Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X\_{t\_{0}}=p^{-1}(t\_{0})$, for a $t\_{0}\in T$, has only canonical singularities of index $1$. Is there a neighborhood $U$ of $X\_{t\_{0}}$ such that any variety $X\_{t}$ belonging to $U$ has at worst canonical singularities? Any variety is over $\mathbb{C}$ here. 



Deformation of Canonical singularities
When the base of the deformation is smooth the answer is *yes*. This was proven by Kawamata in his paper *[Deformations of canonical singularities](http://www.ams.org/journals/jams/1999-12-01/S0894-0347-99-00285-4/)*, Journal of the American Mathematical Society **12** (1999), 85-92.


For the reader's convenience, let me write here the precise statement of Kawamata's result.



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> **Theorem (Kawamata).** Let $\pi \colon X \to S$ be a flat morphism from a germ of a variety $(X, \, x\_0)$ to a germ of a smooth curve $(S, \, s\_0)$, whose special fiber $X\_0 = \pi^{-1}(s\_0)$ has only canonical singularities. Then $X$ has only canonical singularities. In particular, the fibers $X\_s = \pi^{-1}(s)$ have only canonical singularities. 
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Note that your assumption about the index of the singularities of $X\_{0}$ is unnecessary. 


