For semistable fibrations with general type fibers, a Kodaira-Spencer class supported on a suitable non-movable, foliation-invariant divisor forces the general fibers to be birational.
van Straten Infinitesimal deformations of double covers of smooth algebraic varieties, Math
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On supported deformations and birational isotriviality
For semistable fibrations with general type fibers, a Kodaira-Spencer class supported on a suitable non-movable, foliation-invariant divisor forces the general fibers to be birational.