Vanishing sequences and Okounkov bodies

TL;DR

This paper introduces the concept of vanishing sequences along real valuations for line bundle sections, proves an equidistribution result for large powers, and explores the properties of the associated limit measures and functions.

Contribution

It extends the theory of vanishing sequences and Okounkov bodies by establishing equidistribution results and analyzing limit measures and functions in this context.

Findings

Proved equidistribution of vanishing sequences for large line bundle powers

Described the limit measure in terms of restricted volumes for divisorial valuations

Demonstrated discontinuity of the associated concave function at boundary points

Abstract

We define and study the vanishing sequence along a real valuation of sections of a line bundle on a projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure. In particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.