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My question concerns with Theorem 3.1 in the paper "A canonical bundle formula" by Fujino and Mori.

The theorem claims the following:

Suppose $X \to C$ is a fiberation whose general fiber $F$ has Kodaira dimension $0$ and suppose $b$ is the least integer such that $|bK_F| \neq 0$. Let $E \to F$ be the cyclic cover associated to the $b$-th root of the unique element of $|bK_F|$, and $\bar E$ be a smooth model of $E$. Let $B=h^{\dim E}(E, \mathbb C)$ be the middle Betti number, and $N={\rm lcm}\{y \in \mathbb Z_{> 0} \mid \phi(y) \leq B\}$, where $$\phi(y):=\#\{z \in \mathbb Z_{> 0}\mid z \leq y, {\rm gcd}(z,y)=1\}$$ is the Euler function. Then in the canonical bundle formula, the semistable part $L^{ss}_{X/C}$ (see Corollary 2.5 in loc.cit.) satisfies the property that $NL^{ss}_{X/C}$ is an integral divisor.

This result is proved roughly along the following lines (see 3.2-3.8 in loc.cit.):

  1. Cutting by hypersurfaces, we can assume $C$ is a curve.

  2. Let $\bar Z \to C$ be a smooth model such that the general fiber is $\bar E$.

  3. Let $\pi: C' \to C$ be a Galois cover with Galois group $G$, and $\bar Z \times_C C'$ has semistable resolution $\bar h': \bar Z' \to C'$.

  4. One knows that $\pi^*L_{Z/C}^{ss}$ is a Weil divisor and $\mathcal{O}(NL_{Z/C}^{ss})=\mathcal (\pi_*\mathcal{O}(N\pi^*L_{Z/C}^{ss}))^{G}$.

  5. For $P' \in C'$, let $G_{P'}$ be the stabilizer of $G$ and $G_{P'}$ acts on $\mathcal{O}(\pi^*L_{Z/C}^{ss})$ through a character $\chi_{P'}: G_{P'} \to \mathbb C^*$.

  6. Let $e$ be the ramification index at $P'$, then there exists a homomorphism $G_{P'} \to \mu_{e}$, where $\mu_e$ is the group of $e$-th root of unity. Let $H'$ be the canonical extension of $\mathcal{O}_{C_o'}\otimes (R^{m}(\bar{h}_o')_*{\mathbb C_{\bar{Z_o}'}})$, where $m=\dim E$ and $\bar C_o'=C'\backslash \{P'\}$, etc. Then there exists a $\mu_e$-equivariant injection $$\mathcal O(\pi^*L_{Z/C}^{ss}) \subset H' \otimes \mathbb C(P').$$

Finally, it is claimed that if $l$ is the order of the character $\chi_{P'}$, then $\phi(l) \leq B$. Form this fact, it is easy to get the desired result.

I could understand Step 1-5, and more or less understand Step 6. However, I don't know how to deduce $\phi(l) \leq B$.

Another place uses this result appears in "Boundedness of log Calabi-Yau pairs of Fano type" by Hacon and Xu. Where it wrote (in page 9) "The eigenvalues of the corresponding action on $H^m(F', \mathbb Z)\otimes \mathbb C$ are always algebraic integers, thus we conclude $\phi(r) \leq \dim_{\mathbb C}(H^m(F', \mathbb Z)\otimes \mathbb C) = l$. " I don't see the relevant of algebraic integers to this question. Certainly $\mu_4$ can act on $\mathbb C$ by scaling, but $\phi(4)=2>\dim \mathbb C$.

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