Timeline for Quotient of complex manifold by a free and locally proper action (difficulty with reading German)
Current License: CC BY-SA 3.0
6 events
when toggle format | what | by | license | comment | |
---|---|---|---|---|---|
Apr 21, 2017 at 6:15 | history | edited | HLC | CC BY-SA 3.0 |
added 11 characters in body
|
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 |