Multiplicity of holomorphic functions
type: Item
instance of: scholarly article
title: Multiplicity of holomorphic functions
zbMATH Open document ID: 0948.32020
author: Charles Favre
published in: Mathematische Annalen
publication date: 2000-03-23T00:00:00Z
review text: Let \(f:P^2\to P^2\) be a rational map of maximal rank and of degree \(d\). Denote by \(I(f)\) the (finite) set of points of indeterminancy of \(f\). The map \(f\) is said to be algebraically stable if \(f^{-n}(I(f))\) is finite for any \(n\geq 0\). Under this condition, it is possible to define an invariant positive closed (1,1)-current \(T(f)=\lim_{n \to\infty} T_n(f)\), where \(T_n(f)= d^{-n}(f^n)^*w\) and \(w\) is the Fubini-Study form. Let \(\nu(T(f),z)\) denote the Lelong number of \(T(f)\) at \(z\) and \(I(f^\infty)= \cup_{n\geq 0}f^{-n} (I(f^n))\). Theorem 1. \(\nu(T(f),z)>0\) if and only if \(z\in I(f^\infty)\). Theorem 3. There exists a function \(\mu_\infty: I(f^\infty) \to\mathbb{R}^*_+\) to the positive reals of total mass \(\sum_{z\in I(f^\infty)}\mu_\infty (z)=1\) such that \[ \lim_{n\to \infty} T_n(f)\wedge T_n(f)=\sum_{z\in I(f^\infty)} \mu_\infty(z) \delta_z. \]
Mathematics Subject Classification ID: 32H50
zbMATH DE Number: 1423135
zbMATH Keywords: iteration
zbMATH Keywords: rational map
zbMATH Keywords: Lelong number
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
full work available at URL: s002080050016
OpenAlex ID: W2083713534
DOI: 10.1007/S002080050016
description: scientific article; zbMATH DE number 1423135