We say that a set of varieties $S$ lives in a **bounded family** if there exists a projective morphism $\mathcal{X} \to T$ between varieties of finite type, such that for any $X \in S$, there exists a closed point $t \in T$, such that its fibre $\mathcal{X}\_t$ is isomorphic to $X$.


It seems that the following two facts related to bounded families used quite often without proof in the literature of birational geometry:


(1) If $S$ is a bounded family of $\mathbb{Q}$-Gorenstein varieties (we can assume the elements in $S$ are normal, have log terminal singularities), then there exists a universal $m$, such that for any $X \in S$, $mK\_X$ is a Cartier divisor, i.e. the Gorenstein index is bounded.


(2) Granted (1) is true, then the volume (suppose $n=\dim X$)
$${\rm vol}(mK\_X): = \lim\_{k \to +\infty}\frac{n! h^0(X, kmK\_X)}{k^n}$$ is bounded for any $X \in S$.


If I know the universal family $\mathcal{X}$ is $\mathbb Q$-Gorenstein, then the above two results follows easily. But I don't know if we can assume this because such universal families typically come from Hilbert schemes (I don't know if Hilbert schemes are $\mathbb Q$-Gorenstein or not). The closest result related (1) which I can find is Theorem B.1 in the paper [Log canonical thresholds on varieties with bounded singularities](https://arxiv.org/pdf/1004.3336v2.pdf). But it requires more than what I have: the fibres are required to be normal.


However, I do think one needs something extra which natural comes from the construction of bounded family in order to make (1) and (2) holds. Any suggestion?



Finiteness of Gorenstein indexes and volumes for varieties in a bounded family
By Noetherian induction, it suffices to show that indicies and volumes are bounded over an open subset of any irreducible component of $T$. 
We may assume that $T$ is smooth and there is a dense set $\{t\_i\}\subset T$ such that the corresponding fibers $\mathcal X\_{t\_i}$ are normal. By EGA IV Theorem 12.2.4(iv), after shrinking $T$, we may assume that $\mathcal X$ is normal. 
Since $\mathcal X\_{t\_i}$ is log terminal, it is $\mathbb Q$-Gorenstein (by definition) and has rational singularities (see eg. ́Koll\'ar-Mori). By Theorem B.1 in the paper Log canonical thresholds on varieties with bounded singularities <https://arxiv.org/pdf/1004.3336.pdf>, $X$ is $\mathbb Q$-Gorenstein. Thus (1) holds.
Let $\nu:\mathcal X'\to \mathcal X$ be a log resolution and write $K\_{\mathcal X'}+B'=\nu ^\* K\_{\mathcal X}+E$ where $B'$ and $E$ are effective with no common components. Shrinking $T$ we may assume that $(\mathcal X',B')$ is log smooth over $T$ (so that every stratum of the support of $B'$ is smooth over $T$) and that $\mathcal X '\_t\to \mathcal X \_t$ is a log resolution for every $t\in T$. Note that since $\lfloor B'\_{t\_i}\rfloor =0$ then also $\lfloor B'\rfloor =0$. By Theorem 4.2 in <https://arxiv.org/pdf/1412.1186.pdf>, $h^0(mK\_{\mathcal X \_t})=h^0(m(K\_{\mathcal X\_{t}}+B\_t))$ is independent of $t\in T$. In particular ${\rm vol}(K\_{\mathcal X \_t})$ is independent of $t\in T$. Thus (2) also holds.


