I'm interested in the intersection homology of toric varieties associated to a polytope $P$ with proper faces F, and a subdivision $P'$ of P. Let $X_P$ be the toric variety associated to the polytope $P$ by taking the fan of cones over faces of $P$.

Some basic facts to begin with: the dimensions of the intersection homology groups are are combinatorial invariants of the polytope P, and are encoded in the $h$-polynomial $h(P,t)$. Moreover the $h$-polynomial for a non-simplicial polytope $P$ is described using the $g(t,F)$-polynomials of all of the proper faces $F \subset P$; these $g(t,F)$ polynomials may be thought of as encoding the primitive part of the intersection homology $IH(\overline{O}_F)$ of the orbit closure $\overline{O}_F \subset X_P$ associated to the face $F$. (I am taking this description from De Cataldo and Migliorini, *The decomposition theorem, perverse sheaves and the topology of algebraic maps* ([link at AMS site](https://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/))).

Moreover, as discussed by Goresky and in papers of Braden and MacPherson and  (and I think Bernstein and Lunts should be credited here as well), the proof of this formula for the $h(t,P)$ polynomial involving the $g(t,F)$ polynomials passes through the following facts:

Let $P'$ be a subdivision of the polytope $P$, obtained by subdividing some face $F$, say. We then obtain a dominant map $\pi: X_{P'} \rightarrow X_{P}$, which is an isomorphism away from $\overline{O}_F \subset X_P$. We examine $\pi_* (IC_{X_{P'}})$ through the lens of the decomposition theorem, $$\pi_*(IC_{X_{P'}})= \bigoplus_{O_F} IC_{\overline{O_F}}(L_i)[a_i].$$

Moreover, according to de Cataldo and Migliorini, all the local systems $L_i$ appearing in the toric case should be trivial, which simplifies life.

As far as I understand, the question then becomes 

"How much of $\pi_*(IC_{X_{P'}})$ belongs to $IC_{X_{P}}$?" The answer appears to be "The primitive cohomology of the fiber $\pi^{-1}(\overline{O}_F)$ belongs to $IC_{X_P}$." This may not be totally true but this is what I can understand from Goresky's notes (https://www.math.ias.edu/~goresky/math2710/lecture15.pdf#zoom=100, last page).

This brings up a naive guess. Namely, I wish to compare $h(t,P)$ with $h(t,P')$. Geometrically, the following seems to make sense to me: take $IH(\overline{O}_F)$ to be the intersection homology of the image of the exceptional divisor of $\pi$. Shifting and taking dimensions, we get the polynomial $h_{b}$. Then we take the NONprimitive terms of the intersection homology of the fiber $\pi^{-1}(p)$ for $p \in O_F$; after shifting and only remembering dimensions, this produces the polynomial $h_f$. The naive guess then is that $$h(t,P')-h(t,P)=h_f \cdot h_b$$. 

How far is this naive guess from being true?