On the degeneration of the Fr\"olicher spectral sequence and small deformations

We study the behavior of the degeneration at the second step of the Fr\"olicher spectral sequence of a $\mathcal{C}^{\infty}$ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result \textit{\`a la Kodaira-Spencer} for the dimension of the second step of the Fr\"olicher 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. We also provide an example in which we show that, if we omit that hypothesis, we lose the openness of the degeneration.