Proof. Let $f\colon X' \rightarrow X$ be a log resolution of $(X,B)$ , where $\mathbf {M}$ descends. For the sake of simplifying notation, we will use $M'$ to denote $\mathbf {M}_{X'}$ . By Hironaka’s theorem, we can assume that f is obtained by blowing up loci of codimension at least two, and that there exists an effective f-exceptional divisor $C'$ , such that $-C'$ is f-ample. We define $B'$ via the identity

$$ \begin{align*} {K_{X'}} + B' + M' = f^{\ast} ({K_{X}} + B + M ). \end{align*} $$

In view of this definition, we can decompose $B'$ as $B'=f_*^{-1} \lbrace B \rbrace + E^+ + F' - G'$ , where

• • $E^+$ denotes the (not necessarily f-exceptional) divisors with generalized log discrepancy at most $0$ with respect to $(X, B, \mathbf {M})$ ;

• • $F'$ the sum of all f-exceptional divisors with generalized log discrepancy in $(0, 1]$ ; and,

• • $G'$ the sum of all f-exceptional divisors with generalized log discrepancy $>1$ .

We define

. Letting H be a sufficiently ample divisor on X, for all $\epsilon , \nu , \tau \in \mathbb {R}$ ,

(1) $$ \begin{align} E' + (1 + \nu)F' + \tau(-C'+f^{\ast} H)+M' = (1-\epsilon \tau)E' + (1+ \nu)F' + \tau(\epsilon E' - C' + f^{\ast} H) + M', \end{align} $$

and for any $0< \epsilon \ll \tau $ and $\epsilon \ll 1$ , both $\tau (-C'+f^{\ast } H)+M'$ and $\tau (\epsilon E' - C' + f^{\ast } H) + M'$ are ample over X. For any such choice of $\epsilon $ and $\tau $ , we can choose divisors $H^{\prime }_{1,\tau } \sim _{\mathbb {R},X} \tau (-C'+f^{\ast } H)+M'$ and $H^{\prime }_{2,\tau , \epsilon } \sim _{\mathbb {R},X} \tau (\epsilon E' - C' + f^{\ast } H) + M'$ , such that $B'+H^{\prime }_{1,\tau }+H^{\prime }_{2,\tau , \epsilon }$ has simple normal crossing support, and $\lfloor H^{\prime }_{1,\tau } \rfloor = \lfloor H^{\prime }_{2,\tau , \epsilon } \rfloor =0$ . Thus, if $0 < \tau <1 $ and $0 < \nu \ll 1$ , the pair

is klt. By [Reference Birkar, Cascini, Hacon and McKernan6], we can run a $(K_{X'} + \Delta ^{\prime }_{\epsilon , \tau , \nu })$ -MMP over X that terminates with a relative $\mathbb {Q}$ -factorial minimal model

$$ \begin{align*} f^m_{\epsilon,\tau,\nu} \colon (X^m_{\epsilon,\tau,\nu}, \Delta^m_{\epsilon,\tau,\nu}) \rightarrow X. \end{align*} $$

By (1), $f^m_{\epsilon ,\tau ,\nu }$ is also a minimal model over X for the pair

As the dlt property is preserved under steps of the MMP [Reference Kollár and Mori27, Corollary 3.44], both $(X^m_{\epsilon ,\nu , \tau }, \Delta ^m_{\epsilon ,\nu , \tau })$ and $(X^m_{\epsilon ,\nu , \tau }, \Gamma ^m_{\nu , \tau })$ are dlt pairs, where $\Delta ^m_{\epsilon ,\nu , \tau }$ (respectively, $\Gamma ^m_{\nu , \tau }$ ) is the pushforward of $\Delta ^{\prime }_{\epsilon ,\nu , \tau }$ (respectively, $\Gamma ^{\prime }_{\nu , \tau }$ ) on $X^m_{\epsilon , \nu , \tau }$ . Hence, the pair $(X^m_{\epsilon ,\nu , \tau },B^m_{\epsilon ,\nu , \tau })$ is dlt, where $B^m_{\epsilon ,\nu , \tau }$ is the pushforward of $f^{-1}_* \lbrace B \rbrace +E' +F'$ on $X^m_{\epsilon ,\nu , \tau }$ .

To simplify the notation, we will denote by $A^m$ the strict transform on $X_{\epsilon ,\nu , \tau }^m$ of any divisor A on $X'$ . In particular, we will denote $\mathbf {M}_{X^m_{\epsilon , \nu , \tau }}$ by $M^m$ . Then, we define

The divisor N is $f^m_{\epsilon ,\nu , \tau }$ -nef, while T is $f^m_{\epsilon ,\nu , \tau }$ -trivial, and

so that $-D$ is $f_{\epsilon ,\nu , \tau }^m$ -nef and $f_{\epsilon ,\nu , \tau \ast }^m D \geq 0$ . Therefore, by the negativity lemma [Reference Kollár and Mori27, Lemma 3.39], D is effective.

As $C'$ , $E^+-E'$ , $F'$ and $G'$ are independent of ${\epsilon ,\nu , \tau }$ , if we choose $0 < \epsilon \ll \tau \ll \nu \ll 1$ , the $(K_{X'} + \Delta ^{\prime }_{\epsilon , \nu , \tau })$ -MMP contracts $F'$ and $G'$ , as

$$ \begin{align*} K_{X'} + \Delta^{\prime}_{\epsilon, \nu, \tau} \sim_{\mathbb{R}, f} G' + \nu F' -\tau C' -(E^+-E'). \end{align*} $$

Indeed, as the effective divisors $G'$ , $F'$ , and $E^+-E'$ share no prime components, $\operatorname {\mathrm {Supp}}(C') \subset \operatorname {\mathrm {Supp}}(E'+F'+G')$ , and $\tau \ll \nu $ , for every prime divisor $P'$ on $X' \ \mu _{P'} (G' + \nu F' -\tau C' -(E^+-E'))> 0$ if and only if $\mu _{P'} (G' + \nu F' -(E^+-E'))>0$ . We fix once and for all such a choice of the coefficients $\epsilon , \tau , \nu $ , and we drop the dependence from ${\epsilon ,\nu , \tau }$ in our notation.

The generalized pair $(X^m,B^m+M^m)/Z$ is generalized log canonical. In fact, $f^{\ast } H-C'$ is ample, as H is assumed to be sufficiently ample; picking $0 \leq A' \sim _{\mathbb {Q}} \tau (f^{\ast } H - C')$ a general element in its $\mathbb {Q}$ -linear equivalence class, so that $(X',f_*^{-1} \lbrace B \rbrace + E' + (1+\nu )F' + A')$ is dlt by Bertini’s theorem, the generalized pair $(X',f_*^{-1} \lbrace B \rbrace + E' + (1+\nu )F' + A', \mathbf {M})/Z$ is generalized dlt, and each step in the $(K_{X'} + \Delta ^{\prime }_{\epsilon , \nu , \tau })$ -MMP leading to $X^m$ is a $({K_{X'}} + f_*^{-1} \lbrace B \rbrace + E' + (1+\nu )F' + A'+M')$ -negative contraction. Therefore, the generalized pair $(X^m,B^m+A^m,\mathbf {M})/Z$ is generalized log canonical. As $X^m$ is $\mathbb {Q}$ -factorial, then $(X^m,B^m, \mathbf {M})/Z$ is generalized log canonical. Thus, $(X^m,B^m+M^m)$ has all the claimed properties, besides the fact that it may not be generalized dlt.

To conclude, it suffices to substitute $(X^m,B^m+M^m)$ with a generalized dlt model, which exists by [Reference Birkar5, 2.13.(2)] as $(X^m,B^m+M^m)$ is log canonical. Passing to such model only extracts divisors with generalized log discrepancy $0$ with respect to $(X^m,B^m+M^m)$ . This completes the proof of the theorem.

Proof. Let $\pi \colon X' \rightarrow X$ be as in the statement. Let H be a q-ample divisor that is fully supported on $B^{=1}$ . Since X is $\mathbb {Q}$ -factorial, by [Reference Kollár and Mori27, Lemma 2.62], there is an effective divisor F that is fully supported on the $\pi $ -exceptional divisors, such that $-F$ is $\pi $ -ample. Thus, for $0 < \epsilon \ll 1$ , $\pi ^{\ast } H-\epsilon F$ is ample over Z. By assumption, the $\pi $ -exceptional divisors all have center on $\operatorname {\mathrm {Supp}} (H ) = \operatorname {\mathrm {Supp}} (B^{=1})$ . Therefore, if $\epsilon $ is small enough, $\pi ^{\ast } H-\epsilon F$ is effective. By definition of a dlt model, $\operatorname {\mathrm {Supp}} \mathrm {Exc}(\pi ) \subset \operatorname {\mathrm {Supp}}((B')^{=1})$ . Thus, $\operatorname {\mathrm {Supp}}(\pi ^{\ast } H-\epsilon F) = \operatorname {\mathrm {Supp}} ((B')^{=1})$ , and the claim follows.

Proof. This follows at once, since $\pi $ only extracts divisors appearing in $\operatorname {\mathrm {Supp}}((B^m)^{=1})$ and the condition that $(X,B^{<1} + (1-\epsilon )B^{=1},\mathbf {M})/Z$ is generalized klt implies that $\pi (\mathrm {Exc}(\pi )) \subset B^{=1}$ .

Sketch of proof.

Throughout the proof, S will be a quasiprojective variety, with no further assumption. For the reader’s convenience, we subdivide the proof into several steps.

Step 1: In this step, we show that the statement of the theorem holds if we assume that $\mathbf {M}$ is b-semiample.

Let $X'$ be a model where $\mathbf {M}$ descends. For brevity, we set . Let $h \colon X' \rightarrow T$ be the induced morphism. Let $\mathcal {U} \subset |M'|_{\mathbb {Q}}$ be the set consisting of $\mathbb {Q}$ -divisors $0 \leq \Delta ' \sim _{\mathbb {Q}} M'$ , such that $(X',B'+\Delta ')$ is sublog canonical over the generic point of T. As $M'$ is semiample, $\mathcal {U}$ is nonempty. Given $\Delta ' \in \mathcal {U}$ , we can apply Theorem 2.17 to $(X',B'+\Delta ') \rightarrow T$ , thus obtaining a $\mathbb {Q}$ -b-Cartier b-divisor ${\mathbf N}^{\Delta '}$ that is b-nef over S. As discussed in [Reference Filipazzi10, Remark 4.8], we have

$$ \begin{align*} \quad \mathbf{B} = \inf_{\Delta' \in \mathcal{U}} \mathbf{B}^{\Delta'}, \; {\mathbf N}= \sup_{\Delta' \in \mathcal{U}} {\mathbf N}^{\Delta'}. \end{align*} $$

We wish to apply weak semistable reduction to argue that ${\mathbf N}$ is a $\mathbb {Q}$ -b-Cartier b-divisor that descends to a model satisfying explicit properties (see the proof of [Reference Filipazzi10, Theorem 4.13]). The proof of the b-nefness of ${\mathbf N}$ over S goes then through as explained in the proof of [Reference Filipazzi10, Theorem 4.15]. One first shows that the statement is true if $\dim T = 2$ , cf. [Reference Filipazzi10, Remark 4.14]. To conclude, we reduce to the case of dimension $2$ by taking general hyperplane cuts on T, cf. the proof of [Reference Filipazzi10, Theorem 4.15].

Step 2: In this step, we show that the theorem holds when $\mathbf {M}$ is b-semiample over T.

Let $X'$ and $M'$ be the objects constructed in Step 1. Let A be a divisor on T which is ample over S. Since $M'$ is nef over S, and hence over T, and it is semiample over T, $M'+ \epsilon h^{\ast } A$ is semiample over S for any $\epsilon> 0$ , cf. [Reference Filipazzi10, Proposition 4.7]. Fix $\epsilon> 0$ with $\epsilon \in \mathbb {Q}$ . Let H be an ample divisor on S and write $p \colon X' \rightarrow S, \; s\colon T \rightarrow S$ . Hence, for $n=n(\epsilon ) \gg 0, \; M'+\epsilon h^{\ast } A + n p^{\ast } H$ is semiample. We consider the generalized subpairs $(X',B',\mathbf {M} +\epsilon \overline { g^{\ast } A})/S$ and $(X',B',\mathbf {M} +\epsilon \overline {g^{\ast } A} + n \overline {p^{\ast } H})/S$ . Let $\mathbf {B}^{\epsilon }$ , ${\mathbf N} ^{\epsilon }$ and $\mathbf {B}^{\epsilon ,n}$ , ${\mathbf N} ^{\epsilon ,n}$ be the b-divisors, respectively, induced on T. By Remark 2.5, the generalized discrepancies of $(X,B,\mathbf {M})/S$ , $(X',B',\mathbf {M} +\epsilon \overline {g^{\ast } A})/S$ , and $(X',B',\mathbf {M} +\epsilon \overline { g^{\ast } A }+ n \overline {p^{\ast } H})/S$ agree by construction. Thus, we have

$$ \begin{align*} \mathbf{B} = \mathbf{B}^{\epsilon} = \mathbf{B}^{\epsilon,n} , \qquad {\mathbf N} {^{\epsilon,n}} = {\mathbf N} ^{\epsilon} + n \overline{s^{\ast} H}, \qquad {\mathbf N} ^{\epsilon} = {\mathbf N} + \epsilon \overline{A}. \end{align*} $$

Therefore, if for some $(\epsilon ,n)$ the b-divisor ${\mathbf N} {^{\epsilon ,n}}$ is $\mathbb {Q}$ -b-Cartier, then so are ${\mathbf N}$ and ${\mathbf N} ^{\epsilon '}$ for any $\epsilon '>0$ and all these b-divisors descend to the same model of T. We can then apply Step 1 to the generalized pair $(X',B',\mathbf {M} + \epsilon \overline {g^{\ast } A} + n\overline { p^{\ast } H})/S$ together with the morphism $h \colon X' \rightarrow T$ and obtain $\mathbb {Q}$ -b-Cartier b-divisor ${\mathbf N}^{\epsilon ,n}$ which is b-nef $/S$ . Since we have ${\mathbf N}^{\epsilon ,n} = {\mathbf N}^{\epsilon } + n\overline {s^{\ast } H}$ and $\overline {s^{\ast } H}$ is trivial over S, ${\mathbf N} ^{\epsilon }$ is a $\mathbb {Q}$ -b-Cartier b-divisor that is b-nef $/S$ . We let $\epsilon>0$ vary and approach 0. Thus, as ${\mathbf N}$ is a limit of $\mathbb {Q}$ -b-Cartier b-divisors that all descend to the same model of T and are all b-nef over S, then ${\mathbf N}$ is a $\mathbb {Q}$ -b-Cartier b-divisor that is b-nef over S.

Step 3: In this step, we show that the statement of the theorem holds when X is $\mathbb {Q}$ -factorial klt, $\rho (X/T)=1$ and $\mathbf {M}_X$ is relatively ample over T.

The proof of this case goes through as in [Reference Filipazzi10, Lemma 5.2] and reduces to Step 2. More precisely, since $\mathbf {M}$ is b-nef over S and ${\mathbf {M}_{X}}$ is ample over T, we can approximate $\mathbf {M}$ with b-divisors that are b-nef over S and b-semiample over T. The fact that X is a $\mathbb {Q}$ -factorial klt variety allows us to move the difference between $\mathbf {M}$ and its approximation to the boundary part of a generalized pair. So, we can regard $(X,B,\mathbf {M})$ as being approximated by generalized subpairs to which Step 2 can be applied. Since the boundary parts of these approximations can be arranged to have the same support, we can apply the observation in Step 1 about the explicit description of the model where the b-divisors descend. This allows us to conclude, as $\mathbf {N}$ is realized as a limit of $\mathbb {Q}$ -b-Cartier b-divisors that descend to the same model, and hence it is $\mathbb {Q}$ -b-Cartier itself.

Step 4: In this step, we show that the statement of the theorem holds in its full generality.

By Theorem 2.9, we can assume that X is $\mathbb {Q}$ -factorial and $(X,B^h)$ is dlt. If $\mathbf {M}_X$ is numerically trivial along the generic fiber of $X \rightarrow T$ , we can reduce to Theorem 2.17. If $\mathbf {M}_X$ is not numerically trivial along the generic fiber of $X \rightarrow T$ , $(X,B^h)$ is not pseudo-effective over T. Thus, we can run a $({K_{X}} + B^h)$ -MMP over T with scaling, which terminates with a Mori fiber space $X" \rightarrow U$ . Thus, we can apply Step 3 to $(X",B",\mathbf {M})/S$ and $X" \rightarrow U$ . By [Reference Filipazzi10, Lemma 5.1], we can then conclude inductively.

Proof. Let $X^r \rightarrow X$ be a resolution of X, such that $\mathbf {M}$ descends to $X^r$ . Let $X^f \rightarrow T^f$ be a flattening of $X^r \rightarrow T$ , see [Reference Raynaud and Gruson33, Théorème 5.2.2], and let $\widetilde {T}$ be a resolution of $T^f$ . Let $\widetilde {X}$ be the normalization of the main component of $X^f\times _{T^f}\widetilde {T}$ . This ensures that $\widetilde {X}\to \widetilde {T}$ still has equidimensional fibers. Since ${\mathbf {M}_{X}}$ is torsion along the generic fiber of $X \rightarrow T$ , the same holds true for ${\mathbf {M}_{\widetilde {X}}}$ along the generic fiber of $\widetilde {X}\to \widetilde {T}$ by the negativity lemma. Thus, by the proof of [Reference Witaszek37, Lemma 2.18], under our assumptions,Footnote ¹ it follows that ${\mathbf {M}_{{\widetilde {X}}}} \sim _{\mathbb {Q},\widetilde {T}} 0$ . Since $\mathbf {M}$ is the b-Cartier closure of $\mathbf {M}_{\widetilde {X}}$ , the conclusion of the statement follows at once.

Proof. As $\mathbf {M}_{X'}$ is nef and big, and it descends to the model $X'$ fixed before the lemma, there exists an effective divisor E, such that $\mathbf {M}_{X'} \sim _{\mathbb {Q}} A_k+E_k$ , where $A_k$ is ample for (see [Reference Kollár and Mori27, Proposition 2.61]). As E may not be simple normal crossing, it suffices to replace $X'$ with a log resolution of $(X, E)$ and define $A^{\prime }_k$ , $E'$ to be their pullback on the new model.

Proof. Since by construction

$$ \begin{align*} {K_{X_i^m}} +B_i^m+\Gamma_i^m \leq \pi_i^{\ast} ({K_{X}} + B + \Gamma), \end{align*} $$

$(X_i",B_i"+\Gamma _i")$ is also a dlt model for $(X,B+\Gamma )$ . Therefore, by [Reference Nakamura31, Proposition 2.14], $\mathcal {D}((B_1"+\Gamma _1")^{=1})$ and $\mathcal {D}((B_2"+\Gamma _2")^{=1})$ are simple homotopy equivalent to each other, as they compute the dual complex of $(X,B+\Gamma )$ .

Proof of Lemma 2.34.

By construction, the exceptional divisors of $X_i" \rightarrow X_i^m$ appear as divisors on $X'$ . Furthermore, as already observed, a divisor $E \subset X'$ satisfies $a_E(X_i^m,B_i^m+\Gamma _i^m) \leq 0$ if and only if $a_E(X^m_i,B^m_i) \leq 0$ . Therefore, $X_i" \rightarrow X_i^m$ only extracts divisors E with $a_E(X^m_i,B^m_i) \leq 0$ . Hence, the variety $X^{\prime \prime }_i$ provides a dlt model for the dlt pair $(X^m_i,B^m_i)$ . Hence, $\mathcal {D}((B_i"+\Gamma _i")^{=1})$ computes $\mathcal {DMR}(X_i^m,B_i^m)$ . As $(X^m_i,B^m_i)$ is dlt, by [Reference de Fernex, Kollár and Xu8, Proposition 11], $\mathcal {DMR}(X_i^m,B_i^m)$ is a well-defined PL-homeomorphism class. Therefore, $\mathcal {D}((B_i^m)^{=1})$ is PL-homeomorphic to $\mathcal {D}((B_i"+\Gamma _i")^{=1})$ for $i=1,2$ . Thus, $\mathcal {D}((B_i^m)^{=1})$ is simple homotopy equivalent to $\mathcal {D}((B_i"+\Gamma _i")^{=1})$ for $i=1,2$ .

Proof. Let $\pi \colon X' \rightarrow X$ be a log resolution for $(X,B)$ , where $\mathbf {M}$ descends. Thus, we may write

where $E' \geq 0$ , $F' \geq 0$ , $E' \wedge F' = 0$ , $E'-F'$ is $\pi $ -exceptional and $\pi _*^{-1} (B) + E' - F'$ has simple normal crossing support. Thus,

$$ \begin{align*} \pi^{\ast} L + \lceil F' \rceil = {K_{X'}} + \pi_*^{-1} (B) + E' + (\lceil F' \rceil - F') + M' + \pi^{\ast} H. \end{align*} $$

As $M' + \pi ^{\ast } H$ is nef and big, by Kawamata–Viehweg vanishing [Reference Kollár and Mori27, Theorem 2.64], we have $H^i(X',\pi ^{\ast } L + \lceil F' \rceil )=0$ for $i>0$ . Similarly, by [Reference Hacon and Kovács17, Theorem 3.45], we have $R^i\pi _* {\mathcal {O}_{ X'}} (\pi ^{\ast } L + \lceil F' \rceil )=0$ for $i>0$ . Since $\lceil F' \rceil $ is $\pi $ -exceptional, the projection formula implies $\pi _* {\mathcal {O}_{X'}}(\pi ^{\ast } L + \lceil F' \rceil )={\mathcal {O}_{X}} (L)$ . Hence, we conclude that $H^i(X,L)=0$ for $i>0$ .

Proof. If $(X,B, \mathbf {M})$ is generalized klt, then it suffices to take $Z=\overline {X}=X$ . Hence, we can assume that $\operatorname {\mathrm {Nklt}}(X,B, \mathbf {M}) \neq \emptyset $ .

Step 0. In this step, we reduce to the case when the dual complex is connected.

If $\operatorname {\mathrm {Nklt}}(X,B, \mathbf {M})$ is disconnected, the statement follows from Theorem 1.1 and its proof. More precisely, as $(X,B,\mathbf {M})$ is already $\mathbb Q$ -factorial generalized dlt, we may apply the algorithm in the proof of Theorem 1.1 to X: we run a $(K_X+B^{<1}+{\mathbf {M}_{X}})$ -MMP $X \dashrightarrow \overline {X}$ , which will terminate, by construction, with a Mori fiber space $\overline {X} \rightarrow Z$ , the morphism whose existence is claimed in the statement of 4.2. By Theorem 1.1, items (1)–(2) in the statement of 4.2 hold. By construction, $X \dashrightarrow \overline {X}$ is a birational contraction, so item (3) is satisfied. Lastly, this MMP is a $(-B^{=1})$ -MMP, and thus by the results in Section 3, also item (4) is satisfied. Therefore, we can assume that $\operatorname {\mathrm {Supp}}(B^{=1})$ is connected and nonempty.

Step 1. In this step, we construct a birational map $X \dashrightarrow X'$ and a fibration $X' \rightarrow Z$ satisfying properties (1), (3), and (4) in the statement of the theorem.

We run a $({K_{X}} + B^{<1} +M)$ -MMP with scaling of an ample divisor

(3)

and we define , and . As ${K_{X}} + B^{<1} + M \sim _{\mathbb {Q}} -B^{=1}$ , each step of the $({K_{X}} + B^{<1} +M)$ -MMP in (3) is $(B^{=1})$ -positive; moreover, this run of the MMP terminates with a Mori fiber space $q \colon X' \rightarrow Z$ and we have an induced generalized pair $(X', B', \mathbf {M})$ , , such that $K_{X'}+B' +\mathbf {M}_{X'} \sim _{\mathbb {Q}} 0$ . As each step of (3) is $(B^{=1})$ -positive, the indeterminacy locus of the inverse map $\pi _i^{-1} \colon X_{i} \dashrightarrow X_{i-1}$ is contained in $B_i^{=1}$ . This implies that property (4) in the statement of the theorem holds for $(X', B', \mathbf {M})$ ; property (1) is satisfied as q is a Mori fiber space, while (3) holds, by construction, since $X \dashrightarrow X'$ is a birational contraction.

If property (2) holds for $(X', B', \mathbf {M})$ , then the proof of the theorem is completed. Hence, we shall assume that not all generalized log canonical centers of $(X',B', \mathbf {M})$ dominate Z.

Step 2. In this step, assuming property (2) does not hold for $(X', B', \mathbf {M})$ , we construct a birational map $X \dashrightarrow \widetilde {X}$ over Z, a generalized pair $(\widetilde X, \widetilde B, \mathbf {M})$ crepant to $(X, B, \mathbf {M})$ , and a commutative diagram

where $\widetilde {q}$ is a Mori fibration, $\widetilde {Z} \rightarrow Z$ is a birational contraction, and all the log canonical centers of $(\widetilde X, \widetilde B, \mathbf {M})$ that do not dominate Z are fully supported on the pullback of a divisor on $\widetilde Z$ . Moreover, $(\widetilde X, \widetilde B, \mathbf {M})$ satisfies properties (3), (4), and $\widetilde {B}^{\hspace{1pt}=1}$ fully supports an effective divisor that is relatively ample over $\widetilde {Z} \setminus \widetilde {q}(\widetilde {B} {^{=1,v}})$ .

By construction, $(X', B', \mathbf {M})$ is $\mathbb {Q}$ -factorial and generalized dlt away from $B^{\prime =1}$ . We pass to a generalized dlt model $\psi \colon X" \rightarrow X'$ ,

Property (3) holds on $X"$ by construction. The morphism $\psi $ is an isomorphism over $X' \setminus B^{\prime =1}$ , hence, property (4) holds on $X"$ as well. Lemma 2.13 implies that property (1) is also preserved.

Thus, up to substituting $X'$ with $X"$ , we can assume that $(X',B', \mathbf {M})$ is $\mathbb Q$ -factorial and generalized dlt, and that any of its log canonical places is a log canonical place for the pair $(X',B')$ .

To simplify the notation, we write .

By construction, $B^{\prime =1}$ fully supports an effective $\mathbb {Q}$ -divisor H ample over Z, hence, for $0 < \epsilon \ll 1$ , the support of $B^{\prime =1} -\epsilon H$ coincides with that of $B^{\prime =1}$ , and, moreover, $\lfloor B^{\prime =1} -\epsilon H \rfloor =0$ , so that

(4)

and $(X', B^{\prime <1} + (B^{\prime =1} -\epsilon H)^h + \mathbf {M}_{X'} + \epsilon H)$ is generalized klt over Z.

By [Reference Birkar and Zhang7, Lemma 4.4], we can run a $({K_{X'}} + B^{\prime <1} + (B^{\prime =1} -\epsilon H)^h + \mathbf {M}_{X'} + \epsilon H)$ -MMP with scaling of an ample divisor over Z, that terminates with a relatively good minimal model/Z

(5)

We denote by $\widetilde B, \widetilde F, \widetilde H$ the strict transforms of $B', F', H$ on $\widetilde X$ . The morphism $\widetilde Z \rightarrow Z$ is birational, as $\overline F$ is vertical $/Z$ . Since the MMP in (5) is positive for $F'$ by construction, then $\widetilde F> 0$ and $\widetilde F \sim _{\mathbb {Q}, \widetilde q} 0$ . The map $X \dashrightarrow \widetilde X$ satisfies properties (3) and (4) in view of [Reference Kollár and Xu28, Lemma 55] and the fact that $\operatorname {\mathrm {Supp}}(F') \subset \operatorname {\mathrm {Supp}}(B^{\prime =1})$ .

On the other hand, property (1) may no longer hold on $\widetilde X$ . Nevertheless, the following claim holds.

Step 3. In this step, we show that there exist generalized pairs $(\widetilde Z, B_{\widetilde Z}, \mathbf {M}_{\widetilde Z})$ , $(\widetilde Z, \Delta _{\widetilde Z}, \mathbf {M}_{\widetilde Z})$ on $\widetilde Z$ , such that $K_{\widetilde Z}+ B_{\widetilde Z}+ \mathbf {M}_{\widetilde Z} \sim _{\mathbb {Q}} 0$ and $-(K_{\widetilde Z}+ \Delta _{\widetilde Z}+ \mathbf {M}_{\widetilde Z})$ is pseudo-effective.