Iitaka fibrations for dlt pairs polarized by a nef and log big divisor

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