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 other fibres $X\_t $ at worst have log
>  terminal singularities and $X$ at worst have log terminal singularities?. 
>  What about the inverse. If $X$ has log terminal singularity then the
>  central fibre $X\_0$ at worst has log terminal singularity?
> 
> 
> 



Central fibre singularities
It could easily happen that $X\_0$ has log terminal singularities and $X$ is not log terminal. The standard example is if $f:Y\to X$ is a flipping contraction of a 3-fold over a curve $T$ (where the flipping curve $C$ is contained in the central fiber). The issue is that if $Y\_0$ is log terminal, then as $Y\_0\to X\_0$ is $K\_{Y\_0}$ negative birational map of surfaces, then $X\_0$ also has log terminal singularities (and is even Q-factorial). On the other hand $K\_X$ is automatically not Q-Cartier (if it where, then $K\_Y=f^\*K\_X$ and so $K\_Y\cdot C=0$ which is impossible). Note however that if you require $X\_0$ to be canonical, then so is $X$ (see Theorem 1.4 of arXiv:math/9809091). Finally, if you assume that $X$ is log terminal, you can deduce almost nothing for $X\_0$ (it could be reducible or even non-reduced). However, if you assume that $(X,X\_0)$ is plt, then you can deduce that $(K\_X+X\_0)|\_{X\_0}=K\_{X\_0}+B\_0$ where $(X\_0,B\_0)$ is klt (the converse to this statement is also true; it is known as inversion of adjunction; see the book of Kollár-Mori).


