**Background:** I've been trying to read

Baily, Walter L., Jr. The decomposition theorem for V-manifolds. Amer. J. Math. 78 1956

and I have problems with the language used in the paper. Firstly I am not familiar with V-manifolds (or orbifolds). Secondly I want to apply some of the theorems in the paper to modular forms of real weight and I don't know much about the 'modular forms as sections of a line bundle on $\Gamma/\mathbb{H}$ point of view, for $\Gamma$ a Fuchsian group of first kind and $\mathbb{H}$ the upper halfplane.

**My main question is:** Can we view modular forms of arbitrary weight $r$ and multiplier system $v$ as sections of a line bundle $L_{r,v}$ on $\Gamma/\mathbb{H}^*$ viewed as an orbifold?

My main reference for this are Milne's lecture notes on modular forms (Proposition 4.25) where he explains briefly a factor of automorphy (includes real weight) corresponds uniquely to a line bundle $L$ on $\Gamma/\mathbb{H}$ and an isomorphism $\mathbb{H}\times \mathbb{C}\cong p^*(L)=\{(h,l)\in \mathbb{H}\times L|p(h)=\pi(L)\}$, where $p$ is the quotient map $\mathbb{H}\rightarrow \Gamma/\mathbb{H}$ and $\pi$ is the map $L\rightarrow\Gamma/\mathbb{H}$. He then says that for even integer weight $2k$ one can extend this line bundle to a line bundle $L^*$ on $\Gamma/\mathbb{H}^*$ and sections of $L^*$ are modular forms of weight $2k$. Is there any reason for restricting the weight here? To me it seems that one can do this with any factor of automorphy.

The other restriction he makes is that $\Gamma$ should act freely on $\mathbb{H}$. Can one ommit this condition by talking about line bundles on orbifolds the orbifold $\Gamma/\mathbb{H}$ or is there a caveat that I am missing?