Let $P$ be a holomorphic fibre bundle over a closed Riemann surface $\Sigma$, denoted $P \to \Sigma$. Furthermore, I assume that the fibers are diffeomorphic to $\mathbb{CP}^1$.
I am interested in the existence of a global holomorphic section of this bundle.
My idea was to use that fact that this is a projective line bundle over a closed Riemann surface and it therefore arises as the projectivization of a rank $2$ vector bundle $E$, denoted $P=\mathcal{P}(E)$.
Observation: Taking the tensor product $E \otimes L$ with a complex line bundle $L$ does not affect to projectivization, since $\mathcal P(E)=\mathcal{P}(E \otimes L)$.
However, I could not find a reference/theorem to make sure that we can build a global section of $P$ from, say a "special" global section of $E$. Any hints/references for that? Thank you in advance.
Additional thoughts: I was thinking of using cohomology, but that didnt help me for now.
Best Answer
This is just a record of Aphelli's suggestion, as an answer.
Write $ P $ as $ \mathbb{P} (E) $ for a holomorphic rank $ 2 $ vector bundle $ E $ on $ X $. We will use the following theorem of Serre.
Theorem: Let $ X $ be a projective variety and $ E $ a vector bundle on $ X $. Let $ H $ be a very ample line bundle on $ X $. Then there is an integer $ N $ (which depends on $ H $ and $ E $ in general) such that for every $ n \ge N $, the vector bundle $ E \otimes H^{\otimes n } $ is generated by its global sections - For clarification, this means that the global sections restricted to a fiber span the fiber as a vector space.
Remark: The space of global sections of $ E \otimes H^{\otimes n} $ is finite dimensional and for large enough $ n $ as above, this dimension grows asymptotically at the rate of $ n^{\dim X} $. (This is not strictly needed but good to know.)
Noting that a compact Riemann surface is a projective variety, we pick a very ample line bundle $ H $ on $ \Sigma $. Now choose an $ N $ for the pair $ E,H $ according to the theorem. Then $ E \otimes H^{\otimes N} $ has enough global sections. A general global section of the above vector bundle does not vanish on $ \Sigma $ as it is of rank $ 2 $ while $ \Sigma $ has dimension $ 1 $. (For a rigourous explanation of what 'general' means, I recommend reading about Bertini's Lemma)
Pick such a general section $ s $ as above. Note that the fibers of $ \mathbb{P}(E \otimes H^{\otimes N}) \rightarrow \Sigma $ are just projectivizations of the fibers of $ E \otimes H^{\otimes N} $, more or less, by definition. Since $ s $ does not vanish in any fiber, its value in every fiber's projectivization defines a section $ [s] $ of $ \mathbb{P}(E \otimes H^{\otimes N}) \rightarrow \Sigma $.
Finally $ \mathbb{P}(E \otimes H^{\otimes N}) $ is just $ \mathbb{P}(E) = P $ by the remark that tensoring by any line bundle does not change the projective bundle, hence $ [s] $ indeed is a section of $ P \rightarrow \Sigma $ as desired.