Let $G$ be group of $n^{th}$ roots of unity. Assume that $G=<\alpha>$, where $\alpha ^n=1$. Now we define a group action, $G$ act on $\mathbb{C}$, by $(x,z)\mapsto xz$. The Quotient space $\mathbb{C}/G$ is homeomorphic to a cone, that is $\mathbb{C}/G$ is homeomorphic to cone of $S^1 \times [0,\infty)$. The open sets of the $\mathbb{C}/G$ is $\hat{U}=\{Gz : z \in U\}$, where $U$ is an open set in $\mathbb{C}$. So, one can give an complex chart $\phi :\hat{U}\rightarrow U$, by $\phi(Gz)=z$.
My Question, is it possible to give a complex structure $\mathbb{C}/G$? Even it is not a smooth manifolds, I also confused where I was wrong to define the complex charts on $\mathbb{C}/G$.
Thanks!