Positive degree and arithmetic bigness

TL;DR

This paper proves the convergence of truncated Harder-Narasimhan polygons for big Hermitian line bundles and applies this to confirm a conjecture on the arithmetic volume function, providing a method to compute asymptotic polygons.

Contribution

It establishes the convergence and continuity properties of polygons associated with Hermitian line bundles and confirms a conjecture relating to the arithmetic volume function.

Findings

Convergence of truncated Harder-Narasimhan polygons

Arithmetic volume function is a true limit, not just a supremum

Method to compute asymptotic polygons using the volume function

Abstract

We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit. As applications, we prove a conjecture of Moriwaki asserting that the arithmetic volume function is actually a limit instead of a sup-limit, and we show how to compute the asymptotic polygon of a Hermitian line bundle, by using the arithmetic volume function.