While reading the paper Seifert Fibred Homology 3-Spheres and the Yang-Mills Equations on Riemann Surfaces with Marked points by M. Furuta and B. Steer, I stumbled upon the following statement:
Any compact orbifold Riemann surface, with $n\geq 3$ singular points or $n=2$ and $\alpha_1=\alpha_2$ if the genus $g$ is zero, has the form $N/D$, where $N$ is a compact Riemann surface and $D$ is a finite group of diffeomorphisms.
However, there is no comment why this should be true. The paper cites the article of P. Scott on the geometrization conjecture, but on that paper the only thing that is assured is that if this conditions are satisfied, then the orbifold is a global quotient, by some unspecified manifold. However,
How can we deduce that this manifold must be a Riemann surface?
This also leads me to ask,
If $N$ is a compact Riemann surface and $D$ is a finite group of diffeomorphisms, is the quotient $N/D$ a compact Riemann orbifold surface?
Thanks in advance for your answers.