I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on $L$.
Now the space $\mathcal{O}(X)$ is better understood, and since $L\to X$ is holomorphically trivial, we could identify $\mathcal{O}(X)$ with the space of holomorphic sections $\Gamma(X,L)$. So I want to use $\Gamma(X,L)$ to understand $\mathcal{O}(L)$.
I have been trying to find a relation using general constructions, but have not succeeded. The only idea I had is the following very simple one: if $f\in \mathcal{O}(L)$ and $s\in \Gamma (X,L)$, $f\circ s\in \mathcal{O}(X) \cong \Gamma(X,L)$, so $f$ induces a transformation on $\Gamma(X,L)$.
Could anyone point out a direction for me? Any advice is appreciated! Thanks!
Since $L$ is holomorphically trivial, the total space is biholomorphic to $X \times \mathbf{C}$. Fix a biholomorphism, and let $\pi_{1}:L \to X$ and $\pi_{2}:L \to \mathbf{C}$ denote the respective projections.
Every entire function $f$ gives rise to a holomorphic function $f \circ \pi_{2}$ on $L$.
Similarly, every holomorphic function $g$ on $X$ (a.k.a., holomorphic section of $L$) gives rise to a holomorphic function $g \circ \pi_{1}$ on $L$.
The algebra of holomorphic functions on $L$ is "generated by" the functions of these two types, in the same sense that the algebra of holomorphic functions in two variables is generated by holomorphic functions in one variable.
In particular, if $X$ admits no non-constant holomorphic functions, e.g., if $X$ is compact, then holomorphic functions on $L$ correspond to entire functions on $\mathbf{C}$.