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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?

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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.

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  • $\begingroup$ Hello, thanks for the answer. One question: Where you used the two charts? I think only in the step where you wrote that we can assume that $\infty$ is not a pole, right? $\endgroup$ – Marc Jul 5 '16 at 17:10
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    $\begingroup$ @Marc The charts are used implicitly whenever we write down a function using coordinates. For instance, all the functions $p_i$ are described in terms of the chart $(\mathbb{C},z)$. $\endgroup$ – Alex Provost Jul 5 '16 at 17:15
  • $\begingroup$ Thanks :) The second part of the exercise (from the book) is: "Show further that there are no nonzero holomorphic 1-forms on $\mathbb C_{\infty}$." It seems familiar to your lemma above. In our setting we have two differential forms $\omega_1=f_1(z)dz$ and $\omega_2=f_2(z)dz$, one for each of the two charts, such that they are compatible. Now we have to show that the function $F:\mathbb C_{\infty}\rightarrow \mathbb C$ defined as $f_1(z)$ in the first chart and $f_2(z)$ in the second one must be constant, right? This should be exactly your lemma? $\endgroup$ – Marc Jul 5 '16 at 17:52
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EDIT: By multiplying by an appropriate polynomial, we may assume that $\omega$ has poles (at most) at $0$ and $\infty$.

On $\Bbb C-\{0\}$ you now have holomorphic functions $f$ and $g$ (your $f_2$) with $$z^2f(z)=-g(1/z).$$ Since $f$ and $g$ have at worst poles at $0$, this equation tells us that each of their Laurent series has only finitely many nonzero terms.

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  • $\begingroup$ Thanks for the answer. I got one question: Why can they only have a pole at zero? I can't see why $z=2$ is a pole of $f$ and $z=1/2$ is a pole of $g$ can not occur. $\endgroup$ – Marc Jul 5 '16 at 17:12
  • $\begingroup$ Marc, you're right, of course. I forgot the first sentence. Please see the edit. $\endgroup$ – Ted Shifrin Jul 6 '16 at 3:54

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