Pluricomplex charge at weak singularities

2005/10/31 by Jonas Wiklund, Wiklund, Jonas
Mathematics · #32F07 #32W20 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #math.CV #msc:32F07 #msc:32W20

paper · pdf · doi:10.48550/arxiv.math/0510671

arxiv created 2005/10/31 · openalex publication_date 2005/10/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let u be a plurisubharmonic function, defined on a neighbourhood of a point x, such that the complex Monge-Ampère operator is well-defined on u. Suppose also that u has a weak singularity, in the sense that the Lelong number of u at x vanish. Is it true that the residual mass of the measure (dd^cu)ⁿ vanish at x? To our knowledge there is no known example that falsifies the posed question. In this paper some partial results are obtained. We find that for a significant subset of plurisubharmonic functions with well defined Monge-Ampère mass vanishing Lelong number does implies vanishing residual mass of the Monge-Ampère measure.

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