So I'm trying to read Feng Luo's paper "Monodromy Groups of Projective Structures on Punctured Surfaces", but I'm stumped on the first page. A projective structure on a Riemann surface is defined to be an (equivalence class of) atlas of holomorphic coordinate charts whose composition maps are restrictions of projective transformations - in other words, $S = \bigcup_a U_a$ with $\phi_a: U_a \rightarrow V_a \subset \mathbb{C},$ with $\phi_b \circ \phi_a^{-1}$ a Mobius map.
On the other hand, a holomorphic quadratic differential is something that can locally be written as $f(z) dz^2$, such that $f$ is holomorphic if $z$ is a holomorphic coordinate - so, with a holomorphic atlas $U_a$, $\phi_a = z_a$, we have a collection of functions $f_a : V_a \rightarrow \mathbb{C}$ such that $$\frac{f_a}{f_b} = \left( \frac{dz_b}{dz_a} \right)^2 $$ for all $a,b$.
The statement is then that the set of all projective structures on a Riemann surface is an affine space over the vector space of all holomorphic quadratic differentials. So, given a projective structure $\lbrace \phi_a \rbrace_a,$ there is a way to "add" any holomorphic quadratic differential $\lbrace f_a \rbrace_a$ to it, to obtain a new projective structure, such that any two projective structures differ by a holomorphic quadratic differential in this way.
Can anyone help me to see how this works?