1 Introduction
Throughout this paper, we will work over the complex number field $\mathbb {C}$ .
In this paper, we study triples $(X,\Delta ,M)$ such that $(X,\Delta )$ is a projective lc pair and M is a nef $\mathbb {R}$ divisor on X that is log big with respect to $(X,\Delta )$ (see Definition 2.13). By definition, the log bigness coincides with the bigness in the case of klt pairs, and a nef $\mathbb {R}$ divisor D on a normal projective variety V is log big with respect to a projective lc pair $(V,B)$ if and only if $(D^{\mathrm { dim}V})>0$ and $(S\cdot D^{n})>0$ for any ndimensional lc center S of $(V,B)$ . Especially, all ample divisors are nef and log big with respect to any projective lc pair. The triples $(X,\Delta ,M)$ can be regarded as generalized pairs defined by Birkar–Zhang [Reference Birkar and Zhang7]. Generalized pairs are the main objects in the study of lctrivial fibrations. Because the log bigness is one of the special cases of the property of being log abundant (see Definition 2.13), the recent progress of the canonical bundle formula for lctrivial fibrations by Floris–Lazić [Reference Floris and Lazić11] (see also [Reference Hu34, Theorem 1.2] by Hu) implies that the above triples $(X,\Delta ,M)$ appear as the structures on the base varieties of special lctrivial fibrations. The goal of this paper is to develop the theory of the above triples $(X,\Delta ,M)$ from viewpoints of the minimal model theory.
We start with the minimal model theory for the above triples $(X,\Delta ,M)$ . If $(X,\Delta )$ is a klt pair, then we can run a minimal model program (MMP, for short) on $K_{X}+\Delta+M$ and we get a birational contraction $\phi \colon X\dashrightarrow X'$ such that $\phi _{*}(K_{X}+\Delta +M)$ is semiample or $X'$ has the structure of a Mori fiber space with respect to $\phi _{*}(K_{X}+\Delta +M)$ . This fact is a consequence of the celebrated result by Birkar–Cascini–Hacon–M^{c}Kernan [Reference Birkar, Cascini, Hacon and McKernan6]. Even if $(X,\Delta )$ is not klt, the same statement is known by the author and Hu [Reference Hashizume and Hu33] under the assumption of the ampleness of M. In the general case, the minimal model theory for $K_{X}+\Delta +M$ is not known. However, the abundance theorem (more strongly, the effective base point free theorem in the $\mathbb {Q}$ boundary case) was proved by Fujino [Reference Fujino14] if $K_{X}+\Delta +M$ is nef and Cartier. Moreover, Fujino [Reference Fujino16] proved the abundance theorem for $K_{X}+\Delta +M$ in the case where $(X,\Delta )$ is slc, a more general situation than the lc case.
The first main result of this paper is the minimal model theory for $K_{X}+\Delta +M$ .
Theorem 1.1. Let $(X,\Delta )$ be a projective lc pair, and let M be a nef $\mathbb {R}$ divisor on X that is log big with respect to $(X,\Delta )$ . Suppose that M is a finite $\mathbb {R}_{>0}$ linear combination of nef $\mathbb {Q}$ Cartier divisors on X. Then there exists a birational contraction $\phi \colon X\dashrightarrow X'$ , which is a sequence of steps of a $(K_{X}+\Delta +M)$ MMP such that $X'$ satisfies one of the following conditions.

○ $\phi _{*}(K_{X}+\Delta +M)$ is semiample, or

○ There is a contraction $X' \to Z$ to a projective variety Z such that $\mathrm {dim}Z <\mathrm {dim}X'$ , $\phi _{*}(K_{X}+\Delta +M)$ is ample over Z and the relative Picard number is one.
Note that $K_{X'}+\phi _{*}\Delta $ is $\mathbb {R}$ Cartier and $(X',\phi _{*}\Delta )$ is an lc pair.
Theorem 1.1 is proved by generalizing [Reference Hashizume31, Theorem 3.5] to the context of generalized pairs. For the generalization, see Theorem 3.15. To prove Theorem 1.1, we only need to prove the termination of the MMP. It is because the abundance theorem in the situation of Theorem 1.1 was already proved by Fujino [Reference Fujino14]. The proof in [Reference Fujino14] heavily depends on a vanishing theorem for quasilog schemes (see [Reference Fujino17, Theorem 5.7.3]). On the other hand, for the termination of the MMP of Theorem 1.1, we need to discuss the MMP in the framework of generalized pairs (see, for example, [Reference Han and Li25] by Han–Li) and we need the techniques developed in [Reference Hashizume31]. By Theorem 3.15, we also know the existence of a minimal model for generalized lc pairs with a polarization (Theorem 3.17), which was mentioned in [Reference Hashizume32]. We note that we do not need the existence of flips for $\mathbb {Q}$ factorial generalized lc pairs by Hacon–Liu [Reference Hacon and Liu23] in the proof of Theorem 1.1 or Theorem 3.17.
We apply Theorem 1.1 to study the effectivity of the Iitaka fibrations for the above triples $(X,\Delta ,M)$ such that $(X,\Delta )$ is a dlt pair. The effectivity of the Iitaka fibrations for higher dimensional lc pairs was studied by Fujino–Mori [Reference Fujino and Mori20], Viehweg–Zhang [Reference Viehweg and Zhang41], Birkar–Zhang [Reference Birkar and Zhang7] and Hacon–Xu [Reference Hacon and Xu24]. Those results are based on the canonical bundle formula for the Iitaka fibrations [Reference Fujino and Mori20]. If we know the effectivity of the Iitaka fibrations for lc pairs, then we know the existence of an integer m, which is independent of the variety, such that every projective lc pair $(Y,0)$ of fixed dimension with $\kappa (Y,K_{Y})\geq 0$ satisfies $H^{0}(Y, \mathcal {O}_{Y}( m K_{Y}))\neq 0$ . Especially, if $K_{Y}$ is numerically trivial, then $mK_{Y}$ is Cartier. Hence, the effectivity of the Iitaka fibrations for lc pairs is stronger than the boundedness of the Cartier indices of numerically trivial lc varieties of fixed dimension.
In the case of klt pairs whose boundary divisors are big, the effectivity of the Iitaka fibrations is known by Hacon–Xu [Reference Hacon and Xu24]. In the nonklt case, the effectivity of the Iitaka fibrations for lc pairs of log general type was proved by Hacon–M^{c}Kernan–Xu [Reference Hacon, McKernan and Xu21], and their result was generalized to the context of generalized lc pairs (see Definition 2.6) by Birkar–Zhang [Reference Birkar and Zhang7]. The theorem by Birkar–Zhang [Reference Birkar and Zhang7] implies the effectivity of the Iitaka fibrations for $K_{X}+\Delta +M$ of the triples $(X,\Delta ,M)$ such that $K_{X}+\Delta +M$ are big and M are finite $\mathbb {R}_{>0}$ linear combinations of nef $\mathbb {Q}$ Cartier divisors. In this paper, we study the case where $K_{X}+\Delta +M$ has an intermediate Iitaka dimension.
Theorem 1.2. Let d and p be positive integers, and let $\Phi \subset \mathbb {Q}_{\geq 0}$ be a DCC set. Then there exists a positive integer m, depending only on d, p and $\Phi $ , satisfying the following. Let $(X,\Delta )$ be a projective dlt pair, and let M be a nef $\mathbb {Q}$ divisor on X such that

○ $\mathrm {dim}X=d$ ,

○ the coefficients of $\Delta $ belong to $\Phi $ ,

○ $pM$ is Cartier and

○ M is log big with respect to $(X,\Delta )$ and $K_{X}+\Delta +M$ is pseudoeffective.
Then, the linear system $\lfloor lm(K_{X}+\Delta +M)\rfloor $ is not empty and it defines a map birational to the Iitaka fibration for every positive integer l.
More precisely, we prove the existence of a generalized lc pair such that the coefficients of the boundary divisor and the bCartier index of the nef part are controlled.
Theorem 1.3. Let d be a positive integer, and let $\Phi \subset \mathbb {Q}_{\geq 0}$ be a DCC set. Then there exist positive integers n and q and a DCC set $\Omega \subset \mathbb {Q}_{\geq 0}$ , depending only on d and $\Phi $ , satisfying the following. Let $(X,\Delta )$ be a projective dlt pair, and let $M=\sum _{i}\mu _{i}M_{i}$ be a nef $\mathbb {R}$ divisor on X, which is the sum of nef and log big Cartier divisors $M_{i}$ with respect to $(X,\Delta )$ , such that

○ $\mathrm {dim}X=d$ ,

○ the coefficients of $\Delta $ and $\mu _{i}$ belong to $\Phi $ and

○ $K_{X}+\Delta +M$ is pseudoeffective but not big.
Then the variety $Z:={\mathbf {Proj}}\bigoplus _{l \in \mathbb {Z}_{\geq 0}}H^{0}(X,\mathcal {O}_{X}(\lfloor l(K_{X}+\Delta +M)\rfloor ))$ , which is well defined by Theorem 1.1, has the structure of a generalized lc pair $(Z,\Delta _{Z},\mathbf {N})$ such that

○ $H^{0}(X,\mathcal {O}_{X}(\lfloor ln(K_{X}+\Delta +M)\rfloor )) \simeq H^{0}(Z,\mathcal {O}_{Z}(\lfloor ln(K_{Z}+\Delta _{Z}+\mathbf {N}_{Z})\rfloor ))$ for every positive integer l,

○ the coefficients of $\Delta _{Z}$ belong to $\Omega $ and

○ $q \mathbf {N}$ is bCartier.
Theorem 1.2 directly follows from Theorem 1.3 and [Reference Birkar and Zhang7, Theorem 1.3].
We further study the structures of the Iitaka fibrations. In [Reference Li39], Li defined the Iitaka volumes of $\mathbb {Q}$ Cartier divisors on normal projective varieties (see Definition 5.1). In our context, with notations as in Theorem 1.3, the Iitaka volume of $K_{X}+\Delta +M$ coincides with the volume of $K_{Z}+\Delta _{Z}+\mathbf {N}_{Z}$ . The following theorem is the DCC for the Iitaka volumes of $K_{X}+\Delta +M$ .
Theorem 1.4. Let d be a positive integer and $\Phi \subset \mathbb {Q}_{\geq 0}$ a DCC set. Then there exists a DCC set $\Omega \subset \mathbb {Q}_{>0}$ , depending only on d and $\Phi $ , satisfying the following. Let $(X,\Delta )$ be a projective dlt pair, and let $M=\sum _{i}\mu _{i}M_{i}$ be a nef $\mathbb {R}$ divisor on X, which is the sum of nef and log big Cartier divisors $M_{i}$ with respect to $(X,\Delta )$ , such that

○ $\mathrm {dim}X=d$ ,

○ the coefficients of $\Delta $ and $\mu _{i}$ belong to $\Phi $ and

○ $K_{X}+\Delta +M$ is pseudoeffective.
Then the Iitaka volume $\mathrm {Ivol}(K_{X}+\Delta +M)$ is an element of $\Omega $ .
We give remarks on general fibers of the Iitaka fibrations. As shown in the example below, compared to [Reference Hacon and Xu24, Theorem 1.3], we cannot prove a kind of the boundedness of general fibers of the Iitaka fibrations.
Example 1.5 (see Example 4.6)
We put $d=2$ , $p=1$ , and $\Phi =\{1\}$ . We consider the category $\mathcal {C}$ whose objects are the triples $(X,\Delta ,M)$ , where $(X,\Delta )$ is a projective dlt pair and M is a nef Cartier divisor on X which is log big with respect to $(X,\Delta )$ such that $\mathrm {dim}X=2$ , $\Delta $ is a Weil divisor, and $K_{X}+\Delta +M\sim _{\mathbb {Q}}0$ . Then, for some set
$\mathcal {D}$ is unbounded. Indeed, for each positive integer n, defining $X_{n}:=\mathbb {P}_{\mathbb {P}^{1}}(\mathcal {O}_{\mathbb {P}^{1}}\oplus \mathcal {O}_{\mathbb {P}^{1}}(n))$ , then there are divisors $\Delta _{n}$ and $M_{n}$ on $X_{n}$ such that $(X_{n},\Delta _{n},M_{n})$ is an object of $\mathcal {C}$ . However, we see that the set $\{X_{n}\}_{n\geq 1}$ is unbounded. For details, see Example 4.6.
Despite the example, it is possible to prove the effective nonvanishing (and the boundedness of complements) for general fibers of the Iitaka fibrations for $K_{X}+\Delta +M$ .
Theorem 1.6. Let d be a positive integer, and let $\Phi \subset \mathbb {Q}_{\geq 0}$ be a DCC set. Then there exists a positive integer n, depending only on d and $\Phi $ , satisfying the following. Let $(X,\Delta )$ be a projective dlt pair, and let $M=\sum _{i}\mu _{i}M_{i}$ be a nef $\mathbb {R}$ divisor on X, which is the sum of nef and log big Cartier divisors $M_{i}$ with respect to $(X,\Delta )$ such that

○ $\mathrm {dim}X=d$ ,

○ the coefficients of $\Delta $ and $\mu _{i}$ belong to $\Phi $ and

○ $K_{X}+\Delta +M$ is pseudoeffective.
Let $\phi \colon X \dashrightarrow X'$ be finite steps of a $(K_{X}+\Delta +M)$ MMP such that $\phi _{*}(K_{X}+\Delta +M)$ is semiample (see Theorem 1.1), and let F be a general fiber of the contraction induced by $\phi _{*}(K_{X}+\Delta +M)$ . Put $\Delta ^{\prime }_{F}=\phi _{*}\Delta _{F}$ and $M^{\prime }_{F}=\phi _{*}M_{F}$ . Then

○ $n(K_{F}+\Delta ^{\prime }_{F}+M^{\prime }_{F})\sim 0$ , and

○ there is $B^{\prime }_{F} \in nM^{\prime }_{F}$ such that $(F,\Delta ^{\prime }_{F}+\frac {1}{n}B^{\prime }_{F})$ is an lc pair.
Theorem 1.6 is a consequence of the boundedness of complements for a special kind of dlt pairs (Theorem 5.4). In Example 5.5, we show that the log bigness of $M_{i}$ cannot be relaxed to the bigness in Theorem 1.2 and Theorem 1.6. It is not clear that the dlt property can be generalized to log canonicity.
Finally, we discuss the effective finite generation of the generalized log canonical rings for the above $(X,\Delta ,M)$ . A kind of the effective finite generation for projective klt threefolds $(V,B)$ was proved by Cascini–Zhang [Reference Cascini and Zhang8] when $K_{V}+B$ is big or B is nef and big. In this paper, we prove the following theorem.
Theorem 1.7. Let d and p be positive integers, and let v be a positive real number. Then there exists a positive integer m, depending only on d, p and v, satisfying the following. Let $(X,\Delta )$ be a projective dlt pair, let M be a nef $\mathbb {Q}$ divisor on X that is log big with respect to $(X,\Delta )$ and let $\phi \colon X \dashrightarrow X'$ be finite steps of a $(K_{X}+\Delta +M)$ MMP such that

○ $\mathrm {dim}X=d$ ,

○ $pM$ is Cartier and $p\phi _{*}(K_{X}+\Delta +M)$ is nef and Cartier and

○ the Iitaka volume $\mathrm {Ivol}(K_{X}+\Delta +M)$ is less than or equal to v.
Putting $R_{l}=H^{0}(X,\mathcal {O}_{X}(\lfloor l(K_{X}+\Delta +M)\rfloor ))$ for every $l \in \mathbb {Z}_{\geq 0}$ , then

○ $\bigoplus _{l \in \mathbb {Z}_{\geq 0}} R_{lm}$ is generated by $R_{m}$ as a graded $\mathbb {C}$ algebra, and

○ the variety $Z:={\mathbf {Proj}}\bigoplus _{l \in \mathbb {Z}_{\geq 0}}R_{l}$ belongs to a bounded family $\mathfrak {F}$ that depends only on d, p and v.
The proof of the theorem heavily depends on the effective base point free theorem for spacial generalized dlt pairs (Theorem 5.8) and the boundedness of generalized lc pairs in a special case (Lemma 5.10), thus we will need some extra work to remove the upper bound of the Cartier index of $\phi _{*}(K_{X}+\Delta +M)$ in Theorem 1.7.
The contents of the paper are as follows: In Section 2, we collect notations and definitions. In Section 3, we prove Theorem 1.1. In Section 4, we study the effectivity of the Iitaka fibrations for projective dlt pairs polarized by nef and log big divisors, and we prove Theorem 1.2 and Theorem 1.3. In Section 5, we show some boundedness results, and we prove Theorem 1.4, Theorem 1.6 and Theorem 1.7. In Section 6, which is an appendix, we discuss the definition of generalized dlt pairs.
2 Preliminaries
In this section, we collect notations and definitions. We will freely use the notations in [Reference Kollár and Mori38] and [Reference Birkar, Cascini, Hacon and McKernan6].
2.1 Divisors, morphisms and singularities
We collect notations and definitions on divisors, morphisms and singularities of generalized pairs.
We will use the standard definitions of nef $\mathbb {R}$ divisor, ample $\mathbb {R}$ divisor, semiample $\mathbb {R}$ divisor and pseudoeffective $\mathbb {R}$ divisor. All big $\mathbb {R}$ divisors in this paper are $\mathbb {R}$ Cartier. For a morphism $f\colon X\to Y$ and an $\mathbb {R}$ Cartier divisor D on Y, we sometimes denote $f^{*}D$ by $D_{X}$ . For a prime divisor P over X, the image of P on X is denoted by $c_{X}(P)$ . A projective morphism $f\colon X\to Y$ of varieties is called a contraction if $f_{*}\mathcal {O}_{X} \simeq \mathcal {O}_{Y}$ . For a variety X and an $\mathbb {R}$ divisor $D'$ on it, a log resolution of $(X,D')$ denotes a projective birational morphism $f\colon Y\to X$ from a smooth variety Y such that the exceptional locus $\mathrm {Ex}(f)$ of f is pure codimension one and $\mathrm {Ex}(f)\cup \mathrm { Supp}f_{*}^{1}D'$ is an snc divisor.
Definition 2.1. We say that a subset of $\mathbb{R}$ satisfies the descending chain condition (DCC, for short) if the subset does not contain a strictly decreasing infinite sequence. We say that a subset of $\mathbb{R}$ satisfies the ascending chain condition (ACC, for short) if the subset does not contain a strictly increasing infinite sequence. A subset of $\mathbb{R}$ is called a DCC set (resp. an ACC set) if the subset satisfies the DCC (resp. ACC).
Let a be a real number. Then we define $\lfloor a \rfloor $ to be the unique integer satisfying $ \lfloor a \rfloor \leq a <\lfloor a \rfloor +1$ , and we define $\{a\}$ by $\{a\}=a\lfloor a \rfloor $ .
Let X be a normal variety, and let D be an $\mathbb {R}$ divisor on X. Let $D=\sum _{i} d_{i}D_{i}$ be the prime decomposition. Then we define $\lfloor D \rfloor :=\sum _{i} \lfloor d_{i} \rfloor D_{i}$ , $\{D\}:=\sum _{i} \{d_{i}\}D_{i}$ and $\lceil D \rceil :=\lfloor D \rfloor $ . By definition, we have $D=\lfloor D \rfloor + \{D\}$ .
Definition 2.2 (Hyperstandard set)
Let $\mathfrak {R}\subset \mathbb {R}$ be a subset. Throughout this paper, $\Phi (\mathfrak {R})$ is defined by
and we call it a hyperstandard set associated to $\mathfrak {R}$ .
Definition 2.3 (Asymptotic vanishing order)
Let X be a normal projective variety and D a pseudoeffective $\mathbb {R}$ Cartier divisor on X. Let P be a prime divisor over X. We define asymptotic vanishing order of P with respect to D, which we denote $\sigma _{P}(D)$ , as follows. We take a projective birational morphism $f\colon Y \to X$ such that P appears as a prime divisor on Y. When D is big, we define $\sigma _{P}(D)$ by
When D is not necessarily big, $\sigma _{P}(D)$ is defined by
for an ample $\mathbb {R}$ divisor A on X. It is easy to see that $\sigma _{P}(D)$ is independent of $f\colon Y\to X$ . We have $\sigma _{P}(D)\geq 0$ , and $\sigma _{P}(D)$ is also independent of A ([Reference Nakayama40, III, 1.5 (2) Lemma]). We can easily check
.
Definition 2.4 (Negative part of Nakayama–Zariski decomposition)
For X and D as in Definition 2.3, the negative part of Nakayama–Zariski decomposition of D, denoted by $N_{\sigma }(D)$ , is defined by
Note that $N_{\sigma }(D)$ is not necessarily $\mathbb {R}$ Cartier in this paper.
When X is smooth, the definition of $N_{\sigma }(D)$ coincides with [Reference Nakayama40, III, 1.12 Definition].
We refer to [Reference Hashizume31, Remark 2.3] for basic properties of asymptotic vanishing order and the negative part of Nakayama–Zariski decomposition.
Definition 2.5 (bdivisor)
Let X be a normal variety. Then an $\mathbb {R}$ bdivisor $\mathbf {D}$ on X is a (possibly infinite) $\mathbb {R}$ linear combination of divisorial valuations $v_{P}$
such that, for every projective birational morphism $Y\to X$ , the set
is a finite set. When $r_{P}\in \mathbb {Q}$ for all P, we call $\mathbf {D}$ a $\mathbb {Q}$ bdivisor. For a projective birational model Y of X, the trace of an $\mathbb {R}$ bdivisor $\mathbf {D}=\sum _{P} r_{P}v_{P}$ on Y, which we denote $\mathbf {D}_{Y}$ , is defined by
By definition, $\mathbf {D}_{Y}$ is an $\mathbb {R}$ divisor on Y.
Let $\mathbf {D}$ be an $\mathbb {R}$ bdivisor on X. If there exists an $\mathbb {R}$ Cartier divisor D on a projective birational model Y of X such that
then we say that $\mathbf {D}$ is $\mathbb {R}$ bCartier, we say that $\mathbf {D}$ descends to Y and we denote $\mathbf {D}=\overline {D}$ . When $\mathbf {D}=\overline {D}$ and D is $\mathbb {Q}$ Cartier, we say that $\mathbf {D}$ is $\mathbb {Q}$ bCartier. When $\mathbf {D}=\overline {\mathbf {D}_{Y'}}$ and $\mathbf {D}_{Y'}$ is Cartier for some projective birational model $Y'$ , we say that $\mathbf {D}$ is bCartier.
Suppose that X has a projective morphism $X\to Z$ to a variety Z, and suppose in addition that $\mathbf {D}=\overline {\mathbf {D}_{Y'}}$ and $\mathbf {D}_{Y'}$ is nef over Z (resp. big over Z) for some projective birational model $Y'$ of X. Then, we say that $\mathbf {D}$ is bnef $/Z$ (resp. bbig $/Z$ ).
Let X and $X'$ be normal varieties which are projective over a variety Z, and let $\phi \colon X \dashrightarrow X'$ be a birational map over Z. Let $\mathbf {D}$ and $\mathbf {D}'$ be $\mathbb {R}$ bdivisors on X and $X'$ , respectively. Then, we say that $\mathbf {D}=\mathbf {D}'$ by $\phi $ if $\mathbf {D}_{Y}=\mathbf {D}^{\prime }_{Y}$ for all projective birational morphisms $f \colon Y \to X$ and $f' \colon Y \to X'$ such that $f'=\phi \circ f$ .
Definition 2.6 (Singularities of generalized pairs)
A generalized pair $(X,\Delta ,\mathbf {M})/Z$ consists of

○ a projective morphism $X\to Z$ from a normal variety to a variety,

○ an effective $\mathbb {R}$ divisor $\Delta $ on X and

○ a bnef $/Z \ \mathbb {R}$ bCartier $\mathbb {R}$ bdivisor $\mathbf {M}$ on X
such that $K_{X}+\Delta +\mathbf {M}_{X}$ is $\mathbb {R}$ Cartier. When $\mathbf {M}=0$ , the generalized pair $(X,\Delta ,\mathbf {M})/Z$ is a usual pair $(X,\Delta )$ with $X\to Z$ . When Z is a point, we simply denote $(X,\Delta ,\mathbf {M})$ .
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized pair and P a prime divisor over X. Let $f \colon Y \to X$ be a projective birational morphism such that $\mathbf {M}$ descends to Y and P appears as a divisor on Y. Then there is an $\mathbb {R}$ divisor $\Gamma $ on Y such that
Then the generalized log discrepancy $a(P,X,\Delta +\mathbf {M}_{X})$ of P with respect to $(X,\Delta ,\mathbf {M})/Z$ is defined to be $1\mathrm { coeff}_{P}(\Gamma )$ . When $\mathbf {M}=0$ , the generalized log discrepancy $a(P,X,\Delta )$ coincides with the log discrepancy of P with respect to the usual pair $(X,\Delta )$ .
A generalized pair $(X,\Delta ,\mathbf {M})/Z$ is called a generalized klt (resp. generalized lc) pair if $a(P,X,\Delta +\mathbf {M}_{X})>0$ (resp. $a(P,X,\Delta +\mathbf {M}_{X})\geq 0$ ) for all prime divisors P over X. A generalized lc center of $(X,\Delta ,\mathbf {M})/Z$ is the image on X of a prime divisor P over X satisfying $a(P,X,\Delta +\mathbf {M}_{X})=0$ .
A generalized pair $(X,\Delta ,\mathbf {M})/Z$ is a generalized dlt pair if it is generalized lc and for any generic point $\eta $ of any generalized lc center of $(X,\Delta ,\mathbf {M})/Z$ , $(X,\Delta )$ is log smooth near $\eta $ and $\mathbf {M}$ descends to X on a neighborhood of $\eta $ .
Lemma 2.7. Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized dlt pair such that Z is quasiprojective, and let S be a component of $\lfloor \Delta \rfloor $ . Then there is a plt pair $(X,B)$ such that $S= \lfloor B \rfloor $ . In particular, S is normal, and $(X,B')$ is a klt pair for some $B'$ .
Proof. In this proof, we will use the notion of generalized sublc pair, whose definition is the same as that of generalized lc pairs except that the boundary part is not necessarily effective. This is an analogous definition of sublc pairs.
First, we reduce the lemma to the case, where $\lfloor \Delta \rfloor =S$ . We can take an open subset $U\subset X$ such that $(U,\Delta _{U})$ is log smooth, U contains the generic points of all generalized lc centers of $(X,\Delta ,\mathbf {M})/Z$ and $\mathbf {M}_{U}$ descends to U. We take a log resolution $f \colon Y \to X$ of $(X,\Delta )$ such that

○ f is an isomorphism over U, and

○ $E$ is fample for some effective fexceptional divisor E on Y.
We note that we do not assume $\mathbf {M}_{Y}$ to be nef over Z. We may write
Since U contains the generic points of all generalized lc centers of $(X,\Delta ,\mathbf {M})/Z$ , there is a real number $t>0$ such that $(Y, \Gamma +tE,\mathbf {M})/Z$ is generalized sublc. Take an ample divisor A on X such that $\frac {t}{2}E+f^{*}A$ is ample. By perturbing the coefficients of $\Gamma $ with $\frac {t}{2}E+f^{*}A$ and pushing down, we get an $\mathbb {R}$ divisor $G \geq 0$ on X such that

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

○ $(U,G_{U})$ is log smooth,

○ $\lfloor G \rfloor =S$ and

○ writing $K_{Y}+G_{Y}+\mathbf {M}_{Y}=f^{*}(K_{X}+G+\mathbf {M}_{X})$ , then $(Y, G_{Y}, \mathbf {M})/Z$ is a generalized sublc pair whose generalized lc centers intersect $f^{1}(U)$ .
In this way, we get a generalized dlt pair $(X, G, \mathbf {M})/Z$ such that $\lfloor G \rfloor =S$ .
In the case where $\lfloor \Delta \rfloor =S$ , we see that $(U,\Delta _{U})$ is plt. Hence, $a(P,X,\Delta +\mathbf {M}_{X})>0$ for all prime divisors P over X except $P=S$ . Let $g \colon Y' \to X$ be a log resolution of $(X,\Delta )$ such that

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

○ $F'$ is gample for some gexceptional divisor $F'$ .
Let H be an ample divisor on X such that $F'+g^{*}H$ is ample. Since $\mathbf {M}_{Y'}$ is nef over Z and $a(F_{i},X,\Delta +\mathbf {M}_{X})>0$ for every component $F_{i}$ of $F'$ , we can apply the argument of perturbation of coefficients. We get a plt pair $(X,B)$ such that $\lfloor B \rfloor =S$ .
We introduce properties of divisorial adjunction for generalized pairs.
Remark 2.8 (Divisorial adjunction)
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair, let S be a component of $\lfloor \Delta \rfloor $ with the normalization $S^{\nu }$ and let $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})/Z$ be a generalized lc pair defined with divisorial adjunction for generalized pairs.

○ If $p\mathbf {M}$ is bCartier, then $p \mathbf {N}$ is bCartier.

○ We fix a DCC set $\Lambda \subset \mathbb {R}_{\geq 0}$ . By [Reference Birkar and Zhang7, Proposition 4.9], there is a DCC set $\Omega \subset \mathbb {R}_{\geq 0}$ , depending only on $\Lambda $ , such that if the coefficients of $\Delta $ belong to $\Lambda $ and $\mathbf {M}$ is the sum of finitely many bnef $/Z$ bCartier bdivisors with the coefficients in $\Lambda $ , then the coefficients of $\Delta _{S^{\nu }}$ belong to $\Omega $ .

○ We fix a finite set of rational numbers $\mathfrak {R}\subset [0,1]$ and a positive integer p. By [Reference Birkar4, Lemma 3.3], there is a finite set of rational numbers $\mathfrak {S}\subset [0,1]$ , depending only on $\mathfrak {R}$ and p, such that if the coefficients of $\Delta $ belong to $\Phi (\mathfrak {R})$ and $p \mathbf {M}$ is bCartier, then the coefficients of $\Delta _{S^{\nu }}$ belong to $\Phi (\mathfrak {S})$ .
Definition 2.9 (Restriction morphism)
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair, and let S be a component of $\lfloor \Delta \rfloor $ with the normalization $S^{\nu }$ . With divisorial adjunction for generalized pairs we define a generalized lc pair $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})/Z$ . Suppose that $p \mathbf {M}$ is bCartier. Let $\tau \colon S^{\nu }\to S \hookrightarrow X$ be the natural morphism. With the idea of [Reference Birkar4, 3.1], we will define a morphism
We take a log resolution $f\colon Y \to X$ of $(X,\Delta )$ such that $\mathbf {M}$ descends to Y, and we put $T = f^{1}_{*}S$ . Note that $p \mathbf {M}_{Y}$ is Cartier since $p \mathbf {M}$ is bCartier. Hence, we may replace $p \mathbf {M}_{Y}$ by a linear equivalence class so that $\mathrm {Supp}\mathbf {M}_{Y}$ does not contain T. We can write
and we have
Thus, we obtain a natural morphism
where $f_{T}\colon T \to S^{\nu }$ and $\tau \colon S^{\nu } \to S \to X$ are the natural morphisms. With the relation
we have
By construction of divisorial adjunction for generalized pairs, we have
as $\mathbb {R}$ divisors. Since $p \mathbf {M}_{Y}_{T}$ is a Weil divisor, we see that
From this fact, we see that
By equations (*), (**) and (***), we can define the desired morphism.
Definition 2.10 (Models, cf. [Reference Han and Li25])
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair, and let $(X',\Delta ',\mathbf {M}')/Z$ be a generalized pair. Let $\phi \colon X\dashrightarrow X'$ be a birational map over Z.
We say that $(X',\Delta ',\mathbf {M}')/Z$ is a generalized log birational model of $(X,\Delta ,\mathbf {M})/Z$ if $\mathbf {M}=\mathbf {M}'$ by $\phi $ and $\Delta '=\phi _{*}\Delta +\sum _{i}E_{i}$ , where $E_{i}$ runs over $\phi ^{1}$ exceptional prime divisors.
We say that a generalized log birational model $(X',\Delta ',\mathbf {M}')/Z$ of $(X,\Delta ,\mathbf {M})/Z$ is a weak generalized log canonical model (weak generalized lc model, for short) over Z if

○ $K_{X'}+\Delta '+\mathbf {M}^{\prime }_{X'}$ is nef over Z, and

○ $a(D, X, \Delta +\mathbf {M}_{X}) \leq a(D, X', \Delta '+\mathbf {M}^{\prime }_{X'})$ for every prime divisor D on X which is exceptional over $X'$ .
We say that a weak generalized lc model $(X',\Delta ',\mathbf {M}')/Z$ of $(X,\Delta ,\mathbf {M})/Z$ is a minimal model over Z if the inequality on generalized log discrepancies is strict.
We say that a minimal model $(X',\Delta ',\mathbf {M}')/Z$ of $(X,\Delta ,\mathbf {M})/Z$ is a good minimal model over Z if $K_{X'}+\Delta '+\mathbf {M}^{\prime }_{X'}$ is semiample over Z.
2.2 Abundant divisors
We define the invariant Iitaka dimension and the numerical dimension ([Reference Nakayama40]) for $\mathbb {R}$ Cartier divisors on normal projective varieties, then we define abundant divisors, log abundant divisors and log big divisors.
Definition 2.11 (Invariant Iitaka dimension)
Let X be a normal projective variety, and let D be an $\mathbb {R}$ Cartier divisor on X. We define the invariant Iitaka dimension of D, denoted by $\kappa _{\iota }(X,D)$ , as follows ([Reference Choi9, Definition 2.2.1]; see also [Reference Fujino17, Definition 2.5.5]): If there is an $\mathbb {R}$ divisor $E\geq 0$ such that $D\sim _{\mathbb {R}}E$ , set $\kappa _{\iota }(X,D)=\kappa (X,E)$ . Here, the righthand side is the usual Iitaka dimension of E. Otherwise, we set $\kappa _{\iota }(X,D)=\infty $ .
Let $X\to Z$ be a projective morphism from a normal variety to a variety and D an $\mathbb {R}$ Cartier divisor on X. Then the relative invariant Iitaka dimension of D, denoted by $\kappa _{\iota }(X/Z,D)$ , is similarly defined: If there is an $\mathbb {R}$ divisor $E\geq 0$ such that $D\sim _{\mathbb {R},Z}E$ , then we set $\kappa _{\iota }(X/Z,D)=\kappa _{\iota }(F,D_{F})$ , where F is a sufficiently general fiber of the Stein factorization of $X\to Z$ , and otherwise we set $\kappa _{\iota }(X/Z,D)=\infty $ .
Definition 2.12 (Numerical dimension)
Let X be a normal projective variety, and let D be an $\mathbb {R}$ Cartier divisor on X. We define the numerical dimension of D, denoted by $\kappa _{\sigma }(X,D)$ , as follows ([Reference Nakayama40, V, 2.5 Definition]): For any Cartier divisor A on X, we set
if $\mathrm {dim}H^{0}(X,\mathcal {O}_{X}(\lfloor mD \rfloor +A))>0$ for infinitely many $m\in \mathbb {Z}_{>0}$ , and otherwise we set $\sigma (D;A):=\infty $ . Then, we define
Let $X\to Z$ be a projective morphism from a normal variety to a variety, and let D be an $\mathbb {R}$ Cartier divisor on X. Then, the relative numerical dimension of D over Z is defined by $\kappa _{\sigma }(F,D_{F})$ , where F is a sufficiently general fiber of the Stein factorization of $X\to Z$ . Then $\kappa _{\sigma }(F,D_{F})$ does not depend on the choice of F, so the relative numerical dimension is welldefined. In this paper, we denote $\kappa _{\sigma }(F,D_{F})$ by $\kappa _{\sigma }(X/Z,D)$ .
We refer to [Reference Hashizume31, Remark 2.14] (see also [Reference Nakayama40, V, 2.7 Proposition], [Reference Hashizume and Hu33, Remark 2.8]) for basic properties of the invariant Iitaka dimension and the numerical dimension.
Definition 2.13 (Abundant divisor, log abundant divisor, log big divisor)
Let $X \to Z$ be a projective morphism from a normal variety to a variety, and let D be an $\mathbb {R}$ Cartier divisor on X. We say that D is abundant over Z if $\kappa _{\iota }(X/Z,D)=\kappa _{\sigma }(X/Z,D)$ .
Let $X\to Z$ and D be as above, and let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair. We say that D is log abundant over Z with respect to $(X,\Delta ,\mathbf {M})/Z$ if D is abundant over Z and for any generalized lc center S of $(X,\Delta ,\mathbf {M})/Z$ with the normalization $S^{\nu }\to S$ , the pullback $D_{S^{\nu }}$ is abundant over Z.
We say that D is log big over Z with respect to $(X,\Delta ,\mathbf {M})/Z$ if D is big over Z and for any generalized lc center S of $(X,\Delta ,\mathbf {M})/Z$ with the normalization $S^{\nu }\to S$ , the pullback $D_{S^{\nu }}$ is big over Z.
When Z is a point in the above definitions, we remove ‘over Z’ in each terminology.
Lemma 2.14 (cf. [Reference Hashizume and Hu33, Lemma 2.11], [Reference Nakayama40], [Reference Fujino18])
Let $(X,\Delta ,\mathbf {M})$ be a generalized lc pair such that $K_{X}+\Delta +\mathbf {M}_{X}$ is abundant and there is an effective $\mathbb {R}$ divisor D on X such that $D\sim _{\mathbb {R}}K_{X}+\Delta +\mathbf {M}_{X}$ . Let $X\dashrightarrow V$ be the Iitaka fibration associated to D. We take a log resolution $f\colon Y\to X$ of $(X,\Delta )$ such that $\mathbf {M}$ descends to Y and the induced map $Y\dashrightarrow V$ is a morphism. Let $(Y,\Gamma ,\mathbf {M})$ be a generalized lc pair such that
for some effective fexceptional $\mathbb {R}$ divisor E. Then $\kappa _{\sigma }(Y/V,K_{Y}+\Gamma +\mathbf {M}_{Y})=0$ .
Proof. The argument in [Reference Hashizume and Hu33, Proof of Lemma 2.11] works with no changes because we may apply the discussion in [Reference Fujino18, Section 3].
Lemma 2.15. Let $(X,\Delta )$ be a projective lc pair and $M=\sum _{i}\mu _{i}M_{i}$ a finite $\mathbb {R}_{>0}$ linear combination of nef Cartier divisors $M_{i}$ on X such that M is log big with respect to $(X,\Delta )$ and $\mu _{i}> 2\cdot \mathrm {dim}X$ for all i. Then $K_{X}+\Delta +M$ is log big with respect to $(X,\Delta )$ .
Proof. We need to prove that if S is X or an lc center of $(X,\Delta )$ with the normalization $S^{\nu }$ , then $(K_{X}+\Delta +M)_{S^{\nu }}$ is big. Suppose by contradiction that there is an lc center S of $(X,\Delta )$ or $S=X$ such that $(K_{X}+\Delta +M)_{S^{\nu }}$ is not big. By replacing M with $(1t)M$ for some $0<t \ll 1$ , we may assume that $(K_{X}+\Delta +M)_{S^{\nu }}$ is not pseudoeffective.
Taking a dlt blowup of $(X,\Delta )$ and applying Ambro’s canonical bundle formula as in [Reference Fujino and Gongyo19, Corollary 3.2], we get a generalized lc pair $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})$ such that
Let $(Y,\Gamma , \mathbf {N})$ be a $\mathbb {Q}$ factorial generalized dlt model of $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})$ , and let $M_{Y}$ be the pullback of $M_{S^{\nu }}$ to Y. Then $M_{Y}$ is big and $K_{Y}+\Gamma +\mathbf {N}_{Y}+M_{Y}$ is not pseudoeffective.
By running a $(K_{Y}+\Gamma +\mathbf {N}_{Y}+M_{Y})$ MMP with scaling of an ample divisor, we get a birational contraction $\phi \colon Y \dashrightarrow Y'$ to a normal projective variety $Y'$ which has the structure of a Mori fiber space $Y' \to V$ with respect to $\phi _{*}(K_{Y}+\Gamma +\mathbf {N}_{Y}+M_{Y})$ . By the length of extremal rays (cf. [Reference Han and Li25, Proposition 3.17]), the birational transform of $M_{Y}$ is numerically trivial with respect to the extremal contraction in each step of the MMP. Thus, $\phi _{*}M_{Y}$ is big and $\phi _{*}M_{Y} \sim _{\mathbb {R}, V}0$ , a contradiction because $\mathrm {dim}Y'>\mathrm {dim}V$ .
In this way, we see that $(K_{X}+\Delta +M)_{S^{\nu }}$ is big, and therefore $K_{X}+\Delta +M$ is log big with respect to $(X,\Delta )$ .
The following lemma plays a crucial role in Section 4 and Section 5.
Lemma 2.16. Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair with a morphism $\pi \colon X\to Z$ such that

○ $\Delta $ is a $\mathbb {Q}$ divisor and $\mathbf {M}$ is a $\mathbb {Q}$ bCartier $\mathbb {Q}$ bdivisor,

○ $K_{X}+\Delta +\mathbf {M}_{X}$ is nef over Z and

○ there is a log resolution $f\colon Y \to X$ of $(X,\Delta )$ such that

– $\mathbf {M}$ descends to Y, and

– writing $K_{Y}+\Gamma +\mathbf {M}_{Y}=f^{*}(K_{X}+\Delta +\mathbf {M}_{X})+E$ , where $\Gamma \geq 0$ and $E \geq 0$ have no common components, then $\mathbf {M}_{Y}$ is log big over Z with respect to $(Y,\Gamma )$ .

Let S be a component of $\lfloor \Delta \rfloor $ , let $S^{\nu }$ be the normalization of S and let $(S^{\nu },\Delta _{S^{\nu }},\mathbf {N})/Z$ be a generalized lc pair with a morphism $\pi _{S^{\nu }}\colon S^{\nu }\to Z$ defined with divisorial adjunction for $(X,\Delta ,\mathbf {M})/Z$ and S. Let p be a positive integer such that $p\Delta $ is a Weil divisor and $p\mathbf {M}_{Y}$ is Cartier. Then, the morphism
induced by Definition 2.9 is surjective for every positive integer l.
Proof. The idea is very similar to [Reference Fujino12]. We fix l.
We put
Since $K_{X}+\Delta +\mathbf {M}_{X}$ is nef over Z and $\mathbf {M}_{Y}$ is nef and log big over Z with respect to $(Y,\Gamma )$ , it follows that L is nef and log big over Z with respect to $(Y,\Gamma )$ . We have
Since the coefficients of $p\Delta $ are integers and $p\mathbf {M}_{Y}$ is Cartier, we can write
with an effective fexceptional $\mathbb {Q}$ divisor $E'$ . Put $T=f^{1}_{*}S$ . By restricting sides of equation (2) to T and taking the round down, we have
We define $\Gamma ':=\Gamma T.$ By the definition, $(Y,T+\Gamma ')$ is a log smooth lc pair and L is nef and log big over Z with respect to $(Y,T+\Gamma ')$ . The relations (1) and (2) show
We define a Weil divisor F on Y so that
for every prime divisor P. We put
By the definition, we can check that B is a boundary divisor, $\lfloor B \rfloor \leq \lfloor \Gamma ' \rfloor $ , and G is an effective fexceptional Weil divisor. Since $\lfloor B \rfloor \leq \lfloor \Gamma ' \rfloor $ , the divisor L is nef and log big over Z with respect to $(Y,T+B)$ . Furthermore, the relation (4) implies
We put $D=lp f^{*}(K_{X}+\Delta +\mathbf {M}_{X})$ . Since G is fexceptional, we have
By construction, $(Y,B)$ is a log smooth lc pair and L is nef and log big over Z with respect to $(Y,B)$ . Thus, the Kodaira type vanishing theorem [Reference Fujino12, Lemma 1.5] implies
From these facts, the morphism
is surjective, where $f_{T}\colon T \to S^{\nu }$ is the natural morphism. Since $\lfloor D \rfloor _{T}=\lfloor D_{T} \rfloor $ by equation (3), a natural morphism
is defined. From these facts, we obtain the diagram
and therefore the morphism
is surjective.
3 Minimal model theory for generalized pairs
The goal of this section is to prove Theorem 1.1. All the arguments in this section are very similar to those in [Reference Hashizume and Hu33], [Reference Hashizume29], [Reference Hashizume31] and [Reference Hashizume32].
The following results will be used without any mention.
Lemma 3.1. Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair and $(X',\Delta ',\mathbf {M})/Z$ a generalized lc pair with a birational morphism $X' \to X$ such that $(X',\Delta ',\mathbf {M})/Z$ is a generalized log birational model of $(X,\Delta ,\mathbf {M})/Z$ as in Definition 2.10. Then $(X,\Delta ,\mathbf {M})/Z$ has a minimal model (resp. a good minimal model) if and only if $(X',\Delta ',\mathbf {M})/Z$ has a minimal model (resp. a good minimal model).
Lemma 3.2. Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair and $(X,\Delta ,\mathbf {M})\dashrightarrow (X',\Delta ',\mathbf {M})$ finite steps of a $(K_{X}+\Delta +\mathbf {M}_{X})$ MMP over Z. Then $(X',\Delta ',\mathbf {M})/Z$ is a generalized lc pair, and $(X,\Delta ,\mathbf {M})/Z$ has a minimal model (resp. a good minimal model) if and only if $(X',\Delta ',\mathbf {M})/Z$ has a minimal model (resp. a good minimal model).
Lemma 3.3 (cf. [Reference Han and Li25, Theorem 4.1])
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair such that Z is quasiprojective, $(X,0)$ is a $\mathbb {Q}$ factorial klt pair and $\mathbf {M}$ is a finite $\mathbb {R}_{>0}$ linear combination of bnef $/Z \mathbb {Q}$ bCartier $\mathbb {Q}$ bdivisors. If $(X,\Delta ,\mathbf {M})/Z$ has a minimal model, then any sequence of steps of a $(K_{X}+\Delta +\mathbf {M}_{X})$ MMP over Z with scaling of an ample divisor terminates with a minimal model.
3.1 Auxiliary results
We collect results used in this section. Note that all results in this subsection are known in the case of usual pairs.
Lemma 3.4 (cf. [Reference Hashizume and Hu33, Lemma 2.16])
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair such that Z is quasiprojective and $(X,B)$ is a dlt pair for some B. Let $\mathcal {T}$ be an empty set or a finite set of exceptional prime divisors over X such that $0< a(P,X,\Delta +\mathbf {M}_{X})< 1$ for any $P\in \mathcal {T}$ . Then there is a $\mathbb {Q}$ factorial variety $\widetilde {X}$ and a projective birational morphism $f\colon \widetilde {X} \to X$ such that fexceptional prime divisors are exactly elements of $\mathcal {T}$ .
Proof. Replacing B, we may assume that $(X,B)$ is klt. We pick a real number $0<t\ll 1$ so that the generalized klt pair $\bigl (X,(1t)\Delta +tB,(1t)\mathbf {M}\bigr )/Z$ satisfies
for any $P\in \mathcal {T}$ . Replacing $(X,\Delta ,\mathbf {M})/Z$ with $\bigl (X,(1t)\Delta +tB,(1t)\mathbf {M}\bigr )/Z$ , we may assume that $(X,\Delta ,\mathbf {M})/Z$ is generalized klt. By the perturbation of coefficients, we can find $\Delta '$ such that $(X,\Delta ')$ is klt and $0<a(P,X,\Delta ')<1$ for any $P\in \mathcal {T}$ , where $a(\,\cdot \,,X,\Delta ')$ is the log discrepancy. Then the lemma follows from [Reference Hashizume and Hu33, Lemma 2.16].
Lemma 3.5 (cf. [Reference Hashizume31, Lemma 2.6])
Let $(X,\Delta ,\mathbf {M})$ and $(X',\Delta ',\mathbf {M}')$ be generalized dlt pairs with a birational map $\phi \colon X\dashrightarrow X'$ by which $\mathbf {M}=\mathbf {M}'$ . Let S and $S'$ be generalized lc centers of $(X,\Delta ,\mathbf {M})$ and $(X',\Delta ',\mathbf {M}')$ , respectively, such that $\phi $ is an isomorphism near the generic point of S and the restriction $\phi _{S}$ defines a birational map $\phi _{S}\colon S\dashrightarrow S'$ . Suppose that $K_{X}+\Delta +\mathbf {M}_{X}$ is pseudoeffective. Suppose in addition that

○ $a(D',X',\Delta '+\mathbf {M}^{\prime }_{X'})\leq a(D',X,\Delta +\mathbf {M}_{X})$ for every prime divisor $D'$ on $X'$ , and

○ $\sigma _{P}(K_{X}+\Delta +\mathbf {M}_{X})=0$ for every prime divisor P over X such that $c_{X}(P)\cap S \neq \emptyset $ and $a(P,X,\Delta +\mathbf {M}_{X})< 1$ , where $\sigma _{P}(\;\cdot \;)$ is as in Definition 2.3.
Let $(S,\Delta _{S},\mathbf {N})$ and $(S',\Delta _{S'}, \mathbf {N})$ be generalized dlt pairs which are constructed by applying divisorial adjunctions for generalized pairs to $(X,\Delta ,\mathbf {M})$ and $(X',\Delta ',\mathbf {M}')$ repeatedly. Then
for all prime divisors Q on $S'$ .
Proof. We closely follow [Reference Hashizume31, Proof of Lemma 2.6].
Taking an appropriate common log resolution $f \colon Y \to X$ and $f'\colon Y \to X'$ of the birational map $(X,\Delta )\dashrightarrow (X',\Delta ')$ , we may find a subvariety T of Y which is birational to S and $S'$ such that the induced morphisms $T \to S$ and $T \to S'$ form a common resolution of $\phi _{S}$ . Note that $\mathbf {M}$ does not necessarily descend to Y. We may write
such that $G_{+}\geq 0$ and $G_{}\geq 0$ have no common components. Then $G_{+}$ is $f'$ exceptional by the first condition of Lemma 3.5. Since $K_{X}+\Delta +\mathbf {M}_{X}$ is pseudoeffective, we see that $K_{X'}+\Delta '+\mathbf {M}^{\prime }_{X'}$ is pseudoeffective. We have $\mathrm {Supp}G_{+} \not \supset T$ and $\mathrm {Supp}G_{}\not \supset T$ since S and $S'$ are generalized lc centers of $(X,\Delta ,\mathbf {M})$ and $(X',\Delta ',\mathbf {M}')$ , respectively.
We can write
where all components of $G_{0}$ intersect T and $G_{1}_{T} = 0$ . Pick any component E of $G_{0}$ . Then $\mathrm {ord}_{E}(G_{})=0$ . We have
therefore $\sigma _{P}(K_{X}+\Delta +\mathbf {M}_{X})>0$ . By the second condition of Lemma 3.5, we see that $a(E,X,\Delta +\mathbf {M}_{X})\geq 1$ , hence we have
By the same argument as in [Reference Hashizume31, Proof of Lemma 2.5], we see that $E_{T}$ is exceptional over $S'$ . Therefore, $G_{0}_{T}$ is exceptional over $S'$ . This implies that
for every prime divisor Q on $S'$ . Since we have $G_{1}_{T}=0$ and
the inequality
holds for every prime divisor Q on $S'$ . Therefore, Lemma 3.5 holds.
Lemma 3.6 (cf. [Reference Hacon, McKernan and Xu22, Lemma 5.3], [Reference Hashizume31, Lemma 2.25])
Let $(X,\Delta ,\mathbf {M})$ be a generalized lc pair such that $\mathbf {M}$ is a finite $\mathbb {R}_{>0}$ linear combination of bnef $\mathbb {Q}$ bCartier $\mathbb {Q}$ bdivisors. Let $(Y,\Gamma ,\mathbf {M})$ be a generalized lc pair with a projective birational morphism $f\colon Y\to X$ . Suppose that $K_{X}+\Delta +\mathbf {M}_{X}$ is pseudoeffective and all prime divisors D on Y satisfy
Then $(X,\Delta ,\mathbf {M})$ has a minimal model (resp. a good minimal model) if and only if $(Y,\Gamma ,\mathbf {M})$ has a minimal model (resp. a good minimal model).
Proof. Let $g \colon W \to Y$ be a log resolution of $(Y,\Gamma )$ such that $f\circ g\colon W \to X$ is a log resolution of $(X,\Delta )$ . Let $(W, \Delta _{W},\mathbf {M})$ and $(W, \Gamma _{W},\mathbf {M})$ be generalized log birational models of $(X,\Delta ,\mathbf {M})$ and $(Y,\Gamma ,\mathbf {M})$ as in Definition 2.10, respectively. We may write
for some $E_{W}\geq 0$ which is exceptional over X.
By [Reference Hashizume31, Remark 2.3 (3)], all prime divisors P on W satisfy
If P is not exceptional over X, then we have
If P is exceptional over X but not exceptional over Y, then we have
If P is exceptional over Y, then we have
In any case, by simple computations using the hypothesis about the relations between generalized log discrepancies, we see that
Thus, we see that $(W, \Delta _{W},\mathbf {M})$ and $(W, \Gamma _{W},\mathbf {M})$ satisfy the hypothesis of Lemma 3.6. By Lemma 3.1, we may replace $(X,\Delta ,\mathbf {M})$ and $(Y,\Gamma ,\mathbf {M})$ with $(W, \Delta _{W},\mathbf {M})$ and $(W, \Gamma _{W},\mathbf {M})$ , respectively. By the replacement, we may assume that $X=Y$ and $(X,0)$ is a $\mathbb {Q}$ factorial klt pair.
We may carry out [Reference Hacon, McKernan and Xu22, Proof of Lemma 5.3] in the framework of generalized pairs. It is because we may use the argument of the length of extremal rays [Reference Han and Li25, Proposition 3.17] and the result of the termination of MMP [Reference Han and Li25, Theorem 4.1]. By the same argument as in [Reference Hacon, McKernan and Xu22, Proof of Lemma 5.3], we see that Lemma 3.6 holds.
Lemma 3.7 (cf. [Reference Hashizume29, Lemma 2.15])
Let $(X,\Delta ,\mathbf {M})$ be a generalized lc pair such that $\mathbf {M}$ is a finite $\mathbb {R}_{>0}$ linear combination of bnef $\mathbb {Q}$ bCartier $\mathbb {Q}$ bdivisors. Let $(Y,\Gamma ,\mathbf {M})$ be a generalized lc pair with a projective birational morphism $f\colon Y\to X$ . Suppose that we can write
such that E is effective and fexceptional. Then $(X,\Delta ,\mathbf {M})$ has a minimal model (resp. a good minimal model) if and only if $(Y,\Gamma ,\mathbf {M})$ has a minimal model (resp. a good minimal model).
Proof. Replacing $(Y,\Gamma ,\mathbf {M})$ by a $\mathbb {Q}$ factorial generalized dlt model, we may assume that $(Y,\Gamma ,\mathbf {M})$ is a $\mathbb {Q}$ factorial generalized dlt pair. Running a $(K_{Y}+\Gamma +\mathbf {M}_{Y})$ MMP over X and replacing $(Y,\Gamma ,\mathbf {M})$ by a crepant generalized dlt model of $(X,\Delta ,\mathbf {M})$ , we may assume $E=0$ . Then Lemma 3.7 follows from Lemma 3.6.
Lemma 3.8 (cf. [Reference Hashizume31, Lemma 2.22])
Let $(X,\Delta ,\mathbf {M})/Z$ be a generalized lc pair such that Z is quasiprojective. Let S be a subvariety of X. We denote the morphism $X \to Z$ by $\pi $ . Let
be a sequence of steps of a $(K_{X}+\Delta +\mathbf {M}_{X})$ MMP over Z with scaling of some $\mathbb {R}$ divisor $A\geq 0$ . We define
Suppose that each step of the $(K_{X}+\Delta +\mathbf {M}_{X})$ MMP is an isomorphism on a neighborhood of S and $\mathrm { lim}_{i\to \infty }\lambda _{i}=0$ . Then, for any $\pi $ ample $\mathbb {R}$ divisor H on X and any closed point $x\in S$ , there exists $E\geq 0$ such that $E\sim _{\mathbb {R},Z}K_{X}+\Delta +\mathbf {M}_{X}+H$ and $\mathrm {Supp}E \not \ni x$ . In particular, if Z is a point, then $\sigma _{P}(K_{X}+\Delta +\mathbf {M}_{X})=0$ for every prime divisor P over X such that $c_{X}(P)$ intersects S.
Proof. The argument in [Reference Hashizume31, Proof of Lemma 2.22] works with no changes. Note that [Reference Hashizume31, Proof of Lemma 2.22] does not use any result about the abundance conjecture or the existence of flips for lc pairs.
Lemma 3.9 (cf. [Reference Hashizume31, Lemma 2.26])
Let $(X,\Delta ,\mathbf {M})$ and $(X',\Delta ',\mathbf {M}')$ be generalized lc pairs with a birational map $X\dashrightarrow X'$ such that $\mathbf {M}$ is a finite $\mathbb {R}_{>0}$ linear combination of bnef $\mathbb {Q}$ bCartier $\mathbb {Q}$ bdivisors and $\mathbf {M}=\mathbf {M}'$ by $X\dashrightarrow X'$ . Suppose in addition that

○ $a(P,X,\Delta +\mathbf {M}_{X})\leq a(P,X',\Delta '+\mathbf {M}^{\prime }_{X'})$ for all prime divisors P on X, and

○ $a(P',X',\Delta '+\mathbf {M}^{\prime }_{X'})\leq a(P',X,\Delta +\mathbf {M}_{X})$ for all prime divisors $P'$ on $X'$ .
Then $K_{X}+\Delta +\mathbf {M}_{X}$ is abundant if and only if $K_{X'}+\Delta '+\mathbf {M}^{\prime }_{X'}$ is abundant. Furthermore, $(X,\Delta ,\mathbf {M})$ has a minimal model (resp. a good minimal model) if and only if $(X',\Delta ',\mathbf {M}')$ has a minimal model (resp. a good minimal model).
Proof. We closely follow [Reference Hashizume31, Proof of Lemma 2.26]. Let $f\colon Y\to X$ and $f'\colon Y \to X'$ be a common log resolution of $(X,\Delta )\dashrightarrow (X',\Delta ')$ such that $\mathbf {M}$ descends to Y. We define an $\mathbb {R}$ divisor $\Gamma $ on Y by
where D runs over prime divisors on Y. Then $\Gamma $ is an effective snc $\mathbb {R}$ divisor, $(Y,\Gamma , \mathbf {M})$ is generalized lc and there exist an fexceptional $\mathbb {R}$ divisor $E\geq 0$ and an $f'$ exceptional $\mathbb {R}$ divisor $E'\geq 0$ such that
Lemma 3.10 (cf. [Reference Hashizume29, Proposition 3.3])
Let $(X,\Delta ,\mathbf {M})$ be a generalized lc pair such that $\mathbf {M}$ is a finite $\mathbb {R}_{>0}$ linear combination of bnef $\mathbb {Q}$ bCartier $\mathbb {Q}$ bdivisors. Let $\pi \colon X \to Z$ be a contraction to a normal projective variety Z. Suppose that

○ $\kappa _{\iota }(X/Z,K_{X}+\Delta +\mathbf {M}_{X})=\kappa _{\sigma }(X/Z,K_{X}+\Delta +\mathbf {M}_{X})=0$ ,

○ any generalized lc center of $(X,\Delta ,\mathbf {M})$ dominates Z and

○ $\kappa _{\sigma }(X,K_{X}+\Delta +\mathbf {M}_{X})=\mathrm {dim}Z$ .
Then $(X,\Delta ,\mathbf {M})$ has a good minimal model.
Proof. The argument in [Reference Hashizume29, Proof of Proposition 3.3] works with no changes because we can use the generalized canonical bundle formula [Reference Han and Liu26] instead of the canonical bundle formula [Reference Fujino and Gongyo19]. So we only outline the proof.
By taking a log resolution of $(X,\Delta )$ and applying weak semistable reduction ([Reference Abramovich and Karu2]), we may assume that $(X,0)$ is $\mathbb {Q}$ factorial klt and all fibers of $\pi $ have the same dimensions.
We run a $(K_{X}+\Delta +\mathbf {M}_{X})$ MMP over Z with scaling of an ample divisor. Applying the negativity lemma for very exceptional divisors ([Reference Birkar3, Section 3]) and replacing $(X,\Delta ,\mathbf {M})$ , we may assume $K_{X}+\Delta +\mathbf {M}_{X}\sim _{\mathbb {R},Z}0$ (but we lose the property of being equidimensional of $X \to Z$ ). We used the first condition of Lemma 3.10 for the reduction.
By generalized canonical bundle formula [Reference Han and Liu26, Theorem 1.2] and the second condition of Lemma 3.10, there is a generalized klt pair $(Z,\Delta _{Z},\mathbf {N})$ such that
By the third condition of Lemma 3.10, we see that $K_{Z}+\Delta _{Z}+\mathbf {N}_{Z}$ is big, so $(Z,\Delta _{Z},\mathbf {N})$ has a good minimal model $(Z', \Delta _{Z'}, \mathbf {N})$ by [Reference Birkar, Cascini, Hacon and McKernan6]. Since $(Z,\Delta _{Z},\mathbf {N})$ is generalized klt, the birational map $Z \dashrightarrow Z'$ is a birational contraction. Hence, we can find an open subset $U' \subset Z'$ such that and $Z'\dashrightarrow Z$ is an isomorphism on $U'$ .
Let $f\colon Y \to X$ be a log resolution of $(X,\Delta )$ such that $\mathbf {M}$ descends to Y and the map $\pi _{Y} \colon Y\dashrightarrow Z'$ is a morphism. Let $(Y,\Gamma ,\mathbf {M})$ be a generalized log birational model of $(X,\Delta ,\mathbf {M})$ as in Definition 2.10. Then
for some $E \geq 0$ and $F\geq 0$ such that E is exceptional over X and . Then $(Y,\Gamma ,\mathbf {M})$ has a good minimal model over $U'$ .
Running a $(K_{Y}+\Gamma +\mathbf {M}_{Y})$ MMP over $Z'$ with scaling of an ample divisor, we get a birational contraction $\phi \colon Y \dashrightarrow Y'$ over $Z'$ with the morphism $\pi ' \colon Y' \to Z'$ such that
is the limit of movable divisors over $Z'$ and $\phi _{*}E_{\pi ^{\prime 1}(U')}=0$ . Since , applying the negativity lemma for very exceptional divisors ([Reference Birkar3, Section 3]) to $\phi _{*}E+\phi _{*}F$ , we have $\phi _{*}E+\phi _{*}F=0$ . Thus, we have
and the righthand side is semiample. From the fact, we see that $(Y,\Gamma ,\mathbf {M})$ has a good minimal model. So $(X,\Delta ,\mathbf {M})$ has a good minimal model.
3.2 Generalized abundance
In this subsection, we study the property of being log abundant for special generalized lc pairs under MMP.
Theorem 3.11. Let $(X,\Delta )$ be a projective lc pair, and let $\pi \colon X\to Z$ be a projective surjective morphism to a normal projective variety Z. Suppose that there is an effective $\mathbb {R}$ Cartier divisor C on X such that

○ the pair $(X,\Delta +tC)$ is lc for some $t>0$ , and

○ $K_{X}+\Delta +C \sim _{\mathbb {R},Z}0$ .
Let $A_{Z}$ be a big and semiample $\mathbb {R}$ divisor on Z such that the pullback of $A_{Z}$ to $\pi (S)^{\nu }$ is big for any lc center S of $(X,\Delta )$ , where $\pi (S)^{\nu }$ is the normalization of $\pi (S)$ . Then, $K_{X}+\Delta +\pi ^{*}A_{Z}$ is abundant.
We will use Lemma 3.12 below to prove Theorem 3.11.
Lemma 3.12. Assume Theorem 3.11 for all projective lc pairs of dimension at most $n1$ . Let $(X,\Delta )$ be a projective lc pair, let $\pi \colon X\to Z$ be a morphism and let C and $A_{Z}$ be $\mathbb {R}$ Cartier divisors as in Theorem 3.11 such that $\mathrm {dim}X\leq n 1$ . Let $f\colon Y \to X$ be a log resolution of $(X,\Delta )$ , and let $\Gamma \geq 0$ be an $\mathbb {R}$ divisor on Y such that $(Y,\Gamma )$ is an lc pair and the effective part of the divisor $K_{Y}+\Gamma f^{*}(K_{X}+\Delta )$ is fexceptional. Then, $K_{Y}+\Gamma +f^{*}\pi ^{*}A_{Z}$ is abundant.
Proof. The argument in [Reference Hashizume and Hu33, Proof of Lemma 5.2] or [Reference Hashizume32, Proof of Lemma 3.6] works with no changes.
Proof of Theorem 3.11
The argument of [Reference Hashizume and Hu33, Proof of Theorem 5.4] works with no changes. We only outline the proof. We prove Theorem 3.11 by induction on $\mathrm {dim}X$ .
We put $A= \pi ^{*}A_{Z}$ . We may assume that $K_{X}+\Delta +A$ is pseudoeffective and