Toric chordality

TL;DR

This paper introduces toric chordality, a new concept extending graph chordality to higher skeleta of simplicial complexes, enabling advanced results in polytope theory and combinatorial geometry.

Contribution

It defines toric chordality, explores its properties, and applies it to prove the balanced generalized lower bound conjecture and other key theorems in polytope theory.

Findings

Toric chordality generalizes graph chordality to higher dimensions.

It leads to a higher Dirac propagation principle.

The technique provides a new proof of the generalized lower bound theorem.

Abstract

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.