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?