In this mathoverflow question, it is asserted that diffeomorphic Kähler manifolds with different Hodge numbers cannot be deformation equivalent.
Why is that true?
I know that if $\mathcal X \to B$ is a deformation, and the central fiber $X_0$ is Kähler, then small deformations over a neighbourhood $U \subset B$ will be Kähler with the same Hodge numbers. This means that for the Hodge numbers to jump, I would have to pass a non-Kähler deformation. But a priori it could still happen that for some $t \in B \setminus \overline U$, the manifold $X_t$ is Kähler and has different Hodge numbers? Are there examples of this happening?