scholarly journals Polynomial convexity, special polynomial polyhedra and the pluricomplex Green function for a compact set in Cn

2009 ◽  
Vol 91 (4) ◽  
pp. 364-383 ◽  
Author(s):  
Stéphanie Nivoche
Keyword(s):  
Green Function ◽  
Compact Set ◽  
Polynomial Convexity ◽  
Special Polynomial ◽  
Pluricomplex Green Function

scholarly journals Boundary behavior of the Bergman metric

2002 ◽  
Vol 168 ◽  
pp. 27-40 ◽  
Author(s):  
Bo-Yong Chen
Keyword(s):  
Green Function ◽  
Pseudoconvex Domain ◽  
Boundary Point ◽  
Boundary Behavior ◽  
Sufficient Conditions ◽  
Bergman Metric ◽  
Hölder Continuous ◽  
Pluricomplex Green Function ◽  
Peak Function

AbstractLet Ω be a bounded pseudoconvex domain in Cn. We give sufficient conditions for the Bergman metric to go to infinity uniformly at some boundary point, which is stated by the existence of a Hölder continuous plurisubharmonic peak function at this point or the verification of property (P) (in the sense of Coman) which is characterized by the pluricomplex Green function.

scholarly journals LIMITS OF MULTIPOLE PLURICOMPLEX GREEN FUNCTIONS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250065 ◽  
Author(s):  
JÓN I. MAGNÚSSON ◽  
ALEXANDER RASHKOVSKII ◽  
RAGNAR SIGURDSSON ◽  
PASCAL J. THOMAS
Keyword(s):  
Green Function ◽  
Uniform Convergence ◽  
Complete Intersection ◽  
Green Functions ◽  
Ideal Formula ◽  
Vanishing Ideal ◽  
Pluricomplex Green Function ◽  
Samuel Multiplicity ◽  
Hyperconvex Domain ◽  
Local Uniform Convergence

Let Ω be a bounded hyperconvex domain in ℂn, 0 ∈ Ω, and Sε a family of N poles in Ω, all tending to 0 as ε tends to 0. To each Sε we associate its vanishing ideal [Formula: see text] and pluricomplex Green function [Formula: see text]. Suppose that, as ε tends to 0, [Formula: see text] converges to [Formula: see text] (local uniform convergence), and that (Gε)ε converges to G, locally uniformly away from 0; then [Formula: see text]. If the Hilbert–Samuel multiplicity of [Formula: see text] is strictly larger than its length (codimension, equal to N here), then (Gε)ε cannot converge to [Formula: see text]. Conversely, if [Formula: see text] is a complete intersection ideal, then (Gε)ε converges to [Formula: see text]. We work out the case of three poles.

THE PLURICOMPLEX GREEN FUNCTION ON PSEUDOCONVEX DOMAINS WITH A SMOOTH BOUNDARY

2000 ◽  
Vol 11 (04) ◽  
pp. 509-522 ◽  
Author(s):  
GREGOR HERBORT
Keyword(s):  
Green Function ◽  
Boundary Point ◽  
Additional Assumption ◽  
Smooth Boundary ◽  
Pseudoconvex Domains ◽  
Exceptional Set ◽  
Exhaustion Function ◽  
Pluricomplex Green Function ◽  
Pluripolar Set ◽  
Hyperconvex Domain

In this article we deal with the behavior of the pluricomplex Green function GD(·;w), of a pseudoconvex domain D in [Formula: see text], when the pole tends to a boundary point. In [7], it was shown that, given a boundary point w0 of a hyperconvex domain D, then there is a pluripolar set E⊂D, such that lim sup w→w0 GD(z;w)=0 for z∈D\E. Under an additional assumption on D, that can be viewed as natural, one can avoid the pluripolar exceptional set. Our main result is that on a bounded domain [Formula: see text] that admits a Hoelder continuous plurisubharmonic exhaustion function ρ:D→[-1,0), the pluricomplex Green function GD(·,w) tends to zero uniformly on compact subsets of D, if the pole w tends to a boundary point w0 of D.

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