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 $h(t,P)$. 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 $IH(X_F)$, I think. (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)).
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 $\pi$. 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 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?
