Let $X=\mathbb C_{\infty}$ be the Riemann sphere with the local coordinates $\{z\ ,1/z\}$. I want to show the following two statements:
i) There does not exist any non-vanishing holomorphic 1-form on $X$.
ii) Where are the poles and zeros of the meromorphic 1-forms $dz$ and $d/z$? Also determine their orders.
My attempt:
i) Let $w$ be a non-vanishing 1-form on $X$. Then we can write $w=f(z)dz$ in the coordinate $z$ for a holomorphic function $f$.
In the other chart we have then $w=f(\frac{1}{z})(-\frac{1}{z^2})d/z$. Now the laurent-series of $f$ around $0$ has only non-negative exponents, hence the above function has a pole in $0$, which is a contradiction to the assumption that $f$ is holomorphic.
ii) For $w=1 dz$: $1$ has no zeros or poles in $\mathbb C$.
Lets consider $\infty:$ In the other char we have $w=-1/z^2$ which has a pole of order two in zero, hence we have $ord_{\infty}w=-2$ and $ord_p(w)=0$ for $p\in\mathbb C$.
For $w=dz/z$: $1/z$ has only a pole (of order $1$) in zero. In the other chart we have $w=-1/z$ which has also just a pole of order 1 in zero. Hence we have $ord_0(w)=-1$, $ord_{\infty}(w)=-1$ and $ord_p(w)=0$ otherwise.
Since I am a beginner I would like if someone could check my solutions.
Thanks in advance!:)
Best Answer
First, for these kind of questions, the book of Rick Miranda, Algebraic curves and Riemann surfaces is really well done and have lot of details.
Remark : maybe for notation this is clearer to write $\frac{1}{z}$ as $w$ and a differential form as $\omega$ (for example, when you change of coordinate you can write $\omega' = -\frac{dw}{w}$ or something like this, instead of using $z$ for two different coordinates).
Else, your computations seems all ok to me !