Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?


An example of varying Hodge diamonds is given in the paper "Complex parallelisable manifolds and their small deformations".



Infinitely many deformation equivalent Hodge diamonds
No, because $h^i(\Omega\_{X,s}^j)$ is upper semicontinuous in the analytic Zariski topology. So it can attain only a finite number of possible values over $S$.


