Envelopes of positive metrics with prescribed singularities

2012/10/08 by Julius Ross, Ross, Julius, David Witt Nyström +2 · 1 citation
Mathematics · #14C30 #32U15 #32W20 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #math.CV #msc:14C30 #msc:32U15 #msc:32W20

paper · pdf · doi:10.48550/arxiv.1210.2220

32 pages. v2 expositional changes. Final version, to appear in Annales de la Faculté des Sciences de Toulouse

openalex publication_date 2012/10/08 · arxiv created 2016/07/15 · arxiv updated 2016/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate envelopes of positive metrics with a prescribed singularity type. First we generalise work of Berman to this setting, proving C^1,1 regularity of such envelopes, showing their Monge-Ampere measure is supported on a certain "equilibrium set" and connecting with the asymptotics of the partial Bergman functions coming from multiplier ideals. We investigate how these envelopes behave on certain products, and how they relate to the Legendre transform of a test curve of singularity types in the context of geodesic rays in the space of Kähler potentials. Finally we consider the associated exhaustion function of these equilibrium sets, connecting it both to the Legendre transform and to the geometry of the Okounkov body.

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