I am referring to page 300 of Okonek et al. book "Vector Bundles on Complex Projective Spaces" (1988.)

Let $G=GL_n(\mathbb{C})$ act holomorphically and freely on a complex manifold $X$. Define the mapping

$\gamma:X\times G\longrightarrow X\times X$

by sending $(x,g)$ to $(x,gx)$. Given that $\gamma$ is an isomorphism onto a closed analytic subspace of $X\times X$,

Why is $X/G$ also a complex manifold?

Why is $X\rightarrow X/G$ a $G$-principal bundle (in the complex analytic category)?

Apparently, this result can be found in the two papers:

• Holmann, H.: Komplexe Raume mit komplexen Transformationsgruppen. Math Ann. 150, 1963 (p.359),

• Holmann, H.: Quotienten komplexer Raume. Math Ann 142, 1961 (p.433.)

However, Germans is so hard for me and I would like to understand at least the rough idea. My feeling is that this should be something standard nowadays?

I know the condition on the image of $\gamma$ is basically to ensure separatedness/Hausdorffness of $X/G$ so maybe the first question is not hard but for the second question I really have no idea.