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Let $X$ be a complex manifold with an action of $G=GL(n,\mathbb{C})$ which is free and locally proper (each point of $X$ has a $G$-invariant neighborhood on which $G$ acts properly.)

Satz 24 of the paper

H. Holmann, Quotienten komplexer Ra ̈ume, Math. Ann., 142 (1961), pp. 407–440

asserts that $X/G$ is a complex manifold.

Why is it true?

The biggest issue is, since I cannot read German, I don't know if by "free" he means set-theorecial free or scheme-theoretical free (the latter is in the sense of Mumford's GIT book.)

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    $\begingroup$ 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). $\endgroup$
    – nfdc23
    Commented Apr 21, 2017 at 5:10
  • $\begingroup$ @nfdc23 But the action here is only locally proper. Will that proposition still apply? $\endgroup$
    – HLC
    Commented Apr 21, 2017 at 5:23
  • $\begingroup$ 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? $\endgroup$
    – nfdc23
    Commented Apr 21, 2017 at 5:52

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Assuming that by $GL(V)$ you mean $GL(\mathbb{C})$, then this question concerns the action of a non-compact complex Lie group on a smooth complex manifold, such that

the action is free

the action is locally-proper

and the question is whether the quotient space $X/G$ can be equipped with a complex manifold structure.

This appears to be true, a roadmap to (though not the details of) a proof being given in Lemma 3.1 of

Miebach. C., Oeljeklaus, K. On proper $\mathbb{R}$-action on hyperbolic Stein manifolds. Documenta Mathematica 14 (2009) 673–689

which is freely accessible online.

Incidentally, it might be instructive if someone could provide an example of a locally-proper non-proper action of a complex Lie group on a complex manifold. All such examples I know are actions of real Lie groups on real manifolds.

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  • $\begingroup$ Yes, I mean $GL(n,\mathbb{C})$. Thank you for the paper. $\endgroup$
    – HLC
    Commented Apr 21, 2017 at 6:22

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