In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation on a compact manifold is not necessarily transverse to some compact Riemann surface. Indeed, the existence of such a compact transverse section often implies several restrictions on the foliation..."

My question is simply "What restrictions?" This is not really discussed in their paper nor the references, as they concentrate on real transversals. Also, most results I have seen are local, I'm interested in the global statement as phrased by Scardua.

I'd be happy to even concentrate on the case of smooth projective surfaces and smooth transverse algebraic curves. The only general result of which I am aware is the result by Bogomolov/McQuillan in "Rational Curves on Foliated Varieties" and the follow-up article by Kebekus/Conde/Toma "Rationally connected foliations after Bogomolov and McQuillan", that if the transverse curve has positive self-intersection (i.e. is an ample divisor), then the foliation is a fibration with rational leaves. (I'm phrasing it in the case of dimension 2, similar results also hold in higher dimension, giving rational connectedness of the leaves, and with weaker hypotheses as to the nature of the curve and foliation singularities, amount of tangency, etc).

But how about everywhere transverse curves with non-positive self-intersection? What restrictions does this give on the foliation?

everywheretransverse to the foliation $\mathcal{F}$? If so, then there is an obvious topological condition: the natural map from the direct sum $\mathcal{F}|_C \oplus T_C$ to the restricted tangent bundle $T_X|_C$ must be an isomorphism. This can easily fail, e.g., if $X$ is a complex torus containing no elliptic curves and $\mathcal{F}$ is any codimension one foliation on $X$. $\endgroup$oneexample of a foliation on a general type surface of Picard number 2 that has no transversal, however, I doubt that is what you are looking for. $\endgroup$