Okounkov bodies of filtered linear series

TL;DR

This paper introduces a new way to analyze filtered linear series using concave functions on Okounkov bodies, leading to approximation theorems and applications in Arakelov geometry.

Contribution

It develops a framework connecting filtrations of linear series with concave functions on Okounkov bodies, extending Fujita approximation and constructing arithmetic Okounkov bodies.

Findings

Established a concave function law describing filtration jumps.

Proved a Fujita-type approximation theorem for filtered linear series.

Constructed arithmetic Okounkov bodies and proved the existence of arithmetic volume.

Abstract

We associate to a filtration of a graded linear series of a big line bundle a concave function on the Okounkov body whose law with respect to Lebesgue's measure describes the asymptotic distribution of the jumps of the filtration. As a consequence we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to filtrations by minima in the usual context of Arakelov geometry, thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and the arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain by a variant of this construction a short proof of the existence of the sectional capacity.