On Volumes of Arithmetic Line Bundles II

TL;DR

This paper develops a convex function on the Okounkov body for hermitian line bundles over arithmetic varieties, linking its integral to the Euler characteristic growth, extending Nystrom's recent work.

Contribution

It introduces a new convex function on the Okounkov body for hermitian line bundles, providing a global perspective on Euler characteristic growth.

Findings

Constructed a convex continuous function on the Okounkov body.

Linked the integral of this function to Euler characteristic growth.

Extended Nystrom's recent local work to a global setting.

Abstract

For a hermitian line bundle over an arithmetic variety, we construct a convex continuous function on the Okounkov body associated to the generic fibre of the line bundle. The integration of the continuous function gives the growth of the Euler characteristic of the hermitian line bundle. It is the global version of the recent work of Nystrom.