Exponential estimates for plurisubharmonic functions
type: Item
instance of: scholarly article
title: Exponential estimates for plurisubharmonic functions
zbMATH Open document ID: 1211.32021
author: Tien-Cuong Dinh
author: Viêt Anh Nguyên
author: Nessim Sibony
published in: Journal of Differential Geometry
publication date: 2010-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/0801.1983
full work available at URL: https://projecteuclid.org/euclid.jdg/1279114298
review text: Let \(X\) be a complex manifold of dimension \(k\). A positive measure \(\mu\) on \(X\) is said to be locally moderate if, for any open set \(U\subset X\), any compact set \(K\subset U\), and any compact family \(\mathcal F\) of plurisubharmonic functions (p.s.h. for short) on \(U\), there are constants \(\alpha>0\) and \(c>0\) such that \(\int_Ke^{-\alpha\psi}\,d\mu\leq c\)\ for \(\psi\in\mathcal F\). Fix a smooth strictly positive (1.1)-form \(\omega\) on \(X\). A positive closed current \(S\) of bidegree \((p,p)\) on \(X\) is said to be locally moderate if its trace measure \(\sigma_S\):=\(S\wedge\omega^{k-p}\) is locally moderate. If \(u\) is a continuous real-valued function on the support of \(S\), then the multiplication \(uS\) defines a current on \(X\), so the current \(dd^c(uS)\) is also well-defined. The function \(u\) is \(S\)-p.s.h. if \(dd^c(uS)\) is a positive current. One of the main results of the paper is: Theorem 1.1. Let \(S\) be a locally moderate positive closed \((p,p)\)-current on \(X\). If \(u\) is a Hölder continuous \(S\)-p.s.h. function, then \(dd^c(uS)\) is locally moderate. In particular, if \(R\) is a positive closed (1,1)-current with Hölder continuous local potentials, then \(R\wedge S\) is locally moderate. This yields the following Corollary 1.2. Let \(u\) be a Hölder continuous p.s.h. function on \(X\). Then the Monge-Ampère currents \((dd^cu)^p\), \(1\leq p\leq k\), are locally moderate. The authors apply their result to dynamics. Given a non-invertible holomorphic endomorphism \(f\) of the projective space \(\mathbb P^k\) that is induced by a homogeneous polynomial endomorphism of degree \(d\geq 2\) on \(\mathbb C^{k+1}\), let \(T:=\lim_{n\to\infty}d^{-n}(f^n)^*(\omega_{FS})\), where \(\omega_{FS}\) is the normalized Fubini-Study form. Then \(T\) is a positive closed (1,1)-current of mass 1 on \(\mathbb P^k\) and one can define \(T^p\):=\(T\wedge\dots\wedge T\) (\(p\) times). The currents \(T^p\) are the Green currents associate…
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 37A50
zbMATH DE Number: 5774388
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: Monge-Ampère measures
zbMATH Keywords: stochastic dynamics
Wikidata QID: Q115171286
MaRDI profile type: Publication
OpenAlex ID: W1521717632
arXiv ID: 0801.1983
DOI: 10.4310/JDG/1279114298
description: scientific article