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arXiv · 2106.13395

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

Yueqiao Wu
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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.

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Yueqiao Wu. 2021-06-25. Volume and Monge-Amp\`ere energy on polarized affine varieties. https://doi.org/10.1007/s00209-021-02935-z

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