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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.

ralleee
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1 Answers1

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$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Hol}{\mathscr{O}}$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 $\Hol(m + n) \simeq \Hol(m) \otimes \Hol(n)$.)

Andrew D. Hwang
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