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Complexification of proper Hamiltonian \(G\)–spaces. (English) Zbl 1072.32006

For a given manifold \(M\) a Stein tube is a triple \(\left( X,i,\sigma \right) \) with \(X\) a Stein manifold, \(i:M\hookrightarrow X\) a real analytic totally real embedding and \(\sigma :X\rightarrow X\) an antiholomorphic involution with \(\operatorname{Fix}\sigma =M\) and \(M\) a strong deformation retract. P. Heiner, A. T. Huckleberry and F. Loose proved [J. Reine Angew. Math. 455, 123-140 (1994; Zbl 0803.53042)] the following Stein complexification: Let \(\tau \) be a closed 2-form on \(M\). There exists a Stein tube and a Kähler form \(\omega \) on \(X\) such that \(i^{\ast }\omega =\tau \). The aim of present paper is to prove the following equivariant form of a previous result: Let \(G\) be a real Lie group with finitely many components acting smooth and properly on \(M\) and let \(\tau \) be a closed \(G\)-invariant 2-form on \(M\). There exists a Stein \(G\)-tube and a \(G\)-invariant Kähler form \(\omega \) on \(X\) with \(i^{\ast }\omega =\tau \). A Stein \(G\)-tube is a Stein tube with \(G\) acting properly on \(X\) by holomorphic transformations so that the embedding \(i\), the involution \(\sigma \) and the strong deformation retract are \(G\)-invariant. It is shown that the construction is canonical up to local \(G\)-equivariant diffeomorphisms around \(M\).

MSC:

32E10 Stein spaces
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
32M05 Complex Lie groups, group actions on complex spaces
57S20 Noncompact Lie groups of transformations

Citations:

Zbl 0803.53042
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