Does there exist a family of compact complex manifolds over unit disk such that the Hodge numbers are not constant in the family? 


The answer is manifestly positive in complex dimension 1. 


It is known, I think, that if there a Kaehler fiber, then Hodge numbers must be constant in a small neighbourhood thereof.


Note that [this question](https://mathoverflow.net/questions/42709/question-about-hodge-number) asks about topological invariance of Hodge numbers, while we are asking about deformation-equivalence invariance.



Constancy of Hodge numbers in a family of compact complex manifolds
Nakamura has constructed a family of compact complex threefolds such that the Hodge number $h^{p,q}$ is *not* deformation invariant for $p+q>0$. They are obtained by deforming an "Iwasawa manifold" $\mathbb{C}^3/\Gamma $. See [*Complex parallelisable manifolds and their small deformations*](https://projecteuclid.org/download/pdf_1/euclid.jdg/1214432677), J. Diff. Geom. 10 (1975), 95-112. 


