Meromorphic quotients for some holomorphic G-actions (version 2 bis)
Abstract
Using mainly tools from [B.13] and [B.15] we try to make a first step to obtain a " Transcendental Geometric Invariant Theory " , that is to say to study conditions for the existence of " meromorphic quotients " for a holomorphic actions of a complex Lie group G on a reduced complex space X. In this article we give necessary and sufficient conditions [H.1] [H.2] and [H.3] on the G−orbits' configuration in X in order that a holomorphic action of a connected complex Lie group G on a reduced complex space X admits a strongly quasi-proper meromorphic quotient. Under these conditions a canonical (minimal) such quotient exists and it factorizes in a canonical way any G−invariant meromorphic map defined on X. In order to show how these conditions can be used, we apply this characterization to obtain that, when G = K.B with B a closed complex subgroup of G and K a real compact subgroup of G, the existence of a strongly quasi-proper meromorphic quotient for the B−action implies the existence of a strongly quasi-proper mero-morphic quotient for the G−action on X, assuming moreover that the action of B on X satisfies the condition [H.1str] on a G−invariant dense subset; we prove also that this last condition is automatically satisfied for G when K normalizes B and when [H.1str] [H.2] and [H.3] are satisfied for B. We also give a similar result when the connected complex Lie group has the form G = K.A.K where A is a closed connected complex subgroup and K is a compact (real) subgroup assuming that the A−action satisfies the hypothesis [H.1str] on a G−invariant open set Ω 1 , the hypothesis [H.2] on a G−invariant open set Ω 0 ⊂ Ω 1 and [H.3]. We prove the existence of a natural holomorphic map between the two meromorphic quotients of X for the actions of B and G (resp. of A and G) when they exist and we discuss the properness of this map.
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