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

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.