I know that all fibers in a analytic fibration (proper, holomorpic) are homeomorphic, and if the fibers are Kählerian manifolds, then they have equal Hodge numbers.
Could it happen however that a manifold admits different Kählerian structures for which the Hodge numbers differ?
An Explicit Example. Let $\displaystyle\Sigma\equiv\mathbb{CP}^2\#8\overline{\mathbb{CP}^2}$ be the blow-up of complex projective plane $\mathbb{CP}^2$ at $8$ points in general position, this is a Del Pezzo surface (proof and definition), so it is an algebraic complex surface; in particular, it is a compact Kähler surface. By theorem 1 in D. Kotschick (1989) - On manifolds homeomorphic to $\displaystyle\mathbb{CP}^2\#8\overline{\mathbb{CP}^2}$, Invent. Math., 95 3, 591–600, $\Sigma$ is homeomorphic but not diffeomorphic to the Barlow surface, which is another compact Kähler surface.