## 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

Zbl 0803.53042
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