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Let $L$ be the (holomorphic) line bundle on $\mathbb{P}^1$ for which the glueing function from $\mathbb{P}^1\setminus \{\infty\}$ to $\mathbb{P}^1\setminus \{0\}$ on the overlap $\mathbb{C}^\times = \mathbb{P}^1\setminus \{0,\infty\}$ is given by multiplication with $g(z) = \exp(1/z)$.

A global (holomorphic) section of $L$, say $s$, has to satisfy $\exp(1/x) \cdot s(x) = s(1/x)$ for all nonzero complex numbers $x$. Surely such an $s$ does not exist. In fact, there does not even exist a meromorphic such $s$, as far as I can tell.

But, how can a line bundle $L$ not have a meromorphic section?

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tl; dr: This line bundle actually admits a non-vanishing holomorphic section, and is therefore trivial.


The equation $\exp(1/x) \cdot s(x) = s(1/x)$ is an equation for the section $s$, but in our covering coordinate charts $s$ is represented by a pair of functions, not by a single function.

In more detail, a meromorphic section of $L$ corresponds to a pair of meromorphic functions $s_{0}$ and $s_{1}$ satisfying $$ \exp(1/x) \cdot s_{0}(x) = s_{1}(1/x),\quad x \neq 0. $$ We may take $s_{0}$ to be the constant function $1$ and $s_{1} = \exp$. These functions are holomorphic and non-vanishing, so together they constitute a non-vanishing holomorphic section of $L$.

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