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Apr 21, 2017 at 6:15 history edited HLC CC BY-SA 3.0
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Apr 21, 2017 at 5:52 comment added nfdc23 By the definition you gave for "locally proper", $X$ is covered by $G$-stable open subsets $U_i$ on which the action is proper, so likewise on all $G$-stable open subsets of each $U_i$. Thus, the Bourbaki result yields complex manifolds $U_i/G$ and $(U_i \cap U_j)/G$ with the natural maps $(U_i \cap U_j)/G \rightarrow U_i/G$ open immersions satisfying the triple overlap condition to make a gluing, and one checks this gives the desired $X/G$ (with the desired properties). Am I overlooking something?
Apr 21, 2017 at 5:42 answer added Peter Heinig timeline score: 2
Apr 21, 2017 at 5:23 comment added HLC @nfdc23 But the action here is only locally proper. Will that proposition still apply?
Apr 21, 2017 at 5:10 comment added nfdc23 In analytic geometry over $\mathbf{R}$ and $\mathbf{C}$ there are no such "scheme-theoretic" issues, informally due to being in characteristic 0. For actual rigorous proofs to justify that informal idea (and in particular to address $X/G$ being naturally a complex manifold in the setting of interest), see: Bourbaki, Lie Groups and Lie Algebras, Chapter III, section 1.5, Corollary to Prop. 9 and then Prop. 10 (this all applies in the $C^{\infty}$-category, as well as in the real-analytic and complex-analytic categories).
Apr 21, 2017 at 3:31 history asked HLC CC BY-SA 3.0