A criterion of algebraicity for Lelong classes and analytic sets
type: Item
instance of: scholarly article
title: A criterion of algebraicity for Lelong classes and analytic sets
zbMATH Open document ID: 1016.32015
DOI: 10.1007/BF02392783
author: Ahmed Zeriahi
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: Let \(Y\) be a local and irreducible analytic subset of dimension \(n\) in \(\mathbb C^N\). Let \(\mathcal L(Y)\) denote the closure in \(L^1_{\text{loc}}(Y)\) of the class \(\mathcal L _Y\) of plurisubharmonic functions of ``restricted logarithmic growth'' on \(Y\) defined by \[ \mathcal L_Y:=\text{PSH}(Y)\cap \{v|Y: v\in \mathcal L(\mathbb C^N)\}. \] Here PSH\((Y)\) denotes the class of all functions \(v:Y\to [-\infty,+\infty)\) that are locally bounded above on \(Y\) and plurisubharmonic on \(Y_{\text{reg}}\) the set of regular points of \(Y\); \(\mathcal L(\mathbb C^N)\) is the usual class of plurisubharmonic functions of \(N\) complex variables with logarithmic growth at infinity. Let \(\mathcal A(Y) = \bigcup_{d\geq 1}\mathcal A_d(Y)\) be the graded algebra of holomorphic functions on \(Y\), where \[ \mathcal A_d(Y):= \{f|Y:f \text{ is a polynomial of degree at most } d\},\quad d\geq 1. \] Let \(U\) be a coordinate neighborhood of a point \(a\in Y_{\text{reg}}\) such that there exists a homeomorphism \(\Phi:\overline U \mapsto \overline {\Delta}^n\) onto the closed unit polydisc \(\overline {\Delta}^n\) which is holomorphic on \(U\), and \(\Phi (a) =0\). Put \(\overline U_s: \Phi^{-1}(\overline {\Delta}^n_s)\), where \({\Delta}^n_s\) is the open polydisc with center \(0\) and radius \(0<s<1\). Let \(w\in\text{PSH}(U)\). The number \(\nu(w;a):= \lim_{s\to 0} \frac {\sup_{\overline{U}_s}w }{\log s}\) does not depend on the choice of the coordinate system and is called Lelong number of \(w\) at the point \(a\). Put \(\nu_{\mathcal L(Y)}(a):=\sup\{\nu (w,a); w\in \mathcal L(Y)\}\). Then the number \(\nu(Y):= \inf\{\nu_{\mathcal L(Y)}(a); a\in Y_{\text{reg}}\} \) is called minimal graded Lelong number of \(Y\). For each point \(a\in Y_{\text{reg}}\) define the number \(\mu_{\mathcal A(Y)}(a):= \sup \{m_f(a)/d\); \(f\in \mathcal A_d(Y)\), \(f\neq 0\), \(d\geq 1\}\), where \(m_f(a)\) denotes the order of vanishing of a holomorphic function \(f\neq 0\) at \(a\). The n…
Mathematics Subject Classification ID: 32U25
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 32B99
Mathematics Subject Classification ID: 32U15
zbMATH DE Number: 1541220
zbMATH Keywords: global extremal function
zbMATH Keywords: Lelong class
zbMATH Keywords: Lelong number
zbMATH Keywords: algebraicity criterion
reviewed by: Józef Siciak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1541220