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Is a holomorphic family whose fibers are all smooth locally trivial? –
mathoverflow.net

Let $\pi : X \to B$ be a proper, surjective holomorphic submersion, where
both $X$ and $B$ are compact Kahler manifolds. Assume that all the fibers
$X_b = \pi^{-1}(b)$ are smooth. Is the family $\pi : …