2.1 Special notation

Notation 2.2. Let $f: X\dashrightarrow X'$ be a birational map between normal varieties. We denote by $\operatorname{Exc}(f)$ the reduced divisor supported on the codimension one part of the exceptional locus of f.

Notation 2.3. Let X be a normal variety and D,D’ two $\mathbb{R}$ -divisors on X. We define $D\wedge D':=\sum_P\min\{\operatorname{mult}_PD,\operatorname{mult}_PD'\}P$ where the sum runs through all the prime divisors P on X. We denote by $\operatorname{Supp} D$ the reduced divisor supported on D.

Notation 2.5. A general choice of a real number a is a choice of a real number such that $a\not\in\mathbb Q(\Gamma_0)$ for a finite set $\Gamma_0\subset\mathbb R$ . Here $\mathbb Q(\Gamma_0)$ is the field extension of $\mathbb Q$ by elements in $\Gamma_0$ . We also say that a is general in $\mathbb R/\mathbb Q$ .

2.2 Foliations

1. (1) $\mathcal{F}$ is saturated in $T_X$ , i.e. $T_X/\mathcal{F}$ is torsion free; and

2. (2) $\mathcal{F}$ is closed under the Lie bracket.

The rank of the foliation $\mathcal{F}$ is the rank of $\mathcal{F}$ as a sheaf and is denoted by $\operatorname{rank}\mathcal{F}$ . The co-rank of $\mathcal{F}$ is $\dim X-\operatorname{rank}\mathcal{F}$ . The canonical divisor of $\mathcal{F}$ is a divisor $K_\mathcal{F}$ such that $\mathcal{O}_X(-K_{\mathcal{F}})\cong\mathrm{det}(\mathcal{F})$ . If $\mathcal{F}=0$ , then we say that $\mathcal{F}$ is a foliation by points.

Given any dominant map $h: Y\dashrightarrow X$ , we denote by $h^{-1}\mathcal{F}$ the pullback of $\mathcal{F}$ on Y as constructed in [Reference DruelDru21, 3.2] and say that $h^{-1}\mathcal{F}$ is induced by $\mathcal{F}$ . Given any birational map $g: X\dashrightarrow X'$ , we denote by $g_*\mathcal{F}:=(g^{-1})^{-1}\mathcal{F}$ the pushforward of $\mathcal{F}$ on X’ and also say that $g_*\mathcal{F}$ is induced by $\mathcal{F}$ . We say that $\mathcal{F}$ is an algebraically integrable foliation if there exists a dominant map $f: X\dashrightarrow Z$ such that $\mathcal{F}=f^{-1}\mathcal{F}_Z$ , where $\mathcal{F}_Z$ is the foliation by points on Z, and we say that $\mathcal{F}$ is induced by f.

A subvariety $S\subset X$ is called $\mathcal{F}$ -invariant if for any open subset $U\subset X$ and any section $\partial\in H^0(U,\mathcal{F})$ , we have $\partial(\mathcal{I}_{S\cap U})\subset \mathcal{I}_{S\cap U}$ , where $\mathcal{I}_{S\cap U}$ is the ideal sheaf of $S\cap U$ . For any prime divisor P on X, we denote $\epsilon_{\mathcal{F}}(P):=1$ if P is not $\mathcal{F}$ -invariant and $\epsilon_{\mathcal{F}}(P):=0$ if P is $\mathcal{F}$ -invariant. For any prime divisor E over X, we define $\epsilon_{\mathcal{F}}(E):=\epsilon_{\mathcal{F}_Y}(E)$ where $h: Y\dashrightarrow X$ is a birational map such that E is on Y and $\mathcal{F}_Y:=h^{-1}\mathcal{F}$ . For any $\mathbb{R}$ -divisor D on X, we denote by $D^{\mathcal{F}-\mathrm{ninv}}$ the reduced divisor supported on the union of non- $\mathcal{F}$ -invariant components of D.

1. (1) $\mu^{-1}\mathcal{F}$ is induced by f’; and

2. (2) V’ is contained in a fiber of f’ and $\mu(V')=V$ .

If $\mathcal{F}=T_X$ , then we may drop $\mathcal{F}$ and say that $(X,B)/U$ is a pair. If U is not important, then we may drop U. If $\mathcal{F}$ is algebraically integrable, then we say that $(X,\mathcal{F},B)$ is algebraically integrable. If X is $\mathbb{Q}$ -factorial, then we say that $(X,\mathcal{F},B)$ is $\mathbb{Q}$ -factorial. If we allow B to have negative coefficients, then we shall add the prefix ‘sub-’. If B is a $\mathbb{Q}$ -divisor then we may add the prefix ‘ $\mathbb Q$ -’.

Definition 2.9 (Singularities). Let $(X,\mathcal{F},B)$ be a foliated triple. For any prime divisor E over X, let $f: Y\rightarrow X$ be a birational morphism such that E is on Y, and suppose that

$$K_{\mathcal{F}_Y}+B_Y:=f^*(K_\mathcal{F}+B),$$

where $\mathcal{F}_Y:=f^{-1}\mathcal{F}$ . We define $a(E,\mathcal{F},B):=-\operatorname{mult}_EB_Y$ to be the discrepancy of E with respect to $(X,\mathcal{F},B)$ . If $\mathcal{F}=T_X$ , then we define $a(E,X,B):=a(E,\mathcal{F},B)$ which is the usual discrepancy for pairs.

We say that $(X,\mathcal{F},B)$ is lc (respectively, klt) if $a(E,\mathcal{F},B)\geq -1$ (respectively, $\gt -1$ ) for any prime divisor E over X. For foliated sub-triples, we define singularities in the same way and we shall add the prefix ‘sub-’ for the descriptions of singularities.

An lc place of $(X,\mathcal{F},B)$ is a prime divisor E over X such that $a(E,\mathcal{F},B)=-\epsilon_{\mathcal{F}}(E)$ . An lc center of $(X,\mathcal{F},B)$ is the center of an lc place of $(X,\mathcal{F},B)$ on X.

We remark that our definition of lc and klt singularities has some differences compared with the classical definition [Reference McQuillanMcQ08, Reference Cascini and SpicerCS20, Reference Ambro, Cascini, Shokurov and SpicerACSS21, Reference Cascini and SpicerCS21, Reference Chen, Han, Liu and XieCHLX23], where the $-1$ in the inequality is replaced with $-\epsilon_{\mathcal{F}}(E)$ . The next lemma shows that the two definitions on lc coincide so we are free to use results on ‘lc foliations’ in literature. Moreover, there are good reasons why we refine the definition of klt singularities. We refer the readers to Remark 10.1 for details.

1. (1) $(X,\mathcal{F},B)$ is sub-lc;

2. (2) $a(E,\mathcal{F},B)\geq -\epsilon_{\mathcal{F}}(E)$ for any prime divisor E over X.

2.3 Special algebraically integrable foliations

Let X be a normal quasi-projective variety, B an $\mathbb{R}$ -divisor on X, and $\mathcal{F}$ an algebraically integrable foliation on X. A foliated log resolution of $(X,\mathcal{F},B)$ is a birational morphism $h: X'\rightarrow X$ such that

$$(X',\mathcal{F}':=h^{-1}\mathcal{F},B':=h^{-1}_*B+\operatorname{Exc}(h))$$

is foliated log smooth. The existence of foliated log resolutions for any such $(X,\mathcal{F},B)$ is guaranteed by [Reference Chen, Han, Liu and XieCHLX23, Lemma 6.2.4].

1. (1) $\mathcal{F}$ is induced by f and G is an $\mathcal{F}$ -invariant divisor;

2. (2) f(G) is of pure codimension 1, (Z,f(G)) is log smooth, and $G=f^{-1}(f(G))$ ;

3. (3) for any closed point $z\in Z$ and any reduced divisor $\Sigma\ge f(G)$ on Z such that $(Z,\Sigma)$ is log smooth near z, $(X,B+G+f^*(\Sigma-f(G)))$ is lc over a neighborhood of z.

We say that f, Z, and G are associated with $(X,\mathcal{F},B)$ .

Proposition 2.15 (Cf. [Reference Ambro, Cascini, Shokurov and SpicerACSS21, Proposition 3.6] and [Reference Chen, Han, Liu and XieCHLX23, Proposition 7.3.6]). Let $(X,\mathcal{F},B)$ be a foliated triple. Let $G\geq 0$ be a reduced divisor on X and $f: X\rightarrow Z$ an equidimensional contraction, such that $(X,\mathcal{F},B;G)/Z$ satisfies Property $(*)$ and B is horizontal $/Z$ . Then

$$K_{\mathcal{F}}+B\sim_{Z}K_X+B+G.$$

1. (1) We have that $(X,\mathcal{F},B;G)/Z$ satisfies Property $(*)$ .

2. (2) We have that f is equidimensional.

3. (3) there exists an $\mathbb{R}$ -Cartier $\mathbb{R}$ -divisor $D\geq 0$ on X, such that $\operatorname{Supp}\{B\}\subset\operatorname{Supp} D$ , and for any reduced divisor $\Sigma\geq f(G)$ such that $(Z,\Sigma)$ is log smooth,

   $$(X,B+D+G+f^*(\Sigma-f(G)))$$
   is quotient divisorial log terminal (qdlt) (cf. [Reference de Fernex, Kollár and XudFKX17, Definition 35]).

4. (4) For any lc center of $(X,\mathcal{F},B)$ with generic point $\eta$ , over a neighborhood of $\eta$ :

   1. (a) $\eta$ is the generic point of an lc center of $(X,\mathcal{F},\lfloor B\rfloor)$ ; and

   2. (b) $f: (X,B+G)\rightarrow (Z,f(G))$ is a toroidal morphism.

If $(X,\mathcal{F},B;G)/Z$ is ACSS and G is super $/Z$ , then we say that $(X,\mathcal{F},B;G)/Z$ is super ACSS. If $(X,\mathcal{F},B;G)/Z$ is (super) ACSS, then we say that $(X,\mathcal{F},B)/Z$ and $(X,\mathcal{F},B)$ are (super) ACSS.

2.4 Birational maps in MMP

1. (1) $\phi$ does not extract any divisor;

2. (2) $D':=\phi_*D$ is $\mathbb{R}$ -Cartier;

3. (3) there exists a resolution of indeterminacy $p: W\rightarrow X$ and $q: W\rightarrow X'$ , such that

   $$ p^*D=q^*D'+F, $$
   where $F\geq 0$ (respectively, $F\geq 0$ and $\operatorname{Supp} p_*F=\operatorname{Exc}(\phi)$ , $F=0$ , $0\geq F$ , $0\geq F$ and $\operatorname{Supp} p_*F=\operatorname{Exc}(\phi)$ ).

1. (1) We say that X’ is a minimal model $/U$ of D if D’ is nef $/U$ .

2. (2) We say that X’ is a good minimal model $/U$ of D if D’ is semi-ample $/U$ .

3. (3) A contraction $/U$ $f: X'\rightarrow Z$ is called a Mori fiber space $/U$ of D if f is a D’-Mori fiber space $/U$ .

Since $\phi$ is $H_1$ -trivial, q is $p^*H_1$ -trivial. Therefore, q only contracts $p^*H_1$ -trivial extremal rays in $\overline{NE}(W/U)$ , so q only contracts $p^*H_2$ -trivial extremal rays in $\overline{NE}(W/U)$ . Thus, q is $p^*H_2$ -trivial, so $\phi$ is $H_2$ -trivial.

Proof. Let A’ and B’ be the images of A,B on X’, respectively. Then $A'+tB'$ is $\mathbb{R}$ -Cartier. By [Reference Han, Liu and ShokurovHLS24, Lemma 5.3], A,B,A’,B’ are $\mathbb{R}$ -Cartier. Let $p: W\rightarrow X$ and $q: W\rightarrow X'$ be a resolution of indeterminacy, then

$$p^*A+tp^*B=p^*(A+tB)=q^*(A'+tB')=q^*A'+tq^*B'.$$

By [Reference Han, Liu and ShokurovHLS24, Lemma 5.3], $p^*A=q^*A'$ and $p^*B=q^*B'$ . The lemma follows.

2.5 Relative Nakayama–Zariski decompositions

1. (1) For any exceptional $/X$ $\mathbb{R}$ -Cartier $\mathbb{R}$ -divisor $E\ge0$ and any prime divisor P on Y, we have

   $$ \sigma_P(Y/U,f^*D+E)=\sigma_P(Y/U,f^*D)+\operatorname{mult}_PE. $$

2. (2) For any exceptional $/X$ $\mathbb{R}$ -Cartier $\mathbb{R}$ -divisor $E\ge0$ on Y, we have

   $$N_{\sigma}(X/U,D)=f_*N_{\sigma}(Y/U,f^*D+E).$$

3. (3) The set $\operatorname{Supp} N_{\sigma}(X/U,D)$ coincides with the divisorial part of ${\textbf{B}}_-(X/U,D)$ .

1. (1) The divisors contracted by $\phi$ are contained in $\operatorname{Supp} N_{\sigma}(X/U,D)$ .

2. (2) If D’ is movable $/U$ , then $\operatorname{Supp} N_{\sigma}(X/U,D)$ is the set of all $\phi$ -exceptional divisors.

Proof. Let $p: W\rightarrow X$ and $q: W\rightarrow X'$ be a resolution of indeterminacy. Then

$$p^*D=q^*D'+E$$

for some $E\geq 0$ that is exceptional $/X'$ and $\operatorname{Supp} E$ contains the strict transforms of all $\phi$ -exceptional divisors on W. By Lemma 2.24(1),

$$\operatorname{Supp} E\subset\operatorname{Supp} N_\sigma(W/U,q^*D'+E)=\operatorname{Supp} N_\sigma(W/U,p^*D),$$

By Lemma 2.24(2), $\operatorname{Supp} p_*E\subset\operatorname{Supp} N_\sigma(X/U,D)$ . Therefore, any $\phi$ -exceptional divisor is contained in $\operatorname{Supp} N_\sigma(X/U,D)$ .

If D’ is movable $/U$ , then by Lemma 2.24(2), $q_*N_\sigma(W/U,q^*D'+E)=0$ . Thus,

$$\operatorname{Supp} N_\sigma(W/U,q^*D'+E)$$

is q-exceptional. By Lemma 2.24(2) again, we have $\operatorname{Supp} N_\sigma(X/U,D)=\operatorname{Supp} p_*N_\sigma(W/U, q^*D'+E)$ , whose components are all $\phi$ -exceptional. Condition (2) follows from condition (1).

2.6 Generalized pairs and generalized foliated quadruples

For $\boldsymbol{b}$ -divisors and generalized pairs, we follow the notation and definitions in [Reference Birkar and ZhangBZ16, Reference Hacon and LiuHL23]. For generalized foliated quadruples, we shall follow [Reference Chen, Han, Liu and XieCHLX23].

Generalized pairs and generalized foliated quadruples are very technical concepts. To make the statements in this paper more concise, for most results whose proofs for generalized foliated quadruples are similar to the proofs for foliated triples, we only prove the foliated triple version and do not prove the generalized foliated quadruple version. We state the corresponding generalized foliated quadruple version in Appendix A. We freely use results from [Reference Chen, Han, Liu and XieCHLX23] on generalized foliated quadruples.

We need some results on NQC $\mathbb{R}$ -divisors which are related to generalized pairs and generalized foliated quadruples.

We say that D is NQC $/U$ if $D=\sum d_iD_i$ , where each $d_i\geq 0$ and each $D_i$ is a nef $/U$ Cartier divisor. We say that ${\textbf{M}}$ is NQC $/U$ if ${\textbf{M}}=\sum \mu_i{\textbf{M}}_i$ , where each $\mu_i\geq 0$ and each ${\textbf{M}}_i$ is a nef $/U$ Cartier $\boldsymbol{b}$ -divisor.

Let $L:=D-(K_\mathcal{F}+B)$ . Since ample $/U$ is an open condition, there exists an open set $V\ni\boldsymbol{r}$ in $\mathbb R^c$ , such that $\frac{1}{2}L+D(\boldsymbol{v})-D$ is ample $/U$ for any $\boldsymbol{v}\in V$ .

By [Reference Chen, Han, Liu and XieCHLX23, Theorem 2.3.1], there exist finitely many $\big(K_{\mathcal{F}}+B+\frac{1}{2}L\big)$ -negative extremal rays $/U$ $R_1,\dots,R_l$ , and $R_j=\mathbb R_+[C_j]$ for some curve $C_j$ . Since D is nef, $D\cdot C_j\geq 0$ for each j. Thus, possibly shrinking V, we may assume that for any $\boldsymbol{v}\in V$ , we have that $D(\boldsymbol{v})\cdot C_j\gt 0$ for any j such that $D\cdot C_j\gt 0$ . Since $r_1,\dots,r_c$ are linearly independent over $\mathbb Q$ , for any j such that $D\cdot C_j=0$ , we have $D(\boldsymbol{v})\cdot C_j=0$ for any $\boldsymbol{v}\in\mathbb R^c$ . Therefore, $D(\boldsymbol{v})\cdot C_j\geq 0$ for any j and any $\boldsymbol{v}\in V$ .

By the cone theorem [Reference Chen, Han, Liu and XieCHLX23, Theorem 2.3.1], for any curve C on X, we may write $[C]=\eta+\sum_{i=1}^l a_i[C_i]$ where $a_1,\dots,a_l\geq 0$ and $\eta\in\overline{NE}(X/U)_{K_{\mathcal{F}}+B+\frac{1}{2}L\geq 0}$ . For any $\boldsymbol{v}\in V$ , since

$$D(\boldsymbol{v})\cdot \eta=\big(K_\mathcal{F}+B+\tfrac{1}{2}L\big)\cdot\eta+\big(\tfrac{1}{2}L+D(\boldsymbol{v})-D\big)\cdot \eta\geq 0,$$

$D(\boldsymbol{v})\cdot C\geq 0$ . Therefore, $D(\boldsymbol{v})$ is nef $/U$ for any $\boldsymbol{v}\in V$ . We let $\boldsymbol{v}_1,\dots,\boldsymbol{v}_{c+1}\in V\cap\mathbb Q^c$ be rational points such that $\boldsymbol{r}$ is in the interior of the convex hull of $\boldsymbol{v}_1,\dots,\boldsymbol{v}_{c+1}$ . Then there exist positive real numbers $a_1,\dots,a_{c+1}$ such that $\sum_{i=1}^{c+1}a_i=1$ and $\sum_{i=1}^{c+1}a_i\boldsymbol{v}_i=\boldsymbol{r}$ . Since $D(\boldsymbol{v}_i)$ is a nef $/U$ $\mathbb{Q}$ -divisor for each i and $D=\sum_{i=1}^{c+1}a_iD(\boldsymbol{v}_i)$ , D is NQC $/U$ .

Let $d:=\dim X$ . We show that $\delta_0:={\epsilon}/({2d+\epsilon})$ satisfies our requirements. Let R be a $(K_{\mathcal{F}}+B+(1-\delta)D)$ -non-positive extremal ray $/U$ . If R is not $(K_{\mathcal{F}}+B+D)$ -trivial, then R is $(K_{\mathcal{F}}+B+D)$ -positive, hence $(K_{\mathcal{F}}+B)$ -negative. By the length of extremal rays [Reference Chen, Han, Liu and XieCHLX23, Theorem 2.3.1], R is spanned by a curve C such that $\pi(C)=\{pt\}$ and $0\lt -(K_{\mathcal{F}}+B)\cdot C\leq 2d$ . Thus, for any $\delta\in (0,\delta_0)$ ,

\begin{align*} 0&\ge(K_{\mathcal{F}}+B+(1-\delta)D)\cdot C=(1-\delta)(K_{\mathcal{F}}+B+D)\cdot C+\delta(K_{\mathcal{F}}+B)\cdot C\\ &\geq (1-\delta)\epsilon-2d\delta \gt \epsilon-(2d+\epsilon)\delta_0=0,\end{align*}

which is not possible.

3.1 Core models for two contractions

The variety $\bar X$ is called the core model of (h,f) associated with $(\bar h,\bar f)$ .

Moreover, for any dominant map $\phi: X'\dashrightarrow X''$ such that K(X”) is algebraically closed in K(X’) (e.g. $\phi$ is a birational map or a contraction), and any contractions $h'': X''\rightarrow X$ and $f'': X''\rightarrow Z$ such that $h''\circ\phi=h$ and $f''\circ\phi=f$ , $\bar X$ is the core model of (h”,f”) associated with $(\bar h,\bar f)$ .

Step 2. In this step we show that condition (1) holds for any ample $\mathbb{R}$ -divisor. Suppose that there exists an ample $\mathbb{R}$ -divisor H on X such that $\bar h^*H$ is not ample $/Z$ . Then by applying Step 1 to $H,\bar h,\bar f$ , there exist contractions $g': \bar X\rightarrow\bar X'$ , $\bar h': \bar X'\rightarrow X$ , and $\bar f': \bar X'\rightarrow Z$ such that $\bar h'^*H$ is ample $/Z$ . Since $\bar h^*H$ is not ample $/Z$ , g’ is not an isomorphism. Since g’ is a contraction $/Z$ and $\bar h^*A=g'^*\bar h'^*A$ is ample $/Z$ , g’ is finite and, thus, an isomorphism, which is a contradiction.

Step 3. In this step we prove the moreover part. There exist a birational morphism $p: W\rightarrow X'$ from a normal quasi-projective variety W and a projective surjective morphism $q: W\rightarrow X''$ such that $q=\phi\circ p$ . Since K(X”) is algebraically closed in K(X’), q is a contraction. Let $\bar X''$ be the core model of (h”,f”) associated with $(\bar h'',\bar f'')$ and $g'': X''\rightarrow\bar X''$ the induced contraction. Since both $g\circ p$ and $g''\circ q$ are ample models $/Z$ of $(h\circ p)^*A=(h''\circ q)^*A$ , the ample model $/Z$ of $h^*A$ is isomorphic to the ample model $/Z$ of $h''^*A$ . Thus, $\bar X$ is the core model of (h”,f”) associated with $(\bar h,\bar f)$ .

3.2 Core models for algebraically integrable foliations

1. (1) $\mathcal{F}':=h^{-1}\mathcal{F}$ and $B':=h^{-1}_*(B^{\mathcal{F}-\mathrm{ninv}}\wedge\operatorname{Supp} B)+\operatorname{Exc}(h)^{\mathcal{F}'-\mathrm{ninv}}$ ;

2. (2) $(X',\mathcal{F}',B')$ is lc;

3. (3) $a(E,\mathcal{F},B)\leq-\epsilon_{\mathcal{F}}(E)$ for any h-exceptional prime divisor E;

4. (4) $K_{\mathcal{F}'}+B'\sim_ZK_{X'}+B'+G$ ;

5. (5) $(X',\mathcal{F}',B';G)/Z$ satisfies Property $(*)$ .

We say that $h: (X',\mathcal{F}',B',G)/Z\rightarrow (X,\mathcal{F},B)$ is an ACSS modification if it is a simple modification and $(X',\mathcal{F}',B';G)/Z$ is ACSS.

We say that $h: (X',\mathcal{F}',B',G)/Z\rightarrow (X,\mathcal{F},B)$ is a core modification if it is a simple modification and $h^*A$ is ample $/Z$ for any ample $\mathbb{R}$ -divisor A on X.

If $h: (X',\mathcal{F}',B',G)/Z\rightarrow (X,\mathcal{F},B)$ is a simple (respectively, ACSS, core) modification, then we also say that h is a simple (respectively, ACSS, core) modification of $(X,\mathcal{F},B)$ .

We say that $(X',\mathcal{F}',B';G)/Z$ , $(X',\mathcal{F}',B')/Z$ , and $(X',\mathcal{F}',B')$ are simple (respectively, ACSS, core) models of $(X,\mathcal{F},B)$ . Moreover, we say that h is:

1. (1) $\mathbb{Q}$ -factorial if X’ is $\mathbb{Q}$ -factorial;

2. (2) strict if the $\mathcal{F}$ -invariant part of $\operatorname{Supp}\operatorname{Exc}(h)$ is contained in $\operatorname{Supp} G$ ;

3. (3) super if G is super $/Z$ .

We frequently use the following result on the existence of ACSS models.

Assume that f is equidimensional. Then:

1. (1) $K_{\mathcal{F}'}+B'=g^*(K_{\bar{\mathcal{F}}}+\bar B)$ ;

2. (2) $K_{X'}+B'+G=g^*(K_{\bar X}+\bar B+\bar G)$ ;

3. (3) $\bar h: (\bar X,\bar{\mathcal{F}},\bar B;\bar G)/Z\rightarrow (X,\mathcal{F},B)$ is a core model;

4. (4) if $h: (X',\mathcal{F}',B';G)/Z\rightarrow (X,\mathcal{F},B)$ is strict (respectively, super), then $\bar h: (\bar X,\bar{\mathcal{F}},\bar B;\bar G)/ Z\rightarrow (X,\mathcal{F},B)$ is strict (respectively, super).

Definition 3.2(1): By our construction, $\bar{\mathcal{F}}=\bar h^{-1}\mathcal{F}$ and $\bar B=\bar h^{-1}_*B+\operatorname{Exc}(\bar h)^{\bar{\mathcal{F}}-\mathrm{ninv}}$ .

Definition 3.2(2): By condition (1), $K_{\bar{\mathcal{F}}}+\bar B=\bar h^*(K_{\mathcal{F}}+B)$ . Since $(X,\mathcal{F},B)$ is lc, $(\bar X,\bar{\mathcal{F}},\bar B)$ is lc.

Definition 3.2(3): Since any $\bar h$ -exceptional divisor is also h-exceptional, it follows from our construction.

Definition 3.2(4): Thus follows from condition (1) and (2).

Definition 3.2(5): We only need to check Definition 2.14 for $(\bar X,\bar{\mathcal{F}},\bar B,\bar G)/Z$ . Definition 2.14(1): Since $\mathcal{F}'$ is induced by f, $\bar{\mathcal{F}}$ is induced by $\bar f$ . Since G is $\mathcal{F}'$ -invariant, $\bar G$ is $\bar{\mathcal{F}}$ -invariant. Definition 2.14(2): $(Z,f(G)=\bar f(\bar G))$ is log smooth by assumption. Since $G=f^{-1}(f(G))$ , $\bar G=\bar f^{-1}(\bar f(G))$ . Definition 2.14(3): For any closed point $z\in Z$ and any reduced divisor $\Sigma\geq\bar f(\bar G)$ on Z such that $(Z,\Sigma)$ is log smooth near z, $(X',B'+f^*(\Sigma-\bar f(\bar G)))$ is lc over a neighborhood of z. By condition (2),

$$K_{X'}+B'+f^*(\Sigma-f(G))=g^*(K_{\bar X}+\bar B+\bar f^*(\Sigma-\bar f(\bar G))),$$

so $(\bar X,\bar B+\bar f^*(\Sigma-\bar f(\bar G)))$ is lc over a neighborhood of z.

1. (4) Thus follows from the definitions of being strict or super.

Y, yes; N, not necessarily true; $\bullet$ , holds when X’ is $\mathbb{Q}$ -factorial.

Remark 3.7. ‘ $(*)$ -models’ are defined in [Reference Ambro, Cascini, Shokurov and SpicerACSS21, Reference Cascini and SpicerCS25a] and ‘great ACSS models’ are defined in [Reference Chen, Han, Liu and XieCHLX23]. We do not need these models in this paper. Nevertheless, we provide the readers with a description of the properties of different types of models in Table 1. Note that ‘ $(*)$ -models’ are defined differently in [Reference Ambro, Cascini, Shokurov and SpicerACSS21] and [Reference Cascini and SpicerCS25a].

Table 1:

Different types of simple models.

Table 1: Long description

The table compares different types of simple models across six properties: Q-factorial, Strict, Super, X' to Z equidim, (X', B' + G), and X'. It consists of six rows and six columns. The rows are labeled as Simple models, Core models, (*)-models [ACSS21], ACSS models, (*)-models [CS25a], and Great ACSS models. The columns are labeled with the properties mentioned. Each cell contains values indicating the presence or absence of specific properties, such as N for no, Y for yes, and various abbreviations like lc, klt, and qdlt. Notable trends include the Great ACSS models having the most properties marked as present, while Simple and Core models have fewer properties marked.

4.1 Definitions of minimal models and Mori fiber spaces

• – For models requiring good singularities (e.g. $\mathbb{Q}$ -factorial ACSS), we always keep the word ‘log’. We always allow extraction of lc centers when considering these models.

• – For models without these strict singularity conditions (e.g. only requiring lc), we shall not use the word ‘log’. Moreover, if we allow extraction of lc centers, then we shall add the prefix ‘bs-’ or write ‘in the sense of Birkar and Shokurov’.

Definition 4.2 (Log birational model). Let $(X,\mathcal{F},B)/U$ be a foliated triple, $\phi: X\dashrightarrow X'$ a birational map over U, $E:=\operatorname{Exc}(\phi^{-1})$ the reduced $\phi^{-1}$ -exceptional divisor, and $\mathcal{F}':=\phi_*\mathcal{F}$ . Assume that $a(D,\mathcal{F},B)\leq-\epsilon_{\mathcal{F}}(D)$ for any component D of E. We let

$$B':=\phi_*B+\sum_D(-a(D,\mathcal{F},B))E$$

and say that $(X',\mathcal{F}',B')/U$ is a log birational model of $(X,\mathcal{F},B)/U$ , where the sum runs through all components of E.

1. (1) We say that $(X',\mathcal{F}',B')/U$ is a bs-weak lc model or weak lc model in the sense of Birkar and Shokurov of $(X,\mathcal{F},B)/U$ , if for any prime divisor D on X which is exceptional over X’,

   $$a(D,\mathcal{F},B)\leq a(D,\mathcal{F}',B').$$

2. (2) We say that $(X',\mathcal{F}',B')/U$ is a bs-minimal model or minimal model in the sense of Birkar and Shokurov of $(X,\mathcal{F},B)/U$ , if for any prime divisor D on X which is exceptional over X’,

   $$a(D,\mathcal{F},B)<a(D,\mathcal{F}',B').$$

3. (3) We say that $(X',\mathcal{F}',B')/U$ is a bs-semi-ample model or semi-ample model in the sense of Birkar and Shokurov of $(X,\mathcal{F},B)/U$ if it is a bs-weak lc model of $(X,\mathcal{F},B)/U$ and $K_{\mathcal{F}'}+B'$ is semi-ample $/U$ .

4. (4) We say that $(X',\mathcal{F}',B')/U$ is a bs-good minimal model or good minimal model in the sense of Birkar and Shokurov of $(X,\mathcal{F},B)/U$ if it is a bs-minimal model of $(X,\mathcal{F},B)/U$ and $K_{\mathcal{F}'}+B'$ is semi-ample $/U$ .

If, in addition, the induced birational map $X\dashrightarrow X'$ does not extract any divisor, then we say remove the initial ‘bs-’ or the phrase ‘in the sense of Birkar and Shokurov’ in the previous definitions.

1. (5) We say that $(X',\mathcal{F}',B')/U$ is a log minimal model of $(X,\mathcal{F},B)/U$ if it is a bs-minimal model of $(X,\mathcal{F},B)$ and $(X',\mathcal{F}',B')$ is $\mathbb{Q}$ -factorial ACSS.

2. (6) We say that $(X',\mathcal{F}',B')/U$ is a good log minimal model of $(X,\mathcal{F},B)/U$ if it is a log minimal model of $(X,\mathcal{F},B)$ and $K_{\mathcal{F}'}+B'$ is semi-ample $/U$ .

We remark that, similar to [Reference Chen, Han, Liu and XieCHLX23], the definition of ‘log minimal model’ in our paper does not coincide with the classical definition with $\mathcal{F}=T_X$ as $\mathbb{Q}$ -factorial ACSS is equivalent to $\mathbb{Q}$ -factorial qdlt instead of $\mathbb{Q}$ -factorial dlt when $\mathcal{F}=T_X$ .

1. (1) We say that $(X',\mathcal{F}',B')\rightarrow Z$ is a bs-Mori fiber space, or a Mori fiber space in the sense of Birkar and Shokurov of $(X,\mathcal{F},B)/U$ , if for any prime divisor D on X which is exceptional over X’,

   $$a(D,\mathcal{F},B)<a(D,\mathcal{F}',B').$$

2. (2) We say that $(X',\mathcal{F}',B')\rightarrow Z$ is a Mori fiber space of $(X,\mathcal{F},B)/U$ if $(X',\mathcal{F}',B')\rightarrow Z$ is a bs-Mori fiber space of $(X,\mathcal{F},B)/U$ and the induced birational map $X\dashrightarrow X'$ does not extract any divisor.

3. (3) We say that $(X',\mathcal{F}',B')\rightarrow Z$ is a log Mori fiber space of $(X,\mathcal{F},B)/U$ if it is a bs-Mori fiber space of $(X,\mathcal{F},B)/U$ and $(X',\mathcal{F}',B')$ is $\mathbb{Q}$ -factorial ACSS.

We also remark that we do not have any requirement on the singularities of $(X,\mathcal{F},B)$ and $(X',\mathcal{F}',B')$ in Definition 4.3(1)–(4) and Definitions 4.4(1)–(2). This is because in many cases, we want to consider a generalized foliated quadruple polarized by an ample divisor A. Due to the failure of Bertini-type theorems for foliations, usually the only thing we can do is to consider a generalized foliated quadruple structure $(X,\mathcal{F},B,\bar A)$ , i.e. we let A be the nef part. However, this is inconvenient when the foliation is associated with some other pair structure, as many theorems on pairs consider structures of the form $(X,B+A)$ instead. Therefore, if we do not have any singularity restrictions on the models, then using $(X,\mathcal{F},B+A)$ will bring us more flexibility when applying results of usual pairs.

4.2 Basic properties of models

In this subsection we prove several basics properties on models of foliated triples. We remark that results in this section works for any foliated triples without any requirement on algebraic integrability nor singularities, so we expect results in this subsection to be useful for further applications, particularly to non-algebraically integrable foliations.

Lemma 4.6 (Cf. [Reference BirkarBir12, Remark 2.6] and [Reference Hacon and LiuHL23, Lemma 3.4]). Let $(X,\mathcal{F},B)/U$ be a foliated triple and let $(X',\mathcal{F}',B')/U$ a bs-weak lc model of $(X,\mathcal{F},B)/U$ associated with the birational map $\phi: X\dashrightarrow X'$ . Let $p: W\rightarrow X$ and $q: W\rightarrow X'$ be birational morphisms such that $q=\phi\circ p$ . Assume that

$$p^*(K_\mathcal{F}+B)=q^*(K_{\mathcal{F}'}+B')+E,$$

then $E\geq 0$ and is exceptional $/X'$ .

Proof. For any prime divisor D that is an irreducible component of E,

$$\operatorname{mult}_DE=a(D,\mathcal{F}',B')-a(D,\mathcal{F},B).$$

Therefore, if D is not exceptional $/X$ , then:

• – if D is not exceptional $/X'$ , then $\operatorname{mult}_DE=0$ by Definition 4.2;

• – if D is exceptional $/X'$ , then $\operatorname{mult}_DE\geq 0$ by Definition 4.3(1).

Therefore, $p_*E\geq 0$ . Since $K_{\mathcal{F}'}+B'$ is nef $/U$ , $q^*(K_{\mathcal{F}'}+B')$ is nef $/X$ , hence E is anti-nef $/X$ . By the negativity lemma, $E\geq 0$ .

If E is not exceptional $/X'$ , then there exists a component D of E that is not exceptional $/X'$ . If D is not exceptional $/X$ , then $\operatorname{mult}_DE=0$ by Definition 4.2, a contradiction. Thus, D is exceptional over X. In particular, $\phi$ extracts D. Since $(X',\mathcal{F}',B')/U$ is a log birational model of $(X,\mathcal{F},B)$ ,

$$a(D,\mathcal{F}',B')=a(D,\mathcal{F},B),$$

which implies that $\operatorname{mult}_DE=0$ , a contradiction.

1. (1) $h_1^*(K_{\mathcal{F}_1}+B_1)=h_2^*(K_{\mathcal{F}_2}+B_2);$

2. (2) if $K_{\mathcal{F}_2}+B_2$ is semi-ample $/U$ , then $K_{\mathcal{F}_1}+B_1$ is semi-ample $/U$ ;

3. (3) if $K_{\mathcal{F}_2}+B_2$ is ample $/U$ , then $\phi$ is a morphism.

Proof. Let $\phi_1: X\dashrightarrow X_1$ and $\phi_2: X\dashrightarrow X_2$ be the induced birational maps. Possibly replacing W with a higher model, we may assume that the induced birational map $h: W\rightarrow X$ is a morphism. Let

$$E_i:=h^*(K_X+B)-h_i^*(K_{X_i}+B_i)$$

for $i\in\{1,2\}$ . By Lemma 4.6, $E_i\geq 0$ and is exceptional over $X_i$ for $i\in\{1,2\}$ . Thus, $h_{1,*}(E_2-E_1)\geq 0$ and $E_1-E_2$ is nef $/X_1$ , and $h_{2,*}(E_1-E_2)\geq 0$ and $E_2-E_1$ is nef $/X_2$ . By the negativity lemma, $E_2-E_1\geq 0$ and $E_1-E_2\geq 0$ . Thus, $E_1=E_2$ , which implies condition (1). Condition (2) immediately follows from condition (1). By condition (1), if $K_{\mathcal{F}_2}+B_2$ is ample $/U$ , then $h_2: W\rightarrow X_2$ is the ample model $/U$ of $h^*(K_{\mathcal{F}_1}+B_1)$ , hence $\phi$ is the ample model $/U$ of $K_{\mathcal{F}_1}+B_1$ . Since $K_{\mathcal{F}_1}+B_1$ is semi-ample $/U$ , $\phi$ is a morphism. This implies condition (3).

Lemma 4.8. Let r be a positive real number. Let $(X,\mathcal{F}_1,B_1)/U$ and $(X,\mathcal{F}_2,B_2)/U$ be two foliated triples such that

$$K_{\mathcal{F}_2}+B_2\equiv_U r(K_{\mathcal{F}_1}+B_1).$$

Let $(X',\mathcal{F}'_1,B_1')/U$ be a weak lc model (respectively, minimal model) of $(X,\mathcal{F}_1,B_1)/U$ with induced birational map $\phi: X\dashrightarrow X'$ . Let $\mathcal{F}_2':=\phi_*\mathcal{F}$ and $B_2':=\phi_*B_2$ . Then $(X',\mathcal{F}_2',B_2')/U$ is a weak lc model (respectively, minimal model) of $(X,\mathcal{F}_2,B_2)/U$ .

If $(X',\mathcal{F}'_1,B_1')/U$ is a semi-ample model (respectively, good minimal model) of $(X,\mathcal{F}_1,B_1)/U$ and

$$K_{\mathcal{F}_2}+B_2\sim_{\mathbb R,U} r(K_{\mathcal{F}_1}+B_1),$$

$(X',\mathcal{F}'_2,B_2')/U$ is a semi-ample model (respectively, good minimal model) of $(X,\mathcal{F}_2,B_2)/U$ .

Proof. Let $p: W\rightarrow X$ and $q: W\rightarrow X'$ be a resolution of indeterminacy. By Lemma 4.6,

$$p^*(K_{\mathcal{F}_1}+B_1)=q^*(K_{\mathcal{F}_1'}+B_1')+E$$

for some $\mathbb{R}$ -divisor $E\geq 0$ that is exceptional $/X'$ . Then

$$p^*(K_{\mathcal{F}_2}+B_2)\equiv_U rq^*(K_{\mathcal{F}_1'}+B_1')+rE,$$

so

$$K_{\mathcal{F}_2'}+B_2'=q_*p^*(K_{\mathcal{F}_1}+B_1)\equiv q_*(rq^*(K_{\mathcal{F}_1'}+B_1')+rE)=r(K_{\mathcal{F}_1'}+B_1')$$

is nef $/U$ . Moreover, if $K_{\mathcal{F}_1'}+B_1'$ is semi-ample $/U$ and $K_{\mathcal{F}_2}+B_2\sim_{\mathbb R,U} r(K_{\mathcal{F}_1}+B_1)$ , then

$$K_{\mathcal{F}_2'}+B_2'\sim_{\mathbb R,U}r(K_{\mathcal{F}_1'}+B_1')$$

is semi-ample $/U$ .

We have

$$p^*(K_{\mathcal{F}_2}+B_2)\equiv_U q^*(K_{\mathcal{F}_2'}+B_2')+rE.$$

Therefore, for any prime divisor D on X which is exceptional over X’,

$$a(D,\mathcal{F}_2',B_2')-a(D,\mathcal{F}_2,B_2)=-\operatorname{mult}_Dp_*(rE)=r(a(D,\mathcal{F}_1',B_1')-a(D,\mathcal{F}_1,B_1)).$$

Therefore, $a(D,\mathcal{F}_2,B_2)\leq\text{(respectively, }<\text{) }a(D,\mathcal{F}_2',B_2')$ if and only if $a(D,\mathcal{F}_1,B_1)\leq\text{(respectively, }<\text{) }a(D,\mathcal{F}_1',B_1')$ . The lemma follows immediately from the definitions.

4.3 Models under foliated log resolutions

From now on, we focus on different models of foliations that are algebraically integrable and lc. We first study the relationship between different types of models and foliated log resolutions. Of course, we expect results in this section to hold in greater generalities provided that there is a proper definition of ‘foliated log resolution’ for non-algebraically integrable foliations.

We first recall the following result.

See [Reference BirkarBir12, Definition 3.1] for the definition of very exceptional divisors. In particular, exceptional divisors coincide with very exceptional divisors in the context of birational morphisms.

1. (1) $K_{\mathcal{F}'}+B'=h^*(K_\mathcal{F}+B)+E$ ;

2. (2) $(X',\mathcal{F}',B')$ is foliated log smooth and lc;

3. (3) E is h-exceptional;

4. (4) for any h-exceptional prime divisor D such that

   $$a(D,X,B)\gt-\epsilon_{\mathcal{F}}(D),$$
   D is a component of E.

We say that $(X',\mathcal{F}',B')$ is a foliated log smooth model of $(X,\mathcal{F},B)$ .

Then any bs-weak lc model (respectively, bs-minimal model, bs-semi-ample model, bs-good minimal model, log minimal model, good log minimal model) of $(W,\mathcal{F}_W,B_W)/U$ is a bs-weak lc model (respectively, bs-minimal model, bs-semi-ample model, bs-good minimal model, log minimal model, good log minimal model) of $(X,\mathcal{F},B)/U$ .

Proof. We let $h: W\rightarrow X$ be the induced birational morphism. We may write

$$K_{\mathcal{F}_W}+B_W=h^*(K_{\mathcal{F}}+B)+E$$

for some $E\geq 0$ that is h-exceptional, and $D\subset\operatorname{Supp} E$ for any h-exceptional prime divisor D such that $a(D,X,B)\gt-\epsilon_{\mathcal{F}}(E)$ .

Claim 4.12. Let $(X',\mathcal{F}',B')/U$ be a bs-weak lc model of $(W,\mathcal{F}_W,B_W)/U$ . Then

$$a(D,\mathcal{F},B)\leq a(D,\mathcal{F}',B')$$

for any prime divisor D over X.

Proof. Let $\phi_W: W\dashrightarrow X'$ be the induced birational map, and let $p: V\rightarrow W$ and $q: V\rightarrow X'$ be a common resolution such that $q=\phi_W\circ p$ . By Lemma 4.6,

$$p^*(K_{\mathcal{F}_W}+B_W)=q^*(K_{\mathcal{F}'}+B')+F$$

for some $F\geq 0$ that is exceptional over X’. Then we have

$$p^*h^*(K_{\mathcal{F}}+B)=q^*(K_{\mathcal{F}'}+B')+F-p^*E,$$

so

$$p^*E-F\sim_{\mathbb{R},X}q^*(K_{\mathcal{F}'}+B')$$

is nef $/X$ . Since $h_*p_*(F-p^*E)=h_*p_*F\geq 0$ , by the negativity lemma, $F\geq p^*E$ . Thus, $a(D,\mathcal{F},B)\leq a(D,\mathcal{F}',B')$ for any prime divisor D over X.

Proof of Lemma 4.11 continued. First we prove the bs-weak lc model case. Let $(X',\mathcal{F}',B')/U$ be a bs-weak lc model of $(W,\mathcal{F}_W,B_W)/U$ with induced birational map $\phi_W: W\dashrightarrow X'$ . By Claim 4.12, we only need to show that $(X',\mathcal{F}',B')/U$ is a log birational model of $(X,\mathcal{F},B)/U$ .

Let $\phi: X\dashrightarrow X'$ be the induced morphism and

$$\tilde B':=\phi_*B+\operatorname{Exc}(\phi^{-1})^{\mathcal{F}'-\mathrm{ninv}},$$

then we only need to show that $B'=\tilde B'$ . Since $(X',\mathcal{F}',B')/U$ is a bs-weak lc model of $(W,\mathcal{F}_W,B_W)/U$ , we have

$$B'=(\phi_W)_*B_W+\operatorname{Exc}(\phi_W^{-1})^{\mathcal{F}'-\mathrm{ninv}}.$$

Let D be a prime divisor on X’. There are three cases.

Case 1. The divisor D is not exceptional over X. In this case,

$$-\operatorname{mult}_D\tilde B'=a(D,\mathcal{F}',\tilde B')=a(D,\mathcal{F},B)=a(D,\mathcal{F}_W,B_W)=a(D,\mathcal{F}',B')=-\operatorname{mult}_DB',$$

so $\operatorname{mult}_DB'=\operatorname{mult}_D\tilde B'$ .

Case 2. The divisor D is exceptional over W. In this case, D is a component of $\operatorname{Exc}(\phi_W^{-1})$ and a component of $\operatorname{Exc}(\phi^{-1})$ , hence

$$\operatorname{mult}_DB'=\epsilon_{\mathcal{F}}(D)=\operatorname{mult}_DB''.$$

Case 3. The divisor D is exceptional over X but not exceptional over W. In this case,

$$-\operatorname{mult}_DB'=a(D,\mathcal{F}',B')=a(D,\mathcal{F}_W,B_W).$$

Since $E\geq 0$ , $a(D,\mathcal{F}_W,B_W)\leq a(D,\mathcal{F},B).$ By Claim 4.12, $a(D,\mathcal{F},B)\leq a(D,\mathcal{F}',B').$ Thus,

$$-\operatorname{mult}_DB'=a(D,\mathcal{F},B)=a(D,\mathcal{F}',B')=a(D,\mathcal{F}_W,B_W).$$

By Definition 4.10(4), $a(D,\mathcal{F},B)=-\epsilon_{\mathcal{F}}(D),$ which implies that

$$\operatorname{mult}_DB'=\epsilon_{\mathcal{F}}(D)=\operatorname{mult}_D\operatorname{Exc}(\phi^{-1})^{\mathcal{F}'-\mathrm{ninv}}=\operatorname{mult}_D\tilde B'.$$

Thus, $B'=\tilde B'$ , so $(X',\mathcal{F}',B')/U$ is a log birational model of $(X,\mathcal{F},B)/U$ , and we are done for the bs-weak lc model case.

Next we prove the bs-minimal model case. Suppose that $(X',\mathcal{F}',B')/U$ be a bs-minimal model of $(W,\mathcal{F}_W,B_W)/U$ . For any prime divisor D on X which is exceptional over X’, $h^{-1}_*D$ is a prime divisor on W which is exceptional over X’. Thus,

$$a(D,\mathcal{F},B)=a(D,\mathcal{F}_W,B_W)<a(D,\mathcal{F}',B').$$

The bs-minimal model case immediately follows from the bs-weak lc model case.

The bs-semi-ample model, bs-good minimal model, log minimal model, and good log minimal model cases follow immediately from the bs-weak lc model and the bs-minimal model cases.

4.4 Models under pullbacks

Then we may run a $(K_{\mathcal{F}_W}+B_W)$ -MMP $/X'$ with scaling of an ample $/X'$ $\mathbb{R}$ -divisor which terminates with a good minimal model $(Y,\mathcal{F}_Y,B_Y)/X'$ of $(W,\mathcal{F}_W,B_W)/X'$ such that

$$K_{\mathcal{F}_Y}+B_Y=q^*(K_{\mathcal{F}'}+B'),$$

where $q: Y\rightarrow X'$ is the induced morphism. In particular, $(Y,\mathcal{F}_Y,B_Y)/U$ is a log minimal model of $(W,\mathcal{F}_W,B_W)/U$ .

Proof. Let $h: W\rightarrow X$ be the induced birational morphism. We have

$$K_{\mathcal{F}_W}+B_W=h^*(K_\mathcal{F}+B)+E$$

for some $E\geq 0$ that is exceptional $/X$ . By Lemma 4.6, we have

$$h^*(K_{\mathcal{F}}+B)=\phi_W^*(K_{\mathcal{F}'}+B')+F,$$

where $F\geq 0$ is exceptional $/X'$ . Thus,

$$K_{\mathcal{F}_W}+B_W\sim_{\mathbb R,X'}F+E.$$

Assume that D is not exceptional over X’. Since $(X',\mathcal{F}',B')/U$ is a log birational model of $(X,\mathcal{F},B)/U$ and $(X,\mathcal{F},B)$ is lc, $a(D,\mathcal{F}',B')=-\epsilon_{\mathcal{F}}(E)$ . Since $F\geq 0$ , $a(D,\mathcal{F},B)\leq a(D,\mathcal{F}',B')$ . Thus, $a(D,\mathcal{F},B)=-\epsilon_{\mathcal{F}}(E)$ , hence D is not a component of E, a contradiction.

Proof of Lemma 4.13 continued. By Claim 4.14, $F+E$ is exceptional over X’. By Theorem 4.9, we may run a $(K_{\mathcal{F}_W}+B_W)$ -MMP $/X'$ with scaling of an ample $/X'$ divisor, which terminates with a good minimal model $(Y,\mathcal{F}_Y,B_Y)/X'$ of $(W,\mathcal{F}_W,B_W)/X'$ such that $K_{\mathcal{F}_Y}+B_Y\sim_{\mathbb{R},X'}0$ . In particular, $(Y,\mathcal{F}_Y,B_Y)$ is $\mathbb{Q}$ -factorial ACSS, and $a(D,\mathcal{F}_W,B_W)<a(D,\mathcal{F}_Y,B_Y)$ for any prime divisor D on W that is exceptional $/Y$ . By the negativity lemma,

$$K_{\mathcal{F}_Y}+B_Y=q^*(K_{\mathcal{F}'}+B').$$

The lemma follows.

Lemma 4.16 Let $(X,\mathcal{F},B)/U$ and $(Y,\mathcal{F}_Y,B_Y)/U$ be two lc algebraically integrable foliated triples, and let $f: Y\rightarrow X$ be a birational morphism such that

$$K_{\mathcal{F}_Y}+B_Y=f^*(K_\mathcal{F}+B)+E$$

for some $E\geq 0$ that is exceptional $/X$ and $f_*\mathcal{F}_Y=\mathcal{F}$ . Then we have the following.

1. (1) Any bs-weak lc model of $(X,\mathcal{F},B)/U$ is a bs-weak lc model of $(Y,\mathcal{F}_Y,B_Y)/U$ .

2. (2) If $(X,\mathcal{F},B)/U$ has a bs-weak lc model (respectively, bs-semi-ample model), then $(Y,\mathcal{F}_Y,B_Y)/U$ has a log minimal model (respectively, good log minimal model).

Proof. (1) Let $(X',\mathcal{F}',B')/U$ be a bs-weak lc model of $(X,\mathcal{F},B)/U$ , $\phi: X\dashrightarrow X'$ the induced birational map, and $\phi_Y:=\phi\circ f$ . Let $p: W\rightarrow Y$ and $q: W\rightarrow X'$ be a resolution of indeterminacy, and let $h:=f\circ p$ . By Lemma 4.6,

$$h^*(K_\mathcal{F}+B)=q^*(K_{\mathcal{F}'}+B')+F$$

for some $F\geq 0$ that is exceptional over X’. Thus,

$$p^*(K_{\mathcal{F}_Y}+B_Y)=q^*(K_{\mathcal{F}'}+B')+p^*E+F.$$

Thus, $a(D,\mathcal{F}_Y,B_Y)\leq a(D,\mathcal{F}',B')$ for any prime divisor D over X’. In particular, if $a(D,\mathcal{F}',B')=-\epsilon_{\mathcal{F}}(D)$ , then $a(D,\mathcal{F}_Y,B_Y)=-\epsilon_{\mathcal{F}}(D)$ .

Since $(X',\mathcal{F}',B')/U$ is a log birational model of $(X,\mathcal{F},B)/U$ and $(X,\mathcal{F},B)$ is lc,

$$B'=\phi_*B+\operatorname{Exc}(\phi^{-1})^{\mathcal{F}'-\mathrm{ninv}}.$$

Let

$$\tilde B':=(\phi_Y)_*B_Y+\operatorname{Exc}(\phi_Y^{-1})^{\mathcal{F}'-\mathrm{ninv}}.$$

For any prime divisor D on X’, there are two cases:

Case 1. The divisor D is not exceptional over X. In this case,

$$\operatorname{mult}_DB'=a(D,\mathcal{F}',B')=a(D,\mathcal{F},B)=a(D,\mathcal{F}_Y,B_Y)=a(D,\mathcal{F}',\tilde B')=-\operatorname{mult}_D\tilde B',$$

so $\operatorname{mult}_DB'=\operatorname{mult}_D\tilde B'$ .

Case 2. The divisor D is exceptional over X. In this case,

$$a(D,\mathcal{F}',B')=-\operatorname{mult}_DB'=-\epsilon_{\mathcal{F}}(D).$$

Since $a(D,\mathcal{F}_Y,B_Y)\leq a(D,\mathcal{F}',B')$ and $(Y,\mathcal{F}_Y,B_Y)$ is lc, $a(D,\mathcal{F}_Y,B_Y)=-\epsilon_{\mathcal{F}}(D)$ . Therefore, if D is not exceptional over Y, then

$$\operatorname{mult}_D\tilde B'=\operatorname{mult}_DB_Y=-a(D,\mathcal{F}_Y,B_Y)=\epsilon_{\mathcal{F}}(D)=\operatorname{mult}_DB',$$

and if D is exceptional over Y, then

$$\operatorname{mult}_D\tilde B'=\operatorname{mult}_D\operatorname{Exc}(\phi_Y^{-1})^{\mathcal{F}'-\mathrm{ninv}}=\epsilon_{\mathcal{F}}(D)=\operatorname{mult}_DB'.$$

Thus, $B'=B''$ , hence $(X',\mathcal{F}',B')/U$ is a log birational model of $(Y,\mathcal{F}_Y,B_Y)/U$ . Since $K_{\mathcal{F}'}+B'$ is nef $/U$ , and $a(D,\mathcal{F}_Y,B_Y)\leq a(D,\mathcal{F}',B')$ for any prime divisor D over X’, $(X',\mathcal{F}',B')/U$ is a bs-weak lc model of $(Y,\mathcal{F}_Y,B_Y)/U$ , and we get condition (1).