Toric chordality

We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil–McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of graph chordality to higher skeleta of simplicial complexes with a coordinatization over characteristic 0, leading us to a far-reaching generalization of Kalai's work on applications of rigidity of frameworks to polytope theory. In contrast to “homological” chordality, the notion that is usually studied as a higher-dimensional analogue of graph chordality, we will show that toric chordality has several advantageous properties and applications. ∘ Most strikingly, we will see that toric chordality allows us to introduce a higher version of Dirac's propagation principle.∘ Aside from the propagation theorem, we also study the interplay with the geometric properties of the simplicial chain complex of the underlying simplicial complex, culminating in a quantified version of the Stanley–Murai–Nevo generalized lower bound theorem.∘ Finally, we apply our technique to give a simple proof of the generalized lower bound theorem in polytope theory and∘ prove the balanced generalized lower bound conjecture of Klee and Novik.