2
$\begingroup$

What are global sections of holomorphic line bundles $\mathcal{O}(n)$ over Riemann sphere?

We define $\mathcal{O}(n)=\mathcal{O}(-1)^{\otimes (-n)}$ for $n<0$ and $\mathcal{O}(n)=(\mathcal{O}(-1)^{\otimes n})^*$ for $n>0$, $\mathcal{O}(0)=\mathbb{P}^1(\mathbb{C})\times\mathbb{C}$, where $\mathcal{O}(-1)$ is the tautological holomorphic line bundle over $\mathbb{P}^1(\mathbb{C})$.

I think I understand how to obtain global sections in the case $n=-1$: if $s$ is a section then when we compose it with $\mathcal{O}(-1)\hookrightarrow \mathbb{P}^1(\mathbb{C})\times\mathbb{C}^2\rightarrow\mathbb{C}^2$, we see that $s$ is constant $c$ and since $s(x)=(x,c)$ for all $x$, we have $c=0$.

But I don't know how to take tensors in.

$\endgroup$
2
$\begingroup$

$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Hol}{\mathscr{O}}$Cover the projective line by charts $U_{0} = \Cpx$ and $U_{1} = \Cpx$, with respective coordinates $z = 1/w$ and $w = 1/z$.

The transition function for $\Hol(n)$ is $g_{01}(z) = 1/z^{n} = w^{n}$. A holomorphic section of $\Hol(n)$ is an entire power series $$ \sigma_{0}(z) = \sum_{k=0}^{\infty} a_{k} z^{k} $$ such that $$ \sigma_{1}(w) = g_{01}(z) \sigma_{0}(z) = w^{n}\sum_{k=0}^{\infty} a_{k} w^{-k} = \sum_{k=0}^{\infty} a_{k} w^{n-k} $$ is entire. That is, a section of $\Hol(n)$may be viewed as an affine polynomial $$ \sigma_{0}(z) = \sum_{k=0}^{n} a_{k} z^{k} = \sum_{k=0}^{n} a_{k} w^{n-k} $$ of degree at most $n$, or as a homogeneous polynomial $$ \sum_{k=0}^{n} a_{k} z^{k} w^{n-k} $$ of degree $n$.

(This is consistent with the fact that if $s_{m}$ and $s_{n}$ are sections of $\Hol(m)$ and $\Hol(n)$, respectively, then $s_{m}\, s_{n}$ is a section of $\Hol(m + n) \simeq \Hol(m) \otimes \Hol(n)$.)

$\endgroup$

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.