TITLE: Existence transverse sections in $\mathbb{CP}^1$-bundles over compact Riemann surfaces
QUESTION [5 upvotes]: We have that every holomorphic $\mathbb{CP}^1$-bundle on a compact Riemann surface admits a holomorphic section, due Tsen and as found in Compact Compact Surfaces of Barth, Peters and Van de Ven, for example. 
If I add to this bundle a meromorphic flat connection (equivalently a foliation $\mathcal{F}$ which is transversal to the fibers except in finitely many fibers) is it possible to find a transversal section to the foliation $\mathcal{F}$?

REPLY [3 votes]: A foliation on a $\mathbb P^1$-bundle over a curve of genus $g$ everywhere transverse to the fibers determines, and is completely determined, by its monodromy representation. The existence of a section everywhere transverse to the foliation is equivalent to endow the curve with a projective structure. Therefore, in the particular case of foliations everywhere transverse to the fibration you are asking for the possible monodromy representations of projective structures on it.
This problem has been completely solved by Gallo-Kapovitch-Marden. For a survey on this circle of ideas, see this paper by Loray-Marin.