Kählerian potentials and convexity properties of the moment map
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 \(X\) is said to be regular if each \(G\)-orbit in \(X\) has an open dense neighborhood which is \(G\)-equivariantly the locally biholomorphic image of an irreducible complex \(G\)-space \(U\) such that the \(G\)-action on \(U\) is a restriction of an algebraic action. For example, if \(X\) is Stein or an algebraic \(G\)-variety, then the \(G\)-action is regular. Convexity Theorem. If the \(G\)-action is regular, then \(\mu (X)_+\) is convex and every \(\mu\)-fiber is connected. Our approach utilizes convexity properties of invariant plurisubharmonic functions. The essential role of plurisubharmonicity in the context of holomorphic group actions was underlined in Ann. Inst. Fourier 35, 59-97 (1985; Zbl 0563.32013) by \textit{J. J. Loeb}. This was in turn influenced by the work of Rothaus and Lassalle.
scientific article; zbMATH DE number 931984