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Invariant meromorphic functions on Stein spaces. (English. French summary) Zbl 1270.32005

Let \(G\) be a complex reductive Lie group and let \(X\) be a Stein \(G\)-space, then following D. M. Snow [Math. Ann. 259, 79–97 (1982; Zbl 0509.32021)] and P. Heinzer [Math. Ann. 289, No.4, 631–662 (1991; Zbl 0728.32010)], the geometric quotient \(\pi: X\to Q\) for the action of \(G\) exists; furthermore \(Q\) is Stein. However, \(Q\) is not algebraic in general. However if \(X\) is, furthermore, an \(H\)-irreducible Stein \(G\)-space, where \(H\) is an algebraic subgroup of \(G\), then in Section 6.3, the authors find a \(G\)-equivariant map \(\phi: X\to Y\) into an \(H\)-irreducible affine variety \(Y\). In view of the Theorem by M. Rosenlicht [ Am. J. Math. 78, 401–443 (1956; Zbl 0073.37601)] (Theorem 2.3), which is the main motivation of this paper, the authors show that (Section 6) there exists an algebraically Zariski-open \(H\)-irreducible subset \(\Omega_Y\) of \(Y\) which admits an algebraic geometric quotient \(p_Y:\Omega_Y\to \Omega_Y/H\).
The above-mentioned preparations are the main ingredients of the following main result of the paper:
Main Theorem. Let \(H<G\) be an algebraic subgroup of a complex-reductive Lie group \(G\) and let \(X\) be a Stein \(G\)-space. Then there exist an \(H\)-invariant Zariski-open dense subset \(\Omega\) in \(X\) and a holomorphic map \(p:\Omega\to Q\) to a Stein space \(Q\) such that
(1) \(p\) is a geometric quotient for the \(H\)-action on \(\Omega\),
(2) \(p\) is universal with respect to \(H\)-stable analytic subsets of \(\Omega\),
(3) \(p\) is a submersion and realizes \(\Omega\) as a topological fibre bundle over \(Q\),
(4) \(p\) extends to a weakly meromorphic map from \(X\) to \(Q\),
(5) \(p\) induces an isomorphism between the fields \({\mathcal M}_X(X)^H\) and \({\mathcal M}_Q(Q)\) of meromorphic functions, and
(6) the \(H\)-invariant meromorphic functions on \(X\) separate the \(H\)-orbits in \(\Omega\).
The last 3 properties in the Main Theorem are carried out in Section 7.
Reviewer: Hans Weber (Udine)

MSC:

32M05 Complex Lie groups, group actions on complex spaces
32Q28 Stein manifolds
32A20 Meromorphic functions of several complex variables
14L30 Group actions on varieties or schemes (quotients)
22E46 Semisimple Lie groups and their representations
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References:

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