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) |
|---|---|
| Journal | Potential Analysis |
| DOIs | |
| State | Published - Jan 1 2019 |
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Keywords
- Martin boundary
- Pluripotential theory
- Plurisubharmonic functions
ASJC Scopus subject areas
- Analysis
Cite this
(Pluri)Potential Compactifications. / Poletsky, Evgeny Alexander.
In: Potential Analysis, 01.01.2019.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - (Pluri)Potential Compactifications
AU - Poletsky, Evgeny Alexander
PY - 2019/1/1
Y1 - 2019/1/1
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
UR - http://www.scopus.com/inward/record.url?scp=85060848074&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85060848074&partnerID=8YFLogxK
U2 - 10.1007/s11118-019-09766-y
DO - 10.1007/s11118-019-09766-y
M3 - Article
AN - SCOPUS:85060848074
JO - Potential Analysis
JF - Potential Analysis
SN - 0926-2601
ER -