Volume and Monge-Amp\`ere energy on polarized affine varieties

TL;DR

This paper introduces a new way to measure the volume of filtrations on polarized affine varieties using Monge-Ampère energy, linking algebraic and analytic perspectives in non-Archimedean geometry.

Contribution

It defines the Monge-Ampère energy for filtrations on polarized affine varieties and proves its equivalence to the volume, extending previous concepts to a broader setting.

Findings

Monge-Ampère energy matches the volume for finitely generated filtrations.

The framework recovers known functionals for test configurations.

Establishes a bridge between algebraic filtrations and non-Archimedean pluripotential theory.

Abstract

Let $(X, \xi)$ be a polarized affine variety, i.e. an affine variety $X$ with a (possibly irrational) Reeb vector field $\xi$. We define the volume of a filtration of the coordinate ring of $X$ in terms of the asymptotics of the average of jumping numbers. When the filtration is finitely generated, it induces a Fubini-Study function $\varphi$ on the Berkovich analytification of $X$. In this case, we define the Monge-Amp\`ere energy for $\varphi$ using the theory of forms and currents on Berkovich spaces developed by Chambert-Loir and Ducros, and show that it agrees with the volume of the filtration. In the special case when the filtration comes from a test configuration, we recover the functional defined by Collins-Sz\'ekelyhidi and Li-Xu.