(Pluri)Potential Compactifications

Evgeny A. Poletsky

doi:10.1007/s11118-019-09766-y

(Pluri)Potential Compactifications

Evgeny A. Poletsky

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L¹(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

Original language	English (US)
Pages (from-to)	231-245
Number of pages	15
Journal	Potential Analysis
Volume	53
Issue number	1
DOIs	https://doi.org/10.1007/s11118-019-09766-y
State	Published - Jun 1 2020

Keywords

Martin boundary
Pluripotential theory
Plurisubharmonic functions

ASJC Scopus subject areas

Analysis

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title = "(Pluri)Potential Compactifications",

abstract = "Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.",

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N2 - Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

AB - Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

KW - Martin boundary

KW - Pluripotential theory

KW - Plurisubharmonic functions

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JO - Potential Analysis

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(Pluri)Potential Compactifications

Abstract

Keywords

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