Proof. We first show that $\Sigma _q$ satisfies the ascending chain condition. Suppose in $\Sigma _q$ we can find an increasing sequence

$$\begin{align*}\frac{b_1^+-b_1}{m_1} < \frac{b_2^+-b_2}{m_2} < \cdots < \frac{b_k^+-b_k}{m_k} < \cdots \end{align*}$$

Since $b_k^+ \in \{i/q \colon 0\leq i\leq q\}$ has only finitely many choices, we may assume, by passing to a subsequence, that all $b_k^+$ are the same and equal to the number $b^+$ . Furthermore, note that

$$\begin{align*}m_k < \frac{m_1(b_k^+-b_k)}{b_1^+-b_1} \leq \frac{m_1}{b^+-b_1}. \end{align*}$$

The second inequality holds as $1\geq b_k^+ \geq b_k^+-b_k$ . Hence, the sequence $m_k$ is bounded above. Thus, by passing to a subsequence we may assume that $m_k = m$ for all k. Now, we obtain a decreasing sequence

$$\begin{align*}b_1> b_2 > \cdots > b_k > \cdots, \end{align*}$$

which violates the descending chain condition of $\Lambda $ . Thus, $\Sigma _q$ satisfies the ascending chain condition.

Let $a\neq 0$ be an accumulation point of $\Sigma _q$ . Since $\Sigma _q$ satisfies the ascending chain condition, we may find a sequence

$$\begin{align*}\frac{b_1^+-b_1}{m_1} \geq \frac{b_2^+-b_2}{m_2} \geq \cdots \geq \frac{b_k^+-b_k}{m_k} \geq \cdots \end{align*}$$

whose limit is a. We may assume that $b_k^+ = b^+$ for all k. Then

$$\begin{align*}m_k \leq \frac{b_k^+-b_k}{a} \leq \frac{1}{a}. \end{align*}$$

The second inequality holds as $1\geq b_k^+ \geq b_k^+-b_k$ . Hence, the sequence $m_k$ is bounded above. By passing to a subsequence, we may assume that $m_k = m$ for all k. Since $\Lambda $ has rational accumulation points,

$$\begin{align*}a = \lim_{k\to \infty} \frac{b^+-b_k}{m} = \frac{b^+}{m} - \frac{1}{m} \lim_{k\to\infty} b_k \in \mathbb{Q}, \end{align*}$$

as desired.

Proof. Up to shrinking Z around z, we can find a generalized log Calabi–Yau pair $(X,B+\Gamma ,\mathbf {M})$ over Z that computes the absolute coregularity of $(X,B,\mathbf {M})$ over z. Let $\Gamma _Y=f_*\Gamma $ . Since $(X,B+\Gamma ,\mathbf {M})$ is generalized log Calabi–Yau over Z, we conclude that $(Y,B_Y+\Gamma _Y,\mathbf {M})$ is generalized log canonical. As $(X,B+\Gamma ,\mathbf {M})$ and $(Y,B_Y+\Gamma _Y,\mathbf {M})$ are crepant equivalent, we conclude that they have the same coregularity. Hence, we deduce that

$$\begin{align*}\hat{\mathrm{coreg}}_z(Y,B_Y,\mathbf{M}) \leq \hat{\mathrm{coreg}}_z(X,B,\mathbf{M}). \end{align*}$$

On the other hand, up to shrinking Z around z, we can find an effective divisor $D_Y$ on Y that computes the absolute coregularity of $(Y,B_Y,\mathbf {M})$ over z. Let $p\colon W \rightarrow X$ and $q\colon W\rightarrow Y$ be a common resolution. Write

$$\begin{align*}K_X+B+D+{\mathbf{M}_{X}} = q_*p^*(K_Y+B_Y+D_Y+{\mathbf{M}_{Y}}). \end{align*}$$

By the negativity lemma, we know that D is an effective divisor. Hence, $(X,B+D,\mathbf {M})$ is a generalized log Calabi–Yau pair over Z. As above, we conclude that the absolute coregularity of $(X,B,\mathbf {M})$ over $z\in Z$ is at most the absolute coregularity of $(Y,B_Y,\mathbf {M})$ over $z\in Z$ . This finishes the proof.

Proof. By passing to a generalized dlt modification, we may assume that both generalized pairs $(X,B,\mathbf {M})$ and $(S,B_S,\mathbf {N})$ are generalized dlt. Since any minimal generalized log canonical center of $(S,B_S,\mathbf {N})$ is a minimal generalized log canonical center of $(X,B,\mathbf {M})$ , we have that $\mathrm {coreg}(S,B_S,\mathbf {N})\geq \mathrm {coreg}(X,B,\mathbf {M})$ . On the other hand, let W be a minimal generalized log canonical center of $(X,B,\mathbf {M})$ whose image on Z contains z. By [Reference Filipazzi and Svaldi16, Theorem 1.4], $(S,B_S,\mathbf {N})$ admits a generalized log canonical center $W_S$ that is $\mathbb {P}^1$ -linked to W, thus $\dim W_S =\dim W$ . This implies that $\mathrm {coreg}(X,B,\mathbf {M})\geq \mathrm {coreg}(S,B_S,\mathbf {N})$ .

Proof. Define and let $\mathbf {P}$ denote its b-Cartier closure. Then, $\mathbf {P}$ is a b-nef $\mathbb {Q}$ -Cartier divisor. We can apply Lemma 2.23 to the generalized log Calabi–Yau pair $(X,B,\mathbf {M}+ \mathbf {P})$ . Therefore, $\mathrm {coreg} (S,B_S, \mathbf {N}+ \mathbf {P}|_{S} )=\mathrm {coreg} (X,B, \mathbf {M} + \mathbf {P})$ . Since $\mathbf {P}$ descends on S, we conclude that $\mathrm {coreg}(S,B_S,\mathbf {N}+\mathbf {P}|_{S})=\mathrm {coreg}(S,B_S,\mathbf {N})$ . Hence, $\mathrm {coreg}(S,B_S, \mathbf {N})=\mathrm {coreg}(X,B, \mathbf {M})$ .

Proof. We follow the proof of [Reference Birkar2, Lemma 3.2].

Since G is the pull-back of a general hyperplane section on Z, $(X,B+G)$ is log canonical. Thus, by adjunction, $(G,B_G)$ is log canonical and log Calabi–Yau over H. Moreover, every log canonical center of $(G,B_G)$ is a component of the intersection of a log canonical center of $(X,B+G)$ and G, and hence must dominate H. By 2.23, we have an equality

$$\begin{align*}\text{coreg}(G,B_G) = \text{coreg}(X,B) \leq c. \end{align*}$$

Denote the map $G\to H$ by $\psi $ . Let t be the log canonical threshold of $\pi ^*D$ with respect to $(X,B)$ over the generic point of D. Then there is a non-klt center W of $(X,B+t\pi ^*D)$ which dominates D and the pair $(X,B+t\pi ^*D)$ is lc over the generic point of D. Since G is a general pull-back, $(X,B+G+t\pi ^*D)$ is also lc over the generic point of D. By inversion of adjunction [Reference Kawakita27], there exists a component of $G\cap W$ which is a non-klt center of $(G,B_G+t\pi ^*C)$ and $(G,B_G+t\pi ^*C)$ is lc near the generic point of C. Thus, t is the log canonical threshold of $\psi ^*C$ with respect to $(G,B_G)$ . In particular, we have

$$\begin{align*}\operatorname{\mathrm{coeff}}_D(B_Z) = 1-t = \operatorname{\mathrm{coeff}}_C(B_H).\\[-38pt] \end{align*}$$

Proof. Let $(X,B,\mathbf {M})$ be a generalized pair as in the statement. Then, we first consider a $\mathbb {Q}$ -factorial generalized dlt modification and then apply Theorem 2.28 to such modification. Call $(\overline {X},\overline {B},\mathbf {M})$ the resulting model and $\pi \colon \overline {X} \rightarrow Z$ the morphism claimed in Theorem 2.28. Since $(X,B,\mathbf {M})$ has coregularity c, then so does $(\overline {X},\overline {B},\mathbf {M})$ . In particular, $(\overline {X},\overline {B},\mathbf {M})$ has a c-dimensional generalized log canonical center. Then, by (2) in Theorem 2.28, it follows that Z has dimension at most c. By (1) in Theorem 2.28, $\overline {B}$ fully supports a $\pi $ -ample divisor, which we will denote by $\overline {H}$ . Now, let $(Y,B_Y,\mathbf {M})$ be a $\mathbb {Q}$ -factorial generalized dlt modification of $(\overline {X},\overline {B},\mathbf {M})$ , with morphism $p \colon Y \rightarrow \overline {X}$ . We denote by $\pi _Y$ the induced morphism $\pi _Y \colon Y \rightarrow Z$ . Note that every generalized log canonical center of $(Y,B_Y,\mathbf {M})$ dominates Z. Then, (2) in Definition 2.27 holds. By [Reference Filipazzi and Svaldi16, Remark 4.3], we have $\mathrm {Nklt}(\overline {X},\overline {B},\mathbf {M})=\operatorname {\mathrm {Supp}}\lfloor \overline {B}\rfloor $ . Since p only extracts generalized log canonical places, it then follows that every p-exceptional divisor has positive coefficients in and that $\operatorname {\mathrm {Supp}} H_Y= \operatorname {\mathrm {Supp}} \lfloor B_Y \rfloor $ . Then, it follows that $\lfloor B_Y \rfloor $ fully supports a $\pi _Y$ -big and $\pi _Y$ -semiample divisor. Thus, (3) in Definition 2.27 holds. The statements (1) and (3) in Definition 2.27 hold by construction. We conclude that $(Y,B_Y,\mathbf {M})$ , together with $\pi _Y$ , are a Kollár–Xu model of $(X,B,\mathbf {M})$ . Lastly, $(Y,B_Y,\mathbf {M})$ has coregularity c, since it is crepant to a generalized pair of coregularity c.

Proof. By the global ascending chain condition (ACC) [Reference Birkar and Zhang7, Theorem 1.6], we may assume that $\Lambda $ is a finite set of rational numbers. The statement is clear in dimension $1$ . We proceed by induction on the dimension. If $\mathbf {M}=0$ , then the statement follows from the conjecture. Since $(X,B,\mathbf {M})$ is generalized klt, it admits a small $\mathbb {Q}$ -factorialization. Therefore, we may assume that X is $\mathbb {Q}$ -factorial. By the ACC for generalized log canonical thresholds, we may assume $(X,B,\mathbf {M})$ is $\epsilon $ -log canonical for some $\epsilon $ that only depends on d, p and $\Lambda $ . Then, it follows that X is itself $\epsilon $ -log canonical. We run a $K_X$ -MMP which terminates with a Mori fiber space $X\dashrightarrow X'\rightarrow Z$ . If Z is zero-dimensional, then $X'$ belongs to a bounded family by [Reference Birkar4]. By [Reference Filipazzi and Moraga14, Theorem 1.2] $(K_{X'}+B'+{\mathbf {M}_{X'}})$ admits an I-complement for some I that only depends on $\Lambda ,d$ and p. Since $K_{X'}+B'+{\mathbf {M}_{X'}}\sim _{\mathbb {Q}} 0$ , we conclude that $I(K_{X'}+B'+{\mathbf {M}_{X'}})\sim 0$ , so the statement follows for X as well. Now, assume that Z is positive-dimensional. We write $\pi '\colon X'\rightarrow Z$ for the corresponding contraction. By [Reference Filipazzi and Moraga14, Lemma 5.4], we can write

$$\begin{align*}q(K_{X'}+B'+{\mathbf{M}_{X'}}) \sim q \pi^*(K_Z+B_Z+{\mathbf{N}_{Z}}), \end{align*}$$

where the coefficients of $B_Z$ belong to $\Omega $ which satisfies the DCC and only depends on $\Lambda ,d$ and p. Furthermore, q only depends on $\Lambda ,d$ and p and $q\mathbf {N}$ is Cartier where it descends. The generalized pair $(Z,B_Z, \mathbf {N})$ is generalized log canonical since it comes from the generalized canonical bundle formula [Reference Filipazzi11, Theorem 1.4]. By induction on the dimension, we know that $I_0(K_Z+B_Z+{\mathbf {N}_{Z}})\sim 0$ for some $I_0$ that only depends on $\Lambda ,d$ and p. Thus, we conclude that $I(K_X+B+{\mathbf {M}_{X}})\sim 0$ , where $I=\mathrm {lcm}(q,I_0)$ .

Proof. Let $(X,B,\mathbf {M})$ , S, $f \colon X \rightarrow W$ and be as in the statement. First, we show that we can apply the inductive hypothesis to S.

Since $(X,\lfloor B \rfloor )$ is dlt, S is normal. Furthermore, we have $\mathrm {Nklt}(X,S)=S$ . Since $-({K_{X}} + S)$ is f-ample, by the connectedness principle [Reference Filipazzi and Svaldi16], it follows that $S \rightarrow W$ has connected fibers. Now, let $(S,B_S,\mathbf {N})$ be the generalized pair induced on S by generalized divisorial adjunction. By Lemma 2.23, $(S,B_S,\mathbf {N})$ has coregularity c. Furthermore, by Lemma 2.17 and Lemma 2.16, it satisfies the assumptions of Theorem 6 $(d-1,c)$ with constant I. Thus, we have

(2.1) $$ \begin{align} I ({K_{S}} + B_S + {\mathbf{N}_{S}}) \sim 0. \end{align} $$

By [Reference Figueroa, Moraga and Peng10, Theorem 3.1], the coefficients of $B_S$ belong to a finite set only depending on $\lambda $ and c. In particular, they are divisible by I, as so are the coefficients of ${\mathbf {N}_{S}}$ . Then, these coefficients control the coefficients of $\mathrm {Diff}_S(0)$ , as we explain in what follows. By [Reference Kollár29, 3.35], along the codimension 2 points of X contained in S, X has cyclic singularities.

Then, given a prime divisor P in S, an étale local neighbourhood of a general point $p \in P$ is isomorphic to

$$\begin{align*}(p \in (X, B, \mathbf{M}))\simeq (0 \in (\mathbb{A}^2=(x,y), (x=0) + c(y=0)))/(\mathbb{Z}/m\mathbb{Z})\times \mathbb{A}^{\dim X -2}, \end{align*}$$

where $Z \simeq (\mathbb {A}^2=(x,y))/(\mathbb {Z}/m\mathbb {Z})$ , $S=(x=0)$ and $S'=(y=0)$ . Since the class group of Z is generated by $S'$ , there exists an integer $\mu $ such that

(2.2) $$ \begin{align} I(K_X+B+{\mathbf{M}_{X}}) \sim \mu S'. \end{align} $$

By adjunction, $S'|_S \sim _{\mathbb {Q}} \frac {1}{m}\{0\}$ . We also have that $I(K_X+B+{\mathbf {M}_{X}})|_S\sim 0$ , as the denominators of the coefficients of $B_S$ and ${\mathbf {N}_{S}}$ divide I and hence it is a Cartier divisor on a smooth germ.

Then, we can write

$$\begin{align*}0 \sim I(K_X+B+M)|_S \sim \mu S'|_S \sim \frac{\mu}{m}\{0\}. \end{align*}$$

We conclude that m divides $\mu $ . In particular, we have that $\mu S$ is a Cartier divisor, as the Cartier index of any Weil divisor of X through $\{0\}$ divides m. By the linear equivalence (2.2), we conclude that the divisor $I(K_X+B+M)$ is Cartier at the generic point of P. Note that this argument is independent of P, so we conclude that $I(K_X+B+M)$ is Cartier at the generic point of every divisor on S. Thus, $I(K_X+B+M)$ is Cartier along a set U that contains the generic point of every divisor of S.

Then, by [Reference Birkar3, 2.41 and Lemma 2.42], we have the following short exact sequence

(2.3) $$ \begin{align} 0 \rightarrow {\mathcal{O}_{X}}(I(K_{X}+B + {\mathbf{M}_{X}})-S) \rightarrow {\mathcal{O}_{X}}(I(K_{X}+B+ {\mathbf{M}_{X}})) \rightarrow {\mathcal{O}_{S}}(I(K_{S}+B_{S}+ {\mathbf{N}_{S}})) \rightarrow 0. \end{align} $$

Since $I(K_{X}+B + {\mathbf {M}_{X}})-S\sim _{\mathbb {Q},f} -S$ , the divisor $-S$ is f-ample and $\dim W < \dim X$ , we have

$$\begin{align*}f_*{\mathcal{O}_{X}}(I(K_{X}+B + {\mathbf{M}_{X}})-S)=0. \end{align*}$$

Similarly, we write

$$\begin{align*}I(K_{X}+B + {\mathbf{M}_{X}})-S\sim_{\mathbb{Q},f} -S \sim_{\mathbb{Q},f} K_X +(B-S+{\mathbf{M}_{X}}). \end{align*}$$

Note that X is klt and $B-S+{\mathbf {M}_{X}} = B^{<1}+{\mathbf {M}_{X}} + (B^{=1} - S)$ is f-ample since f is a Mori fiber space and the divisor $B^{<1}+{\mathbf {M}_{X}}$ is f-ample. Thus, by the relative version of Kawamata–Viehweg vanishing, we have

$$\begin{align*}R^1f_*{\mathcal{O}_{X}}(I(K_{X}+B + {\mathbf{M}_{X}})-S)=0. \end{align*}$$

Therefore, by pushing forward (2.3) via f, we obtain

$$\begin{align*}f_* {\mathcal{O}_{X}}(I(K_{X}+B+ {\mathbf{M}_{X}})) \simeq f_* {\mathcal{O}_{S}}(I(K_{S}+B_{S}+ {\mathbf{N}_{S}})). \end{align*}$$

Now, taking global sections, we have

(2.4) $$ \begin{align} H^0(X, {\mathcal{O}_{X}}(I(K_{X}+B+ {\mathbf{M}_{X}}))) = H^0(S, {\mathcal{O}_{S}}(I(K_{S}+B_{S}+ {\mathbf{N}_{S}})))=H^0(S, {\mathcal{O}_{S}}) \neq 0. \end{align} $$

By Lemma [Reference Filipazzi, Mauri and Moraga13, Lemma 3.1], (2.4) implies that $I(K_{X}+B+ {\mathbf {M}_{X}}) \sim 0$ .

Proof. Let $(X,B)$ , S, $f \colon X \rightarrow W$ and be as in the statement. We will proceed in several steps.

Step 1: In this step, we observe that $\dim X - \dim W=1$ , f is generically a ${\mathbb {P}^{1}}$ -fibration and $B_{\mathrm {hor}}=S$ .

Since $(X,B)$ is log canonical and X is $\mathbb {Q}$ -factorial, then $(X,S)$ is log canonical. By considering a general fiber of f, the restriction of S induces a disconnected ample divisor. Therefore, by [Reference Hartshorne25, Exercise III.11.3], it follows that the general fiber of f is a curve. By the log Calabi–Yau condition and the fact that $0 \neq B_{\mathrm {hor}} \geq S$ , it follows that f is generically a ${\mathbb {P}^{1}}$ -fibration. Since $S \rightarrow W$ does not have connected fibers and $\deg K_{{\mathbb {P}^{1}}}=-2$ , it follows that S is the only component of $\operatorname {\mathrm {Supp}} B$ that dominates W, that is, we have $B_{\mathrm {hor}} =S$ . In particular, we may find a nonempty open subset $U \subseteq W$ such that $K_X+S \sim _{\mathbb {Q}} 0 /U$ .

Step 2: In this step, we show that $\dim W> 0$ .

By Step 1, $X \rightarrow W$ is generically a ${\mathbb {P}^{1}}$ -fibration, and S determines two distinct points on the geometric generic fiber of f. Thus, if $\dim W =0$ , it would follow that $(X,B) \simeq ({\mathbb {P}^{1}},\{ 0 \} + \{\infty \})$ , with S identified with $\{ 0 \} + \{\infty \}$ . Since we assumed that S is a prime divisor, this leads to the sought contradiction.

Step 3: In this step, we show that we may replace X birationally so that $(X,S)$ is plt, $K_X +S \sim _{\mathbb {Q}} 0/W$ and f is a Mori fiber space.

Since $(X,B)$ is log canonical and X is $\mathbb {Q}$ -factorial, then $(X,S)$ is log canonical. Let $X'$ be a $\mathbb {Q}$ -factorial dlt modification of $(X,S)$ , and let $S'$ denote the strict transform of S. In particular, $(X',S')$ is plt. By [Reference Hacon and Xu23, Theorem 1.1], $(X',S')$ has a relatively good minimal model over W, which we denote by $(\tilde X, \tilde S)$ . Since $S'$ dominates W and $\dim X - \dim W =1$ , $S'$ is relatively big over W. Therefore, $S'$ cannot be contracted on $\tilde X$ , and therefore $\tilde S$ is a divisor. Since $(X',S')$ is plt, then so is $(\tilde X,\tilde S)$ . We denote by $\tilde W$ the relatively ample model, which is birational to W. Now, we run a ${K_{\tilde X}}$ -MMP with scaling over $\tilde W$ , which terminates with a Mori fiber space $\hat f \colon \hat X \rightarrow \hat W$ over $\tilde W$ . Since $\dim X -\dim W =1$ , it follows that $\hat W \rightarrow \tilde W$ is birational. As before, since $\tilde S$ dominates $\tilde W$ and is relatively big over $\tilde W$ as $\dim X - \dim W =1$ , it follows that $\tilde S$ cannot be contracted on $\hat X$ . Let $\hat S$ be its strict transform on $\hat X$ . Lastly, as ${K_{\tilde X}} + \tilde S \sim _{\mathbb {Q}} 0 /\tilde W$ and $\tilde S$ is the only log canonical place of $(\tilde X,\tilde S)$ , $\hat S$ is the only log canonical place of $(\hat X,\hat S)$ . But then, since $\hat S$ is a divisor on $\hat X$ , it follows that $(\hat X,\hat S)$ is plt. In particular, $\hat S$ is normal. Thus, by [Reference Filipazzi, Mauri and Moraga13, Corollary 3.3], up to replacing X, S and Z with $\hat X$ , $\hat S$ and $\hat Z$ , respectively, in the following of the proof, we may further assume that $(X,S)$ is plt, $K_X +S \sim _{\mathbb {Q}} 0/W$ and f is a Mori fiber space.

Step 4: In this step, we introduce a suitable pair structure on the base W.

Let $(S,B_S)$ denote the pair induced by adjunction from $(X,B)$ to S. By Lemma 2.23, $(S,B_S)$ has coregularity c. Then, by Lemma 2.17, the inductive hypothesis applies to $(S,B_{S})$ for the same value of I. We also let $(S,\mathrm {Diff}_S(0))$ be the pair structure induced from $(X,S)$ to S. By Step 1, $f_S \colon S \rightarrow W$ is generically 2:1 and hence Galois. By [Reference Kollár29, Proposition 4.37.(3)], $(S,\mathrm {Diff}_S(0))$ is invariant under the rational Galois involution. Then, since ${K_{X}} + S \sim _{\mathbb {Q}} 0 /W$ and f is a Mori fiber space, it follows that $B-S$ is the pull-back of a $\mathbb {Q}$ -divisor on W. Then, it follows that also $(S,B_S)$ is Galois invariant. Then, by considering the Stein factorization of $S \rightarrow W$ and descending the pair structure thanks to the fact that ${K_{S}} +B_S \sim _{\mathbb {Q}} 0$ , it follows that we can induce a pair structure $(W,B_W)$ such that $f_S^*({K_{W}} + B_W)={K_{S}} + B_S$ . Since $(S,B_S)$ has coregularity c, by [Reference Filipazzi, Mauri and Moraga13, Proposition 3.11], then so does $(W,B_W)$ . Furthermore, since $(S,B_S)$ has coefficients in $D_\lambda $ , then so does $(W,B_W)$ . Indeed, at the codimension 1 points of W where $S \rightarrow W$ is étale, we will have the same coefficients on $B_W$ and $B_S$ for the corresponding divisors. Then, we can consider a prime divisor $Q \subset W$ such that $S \rightarrow W$ ramifies of order 2 at the generic point of Q. Then, over the generic point of Q, by the Riemann–Hurwitz formula, we have

$$\begin{align*}{K_{S}}+ cP=f_S^*\left(K_W+\frac{1}{2}Q+\frac{c}{2}Q \right), \end{align*}$$

where P is the unique prime divisor in S dominating Q, in other words $\operatorname {\mathrm {coeff}}_P(B_S)= c \in D_\lambda $ and $\operatorname {\mathrm {coeff}}_Q(B_W)=\frac {1+c}{2}$ . By the definition of $D_\lambda $ , we must have $c=\frac {m-1+a\lambda ^{-1}}{m}$ for some $m \in \mathbb {Z}_{>0}$ and $a \in \mathbb {Z}_{\geq 0}$ , and it follows that

$$\begin{align*}\frac{1}{2}+\frac{m-1+a\lambda^{-1}}{m}=\frac{2m-1+a\lambda^{-1}}{2m} \in D_\lambda. \end{align*}$$

Thus, by the inductive hypothesis, we have

(2.5) $$ \begin{align} I(K_W+B_W)\sim 0. \end{align} $$

Step 5: In this step, we introduce a suitable generalized pair structure on W, and we compare it with $(W,B_W)$ .

By the canonical bundle formula, the lc-trivial fibration $f \colon (X,B) \rightarrow W$ induces a generalized pair structure $(W,\Delta _W,\mathbf {N})$ on W. By [Reference Prokhorov and Shokurov38, § 7.5, (7.5.5)] and the fact that the generic fiber of $f \colon (X,B) \rightarrow W$ is a conic with two points, we have

(2.6) $$ \begin{align} {K_{X}} + B \sim f^*({K_{W}} + \Delta_W+ {\mathbf{N}_{W}}). \end{align} $$

Furthermore, the representatives of the b-divisor $\mathbf {N}$ are determined up to $\mathbb {Z}$ -linear equivalence.

By [Reference Fujino and Gongyo20, proof of Theorem 1.1], in an lc-trivial fibration $(Y,\Gamma ) \rightarrow C$ , the total space of the fibration and a minimal (with respect to inclusion) log canonical center $\Xi $ dominating the base C induce the same generalized pair on the base space. More precisely, they induce the same boundary divisor as $\mathbb {Q}$ -Weil divisor and the same moduli divisor as $\mathbb {Q}$ -b-divisor class. Yet, this comparison is possible after the base change induced by the Stein factorization of $\Xi \rightarrow C$ , as $\Xi \rightarrow C$ may not have connected fibers. We observe that the identification between the boundary divisors can also be obtained by inversion of adjunction together with the connectedness principle.

In our situation, this implies that $(X,B)$ and $(S,B_S)$ induce the same generalized pair on W, up to pull-back to the Stein factorization on $S \rightarrow W$ . In this case, S is generically a 2:1 cover of W. Thus, it follows that $(W,\Delta _W,\mathbf {N})$ and $(W,B_W)$ agree once pulled back to S. By construction, we have $f_S^{\prime *}({K_{W}} + B_W)={K_{S}} + B_S$ , and the moduli b-divisor is trivial. In turn, this implies that $\Delta _W=B_W$ and $f_S^* {\mathbf {N}_{W}} \sim 0$ . As for the moduli b-divisor, we only claim $\mathbb {Z}$ -linear equivalence, as a representative of the b-divisorial class can be replaced in the $\mathbb {Z}$ -linear equivalence class.

Let $S'$ denote the Stein factorization of $S \to W$ , with induced morphism $f_{S'} \colon S' \rightarrow W$ . Then, $f_{S'}$ is a finite Galois morphism of degree 2, $f_{S'}^* {\mathbf {N}_{W}} \sim 0$ . By construction, $f_{S'}^* {\mathbf {N}_{W}}$ is Galois invariant since it is the $\mathbb {Q}$ -Cartier pull-back of a $\mathbb {Q}$ -divisor on W via the finite morphism $f_{S'}$ . We observe that this implies that $2 {\mathbf {N}_{W}}$ is a $\mathbb {Z}$ -divisor and that $2f_{S'}^* {\mathbf {N}_{W}}$ is the integral pull-back of the integral divisor $2 {\mathbf {N}_{W}}$ .

Now, let s be a trivializing section of $f_{S'}^* {\mathbf {N}_{W}}$ , and let $\tau $ be the nontrivial element in the Galois group of $S' \rightarrow W$ . By the invariance of $f_{S'}^* {\mathbf {N}_{W}}$ , we have that $f_{S'}^* {\mathbf {N}_{W}} + \tau ^*f_{S'}^* {\mathbf {N}_{W}}=2f_{S'}^* {\mathbf {N}_{W}}$ . But then, $s \otimes \tau ^*s$ is a Galois invariant trivializing section of $2f_{S'}^* {\mathbf {N}_{W}}$ . Then, this section descends to W, thus implying that

(2.7) $$ \begin{align} 2{\mathbf{N}_{W}} \sim 0. \end{align} $$

Step 6: In this step, we conclude the proof.

Combining the previous steps and using the fact that $2|I$ , we have

$$ \begin{align*} I({K_{X}} + B) &\sim If^*({K_{W}} + B_W + {\mathbf{N}_{W}})\\ &\sim If^*({K_{W}}+B_W) + If^* {\mathbf{N}_{W}}\\ &\sim f^*(I({K_{W}} + B_W)) + f^* I{\mathbf{N}_{W}}\\ &\sim f^*0 + f^*0 \sim 0, \end{align*} $$

where the first linear equivalence follows from Equation (2.6), the second one follows from the fact that $K_W+B_W$ is $\mathbb {Q}$ -Cartier, the third one follows from the definition of pull-back of $\mathbb {Q}$ -divisors, while the last line follows from Equations (2.5) and (2.7). This concludes the proof.