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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".

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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 $S$.

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