Newton-Okounkov bodies and complexity functions

TL;DR

This paper investigates the relationship between the holonomicity of complexity functions and the polyhedrality of Newton-Okounkov bodies, revealing that one does not necessarily imply the other across different flags.

Contribution

It demonstrates that the holonomicity of complexity functions does not universally predict the polyhedral structure of Newton-Okounkov bodies for all flags.

Findings

Holonomicity of complexity functions is not a reliable predictor of Newton-Okounkov body polyhedrality.

Counterexamples show the independence of these properties across flags.

The results clarify the limitations of using complexity functions to infer geometric structures.

Abstract

We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.