Several special complex structures and their deformation properties

We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the K\"ahler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a K\"ahlerian family is contained in the closure of the Gauduchon cone for this fiber.