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On Ehrhart theory for tropical vector bundles

Suhyon Chong, Kiumars Kaveh

Abstract

The notion of a tropical vector bundle on a toric variety was recently introduced by Khan-Maclagan and Kaveh-Manon. In this paper, we study the Euler characteristic and rank of global sections for tropical vector bundles. We associate a convex chain (a finite integer linear combination of indicator functions of convex polytopes) to a tropical vector bundle encoding its Euler characteristic. We then see that the Khovanskii-Pukhlikov theory of convex chains gives a combinatorial Hirzebruch-Riemann-Roch theorem for tropical vector bundles. This, in particular, applies to toric vector bundles. Also, we extend Klyachko's resolution of a toric vector bundle by split toric vector bundles to tropical vector bundles. As shown by Kaveh-Manon, every matroid comes with a tautological tropical vector bundle. We answer positively a question posed by Kaveh-Manon about equality of Euler characteristic with rank of space of global sections (in other words, vanishing of higher cohomologies) for the tautological bundle of a matroid.

On Ehrhart theory for tropical vector bundles

Abstract

The notion of a tropical vector bundle on a toric variety was recently introduced by Khan-Maclagan and Kaveh-Manon. In this paper, we study the Euler characteristic and rank of global sections for tropical vector bundles. We associate a convex chain (a finite integer linear combination of indicator functions of convex polytopes) to a tropical vector bundle encoding its Euler characteristic. We then see that the Khovanskii-Pukhlikov theory of convex chains gives a combinatorial Hirzebruch-Riemann-Roch theorem for tropical vector bundles. This, in particular, applies to toric vector bundles. Also, we extend Klyachko's resolution of a toric vector bundle by split toric vector bundles to tropical vector bundles. As shown by Kaveh-Manon, every matroid comes with a tautological tropical vector bundle. We answer positively a question posed by Kaveh-Manon about equality of Euler characteristic with rank of space of global sections (in other words, vanishing of higher cohomologies) for the tautological bundle of a matroid.
Paper Structure (14 sections, 33 theorems, 114 equations, 2 figures)

This paper contains 14 sections, 33 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Klyachko classification of toric vector bundles
  4. Matroids
  5. Tropical vector bundles
  6. Sheaf of sections
  7. Khovanskii-Pukhlikov Hirzebruch-Riemann-Roch theorem for polytopes
  8. Euler characteristic of a tropical vector bundle
  9. Convex chain associated to a tropical vector bundle
  10. Construction of convex chain using a split resolution
  11. Euler characteristic of a tropical vector bundle
  12. Global sections and Euler characteristic of the tautological bundle of a matroid
  13. The map $\Phi_\mathcal{M}$ and the bundle $\mathcal{E}_\mathcal{M}$
  14. Vanishing of higher cohomologies of tautological bundle

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a tropical vector bundle on a toric variety $X_\Sigma$ with fan $\Sigma$, and let $\alpha_{\mathcal{E}}$ be the associated convex chain. Then for every character $u$ of the torus,

Figures (2)

  • Figure 1: The Fano plane.
  • Figure 2: The parliament of polytopes for $\mathcal{E}$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Vanishing of higher cohomologies of a tautological bundle
  • Remark 1.5
  • Proposition 2.1
  • Theorem 2.2: Klyachko
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • ...and 64 more