Solutions to degenerate complex Hessian equations
type: Item
instance of: scholarly article
title: Solutions to degenerate complex Hessian equations
zbMATH Open document ID: 1288.53072
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2013-12-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.2436
reviewed by: Maria Falcitelli
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6236655
zbMATH Keywords: Kähler manifold
zbMATH Keywords: \((\omega,m)\)-subharmonic
zbMATH Keywords: Hessian operator
zbMATH Keywords: weak solution
MaRDI profile type: Publication
OpenAlex ID: W2964312849
arXiv ID: 1202.2436
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DOI: 10.1016/J.MATPUR.2013.03.002
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author: Hoang Chinh Lu
review text: Given a compact Kähler manifold \((X,\omega)\) of complex dimension \(n\), let \(dd^c\) be the usual real differential operators. Fixed an integer \(m\), \(1\leq m\leq n\), one studies the degenerate complex Hessian equations of the form NEWLINE\[NEWLINE(\omega+ dd^c\varphi)^m\wedge \omega^{n-m}= F(x,\varphi)\omega^n,\tag{1}NEWLINE\]NEWLINE where the density \(F: X\times\mathbb{R}\to \mathbb{R}_+\) satisfies suitable conditions. Equation (1) is a generalization of both Laplace and Monge-Ampère equations.NEWLINENEWLINE Firstly the author introduces the class of \((\omega,m)\)-subharmonic functions and the concepts of \((\omega,m)\)-capacity for Borel subsets of \(X\) and of quasi-uniform convergence. This allows to define a suitable class of \((\omega,m)\)-subharmonic functions on which the complex Hessian operator is well-defined and continuous under quasi-uniform convergence. This class is denoted by \({\mathcal P}_m(X,\omega)\). Under suitable conditions for \(F\), the author proves existence and unicity, up to an additive constant, of a solution \(\varphi\) of (1) which belongs to \(\mathbb{P}_m(X,\omega)\cap{\mathcal C}^0(X)\). In particular, one obtains the Hölder continuity of the solution when \(X\) is homogeneous and \(\omega\) invariant under the Lie group action.
description: scientific article; zbMATH DE number 6236655