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Given the finite group $G_n\subset \mathbb{C}^\times$ of the $n$-th roots of 1, I know that, even if the action of $G_n$ on $\mathbb{C}$ is not properly discontinuous, $\mathbb{C}/G_n$ has a natural structure of complex manifold.

How are these charts defined on $\mathbb{C}/G_n$?

Shaun
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Watanabe
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    An action of a finite group is always properly discontinuous, isn't it? – freakish Feb 25 '20 at 14:13
  • It is properly discontinuous on $\mathbb{C}^\times$ but not on $\mathbb{C}$. In this case, for every open neighborhood of $0\in \mathbb{C}$, we have that $0\in g_i.U\cap U={g_iz:z\in U}\cap U\neq \emptyset$. – Watanabe Feb 25 '20 at 14:22
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    The general theorem uses a slightly looser notion of proper discontinuity, which does indeed apply to your example. Namely: Given a group $G$ acting on $\mathbb C$, if each element of $G$ acts as a biholomorphism of $\mathbb C$, and if each $z \in \mathbb C$ has a neighborhood $U \subset \mathbb C$ such that the set ${g \in G \mid g \cdot U \cap U \ne \emptyset}$ is finite, then $\mathbb C / G$ has a natural structure of a complex manifold. – Lee Mosher Feb 25 '20 at 16:22
  • https://math.stackexchange.com/questions/160170/riemann-surface-and-discontinuous-group-action?rq=1 and https://math.stackexchange.com/questions/3438151/finite-group-quotient-of-complex-manifolds/3499792#3499792 – Moishe Kohan Feb 25 '20 at 17:58

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