Rigidity of certain holomorphic foliations

There is a well-known rigidity theorem of Y. Ilyashenko for (singular) holomorphic foliations in $\mathbb{CP}^2$ and also the extension given by Gómez-Mont and Ortíz-Bobadilla (1989). Here we present a different generalization of the result of Ilyashenko: some cohomological and (generic) dynamical conditions on a foliation $\mathcal{F}$ on a fibred complex surface imply the d-rigidity of $\mathcal{F}$, i.e. any topologically trivial deformation of $\mathcal{F}$ is also analytically trivial. We particularize this result to the case of ruled surfaces. In this context, the foliations not verifying the cohomological hypothesis above were completely classified in an earlier work by X. Gómez-Mont (1989). Hence we obtain a (generic) characterization of non-d-rigid foliations in ruled surfaces. We point out that the widest class of them are Riccati foliations.