Proof. Since being dlt, the property in (2), and being log smooth are local properties, we may work over a neighborhood of a generic point \(\eta\) of an lc center of \((X,B,{\boldsymbol{M}})\). (2)\(\Rightarrow\) (1) immediately becomes obvious, so we only need to prove (1)\(\Rightarrow\) (2).

By Definition 61, there exists a neighborhood \(V\) of \(\eta\) such that \((V,B|_V)\) is log smooth and \(\eta\) is an lc center of \((V,B|_V)\). Since \((V,B|_V)\) is log smooth, \({\boldsymbol{M}}_X|_V\) is \(\mathbb{R}\)-Cartier. We let \({\boldsymbol{M}}^V:={\boldsymbol{M}}|_V\) be the restricted \(\boldsymbol{b}\)-divisor of \({\boldsymbol{M}}\) on \(V\), then \({\boldsymbol{M}}^V\) is nef\(/X\) and \({\boldsymbol{M}}^V_V={\boldsymbol{M}}_X|_V\). Suppose that \(h: V'\rightarrow V\) is a resolution of \(V\) such that \({\boldsymbol{M}}^V\) descends to \(V'\) and there exists a prime divisor \(E\) on \(V'\) such that \(\operatorname{center}_{V}E=\bar\eta\) and \(E\) is an lc place of \((V,B|_V)\). By the negativity lemma, \[{\boldsymbol{M}}^V_{V'}=h^*{\boldsymbol{M}}^V_V-F\] for some \(F\geq 0\), such that either \(F=0\) over \(\bar\eta\) or \(\operatorname{Supp}F=\operatorname{Supp}h^{-1}(\bar\eta)\). Since \((X,B,{\boldsymbol{M}})\) is lc, \((V,B|_V,{\boldsymbol{M}}^V)\) is lc. Thus \(F=0\) over \(\bar\eta\). Possibly shrinking \(V\), we may assume that \({\boldsymbol{M}}\) descends to \(V\). The lemma follows. ◻

Proof. By [4], there exists \(0\leq\Delta\sim_{\mathbb{R},U}B+A+{\boldsymbol{M}}_X\) such that \((X,\Delta)\) is klt. By [29], any \((K_X+\Delta)\)-MMP\(/U\) with scaling of an ample\(/U\) \(\mathbb{R}\)-divisor terminates with either a Mori fiber space of \((X,\Delta)/U\) or a semi-good minimal model of \((X,\Delta)/U\). The lemma follows. ◻

Proof. Possibly rescaling \(A\), we may assume that \(K_X+B+A+{\boldsymbol{M}}_X\) is nef\(/U\). We first prove that we may run this MMP by induction on \(i\). Let \(\lambda_0:=1\) and suppose that there is already a sequence of steps of a \((K_X+B+{\boldsymbol{M}}_X)\)-MMP\(/U\) with scaling of \(A\) \[(X,B,{\boldsymbol{M}}):=(X_1,B_1,{\boldsymbol{M}})\dashrightarrow (X_2,B_2,{\boldsymbol{M}})\dashrightarrow\dots\dashrightarrow (X_k,B_k,{\boldsymbol{M}})\] for some \(k\ge1\), such that \(\lambda_i\geq\lambda_{i+1}\) for any \(i\leq k-2\). If \(K_{X_k}+B_k+{\boldsymbol{M}}_{X_k}\) is nef\(/U\), then we are done, so we may assume that \(K_{X_k}+B_k+{\boldsymbol{M}}_{X_k}\) is not nef\(/U\). Since nef\(/U\) is a closed condition, \(\lambda_k>0\). Since \(K_X+B+\lambda_0A+{\boldsymbol{M}}_X\) is nef\(/U\) and \(K_{X_{k-1}}+B_{k-1}+\lambda_{k-1}A_{k-1}+{\boldsymbol{M}}_{X_{k-1}}\) is nef\(/U\) when \(k\geq 2\), \(K_{X_{k}}+B_{k}+\lambda_{k-1}A_{k}+{\boldsymbol{M}}_{X_{k}}\) is nef, hence \(\lambda_{k-1}\geq\lambda_{k}\).

By [4], there exists a klt pair \((X,\Delta)\) such that \[K_X+\Delta\sim_{\mathbb{R},U}K_X+B+{\boldsymbol{M}}_X+\frac{\lambda_k}{2}A.\] Possibly replacing \(A\), we may assume that \((X,\Delta+(1-\frac{\lambda_k}{2})A)\) is lc. Then we have an induced sequence of steps of a \((K_X+\Delta)\)-MMP\(/U\) with scaling of \((1-\frac{\lambda_k}{2})A\) \[(X,\Delta):=(X_1,\Delta_1)\dashrightarrow (X_2,\Delta_2)\dashrightarrow\dots\dashrightarrow (X_k,\Delta_k),\] such that \(K_{X_k}+\Delta_k\) is not nef. Let \((X_k,\Delta_k)\dashrightarrow (X_{k+1},\Delta_{k+1})\) be the next step of the \((K_X+\Delta)\)-MMP\(/U\) with scaling of \((1-\frac{\lambda_k}{2})A\). Then the induced birational map \(X_k\dashrightarrow X_{k+1}\) is a step of a \((K_X+B+{\boldsymbol{M}}_X)\)-MMP\(/U\) with scaling of \(A\).

It remains to prove that if this MMP does not terminate, then \(\lim_{i\rightarrow+\infty}\lambda_i=0\). Suppose that \(\lambda:=\lim_{i\rightarrow+\infty}\lambda_i>0\). Then \[\begin{align} \left(X,B+\frac{\lambda}{2}A,{\boldsymbol{M}}\right):=&\left(X_1,B_1+\frac{\lambda}{2}A_1,{\boldsymbol{M}}\right)\dashrightarrow\left(X_2,B_2+\frac{\lambda}{2}A_2,{\boldsymbol{M}}\right)\dashrightarrow\\ \dots\dashrightarrow&\left(X_i,B_i+\frac{\lambda}{2}A_i,{\boldsymbol{M}}\right)\dashrightarrow\dots \end{align}\] is an infinite sequence of steps of a \((K_X+B+\frac{\lambda}{2}A+{\boldsymbol{M}}_X)\)-MMP\(/U\), which contradicts Lemma 65. ◻

Proof. By [35], possibly replacing \(A\) with a general element in \(|A/U|_{\mathbb{R}}\), there exists an lc pair \((X,\Delta+\frac{1}{2}A)\) such that \(0\leq \Delta\sim_{\mathbb{R}}B+\frac{1}{2}A+{\boldsymbol{M}}_X\). By [47] and [31], any \((K_X+\Delta+\frac{1}{2}A)\)-MMP\(/U\) with scaling of an ample\(/U\) \(\mathbb{R}\)-divisor terminates. Thus any \((K_X+B+A+{\boldsymbol{M}}_X)\)-MMP\(/U\) with scaling of an ample\(/U\) \(\mathbb{R}\)-divisor terminates. The remaining part of the proposition follows from [15] and [12]. ◻

Proof. We let \(f: Y\rightarrow X\) be a resolution of \(X\). Let \(P_Y:=P(Y,f^*D)\) and \(N_Y:=N(Y,f^*D)\) be the positive and negative parts of the Nakayama-Zariski decomposition of \(f^*D\) respectively, and let \(P:=P(X,D)\) and \(N:=N(X,D)\) be the positive and negative parts of the Nakayama-Zariski decomposition of \(D\) respectively. Since \(D\) is movable, by [15], \(N=0\). By [15], \(f_*N_Y=N\), so \(N_Y\) is exceptional\(/X\). Since \(\kappa_\sigma(f^*D)=\kappa_\sigma(D)=0\), by [71], \(P_Y\equiv 0\). Thus \(D=f_*D_Y=f_*(P_Y+N_Y)\equiv 0.\) ◻

Proof. By Lemma 66, we may run a \((K_X+B+{\boldsymbol{M}}_X)\)-MMP with scaling of \(A\). Let \[(X,B,{\boldsymbol{M}}):=(X_1,B_1,{\boldsymbol{M}})\dashrightarrow (X_2,B_2,{\boldsymbol{M}})\dashrightarrow\dots\dashrightarrow (X_i,B_i,{\boldsymbol{M}})\dashrightarrow\dots\] be any such MMP with scaling numbers \(\lambda_i\ge\lambda_{i+1}\). If this MMP does not terminate, then \(\lim_{i\rightarrow+\infty}\lambda_i=0\) by Lemma 66. There exists a positive integer \(m\) such that \(X_i\dashrightarrow X_{i+1}\) is a flip for any \(i\geq m\). We may denote by \(\phi_i: X_m\dashrightarrow X_i\) the induced birational contraction and \(A_i\) the strict transform of \(A\) on \(X_i\) for any \(i> m.\) Since \(K_{X_i}+B_i+\lambda_iA_i+{\boldsymbol{M}}_{X_i}\) is nef for each \(i\), \[K_{X_m}+B_m+{\boldsymbol{M}}_{X_m}=\lim_{i\rightarrow+\infty}(\phi_i^{-1})_*(K_{X_i}+B_i+\lambda_iA_i+{\boldsymbol{M}}_{X_i})\] is movable. Moreover, since \(\kappa_{\sigma}(K_X+B+{\boldsymbol{M}}_X)=0\), \(\kappa_{\sigma}(K_{X_m}+B_m+{\boldsymbol{M}}_{X_m})=0\). By Lemma 68, \(K_{X_m}+B_m+{\boldsymbol{M}}_{X_m}\equiv 0\), a contradiction. Thus this MMP terminates with a model \((X',B',{\boldsymbol{M}})\) such that \(K_{X'}+B'+{\boldsymbol{M}}_{X'}\) is nef and \(\kappa_{\sigma}(K_{X'}+B'+{\boldsymbol{M}}_{X'})=0\). By Lemma 68 again, \(K_{X'}+B'+{\boldsymbol{M}}_{X'}\equiv 0\). Moreover, if \(\kappa_{\iota}(K_X+B+{\boldsymbol{M}}_X)=0\), then \(\kappa_{\iota}(K_{X'}+B'+{\boldsymbol{M}}_{X'})=0\), hence \(K_{X'}+B'+{\boldsymbol{M}}_{X'}\sim_{\mathbb{R}}0\). ◻

Proof. By [4], \((S,B_S,{\boldsymbol{M}}^S)\) is dlt.

(1) Let \(f: \tilde{X}\rightarrow X\) be a log resolution of \((X,\operatorname{Supp}B)\) such that \({\boldsymbol{M}}\) descends to \(\tilde{X}\). Let \(K_{\tilde{X}}+\tilde{B}+{\boldsymbol{M}}_{\tilde{X}}:=f^*(K_X+B+{\boldsymbol{M}}_X)\) and let \(\tilde{S}\) be the strict transform of \(S\) on \(\tilde{X}\), then \(f|_{\tilde{S}}\) is a log resolution of \((S,\operatorname{Supp}B_S)\) such that \({\boldsymbol{M}}^S\) descends to \(\tilde{S}\). We have \[f|_{\tilde{S}}^*(K_S+B_S+{\boldsymbol{M}}^S_S)=K_{\tilde{S}}+B_{\tilde{S}}+{\boldsymbol{M}}^S_{\tilde{S}}:=(K_{\tilde{X}}+\tilde{B}+{\boldsymbol{M}}_{\tilde{X}})|_{\tilde{S}}.\] Let \(W_S\) be an lc center of \((S,B_S,{\boldsymbol{M}}^S)\). Then \(W_S\) is the image of an lc center \(W_{\tilde{S}}\) of \((\tilde{S},B_{\tilde{S}},{\boldsymbol{M}}^S)\) in \(S\). Since \((\tilde{X},\tilde{B})\) is log smooth and \({\boldsymbol{M}}\) descends to \(\tilde{X}\), \(W_{\tilde{S}}\) is also an lc center of \((\tilde{X},\tilde{B},{\boldsymbol{M}})\) which is contained in \(\tilde{S}\), so \(W:=f(W_{\tilde{S}})\) is an lc center of \((X,B,{\boldsymbol{M}})\) which is contained in \(S\). It is clear that \(W_S=W\) under the natural inclusion \(S\rightarrow X\). This implies (1).

(2) Let \(W\) be an lc center of \((X,B,{\boldsymbol{M}})\) that is contained in \(S\). Since \((X,B,{\boldsymbol{M}})\) is dlt, by Lemma 62, possibly shrinking \(X\) to a neighborhood of the generic point of \(W\), we may assume that \((X,B)\) is log smooth and \({\boldsymbol{M}}\) descends to \(X\). Thus \(W\) is an lc center of \((X,B)\), \(K_S+B_S=(K_X+B)|_S\), and \({\boldsymbol{M}}^S\) descends to \(S\). Since \((X,B)\) is log smooth, \(W\) is an lc center of \((S,B_S)\), hence an lc center of \((S,B_S,{\boldsymbol{M}}^S)\). This implies (2). ◻

Proof. Step 1. In this step, we show that the theorem holds over an étale neighborhood \((s'\in U')\rightarrow (s\in U)\) such that \(k(s)=k(s')\). We use induction on \(\dim X\) and on \(\dim U\).

If \(f^{-1}(s)\cap\lfloor B\rfloor\) is disconnected, then by [7], after an étale base change, there are exactly two non-trivial lc centers of \((X,B,{\boldsymbol{M}})\) intersecting \(f^{-1}(s)\), and they are \(\mathbb{P}^1\)-linked with each other. We are done in this case.

If \(f^{-1}(s)\cap\lfloor B\rfloor\) is connected, then we let \(D_1,\dots,D_r\) be the irreducible components of \(\lfloor B\rfloor\). By passing to an étale neighborhood of \(s\in S\) without changing \(k(s)\), we may assume that each \(D_i\) has connected fiber over \(s\), and every lc center of \((X,B,{\boldsymbol{M}})\) intersects \(f^{-1}(s)\) (cf. [72]). Possibly reordering indices, we may assume that \(Z\subset D_1\), \(W\subset D_r\), and \[f^{-1}(s)\cap D_i\cap D_{i+1}\not=\emptyset\] for any \(1\leq i\leq r-1\). Let \((D_i,B_{D_i},{\boldsymbol{M}}^{D_i})\) be the g-pair induced by adjunction \[K_{D_i}+B_{D_i}+{\boldsymbol{M}}^{D_i}_{D_i}:=(K_X+B+{\boldsymbol{M}}_X)|_{D_i}\] for each \(i\).

Proof of Theorem 77 continued. By Claim 78 applied to \(i=r\), the theorem holds over an étale neighborhood \((s'\in U')\rightarrow (s\in U)\) such that \(k(s)=k(s')\) under the induction hypothesis of \(\dim X\) and \(\dim U\).

Step 2. We show that the étale base change was not necessary and conclude the proof of the theorem. Let \[X\xrightarrow{\tilde{f}}\tilde{U}\rightarrow U\] be the Stein factorization of \(f\). Since \(f^{-1}(s)\) is connected, there exists a unique pre-image \(\tilde{s}\in\tilde{U}\) of \(s\). Let \(Z_i\) be the minimal elements of \(\mathcal{S}\). Since lc centers commute with étale base change, we see that there is a unique irreducible subvariety \[\tilde{s}\in \tilde{V}\subset U\] such that \(\tilde{V}=\tilde{f}(Z_i)\) for each \(i\).

Let \(\tilde{v}\) be the generic point of \(\tilde{V}\). By Step 1 and induction hypothesis, the theorem holds after an étale base change \[\tilde{\pi}: (\tilde{v}'\in \tilde{U}')\rightarrow (\tilde{v}\in\tilde{U}).\] Since \(\tilde{f}\) has connected fibers, \(\tilde{\pi}\) induces an isomorphism of the fibers \[\tilde{\pi}: (\tilde{f}')^{-1}(\tilde{v}')\cong\tilde{f}^{-1}(\tilde{v}).\] Thus \(Z_i\) canonically lift to \(Z_i'\cong Z_i\) and the \(\mathbb{P}^1\)-links\(/U\) between the \(Z_i'\) descend to \(\mathbb{P}^1\)-links\(/U\) between the \(Z_i\). ◻

Proof. By Definition-Lemma 63, possibly replacing \((X,B,{\boldsymbol{M}})\) with a dlt model, we may assume that \((X,B,{\boldsymbol{M}})\) is dlt. Since \(f\) is a contraction, \(f^{-1}(z)\) is connected. For any element \(W\in\mathcal{S}_z\) that is minimal with respect to inclusion, there exists an lc center \(Z_W\) of \((X,B,{\boldsymbol{M}})\) that is minimal among all lc centers whose image on \(Z\) is equal to \(W\) with respect to inclusion. By Theorem 77, all such \(Z_W\) are \(\mathbb{P}^1\)-linked\(/Z\) to each other, hence their images on \(Z\) are the same. This proves (1). (2) follows from (1) by considering every étale neighborhood of \(z\).

For any lc centers \(W_1,W_2\) on \(Z\), let \(z\in W_1\cap W_2\) be any point. By (1), there exists a unique element \(W_z\) of \(\mathcal{S}_z\). Then \[z\in W_z\subset W_1\cap W_2,\] so \[W_1\cap W_2=\cup_{z\in W_1\cap W_2}z\subset \cup_{z\in W_1\cap W_2}W_z\subset W_1\cap W_2.\] Therefore, \[W_1\cap W_2=\cup_{z\in W_1\cap W_2}W_z\] is a union of lc centers. We get (3). ◻

Proof. (1) We only need to show that \((Y,B_Y,{\boldsymbol{M}}^Y)\) is dlt, which follows from [4].

(2) There exists an lc center \(V_Y\) of \((Y,B_Y,{\boldsymbol{M}}^Y)\) such that \(f_Y(V_Y)=W_Y\). By Lemma 71, \(V_Y\) is also an lc center of \((X,B,{\boldsymbol{M}})\). Thus \(\pi(W_Y)=f(V_Y)\) is an lc center of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\).

(3) Let \(z\) be the generic point of \(W\). Since the question is étale local, possibly replacing \(Z\) by an étale neighborhood of \(z\) and replacing \(Y\) with its irreducible components, we may assume that \(f^{-1}(z)\cap Y\) is connected, and we only need to show that there exists an lc center \(V_Y\) of \(f_Y: (Y,B_Y,{\boldsymbol{M}}^Y)\rightarrow Z_Y\) such that \(f_Y(V_Y)\) is an irreducible component of \(\pi^{-1}(W)\).

Let \(V_X\) be a minimal lc center of \((X,B,{\boldsymbol{M}})\) which dominates \(W\), i.e., \(V_X\) is minimal in \[\{V\mid V\text{ is an lc center of }(X,B,{\boldsymbol{M}}), V\text{ dominates }W\}\] with respect to inclusion. Then \(f(V_X)=W\). By Theorem 77, there exists an lc center \(V_Y\subset Y\) of \((X,B,{\boldsymbol{M}})\) that is \(\mathbb{P}^1\)-linked\(/Z\) to \(V_X\). By Lemma 71, \(V_Y\) is also an lc center of \((Y,B_Y,{\boldsymbol{M}}^Y)\). Thus \(f_Y(V_Y)\subset Z_Y\) is an lc center of \(f_Y: (Y,B_Y,{\boldsymbol{M}}^Y)\rightarrow Z_Y\). Moreover, since \(V_Y\) is \(\mathbb{P}^1\)-linked\(/Z\) to \(V_X\), \((f|_Y)(V_Y)=f(V_X)=W\). Thus \(f_Y(V_Y)\) is an irreducible component of \(\pi^{-1}(W)\) and we are done. ◻

Proof. By Lemma 79, \(V\cap S\) is a union of lc centers of \((X,B,{\boldsymbol{M}})\). In particular, \(\nu(W)\) is an lc center of \((X,B,{\boldsymbol{M}})\). Possibly replacing \(V\) with \(\nu(W)\), we may assume that \(V\subset S\) and \(\nu(W)=S\). We let \(f: Y\rightarrow X\) be a dlt modification of \((X,B,{\boldsymbol{M}})\), \[K_Y+B_Y+{\boldsymbol{M}}_Y:=f^*(K_X+B+{\boldsymbol{M}}_X),\] and let \(S_Y:=f^{-1}_*S\). We let \[K_{S_Y}+B_{S_Y}+{\boldsymbol{M}}^S_{S_Y}:=(K_Y+B_Y+{\boldsymbol{M}}_Y)|_{S_Y}\] and let \(f_S: S_Y\rightarrow S^\nu\) be the induced morphism such that \(\nu\circ f_S=f|_{S_Y}\). Since \(f: (Y,B_Y,{\boldsymbol{M}})\rightarrow X\) is a dlt crepant log structure and \(V\) is an lc center of \(f: (Y,B_Y,{\boldsymbol{M}})\rightarrow X\), by Lemma 80(1)(3), any irreducible component of \(\nu^{-1}(V)\) is an lc center of \(f_S: (S_Y,B_{S_Y},{\boldsymbol{M}}^S)\rightarrow S\). In particular, \(W\) is an lc center of \(f_S: (S_Y,B_{S_Y},{\boldsymbol{M}}^S)\rightarrow S\). Since \[K_{S_Y}+B_Y+{\boldsymbol{M}}^S_{S_Y}=f_S^*(K_{S^\nu}+B_S+{\boldsymbol{M}}^S_{S^\nu}),\] \(W\) is an lc center of \((S^\nu,B_S,{\boldsymbol{M}}^S)\). ◻

Proof. We let \(V_k:=\cap_{i=1}^kS_i\) for each \(1\le k\le n-1\). Let \(W\) be a component of \(V\) with normalization \(W^\nu\). Then for each \(1\le k\le n-1\), there exists an irreducible component \(W_k\) of \(V_k\) such that \[W:=W_n\subset W_{n-1}\subset\dots\subset W_1=S_1.\] We let \(W_k^\nu\) be the normalization of \(W_k\) for each \(k\) and let \((W_0,B_0,{\boldsymbol{M}}^{(0)}):=(X,B,{\boldsymbol{M}})\). By Lemmas 81 and 82, we may repeatedly apply adjunction and get lc g-pairs \((W_k^\nu,B_k,{\boldsymbol{M}}^{(k)})\), such that for each \(1\leq k\leq n\)

• \(W_k\) is an lc center of \((W_{k-1}^\nu,B_{k-1},{\boldsymbol{M}}^{(k-1)})\), and

• \(K_{W_k^\nu}+B_k+{\boldsymbol{M}}^{(k)}_{W_k^\nu}=\left(K_{W_{k-1}^\nu}+B_{k-1}+{\boldsymbol{M}}^{(k-1)}_{W_{k-1}^\nu}\right)|_{W_k^\nu}=(K_X+B+{\boldsymbol{M}}_X)|_{W_k^\nu}.\)

Let \[t_{i,W}:=\sup\{t\geq 0\mid (X,B+tD_i,{\boldsymbol{M}})\text{ is lc near }W\}\] for \(i=1,2\). Then there exists an lc center \(T_i\subset\operatorname{Supp}D_i\) such that \(T_i\cap W\not=\emptyset\). By Lemma 79, we may assume that \(T_i\subset W\) for each \(i\). By Lemma 81, \((W^\nu,B_n+t_{i,W}D_i|_{W^\nu},{\boldsymbol{M}}^{(n)})\) is lc for each \(i\). By Lemma 82, there exists an lc center \(T_{W,i}\) of \((W^\nu,B_n+t_{i,W}D_i|_{W^\nu},{\boldsymbol{M}}^{(n)})\) such that \(T_{W,i}\subset\operatorname{Supp}D_i|_{W^\nu}\). Since \(D_1|_{W^\nu}=D_2|_{W^\nu}\), \[t_{1,W}=\operatorname{lct}\left(W^\nu,B_n,{\boldsymbol{M}}^{(n)};D_1|_{W^\nu}\right)=\operatorname{lct}\left(W^\nu,B_n,{\boldsymbol{M}}^{(n)};D_2|_{W^\nu}\right)=t_{2,W}.\] Therefore, \(t_1=t_2\). ◻

Proof. First we prove the case when \((X,B,{\boldsymbol{M}})\) is a \(\mathbb{Q}\)-g-pair.

By Definition-Lemma 63, there exists a birational morphism \(f: Y\rightarrow X\) satisfying the following. Let \(E\) be the reduced \(f\)-exceptional divisor and \(B_Y:=f^{-1}_*(B\wedge\operatorname{Supp}B)+E\), then

1. \((Y,B_Y,{\boldsymbol{M}})\) is \(\mathbb{Q}\)-factorial dlt,

2. \(a(F,X,B,{\boldsymbol{M}})\leq 0\) for any prime \(f\)-exceptional divisor \(F\).

We let \[K_Y+\bar B_Y+{\boldsymbol{M}}_Y:=f^*(K_X+B+{\boldsymbol{M}}_X)\] and let \(S_Y\) be the strict transform of \(S\) on \(Y\). Let \((S_Y,B_{S_Y},{\boldsymbol{M}}^S)/U\) and \((S_Y,\bar B_{S_Y},{\boldsymbol{M}}^S)/U\) be the g-pairs induced by adjunction \[K_{S_Y}+B_{S_Y}+{\boldsymbol{M}}^S_{S_Y}=(K_Y+B_Y+{\boldsymbol{M}}_Y)|_{S_Y}\] and \[K_{S_Y}+\bar B_{S_Y}+{\boldsymbol{M}}^S_{S_Y}=(K_Y+\bar B_Y+{\boldsymbol{M}}_Y)|_{S_Y}\] respectively. We let \(Q:=\bar B_Y-B_Y\).

Let \(A\) be an ample divisor on \(Y\) such that \(K_Y+B_Y+{\boldsymbol{M}}_Y+A\) is nef. We may run a \((K_Y+B_Y+{\boldsymbol{M}}_Y)\)-MMP\(/X\) with scaling of \(A\) \[(Y,B_Y,{\boldsymbol{M}}):=(X_0,B_0,{\boldsymbol{M}})\dashrightarrow (X_1,B_1,{\boldsymbol{M}})\dashrightarrow\dots\dashrightarrow (X_n,B_n,{\boldsymbol{M}})\dashrightarrow\dots.\] Let \(S_i,A_i,Q_i,\bar B_i\) be the images of \(S_Y,A,Q,\bar B_Y\) on \(X_i\) for each \(i\), \(f_i: X_i\rightarrow X\) the induced birational morphism, and \[\lambda_i:=\inf\{t\geq 0\mid K_{X_i}+B_i+tA_i+{\boldsymbol{M}}_{X_i}\text{ is nef}/X\}\] the scaling numbers. Then \(K_{X_i}+\bar B_i+{\boldsymbol{M}}_{X_i}=f_i^*(K_X+B+{\boldsymbol{M}}_X)\) and \(\bar B_i=B_i+Q_i\) for any \(i\). Let \[K_{S_i}+B_{S_i}+{\boldsymbol{M}}^S_{S_i}:=(K_{X_i}+B_i+{\boldsymbol{M}}_{X_i})|_{S_i}\] and \[K_{S_i}+\bar B_{S_i}+{\boldsymbol{M}}^S_{S_i}:=(K_{X_i}+\bar B_i+{\boldsymbol{M}}_{X_i})|_{S_i}\] for any \(i\). Then \(\bar B_{S_i}=B_{S_i}+Q_i|_{S_i}\). Moreover, there exists a birational morphism \(g_i: S_i\rightarrow S\) such that \[K_{S_i}+\bar B_{S_i}+{\boldsymbol{M}}^S_{S_i}=g_i^*(K_S+B_S+{\boldsymbol{M}}^S_S).\] Thus \((S_i,\bar B_{S_i},{\boldsymbol{M}}^S)\) is lc. Since \((Y,B_Y,{\boldsymbol{M}})\) is dlt, \((X_i,B_i,{\boldsymbol{M}})\) is dlt. By Lemma 71, \((S_i,B_{S_i},{\boldsymbol{M}}^S)\) is dlt. By Lemma 71, any lc center of \((X_i,B_i,{\boldsymbol{M}})\) is an lc center of \((S_i,B_{S_i},{\boldsymbol{M}}^S)\). Since all components of \(Q_i\) are lc centers of \((X_i,B_i,{\boldsymbol{M}})\) and \((S_i,\bar B_{S_i},{\boldsymbol{M}}^S)\) is lc, \(\operatorname{Supp}Q_i\) does not intersect \(S_i\) for any \(i\).

We pick a non-negative integer \(m\) in the following way. If the \((K_Y+B_Y+{\boldsymbol{M}}_Y)\)-MMP\(/X\) terminates, then we let \(m\) be the index so that \((X_m,B_m,{\boldsymbol{M}})/X\) is a log minimal model of \((Y,B_Y,{\boldsymbol{M}})/X\) for some non-negative integer \(m\). If the \((K_Y+B_Y+{\boldsymbol{M}}_Y)\)-MMP\(/X\) does not terminate, then by Lemma 66, \(\lim_{i\rightarrow+\infty}\lambda_i=0\), so by special termination (cf. [15]), we may pick a positive integer \(m\), such that \(S_i\dashrightarrow S_{i+1}\) is an isomorphism in codimension \(1\) for any \(i\geq m\). We let \(I\geq 2\) be any sufficiently divisible positive integer satisfying the following

• \(IQ\) is a Weil divisor.

• \[(f_m)_*\mathcal{O}_{X_m}(A_m-IQ_m)\subset (f_m)_*\mathcal{O}_{X_m}(A_m)\] are contained in \[\mathcal{I}_{f_m(\operatorname{Supp}Q)}\cdot (f_m)_*\mathcal{O}_{X_m}(A_m).\]

• If \((X_m,B_m,{\boldsymbol{M}})/X\) is a log minimal model of \((Y,B_Y,{\boldsymbol{M}})/X\) and \(m\geq 2\), then \(\lambda_{m-1}>\frac{1}{I}\).

• If the \((K_Y+B_Y+{\boldsymbol{M}}_Y)\)-MMP\(/X\) does not terminate, then \(\lambda_m>\frac{1}{I}\).

Since \(S_i\dashrightarrow S_{i+1}\) is an isomorphism in codimension \(1\) for any \(i\geq m\), for any \(j\geq m\), we have \[(f_j)_*\mathcal{O}_{X_j}(A_j-IQ_j)=(f_m)_*\mathcal{O}_{X_m}(A_m-IQ_m).\] Since \(\operatorname{Supp}Q_i\) does not intersect \(S_i\) for any \(i\), we have an induced homomorphism \[(f_i)_*\mathcal{O}_{X_i}(A_i-IQ_i)\rightarrow(f_m|_{S_m})_*\mathcal{O}_{S_m}(A_i)=(f_i|_{S_i})_*\mathcal{O}_{S_i}(A_i)\] which is not surjective. Therefore, \[R^1(f_i)_*\mathcal{O}_{X_i}(A_i-IQ_i-S_i)\not=0\] for any \(i\geq m\).

We let \(l:=m\) if \((X_m,B_m,{\boldsymbol{M}})/X\) is a log minimal model of \((Y,B_Y,{\boldsymbol{M}})/X\), and let \(l\) be the unique positive integer such that \(\lambda_{l-1}>\frac{1}{I}\geq\lambda_{l}\) if the \((K_Y+B_Y+{\boldsymbol{M}}_Y)\)-MMP\(/X\) does not terminate. Then \(l\geq m\), \[X_0\dashrightarrow X_1\dashrightarrow X_{l}\] is also a sequence of steps of a \((K_{X_0}+B_0+{\boldsymbol{M}}_{X_0}+\frac{1}{I}A)\)-MMP\(/X\) with scaling of \(A\), and \[K_{X_{l}}+B_{l}+{\boldsymbol{M}}_{X_{l}}+\frac{1}{I}A_{l}\] is nef\(/X\). Since \(X_0\) is \(\mathbb{Q}\)-factorial klt, we may pick \[0\leq \Delta_0\sim_{\mathbb{Q}}B_0-S_0+{\boldsymbol{M}}_{X_0}+\frac{1}{I}A\] such that \((X_0,\Delta_0)\) is klt and \((X_0,S_0+\Delta_0)\) is plt. We let \(\Delta_l\) be the image of \(\Delta_0\) on \(X_l\), then \((X_l,S_l+\Delta_l)\) is plt, so \((X_l,\Delta_l)\) is klt. Then

\[A_l-IQ_l-S_l\sim_{\mathbb{Q},X}K_{X_l}+\Delta_l+(I-1)\left(K_{X_l}+B_l+{\boldsymbol{M}}_{X_l}+\frac{1}{I}A_l\right),\] so by the relative Kawamata-Viehweg vanishing [73], \[R^1(f_{l})_*\mathcal{O}_{X_{l}}(A_{l}-IQ_{l}-S_{l})\not=0,\] a contradiction. We are done with the case when \((X,B,{\boldsymbol{M}})\) is a \(\mathbb{Q}\)-g-pair.

Now we prove the case when \((X,B,{\boldsymbol{M}})\) is not necessarily a \(\mathbb{Q}\)-g-pair, hence conclude the proof of the theorem. There exist real numbers \(r_1,\dots,r_c\) such that \(1,r_1,\dots,r_c\) are linearly independent over \(\mathbb{Q}\), \(\boldsymbol{r}:=(r_1,\dots,r_c)\in\mathbb{R}^c\), and \(\mathbb{Q}\)-linear functions \(s_1,\dots,s_p,t_1,\dots,t_q\), such that \[B=\sum_{i=1}^ps_i(1,\boldsymbol{r})B_i,{\boldsymbol{M}}=\sum_{i=1}^qt_i(1,\boldsymbol{r}){\boldsymbol{M}}_i,\] where \(B_i\geq 0\) are distinct Weil divisors and \({\boldsymbol{M}}_i\) are nef\(/X\) \(\boldsymbol{b}\)-Cartier \(\boldsymbol{b}\)-divisors. Let \[B(\boldsymbol{v}):=\sum_{i=1}^ps_i(1,\boldsymbol{v})B_i\text{ and }{\boldsymbol{M}}(\boldsymbol{v}):=\sum_{i=1}^qt_i(1,\boldsymbol{v}){\boldsymbol{M}}_i,\] for any \(\boldsymbol{v}\in\mathbb{R}^c\).

Since the coefficients of divisors under adjunction are transformed via \(\mathbb{Q}\)-linear functions, there are \(\mathbb{Q}\)-linear functions \(s'_1,\dots,s'_{p'},t'_1,\dots,t'_{q'}\), distinct Weil divisors \(B_{S_i}\geq 0\), and nef\(/X\) \(\boldsymbol{b}\)-Cartier \(\boldsymbol{b}\)-divisors \({\boldsymbol{M}}^S_i\), \[B_S(\boldsymbol{v}):=\sum_{i=1}^ps_i(1,\boldsymbol{v})B_{S,i},\text{ and }{\boldsymbol{M}}^S(\boldsymbol{v}):=\sum_{i=1}^qt_i(1,\boldsymbol{v}){\boldsymbol{M}}^S_i,\] such that \[K_{S^\nu}+B_S(\boldsymbol{v})+{\boldsymbol{M}}^S(\boldsymbol{v})_{S^\nu}=(K_X+B(\boldsymbol{v})+{\boldsymbol{M}}(\boldsymbol{v})_X)|_{S^\nu}\] for any \(\boldsymbol{v}\in\mathbb{R}^c\). Since \[(S^\nu,B_S=B_S(\boldsymbol{r}),{\boldsymbol{M}}^S={\boldsymbol{M}}^S(\boldsymbol{r}))\] is lc, there exists an open neighborhood \(U\ni\boldsymbol{r}\) of \(\mathbb{R}^c\) such that \[(S^\nu,B_S(\boldsymbol{v}),{\boldsymbol{M}}^S(\boldsymbol{v}))\] is lc for any \(\boldsymbol{v}\in U\). By the \(\mathbb{Q}\)-g-pair case, \[(X,B(\boldsymbol{v}),{\boldsymbol{M}}(\boldsymbol{v}))\] is lc for any \(\boldsymbol{v}\in U\cap\mathbb{Q}\). Thus \[(X,B=B(\boldsymbol{r}),{\boldsymbol{M}}={\boldsymbol{M}}(\boldsymbol{r}))\] is lc by continuity of log discrepancies. ◻

Proof. When \(\dim X=1\) the proposition is trivial, so we may assume that \(\dim X\geq 2\). We may also assume that \(n\geq 1\), otherwise there is nothing left to prove.

Let \(B_{n+1},\dots,B_{n+\dim\bar X}\) be general hyperplane sections on \(X\) and let \(b_i:=1\) when \(i\geq n+1\). Possibly replacing \(x\) with \(\bar x\cap\cap_{i=1}^{\dim X}H_i\) and \(B\) with \(\sum_{i=1}^{n+\dim\bar x}b_iB_i+D\), we may assume that \(x\) is a closed point.

First we prove the proposition under conditions (1) or (2). Possibly adding general hyperplane sections which pass through \(x\), we may assume that \(x\) is an lc center of \((X,B,{\boldsymbol{M}})\). Let \(E\) be an lc place of \((X,B,{\boldsymbol{M}})\) such that \(\operatorname{center}_XE=x\).

Proof of Proposition 86 continued. By Claim 87, there exists a contraction \(f: Y\rightarrow X\) of \(E\) such that \(-E\) is ample\(/X\). We let \(B_{i,Y},D_Y,B_Y\) be the strict transforms of \(B_{i},D,B\) on \(Y\) respectively. Since \(x\in\operatorname{Supp}B_i\) for each \(i\), \(\operatorname{mult}_EB_i>0\) for each \(i\), so \(B_{i,Y}\) is ample\(/X\) for each \(i\). We let \(E^\nu\) be the normalization of \(E\), \({\boldsymbol{M}}^E:={\boldsymbol{M}}|_{E^\nu}\), and let \[K_{E^\nu}+B_E+{\boldsymbol{M}}^E_{E^\nu}=(K_Y+B_Y+E+{\boldsymbol{M}}_Y)|_{E^\nu}.\] We let \(B_{i,E}:=\operatorname{Supp}(B_{i,Y}|_{E^\nu})\) for each \(i\). Then for any component \(D_{i,j}\) of \(B_{i,E}\), we have \[\operatorname{mult}_{D_{i,j}}B_E=\frac{n_{i,j}-1+\sum_{k=1}^nb_km_{k,i,j}+\gamma_{i,j}}{n_{i,j}}\] for some real number \(\gamma_{i,j}\geq 0\) and non-negative integers \(m_{k,i,j}\), such that \(m_{i,i,j}\not=0\). Since \((E,B_E,{\boldsymbol{M}}^E)\) is lc, \(\operatorname{mult}_{D_{i,j}}B_E\leq 1\), so \[B_E\geq\sum_{i=1}^nb_iB_{i,E}.\] Since \(B_{i,Y}\) is ample\(/X\), \(B_{i,Y}|_{E^\nu}\) is ample, so \(B_{i,E}\) is big. The proposition under conditions (1) or (2) follows from [1].

Now we prove the proposition under condition (3). Let \(S\) be the normalization of an irreducible component of \(B_1\) such that \(x\in S_1\), and let \((S,B_S,{\boldsymbol{M}}^S)\) be the g-pair induced by the adjunction \[K_S+B_S+{\boldsymbol{M}}^S_S:=(K_X+B+D+{\boldsymbol{M}}_X)|_S.\] Since \(x\in\operatorname{Supp}B_i\) for each \(i\), \(B_i|_S\not=0\) for any \(i\geq 2\). Since \(B_i\) is Cartier and \((S,B_S,{\boldsymbol{M}}^S)\) is lc, \(B_i|_S=\operatorname{Supp}(B_i|_S)\) for any \(i\geq 2\), and \[B_S\geq\sum_{i=2}^nB_i|_S.\] Since each \(B_i|_S\) is Cartier, by induction on \(\dim X\), we have \(n\leq\dim X\) and the proposition follows. ◻

Proof. Possibly replacing \(X\) and \(Z\) with high models, we may assume that \({\boldsymbol{M}}\) descends to \(X\), each \(D_i\) is a divisor on \(X\), and each \(D_{Z,i}\) is a divisor on \(Z\). Now the theorem follows from [27], which in turn follows from [74]. We also refer the reader to [75] for a more detailed explanation. ◻

Proof. Let \(f'': X''\rightarrow Z''\) be any contraction that is birationally equivalent to \(f\) such that \(Z''\) is \(\mathbb{Q}\)-factorial and the induced birational map \(h': X''\dashrightarrow X'\) is a morphism. Let \[K_{X''}+B''+{\boldsymbol{M}}_{X''}:=h'^*(K_{X'}+B'+{\boldsymbol{M}}_{X'}),\] then \(K_{X''}+B''+{\boldsymbol{M}}_{X''}=(h\circ h')^*(K_X+B+{\boldsymbol{M}}_X)\). Thus \({\boldsymbol{B}}_{Z''}={\boldsymbol{B}}'_{Z''}\), so \({\boldsymbol{B}}={\boldsymbol{B}}'\). The in particular part follows from the definition of BP stability. ◻

Proof. For any \(f': X'\rightarrow Z'\) that is birationally equivalent to \(f\), such that the induced birational maps \(h: X'\dashrightarrow X\) and \(h_Z: Z'\dashrightarrow Z\) are morphisms, we have \(K_{Z'}+B_{Z'}=h_Z^*(K_{Z}+B_{Z})\). Thus \[\begin{align} {\boldsymbol{N}}_{X'}&=K_{X'}+B'+{\boldsymbol{M}}_{X'}-f'^*(K_{Z'}+B_{Z'})=h^*(K_X+B+{\boldsymbol{M}}_X-f^*(K_Z+B_Z)), \end{align}\] where \(K_{X'}+B'+{\boldsymbol{M}}_{X'}:=h^*(K_X+B+{\boldsymbol{M}}_X)\), and \(B_{Z'}\) is the discriminant part of \(f': (X',B',{\boldsymbol{M}})\rightarrow Z'\). The lemma immediately follows. ◻

Proof. It immediately follows from Lemma 97. ◻

Proof. Let \(h: W\rightarrow X\) be a log resolution of \((X, \operatorname{Supp}B)\) such that \(F\) is a divisor on \(W\) and \({\boldsymbol{M}}\) descends to \(W\), and let \[K_W+B_W+{\boldsymbol{M}}_W:=h^*(K_X+B+{\boldsymbol{M}}_X).\] Let \(C_W:=B_W^{\geq 1}-B_W^{\geq 1}\wedge F\) and let \(a:=a(F,X,B,{\boldsymbol{M}})\). Let \(A_1, A_2\geq 0\) be sufficiently general ample \(\mathbb{R}\)-divisors on \(Y\) such that \[aF-C_W+A_1-A_2\sim_{\mathbb{R}}0.\] Then \((W,\Delta_W:=B_W+aF-C_W+A_1-A_2,{\boldsymbol{M}})\) is lc and \(F\) is the unique lc place of \((W,\Delta_W,{\boldsymbol{M}})\). We let \(\Delta:=h_*\Delta_W\), then \[K_W+\Delta_W+{\boldsymbol{M}}_W=h^*(K_X+\Delta+{\boldsymbol{M}}_X)\] and \((X,\Delta,{\boldsymbol{M}})\) has the required properties. ◻

Proof. By Lemma 96, possibly replacing \((X,B,{\boldsymbol{M}})\) by a crepant model, we may assume that \({\boldsymbol{M}}\) descends to \(X\). Then the theorem follows from the definitions and [27]. ◻

Proof. For any irreducible component \(G\) of \(f^{-1}(z)\), let \(H_1,\dots,H_{\dim G}\) be general very ample divisors on \(X\), \(V:=\cap_{i=1}^{\dim G}H_i\), and \((V,B_V,{\boldsymbol{M}}^V)/U\) the g-pair induced by the adjunction \[K_V+B_V+{\boldsymbol{M}}^V_V:=(K_X+B+{\boldsymbol{M}}_X)|_V.\] Then \((V,B_V+\sum_{i=1}^mf^*D_i|_V,{\boldsymbol{M}}^V)\) is lc, \(G\cap V\) is a closed point, and \(A_i:=f^*D_i|_V\) is Cartier and contains \(G\cap V\) for any \(i\). By Proposition 86, \(m\leq\dim V=d-\dim G\). Thus \(\dim G\leq d-m\). Therefore, the dimension of any irreducible component of \(f^{-1}(z)\) is \(\leq d-m\). By [76], the dimension of any irreducible component of \(f^{-1}(z)\) is \(\geq d-m\). The lemma immediately follows. ◻

Proof. (1) follows from the definition immediately.

(2) Let \(B_Z\) be the discriminant part of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\). By definition and (1), \(\operatorname{Supp}B_Z\ge B_Z\geq\Sigma_Z\). It suffices to show that \(B_Z\le\Sigma_Z\). Let \(P\) be a prime divisor on \(Z\) such that \(P\not\subset\operatorname{Supp}\Sigma_Z\), and let \(z\) be a general closed point in \(P\). Then \((Z,\Sigma_Z+P)\) is log smooth at \(z\). Then \((X,B+f^*P,{\boldsymbol{M}})\) is sub-lc over a neighborhood of \(z\) which implies that \[\sup\{t\mid (X,B+tf^*P,{\boldsymbol{M}})\text{ is sub-lc over the generic point of }P\}=1.\] Thus \(P\not\subset\operatorname{Supp}B_Z\) and hence \(\Sigma_Z=\operatorname{Supp}\Sigma_Z\geq\operatorname{Supp}B_Z\ge B_Z\). This implies (2).

(3) Let \(d:=\dim X\) and \(m:=\dim Z\). Let \(z\in Z\backslash\operatorname{Supp}\Sigma_Z\) be a closed point, and let \(\Sigma_1,\dots,\Sigma_m\) be general hyperplane sections on \(Z\) such that \(z\in \Sigma_i\) for any \(i\). Then \((Z,\Sigma_Z+\sum_{i=1}^m\Sigma_i)\) is log smooth at \(z\). Then \((X,B+\sum_{i=1}^m f^*\Sigma_i,{\boldsymbol{M}})\) is lc over a neighborhood of \(z\). By Lemma 102, the dimension of any irreducible component of \(f^{-1}(z)\) is \(d-m\). This implies (3). ◻

Proof. Let \(k:=\dim Z-\dim V\). Since \((Z,\Sigma)\) is log smooth, there exist irreducible components \(\Sigma_1,\dots, \Sigma_k\) of \(\Sigma\) such that \(V=\bigcap_{i=1}^k \Sigma_i\). By Definition 103(3), for any \(i\) and any general closed point \(z\in\operatorname{Supp}\Sigma_i\), \((X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\) is sub-lc over a neighborhood of \(z\). Thus any irreducible component of \(f^{-1}(\Sigma_i)\) is an lc center of \((X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\). Therefore, any irreducible component of \(f^{-1}(V)\) is an intersection of lc centers of \((X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\). The lemma follows from Lemma 79. ◻

Proof. Since \({\boldsymbol{M}}\) descends to \(X\), by [27], \(f: (X,B)\rightarrow Z\) satisfies Property \((*)\). By Definition 103, \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\) satisfies Property \((*)\). ◻

Proof. Possibly adding components to \(\Sigma_{Z'}\), we may assume that \(\Sigma_{Z'}\) coincides with the image of the vertical\(/Z'\) part of \(\Sigma_{X'}\). We let \(G:=f'^{-1}(\Sigma_{Z'})\) and \[K_{X'}+\tilde{B}'+{\boldsymbol{M}}_{X'}:=h^*(K_X+B+{\boldsymbol{M}}_X),\] then \(G\subset\operatorname{Supp}\Sigma_{X'}\) and \(\operatorname{Supp}\tilde{B}'\subset\operatorname{Supp}\Sigma_{X'}\). We may define an \(\mathbb{R}\)-divisor \(B'\) on \(X'\) as follows: For any prime divisor \(D\) on \(X'\),

• if \(D\) is not a component of \(\operatorname{Supp}\tilde{B}'\) nor \(G\), then \(\operatorname{mult}_DB'=0\),

• if \(D\) is a component of \(G\), then \(\operatorname{mult}_DB'=1\),

• if \(D\) is a component of \(\tilde{B}'\) but is not a component of \(G\), then \(\operatorname{mult}_DB'=\operatorname{mult}_D\tilde{B}'\).

Set \(F':=B'-\tilde{B}'\). Since \(G\) is the vertical\(/Z'\) part of \(\Sigma_{X'}\), the vertical\(/Z'\) part of \(B'\) is equal to \(G=f'^{-1}(\Sigma_{Z'})\). By construction \(\operatorname{Supp}F'\subset G\) is vertical\(/Z'\) and \(\operatorname{Supp}B'\cup\operatorname{Supp}F'\subset\Sigma_{X'}\).

Assume that \((X,B,{\boldsymbol{M}})\) is sub-lc. As \(\operatorname{mult}_DF'=1-\operatorname{mult}_D\tilde{B}'\) if \(D\) is a component of \(G\), one can see that \(\operatorname{mult}_DF'\geq 0\) and hence \(F'\ge0\). By Proposition 106, \(f': (X',B',{\boldsymbol{M}})\rightarrow Z'\) satisfies Property \((*)\). ◻

Proof. Without loss of generality, we may assume that \(\phi\) is a step of a \((K_X+B+{\boldsymbol{M}}_X)\)-MMP\(/Z\).

(1) Let \(\Sigma_Z\) be the discriminant part of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\). Then by assumption \((Z,\Sigma_Z)\) is log smooth. Since the vertical\(/Z\) part of \(B\) is equal to \(f^{-1}(\Sigma_Z)\) and \(\phi\) does not extract any divisor, the vertical\(/Z\) part of \(B_Y\) is equal to \(\phi\circ f^{-1}(\Sigma_Z)=f_Y^{-1}(\Sigma_Z)\). For any reduced divisor \(\Sigma\geq\Sigma_Z\) on \(Z\), \((X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})/U\) is lc. Since \(\phi\) is a step of a \((K_X+B+{\boldsymbol{M}}_X)\)-MMP\(/Z\), \(\phi\) is also a step of a \((K_X+B+f^*(\Sigma-\Sigma_Z)+{\boldsymbol{M}}_X)\)-MMP\(/Z\). Thus \[(Y,B_Y+\phi_*f^*(\Sigma-\Sigma_Z)=B_Y+f_Y^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\] is lc. Therefore, \(f_Y: (Y,B_Y,{\boldsymbol{M}})\rightarrow Z\) satisfies Property \((*)\). By Lemma 104(2), \(\Sigma_Z\) is the discriminant part of \(f_Y: (Y,B_Y,{\boldsymbol{M}})\rightarrow Z\).

(2) Possibly shrinking \(Z\) near \(z\), there exists a reduced divisor \(\Sigma\geq\Sigma_Z\) on \(Z\), such that \((Z,\Sigma)\) is log smooth and \(z\) is a stratum of \(\Sigma\). By Lemma 105, any irreducible component of \(f_Y^{-1}(z)\) is an lc center of \((Y,B_Y+f_Y^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\). By Definition 103(3), \((X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\) is lc. For any irreducible component \(G\) of \(f^{-1}(z)\), let \(D_G\) be an lc place of \((Y,B_Y+f_Y^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\) over the generic point of \(G\). Then \[0\leq a(D_G,X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\leq a(D_G,Y,B_Y+f_Y^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})=0.\] Thus \[a(D_G,X,B+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})=a(D_G,Y,B_Y+f_Y^*(\Sigma-\Sigma_Z))=0,\] so \(\phi^{-1}\) is an isomorphism near the generic point of \(G\).

(3) It immediately follows from (2). ◻

Proof. There are local coordinates systems \(\{x_1,\dots,x_d\}\) and \(\{x_1',\dots,x_d'\}\) near \(o\), such that \(D_i=(x_i=0)\) and \(D_i'=(x_i'=0)\) for each \(i\).

Suppose that \((X,\sum_{i=1}^{d-1}D_i+D_j')\) is not log smooth near \(o\) for any \(j\), then locally analytically near \(o\), \((x_j'=0)\) is contained in the subspace spanned by \(x_1,\dots,x_{d-1}\) for any \(j\). Since locally analytically near \(o\), \(X\) is spanned by \(x_1',\dots,x_d'\), \(X\) is contained in its subspace spanned by \(x_1,\dots,x_{d-1}\), which is absurd. The lemma follows. ◻

Proof. Let \(\Sigma_Z\) be the discriminant part of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\). Then \(\Sigma_Z\) is a reduced divisor, \((Z,\Sigma_Z)\) is log smooth, and the vertical\(/Z\) part of \(B\) is equal to \(f^{-1}(\Sigma_Z)\). Since \(A\) is general ample\(/U\), all components of \(A\) are horizontal\(/U\). Therefore, the vertical part of \(B+A\) is equal to \(f^{-1}(\Sigma_Z)\). By Lemma 104, (2) will follow from (1) and we only need to prove (1).

Let \(\Sigma_0\geq\Sigma_Z\) be a reduced divisor on \(Z\) such that \((Z,\Sigma_0)\) is log smooth near \(z\) and \(z\) is a stratum of \(\Sigma_0\). By Definition 103, \((X,B+f^*(\Sigma_0-\Sigma_Z),{\boldsymbol{M}})\) is lc over a neighborhood of \(z\). Therefore, we may choose \(A\) so that \((X,B+A+f^*(\Sigma_0-\Sigma_Z),{\boldsymbol{M}})\) is lc over a neighborhood of \(z\). By Definition 103 again, we only need to show that, for any reduced divisor \(\Sigma\geq\Sigma_Z\) such that \((Z,\Sigma)\) is log smooth near \(z\), \((X,B+A+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\) is lc over a neighborhood of \(z\).

Possibly adding components to \(\Sigma\), we may assume that \(z\) is an lc center of \((Z,\Sigma)\). We let \(q:=\dim Z\), then \(\Sigma\) and \(\Sigma_0\) both have \(q\) irreducible components near \(z\). Let \(D^0_1,\dots,D^0_q\) be the components of \(\Sigma_0\) and let \(D_1,\dots,D_q\) be the components of \(\Sigma\). Suppose that \(\Sigma\) and \(\Sigma_0\) have \(k\) different components for any integer \(0\leq k\leq q\). Possibly reordering indices, we may assume that \(D^0_i\not=D_i\) for any \(1\leq i\leq k\), and \(D^0_i=D_i\) for any \(k+1\leq i\leq q\). If \(k=0\) then there is nothing left to prove, so we may assume that \(k>0\). By repeatedly applying Lemma 109, possibly reordering indices, we may assume that \[\left(Z,\Sigma_j:=\sum_{i=j+1}^qD_i^0+\sum_{i=1}^jD_i\right)\] is log smooth for any \(0\leq j\leq k\), and \(\Sigma_k=\Sigma\). Moreover, \(\Sigma_j\geq\Sigma_Z\) for each \(j\), and \(\Sigma_j\) and \(\Sigma_{j+1}\) has exactly \(1\) different component. By inductively showing that \((X,B+A+f^*(\Sigma_j-\Sigma_Z),{\boldsymbol{M}})\) is lc for each \(j\), we may assume that \(k=1\).

Let \(V:=\cap_{i=2}^qD_i^0\). Then we have \(D_1|_V=D_1^0|_V=z\). Therefore, for any irreducible component \(W\) of \(f^{-1}(V)=\cap_{i=2}^qf^{-1}(D_i^0)\) with normalization \(W^\nu\), \(f^*D_1|_{W^\nu}=f^*D_i^0|_{W^\nu}\). Let \[R:=B+A+f^*\left(\sum_{i=2}^qD_i^0-\Sigma_Z\right).\] By Lemma 83, \[\begin{align} 1=&\sup\left\{t\geq 0|\left(X,R+tf^*D_1^0,{\boldsymbol{M}}\right)\text{ is lc over a neighborhood of }z\right\}\\ =&\sup\left\{t\geq 0|\left(X,R+tf^*D_1^0,{\boldsymbol{M}}\right)\text{ is lc near }f^{-1}(V)\right\}\\ =&\sup\left\{t\geq 0|\left(X,R+tf^*D_1,{\boldsymbol{M}}\right)\text{ is lc near }f^{-1}(V)\right\}\\ =&\sup\left\{t\geq 0|\left(X,R+tf^*D_1,{\boldsymbol{M}}\right)\text{ is lc over a neighborhood of }z\right\}. \end{align}\] Thus \((X,B+A+f^*(\Sigma-\Sigma_Z),{\boldsymbol{M}})\) is lc over a neighborhood of \(z\), and the proposition follows. ◻

Proof. Let \(\phi_Z: Y\rightarrow Z\) and \(\phi_{Z'}: Y\rightarrow Z'\) be a resolution of indeterminacy of the induced birational map \(Z\dashrightarrow Z'\). Let \(\phi: W\rightarrow X\) and \(\phi': W\rightarrow X'\) be any high enough resolution of indeterminacy of the induced birational map \(X\dashrightarrow X'\) such that \(W\dashrightarrow Y\) is a morphism. Let \[K_W+B_W'+{\boldsymbol{M}}_W:=\phi'^*(K_{X'}+B'+{\boldsymbol{M}}_{X'}).\] Since \((X',B',{\boldsymbol{M}})\) is BP stable\(/Z'\), \((W,B_W',{\boldsymbol{M}})\) is BP stable\(/Y\) and \({\boldsymbol{N}}'\) descends to \(X'\) by Proposition 97. In particular, \(K_W+B_W'+{\boldsymbol{M}}_W\) is nef\(/Z'\). We only need to show that \(\phi^*{\boldsymbol{N}}_X-{\boldsymbol{N}}'_W\geq 0\). Possibly replacing \((X',B',{\boldsymbol{M}})\) and \(Z'\) by \((W,B_W',{\boldsymbol{M}})\) and \(Y\) respectively, we may further assume that both \(h:X'\dashrightarrow X\) and \(h_Z:Z'\dashrightarrow Z\) are morphisms.

Let \(B_Z\) (resp. \(B_{Z'}\)) be the discriminant part of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\) (resp. \(f': (X',B',{\boldsymbol{M}})\rightarrow Z'\)). By Lemma 104(2), \(B_Z\) is a reduced divisor, \((Z,B_Z)\) is log smooth, and the vertical\(/Z\) part of \(B\) is \(f^{-1}(B_Z)\). We may write \[K_{X'}+\tilde{B}'+{\boldsymbol{M}}_{X'}=h^*(K_X+B+ {\boldsymbol{M}}_X)\text{ and }K_{Z'}+\tilde{B}_{Z'}=h_Z^*(K_Z+B_Z)\] for some \(\mathbb{R}\)-divisors \(\tilde{B}'\) and \(\tilde{B}_{Z'}\). Then \[\begin{align} h^*{\boldsymbol{N}}_X-{\boldsymbol{N}}'_{X'}=\tilde{B}'-B'-(f')^*\left(\tilde{B}_{Z'}-B_{Z'}\right) \end{align}\] which is vertical over \(Z\). We claim that \(h_*(h^*{\boldsymbol{N}}_X-{\boldsymbol{N}}'_{X'})\ge0\).

Pick any component \(D\) of \(\operatorname{Supp}h_*(h^*{\boldsymbol{N}}_X-{\boldsymbol{N}}'_{X'})\). Let \(D'\) be the strict transform of \(D\) on \(X'\) and \(D_Z\) the image of \(D\) on \(Z\). There are two possibilities.

Case 1. \(D_Z\) is contained in \(B_Z\).

In this case, since \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\) satisfies Property \((*)\), \(\operatorname{mult}_{D'}\tilde{B}'=\operatorname{mult}_DB=1\) and \((X',\tilde{B}',{\boldsymbol{M}})\) is log stable\(/Z'\). Since \((Z',\tilde{B}_{Z'})\) is sub-lc and \(B_{Z'}\) is the discriminant part of \(f': (X',B',{\boldsymbol{M}})\rightarrow Z'\), \((X',B'+(f')^*(\tilde{B}_{Z'}-B_{Z'}),{\boldsymbol{M}})\) is sub-lc. It follows that \[\operatorname{mult}_{D'}\left(B'+(f')^*\left(\tilde{B}_{Z'}-B_{Z'}\right)\right)\leq 1=\operatorname{mult}_{D'}\tilde{B}',\] so \(\operatorname{mult}_Dh_*(h^*{\boldsymbol{N}}_X-{\boldsymbol{N}}'_{X'})\geq 0.\)

Case 2. \(D_Z\) is not contained in \(B_Z\).

In this case, by assumption and Lemma 104(3), \(f\) is equi-dimensional over the generic point of \(D_Z\). Thus \(D_Z\) is a prime divisor. Since \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\) satisfies Property \((*)\), it holds that \(1=\operatorname{mult}_D(B+f^*D_Z)\). Hence if we denote by \(D_{Z'}\) the strict transform of \(D_Z\) on \(Z'\), then \[1=\operatorname{mult}_D(B+f^*D_Z)=\operatorname{mult}_D\left(h_*\left(\tilde{B}'+(f')^*D_{Z'}\right)\right)=\operatorname{mult}_{D'}\left(\tilde{B}'+(f')^*D_{Z'}\right).\] Let \[b_{D_{Z'}}(X',B',{\boldsymbol{M}};f')=\sup\left\{t\mid \left(X',B'+t(f')^*D_{Z'},{\boldsymbol{M}}\right)\text{ is sub-lc over the generic point of }D_{Z'}\right\}.\] Then \(\operatorname{mult}_{D_{Z'}}B_{Z'}=1-b_{D_{Z'}}(X',B',{\boldsymbol{M}};f')\) and hence \[D_{Z'}-B_{Z'}=b_{D_{Z'}}(X',B',{\boldsymbol{M}};f')D_{Z'}\] near the generic point of \(D_{Z'}\). Moreover, as \(\operatorname{mult}_{D_{Z'}}\tilde{B}_{Z'}=\operatorname{mult}_{D_Z}B_Z=0\), we see that \[\begin{align} \operatorname{mult}_{D'}\left(h^*{\boldsymbol{N}}_X-{\boldsymbol{N}}'_{X'}\right) =&\operatorname{mult}_{D'}\left(\tilde{B}'-B'-(f')^*\tilde{B}_{Z'}+(f')^*B_{Z'}\right)\\ =&\operatorname{mult}_{D'}\left(\tilde{B}'+(f')^*D_{Z'}-B'-(f')^*(D_{Z'}-B_{Z'})\right)\\ =&1-\operatorname{mult}_{D'}\left(B'+(f')^*(D_{Z'}-B_{Z'})\right)\\ =&1-\operatorname{mult}_{D'}\left(B'+b_{D_{Z'}}(X',B',{\boldsymbol{M}};f')\cdot (f')^*D_{Z'}\right)\ge0. \end{align}\] The claim holds. Then the proposition follows from the negativity lemma immediately. We finish the proof. ◻

Proof. Possibly replacing \(X\) with a higher model, we may assume that \(f\) is a morphism. By Definition-Lemma 89, there exists an equidimensional model\(/Z\) of \(f: X\rightarrow Z\), \(f': (X',\Sigma_{X'},{\boldsymbol{M}})\rightarrow (Z',\Sigma_{Z'})\), associated with \(h: X'\rightarrow X\) and \(h_Z: Z'\rightarrow Z\). By [78], there exists a foliation \(\mathcal{F}_{Z'}\) on \(Z'\) such that \((f')^{-1}\mathcal{F}_{Z'}=h^{-1}\mathcal{F}\). We may let \(\mathcal{F}_Z:=(h_Z)_*\mathcal{F}_{Z'}\). ◻

Proof. Let \(\mathcal{F}_Z\) be the foliation by points on \(Z\). Then \(\mathcal{F}_Y:=\tau^{-1}\mathcal{F}_Z\) is the foliation by points on \(Y\). Since \[\mathcal{F}=(\tau\circ f)^{-1}\mathcal{F}_Z=f^{-1}\mathcal{F}_Y,\] \(\mathcal{F}\) is induced by \(f\). ◻

Proof. By [27], \((X,\mathcal{F},B^\mathcal{F})\) is lc. Since \({\boldsymbol{M}}\) descends to \(X\), \((X,\mathcal{F},B^\mathcal{F},{\boldsymbol{M}})\) is lc. ◻

Proof. Let \(h:X'\to X\) be a foliated log resolution of \((X,\mathcal{F},B+S,{\boldsymbol{M}})\) with associated \(f':X'\to Z'\). Let \(\mathcal{F}':=h^{-1}\mathcal{F}\) and \[K_{\mathcal{F}'}+{B}'+{\boldsymbol{M}}_{X'}:=h^*(K_\mathcal{F}+B+{\boldsymbol{M}}_X).\] By Lemma 123, \[(X',\mathcal{F}':=h^{-1}\mathcal{F},\tilde{B}':=(B')^{\geq 0},{\boldsymbol{M}})\] is lc. In particular, \((X',\mathcal{F}',\tilde{B}')\) is lc.

We let \(S':=h^{-1}_*S\) which is normal. There exists an induced birational morphism \(h_S: S'\rightarrow S^\nu\) such that \(\nu\circ h_S=h|_{S'}\). Let \(\mathcal{F}_{S'}\) be the restricted foliation of \(\mathcal{F}\) on \(S'\), then \(\mathcal{F}_{S'}=h_S^{-1}\mathcal{F}_S\). Let \[K_{\mathcal{F}_{S'}}+\tilde{B}_{S'}:=\left(K_{X'}+\tilde{B}'\right)|_{S'}\text{ and }K_{\mathcal{F}_{S'}}+B_{S'}+{\boldsymbol{M}}^S_{S'}:=\left(K_{X'}+B'+{\boldsymbol{M}}_{X'}\right)|_{S'}.\] By [27], \((S',\mathcal{F}_{S'},\tilde{B}_{S'})\) is lc. Since \({\boldsymbol{M}}\) descends to \(X'\), \({\boldsymbol{M}}^S\) descends to \(S'\), so \((S',\mathcal{F}_{S'},\tilde{B}_{S'},{\boldsymbol{M}}^S)\) is lc. Since \(\tilde{B}'\geq B'\), \(\tilde{B}_{S'}\geq B_{S'}\) and thus \[\left(S',\mathcal{F}_{S'},\tilde{B}_{S'},{\boldsymbol{M}}^S\right)\] is sub-lc. By construction, \[K_{\mathcal{F}_{S'}}+B_{S'}+{\boldsymbol{M}}^S_{S'}=h_S^*\left(K_{\mathcal{F}_S}+B_S+{\boldsymbol{M}}^S_{S^\nu}\right),\] so \((S^\nu,\mathcal{F}_S,B_S,{\boldsymbol{M}}^S)\) is sub-lc. This completes the proof. ◻

Proof. We may assume that the induced birational map \(f: X'\dashrightarrow X\) is a morphism. We let \(V:=f(\operatorname{Supp}({\boldsymbol{M}}_{X'}-f^{-1}_*{\boldsymbol{M}}_X))\). Then \(\dim X-\dim V\geq 2\). Since \(H\) is general, \(\dim H-\dim (V\cap H)\geq 2\). Therefore, for any prime divisor \(D\) on \(H\) and identifying \(D\) with its image in \(X\), we have that \({\boldsymbol{M}}\) descends to \(X\) near the generic point of \(D\). The lemma follows immediately. ◻

Proof. By Definition-Theorem 89, there exists an equidimensional model\(/U\) \(f': (X',\Sigma_{X'},{\boldsymbol{M}})\rightarrow (Z,\Sigma_Z)\) of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\) associated with \(h: X'\rightarrow X\) and \(h_Z: Z'\rightarrow Z\), such that \(h\) is a foliated log resolution of \((X,\mathcal{F},B,{\boldsymbol{M}})\) and \(\mathcal{F}':=h^{-1}\mathcal{F}\) is induced by \(f'\). We let \(H':=h^*H\), \[K_{\mathcal{F}'}+B'+{\boldsymbol{M}}_{X'}:=h^*(K_{\mathcal{F}}+B+{\boldsymbol{M}}_X),\text{ and }K_{\mathcal{F}_{H'}}+B_{H'}+{\boldsymbol{M}}^H_{H'}:=\left(K_{\mathcal{F}'}+B'+{\boldsymbol{M}}_{X'}\right)|_{H'}.\] Let \[R(f'):=\sum_{D\mid D\text{ is a prime divisor on }Z'}(f'^*D-f'^{-1}(D))\] be the ramification divisor of \(f'\). Since \(H',H_{Z'}\) are general, we have \[R(f')|_{H'}=R(f')|_{f'^*H_{Z'}}=\sum_{D\mid D\text{ is a prime divisor on }Z'}((f'|_{H'})^*D|_{H_{Z'}}-(f'|_{H'})^{-1}(D|_{H_{Z'}})):=R(f'|_{H'}).\] Therefore, \[K_{\mathcal{F}'}|_{H'}=(K_{X'/Z'}-R(f'))|_{H'}=K_{H'/H_{Z'}}-R(f'|_{H'})=K_{\mathcal{F}_{H'}}.\] Since \({\boldsymbol{M}}^H_H={\boldsymbol{M}}_X|_H\), we have \(B_{H'}=B'|_{H'}\).

(1) Since \(W\) is transverse to \(\mathcal{F}\), \(W':=h^{-1}(W)\) is not tangent to \(\mathcal{F}'\). Thus \(\dim g(W')\geq 1\), so \(H_{Z'}:=h_Z^*H_Z\) intersects \(g(W')\) and \(H_Z\) intersects \(h_Z(g(W'))=f(W)\). Hence \(H\) intersects \(W\).

(3) Since \(H\) is general, near the generic point \(\eta_C\) of \(C\), \(h\) is an isomorphism. Since \(B_{H'}=B'|_{H'}\), \(B|_H=B_H\) near \(\eta_C\). We may write \(B=\sum b_iB_i\) where \(B_i\) are the irreducible components of \(B\), then \[B_H=B|_H=\sum b_i(B_i\cap H)\] near \(\eta_C\). Since \(H\) is general, there exists a unique index \(i\) such that \(B_i\cap H\not=0\) at \(\eta_C\). Then \(B_i\cap H=C\), \(B_i=D\), and hence \(\operatorname{mult}_CB_H=b_i=\operatorname{mult}_DB\).

(2) Assume that \((X,\mathcal{F},B,{\boldsymbol{M}})\) is sub-lc. According to Lemma 123, \((X',\mathcal{F}',\tilde{B}':=(B')^{\geq 0},{\boldsymbol{M}})\) is lc. Let \(K_{\mathcal{F}_{H'}}+\tilde{B}_{H'}:=(K_{\mathcal{F}'}+\tilde{B}')|_{H'}\). By [27], \((H',\mathcal{F}_{H'},\tilde{B}_{H'})\) is lc. Since \({\boldsymbol{M}}\) descends to \(X'\), \[K_{\mathcal{F}_{H'}}+\tilde{B}_{H'}+{\boldsymbol{M}}^H_{H'}=(K_{\mathcal{F}'}+B+{\boldsymbol{M}}_X)|_H\] and \({\boldsymbol{M}}^H\) descends to \(H'\). Thus \((H',\mathcal{F}_{H'},\tilde{B}_{H'},{\boldsymbol{M}}^H)\) is lc. Since \(\tilde{B}'\geq B'\), \(\tilde{B}_{H'}\geq B_{H'}\), and hence \((H',\mathcal{F}_{H'},B_{H'},{\boldsymbol{M}}^H)\) is sub-lc. Since \(H\) is general, one can see that \[K_{\mathcal{F}_{H'}}+B_{H'}+{\boldsymbol{M}}^H_{H'}=(h|_{H'})^*\left(K_{\mathcal{F}_H}+B_H+{\boldsymbol{M}}^H_H\right),\] which implies that \((H,\mathcal{F}_H,B_H,{\boldsymbol{M}}^H)\) is sub-lc. If \((X,\mathcal{F},B,{\boldsymbol{M}})\) is lc, then \((H,\mathcal{F}_H,B_H,{\boldsymbol{M}}^H)\) is lc by (3). ◻

Proof. By [48], \((X,\mathcal{F},B+H)\) is lc over a neighborhood of \(z\). Since \({\boldsymbol{M}}\) descends to \(X\), \((X,\mathcal{F},B+H,{\boldsymbol{M}})\) is lc over a neighborhood of \(z\). ◻

Proof. (1) is obvious. By [69], \(K_{\mathcal{F}_H}=(K_{\mathcal{F}}+H)|_H\), so \(B_H+{\boldsymbol{M}}^H_H=B|_H+{\boldsymbol{M}}_X|_H\). We remark that [69] requires that \(\operatorname{rank}\mathcal{F}\geq 2\), but the same lines of the proof work for the case when \(\operatorname{rank}\mathcal{F}=1\) as well. By Lemma 127, \(B_H=B|_H\) and then (3) follows.

(2) By Definition-Theorem 89, there exists an equidimensional model\(/U\) \(f': (X',\Sigma_{X'},{\boldsymbol{M}})\rightarrow (Z,\Sigma_Z)\) of \(f: (X,B,{\boldsymbol{M}})\rightarrow Z\) associated with \(h: X'\rightarrow X\) and \(h_Z: Z'\rightarrow Z\), such that \(h\) is a foliated log resolution of \((X,\mathcal{F},B,{\boldsymbol{M}})\) and \(\mathcal{F}':=h^{-1}\mathcal{F}\) is induced by \(f'\). We let \[K_{\mathcal{F}'}+B'+{\boldsymbol{M}}_{X'}:=h^*(K_{\mathcal{F}}+B+{\boldsymbol{M}}_X),\] \(H':=h^*H\), \(W':=h^{-1}(W)\), and \(\tilde{B}':=(B')^{\geq 0}\). We let \(z\) be the image of \(W'\) on \(Z'\). Since \((X,\mathcal{F},B,{\boldsymbol{M}})\) is lc, by Lemma 123, \((X',\mathcal{F}',\tilde{B}',{\boldsymbol{M}})\) is lc. Moreover, all components of \(\tilde{B}'\) are horizontal\(/Z\). By Lemma 129, \((X',\mathcal{F}',\tilde{B}'+H',{\boldsymbol{M}})\) is lc over a neighborhood of \(z'\). In particular, \((X',\mathcal{F}',\tilde{B}',{\boldsymbol{M}})\) is lc near \(W'|_{H'}\).

Let \[K_{\mathcal{F}_{H'}}+\tilde{B}_{H'}:=(K_{\mathcal{F}'}+\tilde{B}')|_{H'}\text{ and }K_{\mathcal{F}_{H'}}+B_{H'}:=(K_{\mathcal{F}'}+B')|_{H'}.\] By [27], \((H',\mathcal{F}_{H'},\tilde{B}_{H'})\) is lc near \(W'|_{H'}\). Since \(\tilde{B}'\geq B'\), \(\tilde{B}_{H'}\geq B_{H'}\), so \((H',\mathcal{F}_{H'},B_{H'})\) is sub-lc near \(W'|_{H'}\). Since \({\boldsymbol{M}}\) descends to \(X'\), \({\boldsymbol{M}}^H\) descends to \(H'\), so \((H',\mathcal{F}_{H'},B_{H'},{\boldsymbol{M}}^H)\) is sub-lc near \(W'|_{H'}\). Since \(H\) is general, \[K_{\mathcal{F}_{H'}}+B_{H'}+{\boldsymbol{M}}^H_{H'}=h|_{H'}^*(K_{\mathcal{F}_H}+B_H+{\boldsymbol{M}}^H_H),\] so \((H,\mathcal{F}_H,B_H,{\boldsymbol{M}}^H)\) is sub-lc near \(W|_H\). Finally, if \((X,\mathcal{F},B,{\boldsymbol{M}})\) is lc, then by (3), \(B_H\geq 0\), so \((H,\mathcal{F}_H,B_H,{\boldsymbol{M}}^H)\) is lc near \(W|_H\). ◻

Proof. If \(f(E)\) is a prime divisor, then \(a_{{\operatorname{num}},f}(E,\mathcal{F},B,{\boldsymbol{M}})=-\operatorname{mult}_EB=a(E,\mathcal{F},B,{\boldsymbol{M}})\). Assume that \(E\) is exceptional over \(X\). Write \[K_{\mathcal{F}_Y}+\sum a_iE_i+B_Y+{\boldsymbol{M}}_Y=f^*(K_{\mathcal{F}}+B+{\boldsymbol{M}}_X),\] where \(E_i\) are the prime \(f\)-exceptional divisors, \(\mathcal{F}_Y:=f^{-1}\mathcal{F}\) and \(B_Y:=f^{-1}_*B\). Then \[a_{{\operatorname{num}},f}(E_i,\mathcal{F},B,{\boldsymbol{M}})=-a_i=a(E_i,\mathcal{F},B,{\boldsymbol{M}})\] for any \(i\). Since \(E=E_j\) for some \(j\), the lemma holds. ◻

Proof. We may assume that there exists a morphism \(g: Y'\rightarrow Y\). Let \(E_i\) be the prime \(f'\)-exceptional divisors, \(B_{Y'}:=f'^{-1}_*B-\sum_ia_{{\operatorname{num}},f'}(E_i,\mathcal{F},B,{\boldsymbol{M}})E_i\), and \(B_Y:=g_*B_{Y'}\). Then \((K_{\mathcal{F}_{Y'}}+B_{Y'}+{\boldsymbol{M}}_{Y'})\cdot E_i=0\) for any \(E_i\). Since \(Y\) is smooth, \(K_{\mathcal{F}_Y}+B_Y+{\boldsymbol{M}}_Y\) is \(\mathbb{R}\)-Cartier. By the negativity lemma, we see that \(K_{\mathcal{F}_{Y'}}+B_{Y'}+{\boldsymbol{M}}_{Y'}=g^*(K_{\mathcal{F}_Y}+B_Y+{\boldsymbol{M}}_Y)\). Thus \((K_{\mathcal{F}_Y}+B_Y+{\boldsymbol{M}}_Y)\cdot g_*E_i=0\) for any \(E_i\), so \[a_{{\operatorname{num}},f}(E_i,\mathcal{F},B,{\boldsymbol{M}})=-\operatorname{mult}_{g_*E_i}B_Y=\operatorname{mult}_{E_i}B_{Y'}=a_{{\operatorname{num}},f'}(E_i,\mathcal{F},B,{\boldsymbol{M}})\] for any \(E_i\) such that \(g_*E_i\not=0\). In particular, \(a_{{\operatorname{num}},f}(E,\mathcal{F},B,{\boldsymbol{M}})=a_{{\operatorname{num}},f'}(E,\mathcal{F},B,{\boldsymbol{M}})\). ◻

Proof. It follows from [10]. ◻

Proof. By [41], \(X\) is \(\mathbb{Q}\)-factorial klt and \(C\) is non-singular near \(x\). Possibly shrinking \(X\) near \(x\), we may assume that \(X\) is \(\mathbb{Q}\)-factorial klt, and \(C\) is non-singular. Let \(f: Y\rightarrow X\) be a birational morphism such that \({\boldsymbol{M}}\) descends to \(Y\). By the negativity lemma, \(f^*{\boldsymbol{M}}_X-{\boldsymbol{M}}_Y\geq 0\). This implies that \(\mu=\operatorname{mult}_x((h|_{C_Y})_*(h^*{\boldsymbol{M}}_X-{\boldsymbol{M}}_Y)|_{C_Y})\geq 0\), where \(C_Y:=f^{-1}_*C\). ◻

Proof. By Lemma 141, possibly shrinking \(X\) near \(x\), we may assume that \(X\) is \(\mathbb{Q}\)-factorial klt, \(\mathcal{F}\) is terminal, and \(C\) is non-singular.Then the lemma follows from [42].

Moreover, if \((X,\mathcal{F},\sum_{j=1}^m b_j' B_j,{\boldsymbol{M}})\) is lc, then \(\mu\geq 0\) by Lemma 141. Therefore \(\frac{I-1+\sum_{j=1}^m w_j b_j'}{I}+\mu\geq 0\). The inequality \(\frac{I-1+\sum_{j=1}^m w_j b_j'}{I}+\mu\le1\) follows from 125. ◻

Proof. By Lemma 141, possibly shrinking \(X\) near \(x\), we may assume that \(X\) is \(\mathbb{Q}\)-factorial klt, \(\mathcal{F}\) is terminal, and \(C=C^\nu\) is non-singular.

Let \(f:Y\to X\) be a resolution such that \({\boldsymbol{M}}_k\) descends to \(Y\) for each \(k\). By the negativity lemma, \(h^*{\boldsymbol{M}}_{k,X}-{\boldsymbol{M}}_{k,Y}\geq 0\) for each \(k\).

Since \({\boldsymbol{M}}_{k,X}\) is integral for each \(k\), \(I\operatorname{mult}_x {\boldsymbol{M}}_{k,X}|_C\) is an integer. In particular, \[v_k:=I\left(\operatorname{mult}_x {\boldsymbol{M}}_{k,X}|_C-\operatorname{mult}_x {\boldsymbol{M}}_{k,C}^C\right)=I\operatorname{mult}_x\left(\left(h|_{C_Y})_*(h^*{\boldsymbol{M}}_{k,X}-{\boldsymbol{M}}_{k,Y}\right)|_{C_Y}\right)\geq 0\] is a non-negative integer for each \(k\). Then the lemma follows from Lemma 143.

The moreover part is clear. ◻

Proof. (1) Since \((X,\mathcal{F},B,{\boldsymbol{M}})\) is lc, by Lemma 137, \((X,\mathcal{F},B,{\boldsymbol{M}})\) is num-lc. By Lemma 140, \((X,\mathcal{F},B)\) is num-lc near \(x\). Since \(\mathcal{F}\) is not terminal near \(x\), by [41], \(B=0\) near \(x\). By Lemma 140 again, \({\boldsymbol{M}}\) descends to \(X\) over a neighborhood of \(x\).

(2) By considering a local analytic neighborhood of \(x\) and separate \(C\) into different analytic irreducible components, we may assume that \(y=\nu^{-1}(x)\). (2) follows from [41]. More precisely, we let \(h: Y\rightarrow X\) be the minimal resolution of \(\mathcal{F}\) near \(x\) and let \(C_Y:=h^{-1}_*C\), then we only need to show that \[K_{\mathcal{F}}\cdot C-K_{C^\nu}=h^*K_{\mathcal{F}}\cdot C_Y-K_{C_Y}\] is a positive integer over a neighborhood of \(x\). (2) follows by checking all cases of [41] and applying [20] to \(C_Y\) for each case. ◻

Proof. By Theorem 125, \((S^\nu,\mathcal{F}_S,B_S,{\boldsymbol{M}}^S)\) is sub-lc. It suffices to show that \(B_S\ge0.\) By Propositions 128 and 130, we may cut \(X\) by general elements in base-point-free linear systems and assume that \(\dim X=2\).

If \(\operatorname{rank}\mathcal{F}=0\), then since \((X,\mathcal{F},B,{\boldsymbol{M}})\) is lc, \(B=0\) and \({\boldsymbol{M}}\) descends to \(X\), and the theorem is trivial. If \(\operatorname{rank}\mathcal{F}=2\) then the theorem follows from Lemma 81. In the following we assume that \(\operatorname{rank}\mathcal{F}=1\).

If \(S\) is not \(\mathcal{F}\)-invariant, i.e., \(S\) is transverse to \(\mathcal{F}\). According to [21] and Theorem 125, we can see that\[(K_{\mathcal{F}}+S)|_{S^\nu}=K_{\mathcal{F}_S}.\] It follows that \(B_S\ge0\). Assume that \(S\) is \(\mathcal{F}\)-invariant. We only need to check the coefficients of \(B_S\) near any closed point \(y\) on \(S^\nu\). Let \(x\) be the image of \(y\) in \(S\). If \(\mathcal{F}\) is terminal at \(x\), then the theorem follows from Lemma 143. If \(\mathcal{F}\) is not terminal at \(x\), then the theorem follows from Lemma 144. ◻

Proof. Suppose that we are under the conditions of Theorem 113. According to Theorem 145, we only need to prove that \(B_S\) has the form as in Theorem 113. To this end, by Propositions 128 and 130, we may assume that \(\dim X=2\).

If \(\operatorname{rank}\mathcal{F}=0\), then since \((X,\mathcal{F},B,{\boldsymbol{M}})\) is lc, \(B=0\) and \({\boldsymbol{M}}\) descends to \(X\), and the theorem is trivial. If \(\operatorname{rank}\mathcal{F}=2\), then the theorem follows from the usual precise adjunction formula for lc g-pairs [1]. In what follows we may assume that \(\operatorname{rank}\mathcal{F}=1\).

If \(S\) is not \(\mathcal{F}\)-invariant, then by [21] and Theorem 125, \[(K_{\mathcal{F}}+S)|_{S^\nu}=K_{\mathcal{F}_S}.\] The theorem follows in this case. Thus we may assume that \(S\) is \(\mathcal{F}\)-invariant. Then the statement follows from Lemmas 143 and 144. ◻

Proof. It is clear that (2) implies (1). Thus we only need to prove (1) implies (2).

Let \(\eta\) be the generic point of an lc center of \((X,B,{\boldsymbol{M}})\). Since \((X,B,{\boldsymbol{M}})\) is qdlt, there exists an open subset \(V\subset X\) satisfying Definition 148. In particular, \(\eta\) is an lc center of \((V,B|_V)\) and \({\boldsymbol{M}}_X|_V\) is \(\mathbb{R}\)-Cartier. We let \({\boldsymbol{M}}^V:={\boldsymbol{M}}|_V\) be the restricted \(\boldsymbol{b}\)-divisor of \({\boldsymbol{M}}\) on \(V\), then \({\boldsymbol{M}}^V\) is nef\(/V\) and \({\boldsymbol{M}}^V_V={\boldsymbol{M}}_X|_V\). Suppose that \(h: V'\rightarrow V\) is a resolution of \(V\) such that \({\boldsymbol{M}}^V\) descends to \(V'\) and there exists a prime divisor \(E\) on \(V'\) such that \(\operatorname{center}_{V}E=\bar\eta\) and \(E\) is an lc place of \((V,B|_V)\). By the negativity lemma, \[{\boldsymbol{M}}^V_{V'}=h^*{\boldsymbol{M}}^V_V-F\] for some \(F\geq 0\). Moreover, we have either \(F=0\) over \(\eta\) or \(\operatorname{Supp}F=\operatorname{Supp}h^{-1}(\bar\eta)\). Since \((X,B,{\boldsymbol{M}})\) is lc, \((V,B|_V,{\boldsymbol{M}}^V)\) is lc. Thus \(F=0\) over \(\eta\). Possibly shrinking \(V\), we may assume that \({\boldsymbol{M}}\) descends to \(V\). The lemma follows. ◻

Proof. It is obvious that (1) implies (2). We prove (2) implies (1). Possibly shrinking \(X\) around \(x\), we may assume that \((X,\sum_{i=1}^{d}D_i)\) is a pair. Since \(B\geq\sum_{i=1}^{d}D_i\), \((X,D)\) is lc near \(x\). By [82], \(B=\sum_{i=1}^d D_i\) near \(x\), \((X,B)\) is qdlt near \(x\), and \(\bar x\) is an lc center of \((X,B)\). Since \((X,B,{\boldsymbol{M}})\) is lc, \(x\) is an lc center of \((X,B,{\boldsymbol{M}})\), and \((X,B,{\boldsymbol{M}})\) is qdlt near \(x\). ◻

Proof. By definition, \(\operatorname{Supp}\{B\}\) does not contain any lc center of \((X,B,{\boldsymbol{M}})\). Thus \((X,B+\epsilon D,{\boldsymbol{M}})\) is lc for some positive real number \(\epsilon\). Let \(\delta:=\frac{\epsilon}{2}\), then \((X,B+\delta D,{\boldsymbol{M}})\) is lc, and any lc center of \((X,B+\delta D,{\boldsymbol{M}})\) is an lc center of \((X,B,{\boldsymbol{M}})\). By definition, \((X,B+\delta D,{\boldsymbol{M}})\) is qdlt. ◻

Proof. By Lemma 150, we only need to show that, for any lc center \(S'\) of \((X',B',{\boldsymbol{M}})\) with generic point \(\eta_{S'}\), near \(\eta_{S'}\), we have that \((X',B')\) is \(\mathbb{Q}\)-factorial toroidal and \(S'\) is an lc center of \((X',B')\). Indeed, let \(E\) be an lc place of \((X',B',{\boldsymbol{M}})\) such that \(\operatorname{center}_{X'}E=S'\). By our assumption, \(E\) is also an lc place of \((X,B,{\boldsymbol{M}})\) and \(\phi^{-1}\) is an isomorphism near \(\eta_{S'}\). Then the lemma follows by Lemma 150 again. ◻

Proof. If \((X,\mathcal{F},B,{\boldsymbol{M}})/Z\) satisfies Property \((*)\) (resp. is weak ACSS), then there exists a divisor \(G_0\geq 0\) on \(X\) such that \((X,\mathcal{F},B,{\boldsymbol{M}};G_0)/Z\) satisfies Property \((*)\) (resp. is weak ACSS). We let \(H_1,\dots,H_{2\dim X+1}\) be general elements of a very ample linear system on \(Z\), and let \[G:=G_0+\sum_{i} f^*H_i.\] Then by definition, \((X,\mathcal{F},B,{\boldsymbol{M}};G)/Z\) satisfies Property \((*)\) (resp. is weak ACSS). ◻

Proof. The lemma follows by checking the definitions. ◻

Proof. By definition, \(X\) is \(\mathbb{Q}\)-factorial and there exists an equidimensional toroidal morphism \(f: (X,\Sigma_X,{\boldsymbol{M}})\rightarrow (Z,\Sigma_Z)\) such that \(\operatorname{Supp}B\subset\Sigma_X\). We let \(G\) be the vertical\(/Z\) part of \(\Sigma_X\). By Lemma 157, it suffices to show that \((X,\mathcal{F},B, {\boldsymbol{M}})\) is ACSS. To this end, we may assume that \(B=B^\mathcal{F}=\Sigma_X-G\). By [74], possibly adding general hyperplane sections to \(\Sigma_Z\) and adding their pullbacks to \(\Sigma_X\), we may assume that \(G\) is super\(/Z\). We only need to show that \((X,\mathcal{F},B,{\boldsymbol{M}};G)/Z\) is ACSS. By Proposition 106 and Lemma 123, we can see that \((X,\mathcal{F},B,{\boldsymbol{M}};G)/Z\) satisfies Property \((*)\). Since \(\{B\}=0\), \({\boldsymbol{M}}\) descends to \(X\), and \(f^*(\Sigma-f(G))=f^{-1}(\Sigma-f(G))\), by [74], \[f: (X,B+G+D+f^*(\Sigma-f(G)),{\boldsymbol{M}})\rightarrow (Z,\Sigma)\] is toroidal. In particular, \((X,B+G+D+f^*(\Sigma-f(G)),{\boldsymbol{N}})\) is qdlt. Finally, Definition 155(4) immediately follows from the definition of foliated log smooth. ◻

Proof. Let \[l:=\sup\{s\mid s\geq 0, {\boldsymbol{M}}+s{\boldsymbol{N}}\text{ is nef}/U\}=\max\{s\mid s\geq 0, {\boldsymbol{M}}+s{\boldsymbol{N}}\text{ is nef}/U\}\] since nef is a closed condition. Moreover, \(t\leq l\).

Let \(f: X\dashrightarrow Z\) be a dominant map that induces \(\mathcal{F}\). Let \(g: \bar X\rightarrow Z\) be a birational morphism such that \(f\circ g\) is a morphism and both \({\boldsymbol{M}}\) and \({\boldsymbol{N}}\) descend to \(\bar X\). By Lemma 119, we may assume that \(f\circ g\) is a contraction. Let \(\bar B\) be the reduced divisor supported on \[\operatorname{Supp}(f^{-1}_*B)\cup\operatorname{Supp}(D)\cup\operatorname{Supp}\operatorname{Exc}(g).\] By Definition-Theorem 89, there exists an equidimensional model\(/U\) \(f': (X', \Sigma_{X'})\rightarrow (Z', \Sigma_{Z'})\) of \(f\circ g: (\bar X, \bar B)\rightarrow Z\) associated with \(h: X'\rightarrow\bar X\) and \(h_Z: Z'\rightarrow Z\). Since \({\boldsymbol{M}}\) and \({\boldsymbol{N}}\) descend to \(X'\), let \(\phi:=g\circ h\), \(\mathcal{F}':=\phi^{-1}\mathcal{F}\), and \[K_{\mathcal{F}'}+B'+{\boldsymbol{M}}_{X'}:=\phi^*(K_{\mathcal{F}}+B+{\boldsymbol{M}}_X).\] Then \(\phi\) is a foliated log resolution of \((X,\mathcal{F},\operatorname{Supp}B\cup\operatorname{Supp}D,{\boldsymbol{M}})\), and \(f'\) induces \(\mathcal{F}'\). Since \({\boldsymbol{M}}\) and \({\boldsymbol{N}}\) descend to \(X'\), by Lemma 123, \[\begin{align} t&=\sup\{s\mid 0\leq s\leq l, a(E,X,\mathcal{F},B+sD,{\boldsymbol{M}}+s{\boldsymbol{N}})\geq-\epsilon_{\mathcal{F}}(E)\text{ for any prime divisor } E\text{ on }X\}\\ &=\sup\{s\mid 0\leq s\leq l, a(E,X,\mathcal{F},B+sD,{\boldsymbol{M}}+s{\boldsymbol{N}})\geq-\epsilon_{\mathcal{F}}(E)\text{ for any prime divisor } E\subset\operatorname{Supp}\bar B\}. \end{align}\] Since there are only finitely many components of \(\operatorname{Supp}\bar B\), the lemma follows. ◻

Proof. By assumption, \(\operatorname{Supp}D\) does not contain any lc center of \((X,\mathcal{F},B,{\boldsymbol{M}})\), and \({\boldsymbol{M}}\) descends to \(X\) near the generic point of any lc center of \((X,\mathcal{F},B,{\boldsymbol{M}})\). Thus \({\boldsymbol{N}}\) descends to \(X\) near the generic point of any lc center of \((X,\mathcal{F},B,{\boldsymbol{M}})\). Since \({\boldsymbol{M}}+{\boldsymbol{N}}\) is nef\(/U\), by Lemma 159, there exists a real number \(\gamma_0\in (0,1)\) such that both \((X,\mathcal{F},B+\gamma_0 D,{\boldsymbol{M}})\) and \((X,\mathcal{F},B,{\boldsymbol{M}}+\gamma_0{\boldsymbol{N}})\) are lc. Possibly replacing \(\gamma_0\) with \(\frac{1}{2}\gamma_0\), we may assume that \((X,\mathcal{F},B+\gamma_0 D,{\boldsymbol{M}})\), \((X,\mathcal{F},B,{\boldsymbol{M}}+\gamma_0{\boldsymbol{N}})\), and \((X,\mathcal{F},B,{\boldsymbol{M}})\) have the same lc centers.

By convexity of discrepancies, for any \(\alpha,\beta\in [0,\frac{1}{2}\gamma_0]\), \((X,\mathcal{F},B+\alpha D, {\boldsymbol{M}}+\beta{\boldsymbol{N}})\) is lc with the same lc centers as \((X,\mathcal{F},B,{\boldsymbol{M}})\). In particular, for any lc center of \((X,\mathcal{F},B+\alpha D, {\boldsymbol{M}}+\beta{\boldsymbol{N}})\) with generic point \(\eta\), near \(\eta\), \({\boldsymbol{M}}+\beta{\boldsymbol{N}}\) descends to \(X\), and \(B=\lfloor B+\alpha D\rfloor\). Thus \(f: (X,B+\alpha D+G)\rightarrow (Z,f(G))\) is a toroidal morphism near \(\eta\), and \(\eta\) is the generic point of an lc center of \((X,\mathcal{F},\lfloor B+\alpha D\rfloor)\) over a neighborhood of \(\eta\).

Since \((X,\mathcal{F},B,{\boldsymbol{M}};G)/Z\) is ACSS, there exists an \(\mathbb{R}\)-divisor \(D'\) on \(X\) such that \(\operatorname{Supp}\{B\}\subset\operatorname{Supp}D'\), and a nef\(/X\) \(\boldsymbol{b}\)-divisor \({\boldsymbol{N}}'\), such that \({\boldsymbol{N}}'-\alpha'{\boldsymbol{M}}\) is nef\(/X\) for some \(\alpha'>1\), and for any \(\Sigma\) on \(X\) such that \(\Sigma\geq f(G)\) and \((Z,\Sigma)\) is log smooth, \((X,B+G+D'+f^*(\Sigma-f(G)),{\boldsymbol{N}}')\) is qdlt.

We will show that \[\gamma:=\min\left\{\frac{\gamma_0}{3},\left(\alpha'-1\right)\delta\right\}\] has the required properties. Indeed, As \(\operatorname{Supp}D\subset\operatorname{Supp}\{B\}\subset\operatorname{Supp}D'\), we see that for any \(\alpha\in(0,\gamma)\), it holds that \[D'-\alpha D\ge0\text{ and }\operatorname{Supp}(D'-\alpha D)=\operatorname{Supp}D'\supset\operatorname{Supp}\{B+\alpha D\}.\] By definition, it is enough to prove that \(D'-\alpha D\) and \({\boldsymbol{N}}'\) satisfy Definition 155(3). In particular, we only need to show that there exists a positive real number \(\alpha''>1\) such that \({\boldsymbol{N}}'-\alpha''({\boldsymbol{M}}+\beta{\boldsymbol{N}})\) is nef\(/X\). Take \[\alpha'':=\frac{\alpha'}{1+\frac{\beta}{\delta}},\] then it is clear that \(\alpha''>1\) as \(\beta\le\gamma<(\alpha'-1)\delta\). Then \[{\boldsymbol{N}}'-\alpha''({\boldsymbol{M}}+\beta{\boldsymbol{N}})=\frac{\alpha''}{\alpha'}(1+\frac{\beta}{\delta})({\boldsymbol{N}}'-\alpha'{\boldsymbol{M}})+\frac{\alpha''\beta}{\delta}({\boldsymbol{M}}-\delta{\boldsymbol{N}})\] is nef\(/X\). We may finish the proof. ◻

Proof. By assumption, \(Z\) is smooth and \(B_Z=f(G)\) is the discriminant part of \(f: (X,B+G,{\boldsymbol{M}})\rightarrow Z\). Moreover, \(B_Z\) is reduced by Lemma 104. Then (2) follows from (1) immediately. Thus we only need to prove (1).

Since \(f\) is equidimensional, we have \[K_{\mathcal{F}}=K_{X/Z}-R\] where \(R:=\sum(f^*D-f^{-1}(D))\) and \(D\) runs over the prime divisors on \(Z\). We claim that \(f^*B_Z=R+G\). Indeed, let \(D\) be a prime divisor on \(X\) such that \(D\) is vertical\(/Z\). Since \(f\) is equidimensional, \(D_Z:=f(D)\) is a prime divisor. If \(D_Z\subset\operatorname{Supp}B_Z\), then \(D\subset\operatorname{Supp}G\) and \(\operatorname{mult}_DG=1\). Therefore, \[\begin{align} \operatorname{mult}_Df^*B_Z&=\operatorname{mult}_Df^*D_Z=\operatorname{mult}_Df^{-1}(D_Z)+\operatorname{mult}_D(f^*D_Z-f^{-1}(D_Z))=\operatorname{mult}_DG+\operatorname{mult}_DR. \end{align}\] If \(D_Z\not\subset \operatorname{Supp}B_Z\), then \(\operatorname{mult}_Df^*B_Z=\operatorname{mult}_DG=0\). By assumption, \((X,B+G+f^*D_Z,{\boldsymbol{M}})\) is sub-lc over the generic point of \(D_Z\). In particular, \(\operatorname{mult}_Df^*D_Z=1\) and hence \(\operatorname{mult}_DR=0\). Since \(f^*B_Z\) and \(R+G\) are both vertical\(/Z\), the claim follows.

Since \(f^*B_Z=R+G\), one can see that \[\begin{align} {\boldsymbol{N}}_X&\sim K_X+B+G+{\boldsymbol{M}}_X-f^*(K_Z+B_Z)=K_{X/Z}+B+G+{\boldsymbol{M}}_X-f^*B_Z\\ &=K_{\mathcal{F}}+R+B+G-f^*B_Z=K_{\mathcal{F}}+B+{\boldsymbol{M}}_X. \end{align}\] This completes the proof. ◻

Proof. Since statement (3) is a closed condition and \(M\) is a limit of \(\mathbb{Q}\)-divisors that are nef along general fibers of \(\pi\), we may assume that \(M\) is a \(\mathbb{Q}\)-divisor. Possibly replacing \(M\) with a multiple, we may assume that \(M\) is a Weil divisor.

We let \(X^c\) and \(U^c\) be compactifications of \(X\) and \(U\), such that \(X^c\) and \(U^c\) are normal projective, \(X\) is a dense open subset of \(X^c\), \(U\) is a dense open subset of \(U^c\), and there exists a contraction \(\pi^c: X^c\rightarrow U^c\) such that \(\pi^C|_{X}=\pi\). Let \(M^c,D^c_1,\dots,D^c_d,B^c\) be the closures of \(M,D_1,\dots,D_d,B\) in \(X^c\) respectively, and let \(\mathcal{F}^c\) be the natural extension of \(\mathcal{F}\) in \(X^c\). Then the general fibers of \(\pi^c\) are general fibers of \(\pi\), and \(M^c,D^c_1,\dots,D^c_d\) are \(\mathbb{R}\)-divisors that are nef along general fibers of \(\pi\). Since we only care about properties of general fibers of \(\pi\) and properties near a general closed point \(x\in X\), we may replace \(\pi: X\rightarrow U\) with \(\pi^c: X^c\rightarrow U^c\), \(M,D_1,\dots,D_d,B\) with \(M^c,D^c_1,\dots,D^c_d,B^c\), and \(\mathcal{F}\) with \(\mathcal{F}^c\), and assume that \(\pi\) is a projective morphism between normal projective varieties.

Let \(x\in X\) be a general closed point. Then \(x\) is contained in a general fiber \(F\) of \(\pi\). Let \(q:=\dim U\). Then there exist general hyperplane sections \(H_1,\dots,H_q\) with \(A_i:=\pi^*H_i\), such that \(F=\cap_{i=1}^q\pi^*A_i\). Let \(V_k:=X\cap_{i=1}^kA_i\) and \(W_k:=U\cap_{i=1}^kH_i\) for each \(0\leq k\leq q\). Then we have \[F=V_q\subset V_{q-1}\subset\dots\subset V_0=X\] and \[z:=W_q\subset W_{q-1}\subset\dots\subset W_0=U\] where \(z\) is a closed point. We may inductively define \(\mathcal{F}_{k}\) to be the restricted foliation of \(\mathcal{F}\) on \(V_k\) for each \(k\), and let \(\mathcal{F}_F:=\mathcal{F}_{q}\). We let \(M_k:=M|_{V_k}\), \(B_k:=B|_{V_k}\), \(M_F:=M|_F\), and \(B_F:=B|_F\). Then it is clear that \(M_k|_F=M|_F\), \(B_k|_F=B_F\) for each \(k\), and \(B_{V_k}\geq 0\) for each \(k\). Moreover, since \(H_1,\dots,H_q\) are general hyperplane sections, \(M_k\) is a Weil divisor for each \(k\).

Proof of Theorem 164 continued. It immediately follows from Claim 165 by letting \(k=0\). ◻

Proof. (1) is obvious.

Pick a \((K_{\mathcal{F}}+B+{\boldsymbol{M}}_X)\)-negative extremal ray \(R\). By our assumption, there exists a prime divisor \(E\) on \(X\) such that \(R\) is contained in the image of \(\overline{NE}(E/U)\rightarrow\overline{NE}(X/U)\) and \(\operatorname{mult}_EB=\epsilon_{\mathcal{F}}(E)\). We let \(S\) be the normalization of \(E\). Then there exists a natural surjection \(\overline{NE}(S/U)\rightarrow\overline{NE}(E/U)\), and thus \(R\) is contained in the image of \(\iota: \overline{NE}(S/U)\rightarrow\overline{NE}(E/U)\rightarrow\overline{NE}(X/U)\). We claim that there exists an extremal ray \(R_S\) in \(\overline{NE}(S/U)\) such that \(R=\iota(R_S)\). In fact, there exist extremal rays \(R_i'\) in \(\overline{NE}(S/U)\) such that \(\iota(\sum a_iR_i')=R\) for some \(a_i>0\). Since \(R\) is extremal\(/U\), either \(\iota(R_i')=R\) or \(\iota(R_i')=0\) for each \(i\). Since \(R\not=0\), there exists \(j\) such that \(\iota(R_j')\not=0\). We may take \(R_S:=R_j'\) and the claim holds.

Let \(\mathcal{F}_S\) be the restricted foliation of \(\mathcal{F}\) on \(S\) which is algebraically integrable by Proposition 121, \({\boldsymbol{M}}^S:={\boldsymbol{M}}|_S\), and \[K_{\mathcal{F}_S}+B_S+{\boldsymbol{M}}^S_S:=(K_{\mathcal{F}}+B+{\boldsymbol{M}}_X)|_S.\] Then \(R_S\) is a \((K_{\mathcal{F}}+B+{\boldsymbol{M}}_X)|_S\)-negative extremal ray. By Theorem 145, \((S,\mathcal{F}_S,B_S,{\boldsymbol{M}}^S)/U\) is an lc gfq. Since we assume Theorem 166 in dimension \(\leq d-1\), \(R_S\) is spanned by a rational curve \(C\) such that \(C\) is tangent to \(\mathcal{F}_S\) and \[0<-\left(K_{\mathcal{F}_S}+B_S+{\boldsymbol{M}}^S_S\right)\cdot C\leq 2(d-1).\] We identify \(C\) with its image in \(X\) under the natural inclusion \(S\rightarrow E\rightarrow X\). Then \(C\) spans \(R\) and \[0<-\left(K_{\mathcal{F}_S}+B_S+{\boldsymbol{M}}^S_S\right)\cdot C=-\left(K_{\mathcal{F}}+B+{\boldsymbol{M}}_X\right)\cdot C\leq 2(d-1).\] Moreover, by [27], \(C\) is tangent to \(\mathcal{F}\) and is contracted by \(f\). This implies (2).

For any curve \(C_j'\) such that \([C_j']\in R\), we let \(C''\) be any irreducible component of \(C_j'\). Since \(R\) is extremal, \([C'']\in R\) which implies that \(C\equiv\lambda C''\) for some positive rational number \(\lambda\). Thus \(C''\) is contracted by \(f\) and therefore \(C_j'\) is contracted by \(f\), and we get (3).

Now we may assume that \(f\) is equi-dimensional, \(A\) is an ample\(/U\) \(\mathbb{R}\)-divisor on \(X\), and \(G\) is a divisor associated to \((X,\mathcal{F},B,{\boldsymbol{M}})/Z\). By Lemma 104 and Proposition 161, \((X,B+G,{\boldsymbol{M}})\) is lc and \[K_{\mathcal{F}}+B+{\boldsymbol{M}}_X\sim_{\mathbb{R},Z}K_{X}+B+G+{\boldsymbol{M}}_X.\] By (3), \(R_j\) is a \((K_{X}+B+G+{\boldsymbol{M}}_X)\)-negative extremal ray\(/U\) for any \(j\in\Lambda\).

If \({\boldsymbol{M}}\) is NQC\(/U\), then by [12], \(\Lambda_A\) is a finite set, and thus \(\Lambda\) is a countable set. This implies (4.a) and (4.b). (4.c.i) follows from [13] (see also [14]), and (4.c.ii) follows from [16].

Suppose that \(X\) is \(\mathbb{Q}\)-factorial klt. Possibly adding the pull-back of some very ample divisor to \(A\), we may assume that \(A\) is ample. By [4], there exists an \(\mathbb{R}\)-divisor \(\Delta_A\ge0\) such that \(\Delta_A\sim_{\mathbb{R}}B+G+A+{\boldsymbol{M}}_X\) and \((X,\Delta_A)\) is klt. It follows that \[\Lambda_A=\{j\in\Lambda\mid(K_{\mathcal{F}}+\Delta)\cdot R_j<0\}\] is a finite set by the classical cone theorem (cf. [73], [53]), and \(\Lambda=\cup_{n=1}^{+\infty}\Lambda_{\frac{1}{n}A}\) is a countable set. This implies (4.a) and (4.b). (4.c.i) follows from the classical contraction theorem (cf. [73], [53]) and (4.c.ii) follows from the existence of flips [29].