Finiteness of Gorenstein indexes and volumes for varieties in a bounded family

Question

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. 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?

Answer 1

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.