Plurisubharmonic functions with singularities and affine invariants for finite sets in \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: Plurisubharmonic functions with singularities and affine invariants for finite sets in \(\mathbb C^n\)
zbMATH Open document ID: 1014.32022
author: Dan Coman
author: Stéphanie Nivoche
published in: Mathematische Annalen
publication date: 2002-05-29T00:00:00Z
review text: The authors associate two numbers \(\gamma(S)\geq \widetilde \gamma(S)\) to every finite subset \(S\) of \(\mathbb C^n\) and relate them to algebraic geometric properties of the set \(S\). These numbers are defined as follows: For every \(u\in \text{PSH}(\mathbb C^n)\) let \(\gamma_u=\limsup_{\|z\|\to \infty}(u(z)/\log\|z\|)\), \(\widetilde\gamma(S)=\inf\{\gamma_u: u\in \widetilde E(S)\}\), and \(\gamma(S)=\inf\{\gamma_u: u\in E(S)\}\), where \(\widetilde E(S)\) consists of all \(u\in \text{PSH}(\mathbb C^n)\cap L^\infty_{loc}(\mathbb C^n\setminus S)\) such that \(\gamma_u\in (0,+\infty)\) and \(u(z)-\log\|z-p\|=O(1)\) as \(z\to p\) for all \(p\in S\), and \(E(S)\) is the subclass of \(\widetilde E(S)\) consisting of \(u\) such that \((dd^cu)^n=\sum_{p\in S}\delta_p\), where \(\delta_p\) is the Dirac measure at the point \(p\). Let me only mention two of the many interesting results proved in the paper. If \(|S|\) denotes the number of points in \(S\), then \(|S|^{1/n}\leq\widetilde\gamma(S)\leq\gamma(S)\leq |S|\), and \(\widetilde\gamma(S)=|S|\) if and only if \(S\) is contained in a complex line. If \(V\subset\mathbb C^n\) is an algebraic variety of pure dimension \(m\), then \(|S\cap V|\leq \sum_{p\in S\cap V}\nu(V,p)\leq \widetilde \gamma(S)^m\text{deg} V\), where \(\nu(V,p)\) is the Lelong number of \(V\) at the point \(p\). Many interesting examples are treated in the case \(n=2\). In all the examples \(\widetilde\gamma(S)=\gamma(S)\) and it remains an open question if this equality always holds.
reviewed by: Ragnar Sigurdsson
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 1747292
zbMATH Keywords: pluricomplex Green function
zbMATH Keywords: Monge-Ampère operator
MaRDI profile type: Publication
full work available at URL: s002080100272
OpenAlex ID: W2090844727
DOI: 10.1007/S002080100272
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description: scientific article; zbMATH DE number 1747292