Let $f\colon X\to Y$ be a surjective proper holomorphic fibre space such that $X$ and $Y$ are projective varieties and central fibre $X\_0$ is Calabi-Yau variety with canonical singularities, then can we say that all the fibres $X\_t$ are also Calabi-Yau varieties ? I know that if we replace "Calabi-Yau" with "pseudoeffective" then this this statement is correct.



Fibration when central fibre is a Calabi-Yau variety with canonical singularities
Assume for simplicity that $Y$ is a smooth curve. Since $X\_0$ has canonical, then by Theorem 1.4 of <http://arxiv.org/pdf/math/9809091.pdf>, we may assume that $X$ is canonical. Nearby fibers are then also canonical (see eg. Theorem 4.5.1 <http://arxiv.org/pdf/alg-geom/9601026.pdf>). The proof of Theorem 1.4 actually shows that $O\_X(K\_X+X\_0)\to O\_{X\_0}(K\_{X\_0})$ is surjective and so if $K\_{X\_0}$ is Cartier, then so is $K\_X$ (on a neighborhood of $X\_0$).
By J. Algebr. Geom. 16, No. 1, 1-18 (2007) we even have that $P\_m(X\_y)$ is deformation invariant for $m\geq 1$ so in fact $h^0(K\_{X\_y})>0$ for $y\in Y$. Since $K\_{X\_0}\equiv 0$, then $K\_{X\_y}\equiv 0$ and so $K\_{X\_y}\sim 0$. (One could also conclude using Theorem 1.6 of <http://arxiv.org/pdf/math/9809091.pdf>).


