Local cyclic quotient of a Riemann surface

Let a finite group $H$ of conformal automorphisms of the Riemann sphere fix $p$. The derivative representation at $p$ embeds $H$ into ${\mathbb{C}}^{\times }$, so $H$ is cyclic. Averaging a local coordinate linearizes the action to $z\mapsto \zeta z$, and the quotient has coordinate $u=z^{|H|}$.

For a finite conformal group action, choose a disc around each point that meets only its stabilizer translates. The local cyclic quotient chart $z\mapsto z^e$, where $e$ is the stabilizer order, gives the orbit space a Riemann-surface structure and makes the quotient map holomorphic.