2021/06/30 by Yueqiao Wu
Mathematics · #Affine transformation #Affine variety #Algebraic Geometry and Number Theory #Combinatorics #Filtration (mathematics) #Function (biology) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum mechanics #Variety (cybernetics) #Volume (thermodynamics) #math.AG #math.DG
paper · pdf · doi:10.1007/s00209-021-02935-z
Final version: corrected a few dimensional constants
openalex publication_date 2022/01/30 · arxiv created 2022/02/14 · arxiv updated 2022/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let (X, ξ) be a polarized affine variety, i.e. an affine variety X with a (possibly irrational) Reeb vector field ξ. 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 φ on the Berkovich analytification of X. In this case, we define the Monge-Ampère energy for φ 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-Székelyhidi and Li-Xu.