On Ehrhart theory for tropical vector bundles

TL;DR

This paper develops a combinatorial approach to tropical vector bundles on toric varieties, establishing a Hirzebruch-Riemann-Roch theorem and analyzing cohomology properties, including for tautological bundles of matroids.

Contribution

It introduces a convex chain framework for Euler characteristic, extends Klyachko's resolution to tropical bundles, and confirms vanishing of higher cohomologies for matroid tautological bundles.

Findings

A convex chain encodes the Euler characteristic of tropical vector bundles.

A combinatorial Hirzebruch-Riemann-Roch theorem is established for tropical bundles.

Higher cohomologies vanish for tautological bundles of matroids.

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.