Kählerian potentials and convexity properties of the moment map
type: Item
instance of: scholarly article
title: Kählerian potentials and convexity properties of the moment map
zbMATH Open document ID: 0855.58025
author: P. Heinzner
author: Alan Huckleberry
published in: Inventiones Mathematicae
publication date: 1997-02-11T00:00:00Z
review text: The purpose of the paper is to prove convexity properties of \(\mu\) in the setting of non-compact Kählerian spaces. Let \(K\) be a compact Lie group acting in a Hamiltonian fashion on a connected compact symplectic manifold \(X\) and denote by \(\mu: X\to (\text{Lie } K)^*\) the corresponding equivariant moment map. Then the intersection \(\mu (X)_+\) of the image of the moment map \(\mu: X\to (\text{Lie } K)^*\) with a positive Weyl chamber \(t^*_+\) is convex. Thus \(\mu (X)= K\cdot \mu (X)_+\), where \(\mu (X)_+\) is a natural convex section. This was proved by \textit{M. F. Atiyah} [Bull. Lond. Math. Soc. 14, 1-15 (1982; Zbl 0482.58013)] and \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 491-513 (1982; Zbl 0503.58017)] in the abelian case, i.e., where \(K= T\) is a compact torus, by \textit{D. Mumford} [Appendix to: `A stratification of the null cone via the moment map', by \textit{L. Ness}, Am. J. Math. 106, 1281-1329 (1984; Zbl 0604.14006), pp. 1326-1329] for \(X\) projective algebraic with an integral Kählerian structure and \(K\) not necessarily abelian, and in its final form in the compact symplectic case by \textit{F. Kirwan} [Invent. Math. 77, 547-552 (1984; Zbl 0561.58016)]. Note that if \(X\) is not assumed to be compact then, by removing an appropriate \(K\)-invariant subset from \(X\), one can easily construct non-convex \(\mu\)-images from convex ones. Now, let \(X\) be an irreducible complex space endowed with a holomorphic action of a complex reductive group \(G\). Assume that a maximal compact subgroup \(K\) of \(G\) acts on \(X\) in a Hamiltonian fashion. Let \(X_{\max}:= \{x\in X\); \(\dim K\cdot x\) is maximal\}. Our first main result is the following Internal Convexity Theorem. \(\mu (X\max)_+\) is convex. Thus, whenever \(\mu (X)\) is closed it is likewise convex. For the second main result of this paper we impose a certain condition on the \(G\)-action. Let \(X\) be an irreducible complex \(G\)-space. The \(G\)-action on…
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 931984
zbMATH Keywords: Kählerian potentials
zbMATH Keywords: moment map
zbMATH Keywords: convexity
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: holomorphic group actions
MaRDI profile type: Publication
full work available at URL: s002220050089
OpenAlex ID: W2013531903
DOI: 10.1007/S002220050089
description: scientific article; zbMATH DE number 931984