Finite group quotient of complex manifolds. Let G be a finite group, acting on a complex manifold M holomorphically. Prove that the quotient M/G admits a structure of a complex variety in such a way that the natural map M → M/G is holomorphic.
The followings are my ideas to recruit the construction from algebraic geometry. I am not one hundred per cent sure that it will work.
In step one, I want to prove the analogue of Noether's theorem, namely the finite generation of C-algebras, for the germs of holomorphic functions. I am not sure how to do that. But I guess that there should be a short exact sequence from C-algebras to the germs.
In step two, I have doubts about how to associate the invariant germ to the quotient variety.
Actually, when the dimension is one. Let C be a smooth 1-dimensional affine variety, equipped with an action of a finite group G. Prove that C/G is also smooth.
 A: You are asking two question: General quotients and quotients in dimension 1.  


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*The following is proven in 


H. Cartan, Quotient d'un espace analytique par un groupe d'automorphismes. 
A symposium in honor of S. Lefschetz, Algebraic geometry and topology. pp. 90–102. Princeton University Press, Princeton, N. J. 1957.
Theorem. Suppose that $X$ is a complex manifold, $G$ is a group of holomorphic automorphisms of $X$ acting properly discontinuously on $X$ then the quotient $X/G$ has structure of an analytic space such that the quotient map $X\to X/G$ is analytic. 
The issue is essentially local: Given a point $x\in X$ with (finite) stabilizer $G_x< G$, then the action of $G_x$ near $x$ can be locally holomorphically linearized. Once this is done, one can quote a result from algebraic geometry. 
Cartan's theorem was generalized in 
H. Holmann,  Quotientenräume komplexer Mannigfaltigkeiten nach komplexen Lieschen Automorphismengruppen. Math. Ann. 139 (1960), 383–402 (1960)
for non-properly discontinuous holomorphic actions on general analytic spaces.  


*Suppose now that $X$ is a Riemann surface. 


Then for each $x\in X$, the group $G_x$ is cyclic of order $n$ and, after a local linearization, the action near $z$ is generated by an order $n$ rotation in ${\mathbb C}$, $z\mapsto e^{2\pi i/n}z$. The function $f(z)=z^n$ is $G_x$-invariant and descends to a homeomorphism $h: {\mathbb C}/G_x\to {\mathbb C}$, which is holomorphic away from the projection of $0$. The map $h$ (restricted to a small neighborhood of the image $\bar{x}$ of $x$ in $X/G$) gives a local holomorphic chart near $\bar{x}$. It then an exercise to verify that such local charts define a Riemann surface structure on $X/G$ such that the quotient map $q: X\to X/G$ is holomorphic. 
This is probably described in more detail in some classical books on Riemann surfaces such as Ahlfors and Sario.     
