A numerical characterization of nef adelic divisors

TL;DR

This paper introduces a numerical approach to analyze nef adelic divisors by associating an arithmetic Okounkov body and exploring the properties of the concave transform, extending previous results in special cases.

Contribution

It generalizes the relationship between the infimum of the concave transform and the absolute minimum for vertically nef adelic divisors beyond prior specific cases.

Findings

The infimum of the concave transform equals the absolute minimum for vertically nef divisors.

Extension of Moriwaki's results from curves to higher dimensions.

Generalization of results from toric to more general adelic divisors.

Abstract

To a generically big adelic divisor, we can associate an arithmetic Okounkov body, which is a pair of the geometric Okounkov body and the concave transform of the Green functions. In this paper, we show that the infimum of the concave transform is given by the absolute minimum provided that the divisor is vertically nef. This is a partial generalization of results of Moriwaki (in the curve case) and of Burgos Gil-Moriwaki-Philippon-Sombra (in the toric case).