Proof. In this proof, we will use the notion of generalized sub-lc pair, whose definition is the same as that of generalized lc pairs except that the boundary part is not necessarily effective. This is an analogous definition of sub-lc pairs.

First, we reduce the lemma to the case, where $\lfloor \Delta \rfloor =S$ . We can take an open subset $U\subset X$ such that $(U,\Delta |_{U})$ is log smooth, U contains the generic points of all generalized lc centers of $(X,\Delta ,\mathbf {M})/Z$ and $\mathbf {M}|_{U}$ descends to U. We take a log resolution $f \colon Y \to X$ of $(X,\Delta )$ such that

• ○ f is an isomorphism over U, and

• ○ $-E$ is f-ample for some effective f-exceptional divisor E on Y.

We note that we do not assume $\mathbf {M}_{Y}$ to be nef over Z. We may write

$$ \begin{align*} K_{Y}+\Gamma+\mathbf{M}_{Y}=f^{*}(K_{X}+\Delta+\mathbf{M}_{X}). \end{align*} $$

Since U contains the generic points of all generalized lc centers of $(X,\Delta ,\mathbf {M})/Z$ , there is a real number $t>0$ such that $(Y, \Gamma +tE,\mathbf {M})/Z$ is generalized sub-lc. Take an ample divisor A on X such that $-\frac {t}{2}E+f^{*}A$ is ample. By perturbing the coefficients of $\Gamma $ with $-\frac {t}{2}E+f^{*}A$ and pushing down, we get an $\mathbb {R}$ -divisor $G \geq 0$ on X such that

• ○ $K_{X}+\Delta +\mathbf {M}_{X}+A\sim _{\mathbb {R},Z}K_{X}+G+\mathbf {M}_{X}$ ,

• ○ $(U,G|_{U})$ is log smooth,

• ○ $\lfloor G \rfloor =S$ and

• ○ writing $K_{Y}+G_{Y}+\mathbf {M}_{Y}=f^{*}(K_{X}+G+\mathbf {M}_{X})$ , then $(Y, G_{Y}, \mathbf {M})/Z$ is a generalized sub-lc pair whose generalized lc centers intersect $f^{-1}(U)$ .

In this way, we get a generalized dlt pair $(X, G, \mathbf {M})/Z$ such that $\lfloor G \rfloor =S$ .

In the case where $\lfloor \Delta \rfloor =S$ , we see that $(U,\Delta |_{U})$ is plt. Hence, $a(P,X,\Delta +\mathbf {M}_{X})>0$ for all prime divisors P over X except $P=S$ . Let $g \colon Y' \to X$ be a log resolution of $(X,\Delta )$ such that

• ○ $\mathbf {M}$ descends to $Y'$ , and

• ○ $-F'$ is g-ample for some g-exceptional divisor $F'$ .

Let H be an ample divisor on X such that $-F'+g^{*}H$ is ample. Since $\mathbf {M}_{Y'}$ is nef over Z and $a(F_{i},X,\Delta +\mathbf {M}_{X})>0$ for every component $F_{i}$ of $F'$ , we can apply the argument of perturbation of coefficients. We get a plt pair $(X,B)$ such that $\lfloor B \rfloor =S$ .

Proof. The argument in [Reference Hashizume and Hu33, Proof of Lemma 2.11] works with no changes because we may apply the discussion in [Reference Fujino18, Section 3].

Proof. We need to prove that if S is X or an lc center of $(X,\Delta )$ with the normalization $S^{\nu }$ , then $(K_{X}+\Delta +M)|_{S^{\nu }}$ is big. Suppose by contradiction that there is an lc center S of $(X,\Delta )$ or $S=X$ such that $(K_{X}+\Delta +M)|_{S^{\nu }}$ is not big. By replacing M with $(1-t)M$ for some $0<t \ll 1$ , we may assume that $(K_{X}+\Delta +M)|_{S^{\nu }}$ is not pseudo-effective.

Taking a dlt blowup of $(X,\Delta )$ and applying Ambro’s canonical bundle formula as in [Reference Fujino and Gongyo19, Corollary 3.2], we get a generalized lc pair $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})$ such that

$$ \begin{align*} K_{S^{\nu}}+\Delta_{S^{\nu}}+\mathbf{N}_{S^{\nu}}\sim_{\mathbb{R}}(K_{X}+\Delta)|_{S^{\nu}}. \end{align*} $$

Let $(Y,\Gamma , \mathbf {N})$ be a $\mathbb {Q}$ -factorial generalized dlt model of $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})$ , and let $M_{Y}$ be the pullback of $M|_{S^{\nu }}$ to Y. Then $M_{Y}$ is big and $K_{Y}+\Gamma +\mathbf {N}_{Y}+M_{Y}$ is not pseudo-effective.

By running a $(K_{Y}+\Gamma +\mathbf {N}_{Y}+M_{Y})$ -MMP with scaling of an ample divisor, we get a birational contraction $\phi \colon Y \dashrightarrow Y'$ to a normal projective variety $Y'$ which has the structure of a Mori fiber space $Y' \to V$ with respect to $\phi _{*}(K_{Y}+\Gamma +\mathbf {N}_{Y}+M_{Y})$ . By the length of extremal rays (cf. [Reference Han and Li25, Proposition 3.17]), the birational transform of $M_{Y}$ is numerically trivial with respect to the extremal contraction in each step of the MMP. Thus, $\phi _{*}M_{Y}$ is big and $\phi _{*}M_{Y} \sim _{\mathbb {R}, V}0$ , a contradiction because $\mathrm {dim}Y'>\mathrm {dim}V$ .

In this way, we see that $(K_{X}+\Delta +M)|_{S^{\nu }}$ is big, and therefore $K_{X}+\Delta +M$ is log big with respect to $(X,\Delta )$ .

Proof. The idea is very similar to [Reference Fujino12]. We fix l.

We put

$$ \begin{align*} L=\mathbf{M}_{Y}+(lp-1) f^{*}(K_{X}+\Delta+\mathbf{M}_{X}). \end{align*} $$

Since $K_{X}+\Delta +\mathbf {M}_{X}$ is nef over Z and $\mathbf {M}_{Y}$ is nef and log big over Z with respect to $(Y,\Gamma )$ , it follows that L is nef and log big over Z with respect to $(Y,\Gamma )$ . We have

(1) $$ \begin{align} K_{Y}+\Gamma+L=lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X})+E. \end{align} $$

Since the coefficients of $p\Delta $ are integers and $p\mathbf {M}_{Y}$ is Cartier, we can write

(2) $$ \begin{align} lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X})=\lfloor lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X}) \rfloor + E' \end{align} $$

with an effective f-exceptional $\mathbb {Q}$ -divisor $E'$ . Put $T=f^{-1}_{*}S$ . By restricting sides of equation (2) to T and taking the round down, we have

(3) $$ \begin{align} \lfloor \bigl(lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X})\bigr)\big{|}_{T}\rfloor=\lfloor lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X}) \rfloor|_{T}. \end{align} $$

We define $\Gamma ':=\Gamma -T.$ By the definition, $(Y,T+\Gamma ')$ is a log smooth lc pair and L is nef and log big over Z with respect to $(Y,T+\Gamma ')$ . The relations (1) and (2) show

(4) $$ \begin{align} K_{Y}+T+\Gamma'+L=\lfloor lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X}) \rfloor + E'+E. \end{align} $$

We define a Weil divisor F on Y so that

$$ \begin{align*} \mathrm{coeff}_{P}(F)=\left\{ \begin{array}{l}{0 \qquad (\mathrm{coeff}_{P}(\Gamma'-\{(E+E')\})\geq 0)} \\ {1 \qquad (\mathrm{ coeff}_{P}(\Gamma'-\{(E+E')\})< 0)} \end{array} \right. \end{align*} $$

for every prime divisor P. We put

$$ \begin{align*} B=\Gamma'-\{(E+E')\}+F \qquad \mathrm{and} \qquad G=\lfloor( E + E')\rfloor + F. \end{align*} $$

By the definition, we can check that B is a boundary divisor, $\lfloor B \rfloor \leq \lfloor \Gamma ' \rfloor $ , and G is an effective f-exceptional Weil divisor. Since $\lfloor B \rfloor \leq \lfloor \Gamma ' \rfloor $ , the divisor L is nef and log big over Z with respect to $(Y,T+B)$ . Furthermore, the relation (4) implies

(5) $$ \begin{align} K_{Y}+T+B+L=\lfloor lp f^{*}(K_{X}+\Delta+\mathbf{M}_{X}) \rfloor +G. \end{align} $$

We put $D=lp f^{*}(K_{X}+\Delta +\mathbf {M}_{X})$ . Since G is f-exceptional, we have

$$ \begin{align*} \mathcal{O}_{X}( lp(K_{X}+\Delta+\mathbf{M}_{X}) )\simeq f_{*}\mathcal{O}_{Y}(\lfloor D \rfloor +G). \end{align*} $$

By construction, $(Y,B)$ is a log smooth lc pair and L is nef and log big over Z with respect to $(Y,B)$ . Thus, the Kodaira type vanishing theorem [Reference Fujino12, Lemma 1.5] implies

$$ \begin{align*} R^{1}(\pi \circ f)_{*}\mathcal{O}_{Y}(\lfloor D \rfloor +G-T)=R^{1}(\pi \circ f)_{*}\mathcal{O}_{Y}(K_{Y}+B+L)=0. \end{align*} $$

From these facts, the morphism

$$ \begin{align*} (\pi \circ f)_{*}\mathcal{O}_{Y}(\lfloor D \rfloor +G) \longrightarrow (\pi_{S^{\nu}} \circ f_{T})_{*}\mathcal{O}_{T}\bigl((\lfloor D \rfloor +G)|_{T}\bigr) \end{align*} $$

is surjective, where $f_{T}\colon T \to S^{\nu }$ is the natural morphism. Since $\lfloor D \rfloor |_{T}=\lfloor D|_{T} \rfloor $ by equation (3), a natural morphism

$$ \begin{align*} \mathcal{O}_{S^{\nu}}(\lfloor lp(K_{S^{\nu}}+\Delta_{S^{\nu}}+\mathbf{N}_{S^{\nu}})\rfloor) \longrightarrow f_{T*}\mathcal{O}_{T}(\lfloor D|_{T} \rfloor) \hookrightarrow f_{T*}\mathcal{O}_{T}(\lfloor D \rfloor|_{T} + G|_{T}) \end{align*} $$

is defined. From these facts, we obtain the diagram

and therefore the morphism

$$ \begin{align*} \begin{aligned} \pi_{*}\mathcal{O}_{X}(lp(K_{X}+\Delta+\mathbf{M}_{X})) \longrightarrow \pi_{S^{\nu}*}\mathcal{O}_{S^{\nu}} \bigl(\lfloor lp(K_{S^{\nu}}+\Delta_{S^{\nu}}+\mathbf{N}_{S^{\nu}})\rfloor\bigr) \end{aligned} \end{align*} $$

is surjective.

Proof. Replacing B, we may assume that $(X,B)$ is klt. We pick a real number $0<t\ll 1$ so that the generalized klt pair $\bigl (X,(1-t)\Delta +tB,(1-t)\mathbf {M}\bigr )/Z$ satisfies

$$ \begin{align*} 0< a(P,X,((1-t)\Delta+tB)+(1-t)\mathbf{M}_{X})< 1 \end{align*} $$

for any $P\in \mathcal {T}$ . Replacing $(X,\Delta ,\mathbf {M})/Z$ with $\bigl (X,(1-t)\Delta +tB,(1-t)\mathbf {M}\bigr )/Z$ , we may assume that $(X,\Delta ,\mathbf {M})/Z$ is generalized klt. By the perturbation of coefficients, we can find $\Delta '$ such that $(X,\Delta ')$ is klt and $0<a(P,X,\Delta ')<1$ for any $P\in \mathcal {T}$ , where $a(\,\cdot \,,X,\Delta ')$ is the log discrepancy. Then the lemma follows from [Reference Hashizume and Hu33, Lemma 2.16].

Proof. We closely follow [Reference Hashizume31, Proof of Lemma 2.6].

Taking an appropriate common log resolution $f \colon Y \to X$ and $f'\colon Y \to X'$ of the birational map $(X,\Delta )\dashrightarrow (X',\Delta ')$ , we may find a subvariety T of Y which is birational to S and $S'$ such that the induced morphisms $T \to S$ and $T \to S'$ form a common resolution of $\phi |_{S}$ . Note that $\mathbf {M}$ does not necessarily descend to Y. We may write

$$ \begin{align*} f^{*}(K_{X}+\Delta+\mathbf{M}_{X})=f^{\prime *}(K_{X'}+\Delta'+\mathbf{M}^{\prime}_{X'})+G_{+}-G_{-} \end{align*} $$

such that $G_{+}\geq 0$ and $G_{-}\geq 0$ have no common components. Then $G_{+}$ is $f'$ -exceptional by the first condition of Lemma 3.5. Since $K_{X}+\Delta +\mathbf {M}_{X}$ is pseudo-effective, we see that $K_{X'}+\Delta '+\mathbf {M}^{\prime }_{X'}$ is pseudo-effective. We have $\mathrm {Supp}G_{+} \not \supset T$ and $\mathrm {Supp}G_{-}\not \supset T$ since S and $S'$ are generalized lc centers of $(X,\Delta ,\mathbf {M})$ and $(X',\Delta ',\mathbf {M}')$ , respectively.

We can write

$$ \begin{align*} G_{+}=G_{0}+G_{1}, \end{align*} $$

where all components of $G_{0}$ intersect T and $G_{1}|_{T} = 0$ . Pick any component E of $G_{0}$ . Then $\mathrm {ord}_{E}(G_{-})=0$ . We have

$$ \begin{align*} \begin{aligned} &\sigma_{E}(K_{X}+\Delta+\mathbf{M}_{X})\\ =&\sigma_{E}\bigl(f^{*}(K_{X}+\Delta+\mathbf{M}_{X})\bigr)+\mathrm{ord}_{E}(G_{-}) \qquad \;\;\;\, (\mathrm{ord}_{E}(G_{-})=0) \\ \geq &\sigma_{E}\bigl(f^{*}(K_{X}+\Delta+\mathbf{M}_{X})+G_{-}\bigr) \qquad\qquad\quad\;\;\, (\text{[31, Remark 2.3 (1)]})\\ =&\sigma_{E}\bigl(f^{\prime *}(K_{X'}+\Delta'+\mathbf{M}^{\prime}_{X'})+G_{+}\bigr) \\ =& \sigma_{E}\bigl(f^{\prime *}(K_{X'}+\Delta'+\mathbf{M}^{\prime}_{X'})\bigr)+\mathrm{ord}_{E}(G_{+}) \qquad (\text{[31, Remark 2.3 (3)]}), \end{aligned} \end{align*} $$

therefore $\sigma _{P}(K_{X}+\Delta +\mathbf {M}_{X})>0$ . By the second condition of Lemma 3.5, we see that $a(E,X,\Delta +\mathbf {M}_{X})\geq 1$ , hence we have

$$ \begin{align*} a(E,X',\Delta'+\mathbf{M}^{\prime}_{X'})>a(E,X,\Delta+\mathbf{M}_{X})\geq 1. \end{align*} $$

By the same argument as in [Reference Hashizume31, Proof of Lemma 2.5], we see that $E|_{T}$ is exceptional over $S'$ . Therefore, $G_{0}|_{T}$ is exceptional over $S'$ . This implies that

$$ \begin{align*} \mathrm{coeff}_{Q}(G_{0}|_{T})=0 \end{align*} $$

for every prime divisor Q on $S'$ . Since we have $G_{1}|_{T}=0$ and

$$ \begin{align*} f|_{T}^{*}\bigl((K_{X}+\Delta+\mathbf{M}_{X})|_{S}\bigr)-f'|_{T}^{*} \bigl((K_{X'}+ \Delta'+ \mathbf{M}^{\prime}_{X'})|_{S'}\bigr) = (G_{0}+G_{1})|_{T}-G_{-}|_{T}, \end{align*} $$

the inequality

$$ \begin{align*} a(Q,S',\Delta_{S'}+\mathbf{N}_{S'})- a(Q,S,\Delta_{S}+\mathbf{N}_{S})\leq 0 \end{align*} $$

holds for every prime divisor Q on $S'$ . Therefore, Lemma 3.5 holds.

Proof. Let $g \colon W \to Y$ be a log resolution of $(Y,\Gamma )$ such that $f\circ g\colon W \to X$ is a log resolution of $(X,\Delta )$ . Let $(W, \Delta _{W},\mathbf {M})$ and $(W, \Gamma _{W},\mathbf {M})$ be generalized log birational models of $(X,\Delta ,\mathbf {M})$ and $(Y,\Gamma ,\mathbf {M})$ as in Definition 2.10, respectively. We may write

$$ \begin{align*} K_{W}+\Delta_{W}+\mathbf{M}_{W}=g^{*}f^{*}(K_{X}+\Delta+\mathbf{M}_{X})+E_{W} \end{align*} $$

for some $E_{W}\geq 0$ which is exceptional over X.

By [Reference Hashizume31, Remark 2.3 (3)], all prime divisors P on W satisfy

$$ \begin{align*} \sigma_{P}(K_{W}+\Delta_{W}+\mathbf{M}_{W})=\sigma_{P} (K_{X}+\Delta+\mathbf{M}_{X})+\mathrm{coeff}_{P}(E_{W}). \end{align*} $$

If P is not exceptional over X, then we have

$$ \begin{align*} \mathrm{coeff}_{P}(E_{W})=0, \quad \mathrm{coeff}_{P}(\Delta_{W})=\mathrm{coeff}_{f(g(P))}(\Delta),\quad \mathrm{and} \quad \mathrm{ coeff}_{P}(\Gamma_{W})=\mathrm{coeff}_{g(P)}(\Gamma). \end{align*} $$

If P is exceptional over X but not exceptional over Y, then we have

$$ \begin{align*} \mathrm{coeff}_{P}(E_{W})=a(P,X,\Delta+\mathbf{M}_{X}), \quad \mathrm{coeff}_{P}(\Delta_{W})=1, \quad \mathrm{and} \quad \mathrm{ coeff}_{P}(\Gamma_{W})=\mathrm{coeff}_{g(P)}(\Gamma). \end{align*} $$

If P is exceptional over Y, then we have

$$ \begin{align*} \mathrm{coeff}_{P}(E_{W})=a(P,X,\Delta+\mathbf{M}_{X}), \quad \mathrm{coeff}_{P}(\Delta_{W})=1, \quad \mathrm{and} \quad \mathrm{ coeff}_{P}(\Gamma_{W})=1. \end{align*} $$

In any case, by simple computations using the hypothesis about the relations between generalized log discrepancies, we see that

$$ \begin{align*} 0\leq a(P,W,\Gamma_{W}+\mathbf{M}_{W})-a(P,W,\Delta_{W}+\mathbf{M}_{W})\leq \sigma_{P}(K_{W}+\Delta_{W}+\mathbf{M}_{W}). \end{align*} $$

Thus, we see that $(W, \Delta _{W},\mathbf {M})$ and $(W, \Gamma _{W},\mathbf {M})$ satisfy the hypothesis of Lemma 3.6. By Lemma 3.1, we may replace $(X,\Delta ,\mathbf {M})$ and $(Y,\Gamma ,\mathbf {M})$ with $(W, \Delta _{W},\mathbf {M})$ and $(W, \Gamma _{W},\mathbf {M})$ , respectively. By the replacement, we may assume that $X=Y$ and $(X,0)$ is a $\mathbb {Q}$ -factorial klt pair.

We may carry out [Reference Hacon, McKernan and Xu22, Proof of Lemma 5.3] in the framework of generalized pairs. It is because we may use the argument of the length of extremal rays [Reference Han and Li25, Proposition 3.17] and the result of the termination of MMP [Reference Han and Li25, Theorem 4.1]. By the same argument as in [Reference Hacon, McKernan and Xu22, Proof of Lemma 5.3], we see that Lemma 3.6 holds.

Proof. Replacing $(Y,\Gamma ,\mathbf {M})$ by a $\mathbb {Q}$ -factorial generalized dlt model, we may assume that $(Y,\Gamma ,\mathbf {M})$ is a $\mathbb {Q}$ -factorial generalized dlt pair. Running a $(K_{Y}+\Gamma +\mathbf {M}_{Y})$ -MMP over X and replacing $(Y,\Gamma ,\mathbf {M})$ by a crepant generalized dlt model of $(X,\Delta ,\mathbf {M})$ , we may assume $E=0$ . Then Lemma 3.7 follows from Lemma 3.6.

Proof. The argument in [Reference Hashizume31, Proof of Lemma 2.22] works with no changes. Note that [Reference Hashizume31, Proof of Lemma 2.22] does not use any result about the abundance conjecture or the existence of flips for lc pairs.

Proof. We closely follow [Reference Hashizume31, Proof of Lemma 2.26]. Let $f\colon Y\to X$ and $f'\colon Y \to X'$ be a common log resolution of $(X,\Delta )\dashrightarrow (X',\Delta ')$ such that $\mathbf {M}$ descends to Y. We define an $\mathbb {R}$ -divisor $\Gamma $ on Y by

$$ \begin{align*} \Gamma:=-\sum_{D}\mathrm{min}\{a(D,X,\Delta+\mathbf{M}_{X})-1,a(D,X',\Delta' +\mathbf{M}^{\prime}_{X'})-1,0\}D, \end{align*} $$

where D runs over prime divisors on Y. Then $\Gamma $ is an effective snc $\mathbb {R}$ -divisor, $(Y,\Gamma , \mathbf {M})$ is generalized lc and there exist an f-exceptional $\mathbb {R}$ -divisor $E\geq 0$ and an $f'$ -exceptional $\mathbb {R}$ -divisor $E'\geq 0$ such that

$$ \begin{align*} E+f^{*}(K_{X}+\Delta+\mathbf{M}_{X})=K_{Y}+ \Gamma+\mathbf{M}_{Y}=f^{\prime *}(K_{X'} + \Delta'+\mathbf{M}^{\prime}_{X'})+E'. \end{align*} $$

Lemma 3.9 follows from the relation and Lemma 3.7.

Proof. The argument in [Reference Hashizume29, Proof of Proposition 3.3] works with no changes because we can use the generalized canonical bundle formula [Reference Han and Liu26] instead of the canonical bundle formula [Reference Fujino and Gongyo19]. So we only outline the proof.

By taking a log resolution of $(X,\Delta )$ and applying weak semistable reduction ([Reference Abramovich and Karu2]), we may assume that $(X,0)$ is $\mathbb {Q}$ -factorial klt and all fibers of $\pi $ have the same dimensions.

We run a $(K_{X}+\Delta +\mathbf {M}_{X})$ -MMP over Z with scaling of an ample divisor. Applying the negativity lemma for very exceptional divisors ([Reference Birkar3, Section 3]) and replacing $(X,\Delta ,\mathbf {M})$ , we may assume $K_{X}+\Delta +\mathbf {M}_{X}\sim _{\mathbb {R},Z}0$ (but we lose the property of being equidimensional of $X \to Z$ ). We used the first condition of Lemma 3.10 for the reduction.

By generalized canonical bundle formula [Reference Han and Liu26, Theorem 1.2] and the second condition of Lemma 3.10, there is a generalized klt pair $(Z,\Delta _{Z},\mathbf {N})$ such that

$$ \begin{align*} K_{X}+\Delta+\mathbf{M}_{X}\sim_{\mathbb{R}}\pi^{*}(K_{Z}+\Delta_{Z}+\mathbf{N}_{Z}). \end{align*} $$

By the third condition of Lemma 3.10, we see that $K_{Z}+\Delta _{Z}+\mathbf {N}_{Z}$ is big, so $(Z,\Delta _{Z},\mathbf {N})$ has a good minimal model $(Z', \Delta _{Z'}, \mathbf {N})$ by [Reference Birkar, Cascini, Hacon and McKernan6]. Since $(Z,\Delta _{Z},\mathbf {N})$ is generalized klt, the birational map $Z \dashrightarrow Z'$ is a birational contraction. Hence, we can find an open subset $U' \subset Z'$ such that and $Z'\dashrightarrow Z$ is an isomorphism on $U'$ .

Let $f\colon Y \to X$ be a log resolution of $(X,\Delta )$ such that $\mathbf {M}$ descends to Y and the map $\pi _{Y} \colon Y\dashrightarrow Z'$ is a morphism. Let $(Y,\Gamma ,\mathbf {M})$ be a generalized log birational model of $(X,\Delta ,\mathbf {M})$ as in Definition 2.10. Then

$$ \begin{align*} K_{Y}+\Gamma+\mathbf{M}_{Y}\sim_{\mathbb{R}} \pi_{Y}^{*}(K_{Z'}+\Delta_{Z'}+\mathbf{N}_{Z'})+E+F \end{align*} $$

for some $E \geq 0$ and $F\geq 0$ such that E is exceptional over X and . Then $(Y,\Gamma ,\mathbf {M})$ has a good minimal model over $U'$ .

Running a $(K_{Y}+\Gamma +\mathbf {M}_{Y})$ -MMP over $Z'$ with scaling of an ample divisor, we get a birational contraction $\phi \colon Y \dashrightarrow Y'$ over $Z'$ with the morphism $\pi ' \colon Y' \to Z'$ such that

$$ \begin{align*} \phi_{*}(K_{Y}+\Gamma+\mathbf{M}_{Y}) \sim_{\mathbb{R}}\pi^{\prime *}(K_{Z'}+\Delta_{Z'}+\mathbf{N}_{Z'})+\phi_{*}E+\phi_{*}F \end{align*} $$

is the limit of movable divisors over $Z'$ and $\phi _{*}E|_{\pi ^{\prime -1}(U')}=0$ . Since , applying the negativity lemma for very exceptional divisors ([Reference Birkar3, Section 3]) to $\phi _{*}E+\phi _{*}F$ , we have $\phi _{*}E+\phi _{*}F=0$ . Thus, we have

$$ \begin{align*} \phi_{*} (K_{Y}+\Gamma+\mathbf{M}_{Y}) \sim_{\mathbb{R}}\pi^{\prime *}(K_{Z'}+\Delta_{Z'}+\mathbf{N}_{Z'}), \end{align*} $$

and the right-hand side is semiample. From the fact, we see that $(Y,\Gamma ,\mathbf {M})$ has a good minimal model. So $(X,\Delta ,\mathbf {M})$ has a good minimal model.

Proof. The argument in [Reference Hashizume and Hu33, Proof of Lemma 5.2] or [Reference Hashizume32, Proof of Lemma 3.6] works with no changes.

Proof of Theorem 3.11

The argument of [Reference Hashizume and Hu33, Proof of Theorem 5.4] works with no changes. We only outline the proof. We prove Theorem 3.11 by induction on $\mathrm {dim}X$ .

We put $A= \pi ^{*}A_{Z}$ . We may assume that $K_{X}+\Delta +A$ is pseudo-effective and $\pi $ is a contraction. By replacing A with a general one, we may also assume that $(X,\Delta +A)$ is lc, $\Delta $ and A have no common components and all lc centers of $(X,\Delta +A)$ are lc centers of $(X,\Delta )$ . By taking a dlt blowup of $(X,\Delta +A)$ , we may assume that $(X,\Delta +A)$ is $\mathbb {Q}$ -factorial dlt.

If $(X,\Delta +A)$ is a klt pair, then we follow [Reference Hashizume and Hu33, Proof of Lemma 5.3], and we see that $K_{X}+\Delta +A$ is abundant. More generally, if $K_{X}+\Delta -\epsilon \lfloor \Delta \rfloor +A$ is pseudo-effective for some $\epsilon>0$ , then the argument in [Reference Hashizume and Hu33, Step 1 in the proof of Theorem 5.4] enables us to reduce to the klt case, hence we see that $K_{X}+\Delta +A$ is abundant. From these facts, we may assume that $K_{X}+\Delta -\epsilon \lfloor \Delta \rfloor +A$ is not pseudo-effective for every $\epsilon>0$ . Then there is a component S of $\lfloor \Delta \rfloor $ such that $K_{X}+\Delta -\epsilon S+A$ is not pseudo-effective for every $\epsilon>0$ .

For all $\epsilon '>0$ , we run a $(K_{X}+\Delta -\epsilon ' S+A)$ -MMP and get a birational contraction $X\dashrightarrow X'$ to a Mori fiber space $X'\to \overline {Z}$ . Let $\Delta '$ , $S'$ and $A'$ be the birational transforms of $\Delta $ , S and A on $X'$ , respectively. By taking $\epsilon '>0$ sufficiently small with the aid of the ACC for lc thresholds ([Reference Hacon, McKernan and Xu21, Theorem 1.1]) and the ACC for numerically trivial pairs ([Reference Hacon, McKernan and Xu21, Theorem 1.5]), we see that $(X',\Delta '+A')$ is lc and we have the relation $K_{X'}+\Delta '+A'\sim _{\mathbb {R},\overline {Z}}0$ .

Let $\phi \colon Y \to X$ and $ Y \to X'$ be a common log resolution of the birational map $(X,\Delta )\dashrightarrow (X',\Delta ')$ . Putting $A_{Y}=\phi ^{*}A$ , we can write

(†) $$ \begin{align} K_{Y}+\Gamma+A_{Y}=\phi^{*}(K_{X}+\Delta+A)+E, \end{align} $$

where $\Gamma \geq 0$ and $E\geq 0$ have no common components. We run a $(K_{Y}+\Gamma +A_{Y})$ -MMP over $\overline {Z}$ with scaling of an ample divisor. As in [Reference Hashizume and Hu33, Step 3 in the proof of Theorem 5.4], an appropriate $(K_{Y}+\Gamma +A_{Y})$ -MMP terminates with a good minimal model $(Y',\Gamma '+A_{Y'})$ over $\overline {Z}$ . Let $Y'\to Z'$ be the contraction over $\overline {Z}$ induced by $K_{Y'}+\Gamma '+A_{Y'}$ . Then the induced morphism $Z' \to \overline {Z}$ is birational. We have the following diagram.

We put $T=\phi ^{-1}_{*}S$ , and let $T'$ be the birational transform of T on $Y'$ . By construction, $T'$ dominates $Z'$ . We define projective dlt pairs

$$ \begin{align*} (S,\Delta_{S}), \quad (T,\Gamma_{T}),\quad \mathrm{and} \quad (T',\Gamma_{T'}) \end{align*} $$

with the divisorial adjunctions $K_{S}+\Delta _{S}=(K_{X}+\Delta )|_{S}$ , $K_{T}+\Gamma _{T}=(K_{Y}+\Gamma )|_{T}$ and $K_{T'}+\Gamma _{T'}=(K_{Y'}+\Gamma ')|_{T'}$ , respectively. We put

$$ \begin{align*} A_{S}=A|_{S}, \quad A_{T}=A_{Y}|_{T}, \quad \mathrm{and} \quad A_{T'}=A_{Y'}|_{T'}. \end{align*} $$

By construction, it is sufficient to prove that $K_{Y'}+\Gamma '+A_{Y'}$ is abundant. Since we have $K_{Y'}+\Gamma '+A_{Y'}\sim _{\mathbb {R},Z'}0$ and $T'$ dominates $Z'$ , it is sufficient to prove that

$$ \begin{align*} K_{T'}+\Gamma_{T'}+A_{T'} \sim_{\mathbb{R}}(K_{Y'}+\Gamma'+A_{Y'})|_{T'} \end{align*} $$

is abundant. We take a common log resolution $\tau \colon T'' \to T$ and $\tau '\colon T''\to T'$ of the birational map $(T,\Gamma _{T}) \dashrightarrow (T',\Gamma _{T'})$ . Replacing A with a general member of $|A|_{\mathbb {R}}$ , we may assume that

• ○ $A_{T}\geq 0$ and $A_{T'}\geq 0$ ,

• ○ $\tau ^{*}A_{T}\leq \tau _{*}^{\prime -1}A_{T'}$ (see [Reference Hashizume and Hu33, Lemma 2.1]) and

• ○ $\tau '\colon T''\to T'$ is a log resolution of $(T',\Gamma _{T'}+A_{T'})$ (see [Reference Hashizume and Hu33, Lemma 2.2]).

Let $\pi _{S}\colon S\to Z$ be the restriction of $\pi \colon X\to Z$ to S, and let $\phi _{T} \colon T \to S$ be the birational morphism induced by $\phi \colon Y \to X$ . Now we have the following diagram.

By restricting equation (†) to T, we get

$$ \begin{align*} K_{T}+\Gamma_{T}+A_{T}=\phi_{T}^{*}(K_{S}+\Delta_{S}+A_{S})+E|_{T} \end{align*} $$

such that $\Gamma _{T}$ and $E|_{T}$ are effective. By [Reference Hashizume and Hu33, Lemma 2.4], we see that $\Gamma _{T}+A_{T}$ and $E|_{T}$ have no common components and $E|_{T}$ is $\phi _{T}$ -exceptional. We put $A_{T''}=\tau ^{*}A_{T}$ . Then $A_{T''} \leq \tau ^{\prime -1}_{*}A_{T'}$ . We have

$$ \begin{align*} K_{T''}+\Psi_{T''}+A_{T''}=\tau^{\prime *}(K_{T'}+\Gamma_{T'}+A_{T'})+E_{T''} \end{align*} $$

for some $\Psi _{T''}\geq 0$ and $E_{T''}\geq 0$ such that $(T'', \Psi _{T''}+A_{T''})$ is a log smooth lc pair and $\Psi _{T''}+A_{T''}$ and $E_{T''}$ have no common components.

We may write

$$ \begin{align*} K_{T''}+\Psi_{T''}+A_{T''}=\tau^{*}\phi_{T}^{*}(K_{S}+\Delta_{S}+A_{S})+M-N, \end{align*} $$

with $M\geq 0$ and $N\geq 0$ having no common components. Pick any component Q of M. If Q is not exceptional over S, then we have $a(Q, T'', \Psi _{T''}+A_{T''}) <a(Q,S, \Delta _{S}+A_{S})\leq 1$ . We apply [Reference Hashizume and Hu33, Step 6 in the proof of Theorem 5.4], and we get the following contradiction

$$ \begin{align*} \begin{aligned} a(Q,S, \Delta_{S}+A_{S})=&\mathrm{min}\{1, a(Q,S, \Delta_{S}+A_{S}) \} \qquad \;\;\, (a(Q,S, \Delta_{S}+A_{S})\leq1)\\ =&\mathrm{min}\{1, a(Q, T, \Gamma_{T}+A_{T}) \} \qquad \;\;\,(E|_{T} \text{ is } \phi_{T}\text{-exceptional})\\ \leq &\mathrm{min}\{1, a(Q, T', \Gamma_{T'}+A_{T'}) \} \qquad (\text{[13, Lemma 4.2.10]})\\ =&a(Q, T'', \Psi_{T''}+A_{T''}) \qquad \qquad \quad (\text{[33, Lemma 2.3]})\\ <&a(Q,S, \Delta_{S}+A_{S}). \end{aligned} \end{align*} $$

From this, we see that M is exceptional over S.

By applying the induction hypothesis of Theorem 3.11 to $(S,\Delta _{S}) \to \pi (S)^{\nu }$ , $C|_{S}$ and $A_{Z}|_{\pi (S)^{\nu }}$ , we see that $K_{S}+\Delta _{S}+A_{S}$ is abundant. By applying Lemma 3.12 to $(S,\Delta _{S}) \to \pi (S)^{\nu }$ and $(T'',\Psi _{T''}) \to S$ , we see that $K_{T''}+\Psi _{T''}+A_{T''}$ is abundant. Then $K_{T'}+\Gamma _{T'}+A_{T'}$ is abundant, hence $K_{X}+\Delta +A$ is abundant.

Proof. We can apply Lemma 3.12, and the argument in [Reference Hashizume32, Proof of Theorem 4.1] works with no changes.

Proof. We closely follow [Reference Hashizume31, Subsection 3.1]. We prove Theorem 3.14 in several steps.

Step 1. In this step, we replace $(X,\Delta , \mathbf {M})$ with a crepant $\mathbb {Q}$ -factorial generalized dlt model which has good properties. We follow [Reference Hashizume31, Proof of Proposition 3.2].

Since $K_{X}+\Delta +\mathbf {M}_{X}$ is pseudo-effective and abundant, there is an effective $\mathbb {R}$ -divisor D on X such that $K_{X}+\Delta +\mathbf {M}_{X}\sim _{\mathbb {R}}D$ . We take the Iitaka fibration $X\dashrightarrow V$ associated to D. Then we have

$$ \begin{align*} \mathrm{dim}V=\kappa_{\sigma}(X, K_{X}+\Delta+\mathbf{M}_{X}). \end{align*} $$

We take a log resolution $\overline {f}\colon \overline {X}\to X$ of $(X,\Delta )$ such that $\mathbf {M}$ descends to $\overline {X}$ and the induced map $\overline {X}\dashrightarrow V$ is a morphism. Then we have

$$ \begin{align*} K_{\overline{X}}+\overline{\Delta}+\mathbf{M}_{\overline{X}}=\overline{f}^{*}(K_{X}+\Delta +\mathbf{M}_{X})+\overline{E} \end{align*} $$

for some $\overline {\Delta } \geq 0$ and $\overline {E}\geq 0$ which have no common components. By construction and Lemma 2.14, we have

1. (i) $\kappa _{\sigma }(\overline {X},K_{\overline {X}}+\overline {\Delta }+\mathbf {M}_{\overline {X}})=\mathrm {dim}V$ and $\kappa _{\sigma }(\overline {X}/V,K_{\overline {X}}+\overline {\Delta }+\mathbf {M}_{\overline {X}})=0$ .

Moreover, $K_{\overline {X}}+\overline {\Delta }+\mathbf {M}_{\overline {X}}$ is $\mathbb {R}$ -linearly equivalent to the sum of an effective $\mathbb {R}$ -divisor and the pullback of an ample divisor on V. Therefore, we can find an effective $\mathbb {R}$ -divisor

$$ \begin{align*} \overline{D}\sim_{\mathbb{R}}K_{\overline{X}}+\overline{\Delta}+\mathbf{M}_{\overline{X}} \end{align*} $$

such that $\mathrm {Supp}\overline {D}$ contains all generalized lc centers of $(\overline {X},\overline {\Delta },\mathbf {M})$ which are vertical over V. By applying [Reference Hashizume28, Lemma 2.10] to the morphism $(\overline {X},\overline {\Delta })\to V$ and replacing $(\overline {X},\overline {\Delta })$ , we may assume that

1. (ii) $\overline {\Delta }=\overline {\Delta }'+\overline {\Delta }''$ such that $\overline {\Delta }'$ is effective, $\Delta ''=0$ or $\overline {\Delta }''$ is reduced and vertical over V, and all generalized lc centers of $(\overline {X},\overline {\Delta }-s\overline {\Delta }'', \mathbf {M})$ dominate V for all $s \in (0,1]$ .

By taking a log resolution of $(\overline {X},\overline {\Delta }+\overline {D})$ and replacing $(\overline {X},\overline {\Delta })$ and $\overline {D}$ , we may assume that $(\overline {X},\overline {\Delta }+\overline {D})$ is log smooth. Note that (ii) is preserved after this replacement. Since $ \mathrm {Supp}\overline {\Delta }''$ is a union of generalized lc centers of $(\overline {X},\overline {\Delta },\mathbf {M})$ which are vertical over V, we see that $\mathrm {Supp}\overline {D}\supset \mathrm {Supp}\overline {\Delta }''$ . By decomposing $\overline {D}$ appropriately, we obtain effective $\mathbb {R}$ -divisors $\overline {G}$ and $\overline {H}$ such that

1. (iii) $K_{\overline {X}}+\overline {\Delta }+\mathbf {M}_{\overline {X}}\sim _{\mathbb {R}}\overline {G}+\overline {H}$ ,

2. (iv) $\mathrm {Supp}\overline {\Delta }''\subset \mathrm {Supp}\overline {G}\subset \mathrm {Supp}\lfloor \overline {\Delta } \rfloor $ and

3. (v) no component of $\overline {H}$ is a component of $\lfloor \overline {\Delta } \rfloor $ and $(\overline {X},\overline {\Delta }+\overline {H})$ is log smooth.

We fix a real number $t_{0}\in (0,1)$ such that $\overline {\Delta }-t_{0}\overline {G}\geq 0$ . We pick $t\in (0,t_{0}]$ , and we consider $(\overline {X},\overline {\Delta }-t\overline {G}, \mathbf {M})$ . Then the morphism

$$ \begin{align*} (\overline{X},\overline{\Delta}-t\overline{G}, \mathbf{M})\to V \end{align*} $$

satisfies the following three conditions.

• ○ Any generalized lc center of $(\overline {X},\overline {\Delta }-t\overline {G}, \mathbf {M})$ dominates V,

• ○ $\kappa _{\sigma }(\overline {X},K_{\overline {X}} +\overline {\Delta }-t\overline {G} +\mathbf {M}_{\overline {X}}) = \mathrm {dim}V$ and

• ○ $\kappa _{\iota }(\overline {X}/ V,K_{\overline {X}}+\overline {\Delta }- t\overline {G}+\mathbf {M}_{\overline {X}}) = \kappa _{\sigma }(\overline {X}/V,K_{\overline {X}}+\overline {\Delta }-t\overline {G}+\mathbf {M}_{\overline {X}})=0$ .

Indeed, the first condition follows from (iv) and (ii), and the other two conditions follow from (iii), (i) and [Reference Hashizume and Hu33, Remark 2.8 (1)]. By Lemma 3.10, we see that $(\overline {X},\overline {\Delta }-t\overline {G}, \mathbf {M})$ has a good minimal model for all $t\in (0,t_{0}]$ .

Running a $(K_{\overline {X}}+\overline {\Delta }+\mathbf {M}_{\overline {X}})$ -MMP over X, we get a crepant generalized dlt model

$$ \begin{align*} \widetilde{f}\colon(\widetilde{X},\widetilde{\Delta},\mathbf{M})\to (X,\Delta, \mathbf{M}). \end{align*} $$

Let $\widetilde {G}$ (resp. $\widetilde {H}$ ) be the birational transform of $\overline {G}$ (resp. $\overline {H}$ ) on $\widetilde {X}$ . Then

$$ \begin{align*} K_{\widetilde{X}}+\widetilde{\Delta}+\mathbf{M}_{\widetilde{X}} \sim_{\mathbb{R}}\widetilde{G}+\widetilde{H} \end{align*} $$

by (iii). By (v) and replacing $t_{0}$ , we may assume that $(\overline {X},\overline {\Delta }+t_{0}\overline {H}, \mathbf {M})$ is a $\mathbb {Q}$ -factorial generalized dlt pair, and we may further assume that $\overline {X}\dashrightarrow \widetilde {X}$ is a sequence of steps of a $(K_{\overline {X}}+\overline {\Delta }+t\overline {H}+\mathbf {M}_{\overline {X}})$ -MMP and a sequence of steps of a $(K_{\overline {X}}+\overline {\Delta }-t\overline {G}+\mathbf {M}_{\overline {X}})$ -MMP for all $t\in (0,t_{0}]$ . Then it follows that $(\widetilde {X},\widetilde {\Delta }+t_{0}\widetilde {H},\mathbf {M})$ is a $\mathbb {Q}$ -factorial generalized dlt pair, and $(\widetilde {X},\widetilde {\Delta }-t\widetilde {G},\mathbf {M})$ has a good minimal model for all $t\in (0,t_{0}]$ since $(\overline {X},\overline {\Delta }-t\overline {G}, \mathbf {M})$ has a good minimal model.

Applying [Reference Hashizume31, Lemma 2.4] to $K_{\widetilde {X}}+\widetilde {\Delta }+\mathbf {M}_{\widetilde {X}}$ and $\widetilde {H}$ and replacing $t_{0}$ , we may assume that

$$ \begin{align*} \mathrm{Supp}N_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta}+t\widetilde{H}+\mathbf{M}_{\widetilde{X}}) \end{align*} $$

is independent of the choice of $t\in (0,t_{0}]$ .

Since $K_{\widetilde {X}}+\widetilde {\Delta }+\mathbf {M}_{\widetilde {X}}=\widetilde {f}^{*}(K_{X}+\Delta +\mathbf {M}_{X})$ , it is easy to check that $(\widetilde {X},\widetilde {\Delta },\mathbf {M})$ satisfies all the conditions of Theorem 3.14. Replacing $ (X,\Delta , \mathbf {M})$ by $(\widetilde {X},\widetilde {\Delta },\mathbf {M})$ , we may assume that X is $\mathbb {Q}$ -factorial and there exist effective $\mathbb {R}$ -divisors G and H on X such that

• ○ $K_{X}+\Delta +\mathbf {M}_{X}\sim _{\mathbb {R}}G+H$ ,

• ○ $\mathrm {Supp}G\subset \mathrm {Supp}\lfloor \Delta \rfloor $ and

• ○ there exists a real number $t_{0}>0$ such that the following properties hold for all $t\in (0,t_{0}]$ :

  • – $(X,\Delta +tH, \mathbf {M})$ is generalized dlt and $\mathrm {Supp}N_{\sigma }(K_{X}+\Delta +tH+\mathbf {M}_{X})$ does not depend on t, and

  • – $(X,\Delta -tG, \mathbf {M})$ has a good minimal model.

Step 2. We follow [Reference Hashizume31, Proof of Proposition 3.3].

Pick any real number $t \in (0,t_{0}]$ . Since $(X,\Delta -\tfrac {t}{1+t}G, \mathbf {M})$ has a good minimal model and

$$ \begin{align*} K_{X}+\Delta+tH+\mathbf{M}_{X}\sim_{\mathbb{R}} (1+t)\left(K_{X}+\Delta-\frac{t}{1+t}G+\mathbf{M}_{X}\right), \end{align*} $$

there is a sequence of steps of a $(K_{X}+\Delta +tH+\mathbf {M}_{X})$ -MMP to a good minimal model

$$ \begin{align*} \phi_{t}\colon (X,\Delta+tH, \mathbf{M}) \dashrightarrow (X_{t},\Delta_{t}+tH_{t}, \mathbf{M}). \end{align*} $$

Because $\mathrm {Supp}N_{\sigma }(K_{X}+\Delta +tH+\mathbf {M}_{X})$ is independent of $t \in (0,t_{0}]$ , prime divisors contracted by $\phi _{t}$ are independent of $t \in (0,t_{0}]$ . Therefore, putting

$$ \begin{align*} X_{0}=X_{t_{0}}, \quad \Delta_{0}=\Delta_{t_{0}}, \quad \mathrm{and} \quad H_{0}=H_{t_{0}}, \end{align*} $$

then $X_{0}$ and $X_{t}$ are isomorphic in codimension one for all $t \in (0,t_{0}]$ . From this fact, we see that $(X_{t},\Delta _{t}+tH_{t}, \mathbf {M})$ is a good minimal model of $(X_{0},\Delta _{0}+tH_{0},\mathbf {M})$ .

By applying [Reference Hashizume29, Proof of Lemma 2.14] and [Reference Hashizume31, Proof of Proposition 3.3], we get a sequence of steps of a $(K_{X_{0}}+\Delta _{0}+\mathbf {M}_{X_{0}})$ -MMP with scaling of $t_{0}H_{0}$

$$ \begin{align*} (X_{0},\Delta_{0},\mathbf{M}) \dashrightarrow (X_{1}, \Delta_{1},\mathbf{M}) \dashrightarrow \cdots \dashrightarrow (X_{i}, \Delta_{i},\mathbf{M}) \dashrightarrow \cdots \end{align*} $$

such that

• ○ if we define for each i, then $\mathrm {lim}_{i \to \infty }t_{i}=0$ ,

• ○ for all $t \in [t_{i},t_{i-1}] \cap \mathbb {R}_{>0}$ , the generalized pair $(X_{i}, \Delta _{i}+t H_{i},\mathbf {M})$ is a good minimal model of both $(X, \Delta +t H,\mathbf {M})$ and $(X_{0}, \Delta _{0}+t H_{0},\mathbf {M})$ , and

• ○ the MMP occurs only in $\mathrm {Supp}\lfloor \Delta _{0}\rfloor $ .

By the argument as in [Reference Hashizume31, Step 1 in the proof of Theorem 3.4], it is sufficient to prove the termination of the $(K_{X_{0}}+\Delta _{0}+\mathbf {M}_{X_{0}})$ -MMP.

Step 4. We follow [Reference Hashizume31, Step 2 in the proof of Theorem 3.4].

From now on, we fix a generalized lc center $S_{m}$ of $(X_{m},\Delta _{m},\mathbf {M})$ . Unless otherwise stated, we assume that all integers i appearing in the rest of the proof satisfy $i \geq m$ . For all i, let $S_{i}$ be the generalized lc center of $(X_{i},\Delta _{i}, \mathbf {M})$ such that $X_{m}\dashrightarrow X_{i}$ induces a birational map $S_{m} \dashrightarrow S_{i}$ . By construction of the birational map $X\dashrightarrow X_{i}$ , we can find a generalized lc center S of $(X,\Delta , \mathbf {M})$ such that $X\dashrightarrow X_{i}$ induces a birational map $S\dashrightarrow S_{i}$ . Using the S, we will define the following variety, divisor and inequalities.

1. (a) A birational morphism $\psi \colon T\to S_{m}$ from a projective $\mathbb {Q}$ -factorial variety T such that, for every prime divisor $\bar {D}$ on S, if we have $a(\bar {D},S_{m}, \Delta _{S_{m}}+\mathbf {N}_{S_{m}})<a(\bar {D},S,\Delta _{S}+\mathbf {N}_{S})$ , then $\bar {D}$ is a $\psi $ -exceptional divisor on T,

2. (b) An effective $\mathbb {R}$ -divisor $\Psi $ on T defined by $\Psi =-\sum _{\substack {D}}\bigl (a(D,S,\Delta _{S}+\mathbf {N}_{S})-1\bigr ) D$ , where D runs over all prime divisors on T,

3. (c) The inequality $a(Q,S,(\Delta _{S}+t_{i}H_{S})+\mathbf {N}_{S})\leq a(Q,S_{i},(\Delta _{S_{i}}+t_{i}H_{S_{i}})+\mathbf {N}_{S_{i}})$ for all i and all prime divisors Q over S and

4. (d) The inequality $a(Q',S_{m},\Delta _{S_{m}}+\mathbf {N}_{S_{m}})\leq a(Q',T,\Psi +\mathbf {N}_{T})$ for all prime divisors $Q'$ over $S_{m}$ , in particular, the generalized pair $(T,\Psi , \mathbf {N})$ is generalized lc.

The generalized pair $(X_{i}, \Delta _{i}+t_{i} H_{i},\mathbf {M})$ is a good minimal model of $(X, \Delta +t_{i} H, \mathbf {M})$ , hence the negativity lemma (see also [Reference Fujino13, Proof of Lemma 4.2.10]) implies the inequality

$$ \begin{align*} \begin{aligned} a(Q,S,(\Delta_{S}+t_{i}H_{S})+\mathbf{N}_{S}) \leq a(Q,S_{i}, (\Delta_{S_{i}}+t_{i}H_{S_{i}}) +\mathbf{N}_{S_{i}}) \end{aligned} \end{align*} $$

for all prime divisors Q over S. We have (c).

In this paragraph, we prove that the equality

(⋆) $$ \begin{align} a(\widetilde{D},S_{m},\Delta_{S_{m}}+\mathbf{N}_{S_{m}})= a(\widetilde{D},S,\Delta_{S}+\mathbf{N}_{S}) \end{align} $$

holds for all prime divisors $\widetilde {D}$ on $S_{m}$ . Note that $\widetilde {D}$ is a prime divisor on $S_{i}$ for every i since the birational map $S_{m}\dashrightarrow S_{i}$ is small. For every i, we may apply Lemma 3.5 to the birational map $(X,\Delta ,\mathbf {M})\dashrightarrow (X_{i},\Delta _{i},\mathbf {M})$ and the generalized pairs $(S,\Delta _{S},\mathbf {N})$ and $(S_{i},\Delta _{S_{i}},\mathbf {N})$ because $(X,\Delta ,\mathbf {M})\dashrightarrow (X_{i},\Delta _{i},\mathbf {M})$ obviously satisfies the first condition of Lemma 3.5, and the second condition of Lemma 3.5 follows from the third condition of Theorem 3.14. Thus, we have

$$ \begin{align*} a(\widetilde{D},S_{i},\Delta_{S_{i}}+\mathbf{N}_{S_{i}}) \leq a(\widetilde{D},S,\Delta_{S}+\mathbf{N}_{S}), \end{align*} $$

and therefore we have

$$ \begin{align*} a(\widetilde{D},S_{i},(\Delta_{S_{i}}+t_{i}H_{S_{i}})+\mathbf{N}_{S_{i}}) \leq a(\widetilde{D},S,\Delta_{S}+\mathbf{N}_{S}) \end{align*} $$

for all prime divisors $\widetilde {D}$ on $S_{m}$ . By this inequality and (c), we have

$$ \begin{align*} a(\widetilde{D},S,(\Delta_{S}+t_{i}H_{S})+\mathbf{N}_{S})\leq a(\widetilde{D},S_{i},(\Delta_{S_{i}}+t_{i}H_{S_{i}})+\mathbf{N}_{S_{i}}) \leq a(\widetilde{D},S,\Delta_{S}+\mathbf{N}_{S}). \end{align*} $$

We have $a(\widetilde {D},S_{i},(\Delta _{S_{i}}+t_{i}H_{S_{i}})+\mathbf {N}_{S_{i}})=a(\widetilde {D},S_{m},(\Delta _{S_{m}}+t_{i}H_{S_{m}})+\mathbf {N}_{S_{m}})$ for all i since $\widetilde {D}$ appears as a prime divisor on $S_{m}$ and $S_{i}$ . Since $\mathrm {lim}_{i \to \infty }t_{i}=0$ , we get equation (⋆) for all prime divisors $\widetilde {D}$ on $S_{m}$ by taking the limit $i\to \infty $ .

We put

By equation (⋆), all elements of $\mathcal {C}$ are exceptional over $S_{m}$ . By a basic property of generalized log discrepancies and (c), we have

$$ \begin{align*} a(\bar{D},S, (\Delta_{S}+t_{m}H_{S})+\mathbf{N}_{S}) \leq a(\bar{D},S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}}). \end{align*} $$

This implies

$$ \begin{align*} a(\bar{D},S,(\Delta_{S}+t_{m}H_{S})+\mathbf{N}_{S}) \leq a(\bar{D},S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}})< a(\bar{D},S,\Delta_{S}+\mathbf{N}_{S}) \end{align*} $$

for all $\bar {D}\in \mathcal {C}$ . Since every element of $\mathcal {C}$ is a prime divisor on S, we see that all elements of $\mathcal {C}$ are components of $H_{S}$ . Thus, $ \mathcal {C}$ is a finite set, and $a(\bar {D},S_{m}, \Delta _{S_{m}}+\mathbf {N}_{S_{m}})<1$ for all $\bar {D}\in \mathcal {C}$ because $a(\bar {D},S,\Delta _{S}+\mathbf {N}_{S})\leq 1$ . From these facts, for every $\bar {D}\in \mathcal {C}$ we get the relation

$$ \begin{align*} \begin{aligned} 0\leq &a(\bar{D},S,(\Delta_{S}+t_{0}H_{S})+\mathbf{N}_{S})\\ <&a(\bar{D},S,(\Delta_{S}+t_{m}H_{S})+\mathbf{N}_{S}) \qquad\;\; (\bar{D} \text{ is a component of } H_{S} \text{ and } m\gg 0)\\ \leq &a(\bar{D},S_{m},(\Delta_{S_{m}}+t_{m}H_{S_{m}})+\mathbf{N}_{S_{m}}) \qquad (\text{the inequality (c)})\\ \leq & a(\bar{D},S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}}) \\ <& a(\bar{D},S,\Delta_{S}+\mathbf{N}_{S}) \leq 1\qquad\qquad\qquad\;\;\; (\text{definition of } \mathcal{C}). \end{aligned} \end{align*} $$

In this way, we have $0<a(\bar {D},S_{m}, \Delta _{S_{m}}+\mathbf {N}_{S_{m}})<1$ . By Lemma 3.4, there exists a projective birational morphism $\psi \colon T\to S_{m}$ such that T is $\mathbb {Q}$ -factorial and $\psi ^{-1}$ exactly extracts elements of $\mathcal {C}$ . We have constructed the desired birational morphism as in (a).

Let D be a prime divisor on T. When D is $\psi $ -exceptional, by the definitions of $\mathcal {C}$ and $\psi $ we have

$$ \begin{align*} a(D,S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}})<a(D,S,\Delta_{S}+\mathbf{N}_{S})\leq 1. \end{align*} $$

When D is not $\psi $ -exceptional, from equation (⋆) we see that

$$ \begin{align*} a(D,S,\Delta_{S}+\mathbf{N}_{S})= a(D,S_{m},\Delta_{S_{m}}+\mathbf{N}_{S_{m}})\leq 1. \end{align*} $$

Thus, the relation

(⋆⋆) $$ \begin{align} a(D,S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}})\leq a(D,S,\Delta_{S}+\mathbf{N}_{S})\leq 1. \end{align} $$

holds for all prime divisors D on T. Note that only finitely many prime divisors D on T satisfy $a(D,S,\Delta _{S}+\mathbf {N}_{S})<1$ Therefore, we may define an $\mathbb {R}$ -divisor $\Psi \geq 0$ on T by

$$ \begin{align*} \Psi=-\sum_{\substack {D}}\bigl(a(D,S,\Delta_{S}+\mathbf{N}_{S})-1\bigr) D, \end{align*} $$

where D runs over all prime divisors on T. This is the divisor stated in (b).

Finally, we prove (d). Since T is $\mathbb {Q}$ -factorial, $K_{T}+\Psi +\mathbf {N}_{T}$ is $\mathbb {R}$ -Cartier. By equation (⋆⋆), we obtain

$$ \begin{align*} K_{T}+\Psi +\mathbf{N}_{T}\leq \psi^{*}(K_{S_{m}}+\Delta_{S_{m}}+\mathbf{N}_{S_{m}}). \end{align*} $$

From this, we have

$$ \begin{align*} 0 \leq a(Q',S_{m},\Delta_{S_{m}}+\mathbf{N}_{S_{m}})\leq a(Q',T,\Psi+\mathbf{N}_{T}) \end{align*} $$

for any prime divisor $Q'$ over $S_{m}$ . This shows (d).

Step 6. We follow [Reference Hashizume31, Step 4 in the proof of Theorem 3.4]. With this step, we complete the proof of the special termination of the $(K_{X_{0}}+\Delta _{0}+\mathbf {M}_{X_{0}})$ -MMP in Step 3.

We recall that $S_{m}\dashrightarrow S_{i}$ is small, $\Delta _{S_{i}}+t_{i}H_{S_{i}}$ is equal to the birational transform of $\Delta _{S_{m}}+t_{i}H_{S_{m}}$ on $S_{i}$ , and the divisor

$$ \begin{align*} K_{S_{i}}+\Delta_{S_{i}}+t_{i}H_{S_{i}}+\mathbf{N}_{S_{i}} \sim_{\mathbb{R}} (K_{X_{i}}+\Delta_{i}+t_{i}H_{i}+\mathbf{M}_{X_{i}})|_{S_{i}} \end{align*} $$

is nef. So $(S_{i},\Delta _{S_{i}}+t_{i}H_{S_{i}}, \mathbf {N})$ is a weak generalized lc model of $(S_{m},\Delta _{S_{m}}+t_{i}H_{S_{m}}, \mathbf {N})$ for every i.

We pick an arbitrary prime divisor D on T. By using the generalized pair analogue of [Reference Hashizume31, Remark 2.9 (1)], we see that

$$ \begin{align*} \begin{aligned} &\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}}+t_{i}H_{S_{m}}+\mathbf{N}_{S_{m}})\\ =&a(D,S_{i},(\Delta_{S_{i}}+t_{i}H_{S_{i}})+\mathbf{N}_{S_{i}})-a(D, S_{m},( \Delta_{S_{m}}+t_{i}H_{S_{m}})+\mathbf{N}_{S_{m}}). \end{aligned} \end{align*} $$

From the relation, we have

$$ \begin{align*} \begin{aligned} &\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}}+t_{i}H_{S_{m}}+\mathbf{N}_{S_{m}})\\ =&a(D,S_{i},(\Delta_{S_{i}}+t_{i}H_{S_{i}})+\mathbf{N}_{S_{i}})-a(D, S_{m}, (\Delta_{S_{m}}+t_{i}H_{S_{m}})+\mathbf{N}_{S_{m}})\\ \geq &a(D,S,(\Delta_{S}+t_{i}H_{S})+\mathbf{N}_{S})-a(D, S_{m}, (\Delta_{S_{m}}+t_{i}H_{S_{m}})+\mathbf{N}_{S_{m}}) \qquad (\text{(c) in Step 4)}. \end{aligned} \end{align*} $$

By (b) in Step 4, the equality $a(D,S,\Delta _{S}+\mathbf {N}_{S})=a(D,T,\Psi +\mathbf {N}_{T})$ holds. From these relations and the fact that $\mathrm {lim}_{i \to \infty }t_{i}=0$ , we obtain

$$ \begin{align*} \begin{aligned} &\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}}+\mathbf{N}_{S_{m}})\\ =&\underset{i\to \infty}{\mathrm{lim}} \sigma_{D}(K_{S_{m}}+\Delta_{S_{m}}+t_{i}H_{S_{m}}+\mathbf{N}_{S_{m}}) \qquad \qquad \qquad (\text{[31, Remark 2.3 (2)]})\\ \geq &\underset{i\to \infty}{\mathrm{lim}}\bigl(a(D,S,(\Delta_{S}+t_{i}H_{S})+\mathbf{N}_{S})-a(D, S_{m}, (\Delta_{S_{m}}+t_{i}H_{S_{m}})+\mathbf{N}_{S_{m}})\bigr)\\ =&a(D,S,\Delta_{S}+\mathbf{N}_{S})-a(D, S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}})\\ =&a(D,T,\Psi+\mathbf{N}_{T})-a(D, S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}}) \end{aligned} \end{align*} $$

for all prime divisors D on T. By this relation and (d) in Step 4, we obtain

$$ \begin{align*} 0 \leq a(D,T,\Psi+\mathbf{N}_{T}) - a(D, S_{m}, \Delta_{S_{m}}+\mathbf{N}_{S_{m}}) \leq \sigma_{D}(K_{S_{m}} + \Delta_{S_{m}}+\mathbf{N}_{S_{m}}). \end{align*} $$

Therefore, we may apply Lemma 3.6 to $(S_{m},\Delta _{S_{m}},\mathbf {N})$ and $(T, \Psi ,\mathbf {N})$ . Because $(T, \Psi ,\mathbf {N})$ has a minimal model, $(S_{m},\Delta _{S_{m}},\mathbf {N})$ has a minimal model.

By arguments in the last paragraph of Step 3, we complete the special termination.