I am struggling with the following exercise of Rick Miranda's "Algebraic Curves and Riemann Surfaces" (page 111):
Let $X$ be the Riemann Sphere with local coordinate $z$ in one chart and $w=1/z$ in another chart. Let $\omega$ be a meromorphic $1$-form on $X$. Show that if $\omega=f(z)\,dz$ in the coordinate $z$, then $f$ must be a rational function of $z$.
I have unfortunately no idea how I should begin the proof (since I am new to this topic). Can someone give me a hint?
Edit: The transition map is $T:z\rightarrow 1/z$. We have $\omega=f(z)dz$ in the coordinate $z$. Then we know that $\omega$ transforms into $\omega_2=f_2(w)\,dw$ as follows: $f_2(w)=f(T(w))T'(w)=f(1/w)(-1/w^2)$, but how does this help?
Best Answer
Lemma. Every holomorphic function on a compact Riemann surface is constant.
Proof. Let $f:X \to Y$ be a nonconstant holomorphic mapping between (connected) Riemann surfaces, with $X$ compact. Then $f(X)$ is compact, therefore closed. But it is also open by the open mapping theorem. Therefore by connectedness $Y = f(X)$, and $Y$ is also compact. As $\mathbb{C}$ isn't compact, the claim follows. $\square$
Let's use the coordinate patches $(\mathbb{C},z)$ and $(\mathbb{C}^* \cup \{ \infty \}, 1/z )$. Since $f$ is meromorphic, it has only finitely many poles. We may assume that $\infty$ is not one of them (if it is the case, replace $f$ by $1/f$). Let $a_1,\ldots,a_n$ denote the poles of $f$. At the $i$th pole, $f$ has a principal part $$p_i(z) = \sum_{j=-k_i}^{-1} c_{ij}(z-a_i)^j$$ for some finite $\{k_i\}_{i=1}^n$.
Removing those yields the function $g = f - (p_1 + \cdots + p_n)$ which is holomorphic on all the Riemann sphere. But by the Lemma above, such a function must be constant. Therefore $f = g + (p_1 + \cdots + p_n)$ is rational.