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Let $X$ be a complex manifold with an action of $G=GL(V)$$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.)

Let $X$ be a complex manifold with an action of $G=GL(V)$ 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.)

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.)

Source Link
HLC
HLC
  • 297
  • 1
  • 5

Quotient of complex manifold by a free and locally proper action (difficulty with reading German)

Let $X$ be a complex manifold with an action of $G=GL(V)$ 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.)