Broken Toric Varieties and Cell-Compatible Sheaves

TL;DR

This paper investigates the cohomology of broken toric varieties, establishing a decomposition theorem, spectral sequence degeneration, and explicit Betti number formulas, advancing understanding of their topological and algebraic properties.

Contribution

It introduces a Deligne-type decomposition theorem and explicit Betti number formulas for broken toric varieties, linking their cohomology to polytope complexes.

Findings

Proved a Deligne-type decomposition theorem for broken toric varieties.

Showed degeneration of the Leray-Serre spectral sequence.

Derived explicit formulas for Betti numbers in specific cases.

Abstract

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.