Broken Toric Varieties and Cell-Compatible Sheaves

We study the cohomology of broken toric varieties via the derived push-forward of the constant sheaf to a complex of polytopes, proving a Deligne-type decomposition theorem, degeneration of the associated Leray-Serre spectral sequence, and showing that the Leray filtration on their cohomology is equal to twice the weight filtration. Furthermore, we give an explicit formula for the Betti numbers of some broken toric varieties whose associated complex of polytopes is the $n$-skeleton of a higher dimensional polytope, encompassing some important examples.