I am trying to solve the following exercise from Voisin's Hodge Theory and Complex Algebraic Geometry:
Exercise 2: Let $X$ be compact complex curve (Riemann surface) and let $f$ be a non-constant meromorphic function on $X$.
(a) Show that we can view $f$ as a holomorphic map from $X$ to $\mathbb{P}^1$
For this, I think we can just send the poles of $f$ to the point at infinity.
(b) Let $t$ be a point of $\mathbb{P}^1$. Let $D$ be a disk in $\mathbb{P}^1$ centred at $t$. Using exercise 3(c) of chapter 1, show that $$n_t=\int_{f^{-1}(\partial D)}\frac{1}{2i\pi}\frac{df}{f-t}$$ is equal to the number of points of the fibre $f^{-1}(t)$, counted with the multiplicities given by the order of vanishing $k_x(f-t)$.
The exercise mentioned says that $k_x(f)=\int_D \frac{1}{2i\pi}\frac{df}{f}$, where $D$ is a disk that contains only one pole or zero, namely $x$. I think the exercise will be solved if I can somehow replace $f^{-1}(\partial D)$ by small circles around the fibres of $t$, but am not sure how this could be done.
(c) Show that $n_t$ is independent of $t$
A few questions and ideas: what can we say about $f^{-1}(\partial D)$? I think it should be some 1-dimensional submanifold of $X$ and then maybe one can use something like Stokes' theorem to show that it is homologous to all the small circles around the fibres. But I am not sure whether it is connected and what exactly it bounds (maybe the Jordan curve theorem can be used?). Also, do inverse images commute with boundaries, i.e. $f^{-1}(\partial D)=\partial f^{-1}(D)$? Lastly, what would be a way to show independence of $t$? Intuitively, a small perturbation shouldn't change the integral - perhaps one can differentiate w.r.t. t and get $0$?