We work over the field of complex numbers C. The goal of this paper is to prove the existence of the minimal model program (MMP) for lc algebraically integrable foliations on varieties with mild singularities. One of our main theorems is the following.
Theorem 1.1. Let X be a smooth projective variety and F an algebraically integrable foliation with at worst log canonical singularities on X. Then the cone theorem, the contraction theorem, and the existence of flips hold for K F and thus we can run a K F -MMP.
History on MMP for foliations. The minimal model program for foliations has been extensively studied in the past several years not only due to its importance on the characterization of the ambient variety and its tangent bundle, but also due to its close connection with major conjectures of the classical minimal model program. For example, foliations have played a crucial role in the proof of several key cases of the abundance conjecture for threefolds (cf. [Miy87]).
To prove the existence of the minimal model program, we need at least three ingredients: the cone theorem, the contraction theorem, and the existence of flips. We can only run MMP when all of them are known. For foliations in low dimensions, these three ingredients have been established for surfaces [McQ08,Bru15] and threefolds [CS20,Spi20,CS21,SS22]. Although it is difficult to achieve any of these results in dimension ≥ 4, there are some developments on the minimal model program for foliations induced by dominant rational maps recently, i.e. algebraically integrable foliations. For example, for algebraically integrable foliations, [ACSS21] proved the cone theorem, while [CHLX23] proved the contraction theorem and the existence of flips when the foliation has at worst Q-factorial foliated dlt singularities. These two results together imply the existence of the minimal model program for algebraically integrable foliations with at worst Q-factorial foliated dlt singularities.
MMP for lc foliations on klt varieties. It is known that we can run minimal model program for algebraically integrable foliations with at worst Q-factorial foliated dlt singularities. Q-factorial foliated dlt singularities are usually considered as a foliated version of Q-factorial dlt singularities for usual pairs [CS21,CHLX23], and foliated log smooth singularities are always Q-factorial foliated dlt.
However, Cascini and Spicer [CS23a] pointed out that it is necessary to consider the minimal model program for foliations with singularities which are worse than Q-factorial foliated dlt. One major motivation is that Fano foliations (i.e. foliations with ample anti-canonical divisor -K F ) are never foliated dlt (cf. [AD13, Theorem 5.1]), and they form an important topic in the theory of foliations. This makes [CHLX23] not applicable to Fano foliations.
To resolve this issue, we should consider the minimal model program for algebraically integrable foliations with at worst log canonical (lc) singularities on klt varieties, which is natural and necessary. In this paper, we prove the existence of the minimal model program under this setting.
Theorem 1.2. Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Let R be a (K F + B)-negative extremal ray/U . Then:
(1) (Contraction theorem) There exists a contraction/U cont R : X → T of R.
(2) (Existence of flips) If cont R is a flipping contraction, then the flip/U X + → T associated to R exists.
Theorem 1.2 is known by [CS23a,Theorem 3.2] under the following additional assumptions:
• The termination of Q-factorial klt flips in dimension r = rank F.
• B = ∆ with rational coefficients.
• X is projective, and Q-factorial for statement (2). Theorem 1.2 implies that we can run minimal model program for algebraically integrable foliations with at worst lc singularities on klt varieties. In fact, under the assumptions of Theorem 1.2, we can show that (X, ∆) remains klt after each step of the minimal model program. Therefore, with the help of the (relative) cone theorem [ACSS21,CHLX23], we prove the following result on the existence of minimal model program:
Theorem 1.3 (Existence of minimal model program). Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Then we may run a (K F + B)-MMP/U . Moreover, for any birational map φ : X X + that is a sequence of steps of a (K F + B)-MMP/U , (X + , ∆ + := φ * ∆) is klt.
We remark that when X is Q-factorial, which is the most natural setting when we run MMP, the condition that (X, ∆) is klt for some B ≥ ∆ ≥ 0 is equivalent to the condition that X is klt. In particular, Theorem 1.1 is a direct consequence of Theorems 1.2 and 1.3.
We also remark that Theorem 1.3 is known when (X, F, B)/U is Q-factorial foliated dlt by [CHLX23,Theorem 2.1.1]. Although [CHLX23,Theorem 2.1.1] does not require X to be klt, X is automatically klt by [CHLX23, Theorem 2.1.9]. Therefore, [CHLX23, Theorem 2.1.1] can be viewed as a special case of Theorem 1.3.
Base-point-freeness theorem, good minimal models, and Mori fiber spaces. After the establishment of the cone theorem, the contraction theorem, and the existence of flips, our next goal is to show the existence of good minimal models or Mori fiber spaces. First, we prove the existence of Mori fiber spaces for lc algebraically integrable foliations on klt varieties. Theorem 1.4 (Existence of Mori fiber spaces). Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Assume that K F + B is not pseudoeffective/U .
Then we may run a (K F + B)-MMP/U with scaling of an ample/U R-divisor and any such MMP terminates with a Mori fiber space of (X, F, B)/U .
Next, we deal with the existence of good minimal models. Unfortunately, since we do not know the existence of minimal models for smooth projective varieties in dimension ≥ 5, we cannot prove the existence of minimal models for lc algebraically integrable foliations on klt varieties unconditionally. Nevertheless, we can prove the existence of good minimal models when the boundary divisor contains an ample R-divisor, or when the numerical dimension is 0. Theorem 1.5 (Existence of good minimal models with polarizations). Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Let A be an ample/U R-divisor on X such that K F + B + A is pseudo-effective/U . Then:
(1) We may run a (K F + B + A)-MMP/U with scaling of an ample/U R-divisor and any such MMP terminates with a minimal model of (X, F, B + A)/U . (2) The minimal model in (1) is a good minimal model.
We remark that similar statements for threefold foliations in [CS20, CS21, SS22] usually require that (X, F, B + A) is lc as Bertini-type theorems generally fail. In comparison, we do not need (X, F, B + A) to be lc in Theorem 1.5.
An interesting fact is that we use Theorem 1.5 (1) to prove the following base-point-freeness theorem, which in return gives us Theorem 1.5(2): Theorem 1.6 (Base-point-freeness theorem). Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Let A be an ample/U R-divisor on X such that K F + B + A is nef/U . Then:
(1)
) is globally generated/U for any integer n ≫ 0.
In particular, Theorem 1.6 solves [CS23a, Conjecture 4.1] which further assumes that B = ∆ and (X, F, B) is foliated dlt. We remark that [CHLX23, Theorem A] proved [CS23a, Conjecture 4.1] when (X, F, B) is Q-factorial foliated dlt but the non-Q-factorial case is much more difficult to prove.
An immediate consequence of Theorems 1.5 and 1.6 is the finite generation of the log canonical ring for lc polarized algebraically integrable foliations on klt varieties: Theorem 1.7 (Finite generation of the log canonical rings with polarizations). Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Let A be an ample/U R-divisor on X such that B + A is a Q-divisor. Then the log canonical ring R(X, K F + B + A) := ⊕ +∞ m=0 π * O X (⌊m(K F + B + A)⌋) is a finitely generated O U -algebra.
Another case when we have the existence of good minimal models is when X is projective and the numerical dimension of the foliated triple is 0.
Theorem 1.8. Let (X, F, B) be a projective lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Assume that κ σ (K F + B) = 0.
Then we may run a (K F + B)-MMP with scaling of an ample R-divisor and any such MMP terminates with a minimal model (X min , F min , B min ) of (X, F, B) such that K F min + B min ∼ R 0.
Fano foliations and Mori dream spaces. As a consequence of Theorems 1.4 and 1.5, we show that we can run MMP for any R-Cartier R-divisor on any klt projective variety with an lc algebraically integrable Fano foliation structure, and any such MMP terminates with either a good minimal model or a Mori fiber space. It implies that the ambient variety of an lc algebraically integrable Fano foliation is a Mori dream space if the ambient variety is Q-factorial klt.
Theorem 1.9. Let F be an lc algebraically integrable Fano foliation on a klt projective variety X. Let D be an R-Cartier R-divisor on X. Then:
(1) We may run a D-MMP which terminates with either a good minimal model of D or a Mori fiber space of D. (2) X is a Mori dream space if it is Q-factorial. In particular, D is semi-ample if it is nef, and the section ring R(X, D) is a finitely generated C-algebra if D is a Q-divisor.
Theorem 1.9 also holds in the relative setting and for the foliated Fano type case. We refer the readers to Theorem A.11 for the most general version of Theorem 1.9.
Minimal models in the sense of Birkar-Shokurov. We have studied the MMP for lc algebraically integrable foliations on klt varieties in details in Theorems 1.2, 1.3, 1.4 and 1.5. However, it is also interesting and important to consider the MMP for lc algebraically integrable foliations when ambient varieties are not necessarily klt. They appear in birational geometry naturally. For example, for a locally stable family f : (X, B) → Z, the foliation induced by f is lc, but X may not be klt or even lc (the singularities of X can actually be very bad). Recently, MMP for locally stable families has been established in [MZ23] when they study the wall crossing for moduli of stable pairs. Unfortunately, we are unable to prove the contraction theorem or the existence of flips without any assumption on the ambient variety. Therefore, it can be difficult to talk about minimal models or Mori fiber spaces in this setting. One way to resolve this issue is to study minimal models or Mori fiber spaces in the sense of Birkar-Shokurov, i.e. minimal models or Mori fiber spaces which allow extraction of lc places (cf. [Sho96,Bir12]). We refer the readers to Definition 4.3 for details.
In this paper, we prove the following two results. First, an lc foliated triple polarized with an ample divisor always has a minimal model or a Mori fiber space in the sense of Birkar-Shokurov.
Theorem 1.10. Let (X, F, B)/U be an lc algebraically integrable foliated triple and A ≥ 0 an ample/U R-divisor on X. Then (X, F, B + A)/U has either a minimal model or a Mori fiber space in the sense of Birkar-Shokurov.
Second, when the ambient variety is klt, we show that the existence of a minimal model or a Mori fiber space in the sense of Birkar-Shokurov is equivalent to the termination of minimal model program with scaling of ample divisors. While the latter obviously implies the former, to prove the reverse is highly non-trivial, even for usual pairs (cf. [Bir12, Theorem 1.9(3)]).
Theorem 1.11. Let (X, F, B)/U be an lc algebraically integrable foliated triple. Assume that (X, F, B)/U has a minimal model or a Mori fiber space in the sense of Birkar-Shokurov, and X is potentially klt. Let A be an ample/U R-divisor on X. Then:
(1) Any (K F +B)-MMP/U with scaling of A terminates provided that the (K F +B)-MMP/U with scaling of A exists. (2) If there exists a klt pair (X, ∆) such that B ≥ ∆ ≥ 0, then (X, F, B)/U has a minimal model or a Mori fiber space.
A Shokurov-type polytope. Finally, as an important ingredient in the proof of our main theorems, we prove the existence of a Shokurov-type polytope (cf. [BCHM10, Corollary 1.1.5]) for algebraically integrable foliations.
Theorem 1.12. Let (X, F, B := m i=1 v 0 i B i )/Z be an lc algebraically integrable foliated triple such that K F + B is nef/Z. Let v 0 := (v 0 1 , . . . , v 0 m ). Then there exists an open subset U of the rational envelope of v 0 in R m such that (X, F, m i=1 v i B i ) is lc and
Generalized foliated quadruples. Generalized foliated quadruples play an important role in the proofs of the main theorems in many recent works (cf. [LLM23,LMX24,CHLX23]). In this paper, generalized foliated quadruples are also crucially used due to the failure of Bertini-type theorems for foliations. The generalized foliated quadruple version of all our main theorems holds, although some of them require the nef part of the generalized foliated quadruple to be NQC. We refer the readers to Appendix A for details.
For the convenience of the readers, we avoid using generalized foliated quadruples in the statements and proofs of most of our results. We only essentially use this structure in Theorem 7.2 and its proof, and make remarks on why we need this structure in footnotes therein.
Main difficulties in the proof of the main theorems. Roughly speaking, [CHLX23, Theorem A] established the MMP for algebraically integrable foliations that are Q-factorial foliated dlt (which implies that the ambient variety is klt). However, Q-factorial foliated dlt singularities might be too good to hope for in practice. Actually, [CHLX23] showed that (as conjectured in [ACSS21]) such a foliation is induced by an equidimensional morphism f : X → Z (not just a rational map). Moreover, (even though highly non-trivial) there exists an lc pair structure (X, G) such that K F ∼ f K X + G. Thanks to the cone theorem, the global K F -MMP turns out to be over Z, hence is equivalent to a (K X + G)-MMP/Z whose theory is wellestablished. Thus the authors of [CHLX23] were able to deduce many corresponding results.
In general, things become much more complicated if the foliation is only induced by a rational map, in which case we do not have an associated auxiliary pair structure to work with. One natural idea is to consider the "dlt" modification (whose existence is proved in [ACSS21, CHLX23]) g : X ′ → X such that g -1 F is induced by a fibration f ′ : X ′ → Z ′ with some desired properties. As we have explained, it is very promising that corresponding operations on X ′ can be conducted. The key point is how to descend them back to X. This process is subtle because the modification g is not easy to control. For example, if we want to show the semi-ampleness of K F +A with ample A, the ampleness of A will not be preserved under the modification since perturbation is not allowed due to the failure of Bertini-type theorems. Probably, the best we could hope for is that the pullback of A under g stays ample on the leaves of g -1 F which are exactly (the reduction of) the fibers of f ′ : X ′ → Z ′ . In this case, the restrictions of g on the leaves are finite morphisms, but to descend semi-ampleness under finite morphisms is quite subtle when the schemes (leaves of g -1 F on X ′ and F on X) are not normal (actually the properties of our leaves are even worse). Instead, it turns out that we need to apply very deep and complicated MMP techniques (on X ′ ) to prove the desired semi-ampleness results (e.g. base-point-free theorem) on X.
It is also important to notice that Fano foliations are never foliated dlt, and do not even satisfy Property ( * ). Actually if (X, F, B) is a projective lc foliated triple such that -(K F + B) is ample, then F is not induced by a contraction (cf. [AD13, Theorem 5.1]). Moreover, many Fano foliations are lc, and a lot of work contributed to the classification of lc Fano foliations on smooth projective varieties (cf. [AD13,AD16]). Therefore, it is natural to consider the minimal model program for lc algebraically integrable Fano foliations on smooth projective varieties (e.g. Theorem 1.9).
Idea of the proof. The key idea of the proof is the following observation. Given two contractions h : X ′ → X and f : X ′ → Z between normal quasi-projective varieties, there exists a unique normal quasi-projective variety X satisfying the following:
(1) h and f factor through X.
(2) X is "minimal" among all varieties satisfying (1). In other words, if there exists a variety X ′′ such that h and f factor through X ′′ , then the induced contractions X ′′ → X and X ′′ → Z factor through X.
We call X the core model of (h, f ). It is natural and not difficult to observe the existence of such a X. For example, given two contractions X → Z 0 and Z → Z 0 , then the normalization of the main component of X × Z 0 Z, X, is automatically the core model of ( X → X, X → Z).
In many scenarios, X → Z is viewed as a "base change" of X → Z 0 . For general contractions h and f , the core model of (h, f ) can be viewed as an analogue of such a base change but without a base. Pairs of contractions h : X ′ → X and f : X ′ → Z are very common in the study of algebraically integrable foliations. Given an algebraically integrable foliation F on X, there are many cases when h is a Q-factorial ACSS modification of F (or a ( * )-modification of F) and f is a contraction which induces F ′ := h -1 F. It is usually easier to study F ′ due to its connection with an lc pair structure (X ′ , B ′ + G) (cf. [ACSS21, Proposition 3.6]). We will use
The problem is that, when we study the minimal model program for F (e.g. contraction theorem, existence of flips), we usually need to consider F together with a polarization of an ample divisor A on X. However, A ′ := h * A is only big and nef and not ample. Moreover, (X ′ , B ′ + G + A ′ ) is only an lc pair polarized with a big and nef divisor, and we do not know the existence of good minimal models for such pairs yet. It causes troubles for us to use the minimal model program for (X ′ , B ′ + G + A ′ ) to study the minimal model program for K F + A.
Nevertheless, by using the core models we have introduced, we can resolve this problem. Let X be the core model of (h, f ) with h : X → X, f : X → Z, and g : X ′ → X. Then:
Since g is a contraction over Z, ( X, B + Ḡ) is crepant to (X ′ , B ′ + G) by the negativity lemma and thus it is lc. In particular, X not only has an lc pair structure associated to the foliation F := h-1 F (while X does not necessarily have) but also preserves some information of X via the divisor Ā, at least over Z. The core model X, instead of X ′ , seems to be a more natural object for us to study due to its uniqueness with respect to the base. Moreover, it preserves more core information of X comparing to an arbitrary model X ′ . This is why we call it as a "core model". This is also inspired by [MZ23, Proof of Theorem 1.5]. We study the basic properties of core models and its relationship with foliations in Section 3.
In our case, the MMP/Z for ( X, B + Ḡ + Ā) with scaling of ample divisors behaves nicely. Moreover, by our construction, ( X, B + Ḡ) has a close connection with the induced foliation F on X, and we can show that the MMP/Z for K F + B + Ā with scaling of ample divisors behaves nicely. By the cone theorem for algebraically integrable foliations ([ACSS21, Theorem 3.9]), any (K F + B + Ā)-MMP is a (K F + B + Ā)-MMP/Z. Thus any (K F + B + Ā)-MMP with scaling of ample divisors behaves nicely. Then we can use this fact to study the minimal model program for K F + A with scaling of ample divisors, and hence for K F by adopting the some ideas from [CS23a,CHLX23].
More precisely, the key idea of [CS23a, Proof of Theorem 3.2] is the following:
• We take a ( * )-model" 1 of (X, F, B) which satisfies good properties. Achieving this requires us to run the "first MMP" for a foliated log smooth model. • We run the "second MMP" which contracts the strict transforms of the (K F +B)-negative extremal rays. Termination of flips is needed here. • We run the "third MMP" which contracts the strict transform of the exceptional divisor of the ( * )-modification. Termination of flips is again needed here. We follow the same idea to prove Theorems 1.2 and 1.3, but many arguments are different. We do not need to run the "first MMP" and we shall directly use our core model X. We use properties of the core model X to show that the "second MMP" could terminate. Finally, we can use the termination of MMP with scaling for klt generalized pairs of log general type to get the termination of the "third MMP".
There are several additional things to remark for our proof of Theorems 1.2 and 1.3.
(1) We need the minimal model program for generalized foliated quadruples because Bertinitype theorems generally fail for algebraically integrable foliations. We need the concept of generalized foliated quadruples to consider MMP with scaling of ample divisors in more details. We need results on the minimal model program for generalized foliated quadruples in [CHLX23]. (2) When the boundary B has irrational coefficients, we need to establish the existence of a Shokurov-type polytope (Theorem 1.12) for algebraically integrable foliations in order to show that the "third MMP" terminates. This is done in Section 6. Moreover, if we consider the contraction theorem and the existence of flips for (non-NQC) generalized foliated quadruples instead of foliated triples, the Shokurov-type polytope does not exist. We need to resolve this issue by introducing and studying a special class of nef R-divisors, namely "ϵ-nef R-divisors". See Appendix B for details. (3) When X is not Q-factorial, it can be tricky to run MMP on X since X may not be Qfactorial either. In this case, we need to prove some results on the minimal model program on X ′ . Nevertheless, we can still use X as an auxiliary variety to help us establish the MMP on X ′ , hence the "second MMP" on X ′ . See Section 5 for details. Also, when X is not Q-factorial, the arguments in [CS23a] no longer work for the existence of flips,
1 They are called "Property ( * )-models" in [ACSS21] and the arXiv version of [CS23a]. Cascini suggested us that the name "( * )-models" is better.
and we need an alternative argument. Our choice is to consider the ample model of a special R-divisor over the base of the contraction. In this case, some basic properties on different types of models of foliations need to be proved. See Section 4 for details.
Finally, we say a few more words about the proof of other main theorems. First, by using core models, we establish an MMP on X with scaling of the pullback of an ample divisor on X in Section 5, and show that such a MMP terminates in some cases (Theorem 5.6). Theorem 1.10 follows from the establishment of such a MMP. Theorem 1.8 follows from the establishment of such a MMP, the fact that movable divisors with zero numerical Iitaka dimension are numerically trivial, and the abundance for numerically trivial algebraically integrable foliations.
To prove Theorems 1.4 and 1.5, we need to lift MMP to Q-factorial ACSS models. [CS23a, Remark 3.3] briefly mentions such lifting for flips. We discuss the lifting of the MMP in more details in Section 8, which allows divisorial contractions and non-Q-factorial MMP to be lifted. More importantly, we show that MMP with scaling can also be lifted. This, together with the results in Section 5, implies Theorems 1.4 and 1.5. Theorem 1.9 is a direct consequence of Theorems 1.4 and 1.5, since any D-MMP is a (K F + (-K F + ϵD))-MMP for some 0 < ϵ ≪ 1 such that -K F + ϵD is ample. We remark that we need generalized foliated quadruples again to prove these theorems due to the failure of Bertini-type theorems.
Finally, to prove Theorem 1.11, we need to show that the existence of minimal models in the sense of Birkar-Shokurov for (X, F, B) is equivalent to the existence of minimal models of a pair (X ′ , B ′ + G) which is related to (X, F, B). Therefore, we can use the known results on the existence of minimal models for (X ′ , B ′ + G) to deduce Theorem 1.11. Such a relationship is automatic when we have an equidimensional Property ( * ) structure, but it is more complicated when (X, F, B) does not satisfy Property ( * ). The task is done in Section 4. A key observation is to reinterpret an MMP/U which is also an MMP/Z as an MMP/Z U , where Z U is the core model of (X → U, X → Z). The use of the auxiliary variety Z U will greatly help us transform the (K F + B)-MMP into the (K X ′ + B ′ + G)-MMP and lead to the proof of Theorem 1.11. Sketch of the paper. In Section 2 we recall some preliminary results on algebraically integrable foliations and results in [CHLX23]. In Section 3 we introduce the concept of core models and study its basic properties. In Section 4 we study different types of models for algebraically integrable foliations. In Section 5 we use the concept of core models to study the MMP for Q-factorial ACSS foliated triples polarized with the pullback of an ample divisor and prove Theorem 1.10. In Section 6 we construct a Shokurov-type polytope for algebraically integrable foliations and prove Theorem 1.12. In Section 7 we prove Theorems 1.2 and 1.3. In Section 8 we study the lifting of the minimal model program for algebraically integrable foliations to Q-factorial ACSS models. In Section 9 we prove the rest of our main theorems for foliated triples. In Section 10 we propose and discuss some remaining open problems on the minimal model program for algebraically integrable foliations and prove some results that might be useful for future applications. Finally, Appendices A and B focus on generalized foliated quadruple. In Appendix A, we state and prove the generalized foliated quadruple version of our main theorems. In Appendix B, we introduce the concept of ϵ-nefness, which is a replacement of the Shokurov-type polytope for generalized foliated quadruples.
Acknowledgements. The authors would like to thank Caucher Birkar, Paolo Cascini, Guodu Chen, Christopher D. Hacon, Jingjun Han, Junpeng Jiao, Yuchen Liu, Vyacheslav V. Shokurov, Calum Spicer, Roberto Svaldi, Chenyang Xu, Qingyuan Xue, Ziwen Zhu and Ziquan Zhuang for useful discussions. The third author has been partially supported by NSF research grants no. DMS-1801851 and DMS-1952522, as well as a grant from the Simons Foundation (Award Number: 256202).
We will adopt the standard notations and definitions on MMP in [KM98,BCHM10] and use them freely. For foliations and foliated triples, we adopt the notations and definitions in [CHLX23] which generally align with [CS20, ACSS21, CS21] with possible minor differences. Notation 2.2. Let f : X X ′ be a birational map between normal varieties. We denote by Exc(f ) the reduced divisor supported on the codimension one part of the exceptional locus of f . Notation 2.3. Let X be a normal variety and D, D ′ two R-divisors on X. We define D ∧ D ′ := P min{mult P D, mult P D ′ }P where the sum runs through all the prime divisors P on X. We denote by Supp D the reduced divisor supported on D. Definition 2.4. Let m be a positive integer and v ∈ R m . The rational envelope of v is the minimal rational affine subspace of R m which contains v. For example, if m = 2 and
2 , then the rational envelope of v is (
Notation 2.5. A general choice of a real number a is a choice of a real number such that a ̸ ∈ Q(Γ 0 ) for a finite set Γ 0 ⊂ R. Here Q(Γ 0 ) is the field extension of Q by elements in Γ 0 . We also say that a is general in R/Q.
Definition 2.6 (Foliations, cf. [ACSS21,CS21]). Let X be a normal variety. A foliation on X is a coherent sheaf F ⊂ T X such that (1) F is saturated in T X , i.e. T X /F is torsion free, and
(2) F is closed under the Lie bracket. The rank of the foliation F is the rank of F as a sheaf and is denoted by rank F. The co-rank of (F). If F = 0, then we say that F is a foliation by points.
Given any dominant map h : Y X, we denote by h -1 F the pullback of F on Y as constructed in [Dru21, 3.2] and say that h -1 F is induced by F. Given any birational map g : X X ′ , we denote by g * F := (g -1 ) -1 F the pushforward of F on X ′ and also say that g * F is induced by F. We say that F is an algebraically integrable foliation if there exists a dominant map f : X Z such that F = f -1 F Z , where F Z is the foliation by points on Z, and we say that F is induced by f .
A subvariety S ⊂ X is called F-invariant if for any open subset U ⊂ X and any section ∂ ∈ H 0 (U, F), we have ∂(I S∩U ) ⊂ I S∩U , where I S∩U is the ideal sheaf of S ∩ U . For any prime divisor P on X, we denote ϵ F (P ) := 1 if P is not F-invariant and ϵ F (P ) := 0 if P is F-invariant. For any prime divisor E over X, we define ϵ F (E) := ϵ F Y (E) where h : Y X is a birational map such that E is on Y and F Y := h -1 F. For any R-divisor D on X, we denote by D F the reduced divisor supported on the union of non-F-invariant components of D. Definition 2.7 (Tangent, cf. [ACSS21, Section 3.4]). Let X be a normal variety, F a foliation on X, and V ⊂ X a subvariety. Suppose that F is a foliation induced by a dominant rational map X Z. We say that V is tangent to F if there exists a birational morphism µ : X ′ → X, an equidimensional contraction f ′ : X ′ → Z, and a subvariety V ′ ⊂ X ′ , such that (1) µ -1 F is induced by f ′ , and (2) V ′ is contained in a fiber of f ′ and µ(V ′ ) = V .
Definition 2.8 (Foliated triples). A foliated triple (X, F, B)/U consists of a normal quasiprojective variety X, a foliation F on X, an R-divisor B ≥ 0 on X, and a projective morphism X → U , such that K F + B is R-Cartier.
If F = T X , then we may drop F and say that (X, B)/U is a pair. If U is not important, then we may drop U . If F is algebraically integrable, then we say that (X, F, B) is algebraically integrable. If X is Q-factorial, then we say that (X, F, B) is Q-factorial. If we allow B to have negative coefficients, then we shall add the prefix "sub-". If B is a Q-divisor then we may add the prefix "Q-". Definition 2.9 (Singularities). Let (X, F, B) be a foliated triple. For any prime divisor E over X, let f : Y → X be a birational morphism such that E is on Y , and suppose that
where F Y := f -1 F. We define a(E, F, B) := -mult E B Y to be the discrepancy of E with respect to (X, F, B). If F = T X , then we define a(E, X, B) := a(E, F, B) which is the usual discrepancy for pairs.
We say that (X, F, B) is lc (resp. klt) if a(E, F, B) ≥ -1 (resp. > -1) for any prime divisor E over X. For foliated sub-triples, we define singularities in the same way and we shall add the prefix "sub-" for the descriptions of singularities.
An lc place of (X, F, B) is a prime divisor E over X such that a(E, F, B) = -ϵ F (E). An lc center of (X, F, B) is the center of an lc place of (X, F, B) on X.
We remark that our definition of lc and klt singularities has some differences compared with the classical definition [McQ08, CS20, ACSS21, CS21, CHLX23], where the -1 in the inequality is replaced with -ϵ F (E). The next lemma shows that the two definitions on lc coincide so we are free to use results on "lc foliations" in literature. Moreover, there are good reasons why we refine the definition of klt singularities, and we plan to use this new definition in all future works. We refer the readers to Remark 10.1 for details.
Lemma 2.10. Let (X, F, B) be a foliated sub-triple. The following two conditions are equivalent:
(1) (X, F, B) is sub-lc.
(2) a(E, F, B) ≥ -ϵ F (E) for any prime divisor E over X.
Proof. It is clear that (2) implies (1) so we only need to show that (1) implies (2). Suppose the lemma does not hold. Then there exists a prime divisor E over X such that E is F-invariant and a(E, F, B) < 0. Possibly replacing X by a high model, we may assume that E is on X and X is smooth. Thus E is a component of B and mult E B > 0. This contradicts [CS21, Remark 2.3].
Definition 2.11 (Potentially klt). Let X be a normal quasi-projective variety. We say that X is potentially klt if (X, ∆) is klt for some R-divisor ∆ ≥ 0.
Lemma 2.12. Let (X, B)/U be an lc pair such that X is potentially klt and A an ample/U R-divisor. Then there exists a klt pair (X, ∆) such that ∆ ∼ R,U B + A.
Proof. There exist a klt pair (X, ∆ 0 ) and a real number ϵ > 0 sufficiently small such that
+ϵ∆ 0 +H satisfies our requirements. 2.3. Special algebraically integrable foliations. Definition 2.13 (Foliated log resolutions). We refer the readers to [CHLX23, Definition 6.2.1] or [ACSS21, 3.2. Log canonical foliated pairs] for the definition of being foliated log smooth.
Let X be a normal quasi-projective variety, B an R-divisor on X, and F an algebraically integrable foliation on X. A foliated log resolution of (X, F, B) is a birational morphism h :
is foliated log smooth. The existence of foliated log resolutions for any such (X,
F, B) is guaranteed by [CHLX23, Lemma 6.2.4]. Definition 2.14 (Property ( * ) foliations, [ACSS21, Definition 3.8], [CHLX23, Definition 7.2.2]).
Let (X, F, B) be a foliated triple. Let G ≥ 0 be a reduced divisor on X and f : X → Z a contraction. We say that (X, F, B; G)/Z satisfies Property ( * ) if the following conditions hold.
(1) F is induced by f and G is an
(3) For any closed point z ∈ Z and any reduced divisor
) is lc over a neighborhood of z. We say that f , Z, and G are associated with (X, F, B).
be a foliated triple. Let G ≥ 0 be a reduced divisor on X and f : X → Z an equidimensional contraction, such that (X, F, B; G)/Z satisfies Property ( * ) and B is horizontal/Z. Then
Definition 2.16. Let f : X → Z be a projective morphism between normal quasi-projective varieties and G ≥ 0 an R-divisor on X. We say that G is super/Z if there exist ample Cartier divisors H 1 , . . . , H m on Z such that G ≥ m i=1 f * H i , where m := 2 dim X + 1. Definition 2.17 (ACSS, cf. [CHLX23, Definitions 5.4.2, 7.2.2, 7.2.3]). Let (X, F, B) be an lc foliated triple, G ≥ 0 a reduced divisor on X, and f : X → Z a contraction. We say that (X, F, B; G)/Z is ACSS if the following conditions hold:
(1) (X, F, B; G)/Z satisfies Property ( * ).
(2) f is equidimensional.
(3) There exists an R-Cartier R-divisor D ≥ 0 on X, such that Supp{B} ⊂ Supp D, and for any reduced divisor
is qdlt (cf. [dFKX17, Definition 35]). (4) For any lc center of (X, F, B) with generic point η, over a neighborhood of η, (a) η is the generic point of an lc center of (X, F, ⌊B⌋), and
ACSS and G is super/Z, then we say that (X, F, B; G)/Z is super ACSS. If (X, F, B; G)/Z is (super) ACSS, then we say that (X, F, B)/Z and (X, F, B) are (super) ACSS.
Definition 2.18. Let X → U be a projective morphism from a normal quasi-projective variety to a variety. Let D be an R-Cartier R-divisor on X and φ : X X ′ a birational map/U . Then we say that X ′ is a birational model of X. We say that φ is D-non-positive (resp. D-negative, D-trivial, D-non-negative, D-positive) if the following conditions hold:
(1) φ does not extract any divisor.
(
(3) There exists a resolution of indeterminacy p : W → X and q : W → X ′ , such that
Definition 2.19. Let X → U be a projective morphism from a normal quasi-projective variety to a variety. Let D be an R-Cartier R-divisor on X and f : X → Z a contraction/U . We say that f is a D-Mori fiber space/U if f is a contraction of a D-negative extremal ray/U and dim X > dim Z. If f : X → Z is a D-Mori fiber space/U for some R-Cartier R-divisor D, then we say that f : X → Z is a Mori fiber space/U . If f is obviously a contraction/U or the "/U " property is not important, then we may drop the "/U " in the definitions.
Definition 2.20. Let X → U be a projective morphism from a normal quasi-projective variety to a variety. Let D be an R-Cartier R-divisor on X, φ : X X ′ a D-negative map/U and D ′ := φ * D.
(1) We say that
Lemma 2.21. Let X → U be a projective morphism from a normal quasi-projective variety to a variety and F an extremal face in NE(X/U ). Let H 1 , H 2 be two supporting functions/U of F and φ : X
Proof. Let p : W → X and q : W → X ′ be a resolution of indeterminacy. Then there exists a unique extremal face
Lemma 2.22. Let X → U be a projective morphism from a normal quasi-projective variety to a variety. Let A, B be two R-divisors on X and let t be a real number such that t is general in R/Q and A + tB is R-Cartier. Then A, B are R-Cartier, and any (A + tB)-trivial map φ : X X ′ is A-trivial and B-trivial.
Proof. Let A ′ and B ′ be the images of A, B on X ′ respectively. Then A ′ + tB ′ is R-Cartier. By [HLS19, Lemma 5.3], A, B, A ′ , B ′ are R-Cartier. Let p : W → X and q : W → X ′ be a resolution of indeterminacy, then
By [HLS19, Lemma 5.3], p * A = q * A ′ and p * B = q * B ′ . The lemma follows.
2.5. Relative Nakayama-Zariski decompositions. Definition 2.23. Let π : X → U be a projective morphism from a normal variety to a variety, D a pseudo-effective/U R-Cartier R-divisor on X, and P a prime divisor on X. We define
where the sum runs through all prime divisors on X and consider it as a formal sum of divisors with coefficients in [0, +∞) ∪ {+∞}.
Lemma 2.24 ([LX23a, Lemma 3.4(2)(3), Lemma 3.7(4)]). Let π : X → U be a projective morphism from a normal variety to a variety and D a pseudo-effective/U R-Cartier R-divisor on X. Let f : Y → X be a projective birational morphism. Then:
(1) For any exceptional/X R-Cartier R-divisor E ≥ 0 and any prime divisor P on Y , we have
(2) For any exceptional/X R-Cartier R-divisor E ≥ 0 on Y , we have
(3) Supp N σ (X/U, D) coincides with the divisorial part of B -(X/U, D).
Lemma 2.25. Let X → U be a projective morphism from a normal variety to a variety and φ : X X ′ a birational map/U . Let D be an R-Cartier R-divisor on X such that φ is D-negative and D ′ := φ * D. Then:
(1) The divisors contracted by φ are contained in Supp N σ (X/U, D).
(2) If D ′ is movable/U , then Supp N σ (X/U, D) is the set of all φ-exceptional divisors.
Proof. Let p : W → X and q : W → X ′ be a resolution of indeterminacy. Then
for some E ≥ 0 that is exceptional/X ′ and Supp E contains the strict transforms of all φexceptional divisors on W . By Lemma 2.24 (1),
whose components are all φ-exceptional.
(2) follows from (1).
2.6. Generalized pairs and generalized foliated quadruples. 3.4]). We need this notion to discuss the structures induced by MMP in a more accurate manner. For b-divisors and generalized pairs, we will follow the notations and definitions in [BZ16,HL23]. For generalized foliated quadruples, we shall follow [CHLX23].
Generalized pairs and generalized foliated quadruples are very technical concepts. To make the statements in this paper more concise, for most results whose proofs for generalized foliated quadruples are similar to the proofs for foliated triples, we will only prove the foliated triple version and will not prove the generalized foliated quadruple version. We shall state the corresponding generalized foliated quadruple version in Appendix A. We will freely use results in [CHLX23] on generalized foliated quadruples.
We need some results on NQC R-divisors which are related to generalized pairs and generalized foliated quadruples. Definition 2.27 (NQC). Let X → U be a projective morphism from a normal quasi-projective variety to a variety. Let D be a nef R-Cartier R-divisor on X and M a nef b-divisor on X.
We Lemma 4.4(3)]). Let (X, B)/U be a Q-factorial lc pair and L an NQC/U R-divisor on X. Assume that X is klt. Then there exists a positive real number l 0 such that any sequence of steps of a (K X + B + lL)-MMP/U is L-trivial for any l > l 0 .
Lemma 2.29. Let (X, F, B)/U be an lc algebraically integrable foliated triple and
Proof. We write D = c i=1 r i D i , where r 1 , . . . , r c are linearly independent over Q and each D i is a Cartier divisor. We define
Thus possibly shrinking V , we may assume that for any v ∈ V , we have that D(v) • C j > 0 for any j such that D • C j > 0. Since r 1 , . . . , r c are linearly independent over Q, for any j such that D • C j = 0, we have D(v) • C j = 0 for any v ∈ R c . Therefore, D(v) • C j ≥ 0 for any j and any v ∈ V .
By the cone theorem [CHLX23, Theorem 2.3.1], for any curve C on X, we may write
points such that r is in the interior of the convex hull of v 1 , . . . , v c+1 . Then there exist positive real numbers a 1 , . . . , a c+1 such that c+1 i=1 a i = 1 and c+1 i=1 a
Let (X, F, B)/U be an lc algebraically integrable foliated triple and D an Rdivisor on X such that K F + B + D is NQC/U . Then there exists δ 0 ∈ (0, 1) such that for any δ ∈ (0, δ 0 ), any
Let d := dim X. We show that δ 0 := ϵ 2d+ϵ satisfies our requirements. Let R be a (K
which is not possible.
The goal of this section is to introduce two new types of birational models for algebraically integrable foliations, namely core models and simple models. We shall also recall the definition of ACSS models defined in [CHLX23,DLM23].
Roughly speaking, simple models are birational models of algebraically integrable foliations that are weaker than "( * )-models but still have potentially lc pair structures, while core models are unique simple models which satisfy certain universal property. The use of core models is crucial for the proof of our main theorems.
3.1. Core models for two contractions. Definition-Lemma 3.1. Let X ′ , X, Z be normal quasi-projective varieties and h : X ′ → X, f : X ′ → Z contractions. Then there exists a unique normal quasi-projective variety X up to isomorphisms with two contractions h : X → X and f : X → Z satisfying the following.
(1) For any ample R-divisor A on X, h * A is ample/Z.
(2) There exists a contraction g :
The variety X is called the core model of (h, f ) associated with ( h, f ). Moreover, for any dominant map φ : X ′ X ′′ such that K(X ′′ ) is algebraically closed in K(X ′ ) (e.g. φ is a birational map or a contraction), and any contractions h ′′ : X ′′ → X and f ′′ :
Step 1. In this step we construct h, f , g which satisfy (2) such that (1) holds for a fixed ample R-divisor A, and X is unique. Let A be a fixed ample R-divisor on X. Then h * A is semi-ample, hence semi-ample/Z. Let g : X ′ → X be the ample model/Z of h * A. Then there exists an induced contraction f : X → Z. Since the ample model of h * A is h : X ′ → X, we have an induced contraction h : X → X. We denote it by h. By the uniqueness of ample models, X is unique.
Step 2. In this step we show that (1) holds for any ample R-divisor. Suppose that there exists an ample R-divisor H on X such that h * H is not ample/Z. Then by applying Step 1 to H, h, f , there exist contractions g ′ : X → X′ , h′ : X′ → X, and f ′ : X′ → Z such that h′ * H is ample/Z. Since h * H is not ample/Z, g ′ is not an isomoprhism. Since g ′ is a contraction/Z and h * A = g ′ * h′ * A is ample/Z, g ′ is finite and thus an isomorphism, which is a contradiction.
Step 3. In this step we prove the moreover part. There exist a birational morphism p : W → X ′ from a normal quasi-projective variety W and a projective surjective morphism q : W → X ′′ such that q = φ • p. Since K(X ′′ ) is algebraically closed in K(X ′ ), q is a contraction. Let X′′ be the core model of (h ′′ , f ′′ ) associated with ( h′′ , f ′′ ) and g ′′ : X ′′ → X′′ the induced contraction. Since both g • p and g ′′ • q are ample models/Z of (h • p) * A = (h ′′ • q) * A, the ample model/Z of h * A is isomoprhic to the ample model/Z of h ′′ * A. Thus X is the core model of (h ′′ , f ′′ ) associated with ( h, f ).
Core models for algebraically integrable foliations. Definition 3.2 (Simple modifications). Let (X, F, B) and (X ′ , F ′ , B ′ ) be two algebraically integrable foliated triples and h : X ′ → X a birational morphism. Let f : X ′ → Z be a contraction and G a reduced divisor on X ′ . We say that h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) is a simple modification if the following conditions hold.
(1)
) is a simple (resp. ACSS, core) modification, then we also say that h is a simple (resp. ACSS, core) modification of (X, F, B). Definition 3.3 (Core models and ACSS models). Let (X, F, B) be an algebraically integrable foliated triple and h : (X ′ , F ′ , B ′ , G)/Z → (X, F, B) a simple (resp. ACSS, core) modification.
We say that (X ′ , F ′ , B ′ ; G)/Z, (X ′ , F ′ , B ′ )/Z, and (X ′ , F ′ , B ′ ) are simple (resp. ACSS, core) models of (X, F, B). Moreover, we say that h is
We will frequently use the following result on the existence of ACSS models: Lemma 3.5. Let (X, F, B) be an lc algebraically integrable foliated triple and let h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) be a simple model. Let f : X ′ → Z be the associated contraction and let X be the core model of (h, f ) associated with ( h, f ). Let g : X ′ → X be the induced birational morphism, F := g * F ′ , B := g * B ′ , and Ḡ := g * G.
Assume that f is equidimensional. Then:
(1)
Then
Therefore, B′ = B, so
(2) By Proposition 2.15, we have
Since g is a birational morphism/Z,
By applying the negativity lemma twice, we have
(3) Since X is the core model of (h, f ), we only need to show that h : ( X, F , B; Ḡ)/Z → (X, F, B) is a simple model by checking each condition of Definition 3.
(4) It follows from the definitions of being proper or super.
Remark 3.6. [ACSS21, CS23a] define "( * )-models" and [CHLX23] defines "great ACSS models". We do not need these models in this paper. Nevertheless, we provide the readers with the following table on the properties of different types of models. Note that "( * )-models" in [ACSS21] and [CS23a] are defined differently.
Lemma 3.7. Let (X, F, B) be an lc algebraically integrable foliated triple and let h :
Proof. Let (X, ∆) be a klt pair and let
) is sub-klt for any δ ∈ (0, 1). Since G contains all h-exceptional prime divisors that are F ′ -invariant, and since Supp B ′ contains all h-exceptional prime divisors that are not
In particular, X ′ is potentially klt.
The goal of this section is to introduce and study the basic behaviors of different types of models for foliated triples: weak lc models, minimal models, good minimal models, etc. We will also introduce minimal models in the sense of Birkar-Shokurov and log minimal models for foliations. Results in this section are similar to results in [Bir12, Section 2] and [HL23, Section 3] with some differences as we need to take invariant lc centers into consideration. 4.1. Definitions of minimal models and Mori fiber spaces. Remark 4.1. In the classical definition of models, "log minimal model","good minimal model", or "log terminal model" (cf. [BCHM10,Bir12]) usually requires that the model is Q-factorial dlt. This is because the initial structure on which we start running the MMP is usually Qfactorial dlt. For foliations this is replaced by the condition "Q-factorial ACSS" [CHLX23]. However, since the singularities we are going to come up with in this paper is usually worse than Q-factorial ACSS, we have to change these definitions a little bit in order to deal with worse singularities. On the other hand, we still want to consider models of the objects we study that have nice singularities after possibly extracting some lc places. Considering all these issues, we will slightly change the notations in previous literature and define different models in the following way:
• For models requiring good singularities (e.g. Q-factorial ACSS), we always keep the word "log". We always allow extraction of lc centers when considering these models. • For models without these strict singularity conditions (e.g. only requiring lc), we shall not use the word "log". Moreover, if we allow extraction of lc centers, then we shall add the prefix "bs-" or write "in the sense of Birkar-Shokurov".
Definition 4.2 (Log birational model). Let (X, F, B)/U be a foliated triple, φ : X X ′ a birational map over U , E := Exc(φ -1 ) the reduced φ -1 -exceptional divisor, and
Assume that a(D, F, B) ≤ -ϵ F (D) for any component D of E. We let
and say that (X ′ , F ′ , B ′ )/U is a log birational model of (X, F, B)/U , where the sum runs through all components of E. Definition 4.3 (Minimal models). Let (X, F, B)/U be a foliated triple and (X ′ , F ′ , B ′ )/U a log birational model of (X, F, B)/U such that K F ′ + B ′ is nef/U .
(1) We say that (X ′ , F ′ , B ′ )/U is a bs-weak lc model or weak lc model in the sense of Birkar-Shokurov of (X, F, B)/U , if for any prime divisor D on X which is exceptional over X ′ , a(D, F, B) ≤ a(D, F ′ , B ′ ).
(2) We say that (X ′ , F ′ , B ′ )/U is a bs-minimal model or minimal model in the sense of Birkar-Shokurov of (X, F, B)/U , if for any prime divisor D on X which is exceptional over X ′ , a(D, F, B) < a(D, F ′ , B ′ ). (3) We say that (X ′ , F ′ , B ′ )/U is a bs-semi-ample model or semi-ample model in the sense of Birkar-Shokurov of (X, F, B)/U if it is a bs-weak lc model of (X, F, B)/U and K F ′ + B ′ is semi-ample/U . (4) We say that (X ′ , F ′ , B ′ )/U is a bs-good minimal model or good minimal model in the sense of Birkar-Shokurov of (X, F, B)/U if it is a bs-minimal model of (X, F, B)/U and K F ′ + B ′ is semi-ample/U . If, in addition, the induced birational map X X ′ does not extract any divisor, then we say remove the initial "bs-" or the phrase "in the sense of Birkar-Shokurov" in the previous definitions.
(5) We say that (X ′ ,
We remark that, similar to [CHLX23], the definition of "log minimal model" in our paper does not coincide with the classical definition with F = T X as Q-factorial ACSS is equivalent to Q-factorial qdlt instead of Q-factorial dlt when F = T X . Definition 4.4 (Mori fiber space). Let (X, F, B)/U be a foliated triple and let (X ′ , F ′ , B ′ )/U be a log birational model of (X, F, B)/U . Let f : X ′ → Z be a (K F ′ + B ′ )-Mori fiber space/U .
(1) We say that (X ′ , F ′ , B ′ ) → Z is a bs-Mori fiber space, or a Mori fiber space in the sense of Birkar-Shokurov of (X, F, B)/U , if for any prime divisor D on X which is exceptional over
is a bs-Mori fiber space of (X, F, B)/U and the induced birational map X X ′ does not extract any divisor.
(3) We say that (X ′ , F ′ , B ′ ) → Z is a log Mori fiber space of (X, F, B)/U if it is a bs-Mori fiber space of (X, F, B)/U and (X ′ ,
Remark 4.5. The condition "Q-factorial ACSS" is a condition only for algebraically integrable foliations. Therefore, "log minimal model", "good log minimal model", and "log Mori fiber space" are only well-defined for algebraically integrable foliations. However, Definition 4.3(1-4) and Definitions 4.4(1-2) are well-defined for arbitrary foliations. Therefore, many results in this section also hold for arbitrary foliations.
We also remark that we do not have any requirement on the singularities of (X, F, B) and (X ′ , F ′ , B ′ ) in Definition 4.3(1-4) and Definitions 4.4(1-2). This is because in many cases, we want to consider a generalized foliated quadruple polarized with an ample divisor A. Due to the failure of Bertini-type theorems for foliations, usually the only thing we can do is to consider a generalized foliated quadruple structure (X, F, B, Ā), i.e. we let A be the nef part. However, this is inconvenient when the foliation is associated with some other pair structure, as many theorems on pairs consider structures of the form (X, B + A) instead. Therefore, if we do not have any singularity restrictions on the models, then using (X, F, B + A) will bring us more flexibility when applying results of usual pairs. 4.2. Basic properties of models. In this subsection we prove several basics properties on models of foliated triples. We remark that results in this section works for any foliated triples without any requirement on algebraic integrability nor singularities, so we expect results in this subsection to be useful for further applications, particularly to non-algebraically integrable foliations. Lemma 4.6 (cf. [Bir12, Remark 2.6], [HL23, Lemma 3.4]). Let (X, F, B)/U be a foliated triple and let (X ′ , F ′ , B ′ )/U a bs-weak lc model of (X, F, B)/U associated with the birational map φ : X X ′ . Let p : W → X and q : W → X ′ be birational morphisms such that q = φ • p. Assume that
Proof. For any prime divisor D that is an irreducible component of E,
Therefore, if D is not exceptional/X, then:
By the negativity lemma, E ≥ 0. If E is not exceptional/X ′ , then there exists a component D of E that is not exceptional/X ′ . If D is not exceptional/X, then mult D E = 0 by Definition 4.2, a contradiction. Thus D is exceptional over X. In particular, φ extracts D. Since (X ′ , F ′ , B ′ )/U is a log birational model of (X, F, B), a(D,
which implies that mult D E = 0, a contradiction.
Lemma 4.7 (cf. [Bir12, Remark 2.7], [HL23, Lemma 3.5]). Let (X, F, B)/U be a foliated triple. Let (X 1 , F 1 , B 1 )/U and (X 2 , F 2 , B 2 )/U be two bs-weak lc models of (X, F, B)/U with induced birational maps φ : X 1 X 2 . Let h 1 : W → X 1 and h 2 : W → X 2 be two birational morphisms such that φ • h 1 = h 2 . Then:
Proof. Let φ 1 : X X 1 and φ 2 : X X 2 be the induced birational maps. Possibly replacing W with a higher model, we may assume that the induced birational map h : W → X is a morphism. Let Lemma 4.6, E i ≥ 0 and is exceptional over 2) immediately follows from (1). By (1)
Lemma 4.8. Let r be a positive real number. Let (X, F 1 , B 1 )/U and (X, F 2 , B 2 )/U be two foliated triples such that
/U be a weak lc model (resp. minimal model) of (X, F 1 , B 1 )/U with induced birational map φ : X
2 )/U is a semi-ample model (resp. good minimal model) of (X, F 2 , B 2 )/U . Proof. Let p : W → X and q : W → X ′ be a resolution of indeterminacy. By Lemma 4.6,
We have
). The lemma follows immediately from the definitions. 4.3. Models under foliated log resolutions. From now on, we shall focus on different models of foliations that are algebraically integrable and lc. We first study the relationship between different types of models and foliated log resolutions. Of course, we expect results in this section to hold in greater generalities provided that there is a proper definition of "foliated log resolution" for non-algebraically integrable foliations.
We first recall the following result:
Theorem 4.9 ([CHLX23, Theorem 9.4.1]). Let (X, F, B)/U be a Q-factorial ACSS algebraically integrable foliated triple such that K F + B ∼ R,U E ≥ 0 and E is very exceptional/U . Then we may run a (K F + B)-MMP/U with scaling of an ample/U R-divisor A and any such MMP terminates with a good log minimal model
Definition 4.10 (Foliated log smooth model). Let (X, F, B) be an lc algebraically integrable foliated triple and h : X ′ → X a foliated log resolution of (X, F, B) (cf. Definition 2.13). Let F ′ := h -1 F, and let B ′ ≥ 0 and E ≥ 0 be two R-divisors on W satisfying the following.
(1)
(X ′ , F ′ , B ′ ) is foliated log smooth and lc.
(3) E is h-exceptional.
(4) For any h-exceptional prime divisor D such that
D is a component of E.
We say that (X ′ , F ′ , B ′ ) is a foliated log smooth model of (X, F, B).
Lemma 4.11. Let (X, F, B)/U be an lc algebraically integrable foliated triple. Let (W, F W , B W ) be a foliated log smooth model of (X, F, B).
Then any bs-weak lc model (resp. bs-minimal model, bs-semi-ample model, bs-good minimal model, log minimal model, good log minimal model) of (W, F W , B W )/U is a bs-weak lc model (resp. bs-minimal model, bs-semi-ample model, bs-good minimal model, log minimal model, good log minimal model) of (X, F, B)/U . Proof. We let h : W → X be the induced birational morphism. We may write
for some E ≥ 0 that is h-exceptional, and D ⊂ Supp E for any h-exceptional prime divisor D such that a(D, X, B) > -ϵ F (E). Claim 4.12. Let (X ′ , F ′ , B ′ )/U be a bs-weak lc model of (W, F W , B W )/U . Then
for any prime divisor D over X.
Proof. Let φ W : W X ′ be the induced birational map, and let p : V → W and q : V → X ′ be a common resolution such that q = φ W • p. By Lemma 4.6,
for some F ≥ 0 that is exceptional over X ′ . Then we have Proof of Lemma 4.11 continued. First we prove the bs-weak lc model case. Let (X ′ , F ′ , B ′ )/U be a bs-weak lc model of (W, F W , B W )/U with induced birational map φ W : W X ′ . By Claim 4.12, we only need to show that (X ′ , F ′ , B ′ )/U is a log birational model of (X, F, B)/U . Let φ : X X ′ be the induced morphism and
then we only need to show that B ′ = B′ . Since (X ′ , F ′ , B ′ )/U is a bs-weak lc model of (W, F W , B W )/U , we have
Let D be a prime divisor on X ′ . There are three cases:
Case 1. D is not exceptional over X. In this case,
Case 2. D is exceptional over W . In this case, D is a component of Exc(φ -1 W ) and a component of Exc(φ -1 ), hence mult
Case 3. D is exceptional over X but not exceptional over W . In this case,
By Definition 4.10(4), a(D, F, B) = -ϵ F (D), which implies that
Thus B ′ = B′ , so (X ′ , F ′ , B ′ )/U is a log birational model of (X, F, B)/U , and we are done for the bs-weak lc model case.
Next we prove the bs-minimal model case. Suppose that (X ′ , F ′ , B ′ )/U be a bs-minimal model of (W, F W , B W )/U . For any prime divisor D on X which is exceptional over
The bs-minimal model case immediately follows from the bs-weak lc model case.
The bs-semi-ample model, bs-good minimal model, log minimal model, and good log minimal model cases follow immediately from the bs-weak lc model and the bs-minimal model cases.
Lemma 4.13. Let (X, F, B)/U be an lc algebraically integrable foliated triple and (X ′ , F ′ , B ′ )/U a bs-weak lc model of (X, F, B)/U . Let (W, F W , B W ) be a foliated log smooth model of (X, F, B) such that the induced birational map φ W : W X ′ is a morphism. Then we may run a (K F W + B W )-MMP/X ′ with scaling of an ample/X ′ R-divisor which terminates with a good minimal model
where q : Y → X ′ is the induced morphism. In particular, (Y, F Y , B Y )/U is a log minimal model of (W, F W , B W )/U . Proof. Let h : W → X be the induced birational morphism. We have
for some E ≥ 0 that is exceptional/X. By Lemma 4.6, we have
Proof of Lemma 4.13 continued. By Claim 4.14, F + E is exceptional over X ′ . By Theorem 4.9, we may run a (K F W +B W )-MMP/X ′ with scaling of an ample/X ′ divisor, which terminates with a good minimal model (Y,
The lemma follows.
for some E ≥ 0 that is exceptional/X and f * F Y = F. Then:
(1) Any bs-weak lc model of (X, F, B)/U is a bs-weak lc model of (Y, Proof. (1) Let (X ′ , F ′ , B ′ )/U be a bs-weak lc model of (X, F, B)/U , φ : X X ′ the induced birational map, and φ Y := φ • f . Let p : W → Y and q : W → X ′ be a resolution of indeterminacy, and let h := f • p. By Lemma 4.6,
Since (X ′ , F ′ , B ′ )/U is a log birational model of (X, F, B)/U and (X, F, B) is lc,
For any prime divisor D on X ′ , there are two cases:
Case 1. D is not exceptional over X. In this case,
Case 2. D is exceptional over X. In this case, (2) follows from (1), Lemma 4.15, and Lemma 4.7.
4.5. Minimal models and core models. In this subsection, we shall use core models to study how the (bs-)minimal models of (X, F, B)/U are associated with the (bs-)minimal models of (X, B + G)/U when (X, F, B; G) satisfies Property ( * ). First we recall the following results in [CHLX23] and [HH20]:
Lemma 4.17 (cf. [CHLX23, Lemma 9.2.1]). Let (X, F, B)/U be an lc algebraically integrable foliated triple, G a reduced divisor on X, and f : X → Z a contraction, such that (X, F, B; G)/Z satisfies Property ( * ) and
Then any sequence of steps of a (K F + B)-MMP/U (with scaling of D) is a sequence of steps of a (K X + B + G)-MMP/U (with scaling of D), and any sequence of steps of a (K X + B + G)-MMP/U (with scaling of D) is a sequence of steps of a (K F + B)-MMP/U (with scaling of D). Moreover, any sequence of steps of a (K F + B)-MMP/U or a (K X + B + G)-MMP/U is a sequence of steps of an MMP/Z. Theorem 4.18 ([HH20, Theorem 1.7]). Let (X, B)/U be an lc pair and A an ample/U Rdivisor on X such that (X, B + A) is lc and K X + B + A is nef/U . Assume that (X, B)/U has a Q-factorial bs-minimal model or K X + B is not pseudo-effective/U . Then there exists a sequence of (K X + B)-MMP/U with scaling of A which terminates with either a minimal model or a Mori fiber space of (X, B)/U . In Lemma 4.17, and many results in [CHLX23], we will come up with "MMP/U is always an MMP/Z". If we use the language of core models, then it is essentially saying that "MMP/U is always an MMP/Z U , where Z U is the core model of (X → U, X → Z). We have the following lemmas on showing this fact: Lemma 4.19. Let (X, F, B)/U be an lc algebraically integrable foliated triple. Assume that the associated morphism π : X → U is a contraction, and assume that F is induced by a contraction f : X → Z. Let Z U be the core model of (π, f ). Then:
(1) Any sequence of steps of a (K (2) If K F + B is not nef/U , then there exists a (K F + B)-negative extremal ray/U R. By the (relative) cone theorem of algebraically integrable foliations ([ACSS21, Theorem 3.9],
(3) First we suppose that (X, F, B)/U has a bs-weak lc model (X ′ , F ′ , B ′ )/U . Let (W, F W , B W ) be a foliated log smooth model of (X, F, B)/U such that the induced map W X ′ is a morphism. By Lemma 4.13, we may run a (K Lemma 4.11, (Y, F Y , B Y )/Z U is a log minimal model of (X, F, B)/Z U . This proves the only if part.
Next we prove the if part. Assume that (X, F, B)/Z U has a bs-weak lc model (X ′ , F ′ , B ′ )/Z U . By Lemma 4.15, we may assume that (X ′ , F ′ , B ′ )/Z U is a log minimal model of (X, F, B)/Z U . By Definition-Lemma 3.1, Z U is the core model of (X ′ → U, X ′ → Z). By (2), K F ′ + B ′ is nef/U , so (X ′ , F ′ , B ′ )/U is a bs-weak lc model of (X, F, B)/U . This proves the if part. Lemma 4.20. Let (X, B)/U be a pair associated with contraction π : X → U . Let f : X → Z be a contraction such that B is super/Z. Let Z U be the core model of (π, f ). Then:
(
(2) Any sequence of steps of a (K X + B)-MMP/U is a sequence of steps of a (K X + B)-MMP/Z U . (3) (X, B)/U has a minimal model if and only if (X, B)/Z U has a minimal model.
Proof. Let d := dim X.
(1) Let R be a (K X + B)-negative extremal ray/U . Then R which is spanned by a rational curve C such that 0 < -(K X + B) • C ≤ 2d. We may assume that C is of minimal degree among all rational curves which span R, i.e. for any rational curve
which contradicts the cone theorem. Therefore, f (C) is a point. The contraction of C exists by the usual cone theorem, and it is a contraction/Z and a contraction/U . By the universal property of the core models, the contraction of C is a contraction/Z U .
Therefore, any contraction of a (K X + B)-negative extremal ray/U is a contraction/Z U , so any step of a (K
We may replace (X, B) with (Y, B Y ) and continue this process.
(2) Suppose that (X ′ , B ′ )/Z U is a minimal model of (X ′ , B ′ )/U . Since the induced birational map X X ′ does not extract any divisor and is over Z, B ′ is super/Z. If K X ′ + B ′ is not nef/U , then there exists a step of a (K X ′ + B ′ )-MMP/U . This step cannot be over Z U since K X ′ + B ′ is nef/Z U . This contradicts (1), so (X ′ , B ′ )/U is a minimal model of (X, B)/U . Suppose that (X, B)/U has a minimal model. By Lemma 4.15, (X, B)/U has a log minimal model. By Theorem 4.18, we may run a (K X + B)-MMP/U with scaling of an ample divisor which terminates with a minimal model (X ′ , B ′ )/U of (X, B)/U . By (1) the induced map
The following proposition is crucial for us to prove Theorem 1.11.
Proposition 4.21. Let (X, F, B)/U be an lc algebraically integrable foliated triple. Assume that (X, F, B)/U has a bs-weak lc model. Then there exists an ACSS modification h : Proof. Let (Y, F Y , B Y )/U be a bs-weak lc model of (X, F, B)/U . Let g : W → X be a foliated log resolution of (X, F, B) associated with the equidimensional contraction f W : W → Z, such that the induced birational map W Y is a morphism, F W := g -1 F is induced by f W , and
Let π : X → U be the associated morphism and X → U ′ → U be the Stein factorization of π. Possibly replacing U with U ′ , we may assume that π is a contraction. Let Z U be the core model of (π • g, f W ). By Lemma 4.13, we may run a (K F W + B W )-MMP/U which terminates with a log minimal model (X ′′ , F ′′ , B ′′ )/U of (W, F W , B W )/U . By Lemma 4.19, this MMP is a Lemma 4.20, (W, B W + G W )/U has a minimal model.
By Theorem 4.9, we may run a (K F W + B W )-MMP/X with scaling of an ample/X divisor which terminates with a log minimal model (X ′ , F ′ , B ′ )/X such that K F ′ + B ′ ∼ R,X 0. Let φ : W X ′ be the induced birational map and h : X ′ → X the induced birational morphism, and let G := φ * G W . By our construction, h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) is an ACSS modification that is Q-factorial, proper, and super. By Lemma 4.17, φ is also a (K X + B + G)-MMP/X.
Let p : V → W and q : V → X be a resolution of indeterminacy such that p is a log resolution of (W, B W + G W ) and q is a log resolution of (X Lemma 4.16, (V, ∆ V )/U has a bs-weak lc model. By Lemma 4.11, (X ′ , B + G)/U has a bs-weak lc model. By Lemma 4.15, (X ′ , B + G)/U has a log minimal model.
The goal of this section is to prove Theorem 5.6, which essentially implies Theorem 1.10 and is crucial for the proofs of Theorems 1.2 and 1.3. We first recall the following results on the MMP for usual pairs. Lemma 5.1 (cf. [TX24, Lemma 2.20]). Let (X, B + A)/U be an lc pair such that (X, B) is lc and K X + B + A is nef/U . Then there exists a positive real number ϵ ∈ (0, 1) such that any
Theorem 5.2. Let (X, B)/U be an lc pair and H ≥ 0 an R-divisor on X such that K X + B + H is nef/U and (X, B + H) is lc. Assume that there exists an infinite sequence of (K X + B)-MMP/U with scaling of H with scaling numbers λ i such that lim i→+∞ λ i = λ and λ ̸ = λ i for any i. Then (X, B + λH)/U does not have a bs-minimal model. ). Let (X, B)/U be a Q-factorial pair and A ≥ 0 an R-divisor on X such that B is big/U , (X, B + A) is klt, and K X + B + A is nef/U . Then any (K X + B)-MMP/U with scaling of A terminates with either a minimal model or a Mori fiber space of (X, B)/U . Lemma 5.4. Let (X, B)/U be an lc pair. Let H ≥ 0 be an R-divisor on X such that (X, B + H) is lc and K X + B + H is nef/U . Assume that for any µ ∈ [0, 1],
• either (X, B + µH)/U has a log minimal model, or • K X + B + µH is not pseudo-effective/U . Then there exists a (K X + B)-MMP/U with scaling of H which terminates after finitely many steps.
Proof. Denote by this MMP
Let H i be the image of H on X i for each i, and let
be the i-th scaling number of this MMP for each i.
If λ 1 = 0 then there is nothing left to prove. So we may assume that λ 1 > 0. By Lemma 5.1, we may pick λ ′ 1 ∈ (0, λ 1 ) such that any sequence of a (K
By Theorem 4.18, we may run a (K X + B + λ ′ 1 H)-MMP/U with scaling of a general ample/U divisor A which terminates. We let
be this sequence of the MMP/U . Then this sequence consists of finitely many steps of a (K X +B)-MMP/U with scaling of H, with scaling numbers
H is not pseudo-effective/U , then we have already achieved a (K X k 1 + B k 1 )-Mori fiber space/U and we are done. Otherwise,
We may replace (X, B)/U with (X k 1 , B k 1 )/U and continue this process. If this MMP does not terminate, then we may let λ := lim i→+∞ λ i . Then λ ̸ = λ i for any i, and
. By Theorem 5.2, (X, B + λH) does not have a log minimal model, which contradicts our assumption. Therefore, this MMP terminates and we are done.
Theorem 5.5 ([HH20, Theorem 1.5]). Let (X, B)/U be an lc pair and A an ample/U R-divisor on X such that (X, B + A) is lc. Then (X, B + A)/U has a bs-semi-ample model or a bs-Mori fiber space.
The following theorem is crucial for the proof of our main theorems.
Theorem 5.6. Let (X, F, B)/U be an lc foliated triple and let A, H be two ample/U R-divisors on X. Let h : (X ′ , F ′ , B; G)/Z → (X, F, B) be a simple model of (X, F, B) that is proper and super, H ′ := h * H, and A ′ := h * A. Then:
(1) We may run a (K F ′ +B ′ +H ′ )-MMP/U with scaling of A ′ , say P, such that P terminates with either a minimal model or a Mori fiber space of (X ′ , F ′ , B ′ + H ′ )/U . (2) If X is potentially klt, then P can be any (K
Proof. Possibly replacing A with a multiple, we may assume that K F + B + H + A is nef/U . Let π : X → U be the induced projective morphism and let H U be a sufficiently ample R-divisor on U . Possibly replacing A with A + π * H U and H with H + π * H U , we may assume that A and H are ample. Possibly replacing A and H, we may assume that A, H are general in |A| R and |H| R respectively. In particular, (X ′ , B ′ + H ′ + G) is lc.
Since G is super/Z and (X ′ , F ′ , B ′ )/U is lc, by Lemma 4.17, any (K
Let X be the core model of (h, f ) associated with ( h, f ). Let g : X ′ → X be the induced birational morphism. By Lemma 3.5(4), there exists a core model h : ( X, F , B; Ḡ)/Z → (X, F, B) that is proper and super. Let H := h * H and Ā := h * A. By the definition of core models, H and Ā are ample/Z. Since Ḡ is super/Z, Ḡ ≥
where H i are ample Cartier divisors on Z. Then there exists 0
Step 1. First we prove the theorem when X is potentially klt. By Lemma 3.7, X is potentially klt. Since L is ample, by Lemma 2.12, there exists a klt pair ( X, ∆) such that
Let K X ′ + ∆′ := g * (K X + ∆). Then (X ′ , ∆′ ) is sub-klt. Let 0 < δ ≪ 1 be a real number. Since Supp G contains all g-exceptional prime divisors and (X
is klt. Since g * L is big and nef, there exist ample R-divisors L n and R-divisors
for any positive integer n. Then for any n ≫ 0, (X ′ ,
for some n ≫ 0. By our construction and Lemma 3.5(2),
. By Theorem 5.3, any such MMP terminates.
Step 2. Now we prove the general case. For any real number µ ∈ [0, 1] such that
is pseudo-effective/U , by Lemma 3.5(2),
is pseudo-effective/U . Therefore,
is pseudo-effective/U . Since L is ample and Ā is big and nef/U , L + µ Ā is ample/U . Since H, A, L are general, ( X, B + H + G + ( L + µ Ā))/U is lc. By Theorem 5.5, ( X, B + H + G + ( L + µ Ā))/U has a good minimal model.
Denote by f the contraction X → Z. Since
by Lemmas 4.15 and 4.16,
has a good log minimal model. By Theorem 4.18,
has a minimal model. Since
by Lemma 4.8, (X ′ , B ′ + H ′ + G+ µA ′ )/U has a good minimal model. By Lemma 4.15, (X ′ , B ′ + H ′ + G + µA ′ )/U has a log minimal model. By Lemma 5.4, there exists a (K X ′ + B ′ + H ′ + G)-MMP/U with scaling of A ′ , say P, which terminates. By Lemma 4.17, this MMP/U is also a (K F ′ + B ′ + H ′ )-MMP/U with scaling of A ′ . The theorem follows.
The goal of this section is to prove Theorem 1.12.
Proof of Theorem 1.12. Let B(v
there exists an open subset U 1 ∋ v 0 in the rational polytope of v 0 , such that for any v ∈ U 1 , (X, F, B(v)) is lc. We let c := dim U 1 and let v 1 , . . . , v c+1 be vectors in U 1 ∩Q m such that v 0 is contained in the convex hull U 2 spanned by v 1 , . . . , v c+1 . Then there exist positive real numbers a 1 , . . . , a c+1 such that c+1 i=1 a i v i = v 0 and c+1 i=1 a i = 1. We let I be a positive integer such that I(K F + B(v i )) is Cartier for each i. Let d := dim X and a 0 := min 1≤i≤c+1 {a i }.
Consider the set
We have γ 0 := inf{γ ∈ Γ} > 0. We let U be the interior of the set
We show that U satisfies our requirement. By our construction, (X, F, B(v)) is lc for any v ∈ U so we only need to show that K F + m i=1 v i B i is nef/Z for any v = (v 1 , . . . , v m ) ∈ U . We let R be an extremal ray in NE(X/U ). There are three cases.
• R < 0 for some j. In this case, by the relative cone theorem for algebraically integrable foliations (cf. [CHLX23, Theorem 2.2.1], [ACSS21, Theorem 3.9]), R is spanned by a curve C such that (K F + B(v i ) • C ≥ -2d for any i. Thus
Then for any v ∈ U , there exists v ′ ∈ U 2 such that (2d + γ 0 )v = 2dr + γ 0 v ′ . We have
The theorem follows.
We first prove the contraction theorem when the supporting function is not big, and then prove the contraction and the existence of flips when the supporting function is big.
Proposition 7.1. Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is lc for some B ≥ ∆ ≥ 0. Let R be a (K F + B)-negative extremal ray/U and H R a supporting function/U of R. Suppose that H R is not big/U . Then R is also a (K X +∆)-negative extremal ray/U . In particular, there exists a contraction cont R of R.
Proof. By [CHLX23, Theroem 2.2.1, Lemma 8.4.1], we may assume that
for some ample/U R-Cartier R-divisor A on X. Let π : X → U be the induced projective morphism and X → U ′ → U the Stein factorization of π. Possibly replacing U with U ′ , we may assume that π is a contraction.
Let F be a general fiber of π. Then H F := H R | F is nef but not big. Let q := dim F and A F := A| F , then there exists an integer 0 ≤ k ≤ q -1 such that
and
for any general closed point x ∈ X, there exists a rational curve C x such that x ∈ C x , π(C x ) is a closed point, C x is tangent to F, and
In particular, C x spans R. By Theorem 3.4, there exists an ACSS modification h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) that is Q-factorial and proper. Then G contains any h-exceptional F ′ -invariant prime divisor, and Supp B ′ contains any h-exceptional non-F ′ -invariant prime divisor. In particular, Let ∆ ′ := h -1 * ∆. Since (X, ∆) is lc, we may write
where E + , E -≥ 0 are exceptional/X, and
Let x be a general closed point in X and let C ′ x be the strict transform of C x on X ′ . Let
Therefore, R is a (K X + ∆)-negative extremal ray. The existence of cont R follows from the usual contraction theorem for lc pairs. Finally, we prove the contraction theorem and the existence of flips when the supporting function is big. We remark that generalized foliated quadruples will inevitably be used in the proof of the following theorem. For the convenience of the readers that are not familiar with generalized pairs and/or generalized foliated quadruple, in the following proof, we write footnotes whenever when we have to use generalized foliated quadruples and explain the reasons. We also suggest the readers to consider M = N = 0 throughout the proof.
To prove this theorem, we need to use the concept of generalized foliated quadruples. Nevertheless, we can stick to "NQC generalized foliated quadruples" as the non-NQC case is harder to prove.
Theorem 7.2. Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Let R be a (K F + B + M X )-negative extremal ray/U and A an ample/U R-divisor on X, such that
Proof.
Step 1. We reduce to the case when (
Let ϵ ∈ (0, 1) be a real number such that (K X + ∆ + N X + ϵA) • R ̸ = 0. By Lemma 2.12, possibly replacing M with M + ϵ Ā, N with N + ϵ Ā2 , and A with (1 -ϵ)A, we may assume that (K X + ∆ + N X ) • R ̸ = 0, and there exists a klt pair (X, ∆) such that
and the theorem follows from the contraction theorem and the existence of flips for klt pairs. Thus we may assume that
Step 2. In this step we construct an ACSS model (X ′ , F ′ , B ′ , M) of (X, F, B, M) and run a sequence of steps of MMP φ ′ : X ′ X n for this ACSS model to achieve a model (X n , F n , B n , M).
By Theorem 3.4 (Theorem A.21), there exists an ACSS model h :
We remark that this is the first place where we need to use the structure of generalized foliated quadruples.
Even if M = N = 0 at the beginning, since it may not be possible for us to get an lc foliated triple (X, F, B + A) even if A is general in |A| R/U . Nevertheless, if the readers only care about the case when M = N = 0, then the readers may always assume that M, N are NQC/U as this property is preserved throughout the proof.
Since δ 1 < 1 -λ n , φ ′ is a sequence of steps of a (K
Thus there exists a positive real number l 1 > µ(δ 1 ) satisfying the following:
hence a sequence of steps of a Lemma 4.4(2)], we may run a (K Xn + ∆ n + N Xn + l 1 P Xn )-MMP/U with scaling of an ample/U R-divisor which terminates with a good minimal model ( X, ∆, N + l 1 P)/U of (X n , ∆ n , N + l 1 P)/U , and ( X, ∆, N + l 1 P) is kltfoot_0 . We let ϕ : X n X be the induced birational map. By our construction, ϕ is P Xn -trivial.
Step 4. In this step we show that the induced birational map X X does not extract any divisor and is H R -trivial.
Let ψ := X ′ X and α : X X be the induced birational maps. By our construction, ψ is a sequence of steps of a Lemma 2.25, Exc(ψ) = Supp N . In particular, the induced birational map α : X X does not extract any divisor. Since ψ and h are H ′ R -trivial, α is H R -trivial.
Step 5. In this step we construct the contraction cont R : X → T and prove (1).
Recall that by Step 1, we have (K X + ∆ + N X ) • R > 0. Thus there exists a positive real number c such that cA
R is the only (H R -δ 1 A)-negative extremal ray/U , and δ 1 A + 1 l 1 (K X + ∆ + N X ) and A are ample/U , we have that L R is a supporting function/U of R. By Lemma 2.21, α is L R -trivial.
Let A, H R , and L R be the images of A, H R and L R on X respectively. Then L R is big and nef, and
Since P descends to X n , P Xn is nef. Since ϕ is P Xn -trivial, P X is nef. Since ( X, ∆, N + l 1 P) is klt, ( X, ∆, N) is klt. Since L R is big/U and nef/U and α is L R -trivial, L R is big/U and nef/U . Thus L R + c δ 1 P X is big/U and nef/U . By the base-point-freeness theoremfoot_1 , H R is semiample/U , and O X (mH R ) is globally generated/U if H R is Cartier. We let cont R : X → T and cont R : X → T be the contractions/U induced by H R and H R respectively.
If H R is Cartier, then H ′ R is Cartier. Since φ ′ is a sequence of steps of an MMP of a Q-factorial ACSS generalized foliated quadruples and ϕ is a sequence of steps of an MMP of a klt pair, H R is Cartier. Therefore, O X (m H R ) is globally generated/U for any integer m ≫ 0. Since
) is globally generated/U for any integer m ≫ 0. This implies (1.a).
We prove (1.b). Since L -(K F + B + M X ) is ample/T , by (1.a), O X (mL) is globally generated/T for any m ≫ 0. Thus mL ∼ = cont * R L T,m and (m + 1)L ∼ = cont * R L T,m+1 for line bundles L T,m and L T,m+1 for any m ≫ 0. We may let L T := L T,m+1 -L T,m .
Step 6. We prove (2) and conclude the proof of the theorem.
Since
Since R is the only ((K X + ∆ + N X ) + l 1 (H R -δ 1 A))-negative extremal ray/U , X + is also the ample model/T of K F + B + M X . Since β is (K X + ∆ + N X + l 1 P X )-trivial and l 1 is general in R/Q, by Lemma 2.22, β is (K X + ∆ + N X )-trivial. Since ( X, ∆, N) is klt, (X + , ∆ + , N) is klt, where ∆ + is the image of ∆ on X + . This implies (2.a). (2.b) follows from the definition of a flip.
The proof of (2.c) for the divisorial contraction case is similar to the proof of [KM98, Corollary 3.17], and the proof of (2.c) for the flipping contraction case is similar to [HL23, Theorem 6.1, Step 3]. We omit these proofs. 8. Lifting of the MMP Lemma 8.1. Let (X, F, B)/U be an lc algebraically integrable foliated triple and A an ample/U R-divisor on X. Let P :
be a sequence of steps of a (K F + B)-MMP/U with scaling of A and let A i be the image of A on X i for each i. Let
Suppose that λ n > 0. Then there exists a (K Fn + B n )-negative extremal ray/U R such that
Proof. P is also a sequence of steps of a (K F + B + λ n A)-MMP/U with scaling of A.
By [CHLX23, Lemma 16.1.1], there exists an lc algebraically integrable generalized foliated quadruple (X n , F n , B ′ n , M ′ )/U and an ample/U R-divisor
Xn is NQC/U . Thus there exists a positive real number ϵ 0 , such that for any curve C on X n , either (K
there exists a positive real number δ ∈ 0, ϵ 0 2 dim X+ϵ 0 and a (K
which is not possible. Therefore, (K Fn + B n + λ n A n ) • R = 0 and we are done.
Proposition 8.2. Let (X, F, B)/U be an lc algebraically integrable foliated triple. Let P :
be a (possibly infinite) sequence of (K F + B)-MMP/U . For each i ≥ 0, we let ψ i : X i → T i and ψ + i : X i+1 → T i be the (i + 1)-th step of this MMP and let φ i := (ψ + i ) -1 • ψ i : X i X i+1 be the induced birational map. Let h : (Y, F Y , B Y ; G)/Z → (X, F, B) be an ACSS modification of (X, F, B) that is Q-factorial, proper, and super. Let A be an ample/U R-divisor on X and let A i be the image of A on X i for each i.
Then there exist a (possibly infinite) sequence P Y of birational maps
satisfying the following. Let φ i,Y : Y i Y i+1 be the induced birational map. Then:
(1) For any i ≥ 0, there exist an ACSS modification
proper, and super, such that h 0 = h and G i is the image of G on Y i . (2) For any
the output of this MMP, such that φ i,Y is not the identity map. (4) P Y is a sequence of steps of a (K F Y + B Y )-MMP/U . (5) Suppose that P is an MMP/U with scaling of A. Let A Y := h * A and let A Y i the image of A Y on Y i for each i. Let
be the (i + 1)-th scaling number. Then: (a) φ i,Y is a sequence of steps of a (K Let n be a non-negative integer. We prove the proposition by induction on n and under the induction hypothesis that we have already constructed (Y i , F i , B i ; G i )/U and h i for any i ≤ n and φ i,Y for any i ≤ n-1 which satisfy (1)(2)(3)(5). When n = 0, this follows from our assumption, so we may assume that n > 0. We need to construct φ n,Y , h n+1 , and (Y n+1 , F n+1 , B n+1 ; G n+1 )/U .
We let H n be a supporting function of the extremal ray/U contracted by ψ n and let
with scaling of A. Then L n is ample/T n . Now we run a (K F Yn + B Yn )-MMP/T n with scaling of h * n L n , then this MMP is also a (K F Yn + B Yn + 1 2 h * n L n )-MMP/T i with scaling of h * n L n . By Theorem 5.6, we may choose such an MMP which terminates with a good minimal model (Y n+1 , F Y n+1 , B Y n+1 )/T n of (Y n , F Yn , B Yn )/T n . Since X n+1 is the ample model/T n of K Fn + B n , X n+1 is also the ample model/T n of K F Y n+1 + B Y n+1 , so there exists an induced birational morphism h n+1 : Y n+1 → X n+1 . Since K Yn + B Yn is not nef/T n and K Proof of Theorem 1.3. By Proposition 7.1 and Theorem 7.2, we can run a step of a (K F + B)-MMP/U . By Theorem 7.2(2.a), after a step of the MMP φ : X X ′ that is not a Mori fiber space, (X ′ , ∆ ′ := φ * ∆) is klt. Thus we may continue this process.
Proof of Theorem 1.1. It is a special case of Theorems 1.2 and 1.3. Theorem 9.1. Let (X, F, B)/U be an lc algebraically integrable foliated triple such that (X, ∆) is klt for some B ≥ ∆ ≥ 0. Let A be an ample/U R-divisor on X. Then we may run a (K F + B)-MMP/U with scaling of A.
Proof. It follows from Theorem 1.3 and Lemma 8.1.
Proof of Theorem 1.4. By Theorem 9.1, we can run a (K F + B)-MMP/U with scaling of any ample/U R-divisor A. Let ϵ be a positive real number such that (K F + B + ϵA) is not pseudoeffective/U . Let h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) be an ACSS modification of (X, F, B) that is Qfactorial, proper, and super. By Proposition 8.2, any infinite sequence of steps of a (K F +B+ϵA)-MMP/U with scaling of A induces an infinite sequence of steps of a (K F ′ + B ′ + ϵh * A)-MMP/U with scaling of h * A. By Theorem 5.6, any (K F ′ + B ′ + ϵh * A)-MMP/U with scaling of h * A terminates with a Mori fiber space/U . Thus any (K F + B + ϵA)-MMP/U with scaling of A terminates with a Mori fiber space/U , and the theorem follows.
Proof of Theorem 1.5 (1). By Theorem 9.1 (actually, Theorem A.5), we can run a (K F + B + A)-MMP/U with scaling of any ample/U R-divisor H. Let h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) be an ACSS modification of (X, F, B) that is Q-factorial, proper, and super. By Proposition 8.2, any infinite sequence of steps of a (K F + B + A)-MMP/U with scaling of H induces an infinite sequence of steps of a (K F ′ + B ′ + h * A)-MMP/U with scaling of h * H. By Theorem 5.6, any (K F ′ + B ′ + h * A)-MMP/U with scaling of h * H terminates with a minimal model/U . Thus any (K F + B + A)-MMP/U with scaling of H terminates with a minimal model of (X, F, B + A)/U , and the theorem follows.
Proof of Theorem 1.6. Let H := K F + B + A. By Lemma 2.29, H is NQC/U . Thus we have H = a i H i for some nef/U Cartier divisors H i on X and a i > 0 for each i. Let ϵ 0 := min{a i } and let l > 2 dim X ϵ 0 be an integer. Let 0 < e ≪ 1 be a real number such that
) is globally generated/U for any integer n ≫ 0, and so O X (nH) is globally generated/U for any integer n ≫ 0.
Finally we give the proof of Claim 9.2: Let h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) be an ACSS modification that is Q-factorial, proper and super. Suppose K F + B + A is not pseudo-effective/U , then K F ′ + B ′ + A X ′ is also not pseudo-effective/U . By [CHLX23, Proposition 9.3.2], we may run a (K F ′ + B ′ + A X ′ )-MMP/U which terminates with a Mori fiber space/U , and this MMP is also an MMP/Z. Since
Proof. (Y, π * ∆) is klt by the smoothness of π. Let T be the union of the sections of π corresponding to E L i , 1 ≤ i ≤ m. We will show B Y := (1 -ϵ)T + π * B satisfies our requirements for any 0 < ϵ ≪ 1.
By taking a foliated log resolution of (X, F, B) and considering the base change of π, we can easily see by definition that (Y, π -1 F, T + π * B) is lc if and only if (X, F, B) is lc. Hence (Y, π -1 F, B Y ) is lc as well.
Since K π -1 F + T = π * K F and T is π-ample, we have
Proof of Theorem 1.9 (2). By Theorem 1.6 we have Pic(X
and H be the tautological line bundle O Y (1). Note that H is a big Cartier divisor on Y . It is easy to check that the Cox ring of X is finitely generated if and only if the section ring R(Y, H) is finitely generated. By Lemma 9.3, there is a boundary divisor
for a sufficiently small rational number δ > 0. By Theorem 1.7, R(Y, δH) is finitely generated, and so is R(Y, H). Therefore, the Cox ring of X is also finitely generated and hence X is a Mori dream space.
Proof of Theorem 1.10. Let h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) be an ACSS model of (X, F, B) that is Q-factorial, proper, and super. Let A ′ := h * A. Let H be an ample R-divisor on X and let H ′ := h * H. By Theorem 5.6, we may run a (K
X ′′ which terminates with either a minimal model (X ′′ , F ′′ , B ′′ + A ′′ )/U or a Mori fiber space (X ′′ , F ′′ , B ′′ + A ′′ ) → T of (X ′ , F ′ , B ′ + A ′ )/U , where B ′′ and A ′′ are the images of B ′ and A ′ on X ′′ respectively. Let φ : X X ′′ be the induced birational map. For any prime divisor D that is extracted by φ -1 , D is also extracted by h, so
Therefore, (X ′′ , F ′′ , B ′′ + A ′′ ) is a log birational model of (X, F, B). For any prime divisor D on X that is exceptional/X ′′ , D is also a prime divisor on X ′ that is exceptional/X ′′ , so
Thus either (X ′′ , F ′′ , B ′′ + A ′′ )/U is a bs-minimal model of (X, F, B + A)/U , or (X ′′ , F ′′ , B ′′ + A ′′ ) → T is a bs-Mori fiber space of (X, F, B + A)/U . In particular, (X, F, B + A)/U has a bs-minimal model or a bs-Mori fiber space.
Proof of Theorem 1.11. There exists an ACSS modification h : (X ′ , F ′ , B ′ ; G)/Z → (X, F, B) that is Q-factorial, proper, and super, and (X ′ , B ′ +G)/U has a log minimal model if (X, F, B)/U has a bs-minimal model by Proposition 4.21. Let f : X ′ → Z be the associated contraction and let A ′ := h * A. By Proposition 8.2, any infinite sequence of (K F + B)-MMP/U with scaling of A induces an infinite sequence of (K F ′ + B ′ )-MMP/U with scaling of A ′ .
First we prove (1). Suppose that the MMP does not terminate. We let λ i be the scaling numbers of the (K F + B)-MMP/U with scaling of A and let λ := lim i→+∞ λ i . By Lemma 4.17, any (K F ′ + B ′ )-MMP/U with scaling of A ′ is also a (K X ′ + B ′ + G)-MMP/U with A ′ , and λ is also the limit of the scaling numbers of the (K X ′ + B ′ + G)-MMP/U with A ′ . By Theorem 5.6, we have λ = 0. In particular, λ ̸ = λ i for any i and K F + B is pseudo-effective/U . Thus (X, F, B)/U has a bs-minimal model, so (X ′ , B ′ + G)/U has a log minimal model. This contradicts Theorem 5.2.
(2) follows from (1) and Theorem 9.1.
10. Further discussions 10.1. New definition of foliated klt singularities.
Remark 10.1. We remark that our definitions of lc and klt singularities in Definition 2.9 have some differences with the classical definitions [McQ08, Definition I.1.5], where the -1 is replaced with -ϵ F (E). The other definition is used in most literature (e.g. [CS20, ACSS21, CS21, CHLX23]). Lemma 2.10 shows that our definition of "lc" coincides with the classical definition of "lc". We briefly explain why we change the definition of "klt". This is with several reasons:
(1) The classical "klt" is an empty condition in many scenarios. For any non-trivial algebraically integrable foliation F (F ̸ = T X ), there are a lot of F-invariant divisors, and each of them is an lc place. Therefore, the condition a(E, F, 0) > -ϵ F (E) for any prime divisor E over X as in [McQ08, Definition I.1.5(3)] is a condition that cannot be satisfied by any nontrivial algebraically integrable foliation. Similar issues may appear for foliations with non-trivial algebraic parts.
(2) "Plt" is missing. For example, [CS23b, Theorem 1.1] established the correspondence via adjunction to non-invariant divisors for lc singularities. But the "plt-klt" correspondence is missing. This prevents us to prove a lot of things, e.g. the existence of "pl-flips" for foliations in dimension 4. (3) "Terminal" is also missing. [CS23b, Theorems 1.1, 3.16] can only show that "adjunction of canonical (resp. terminal) singularities to non-invariant divisors is canonical (resp. terminal)". However, for usual pairs, we know that "adjunction of canonical singularities to divisors is terminal". One reason for this is that the definition of "terminal" for foliations requires that the discrepancies of F-invariant divisors are > 0. Thus it is also natural to ask whether we can establish the "canonical-terminal" correspondence if we ignore the invariant lc places. (4) There are substantial differences between non-invariant divisors and invariant divisors and it is very important to use non-invariant divisors to lift sections. This is because we usually have exact sequences of the form
which allow us to lift sections. Here L usually has an lc structure K + B and S is a component of ⌊B⌋. However, if K = K F and S is an F-invariant divisor, then (X, F, B) will not be lc by [CS21, Remark 2.3]. Therefore, non-invariant lc places behave better than lc places in this scenario.
With the above discussion, we also propose the definition of "plt":
Definition 10.2. Let (X, F, B) be a foliated triple. We say that (X, F, B) is plt if a(E, F, B) > -ϵ F (E) for any prime divisor E that is exceptional over X.
We propose the following questions on foliations with klt singularities.
Question 10.3 (cf. [CS23b, Theorem 1.1]). Let (X, F, B) be a (Q-factorial) plt foliated triple and S a component of ⌊B⌋ with normalization S ν . Let F S ν be the restricted foliation of F on S ν and let K F S ν + Diff S ν (F, B) := (K F + B)| S ν . Is (S ν , F S ν , Diff S ν (F, B)) klt? Question 10.4 (cf. [CS23a, Conjecture 4.2(2)]). Let (X, F, B) be a Q-factorial klt algebraically integrable foliated triple such that (X, B) is klt. Is F induced by a contraction? Question 10.5 (cf. [CS21, Theorem 11.3]). Let (X, F, B) be a klt foliated triple such that dim X = 3 and rank F = 2. Is F non-dicritical?
Finally, we remark that the existence of pl-flips is one crucial step towards the existence of flips for usual varieties. With this in mind, we ask the following: Question 10.6 (Pl-flip). Let (X, F, B) be a Q-factorial plt projective foliated triple and f : X → Z a small contraction such that ρ(X/Z) = 1, -(K F + B) is ample/Z, S := ⌊B⌋ is irreducible, and -S is ample/Z. Assume that dim X = 4. Does the flip X + → Z of f exist? 10.2. MMP when the ambient variety is not klt. We still expect that the minimal model program holds for lc algebraically integrable foliations even if the ambient variety is not necessarily klt. We propose the following conjecture:
Conjecture 10.7 (MMP for algebraically integrable foliations). Let (X, F, B)/U be an lc algebraically integrable foliated triple and R a (K F + B)-negative extremal ray. Then:
(1) (Contraction theorem) There exists a contraction/U cont R : X → T of R.
(2) (Existence of flips) If cont R is a flipping contraction, then the flip/U X + → T associated to R exists. (3) (MMP) We can run a (K F + B)-MMP/U . Theorem 1.10 provides some positive evidence towards Conjecture 10.7. In fact, if the "minimal model in the sense of Birkar-Shokurov" in Theorem 1.10 is replaced with "good minimal model in the sense of Birkar-Shokurov", then Conjecture 10.7 will immediately follow.
Another positive evidence for Conjecture 10.7 is the case when F is induced by a locally stable family f : (X, B) → Z. In this case, Conjecture 10.7 is essentially settled in [MZ23, Theorem 1.5] although it is not written in the language of foliations. We reinterpret [MZ23, Theorem 1.5] in the following way and slightly improve this result.
Theorem 10.8 ([MZ23, Theorem 1.5]). Let f : (X, B) → Z be a locally stable family over a normal variety with normal generic fiber and F the foliation induced by f . Let π : X → U be a contraction. Then:
(1) K F = K X/Z and (X, F, B) is lc.
(2) We may run a (K F + B)-MMP/U and any sequence of steps of a (K F + B)-MMP/U is a (K F + B)-MMP/Z.
(2) (Existence of flips) If cont R is a flipping contraction, then the flip/U X + → T associated to R exists.
Proof. It follows from Proposition A.42 and Theorem 7.2.
Theorem A.4 (Theorem 1.3). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Then we may run a (K F + B + M X )-MMP/U . Moreover, for any birational map φ : X X + that is a sequence of steps of a (K F + B + M X )-MMP/U , (X, ∆ + := φ * ∆, N) is klt.
Proof of Theorem 1.3. By Proposition A.42 and Theorem 7.2, we can run a step of a (K F + B)-MMP/U . By Theorem 7.2(2.a), after a step of the MMP φ : X X ′ that is not a Mori fiber space, (X ′ , ∆ ′ := φ * ∆) is klt. Thus we may continue this process.
Theorem A.5 (Theorem 9.1). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Let A be an ample/U R-divisor on X. Then we may run a (K F + B + M X )-MMP/U with scaling of A.
Proof. It follows from Theorem A.4 and Lemma A.44.
Theorem A.6 (Theorem 1.4). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Assume that K F + B + M X is not pseudo-effective/U . Then we may run a (K F + B + M X )-MMP/U with scaling of an ample/U R-divisor and any such MMP terminates with a Mori fiber space/U . Proof. It follows from the same lines of the proof of Theorem 1.4 except that we replace Theorems 5.6, 9.1 and Proposition 8.2 with Theorems A.41, A.5 and Proposition A.45 respectively. Theorem A.7 (Theorem 1.5). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Let A be an ample/U R-divisor on X such that either B ≥ A ≥ 0 or M -N -Ā is nef/U . Then:
(1) We may run a (K F + B + A + M X )-MMP/U with scaling of an ample/U R-divisor and any such MMP terminates with a minimal model of (X, F, B + A, M)/U . (2) The minimal model in (1) is a good minimal model.
Proof. It follows from the same lines of the proof of Theorem 1.5 except that we replace Theorems 5.6, 9.1 and Proposition 8.2 with Theorems A.41, A.5 and Proposition A.45 respectively. Theorem A.8 (Theorem 1.6). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Let A be an ample/U R-divisor on X such that K F + B + A is nef/U . Then:
(1)
) is globally generated/U for any integer n ≫ 0.
Proof. It follows the same proof of Theorem 1.6, except we replace Lemma 2.29 with Lemma A.19.
Theorem A.9 (Theorem 1.7). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N) is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Let A be an ample/U R-divisor on X such that B + A + M X is a Q-divisor. Then the log canonical ring R(X, K F + B + A + M X ) := ⊕ +∞ m=0 π * O X (⌊m(K F + B + A + M X )⌋) is a finitely generated O U -algebra.
Proof. This is an immediate consequence of Theorem A.7(2).
Theorem A.10 (Theorem 1.8). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Assume that κ σ (K F + B + M X ) = 0.
The we may run a (K F + B + M X )-MMP with scaling of an ample R-divisor and any such MMP terminates with a minimal model (X min , F min , B min , M) of (X, F, B, M) such that K F min + B min + M X min ≡ 0. Moreover, if κ ι (K F + B + M X ) = 0, then K F min + B min + M X min ∼ R 0.
Proof. Except the last sentence of the proof where [DLM23, Theorem 1.4] is applied to show that K F min + B min ∼ R 0, the proof of Theorem A.10 follows from the same lines of the proof of Theorem 1.5 by replacing Theorems 5.6, 9.1 and Proposition 8.2 with Theorems A.41, A.5 and Proposition A.45 respectively. In this case, we get a minimal model (X min , F min , B min , M) of (X, F, B, M) such that K F min + B min + M X min ≡ 0. The moreover part is obvious.
Theorem A.11 (Theorem 1.9(1)). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple such that (X, ∆, N)/U is klt, where B ≥ ∆ ≥ 0 and M -N is nef/U . Assume that -(K F + B + M X ) is ample/U . Let D be an R-Cartier R-divisor on X. Then we may run a D-MMP which terminates with either a good minimal model/U of D or a Mori fiber space/U of D.
Proof. It follows from the same lines of the proof of Theorem 1.9 except we replace Theorems 1.4 and 1.5 with Theorems A.6 and A.7 respectively. Theorem A.12 (Theorem 1.10). Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple and A an ample/U R-divisor on X. Assume that either X is potentially klt or M is NQC/U . Then (X, F, B, M + Ā)/U has either a minimal model or a Mori fiber space in the sense of Birkar-Shokurov.
Proof. It is an immediate consequence of Theorem A.41. Note that the proof is even simpler comparing to Theorem A.12 since A does not contribute to any singularity if it is in the nef part rather than the boundary part.
Theorem A.13 (Theorem 1.11). Let (X, F, B, M)/U be an NQC lc algebraically integrable generalized foliated quadruple. Assume that (X, F, B, M)/U has a minimal model or a Mori fiber space in the sense of Birkar-Shokurov and X is potentially klt. Let A be an ample/U R-divisor on X. Then:
(1) Any (K F + B + M X )-MMP/U with scaling of A terminates.
(2) If there exists a klt generalized pair (X, ∆, N) such that B ≥ ∆ ≥ 0 and M-N is nef/U , then (X, F, B, M)/U has a minimal model or a Mori fiber space.
Proof. It follows from the same lines of the proof of Theorem 1.11 except that we replace Lemma 4.17, Propositions 4.21 and 8.2, Theorems 5.2, 5.6, and 9.1 with Lemma A.32, Propositions A.36 and A.45, Theorems A.38, A.41, and A.5 respectively.
Theorem A.14 (Theorem 1.12). Let (X, F, B := m i=1 v 0 i B i , M = n i=1 µ 0 i M i )/Z be an lc algebraically integrable generalized foliated quadruple such that K F + B + M X is nef/Z, each B i ≥ 0 is a Weil divisor, and each M i is a nef/Z Cartier b-divisor.
Let v 0 := (v 0 1 , . . . , v 0 m , µ 0 1 , . . . , µ 0 n ). Then there exists an open subset U of the rational envelope of v 0 in R m+n , such that (X, F, m i=1 v i B i , n i=1 µ i M i ) is lc and K F + m i=1 v i B i + n i=1 µ i M i,X is nef/Z for any (v 1 , . . . , v m , µ 1 , . . . , µ n ) ∈ U . Then λ 0 ≤ 1. By Lemma B.3, there exists a curve C 0 on X such that C 0 ≡ U λ 0 C ′ and π(C 0 ) = {pt}. Then C 0 spans R, and for any curve C ′′ on X so that C ′′ ≡ U λC 0 for some real number λ > 0, we have λ ≥ 1. By our assumption, D • C 0 ≥ ϵ. Therefore,
Lemma B.5. Let d be a positive integer and ϵ a positive real number. Let a 1 , . . . , a k be positive real numbers that are linearly independent over Q. Let δ 0 := ϵ 2(2d+ϵ) . Then there exists function τ : (0, δ 0 ] → R >0 depending only on d, ϵ, and a 1 , . . . , a k satisfying the following.
Let (X, F, B, M)/U be an lc algebraically integrable generalized foliated quadruple and N an NQC/U b-divisor on X, such that (1) (X, F, B, M + N) is lc, (2) K F + B + M X is nef/U , (3) K F + B + M X + N X is ϵ-NQC/U , and (4) N = a i N i , where each N i is a nef/U Cartier b-divisor and each N i,X is Cartier. Then K F + B + M X + (1 -δ)N X is τ (δ)-nef/U for any δ ∈ (0, δ 0 ).
Proof. Let M := max 2d a i 1 ≤ i ≤ k . Consider the set
It is easy to see that Γ 0 is a set whose only accumulation point is +∞. In particular,
In the following, we shall show that τ : δ → min ϵ 2 , δγ 0 satisfies our requirements. Fix δ ∈ (0, δ 0 ). By Lemma B.4, we only need to show that (K F +B +M X +(1-δ)N X ) ≥ τ (δ) for any curve C on X satisfying the following:
• C spans an extremal ray in NE(X/U ).
• For any curve C ′ on X such that C ′ ≡ U λC for some real number λ > 0, we have λ ≥ 1.
This is the fourth time when a generalized pair or a generalized foliated quadruple structure is used.
Actually, we need the base-point-freeness theorem for generalized pairs(Lemma A.43) here.