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Central fibre singularities

Let $f:X\to Y $ be a proper surjective holomorphic fibre space where $X,Y $ are projective varieties. If the central fibre $X_0$ has at worst log terminal singularities, then can we say that all ...
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Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
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