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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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  • $\begingroup$ Do (24) and (25) of arXiv:0805.0576 Hulls and Husks. János Kollár answer your questions (1) and (2)? (In the terminal case, I think q>0) Also, I think that arXiv:0911.0504 Rigidity properties of Fano varieties. Tommaso de Fernex, Christopher Hacon (especially Thm 4.5) answers the last question also in the terminal case. $\endgroup$
    – Hacon
    Commented Jun 19, 2015 at 3:30
  • $\begingroup$ Thank you for your comment. Actually, I forgot to mention that Theorem 12 in Kollár's paper ``Flatness criteria'' also answers my questions (1) and (2). $\endgroup$
    – tarosano
    Commented Jun 21, 2015 at 12:24

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