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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.

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When the base of the deformation is smooth the answer is yes. This was proven by Kawamata in his paper Deformations of canonical singularities, 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.

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.

Note that your assumption about the index of the singularities of $X_{0}$ is unnecessary.

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