On Ehrhart theory for tropical vector bundles

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.