# Line bundle on projective line

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?

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 $$L$$.