3
$\begingroup$

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!:)

$\endgroup$
  • $\begingroup$ How did you assert that the Laurent series of $f$ around 0 has only nonnegative exponents? $\endgroup$ – user319128 Feb 7 '17 at 0:08
2
$\begingroup$

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 !

$\endgroup$
  • $\begingroup$ You're welcome ! $\endgroup$ – user171326 Jul 17 '16 at 22:47
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.