Note on Euler characteristic of a toric vector bundle

TL;DR

This paper extends Ehrhart theory to convex chains associated with toric vector bundles, linking convex geometry with algebraic topology to compute Euler characteristics.

Contribution

It introduces a new method connecting convex chains with toric vector bundles, generalizing the Ehrhart theory for lattice polytopes.

Findings

Convex chains can be associated with toric vector bundles.

Sum of convex chain values on lattice points computes Euler characteristic.

Extends the relationship between line bundles and lattice polytopes.

Abstract

A convex chain is a finite integer linear combination of indicator functions of convex polytopes. Khovanskii-Pukhlikov extend the Ehrhart theory of convex lattice polytopes to the setting of convex chains. Extending the relationship between equivariant line bundles on projective toric varieties and virtual lattice polytopes, we associate a lattice convex chain to a torus equivariant vector bundle on a toric variety and show that sum of values of this convex chain on lattice points gives the Euler characteristic of the bundle.