \lambda$ elements in $W$. For some $\lambda$, they can take resolutions called *small* and use the fact that the Kazhdan-Lusztig polynomial $P_{\tau, \lambda}(q)$ is equal to the Poincaré polynomial in $q^{1/2}$ of the fibre of $\tau P/P$. The resolutions are constructed as a tower of locally trivial fibrations with fibers being Schubert varieties.

My question concerns the theory behind their computation of the Poincaré polynomial. Let $F_q$ be a finite field with $q$ elements. Then they claim that the value of the polynomial at $q$ is given by the number of rational points of the fibre if the varieties are considered over $F_q$ (all varieties are well defined over any field in this case). Why is this true? I guess this derives from a more general and well known theorem about the Poincaré polynomial and counting points over finite fields.