Note on asymptotic behaviour of the canonical ring

TL;DR

This paper explores the asymptotic properties of the canonical ring of a line bundle over complex spaces, utilizing Oukounkov bodies and algebraic reduction to extend understanding beyond projective manifolds.

Contribution

It introduces a method to analyze the asymptotic behaviour of canonical rings on arbitrary compact normal complex spaces using Oukounkov bodies and algebraic reduction.

Findings

Extended asymptotic analysis to non-projective complex spaces

Applied Oukounkov body theory in a new context

Provided insights into the structure of canonical rings

Abstract

The theory of the Oukounkov body is a useful tool for studying the asymptotic behaviour of the canonical ring of a line bundle over a projective manifold. In this note, combined with the algebraic reduction, we study the asymptotic behaviour of the canonical ring of a line bundle over an arbitrary compact normal irreducible complex space.