# 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 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?

• What do you mean when you write "fibre space" at the beginning of the first sentence? Is the morphism locally analytically isomorphic to a product analytic variety with a projection to one factor? – Jason Starr Jul 30 '16 at 20:50
• holomorphic fibre space here means a surjective map $f:X\to Y$ which is holomorphic map and general fiber is connected. You mean fiber bundle? – user21574 Jul 30 '16 at 21:32
• Of course, even if $X_t$ is smooth, $X_0$ may not even be irreducible (so certainly not log terminal). Deformation to the normal cone gives a simple example of this. – Ruadhaí Dervan Jul 30 '16 at 21:49
• Theorem 24 , page 21, when $Y$ is smooth and $f$ is flat morphism, then the first part seems to be correct math.kyoto-u.ac.jp/~fujino/Fujino-Reid.pdf – user21574 Jul 30 '16 at 21:51

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).