Complex codimension one singular foliations and Godbillon–Vey sequences

Let $\mathcal F$ be a codimension one singular holomorphic foliation on a compact complex manifold $M$. Assume that there exists a meromorphic vector field $X$ on $M$ generically transversal to $\mathcal F$. Then, we prove that $\mathcal F$ is the meromorphic pull-back of an algebraic foliation on an algebraic manifold $N$, or $\mathcal F$ is transversely projective (in the sense of [19]), improving our previous work [7].
Such a vector field insures the existence of a global meromorphic Godbillon–Vey sequence for the foliation $\mathcal F$. We derive sufficient conditions on this sequence insuring such alternative. For instance, if there exists a finite Godbillon–Vey sequence or if the Godbillon–Vey 3-form $\omega_0\land\omega_1\land\omega_2$ is zero, then $\mathcal F$ is the pull-back of a foliation on a surface, or $\mathcal F$ is transversely projective (in the sense of [19]). We illustrate these results with many examples.