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Note on Euler characteristic of a toric vector bundle

Suhyon Chong, Shaoyu Huang, Kiumars Kaveh

TL;DR

The paper addresses extending Ehrhart-type lattice-point counting from toric line bundles to toric vector bundles by introducing a convex-chain formalism. It builds a lattice convex chain $\alpha_{\mathcal{E}}$ associated to a toric vector bundle via multi-valued support functions and piecewise-linear valuations, and proves that the equivariant Euler characteristic satisfies $\chi(\mathcal{E})_u=\alpha_{\mathcal{E}}(u)$ for all weights $u$ and $\chi(\mathcal{E})=\sum_{u\in M}\alpha_{\mathcal{E}}(u)$. The line-bundle case is recovered as $\alpha_{\mathcal{L}}=\mathbbm{1}_{P_D}$, and the framework is shown to be invariant under fan refinements, with explicit examples illustrating the method. This work broadens the combinatorial toolkit for toric geometry by linking equivariant Euler characteristics to KP convex-chain theory and paves the way for tropical-toric generalizations of vector bundles.

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.

Note on Euler characteristic of a toric vector bundle

TL;DR

The paper addresses extending Ehrhart-type lattice-point counting from toric line bundles to toric vector bundles by introducing a convex-chain formalism. It builds a lattice convex chain associated to a toric vector bundle via multi-valued support functions and piecewise-linear valuations, and proves that the equivariant Euler characteristic satisfies for all weights and . The line-bundle case is recovered as , and the framework is shown to be invariant under fan refinements, with explicit examples illustrating the method. This work broadens the combinatorial toolkit for toric geometry by linking equivariant Euler characteristics to KP convex-chain theory and paves the way for tropical-toric generalizations of vector bundles.

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.
Paper Structure (11 sections, 15 theorems, 58 equations, 4 figures)

This paper contains 11 sections, 15 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Ehrhart theory
  4. Khovanskii-Pukhlikov theory of convex chains
  5. Toric vector bundles
  6. Toric vector bundles as piecewise linear valuations
  7. Convex chains and toric vector bundles
  8. Convex chain associated to a toric vector bundle
  9. Convex chain associated to a piecewise linear valuation
  10. Euler characteristic as sum of values of a convex chain over lattice points
  11. Examples

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a toric vector bundle on a projective toric variety $X_\Sigma$. The equivariant Euler characteristic of $\mathcal{E}$ is given by In particular,

Figures (4)

  • Figure 1: Polytopes from $h$
  • Figure 2: Values of the convex chain $\alpha$ on lattice points
  • Figure 3: Polytopes from $h$
  • Figure 4: Values of the convex chain $\alpha$ on lattice points

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1: Ehrhart–Macdonald reciprocity
  • Theorem 2.2
  • Corollary 2.3
  • Example 2.4
  • Theorem 2.5: Serre duality
  • Definition 2.6
  • Remark 2.7
  • ...and 20 more