On the degeneration of the Frolicher spectral sequence and small deformations

We study the behavior of the degeneration at the second step of the Frolicher spectral sequence of a C-infinity family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result a la Kodaira-Spencer for the dimension of the second step of the Frolicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure.