Potential theory on complex projective space: Application to characterization of pluripolar sets and growth of analytic varieties
type: Item
instance of: scholarly article
title: Potential theory on complex projective space: Application to characterization of pluripolar sets and growth of analytic varieties
zbMATH Open document ID: 0583.32045
author: Robert Molzon
published in: Illinois Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The purpose of this paper is to characterize the non-locally pluripolar subsets of the projective space \(P({\mathbb{C}}^ n)\) in terms of a capacity on the homogeneous polynomials. If f is a homogeneous polynomial of degree d on \(P({\mathbb{C}}^ n)\) and \(f^*\) is the dual polynomial, the author sets \(K_ d(z,f)=\log \| z\|^ d| f^*(z)|^{- 1}.\) Then basically, the non-locally pluripolar subsets E are those such that, if \[ U_ d(E)=\inf_{\mu \geq 0, \int d\mu =1}\sup_{z\in P({\mathbb{C}}^ n)}\int_{f}K_ d(z,f) d\mu (f), \] then \(\limsup_{d\to \infty}U_ d(E)>0.\) This is closely related to the projective capacity of \textit{H. Alexander} [Recent developments in several complex variables, Proc. Conf., Princeton Univ. 1979, Ann. Math. Stud. 100, 3-27 (1981; Zbl 0494.32001)]. This capacity can be used to obtain average Bezout type theorems for algebraic hypersurfaces.
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 31C15
zbMATH DE Number: 3933467
zbMATH Keywords: growth of analytic varieties
zbMATH Keywords: potential theory
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: locally pluripolar subsets of the projective space
zbMATH Keywords: projective capacity
zbMATH Keywords: Bezout type theorems
reviewed by: Lawrence Gruman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3933467