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!:)

  • $\begingroup$ How did you assert that the Laurent series of $f$ around 0 has only nonnegative exponents? $\endgroup$ – user319128 Feb 7 '17 at 0:08

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 !

  • $\begingroup$ You're welcome ! $\endgroup$ – user171326 Jul 17 '16 at 22:47

Your reasoning is sound! The only (minor) complaints I have are regarding the formatting. For instance, you sometimes forget a differential at the end of a local expression for $\omega$. You should write, for instance,

In the other chart we have $\omega=(−1/z^2) d(1/z)$.

Or, later:

In the other chart we have $\omega=(-1/z) d(1/z)$.

Also, note that $\omega$ is not the same thing as $w$ (assuming you're using Miranda's book). But, regarding the actual mathematics, everything is perfectly fine.

  • $\begingroup$ Thanks for the answer! I just forgot this. And yes, you are right. I am using Rick Mirandas Bock:) Have a nice day. $\endgroup$ – Marc Jul 17 '16 at 17:20

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