# Invariance of anti-plurigenus of Fano varieties with canonical singularities under small deformation?

I want to know the reference for the following theorem (Theorem 5.28. in Kollar-Mori's book "Birational geometry of algebraic varieties"):

Theorem Let $f:X \rightarrow S$ be a proper flat morphism of algebraic varieties such that $f^{-1}(s)$ has only canonical singularities. Then

(1) $\omega_{X/S}^{[q]}$ commutes with base change.
(2) $\\chi (X_s, \omega_{X_s}^{[q]})$ is locally constant.

Kollar-Mori's book refers to Kollar's Ph.D thesis. They don't seem to mention about what is $q$.

Q1 Is $q$ arbitrary integer?
Q2 Are there other references on this theorem?

If $-K_{X_s}$ is ample and $X$ satisfies the hypothesis of the theorem, then this theorem seems to imply that $h^0(X_s, -m K_{X_s})$ is invariant under small deformations. Is it true?

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