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Forum of Mathematics, Sigma (2025), Vol. 13:e39 1–55
doi:10.1017/fms.2024.69
RESEARCH ARTICLE
Complements and coregularity of Fano varieties
Fernando Figueroa 1, Stefano Filipazzi 2, Joaquín Moraga3and Junyao Peng4
1Department of Mathematics, Northwestern University, Evanston, IL 60208, USA; E-mail: fzamora@princeton.edu.
2EPFL, SB MATH-CAG, MA C3 625 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland;
E-mail: stefano.filipazzi@epfl.ch.
3UCLA Mathematics Department, Box 951555, Los Angeles, Los Angeles, CA 90095-1555, USA;
E-mail: jmoraga@math.ucla.edu (corresponding author).
4Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA;
E-mail: junyaop@princeton.edu.
Received: 7 July 2023; Revised: 21 May 2024; Accepted: 4 June 2024
2020 Mathematics Subject Classification: Primary – 14E30, 14B05
Abstract
We study the relation between the coregularity, the index of log Calabi–Yau pairs and the complements of Fano
varieties. We show that the index of a log Calabi–Yau pair (𝑋, 𝐵)of coregularity 1 is at most 120𝜆2,where𝜆is
the Weil index of 𝐾𝑋+𝐵. This extends a recent result due to Filipazzi, Mauri and Moraga. We prove that a Fano
variety of absolute coregularity 0 admits either a 1-complement or a 2-complement. In the case of Fano varieties
of absolute coregularity 1, we show that they admit an N-complement with Nat most 6. Applying the previous
results, we prove that a klt singularity of absolute coregularity 0 admits either a 1-complement or 2-complement.
Furthermore, a klt singularity of absolute coregularity 1 admits an N-complement with Nat most 6. This extends
the classic classification of 𝐴, 𝐷, 𝐸-type klt surface singularities to arbitrary dimensions. Similar results are proved
in the case of coregularity 2. In the course of the proof, we prove a novel canonical bundle formula for pairs with
bounded relative coregularity. In the case of coregularity at least 3, we establish analogous statements under the
assumption of the index conjecture and the boundedness of B-representations.
Contents
1 Introduction 2
1.1 Log Calabi–Yau pairs ................................... 2
1.2 Index and coregularity of log Calabi–Yau pairs ..................... 2
1.3 Fano varieties ....................................... 3
1.4 Complements and coregularity of Fano type varieties .................. 4
1.5 Calabi–Yau pairs of higher coregularity ......................... 5
1.6 Fano varieties of higher coregularity. ........................... 5
1.7 Coregularity and the canonical bundle formula ...................... 6
1.8 Kawamata log terminal singularities ........................... 6
1.9 On the techniques of the article .............................. 7
2 Preliminaries 11
2.1 Divisors, b-divisors and generalized pairs ........................ 11
2.2 Singularities of generalized pairs ............................. 11
© The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative
Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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2F. Figueroa et al.
2.3 Crepant birational maps .................................. 12
2.4 Complements ....................................... 13
2.5 Coefficients under adjunction ............................... 13
2.6 Coregularity of pairs ................................... 15
2.7 Kollár–Xu models for log Calabi–Yau pairs ....................... 18
2.8 Index of generalized klt pairs ............................... 19
2.9 Lifting sections using fibrations .............................. 20
3 Finite coefficients 23
4 Semilog canonical pairs 28
4.1 Lifting complements from nonnormal divisors in fibrations ............... 32
5 Relative complements 34
5.1 Lifting sections from a divisor .............................. 34
5.2 Relative complements ................................... 39
6 Canonical bundle formula 40
7 Proof of the theorems 46
References 54
1. Introduction
Fano varieties and Calabi–Yau varieties are two of the three building blocks of algebraic varieties. In
the former case, the canonical divisor is antiample, while in the latter case it is numerically trivial.
In this article, we study the coregularity of Fano and Calabi–Yau varieties. This invariant measures
the dimension of the dual complexes corresponding to log Calabi–Yau structures on the variety. We
show that if the coregularity is at most two, we can control the index of a Calabi–Yau variety and the
complements of a Fano variety. In [35], the third named author relates various problems about Fano
varieties with the concept of coregularity.
1.1. Log Calabi–Yau pairs
A log Calabi–Yau pair (𝑋, 𝐵)is a projective log canonical pair for which 𝐾𝑋+𝐵≡0. By the abundance
conjecture, which is known in this special case [21], it is also known that 𝐾𝑋+𝐵∼Q0. The index of
(𝑋, 𝐵)is the smallest positive integer Ifor which 𝐼(𝐾𝑋+𝐵)∼0. It is conjectured that the index Iof
(𝑋, 𝐵)admits an upper bound depending on the dimension of Xand the set of coefficients of B.Thisis
known as the index conjecture. For instance, if (𝑋, 𝐵)is two-dimensional and the coefficients of Bare
standard (i.e., of the form 1 −1
𝑚for some 𝑚∈Z>0), then 𝐼(𝐾𝑋+𝐵)∼0forsome𝐼≤66 (see, e.g.,
[26, Theorem 4.11]). The bound 66 is optimal, and it can be obtained by considering nonsymplectic
finite actions on K3 surfaces [34]. In [9], the authors exhibit a sequence of klt Calabi–Yau varieties
𝑋𝑑with index 𝑖𝑑that grows doubly exponentially with the dimension d.Thecoregularity of a log
Calabi–Yau pair (𝑋, 𝐵)is defined to be dim 𝑋−dim D(𝑋, 𝐵)−1. Here, D(𝑋, 𝐵)is the dual complex
of (𝑋, 𝐵). This is a pseudo-manifold that encodes the combinatorial data of log canonical centers of a
dlt modification of (𝑋, 𝐵). The dimension dim D(𝑋, 𝐵)is independent of the chosen dlt modification,
so it is an intrinsic invariant of (𝑋, 𝐵). In the following subsection, we present theorems regarding the
index of log Calabi–Yau pairs of coregularity 0 and 1.
1.2. Index and coregularity of log Calabi–Yau pairs
First, we study log Calabi–Yau pairs of coregularity 0 and 1. We use the language of generalized
pairs as in [3,11]. This gives a larger scope for the theorems and also facilitates inductive argu-
ments. In the case of generalized log Calabi–Yau pairs of coregularity 0, the following theorem is
proved in [13].
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Theorem 1. Let (𝑋, 𝐵, M)be a projective generalized log Calabi–Yau pair1of coregularity 0, and let
be 𝜆a positive integer. Assume that 𝜆(𝐾𝑋+𝐵+M𝑋)is Weil. Then, we have that 2𝜆(𝐾𝑋+𝐵+M𝑋)∼0.
Note that in order to have a linear equivalence 𝐷∼0, the divisor Dmust be Weil. Hence, multiplying
𝐾𝑋+𝐵+M𝑋by 𝜆is indeed needed to compute its index. In [13, Example 7.4], the authors give an
example for which (𝑋, 𝐵)is log Calabi–Yau of coregularity 0, Bis a Weil divisor, 2(𝐾𝑋+𝐵)∼0,
and 𝐾𝑋+𝐵is not linearly equivalent to 0. Hence, the factor 2𝜆in the previous theorem is optimal.
Indeed, the factor 2 is often related to the orientability of the pseudo-manifold D(𝑋, 𝐵)(see [13, §5]).
In [13], the authors use topological methods and birational geometry to prove the previous theorem. In
this article, we recover this statement using birational geometry and the theory of complements.
Our next theorem deals with the index of log Calabi–Yau pairs of coregularity 1. This is a general-
ization of the previous statement to the case of coregularity 1.
Theorem 2. Let (𝑋, 𝐵,M)be a projective generalized log Calabi–Yau pair of coregularity 1and 𝜆be
a positive integer. Assume the two following conditions hold:
◦the generalized pair (𝑋, 𝐵, M)has Weil index 𝜆; and
◦the variety X is rationally connected or M=0.
Then, we have that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0for 𝐼=𝑚𝜆 with 𝑚≤120𝜆.
We emphasize that the previous theorem does not hold if Xis not rationally connected and Mis
nontrivial. For instance, we can let Xbe an elliptic curve and Mbe an I-torsion point in Pic0(𝑋). Then,
we have that 𝐼(𝐾𝑋+M𝑋)∼𝐼M𝑋∼0 is minimal and (𝑋,M)is a generalized log Calabi–Yau pair
of coregularity 1. Note that this is not an issue if Xis rationally connected. In this case, the torsion
of components of the b-nef divisor is controlled by their Weil index. Theorem 1and Theorem 2are
still valid if the coefficients of Bbelong to a set of rational numbers satisfying the descending chain
condition (DCC) condition. This follows from the global ascending chain condition for generalized log
Calabi–Yau pairs with bounded coregularity (see [10, Theorem 2]). Finally, in the case of coregularity
1, we obtain the following statement.
Theorem 3. Let (𝑋, 𝐵, M)be a projective generalized log Calabi–Yau pair of coregularity 1. Assume
that the following conditions hold:
◦the coefficients of B are standard;
◦the divisor 2Mis b-Cartier; and
◦the variety X is rationally connected or M=0.
Then, we have that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0for some 𝐼∈{1,2,3,4,6}.
1.3. Fano varieties
Given a klt Fano variety X, the anticanonical divisor −𝐾𝑋is ample. Hence, the linear system |−𝑚𝐾𝑋|
is basepoint free for msufficiently large and divisible. In particular, we can find an effective divisor
𝐵∈|−𝑚𝐾𝑋|such that the pair (𝑋, 𝐵/𝑚)is klt (Kawamata log terminal) and log Calabi–Yau. This
means that every Fano variety admits a log Calabi–Yau structure. In [3], Birkar showed that an n-
dimensional Fano variety Xadmits an 𝑁(𝑛)-complement, that is, a boundary Bfor which (𝑋, 𝐵)is log
canonical and 𝑁(𝑛)(𝐾𝑋+𝐵)∼0. This can be thought of as an effective log Calabi–Yau structure on
X.In[15], Filipazzi, Moraga and Xu proved that a 3-fold that admits a Q-complement2also admits a
𝑁3-complement.
Let Xbe an n-dimensional Fano variety. We can define the absolute coregularity to be:
ˆcoreg(𝑋)min{coreg(𝑋, 𝐵)| (𝑋, 𝐵)is log Calabi–Yau}.
1A generalized log Calabi–Yau pair is a generalized lc pair (𝑋, 𝐵,M)with 𝐾𝑋+𝐵+M𝑋∼Q0.
2AQ-complement is a boundary Bfor which (𝑋, 𝐵)is lc and 𝐾𝑋+𝐵∼Q0.
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4F. Figueroa et al.
By definition, the absolute coregularity of Xis at most n. It is expected that a Fano variety of absolute
coregularity cadmits an 𝑁(𝑐)-complement (see, e.g., [35, Conjecture 4.1]). Indeed, following the phi-
losophy of Kawamata’s X-method, one expects to lift complements from minimal log canonical centers.
In the case of a Fano variety of absolute coregularity c, we can produce minimal log canonical centers
having dimension c[on a suitable dlt (divisorially log terminal) modification]. In the following subsec-
tion, we present some theorems regarding complements of Fano type varieties of absolute coregularity
0 and 1.
1.4. Complements and coregularity of Fano type varieties
Our main theorem in this direction states that a Fano type variety of absolute coregularity 0 admits a
1-complement or a 2-complement.
Theorem 4. Let (𝑋, 𝐵,M)be a projective generalized Fano type pair of absolute coregularity 0. Assume
that the following conditions hold:
◦the coefficients of B are standard;
◦the b-nef divisor 2Mis b-Cartier.
Then, there exists a boundary 𝐵+≥𝐵satisfying the following conditions:
◦the generalized pair (𝑋, 𝐵+,M)is generalized log canonical;
◦we have that 2(𝐾𝑋+𝐵++M𝑋)∼0; and
◦the equality coreg(𝑋, 𝐵+,M)=0holds.
In the case that 𝐵=M=0, the previous theorem says that, for a Fano variety Xof absolute
coregularity 0, the linear system |−2𝐾𝑋|contains an element with nice singularities. In particular, the
linear system |−2𝐾𝑋|is nonempty. In [42, §8], Totaro investigates Fano varieties with large bottom
weight, which is the smallest positive integer mfor which 𝐻0(𝑋, −𝑚𝐾𝑋)≠0. In particular, [42, Theorem
8.1] implies the existence of a Fano 4-fold that does not admit an m-complement for 𝑚≤1799233. This
shows that the constant 𝑁(4)obtained by Birkar in [3] is at least 1799233. More generally, [42, Theorem
8.1] shows that 𝑁(𝑑)grows at least doubly exponentially with d. In contrast to this, our statement shows
that a Fano variety of absolute coregularity 0 either admits a 1-complement or a 2-complement. In [35,
Example 3.15], the third author gives an example of a Fano surface of absolute coregularity 0 for which
there is no 1-complement. Thus, the previous theorem is sharp. In the case of absolute coregularity 1,
we obtain a similar result.
Theorem 5. Let (𝑋, 𝐵,M)be a projective generalized Fano type pair of absolute coregularity 1. Assume
that the following conditions hold:
◦the coefficients of B are standard;
◦the b-nef divisor 2Mis b-Cartier.
Then, there exists a boundary 𝐵+≥𝐵satisfying the following conditions:
◦the generalized pair (𝑋, 𝐵+,M)is generalized log canonical;
◦we have that 𝑁(𝐾𝑋+𝐵++M)∼0, where 𝑁∈{1,2,3,4,6}; and
◦the equality coreg(𝑋, 𝐵+,M)=1holds.
In Table 1, we summarize the unconditional theorems regarding complements of Fano varieties. The
entry (𝑑, 𝑐)in the table corresponds to the minimum integer 𝑁𝑑,𝑐 for which every d-dimensional Fano
variety of coregularity cadmits at most a 𝑁𝑑,𝑐-complement. In the blank spots, there set of such Fano
varieties is empty. Due to the work of Liu [33], we know that 𝑁2≤101011. However, it is expected that
we can take 𝑁2=66. By the work of Totaro [42], we know that 𝑁𝑑grows at least doubly exponentially
with d.
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Table 1. Dimension, coregularity and complements.
1.5. Calabi–Yau pairs of higher coregularity
In the case of higher coregularity, we need to deal with klt Calabi–Yau varieties of higher dimensions.
We show that controlling the index of log Calabi–Yau pairs of coregularity ccan be reduced to a problem
about c-dimensional klt Calabi–Yau pairs. In order to state our next theorem, we need to introduce two
conjectures about Calabi–Yau pairs. The first one is the boundedness of the index for klt Calabi–Yau
pairs.
Conjecture 1. Let d be a positive integer, and let Λbe a set of rational numbers satisfying the descending
chain condition. There exists a constant 𝐼𝐼(Λ,𝑑), satisfying the following property. For every
projective d-dimensional klt log Calabi–Yau pair (𝑋, 𝐵)such that B has coefficients in Λ, we have that
𝐼(Λ,𝑑)(𝐾𝑋+𝐵)∼0.
The previous conjecture is stated in [5, Conjecture 2.33]. The second conjecture is known as the
boundedness of B-representations. It predicts that the birational automorphism group of a log Calabi–
Yau pair acts on the sections of 𝐼(𝐾𝑋+𝐵)∼0 with bounded order (see, e.g., [18, Conjecture 3.2]).
Conjecture 2. Let d and I be two positive integers. There is a constant 𝑏𝑏(𝑑, 𝐼)satisfying the
following property. For every projective d-dimensional klt log Calabi–Yau pair (𝑋, 𝐵)with 𝐼(𝐾𝑋+𝐵)∼
0, the image of Bir(𝑋, 𝐵)in 𝐺𝐿(𝐻0(𝐼(𝐾𝑋+𝐵))) K∗is finite and has order at most b.
Now, we can state our main theorem about the index of log Calabi–Yau pairs. It shows that the
boundedness of the index of generalized log Calabi–Yau pairs of coregularity ccan be reduced to the
previous two conjectures in dimension c.
Theorem 6. Let c and p be positive integers and Λ⊂Qbe a set satisfying the descending chain condition.
Assume that Conjecture 1and Conjecture 2hold in dimension c. There is a constant 𝐼𝐼(Λ,𝑐,𝑝)
satisfying the following property. Let (𝑋, 𝐵, M)be a projective generalized log Calabi–Yau pair of
coregularity c for which:
◦either X is rationally connected or M=0;
◦the coefficients of B are contained in Λ; and
◦the b-nef divisor 𝑝Mis b-Cartier.
Then, we have that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0.
1.6. Fano varieties of higher coregularity.
In the case of Fano varieties of higher absolute coregularity, we get the boundedness of complements
of Fano type varieties with bounded absolute coregularity subject to the previous conjectures.
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Theorem 7. Let c and p be positive integers and Λ⊂Qbe a closed set satisfying the descending
chain condition. Assume that Conjecture 1and Conjecture 2hold in dimension c. There is a constant
𝑁𝑁(Λ,𝑐,𝑝)satisfying the following. Let (𝑋, 𝐵,M)be a projective generalized Fano type pair of
absolute coregularity c for which:
◦the divisor B has coefficients in Λ;
◦the b-nef divisor 𝑝Mis Cartier where it descends.
Then, there exists a boundary 𝐵+≥𝐵satisfying the following conditions:
◦the generalized pair (𝑋, 𝐵+,M)is generalized log canonical;
◦we have that 𝑁(𝐾𝑋+𝐵++M𝑋)∼0; and
◦the equality coreg(𝑋, 𝐵+,M)=𝑐holds.
We stress that Conjecture 1is known up to dimension 3. On the other hand, Conjecture 2is known
up to dimension 2. In particular, both Theorem 6and Theorem 7hold unconditionally in the case of
coregularity 2.
1.7. Coregularity and the canonical bundle formula
The canonical bundle formula plays a fundamental role in the theory of complements. In many cases,
we need to lift complements from the base of a log Calabi–Yau fibration. We will prove the following
statement that relates the canonical bundle formula with the coregularity.
Theorem 8. Let c and p be nonnegative integers and Λ⊂Qbe a set satisfying the descending chain
condition. Assume that Conjecture 1and Conjecture 2hold in dimension at most 𝑐−1. There exists a set
ΩΩ(Λ,𝑐,𝑝)⊂Qsatisfying the descending chain condition and a positive integer 𝑞𝑞(Λ,𝑐, 𝑝),
satisfying the following property. Let 𝜋:𝑋→𝑍be a Fano type morphism between projective varieties.
Let (𝑋, 𝐵,M)be a projective generalized pair of coregularity c for which:
◦the generalized pair (𝑋, 𝐵, M)is log Calabi–Yau over Z;
◦the coefficients of B belong to Λ;
◦the b-nef divisor 𝑝Mis Cartier where it descends;
◦the b-nef divisor Mis Q-trivial on the general fiber of 𝑋→𝑍;
◦every generalized log canonical center of (𝑋, 𝐵, M)is a log canonical center of (𝑋, 𝐵); and
◦every log canonical center of (𝑋, 𝐵)dominates Z.
Then, we can write
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍),
where the following conditions hold:
◦𝐵𝑍is the discriminant part of the adjunction for (𝑋, 𝐵,M)over Z;
◦the coefficients of 𝐵𝑍belong to Ω; and
◦the b-nef divisor 𝑞Nis b-Cartier.
1.8. Kawamata log terminal singularities
Finally, we show some applications of the previous theorems of this article to the study of klt singularities.
We obtain the following result about klt singularities of absolute coregularity 0.
Theorem 9. Let (𝑋;𝑥)be a klt singularity of absolute coregularity 0. Then, there exists a boundary Γ
through x satisfying the following conditions:
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◦we have that 2(𝐾𝑋+Γ)∼0on a neighborhood of x;
◦the coregularity of (𝑋, Γ)at x is equal to 0.
In particular, the pair (𝑋, Γ;𝑥)is strictly log canonical at x.
Analogously, we obtain a similar result in the context of klt singularities of absolute coregularity 1.
Theorem 10. Let (𝑋;𝑥)be a klt singularity of absolute coregularity 1. Then, there exists a boundary Γ
through x satisfying the following conditions:
◦we have that 𝑁(𝐾𝑋+Γ)∼0for some 𝑁∈{1,2,3,4,6};
◦the coregularity of (𝑋, Γ)at x is equal to 1.
In particular, the pair (𝑋, Γ;𝑥)is strictly log canonical at x.
The two previous theorems generalize the 𝐴, 𝐷, 𝐸 -type classification of klt surface singularities
to higher-dimensional klt singularities. The A-type klt surface singularities are the toric surface sin-
gularities. In the Gorenstein case, these are the 𝐴𝑛-singularities. The A-type singularities are the klt
surface singularities of absolute coregularity 0 that admit a 1-complement. The D-type klt surface
singularities are quotients of toric singularities via an involution. In the Gorenstein case, these are 𝐷𝑛-
singularities. The D-type singularities are the surface singularities of absolute coregularity 0 that admit
a 2-complement but no 1-complement. The E-type klt surface singularities are the exceptional surface
singularities. In the Gorenstein case, these are exactly the 𝐸6,𝐸7and 𝐸8singularities. These are the
klt surface singularities that have absolute coregularity 1. They admit a 3-, 4- or 6-complement but not
a 1-complement or 2-complement. The aforementioned results about complements and coregularity of
two-dimensional klt singularities are proved in [35, Section 3.2].
1.9. On the techniques of the article
The theory of complements was introduced by Shokurov in the early 2000s (see, e.g., [40]), although
these objects already appeared in the work of Keel and McKernan on quasi-projective surfaces [30]. In
this work, complements were called tigers. Since then, it has been understood that vanishing theorems,
the canonical bundle formula and the minimal model program are indispensable tools to produce
complements on a variety (see, e.g., [38,37,30]). Using the aforementioned techniques, the language
of generalized pairs and the boundedness of exceptional Fano varieties, Birkar proved the boundedness
of complements for Fano varieties [3]. Since then, the theory of complements has been expanded to
Fano pairs with more general coefficients [14,24,41], to log canonical Fano varieties [44] and to log
Calabi–Yau 3-folds [15]. In this article, we study the theory of complements through the lens of the
coregularity. The techniques are similar to the ones in the aforementioned papers. However, in order to
obtain novel results, we need to re-prove several parts of this theory keeping track of this new invariant.
The fact that our results are independent of dimension imposes an extra difficulty. At the same time, we
will need to use some recent results regarding the coregularity and its connections to singularities [10]
and Calabi–Yau pairs [13].
Strategy of the proof
In this section, we give a sketch of the proof of the main theorems of this article, namely Theorem 6,
Theorem 7and Theorem 8. The other theorems will be obtained using the same strategy and an analysis
of the coefficients throughout the proof. We write Theorem 𝑋(𝑐)for Theorem Xin coregularity at
most c. Theorem 6(0)follows from [13, Theorem 1], while Theorem 8(0)is trivial. We will prove the
following four statements:
(i) Theorem 7(0)holds;
(ii) Theorem 7(𝑐−1)implies Theorem 8(𝑐);
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8F. Figueroa et al.
(iii) Theorem 8(𝑐)implies Theorem 6(𝑐); and
(iv) Theorem 6(𝑐)and Theorem 8(𝑐)imply Theorem 7(𝑐).
We write Theorem 𝑋(𝑑, 𝑐)for Theorem Xin dimension at most dand coregularity at most c.For
instance, Theorem 7(𝑑, 𝑐)is known by [3, Theorem 1.7]. Thus, we may write 𝑁(Λ,𝑑,𝑐, 𝑝)for the
positive integer provided by Theorem 7(𝑑, 𝑐). We may suppress Λand pfrom the notation whenever
they are clear from the context. Our aim is to show that, once we fix c, there is an upper bound 𝑁(𝑐)
for all 𝑁(𝑑, 𝑐). Similarly, Theorem 8(𝑑, 𝑐)is known due to [3, Proposition 6.3]. We write 𝑞(𝑑, 𝑐)for
the constant provided by Theorem 8(𝑑, 𝑐), and we show that 𝑞(𝑑, 𝑐)is bounded above by a constant
only depending on c. Theorem 6(𝑑, 𝑐)is not known even if we fix the dimension. In this case, the aim
is twofold: to prove the existence of an upper bound 𝐼(𝑑, 𝑐)for fixed dimension dand to show that all
the 𝐼(𝑑, 𝑐)are bounded above in terms of c. The proof of implication (i) is similar to that of (iv). In the
following three subsections, we sketch the proofs of (ii), (iii) and (iv).
A canonical bundle formula
Let (𝑋, 𝐵,M)→𝑍be as in the setting of Theorem 8(𝑑, 𝑐).
First, we show that for every 𝑧∈𝑍closed, we may find a relative 𝑁(𝑐−1)-complement for (𝑋, 𝐵,M)
over z. We pick an effective Cartier divisor Eon Zthrough z.Welettbe the largest positive number
for which (𝑋, 𝐵 +𝑡𝜋∗𝐸, M)has generalized log canonical singularities around z. By the connectedness
theorem, the coregularity of (𝑋, 𝐵 +𝑡𝜋∗𝐸, M)is at most 𝑐−1. Indeed, since all the generalized log
canonical centers of (𝑋, 𝐵, M)are horizontal over Z, introducing a vertical generalized log canonical
center will strictly decrease the coregularity. Taking a dlt modification of (𝑋, 𝐵 +𝑡𝜋∗𝐸, M), we can
produce a new generalized pair (𝑋,𝐵
,M)such that zis contained in the image of a component S
of 𝐵. By perturbing the coefficients, we may assume that the coefficients of 𝐵belong to Λ.We
replace (𝑋, 𝐵,M)by (𝑋,𝐵
,M)and assume there is a vertical divisorial log canonical center S. Notice
that this replacement changes the crepant birational class of the original generalized pair (𝑋, 𝐵,M)in
order to create a new log canonical center. Running a suitable Minimal Model Program (MMP) over
Z, we reduce to the case in which 𝑆→𝜋(𝑆)𝑧is a Fano type morphism. The generalized pair
(𝑆, 𝐵𝑆,M𝑆)obtained by adjunction of (𝑋, 𝐵,M)to Shas dimension at most 𝑑−1 and coregularity at
most 𝑐−1. If 𝑞(𝑆)=𝑧, then we may apply Theorem 7(𝑑−1,𝑐−1)to conclude that (𝑆, 𝐵𝑆,M𝑆)admits
an 𝑁(𝑑−1,𝑐−1)-complement. Since we are assuming Theorem 7(𝑐−1),thisisalsoan𝑁(𝑐−1)-
complement. If dim 𝜋(𝑆)≥1, then we construct an 𝑁(𝑐−1)-complement by induction on the dimension.
In any case, we obtain an 𝑁(𝑐−1)-complement for (𝑆, 𝐵𝑆,M𝑆)around z. Using Kawamata–Viehweg
vanishing, we lift such complement to an 𝑁(𝑐−1)-complement for (𝑋, 𝐵,M)around the fiber of 𝑧∈𝑍.
The details of this proof can be found in §5, where we discuss relative complements. In §5.1, we explain
how to lift complements from divisors.
Now, we can assume the existence of bounded relative 𝑁(𝑐−1)-complements for (𝑋, 𝐵, M)→𝑍.
The existence of bounded relative complements allows us to find qin the statement of Theorem 8(𝑑, 𝑐).
Indeed, we can take 𝑞(𝑑, 𝑐)=𝑁(𝑐−1). The main difficulty is to control the coefficients of N𝑍in the
model where it descends. In order to do so, we will cut the base with hypersurfaces to reduce to the
case in which the base is a curve. Once the base is a curve C, we will study the coefficients of a relative
complement over a closed point 𝑐∈𝐶. Analyzing the coefficients of this relative complement will show
that 𝑞N𝑍is integral. A similar argument on a suitable resolution 𝑍→𝑍proves that 𝑞N𝑍is integral,
where 𝑍is a model on which N𝑍descends. This finishes the proof of Theorem 8(𝑐)using Theorem
7(𝑐−1). The details of this proof are given in §6.
Index of log Calabi–Yau pairs
Let (𝑋, 𝐵,M)be a generalized log Calabi–Yau pair as in Theorem 6(𝑑, 𝑐).By[10, Theorem 2], we may
assume that the set Λin the statement of the theorem is finite. By [16, Theorem 4.2], we can replace
(𝑋, 𝐵,M)by a Kollár–Xu model (see §2.7). We have a Fano type contraction 𝑞:𝑋→𝑍such that all
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the generalized log canonical centers of (𝑋, 𝐵,M)dominate the base Z. Both the index and Weil index
of 𝐾𝑋+𝐵+M𝑋are preserved by the Kollár–Xu model. We will proceed with the proof in three different
cases, depending on the dimension of the base of the Kollár–Xu model and the coefficient sets of Band
M. We argue by induction on the dimension dof X. The base of the induction is the klt case which
follows by Conjecture 1(see, e.g., Lemma 2.30).
Case 1: the moduli part M=0.
In this case, we know that 𝐾𝑋+𝐵∼Q0. We do not assume that Xis rationally connected. We
choose a component Sof 𝐵and run a (𝐾𝑋+𝐵−𝜖𝑆)-MMP. This minimal model program terminates
with a Mori fiber space on which Sis ample over the base. Observe that the variety Smay not be
normal. However, the pair obtained by adjunction (𝑆, 𝐵𝑆)is semilog canonical. In §4, we show that the
statement of Theorem 6(𝑑−1,𝑐)holds for semilog canonical pairs provided it holds for log canonical
pairs. To do so, we use Conjecture 1and Conjecture 2in dimension c. Here, it is crucial that we work
with pairs instead of generalized pairs. Indeed, Conjecture 2is not known for generalized pairs, even in
dimension 2. Hence, we conclude that 𝐼(Λ,𝑑−1,𝑐,0)(𝐾𝑆+𝐵𝑆)∼0.
Thus, in this case, we conclude that the index of (𝑋, 𝐵)is at most 𝐼(Λ,𝑑−1,𝑐,0).
Case 2: the base Zof the Kollár–Xu model is positive-dimensional, the divisor {𝐵}+M𝑋is trivial on
the general fiber of 𝑋→𝑍and the b-nef divisor Mis nontrivial.
In this case, we apply Theorem 8(𝑐). We can write
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍).(1.1)
The variety Zhas dimension at most c. The integer qonly depends on Λ,𝑐and p. The coefficients of 𝐵𝑍
belong to a DCC set that only depends on Λ,𝑐and p. The b-nef divisor 𝑞Nis b-Cartier. The variety Xis
rationally connected, as we are assuming that the b-nef divisor Mis nontrivial. Hence, Zis also rationally
connected. Let 𝑍→𝑍be the model where N𝑍descends. In particular, 𝑍is rationally connected.
Note that in general, N𝑍may have torsion components. However, since 𝑍is rationally connected, the
q-th multiple of such torsion components are linearly equivalent to zero (see [13, Corollary 3.9]). Using
Conjecture 1, we will show that the index of 𝐾𝑍+𝐵𝑍+N𝑍only depends on Λ,𝑐and p. Thus, by the linear
equivalence (1.1), we conclude that the index of (𝑋, 𝐵, M)is bounded above by a constant 𝐼0(Λ,𝑐,𝑝).
Case 3: the divisor {𝐵}+M𝑋is nontrivial on the general fiber of 𝑋→𝑍and the b-nef divisor Mis
nontrivial.
We run a (𝐾𝑋+𝐵)-MMP over Z. Since 𝐾𝑋+𝐵is not pseudo-effective over Z, this minimal
model program terminates with a Mori fiber space 𝑝:𝑋→𝑊over Z. We denote by 𝐵the push-
forwar d of Bon 𝑋. the divisor 𝐾𝑋+𝐵is antiample over W. Since 𝐵is big over Z, the divisor 𝐵
has a component Sthat dominates W. By construction, the general fibers of 𝑆→𝑊are of Fano type. In
this case, Xand 𝑋are rationally connected, as we are assuming that the b-nef divisor Mis nontrivial.
Hence, the image Wof 𝑋is rationally connected. Since a general fiber of 𝑆→𝑊is of Fano type, they
are rationally connected. Thus, Sis rationally connected, being the base and general fibers of 𝑆→𝑊
rationally connected. In particular, if (𝑆, 𝐵𝑆+M𝑆)is the generalized pair obtained by adjunction, then
we know that 𝐼(Λ,𝑑−1,𝑐,𝑝)(𝐾𝑆+𝐵𝑆+M𝑆)∼0. Here, we argued by induction on the dimension
and used Theorem 6(𝑑−1,𝑐). Depending on the dimension of W, we either use Kawamata–Viehweg
vanishing or Kollár’s torsion-free theorem to conclude that 𝐼(Λ,𝑑 −1,𝑐,𝑝)(𝐾𝑋+𝐵+M𝑋)∼0.
Hence, the index of (𝑋, 𝐵,M)is at most 𝐼(Λ,𝑑−1,𝑐,𝑝). These lifting arguments are explained in §5.1.
In summary, a generalized log Calabi–Yau pair (𝑋, 𝐵,M)as in Theorem 6(𝑑, 𝑐)has index at most
max{𝐼0(Λ,𝑐,𝑝),𝐼(Λ,𝑑 −1,𝑐,𝑝),𝐼(Λ,𝑑−1,𝑐,0)}.
Thus, we have that
𝐼(Λ,𝑑,𝑐,𝑝)≤max{𝐼0(Λ,𝑐,𝑝),𝐼(Λ,𝑑−1,𝑐,𝑝),𝐼(Λ,𝑑−1,𝑐,0)}.
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10 F. Figueroa et al.
Hence, there is an upper bound for 𝐼(Λ,𝑑,𝑐,𝑝)which only depends on Λ,𝑐 and p. This finishes the
sketch of the proof of Theorem 6(𝑐)using Theorem 8(𝑐).
Complements on Fano varieties
Let (𝑋, 𝐵,M)be a Fano type pair as in Theorem 7(𝑑, 𝑐).In§3, we reduce to the case in which
Λfinite. This is crucial for lifting complements from divisors (see §5.1).Bytheassumptiononthe
absolute coregularity of (𝑋, 𝐵, M), we may find a generalized log Calabi–Yau structure (𝑋, 𝐵 +Γ,M)
of coregularity c. By dimensional reasons and the assumption on the absolute coregularity of (𝑋, 𝐵,M),
(𝑋, 𝐵 +Γ,M)may be generalized klt only if 𝑑=𝑐and (𝑋, 𝐵, M)is exceptional; this case is settled by
[3, Theorem 1.7] in dimension c. Therefore, in the rest of this sketch, we may assume that (𝑋, 𝐵 +Γ,M)
is not generalized klt. Let (𝑌, 𝐵
𝑌+Γ𝑌+𝐸,M)be a dlt modification of (𝑋, 𝐵 +Γ,M). Here, 𝐵𝑌(resp.
Γ𝑌) is the strict transform of the fractional part of B(resp. Γ), while we set 𝐸=𝐵𝑌+Γ𝑌+𝐸.
Since (𝑋, 𝐵 +Γ,M)is not generalized klt, we have 𝐸≠0. Since Xis of Fano type, it easily follows
that so is Y. In particular, Yis a Mori dream space. We run a −(𝐾𝑌+𝐵𝑌+𝐸+M𝑌)-MMP. Note that
−(𝐾𝑌+𝐵𝑌+𝐸+M𝑌)is a pseudo-effective divisor. Hence, this minimal model program must terminate
with a good minimal model Z.Welet𝐵𝑍and 𝐸𝑍be the push-forwards to Zof 𝐵𝑌and E, respectively. In
order to produce a complement for (𝑋, 𝐵), it suffices to produce a complement for (𝑍, 𝐵𝑍+𝐸𝑍,M𝑍).
Replacing (𝑋, 𝐵, M)by (𝑍, 𝐵𝑍+𝐸𝑍,M𝑍), we may assume that −(𝐾𝑋+𝐵+M𝑋)is semiample and
coreg(𝑋, 𝐵, M)=𝑐. Notice that this reduction does not alter the coefficients set for the boundary part
of (𝑋, 𝐵,M), since the only divisors that may have been introduced in the boundary have coefficient 1.
Furthermore, by the choice of the MMP run, it follows that Ecannot be contracted. In particular, after
this reduction, we may assume that 𝐵≠0. We will proceed in three different cases depending on the
dimension of the ample model Wof the divisor −(𝐾𝑋+𝐵+M𝑋).
Case 1: the dimension of Wis 0.
In this case, we have that 𝐾𝑋+𝐵+M𝑋∼Q0. Hence, producing a complement for (𝑋, 𝐵, M)is the
same as controlling the index of the generalized pair. Thus, the statement follows from Theorem 6(𝑐).
Case 2: the dimension of Wis d.
In this case, we have that −(𝐾𝑋+𝐵+M𝑋)is semiample and big. Furthermore, the round-down 𝐵
is nontrivial. We pass to a suitable birational model of (𝑋, 𝐵,M)where a component Sof 𝐵is of
Fano type. Performing adjunction to S, we obtain a log Fano pair of dimension 𝑑−1 and coregularity
c. Using Theorem 7(𝑑−1,𝑐), we produce an 𝑁(Λ,𝑑−1,𝑐,𝑝)-complement on Sthat can be lifted to
an 𝑁(Λ,𝑑−1,𝑐,𝑝)-complement of (𝑋, 𝐵).
Case 3: The dimension of Wis positive and strictly less than d.
The fibration 𝜋:(𝑋, 𝐵,M)→𝑊is a log Calabi–Yau fibration for (𝑋, 𝐵,M).If{𝐵}+Mis big
over W, then by perturbing the coefficients of Bwe reduce to Case 2. Otherwise, we may replace W
with the ample model of {𝐵}+Mover W. Doing so, we may assume {𝐵}+Mis trivial on the general
fiber of 𝑋→𝑊. If all the generalized log canonical centers of (𝑋, 𝐵)dominate W, then we are in
the situation of Theorem 8(𝑐). The generalized pair (𝑊, 𝐵𝑊,N𝑊)induced on the base is of Fano type
and exceptional. By [3, Theorem 1.7] in dimension cor less, we can find an 𝑁(Ω,𝑐)-complement for
(𝑊, 𝐵𝑊,N𝑊). Here, Ωonly depends on Λ,𝑐 and p. Then, we can pull the complement back via 𝜋to
obtain an 𝑁(Ω,𝑐)-complement for (𝑋, 𝐵,M). Finally, we may assume that {𝐵}+Mis trivial on the
general fiber of 𝑋→𝑊and there is some component 𝑆⊂𝐵that is vertical over W. In this case, 𝐵hor
is big over W. Here, 𝐵hor stands for the sum of the components of Bwhich are horizontal over W.Again,
we can perturb the coefficients of Bto reduce to Case 2.
In summary, a generalized pair (𝑋, 𝐵,M)as in Theorem 7(𝑑, 𝑐)admits an N-complement, where
𝑁≤max{𝑁(Ω,𝑐),𝑁(Λ,𝑑−1,𝑐,𝑝)}. Thus, we have
𝑁(Λ,𝑑,𝑐,𝑝)≤max{𝑁(Ω,𝑐),𝑁(Λ,𝑑−1,𝑐,𝑝)}.
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Hence, there is an upper bound for 𝑁(Λ,𝑑,𝑐,𝑝)which only depends on Λ,𝑐and p. This finishes the
proof of Theorem 7(𝑐)using Theorem 6(𝑐)and Theorem 8(𝑐).
2. Preliminaries
We work over an algebraically closed field Kof characteristic zero. Our varieties are connected and
quasi-projective unless otherwise stated. In this section, we introduce some preliminaries regarding
singularities, Fano varieties, Calabi–Yau pairs and coregularity.
2.1. Divisors, b-divisors and generalized pairs
In this subsection, we recall some basics about b-divisors and generalized pairs.
Definition 2.1. Let Xbe a normal variety. A b-divisor Mon Xis a function which associates any
birational map 𝑋𝑋with an R-divisor M𝑋on 𝑋. The set of divisors {M𝑋:𝑋𝑋}satisfies the
following compatibility condition: If 𝑔:𝑋1→𝑋2is a birational morphism over X, then 𝑔∗M𝑋1=M𝑋2.
We say that a b-divisor Mon X descends on some birational model 𝑋of Xif M𝑋is R-Cartier and M
is equivalent to (𝑋→𝑋, M𝑋). In other words, for any birational map ℎ:𝑌→𝑋over X,wehave
ℎ∗M𝑋=M𝑌. In the previous case, we say that Mis a b-R-Cartier divisor.
Let 𝑋→𝑍be a projective morphism. The b-divisor Mis said to be b-Cartier (resp. b-nef, b-nef/ Z)if
M𝑋is Cartier (resp. nef, relatively nef over Z) on some birational model 𝑋over Xwhere Mdescends.
The b-Cartier closure of an R-Cartier divisor Mis a b-divisor Mwhose trace on every birational
model 𝑓:𝑌→𝑋is 𝑓∗𝑀.
Definition 2.2. Let Xbe a normal variety and 𝜋:𝑋→𝑍be a projective morphism. A generalized pair
on X over Z is a triple (𝑋, 𝐵, M)where
◦Bis an effective R-divisor on X;
◦Mis a b-nef/Zb-R-Cartier on X; and
◦𝐾𝑋+𝐵+M𝑋is R-Cartier.
When Zis a point, we simply call (𝑋, 𝐵,M)ageneralized pair.
2.2. Singularities of generalized pairs
In this subsection, we define the notions of singularities for generalized pairs.
Definition 2.3. Let Xbe a normal variety and (𝑋, 𝐵, M)be a generalized pair on X.LetDbe a divisor
over X. Pick a log resolution 𝑓:𝑋→𝑋of (𝑋, 𝐵)such that Dis a divisor on 𝑋and Mdescends on
𝑋. We can write
𝐾𝑋+𝐵+M𝑋=𝑓∗(𝐾𝑋+𝐵+M𝑋)
for some uniquely determined 𝐵. Define the generalized log discrepancy 𝑎𝐷(𝑋, 𝐵,M)to be 1 −
coeff𝐷(𝐵).
We say that (𝑋, 𝐵, M)is generalized log canonical (resp. generalized klt)if𝑎𝐷(𝑋, 𝐵,M)is nonneg-
ative (resp. positive) for any divisor Dover X.Ageneralized non-klt place (resp. generalized log canon-
ical place)of(𝑋, 𝐵,M)is a prime divisor Dover Xwith 𝑎𝐷(𝑋, 𝐵,M)≤0 (resp. 𝑎𝐷(𝑋, 𝐵,M)=0). A
generalized non-klt center of (𝑋, 𝐵, M)is the image of a generalized non-klt place. We denote the set
of generalized non-klt centers of (𝑋, 𝐵,M)by Nklt(𝑋, 𝐵,M).Ageneralized log canonical center of
(𝑋, 𝐵,M)is the image Zof a generalized non-klt place such that every generalized non-klt place whose
image on Xcontains Zis a generalized log canonical place.
We say that (𝑋, 𝐵, M)is generalized dlt if it is generalized log canonical and satisfies the following
condition: for any generalized log canonical center Vof (𝑋, 𝐵,M), the pair (𝑋, 𝐵)is log smooth around
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12 F. Figueroa et al.
the generic point of Vand Mdescends on Xin a neighborhood of the generic point of V. We say that
(𝑋, 𝐵,M)is generalized plt if it is generalized dlt and every connected component of 𝐵is irreducible.
Let (𝑋, 𝐵,M)be a generalized log canonical pair over a base Z.Let 𝑓:𝑌→𝑋be a birational
morphism and write
𝐾𝑌+𝐵𝑌+M𝑌=𝑓∗(𝐾𝑋+𝐵+M𝑋).
We say that (𝑌, 𝐵
𝑌,M)is a Q-factorial generalized dlt modification of (𝑋, 𝐵, M)if the variety Yis Q-
factorial, (𝑌, 𝐵
𝑌,M)is generalized dlt and every f-exceptional divisor appears in 𝐵𝑌with coefficient 1.
Lemma 2.4 [16, Theorem 2.9]. Every generalized log canonical pair over a base Z has a Q-factorial
generalized dlt modification.
The following lemma states that the singularities of the pair (𝑋, 𝐵)are milder than the singularities
of (𝑋, 𝐵,M).
Lemma 2.5 [7, Remark 4.2.(3)]. Let (𝑋, 𝐵,M)be a generalized log canonical pair over Z. Suppose
𝐾𝑋+𝐵is R-Cartier. Then for any divisor D over X, the log discrepancies satisfy
𝑎𝐷(𝑋, 𝐵,M)≤𝑎𝐷(𝑋, 𝐵).
In particular, the pair (𝑋, 𝐵)is log canonical.
2.3. Crepant birational maps
In this subsection, we recall the notion of crepant birational map and group of crepant birational
automorphisms.
Definition 2.6. Let (𝑋1,𝐵
1,M)and (𝑋2,𝐵
2,M)be generalized pairs over Z. We say that they are crepant
if there exists a common resolution 𝛼1:𝑋→𝑋1and 𝛼2:𝑋→𝑋2, where each 𝛼𝑖is proper, such that
𝐾𝑋+𝐵
1+M𝑋=𝐾𝑋+𝐵
2+M𝑋,
holds, where we have 𝐾𝑋+𝐵
𝑖+M𝑋=𝛼∗
𝑖(𝐾𝑋𝑖+𝐵𝑖+M𝑋𝑖)for 𝑖=1,2.
In the case of pairs, we recall the notion of B-birational map, originally due to Fujino [17, Definition
1.5]. Observe that, for our purposes in later sections, it is important to deal with possibly reducible
varieties.
Definition 2.7. Let (𝑋, Δ)=(𝑋𝑖,Δ𝑖)and (𝑋,Δ)=(𝑋
𝑖,Δ
𝑖)be possibly reducible normal pairs.
We say that 𝑓:𝑋𝑋is a B-birational map if (𝑋,Δ)and (𝑋,Δ)are crepant. That is, Xand 𝑋have
the same number of irreducible components, and there exists a permutation 𝜎of the index set of the
irreducible component such that, for every i, the restriction 𝑓𝑖:𝑋𝑖𝑋
𝜎(𝑖)is birational and (𝑋𝑖,Δ𝑖)is
crepant to (𝑋
𝜎(𝑖),Δ
𝜎(𝑖)).
Definition 2.8. Given a pair (𝑋, Δ)=(𝑋𝑖,Δ𝑖)as in Definition 2.7, we define
Bir(𝑋, Δ){𝑓|𝑓:(𝑋,Δ)(𝑋, Δ)is 𝐵−birational}.
The set Bir(𝑋, Δ)forms a group under composition.
We observe that Definition 2.7 and Definition 2.8 naturally extend to the case of generalized pairs.
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2.4. Complements
In this subsection, we introduce the notion of relative complements.
Definition 2.9. Acontraction is a projective morphism of quasi-projective varieties 𝑓:𝑋→𝑍such
that 𝑓∗O𝑋=O𝑍. Notice that, if Xis normal, then so is Z.Afibration is a contraction 𝑋→𝑍such that
dim 𝑍<dim 𝑋.
Definition 2.10. Let (𝑋, 𝐵)be a pair and 𝑋→𝑍a contraction. We say that a pair (𝑋, 𝐵)is log Fano
(resp. weak log Fano or log Calabi–Yau)over Z if it is log canonical and −(𝐾𝑋+𝐵)is ample over Z
(resp. −(𝐾𝑋+𝐵)is nef and big over Zor 𝐾𝑋+𝐵is R-trivial over Z).
We say that (𝑋, 𝐵)is of Fano type (resp. log Calabi–Yau type)over Z if (𝑋, 𝐵 +Δ)is klt and weak
log Fano (resp. log Calabi–Yau) for some choice of Δ≥0.
If (𝑋,0)is of Fano type (resp. log Calabi–Yau type) over Z, we say that 𝑋→𝑍is a Fano type
morphism (resp. log Calabi–Yau type morphism). If (𝑋, 𝐵)is log Fano (resp. log Calabi–Yau, Fano
type, Calabi–Yau type) over a point, we simply say that (𝑋, 𝐵)is log Fano (resp. log Calabi–Yau, Fano
type, Calabi–Yau type).
Definition 2.11. Let 𝑋→𝑍be a contraction and (𝑋, 𝐵,M)be a generalized pair over Z.LetNbe a
positive integer. An N-complement of 𝐾𝑋+𝐵+M𝑋over a point 𝑧∈𝑍is a divisor 𝐾𝑋+𝐵++M𝑋such
that over some neighborhood of z,wehave:
◦(𝑋, 𝐵+,M)is generalized log canonical;
◦𝑁(𝐾𝑋+𝐵++M𝑋)∼
𝑍0;
◦𝑁Mis b-Cartier; and
◦𝐵+≥𝐵.
If the above conditions hold for 𝐾𝑋+𝐵++M𝑋over every 𝑧∈𝑍, we say that 𝐾𝑋+𝐵++M𝑋is an
N-complement of 𝐾𝑋+𝐵+M𝑋over Z. We say that 𝐾𝑋+𝐵++M𝑋is a Q-complement of 𝐾𝑋+𝐵+M𝑋
over 𝑧∈𝑍(resp. Q-complement of 𝐾𝑋+𝐵+M𝑋over Z)itisaq-complement for some 𝑞∈Z>0.
The following lemma states that complements can be pulled back via 𝐾𝑋-positive birational contrac-
tions (see [3, 6.1.(3)]).
Lemma 2.12. Let (𝑋, 𝐵, M)be a generalized log canonical pair over a base Z. Suppose 𝑓:𝑋𝑋is
a(𝐾𝑋+𝐵+M𝑋)-nonnegative birational contraction over Z. Let 𝐵=𝑓∗𝐵and N be a positive integer.
If 𝐾𝑋+𝐵+M𝑋has an N-complement over 𝑧∈𝑍, then 𝐾𝑋+𝐵+M𝑋also has an N-complement over
𝑧∈𝑍.
The following lemma says that extracting divisors with small log discrepancy from a Fano type
variety preserves the Fano type property (see [3, 6.13.(7)]).
Lemma 2.13. Let 𝑋→𝑍be a contraction. Let X be a Fano type variety over Z and (𝑋, 𝐵)alog
Calabi–Yau pair over Z. Let 𝑓:𝑌→𝑋be a birational morphism. Suppose that every f-exceptional
divisor E satisfies 𝑎𝐸(𝑋, 𝐵)<1. Then Y is of Fano type over Z.
Furthermore, let 𝐾𝑌+𝐵𝑌be the log pull-back of 𝐾𝑋+𝐵.If𝐾𝑌+𝐵𝑌has an N-complement over
𝑧∈𝑍, then (𝑋, 𝐵)also has an N-complement over 𝑧∈𝑍.
2.5. Coefficients under adjunction
In this subsection, we study the coefficients of a pair under adjunction.
Definition 2.14. Let Rbe a set of rational numbers. We define 𝐼Rto be the minimal integer Isuch that
for any 𝑟∈𝑅and 𝑛∈N,wehavethat
𝑛𝐼𝑟≥𝑛(𝐼−1)𝑟.
If there does not exist such an integer, we define 𝐼Rto be 0.
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14 F. Figueroa et al.
Note that when the set Ris finite 𝐼Rexists as the least common multiple of the denominators will
satisfy the previous inequality. When Ris the set of standard coefficients 𝐼R=1. If Ris finite, then 𝐼R
is bounded above by the least common multiple of the rational numbers in Rthat are not standard.
Definition 2.15. Let Λbe a set of real numbers in [0,1]. Define the derived set of Λas
𝐷(Λ)𝑎∈[0,1]|𝑎=𝑚−1+𝜆1+···+𝜆𝑛
𝑚,where 𝑛∈Z≥0,𝑚 ∈Z>0and 𝜆1,...,𝜆
𝑛∈Λ∪{0,1}.
We also define 𝐷𝜆0(Λ)⊂𝐷(Λ)to be the subset in which, in the definition of a, at least one 𝜆𝑖
is equal to 𝜆0.ThesetΛis said to be derived if Λ=𝐷(Λ).If𝜆is a positive integer, then we set
𝐷𝜆𝐷Z1
𝜆∩[0,1].
For instance, the set of standard coefficients S{1−1/𝑚|𝑚∈Z>0}∪{1}is derived. The
following lemmata describe some properties of derived sets.
Lemma 2.16 [22, Proposition 3.4.1]. Let Λbe a set of real numbers in [0,1]. Then 𝐷(Λ)=𝐷(𝐷(Λ)),
that is, 𝐷(Λ)is a derived set.
The following lemma allows us to control the coefficients of the generalized pairs obtained by
divisorial adjunction. The lemma is a special case of [3, Lemma 3.3]; we refer to the proof of [13,
Lemma 3.8] for the details of this adaptation.
Lemma 2.17 [3, Lemma 3.3]. Let (𝑋, 𝐵, M)be a generalized log canonical pair over Z and Λbe a set
of rational numbers in [0,1]. Suppose the coefficients of B and M𝑋belong to Λfor some model 𝑋
where Mdescends. Let S be the normalization of a component of 𝐵.Write
(𝐾𝑋+𝐵+M𝑋)|𝑆∼𝐾𝑆+𝐵𝑆+N𝑆
for the generalized adjunction on S, where 𝐵𝑆is the boundary part and Nthe moduli part. Then the
coefficients of 𝐵𝑆and N𝑆belong to the derived set 𝐷(Λ)for some model 𝑆where Ndescends.
The following lemma is used in the proof of Theorem 6.1 to control the coefficients of the discriminant
part of a log Calabi–Yau fibration over a curve.
Lemma 2.18. Let q be a positive integer. Let Λbe a set of nonnegative rational numbers. Suppose Λ
satisfies the DCC and has rational accumulation points. Then the set
Σ𝑞𝑏+−𝑏
𝑚≥0|𝑞𝑏+∈Z>0,𝑏
+≤1,𝑚 ∈Z>0,𝑏 ∈Λ⊆Q
satisfies the ascending chain condition and has rational accumulation points.
Proof. We first show that Σ𝑞satisfies the ascending chain condition. Suppose in Σ𝑞we can find an
increasing sequence
𝑏+
1−𝑏1
𝑚1
<𝑏+
2−𝑏2
𝑚2
<··· <𝑏+
𝑘−𝑏𝑘
𝑚𝑘
<···
Since 𝑏+
𝑘∈{𝑖/𝑞:0 ≤𝑖≤𝑞}has only finitely many choices, we may assume, by passing to a
subsequence, that all 𝑏+
𝑘are the same and equal to the number 𝑏+. Furthermore, note that
𝑚𝑘<𝑚1(𝑏+
𝑘−𝑏𝑘)
𝑏+
1−𝑏1
≤𝑚1
𝑏+−𝑏1
.
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Forum of Mathematics, Sigma 15
The second inequality holds as 1 ≥𝑏+
𝑘≥𝑏+
𝑘−𝑏𝑘. Hence, the sequence 𝑚𝑘is bounded above. Thus, by
passing to a subsequence we may assume that 𝑚𝑘=𝑚for all k. Now, we obtain a decreasing sequence
𝑏1>𝑏
2>··· >𝑏
𝑘>··· ,
which violates the descending chain condition of Λ. Thus, Σ𝑞satisfies the ascending chain condition.
Let 𝑎≠0 be an accumulation point of Σ𝑞. Since Σ𝑞satisfies the ascending chain condition, we may
find a sequence
𝑏+
1−𝑏1
𝑚1
≥𝑏+
2−𝑏2
𝑚2
≥···≥ 𝑏+
𝑘−𝑏𝑘
𝑚𝑘
≥···
whose limit is a. We may assume that 𝑏+
𝑘=𝑏+for all k. Then
𝑚𝑘≤𝑏+
𝑘−𝑏𝑘
𝑎≤1
𝑎.
The second inequality holds as 1 ≥𝑏+
𝑘≥𝑏+
𝑘−𝑏𝑘. Hence, the sequence 𝑚𝑘is bounded above. By passing
to a subsequence, we may assume that 𝑚𝑘=𝑚for all k. Since Λhas rational accumulation points,
𝑎=lim
𝑘→∞
𝑏+−𝑏𝑘
𝑚=𝑏+
𝑚−1
𝑚lim
𝑘→∞ 𝑏𝑘∈Q,
as desired.
2.6. Coregularity of pairs
In this subsection, we define the coregularity of a generalized pair and prove some of its properties.
Definition 2.19. Let (𝑋, 𝐵, M)be a generalized log canonical pair. Let 𝑓:𝑌→𝑋be a generalized dlt
modification, and write
𝐾𝑌+𝐵𝑌+M𝑌=𝑓∗(𝐾𝑋+𝐵+M𝑋).
Let
𝐵𝑌=𝐸1+𝐸2+···+𝐸𝑟
be a simple normal crossing divisor on Y.
The dual complex D(𝑌,𝐵
𝑌+M𝑌)is a simplicial complex constructed as follows:
◦For every 1 ≤𝑖≤𝑟, there is a vertex 𝑣𝑖in D(𝑌, 𝐵
𝑌+M𝑌)corresponding to the divisor 𝐸𝑖.For
every subset 𝐼⊆{1,2,...,𝑟}and every irreducible component Zof 𝑖∈𝐼𝐸𝑖, there is a simplex 𝑣𝑍
of dimension #𝐼−1 corresponding to Z;
◦For every 𝐼⊆{1,2,...,𝑟}and 𝑗∈𝐼, there is a gluing map constructed as follows. Let 𝑍⊆𝑖∈𝐼𝐸𝑖
be any irreducible component. Let Wbe the unique component of 𝑖∈𝐼\{𝑗}𝐸𝑖containing Z. Them,
the gluing map is the inclusion of 𝑣𝑊into 𝑣𝑍as the face of 𝑣𝑍that does not contain the vertex 𝑣𝑖.
Define the dimension of D(𝑌, 𝐵
𝑌+M𝑌)to be the smallest dimension of the maximal simplex, with
respect to the inclusion, of D(𝑌, 𝐵
𝑌+M𝑌). When D(𝑌,𝐵
𝑌+M𝑌)=∅, set its dimension to be −1.
The dual complex D(𝑌,𝐵
𝑌,M𝑌)depends on the dlt modification Y. However, its PL-
homeomorphism type is independent of the dlt modification (see, e.g., [16, Theorem 1.6]).
Define the dual complex D(𝑋, 𝐵,M)associated to the generalized pair (𝑋, 𝐵, M)as the homeomor-
phism type of the complex D(𝑌,𝐵
𝑌+M𝑌). Thus, for any dlt modification we have
dim D(𝑋, 𝐵,M)=dim D(𝑌, 𝐵
𝑌,M𝑌).
When M=0, we write D(𝑋, 𝐵)instead of D(𝑋, 𝐵, 0)for simplicity.
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16 F. Figueroa et al.
Definition 2.20. Let (𝑋, 𝐵, M)be a generalized log canonical pair over Z. We define its coregularity
to be
coreg(𝑋, 𝐵, M)dim 𝑋−1−dim D(𝑋, 𝐵, M).
Definition 2.21. Let (𝑋, 𝐵, M)be a generalized log canonical pair over Z. We define the absolute
coregularity over Z of (𝑋, 𝐵, M), denoted by ˆcoreg(𝑋/𝑍, 𝐵, M), as follows:
◦if (𝑋, 𝐵+,M)is not a generalized log Calabi–Yau pair over Zfor every divisor 𝐵+≥𝐵,weset
ˆcoreg(𝑋/𝑍, 𝐵, M)to be ∞;
◦otherwise, we set ˆcoreg(𝑋/𝑍, 𝐵,M)to be the smallest value of coreg(𝑋, 𝐵+,M), over all divisors
𝐵+≥𝐵for which (𝑋, 𝐵+,M)is generalized log Calabi–Yau over Z.
Let 𝑧∈𝑍be a point. We define the absolute coregularity of (𝑋, 𝐵,M)over 𝑧∈𝑍, denoted by
ˆcoreg𝑧(𝑋, 𝐵,M)to be the minimum of ˆcoreg(𝜋−1(𝑈)/𝑈, 𝐵,M)where Uruns over all neighborhoods
of 𝑧∈𝑍.
By definition, we have that
ˆcoreg𝑧(𝑋, 𝐵,M)∈{0,...,dim 𝑋,∞}.
If 𝑋→𝑍is the structure morphism of Xand coreg(𝑋, 𝐵,M)=dim 𝑋, then we say that (𝑋, 𝐵, M)is
an exceptional generalized pair.
By the negativity lemma, the coregularity is preserved under certain MMP.
Lemma 2.22. Let (𝑋, 𝐵, M)be a generalized log canonical pair over Z. Let 𝑧∈𝑍be a point. Suppose
𝑓:𝑋𝑌is a (𝐾𝑋+𝐵+M𝑋)-nonnegative birational contraction over Z. Write 𝐵𝑌=𝑓∗𝐵. Then
ˆcoreg𝑧(𝑌, 𝐵
𝑌,M)=ˆcoreg𝑧(𝑋, 𝐵,M).
Proof. Up to shrinking Zaround z, we can find a generalized log Calabi–Yau pair (𝑋, 𝐵 +Γ,M)over
Zthat computes the absolute coregularity of (𝑋, 𝐵,M)over z.LetΓ𝑌=𝑓∗Γ. Since (𝑋, 𝐵 +Γ,M)is
generalized log Calabi–Yau over Z, we conclude that (𝑌, 𝐵
𝑌+Γ𝑌,M)is generalized log canonical.
As (𝑋, 𝐵 +Γ,M)and (𝑌, 𝐵
𝑌+Γ𝑌,M)are crepant equivalent, we conclude that they have the same
coregularity. Hence, we deduce that
ˆcoreg𝑧(𝑌, 𝐵
𝑌,M)≤ ˆcoreg𝑧(𝑋, 𝐵,M).
On the other hand, up to shrinking Zaround z, we can find an effective divisor 𝐷𝑌on Ythat computes
the absolute coregularity of (𝑌, 𝐵
𝑌,M)over z.Let 𝑝:𝑊→𝑋and 𝑞:𝑊→𝑌be a common resolution.
Write
𝐾𝑋+𝐵+𝐷+M𝑋=𝑞∗𝑝∗(𝐾𝑌+𝐵𝑌+𝐷𝑌+M𝑌).
By the negativity lemma, we know that Dis an effective divisor. Hence, (𝑋, 𝐵 +𝐷, M)is a generalized
log Calabi–Yau pair over Z. As above, we conclude that the absolute coregularity of (𝑋, 𝐵, M)over
𝑧∈𝑍is at most the absolute coregularity of (𝑌, 𝐵
𝑌,M)over 𝑧∈𝑍. This finishes the proof.
The following lemma states the coregularity behaves well under adjunction for generalized log
Calabi–Yau pairs.
Lemma 2.23. Let (𝑋, 𝐵, M)be a generalized log Calabi–Yau pair over Z. Let 𝑧∈𝑍be a point. Let S be
the normalization of a component of 𝐵whose image on Z contains z. Let 𝐵𝑆and Nbe the boundary
and moduli parts defined by generalized adjunction, so that (𝐾𝑋+𝐵+M𝑋)|𝑆∼𝐾𝑆+𝐵𝑆+N𝑆.Then,
we have that
coreg(𝑆, 𝐵𝑆,N)=coreg(𝑋, 𝐵,M).
holds after possibly shrinking around 𝑧∈𝑍.
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Proof. By passing to a generalized dlt modification, we may assume that both generalized pairs
(𝑋, 𝐵,M)and (𝑆, 𝐵𝑆,N)are generalized dlt. Since any minimal generalized log canonical center of
(𝑆, 𝐵𝑆,N)is a minimal generalized log canonical center of (𝑋, 𝐵,M), we have that coreg(𝑆, 𝐵𝑆,N)≥
coreg(𝑋, 𝐵, M). On the other hand, let Wbe a minimal generalized log canonical center of (𝑋, 𝐵,M)
whose image on Zcontains z.By[16, Theorem 1.4], (𝑆, 𝐵𝑆,N)admits a generalized log canoni-
cal center 𝑊𝑆that is P1-linked to W, thus dim 𝑊𝑆=dim 𝑊. This implies that coreg(𝑋, 𝐵,M)≥
coreg(𝑆, 𝐵𝑆,N).
See [3, §3] for the construction of generalized adjunction. By altering the pairs, we can get the same
result for pairs with nef anticanonical class.
Lemma 2.24. Let (𝑋, 𝐵, M)be a generalized log canonical pair over Z. Assume that −(𝐾𝑋+𝐵+M𝑋)
nef over Z. Let S be the normalization of a component of 𝐵.Let𝐵𝑆and Nbe the boundary and moduli
parts defined by generalized adjunction so that (𝐾𝑋+𝐵+M𝑋)|𝑆∼𝐾𝑆+𝐵𝑆+N𝑆. Then, we have that
coreg(𝑆, 𝐵𝑆,N)=coreg(𝑋, 𝐵,M).
Proof. Define 𝑃−(𝐾𝑋+𝐵+M𝑋)and let Pdenote its b-Cartier closure. Then, Pis a b-nef Q-
Cartier divisor. We can apply Lemma 2.23 to the generalized log Calabi–Yau pair (𝑋, 𝐵, M+P).
Therefore, coreg(𝑆, 𝐵𝑆,N+P|𝑆)=coreg(𝑋, 𝐵, M+P). Since Pdescends on S, we conclude that
coreg(𝑆, 𝐵𝑆,N+P|𝑆)=coreg(𝑆, 𝐵𝑆,N). Hence, coreg(𝑆, 𝐵𝑆,N)=coreg(𝑋, 𝐵,M).
The following lemma will be used to cut down the dimension of the base Zin a fibration 𝑋→𝑍.
Lemma 2.25. Let (𝑋, 𝐵)be a log canonical pair over Z and 𝜋:𝑋→𝑍be a fibration with dim 𝑍≥2.
Suppose
◦the pair (𝑋, 𝐵)is log Calabi–Yau over Z;
◦𝜙is of Fano type over an open set U of Z;
◦every log canonical center of (𝑋, 𝐵)dominates Z; and
◦the coregularity of (𝑋, 𝐵)is at most c.
Let H be a general hyperplane section of Z and G be the pull-back of H to X. Write
(𝐾𝑋+𝐵+𝐺)|𝐺=𝐾𝐺+𝐵𝐺.
Then we have
◦the pair (𝐺, 𝐵𝐺)is log canonical;
◦the pair (𝐺, 𝐵𝐺)is log Calabi–Yau over H;
◦the induced map 𝐺→𝐻is of Fano type over 𝑈∩𝐻;
◦every log canonical center of (𝐺, 𝐵𝐺)dominates H; and
◦the coregularity of (𝐺, 𝐵𝐺)is at most c.
Furthermore, let 𝐵𝑍and 𝐵𝐻denote the discriminant parts of the adjunction for (𝑋, 𝐵)over Z and
(𝐺, 𝐵𝐺)over H, respectively. Let D be a prime divisor on Z and C a component of 𝐷∩𝐻. Then
coeff𝐷(𝐵𝑍)=coeff 𝐶(𝐵𝐻).
Proof. We follow the proof of [2, Lemma 3.2].
Since Gis the pull-back of a general hyperplane section on Z,(𝑋, 𝐵 +𝐺)is log canonical. Thus, by
adjunction, (𝐺, 𝐵𝐺)is log canonical and log Calabi–Yau over H. Moreover, every log canonical center
of (𝐺, 𝐵𝐺)is a component of the intersection of a log canonical center of (𝑋, 𝐵 +𝐺)and G, and hence
must dominate H.By2.23, we have an equality
coreg(𝐺, 𝐵𝐺)=coreg(𝑋, 𝐵)≤𝑐.
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18 F. Figueroa et al.
Denote the map 𝐺→𝐻by 𝜓.Lettbe the log canonical threshold of 𝜋∗𝐷with respect to (𝑋, 𝐵)over
the generic point of D. Then there is a non-klt center Wof (𝑋, 𝐵 +𝑡𝜋∗𝐷)which dominates Dand the
pair (𝑋, 𝐵 +𝑡𝜋∗𝐷)is lc over the generic point of D. Since Gis a general pull-back, (𝑋, 𝐵 +𝐺+𝑡𝜋∗𝐷)is
also lc over the generic point of D. By inversion of adjunction [27], there exists a component of 𝐺∩𝑊
which is a non-klt center of (𝐺, 𝐵𝐺+𝑡𝜋∗𝐶)and (𝐺, 𝐵𝐺+𝑡𝜋∗𝐶)is lc near the generic point of C. Thus,
tis the log canonical threshold of 𝜓∗𝐶with respect to (𝐺, 𝐵𝐺). In particular, we have
coeff𝐷(𝐵𝑍)=1−𝑡=coeff 𝐶(𝐵𝐻).
Definition 2.26. We say that a log canonical threshold 𝑡=lct((𝑋, 𝐵);Γ)has coregularity cif
coreg(𝑋, 𝐵 +𝑡Γ)=𝑐and the support of Γcontains the image on Xof a c-dimensional log canonical
center on a dlt modification.
2.7. Kollár–Xu models for log Calabi–Yau pairs
In this subsection, we introduce the concept of Kollár–Xu models. Using a theorem due to Filipazzi and
Svaldi, we conclude that every generalized log Calabi–Yau pair admits a Kollár–Xu model.
Definition 2.27. Let (𝑋, 𝐵, M)be a projective generalized log Calabi–Yau pair. We say that (𝑋, 𝐵, M)
is a Kollár–Xu generalized pair if there exists a projective contraction 𝜋:𝑋→𝑍for which the following
conditions are satisfied:
(1) the generalized pair (𝑋, 𝐵, M)is generalized dlt;
(2) every generalized log canonical center of (𝑋, 𝐵,M)dominates Z; and
(3) the divisor 𝐵fully supports a 𝜋-big and 𝜋-semiample divisor.
In particular, the morphism 𝜋:𝑌→𝑍is of Fano type.
Let (𝑋, 𝐵,M)be a generalized log Calabi–Yau pair. Let 𝜋:𝑌𝑋be a birational map. Assume that
𝜋only extracts log canonical places of (𝑋, 𝐵,M)and is an isomorphism over 𝑋\Supp𝐵.If(𝑌, 𝐵
𝑌,M)
is a Kollár–Xu generalized pair, then we say that 𝑌𝑋is a Kollár–Xu model for (𝑋, 𝐵, M).Wemay
also say that (𝑌, 𝐵
𝑌,M), together with 𝜋, defines a Kollár–Xu model for (𝑋, 𝐵, M).
The following theorem is a generalization of [31, Theorem 49] to the context of generalized pairs.
We refer the reader to [16, Theorem 4.2]. It gives a first approximation for the existence of Kollár–Xu
models in the following theorem.
Theorem 2.28. Let (𝑋, 𝐵,M)be a projective Q-factorial generalized dlt log Calabi–Yau pair. Then,
there exists a crepant birational map 𝜙:𝑋𝑋, a generalized pair (𝑋, 𝐵, M)and a morphism
𝜋:𝑋→𝑍such that:
(1) 𝐵fully supports a 𝜋-ample divisor;
(2) every generalized log canonical center of (𝑋, 𝐵, M)dominates Z;
(3) 𝐸⊂Supp𝐵for ever y 𝜙−1-exceptional divisor 𝐸⊂𝑋; and
(4) 𝜙−1is an isomorphism over 𝑋\Supp𝐵.
We observe that the model 𝑋in Theorem 2.28 is not necessarily Q-factorial. However, using
Q-factorial dlt modifications, we construct a Kollár–Xu model.
Theorem 2.29. Let (𝑋, 𝐵,M)be a projective generalized log Calabi–Yau pair. Then, it admits a Kollár–
Xu model 𝑌𝑋. Furthermore, if (𝑋, 𝐵, M)has coregularity c, then so does (𝑌, 𝐵
𝑌,M).
In the context of Theorem 2.29,if𝑐=0, then 𝑍=Spec(K)and Yis a variety of Fano type.
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Proof. Let (𝑋, 𝐵, M)be a generalized pair as in the statement. Then, we first consider a Q-factorial
generalized dlt modification and then apply Theorem 2.28 to such modification. Call (𝑋, 𝐵, M)the re-
sulting model and 𝜋:𝑋→𝑍the morphism claimed in Theorem 2.28. Since (𝑋, 𝐵,M)has coregularity
c, then so does (𝑋, 𝐵, M). In particular, (𝑋, 𝐵, M)has a c-dimensional generalized log canonical cen-
ter. Then, by (2) in Theorem 2.28, it follows that Zhas dimension at most c. By (1) in Theorem 2.28,
𝐵fully supports a 𝜋-ample divisor, which we will denote by 𝐻.Now,let(𝑌, 𝐵
𝑌,M)be a Q-factorial
generalized dlt modification of (𝑋, 𝐵, M), with morphism 𝑝:𝑌→𝑋. We denote by 𝜋𝑌the induced
morphism 𝜋𝑌:𝑌→𝑍. Note that every generalized log canonical center of (𝑌, 𝐵
𝑌,M)dominates Z.
Then, (2) in Definition 2.27 holds. By [16, Remark 4.3], we have Nklt(𝑋, 𝐵, M)=Supp𝐵. Since p
only extracts generalized log canonical places, it then follows that every p-exceptional divisor has posi-
tive coefficients in 𝐻𝑌𝜋∗𝐻and that Supp 𝐻𝑌=Supp𝐵𝑌. Then, it follows that 𝐵𝑌fully supports
a𝜋𝑌-big and 𝜋𝑌-semiample divisor. Thus, (3) in Definition 2.27 holds. The statements (1) and (3) in
Definition 2.27 hold by construction. We conclude that (𝑌, 𝐵
𝑌,M), together with 𝜋𝑌, are a Kollár–Xu
model of (𝑋, 𝐵, M).Lastly,(𝑌, 𝐵
𝑌,M)has coregularity c, since it is crepant to a generalized pair of
coregularity c.
2.8. Index of generalized klt pairs
In this subsection, we reduce the index conjecture for generalized klt pairs to the standard index
conjecture.
Lemma 2.30. Let d and p be two positive integers. Let Λbe a set of rational numbers satisfying
the descending chain condition. Assume Conjecture 1(𝑑)holds. Then, there exists a positive integer
𝐼𝐼(Λ,𝑑,𝑝), satisfying the following. Let (𝑋, 𝐵, M)be a generalized klt Calabi–Yau pair for
which:
◦the variety X has dimension d;
◦the coefficients of B belong to Λ;
◦the divisor 𝑝Mis Cartier where it descends.
Then, we have that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0.
Proof. By the global ascending chain condition (ACC) [7, Theorem 1.6], we may assume that Λis a
finite set of rational numbers. The statement is clear in dimension 1. We proceed by induction on the
dimension. If M=0, then the statement follows from the conjecture. Since (𝑋, 𝐵, M)is generalized
klt, it admits a small Q-factorialization. Therefore, we may assume that Xis Q-factorial. By the ACC
for generalized log canonical thresholds, we may assume (𝑋, 𝐵, M)is 𝜖-log canonical for some 𝜖that
only depends on d,pand Λ. Then, it follows that Xis itself 𝜖-log canonical. We run a 𝐾𝑋-MMP which
terminates with a Mori fiber space 𝑋𝑋→𝑍.IfZis zero-dimensional, then 𝑋belongs to a
bounded family by [4]. By [14, Theorem 1.2] (𝐾𝑋+𝐵+M𝑋)admits an I-complement for some Ithat
only depends on Λ,𝑑and p. Since 𝐾𝑋+𝐵+M𝑋∼Q0, we conclude that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0, so
the statement follows for Xas well. Now, assume that Zis positive-dimensional. We write 𝜋:𝑋→𝑍
for the corresponding contraction. By [14, Lemma 5.4], we can write
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍),
where the coefficients of 𝐵𝑍belong to Ωwhich satisfies the DCC and only depends on Λ,𝑑 and p.
Furthermore, qonly depends on Λ,𝑑 and pand 𝑞Nis Cartier where it descends. The generalized pair
(𝑍, 𝐵𝑍,N)is generalized log canonical since it comes from the generalized canonical bundle formula
[11, Theorem 1.4]. By induction on the dimension, we know that 𝐼0(𝐾𝑍+𝐵𝑍+N𝑍)∼0forsome𝐼0that
only depends on Λ,𝑑and p. Thus, we conclude that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0, where 𝐼=lcm(𝑞, 𝐼0).
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20 F. Figueroa et al.
2.9. Lifting sections using fibrations
In this subsection, we give some lemmata regarding the lifting of sections using fibrations.
Theorem 2.31. Let 𝜆, 𝑑 and c be nonnegative integers. Assume that Theorem 6(𝑑−1,𝑐)holds, and
let 𝐼𝐼(𝐷𝜆,𝑑−1,𝑐,𝜆)be the integer provided by this theorem. Let (𝑋, 𝐵, M)be a d-dimensional
rationally connected generalized log Calabi–Yau pair. Assume that the following conditions hold:
◦XisQ-factorial and (𝑋, 𝐵,M)is generalized dlt;
◦there is a fibration 𝑋→𝑊,whichisa(𝐾𝑋+𝐵)-Mori fiber space;
◦a component 𝑆⊂Supp𝐵is rationally connected and ample over W;
◦the coefficients of B belong to 𝐷𝜆;
◦we have that 𝜆Mis b-Cartier; and
◦the generalized pair (𝑋, 𝐵, M)has coregularity c.
Then, we have that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0.
Proof. Let (𝑋, 𝐵, M),S,𝑓:𝑋→𝑊and 𝐼𝐼(𝐷𝜆,𝑑−1,𝑐,𝜆)be as in the statement. First, we show
that we can apply the inductive hypothesis to S.
Since (𝑋, 𝐵) is dlt, Sis normal. Furthermore, we have Nklt(𝑋, 𝑆)=𝑆. Since −(𝐾𝑋+𝑆)is f-ample,
by the connectedness principle [16], it follows that 𝑆→𝑊has connected fibers. Now, let (𝑆, 𝐵𝑆,N)be
the generalized pair induced on Sby generalized divisorial adjunction. By Lemma 2.23,(𝑆, 𝐵𝑆,N)has
coregularity c. Furthermore, by Lemma 2.17 and Lemma 2.16, it satisfies the assumptions of Theorem
6(𝑑−1,𝑐)with constant I. Thus, we have
𝐼(𝐾𝑆+𝐵𝑆+N𝑆)∼0.(2.1)
By [10, Theorem 3.1], the coefficients of 𝐵𝑆belong to a finite set only depending on 𝜆and c.In
particular, they are divisible by I, as so are the coefficients of N𝑆. Then, these coefficients control the
coefficients of Diff𝑆(0), as we explain in what follows. By [29, 3.35], along the codimension 2 points
of Xcontained in S,Xhas cyclic singularities.
Then, given a prime divisor Pin S, an étale local neighbourhood of a general point 𝑝∈𝑃is
isomorphic to
(𝑝∈(𝑋, 𝐵,M)) (0∈(A2=(𝑥, 𝑦),(𝑥=0)+𝑐(𝑦=0)))/(Z/𝑚Z)×Adim 𝑋−2,
where 𝑍(A2=(𝑥, 𝑦))/(Z/𝑚Z),𝑆=(𝑥=0)and 𝑆=(𝑦=0). Since the class group of Zis generated
by 𝑆, there exists an integer 𝜇such that
𝐼(𝐾𝑋+𝐵+M𝑋)∼𝜇𝑆.(2.2)
By adjunction, 𝑆|𝑆∼Q1
𝑚{0}. We also have that 𝐼(𝐾𝑋+𝐵+M𝑋)|𝑆∼0, as the denominators of the
coefficients of 𝐵𝑆and N𝑆divide Iand hence it is a Cartier divisor on a smooth germ.
Then, we can write
0∼𝐼(𝐾𝑋+𝐵+𝑀)|𝑆∼𝜇𝑆|𝑆∼𝜇
𝑚{0}.
We conclude that mdivides 𝜇. In particular, we have that 𝜇𝑆 is a Cartier divisor, as the Cartier index
of any Weil divisor of Xthrough {0}divides m. By the linear equivalence (2.2), we conclude that the
divisor 𝐼(𝐾𝑋+𝐵+𝑀)is Cartier at the generic point of P. Note that this argument is independent
of P, so we conclude that 𝐼(𝐾𝑋+𝐵+𝑀)is Cartier at the generic point of every divisor on S. Thus,
𝐼(𝐾𝑋+𝐵+𝑀)is Cartier along a set Uthat contains the generic point of every divisor of S.
Then, by [3, 2.41 and Lemma 2.42], we have the following short exact sequence
0→O𝑋(𝐼(𝐾𝑋+𝐵+M𝑋)−𝑆)→O𝑋(𝐼(𝐾𝑋+𝐵+M𝑋)) → O𝑆(𝐼(𝐾𝑆+𝐵𝑆+N𝑆)) → 0.(2.3)
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Since 𝐼(𝐾𝑋+𝐵+M𝑋)−𝑆∼Q,𝑓 −𝑆, the divisor −𝑆is f-ample and dim 𝑊<dim 𝑋,wehave
𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵+M𝑋)−𝑆)=0.
Similarly, we write
𝐼(𝐾𝑋+𝐵+M𝑋)−𝑆∼Q,𝑓 −𝑆∼Q,𝑓 𝐾𝑋+(𝐵−𝑆+M𝑋).
Note that Xis klt and 𝐵−𝑆+M𝑋=𝐵<1+M𝑋+(𝐵=1−𝑆)is f-ample since fis a Mori fiber space and the
divisor 𝐵<1+M𝑋is f-ample. Thus, by the relative version of Kawamata–Viehweg vanishing, we have
𝑅1𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵+M𝑋)−𝑆)=0.
Therefore, by pushing forward (2.3)viaf, we obtain
𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵+M𝑋)) 𝑓∗O𝑆(𝐼(𝐾𝑆+𝐵𝑆+N𝑆)).
Now, taking global sections, we have
𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵+M𝑋))) =𝐻0(𝑆, O𝑆(𝐼(𝐾𝑆+𝐵𝑆+N𝑆))) =𝐻0(𝑆, O𝑆)≠0.(2.4)
By Lemma [13, Lemma 3.1], (2.4) implies that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0.
Theorem 2.32. Let 𝜆, 𝑑 and c be nonnegative integers. Assume that Theorem 6(𝑑−1,𝑐)holds, and let
𝐼𝐼(𝐷𝜆,𝑑 −1,𝑐,0)be the integer provided by this theorem. Assume that I is divisible by 2𝜆.Let
(𝑋, 𝐵)be a d-dimensional log Calabi–Yau pair. Assume that the following conditions hold:
◦XisQ-factorial and klt;
◦there is a fibration 𝑋→𝑊;
◦S is a prime component of 𝐵that is ample over W;
◦(𝑋, 𝐵 −𝑆)is dlt;
◦the morphism 𝑆→𝑊does not have connected fibers;
◦the coefficients of B belong to 𝐷𝜆; and
◦the pair (𝑋, 𝐵)has coregularity c.
Then, we have that 𝐼(𝐾𝑋+𝐵)∼0.
Proof. Let (𝑋, 𝐵),S,𝑓:𝑋→𝑊and 𝐼𝐼(𝐷𝜆,𝑑−1,𝑐,𝜆)be as in the statement. We will proceed in
several steps.
Step 1: In this step, we observe that dim 𝑋−dim 𝑊=1, fis generically a P1-fibration and 𝐵hor =𝑆.
Since (𝑋, 𝐵)is log canonical and Xis Q-factorial, then (𝑋, 𝑆)is log canonical. By considering a
general fiber of f, the restriction of Sinduces a disconnected ample divisor. Therefore, by [25,Exercise
III.11.3], it follows that the general fiber of fis a curve. By the log Calabi–Yau condition and the fact
that 0 ≠𝐵hor ≥𝑆, it follows that fis generically a P1-fibration. Since 𝑆→𝑊does not have connected
fibers and deg 𝐾P1=−2, it follows that Sis the only component of Supp 𝐵that dominates W, that is, we
have 𝐵hor =𝑆. In particular, we may find a nonempty open subset 𝑈⊆𝑊such that 𝐾𝑋+𝑆∼Q0/𝑈.
Step 2: In this step, we show that dim 𝑊>0.
By Step 1, 𝑋→𝑊is generically a P1-fibration, and Sdetermines two distinct points on the geometric
generic fiber of f. Thus, if dim 𝑊=0, it would follow that (𝑋, 𝐵)(P1,{0} + {∞}), with Sidentified
with {0}+{∞}. Since we assumed that Sis a prime divisor, this leads to the sought contradiction.
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22 F. Figueroa et al.
Step 3: In this step, we show that we may replace Xbirationally so that (𝑋, 𝑆)is plt, 𝐾𝑋+𝑆∼Q0/𝑊
and fis a Mori fiber space.
Since (𝑋, 𝐵)is log canonical and Xis Q-factorial, then (𝑋, 𝑆)is log canonical. Let 𝑋be a Q-
factorial dlt modification of (𝑋, 𝑆), and let 𝑆denote the strict transform of S. In particular, (𝑋,𝑆
)is
plt. By [23, Theorem 1.1], (𝑋,𝑆
)has a relatively good minimal model over W, which we denote by
(˜
𝑋, ˜
𝑆). Since 𝑆dominates Wand dim 𝑋−dim 𝑊=1, 𝑆is relatively big over W. Therefore, 𝑆cannot
be contracted on ˜
𝑋, and therefore ˜
𝑆is a divisor. Since (𝑋,𝑆
)is plt, then so is (˜
𝑋, ˜
𝑆). We denote by
˜
𝑊the relatively ample model, which is birational to W.Now,weruna𝐾˜
𝑋-MMP with scaling over ˜
𝑊,
which terminates with a Mori fiber space ˆ
𝑓:ˆ
𝑋→ˆ
𝑊over ˜
𝑊. Since dim 𝑋−dim 𝑊=1, it follows that
ˆ
𝑊→˜
𝑊is birational. As before, since ˜
𝑆dominates ˜
𝑊and is relatively big over ˜
𝑊as dim 𝑋−dim 𝑊=1,
it follows that ˜
𝑆cannot be contracted on ˆ
𝑋.Let ˆ
𝑆be its strict transform on ˆ
𝑋. Lastly, as 𝐾˜
𝑋+˜
𝑆∼Q0/˜
𝑊
and ˜
𝑆is the only log canonical place of (˜
𝑋, ˜
𝑆),ˆ
𝑆is the only log canonical place of (ˆ
𝑋, ˆ
𝑆). But then,
since ˆ
𝑆is a divisor on ˆ
𝑋, it follows that (ˆ
𝑋, ˆ
𝑆)is plt. In particular, ˆ
𝑆is normal. Thus, by [13, Corollary
3.3], up to replacing X,Sand Zwith ˆ
𝑋,ˆ
𝑆and ˆ
𝑍, respectively, in the following of the proof, we may
further assume that (𝑋, 𝑆)is plt, 𝐾𝑋+𝑆∼Q0/𝑊and fis a Mori fiber space.
Step 4: In this step, we introduce a suitable pair structure on the base W.
Let (𝑆, 𝐵𝑆)denote the pair induced by adjunction from (𝑋, 𝐵)to S. By Lemma 2.23,(𝑆, 𝐵𝑆)has
coregularity c. Then, by Lemma 2.17, the inductive hypothesis applies to (𝑆, 𝐵𝑆)for the same valu e
of I.Wealsolet(𝑆, Diff 𝑆(0)) be the pair structure induced from (𝑋, 𝑆)to S.ByStep1, 𝑓𝑆:𝑆→𝑊
is generically 2:1 and hence Galois. By [29, Proposition 4.37.(3)], (𝑆, Diff𝑆(0)) is invariant under the
rational Galois involution. Then, since 𝐾𝑋+𝑆∼Q0/𝑊and fis a Mori fiber space, it follows that 𝐵−𝑆
is the pull-back of a Q-divisor on W. Then, it follows that also (𝑆, 𝐵𝑆)is Galois invariant. Then, by
considering the Stein factorization of 𝑆→𝑊and descending the pair structure thanks to the fact that
𝐾𝑆+𝐵𝑆∼Q0, it follows that we can induce a pair structure (𝑊, 𝐵𝑊)such that 𝑓∗
𝑆(𝐾𝑊+𝐵𝑊)=𝐾𝑆+𝐵𝑆.
Since (𝑆, 𝐵𝑆)has coregularity c,by[13, Proposition 3.11], then so does (𝑊, 𝐵𝑊). Furthermore, since
(𝑆, 𝐵𝑆)has coefficients in 𝐷𝜆, then so does (𝑊, 𝐵𝑊). Indeed, at the codimension 1 points of Wwhere
𝑆→𝑊is étale, we will have the same coefficients on 𝐵𝑊and 𝐵𝑆for the corresponding divisors. Then,
we can consider a prime divisor 𝑄⊂𝑊such that 𝑆→𝑊ramifies of order 2 at the generic point of Q.
Then, over the generic point of Q, by the Riemann–Hurwitz formula, we have
𝐾𝑆+𝑐𝑃 =𝑓∗
𝑆𝐾𝑊+1
2𝑄+𝑐
2𝑄,
where Pis the unique prime divisor in Sdominating Q, in other words coeff𝑃(𝐵𝑆)=𝑐∈𝐷𝜆and
coeff𝑄(𝐵𝑊)=1+𝑐
2. By the definition of 𝐷𝜆,wemusthave𝑐=𝑚−1+𝑎𝜆−1
𝑚for some 𝑚∈Z>0and
𝑎∈Z≥0, and it follows that
1
2+𝑚−1+𝑎𝜆−1
𝑚=2𝑚−1+𝑎𝜆−1
2𝑚∈𝐷𝜆.
Thus, by the inductive hypothesis, we have
𝐼(𝐾𝑊+𝐵𝑊)∼0.(2.5)
Step 5: In this step, we introduce a suitable generalized pair structure on W, and we compare it with
(𝑊, 𝐵𝑊).
By the canonical bundle formula, the lc-trivial fibration 𝑓:(𝑋, 𝐵)→𝑊induces a generalized pair
structure (𝑊,Δ𝑊,N)on W.By[38, §7.5, (7.5.5)] and the fact that the generic fiber of 𝑓:(𝑋, 𝐵)→𝑊
is a conic with two points, we have
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Forum of Mathematics, Sigma 23
𝐾𝑋+𝐵∼𝑓∗(𝐾𝑊+Δ𝑊+N𝑊).(2.6)
Furthermore, the representatives of the b-divisor Nare determined up to Z-linear equivalence.
By [20, proof of Theorem 1.1], in an lc-trivial fibration (𝑌, Γ)→𝐶, the total space of the fibration
and a minimal (with respect to inclusion) log canonical center Ξdominating the base Cinduce the same
generalized pair on the base space. More precisely, they induce the same boundary divisor as Q-Weil
divisor and the same moduli divisor as Q-b-divisor class. Yet, this comparison is possible after the
base change induced by the Stein factorization of Ξ→𝐶,asΞ→𝐶may not have connected fibers.
We observe that the identification between the boundary divisors can also be obtained by inversion of
adjunction together with the connectedness principle.
In our situation, this implies that (𝑋, 𝐵)and (𝑆, 𝐵𝑆)induce the same generalized pair on W,up
to pull-back to the Stein factorization on 𝑆→𝑊. In this case, Sis generically a 2:1 cover of W.
Thus, it follows that (𝑊, Δ𝑊,N)and (𝑊, 𝐵𝑊)agree once pulled back to S. By construction, we have
𝑓∗
𝑆(𝐾𝑊+𝐵𝑊)=𝐾𝑆+𝐵𝑆, and the moduli b-divisor is trivial. In turn, this implies that Δ𝑊=𝐵𝑊and
𝑓∗
𝑆N𝑊∼0. As for the moduli b-divisor, we only claim Z-linear equivalence, as a representative of the
b-divisorial class can be replaced in the Z-linear equivalence class.
Let 𝑆denote the Stein factorization of 𝑆→𝑊, with induced morphism 𝑓𝑆:𝑆→𝑊. Then, 𝑓𝑆is
a finite Galois morphism of degree 2, 𝑓∗
𝑆N𝑊∼0. By construction, 𝑓∗
𝑆N𝑊is Galois invariant since it
is the Q-Cartier pull-back of a Q-divisor on Wvia the finite morphism 𝑓𝑆. We observe that this implies
that 2N𝑊is a Z-divisor and that 2 𝑓∗
𝑆N𝑊is the integral pull-back of the integral divisor 2N𝑊.
Now, let sbe a trivializing section of 𝑓∗
𝑆N𝑊, and let 𝜏be the nontrivial element in the Galois group
of 𝑆→𝑊. By the invariance of 𝑓∗
𝑆N𝑊,wehavethat 𝑓∗
𝑆N𝑊+𝜏∗𝑓∗
𝑆N𝑊=2𝑓∗
𝑆N𝑊. But then, 𝑠⊗𝜏∗𝑠
is a Galois invariant trivializing section of 2 𝑓∗
𝑆N𝑊. Then, this section descends to W, thus implying
that
2N𝑊∼0.(2.7)
Step 6: In this step, we conclude the proof.
Combining the previous steps and using the fact that 2|𝐼,wehave
𝐼(𝐾𝑋+𝐵)∼𝐼𝑓∗(𝐾𝑊+𝐵𝑊+N𝑊)
∼𝐼𝑓∗(𝐾𝑊+𝐵𝑊)+𝐼𝑓∗N𝑊
∼𝑓∗(𝐼(𝐾𝑊+𝐵𝑊)) + 𝑓∗𝐼N𝑊
∼𝑓∗0+𝑓∗0∼0,
where the first linear equivalence follows from Equation (2.6), the second one follows from the fact that
𝐾𝑊+𝐵𝑊is Q-Cartier, the third one follows from the definition of pull-back of Q-divisors, while the
last line follows from Equations (2.5) and (2.7). This concludes the proof.
3. Finite coefficients
In this section, we explain how to reduce the problem of boundedness of complements from pairs with
DCC coefficients to pairs with finite coefficients. First, we prove two lemmata that will be used in the
proof of the main proposition of this section.
Lemma 3.1. Let 𝜙:𝑋→𝑍be a contraction from a projective Q-factorial variety X. Let (𝑋, 𝐵,M)be
a generalized dlt pair over Z. Assume (𝑋, 𝐵, M)is generalized log Calabi–Yau over Z. Assume there
is a component 𝑆⊆𝐵that is vertical over Z. Then, there exists a birational contraction 𝑋𝑋
over Z
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24 F. Figueroa et al.
(𝑋, 𝐵,M)//_______
𝜙
$$
H
H
H
H
H
H
H
H
H
(𝑋,𝐵
,M)
𝜙
zzt
t
t
t
t
t
t
t
t
t
𝑍
satisfying the following conditions:
(i) the generalized pair (𝑋,𝐵
,M)is generalized log canonical;
(ii) the strict transform 𝑆of S in 𝑋is a divisorial generalized log canonical center of (𝑋,𝐵
,M);
(iii) we have that 𝜙−1(𝜙(𝑆)) =𝑆holds set-theoretically; and
(iv) the generalized pair obtained by adjunction of (𝑋,𝐵
,M)to 𝑆is generalized semilog canonical.
Proof. By [36, Lemma 3.5], we may run an MMP for (𝑋, 𝐵 −𝑆, M)over Zwith scaling of an ample
divisor A. By the negativity lemma, the divisor Sis not contracted by this MMP. Furthermore, this
MMP is (𝐾𝑋+𝐵+M𝑋)-trivial. Hence, conditions (i) and (ii) hold for any model in this minimal model
program.
We argue that after finitely many steps condition (iii) holds. Let 𝑋𝑖𝑋𝑖+1be the i-th step of this
MMP and 𝜙𝑖:𝑋𝑖→𝑍be the induced projective morphism. We let 𝜆𝑖the positive real number for which
the birational map 𝑋𝑖𝑋𝑖+1is (𝐾𝑋𝑖+𝐵𝑖−𝑆𝑖+M𝑋𝑖+𝜆𝑖𝐴𝑖)-trivial. Let 𝜆∞=lim𝑖𝜆𝑖.If𝜆∞>0, then
the previous MMP is also an MMP for (𝑋, 𝐵−𝑆+𝜆∞𝐴, M).By[36, Lemma 3.7], this is also an MMP for
a klt pair with big boundary over Zwhich must terminate by [6]. Let 𝑋be the model where this MMP
terminates. In 𝑋,wehavethat−𝑆is nef over Z.So𝑆must be the set-theoretic preimage of 𝜙(𝑆).
From now on, we assume that 𝜆∞=0. Let 𝑊1,...,𝑊
𝑘be the irreducible components of 𝜙−1(𝜙(𝑆))
different than S. Note that every step of the MMP is S-positive. Thus, if the strict transform of any
component 𝑊𝑗is contained in the exceptional locus of 𝑋𝑖𝑋𝑖+1, then the number of components
of 𝜙−1
𝑖(𝜙(𝑆)) drops. Henceforth, it suffices to show that each such component is eventually contained
in the exceptional locus of a step of the MMP. Assume 𝜙(𝑊1)⊆𝜙(𝑆)is maximal among the sets
𝜙(𝑊𝑗)’s with respect to the inclusion. Let 𝑧∈𝜙(𝑊1)be a general point. Up to reordering the 𝑊𝑗’s,
since 𝑋→𝑍has connected fibers, we may assume that 𝜙−1(𝑧)∩𝑊1∩𝑆is nonempty. Hence, for a
general point 𝑤∈𝜙−1(𝑧)∩𝑊1, we can find a curve Csuch that 𝑤∈𝐶,𝐶𝑆, and Cintersects S
nontrivially. In particular, we have that 𝐶⊆Bs−(𝐾𝑋+𝐵−𝑆+M𝑋/𝑍). In particular, since we have
𝑤∈𝐶, it follows that 𝑤∈Bs−(𝐾𝑋+𝐵−𝑆+M𝑋/𝑍). Since wis a general point in 𝜙−1(𝑧)∩𝑊1,
we also get that 𝜙−1(𝑧)∩𝑊1⊆Bs−(𝐾𝑋+𝐵−𝑆+M𝑋/𝑍). Since zis general, we conclude that
𝑊1⊂Bs−(𝐾𝑋+𝐵−𝑆+M𝑋/𝑍).For𝜆1>0 small enough, we have that
𝑊1⊂Bs(𝐾𝑋+𝐵−𝑆+𝜆1𝐴+M𝑋/𝑍).
Since 𝜆∞=0, we conclude that for some ithe birational map 𝑋𝑋𝑖is a minimal model for
(𝑋, 𝐵 −𝑆+𝜆1𝐴, M/𝑍). In particular, 𝑊1must be contained in the exceptional locus of 𝑋𝑋𝑖. Hence,
after finitely many steps of this MMP, condition (iii) is satisfied.
Let 𝑋be a model where condition (iii) holds. By construction, the generalized pair (𝑋,𝐵
,M)is
obtained by a partial run 𝑋𝑋of the MMP for (𝑋, 𝐵 −𝑆, M). In particular, (𝑋,𝐵
−𝑆,M)is
generalized dlt and Q-factorial. Hence, (𝑋,𝐵
−𝜖𝐵) is klt, where 0 <𝜖1. By [19, Example 2.6],
the pair obtained by adjunction of (𝑋,𝐵
−𝜖𝐵+𝜖𝑆)to 𝑆is semilog canonical. In turn, by letting
𝜖→0, it follows that the pair obtained by adjunction of (𝑋,𝐵
)to 𝑆is semilog canonical. Hence, the
generalized pair obtained by adjunction of (𝑋,𝐵
,M)to 𝑆is generalized semilog canonical.
Lemma 3.2. Let c and p be nonnegative integers and Λ⊂Q>0be a closed set satisfying the DCC. There
exists a finite subset RR(𝑐, 𝑝, Λ)⊆Λsatisfying the following. Let (𝑋, 𝐵, M)be a generalized log
canonical pair over Z and 𝑋→𝑍be a fibration for which the following conditions hold:
◦the generalized pair (𝑋, 𝐵, M)is log Calabi–Yau over Z;
◦the generalized pair (𝑋, 𝐵, M)has coregularity c;
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Forum of Mathematics, Sigma 25
◦𝑝Mis b-Cartier; and
◦the coefficients of B belong to Λ.
Then, the coefficients of 𝐵hor belong to R.
Proof. Let (𝑋𝑖,𝐵
𝑖,M𝑖)be a sequence of generalized pairs as in the statement and 𝜙𝑖:𝑋𝑖→𝑍𝑖be the
corresponding contractions. Assume there exist prime divisors 𝑃𝑖⊂𝑋𝑖for which 𝑑𝑖coeff 𝑃𝑖(𝐵𝑖)
is strictly increasing and 𝑃𝑖dominates 𝑍𝑖. Assume that some generalized log canonical center of
(𝑋𝑖,𝐵
𝑖,M𝑖)is ver tical over 𝑍𝑖. We may replace (𝑋𝑖,𝐵
𝑖,M𝑖)with a generalized dlt modification
and assume there is a component 𝑆𝑖⊆𝐵𝑖that is vertical over 𝑍𝑖. Furthermore, up to choosing a
different vertical component possibly dominating a different subset of 𝑍𝑖, we may assume that there is
a generalized log canonical center of (𝑋𝑖,𝐵
𝑖,M𝑖)dimension ccontained in 𝑆𝑖. By Lemma 3.1,upto
losing the dlt property for (𝑋𝑖,𝐵
𝑖,M𝑖), we may assume that 𝑆𝑖is the set-theoretic preimage of 𝜙.Let
𝑊𝑖be the normalization of 𝑆𝑖, and let 𝑊𝑖→𝑍𝑊𝑖be the fibration obtained by the Stein factorization of
𝑊𝑖→𝜙𝑖(𝑆𝑖).Let(𝑊𝑖,𝐵
𝑖,N𝑖)be the generalized pair obtained by generalized adjunction of (𝑋𝑖,𝐵
𝑖,M𝑖)
to 𝑊𝑖. Note that 𝑃𝑖∩𝑆𝑖dominates 𝜙(𝑆𝑖). Hence, there is a component of 𝐵𝑊𝑖with coefficient in 𝐷𝑑𝑖(Λ)
which is horizontal over 𝑍𝑊𝑖(see Lemma 2.17). Observe that the following conditions hold:
◦the generalized pair (𝑊𝑖,𝐵
𝑖,N𝑖)is log Calabi–Yau over 𝑍𝑊𝑖;
◦the generalized pair (𝑊𝑖,𝐵
𝑖,N𝑖)has coregularity c;
◦𝑝N𝑖is b-Cartier;
◦the coefficients of 𝐵𝑊𝑖belong to 𝐷(Λ); and
◦there is a component 𝑄𝑖of 𝐵𝑊𝑖horizontal over 𝑍𝑖whose coefficient belong to 𝐷𝑑𝑖(Λ).
We replace (𝑋𝑖,𝐵
𝑖,M𝑖)with (𝑊𝑖,𝐵
𝑊𝑖,N𝑖)and 𝑃𝑖with 𝑄𝑖. After finitely many replacements, we may
assume that for every ithe generalized log canonical centers of (𝑋𝑖,𝐵
𝑖,M𝑖)are horizontal over 𝑍𝑖.[10,
Theorem 2] applied to the general fiber of 𝑋𝑖→𝑍𝑖implies that the coefficients of 𝐵hor belong to an
ACC set. Thus, we conclude that the coefficients of 𝐵hor belong to a finite set Rwhich only depends on
c,pand Λ.
The proof of the following corollary is verbatim from the previous proof by replacing [10, Theorem 2]
with [13, Corollary 3].
Corollary 3.3. Let (𝑋, 𝐵,M)be a generalized log canonical pair over Z and 𝑋→𝑍be a fibration for
which the following conditions hold:
◦the generalized pair (𝑋, 𝐵, M)is log Calabi–Yau over Z;
◦the generalized pair (𝑋, 𝐵, M)has coregularity 0;
◦2Mis b-Cartier; and
◦the coefficients of B belong to S.
Then, the coefficients of 𝐵hor belong to {1,1
2}.
Notation 3.4. Let Λ⊂Q>0be a closed set of rational numbers satisfying the DCC. Given a natural
number 𝑚∈Z>0, we consider the partition
P𝑚0,1
𝑚,1
𝑚,2
𝑚,...,𝑚−1
𝑚,1
of the interval [0,1]. Denote by 𝐼(𝑏, 𝑚)the interval of P𝑚containing 𝑏∈Λ. Define the number
𝑏𝑚sup{𝑥|𝑥∈𝐼(𝑏, 𝑚)∩Λ}∈Λ.
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26 F. Figueroa et al.
For every positive integer mand every 𝑏∈Λ, we have that 𝑏≤𝑏𝑚as 𝑏∈Λ∩𝐼(𝑏, 𝑚).If𝑏∈Λis fixed
and mdivisible enough, we have that 𝑏=𝑏𝑚.ThesetC𝑚{𝑏𝑚|𝑏∈Λ}is finite, and we have that
the set
Λ=
𝑚∈Z>0
C𝑚
satisfies the DCC. Given a boundary divisor 𝐵≥0 on a quasi-projective variety X, we can write
𝐵=𝑗𝑏(𝑗)𝐵(𝑗)in a unique way such that the 𝐵(𝑗)’s are pairwise different prime divisors on X.Ifthe
coefficients of Bbelong to Λ, we define
𝐵𝑚
𝑗
𝑏(𝑗)
𝑚𝐵(𝑗).
It follows that 𝐵≤𝐵𝑚.
Theorem 3.5. Let c and p be nonnegative integers and Λ⊂Q>0be a set satisfying the DCC with
rational accumulation points. There exists a finite subset RR(𝑐, 𝑝, Λ)⊂ ¯
Λ⊂Q>0satisfying the
following. Let (𝑋, 𝐵,M)be a generalized log canonical pair over Z, 𝑋→𝑍be a contraction and 𝑧∈𝑍
be a point. Assume the following conditions are satisfied:
◦the variety X is of Fano type over Z;
◦the divisor B has coefficients in Λ;
◦𝑝Mis b-Cartier;
◦the generalized pair (𝑋, 𝐵, M)has coregularity c around z; and
◦the divisor −(𝐾𝑋+𝐵+M𝑋)is nef over Z.
There exists a birational transformation 𝑋𝑋over Z and a generalized pair (𝑋,Γ,M)satisfying
the following:
◦the coefficients of Γbelong to R;
◦the pair (𝑋,Γ,M)has coregularity c over z;
◦the divisor −(𝐾𝑋+Γ+M𝑋)is nef over a neighborhood of 𝑧∈𝑍; and
◦if (𝑋,Γ,M)is N-complemented over 𝑧∈𝑍, then (𝑋, 𝐵,M)is N-complemented over 𝑧∈𝑍.
Proof. Let (𝑋, 𝐵, M)be a generalized pair as in the conditions of the theorem. By passing to a
Q-factorial generalized dlt modification, we may assume the considered generalized pairs are gdlt and
Q-factorial. We denote by 𝑚(𝑋, 𝐵,M)the minimal mfor which R=C𝑚satisfies the statement of the
theorem for (𝑋, 𝐵,M). Since 𝐵𝑚=𝐵for mlarge enough, then 𝑚(𝑋, 𝑏,M)is finite. It suffices to show
that 𝑚(𝑋, 𝐵,M)is bounded above by a constant that only depends on c,pand Λ. Assume that this is
not the case. Then, we may find a sequence of generalized pairs (𝑋𝑖,𝐵
𝑖,M𝑖), contractions 𝑋𝑖→𝑍𝑖and
closed points 𝑧𝑖∈𝑍𝑖, satisfying the conditions of the theorem, for which 𝑚𝑖𝑚(𝑋𝑖,𝐵
𝑖,M𝑖)−1is
strictly increasing. In particular, we have that 𝐵𝑖,𝑚𝑖−𝐵𝑖is a nontrivial effective divisor. Let 𝑃𝑖be a prime
component of 𝐵𝑖,𝑚𝑖−𝐵𝑖that intersects the fiber over z. We study how the singularities of (𝑋𝑖,𝐵
𝑖,M𝑖)
over 𝑧𝑖∈𝑍𝑖and the nefness of −(𝐾𝑋𝑖+𝐵𝑖+M𝑖)over 𝑍𝑖change as we increase the coefficient at 𝑃𝑖.
Step 1: For the generalized pair (𝑋𝑖,𝐵
𝑖,M𝑖), we will produce a positive real number 𝑡𝑖which either
computes a log canonical threshold or a pseudo-effective threshold.
For each generalized pair (𝑋𝑖,𝐵
𝑖,M𝑖), we will define a real number 𝑡𝑖as follows. We consider the
generalized pairs
(𝑋𝑖,𝐵
𝑖,𝑡,M𝑖)(𝑋𝑖,𝐵
𝑖+𝑡(coeff𝑃𝑖(𝐵𝑖,𝑚𝑖−𝐵𝑖))𝑃𝑖,M𝑖).(3.1)
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Forum of Mathematics, Sigma 27
Let 𝑡𝑖,0be the largest real number for which the generalized pair (3.1) is generalized log canonical over
𝑧𝑖∈𝑍𝑖and
−(𝐾𝑋𝑖+𝐵𝑖+𝑡(coeff𝑃𝑖(𝐵𝑖,𝑚𝑖−𝐵𝑖))𝑃𝑖+M𝑖,𝑋𝑖)
is nef over a neighborhood of 𝑧𝑖∈𝑍𝑖. Assume that 𝑡𝑖,0<1. Then, for 𝑡>𝑡
𝑖,0close enough to 𝑡𝑖,0one
of the following conditions hold:
(i) the generalized pair (𝑋𝑖,𝐵
𝑖,𝑡,M𝑖)is not generalized log canonical over 𝑧𝑖∈𝑍𝑖;or
(ii) the divisor −(𝐾𝑋𝑖+𝐵𝑖,𝑡 +M𝑖,𝑋𝑖)is not pseudo-effective over a neighborhood of 𝑧𝑖∈𝑍𝑖and
(𝑋𝑖,𝐵
𝑖,𝑡,M𝑖)is generalized log canonical over a neighborhood of 𝑧𝑖∈𝑍𝑖;or
(iii) the divisor −(𝐾𝑋𝑖+𝐵𝑖,𝑡 +M𝑖,𝑋𝑖)is pseudo-effective, but it is not nef over every neighborhood of
𝑧𝑖∈𝑍𝑖and (𝑋𝑖,𝐵
𝑖,𝑡,M𝑖)is generalized log canonical over a neighborhood of 𝑧𝑖∈𝑍𝑖.
Assume that case (i) holds. Then, we set
𝑡𝑖𝑡𝑖,0coeff𝑃𝑖(𝐵𝑖,𝑚𝑖−𝐵𝑖)+coeff𝑃𝑖(𝐵𝑖)∈[coeff 𝑃𝑖(𝐵𝑖),coeff𝑃𝑖(𝐵𝑖,𝑚𝑖)).
We show that 𝑡𝑖computes a generalized log canonical threshold of coregularity at most c. Indeed, we
have that
𝑡𝑖=glct((𝑋𝑖,𝐵
𝑖−coeff𝑃𝑖(𝐵𝑖)𝑃𝑖,M𝑖);𝑃𝑖),
so 𝑡𝑖is a generalized log canonical threshold over 𝑧𝑖∈𝑍𝑖. By construction, the support of 𝑃𝑖contains a
generalized log canonical center of the generalized pair (𝑋𝑖,𝐵
𝑖+𝑡𝑖𝑃𝑖,M𝑖). Set N𝑖−(𝐾𝑋𝑖+𝐵𝑖+𝑡𝑖𝑃𝑖+
M𝑖)as a nef b-divisor over 𝑍𝑖, that is, we set N𝑖to be the b-Cartier closure of −(𝐾𝑋𝑖+𝐵𝑖+𝑡𝑖𝑃𝑖+M𝑖,𝑋𝑖).
Then, we have that
(𝑋𝑖,𝐵
𝑖+𝑡𝑖𝑃𝑖,M𝑖+N𝑖)(3.2)
is a generalized log Calabi–Yau pair over 𝑍𝑖. Furthermore, the generalized log canonical centers of
Equation (3.2) are the same as the generalized log canonical centers of (𝑋𝑖,𝐵
𝑖+𝑡𝑖𝑃𝑖,M𝑖).By[16,
Theorem 1.4], up to replacing (𝑋𝑖,𝐵
𝑖+𝑡𝑖𝑃𝑖,M𝑖)with a generalized dlt modification, the support of 𝑃𝑖
contains a generalized log canonical center of (𝑋𝑖,𝐵
𝑖+𝑡𝑖𝑃𝑖,M𝑖)of dimension at most c. Hence, 𝑡𝑖is a
generalized log canonical threshold of coregularity at most c.
Assume that case (ii) holds. Then, we set
𝑡𝑖𝑡𝑖,0coeff(𝐵𝑖,𝑚𝑖−𝐵𝑖)+coeff 𝑃𝑖(𝐵𝑖)∈[coeff 𝑃𝑖(𝐵𝑖),coeff𝑃𝑖(𝐵𝑖,𝑚𝑖)).
In this case, we can find a Mori fiber space structure 𝑋
𝑖→𝑊𝑖over 𝑍𝑖such that the following conditions
are satisfied:
◦the generalized pair (𝑋
𝑖,𝐵
𝑖−coeff𝑃
𝑖(𝐵
𝑖)𝑃
𝑖+𝑡𝑖𝑃
𝑖,M𝑖)is generalized log Calabi–Yau over 𝑊𝑖;
◦the prime divisor 𝑃
𝑖is ample over 𝑊𝑖.
Note that (𝑋
𝑖,𝐵
𝑖−coeff𝑃
𝑖(𝐵
𝑖)𝑃
𝑖+𝑡𝑖𝐵
𝑖,M𝑖)has coregularity at most c. We have that 𝑡𝑖is the coefficient
of a component of 𝐵
𝑖−coeff𝑃
𝑖(𝐵
𝑖)+𝑡𝑖𝑃
𝑖which is horizontal over 𝑊𝑖.
From now on, we assume that (i) and (ii) do not happen. Assume that (iii) holds. Then, there exists a
birational contraction 𝑋𝑖𝑋
𝑖which is (𝐾𝑋𝑖+𝐵𝑖,𝑡𝑖,0+M𝑖,𝑋𝑖)-trivial. Indeed, this contraction is defined
by the partial −(𝐾𝑋𝑖+𝐵𝑖,𝑡+M𝑖,𝑋𝑖)-MMP with scaling of 𝑃𝑖,for𝑡close enough to 𝑡𝑖,0as in (iii). By
construction, the first scaling factor is 𝑡−𝑡𝑖,0, and since −(𝐾𝑋𝑖+𝐵𝑖,𝑡+M𝑖,𝑋𝑖)is pseudo-effective over
𝑍𝑖and 𝑋𝑖→𝑍𝑖is of Fano type, this MMP terminates with a good minimal model. In particular, at the
last step of this MMP, the scaling factor is 0. Then, 𝑋
𝑖is the outcome of the last step where the scaling
factor is 𝑡−𝑡𝑖,0. In particular, we have that −(𝐾𝑋
𝑖+𝐵
𝑖,𝑡 +M𝑖,𝑋
𝑖)is nef over 𝑍𝑖for 𝑡>𝑡
𝑖,0close enough
to 𝑡𝑖,0. Note that an N-complement of (𝑋
𝑖,𝐵
𝑖,𝑡𝑖,0,M𝑋
𝑖)induces an N-complement of (𝑋𝑖,𝐵
𝑖,𝑡𝑖,0,M𝑖)
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28 F. Figueroa et al.
by pulling back, and so an N-complement of (𝑋𝑖,𝐵
𝑖,M𝑖). Henceforth, we may replace 𝑋𝑖with 𝑋
𝑖and
keep increasing t. Since 𝑋𝑖is of Fano type over 𝑍𝑖, there are only finitely many birational contractions
𝑋𝑖𝑋
𝑖. Therefore, we can replace 𝑋𝑖with 𝑋
𝑖only finitely many times. Thus, after finitely many
birational contractions, we either have that 𝑡𝑖,0=1, that 𝑡𝑖<1 is a log canonical threshold or that 𝑡𝑖<1
is a pseudo-effective threshold.
We assume that 𝑡𝑖,0=1. Then there exists a birational contraction 𝑋𝑖𝑋
𝑖and a generalized log
canonical pair
(𝑋
𝑖,𝐵
𝑖−(coeff 𝑃𝑖𝐵
𝑖+coeff𝑃𝑖𝐵
𝑖,𝑚𝑖)𝑃𝑖,M𝑖)(3.3)
for which the divisor
−(𝐾𝑋
𝑖+𝐵
𝑖−(coeff 𝑃𝑖𝐵
𝑖+coeff𝑃𝑖𝐵
𝑖,𝑚𝑖)𝑃𝑖+M𝑖,𝑋
𝑖)
is nef over 𝑍𝑖. Note that the coefficients of the boundary of the generalized pair (3.3) belong to Λand
the variety 𝑋
𝑖is of Fano type over 𝑍𝑖. The b-nef divisor 𝑝M𝑋
𝑖is b-Cartier. By construction, if the
generalized pair (3.3) admits an N-complement, then so does (𝑋𝑖,𝐵
𝑖,M𝑖). We can replace (𝑋𝑖,𝐵
𝑖,M𝑖)
with the generalized pair (3.3). By doing so, we decrease the number of components of 𝐵𝑖,𝑚𝑖−𝐵𝑖.By
the choice of 𝑚𝑖, the divisor 𝐵𝑖,𝑚𝑖−𝐵𝑖cannot be zero after this replacement. Thus, we may pick a new
component and start increasing its coefficient (to this end, notice that 𝑋
𝑖is Q-factorial by construction).
Note that this process must terminate either with 𝑡𝑖<1 a log canonical threshold or 𝑡𝑖<1 a pseudo-
effective threshold. Otherwise, we contradict the definition of 𝑚𝑖.
Step 2: We show that a subsequence of the 𝑡𝑖’s is strictly increasing.
Up to passing to a subsequence, we may assume that 𝑡𝑖is either strictly increasing, strictly decreasing
or it stabilizes. The condition 𝑡𝑖∈[coeff 𝑃(𝐵𝑖),coeff 𝑃(𝐵𝑖,𝑚𝑖)) implies that 𝑡𝑖must be strictly increasing.
Step 3: We finish the proof of the statement.
If case (i) happens infinitely many times, then we get a contradiction to the ACC for generalized
log canonical thresholds with bounded coregularity [10, Theorem 1]. If case (ii) happens infinitely
many times, then we get a contradiction to Lemma 3.2. In any case, we get a contradiction. Hence, the
sequence 𝑚𝑖has an upper bound.
4. Semilog canonical pairs
In this section, we discuss the index of semilog canonical pairs. We show that to control the index of
a semilog canonical log Calabi–Yau pair of coregularity cit suffices to control the index of dlt log
Calabi–Yau pairs of coregularity c. To prove the main statement of this section, we will need to use the
language of admissible and preadmissible sections. The preliminary results for this section are taken
from [19,17,44].
The following definition is due to Fujino [17, Definition 4.1].
Definition 4.1. Let (𝑋, 𝐵)be a possibly disconnected projective semi-dlt pair of dimension n, and
assume that 𝑁(𝐾𝑋+𝐵)is Cartier. Let (𝑋,𝐵
)be its normalization and 𝐷𝑛⊂𝑋be the normalization
of 𝐵. As usual, we denote by (𝐷𝑛,𝐵
𝐷𝑛)the dlt pair obtained by adjunction of (𝑋,𝐵
)to 𝐷𝑛.We
define the concept of preadmissible and admissible sections in 𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵))) by induction on
the dimension using the two following rules:
1. we say that a section
𝑠∈𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵)))
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Forum of Mathematics, Sigma 29
is preadmissible if 𝑠|𝐷𝑛∈𝐻0(𝐷𝑛,O𝐷𝑛(𝐼(𝐾𝐷𝑛+𝐵𝐷𝑛))) is admissible. This set is denoted by
𝑃𝐴(𝑋, 𝐼(𝐾𝑋+𝐵)); and
2. we say that
𝑠∈𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵))
is admissible if sis preadmissible and 𝑔∗(𝑠|𝑋
𝑖)=𝑠|𝑋
𝑗holds for every crepant birational map
𝑔:(𝑋
𝑖,𝐵
𝑋
𝑖)(𝑋
𝑗,𝐵
𝑋
𝑗), where 𝑋=𝑋
𝑖are the irreducible components of 𝑋.Thesetof
admissible sections is denoted by 𝐴(𝑋, 𝐼 (𝐾𝑋+𝐵)).
The following lemma is due to Gongyo [21, Remark 5.2].
Lemma 4.2. Let (𝑋, 𝐵)be a projective semi-dlt pair for which 𝐼(𝐾𝑋+𝐵)∼0.Let𝜋:(𝑋,𝐵
)→(𝑋, 𝐵)
be its normalization. Then, a section 𝑠∈𝐻0(𝑋, O𝑋(𝐼(𝐾𝑋+𝐵))) is preadmissible (resp. admissible) if
and only if 𝜋∗𝑠∈𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵))) is preadmissible (resp. admissible).
The following lemma allows us to descend linear equivalence from normal varieties to semilog
canonical varieties (see, e.g., [17, Lemma 4.2]).
Lemma 4.3. Let (𝑋, 𝐵)be a projective semilog canonical pair for which 𝐼(𝐾𝑋+𝐵)is an integral
divisor. Let (𝑋,𝐵
)→(𝑋, 𝐵)be its normalization and (𝑌, 𝐵
𝑌)aQ-factorial dlt modification of
(𝑋,𝐵
). Assume that 𝐼(𝐾𝑌+𝐵𝑌)is Cartier. Then, a section 𝑠∈𝑃𝐴(𝐼(𝐾𝑌+𝐵𝑌)) descends to
𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵))). In particular, if we have that 𝐼(𝐾𝑌+𝐵𝑌)∼0, and there exists a nowhere
vanishing section 0≠𝑠∈𝑃𝐴(𝐼(𝐾𝑌+𝐵𝑌)), then we have that 𝐼(𝐾𝑋+𝐵)∼0.
Proof. The first part of the statement is [17, Lemma 4.2]. Now, 𝑠∈𝑃𝐴(𝐼(𝐾𝑌+𝐵𝑌)) is nowhere vanish-
ing, it descends to a nowhere vanishing section of O𝑋(𝐼(𝐾𝑋+𝐵)), thus showing that 𝐼(𝐾𝑋+𝐵)∼0.
In the context of connected dlt pairs, the set of admissible sections is the same as the set of
preadmissible sections (see, e.g., [17, Proposition 4.7]).
Lemma 4.4. Let (𝑋, 𝐵)be a connected projective dlt pair with 𝐵≠0. Assume that 𝐼(𝐾𝑋+𝐵)∼0
and I is even. Then, we have that
𝑃𝐴(𝐼(𝐾𝑋+𝐵)) =𝐴(𝐼(𝐾𝑋+𝐵)).
On the other hand, in the dlt setting, we can lift admissible sections from the boundary to preadmissible
sections on the whole pair (see, e.g., [44, Lemma 3.2.14]).
Lemma 4.5. Assume that (𝑋, 𝐵)is a possibly disconnected projective dlt pair. Assume that 𝐼(𝐾𝑋+𝐵)∼0
and I is even. Assume that
0≠𝑠∈𝐴(𝐵,𝐼(𝐾𝑋+𝐵)|𝐵).
Then, there exists
0≠𝑡∈𝑃𝐴(𝑋, 𝐼(𝐾𝑋+𝐵))
for which 𝑡|𝐵=𝑠.
The following lemma states that the boundedness of indices for klt Calabi–Yau pairs together with the
boundedness of B-representations imply the existence of admissible sections (see, e.g., [44, Proposition
3.2.7]).
Lemma 4.6. Let c be a nonnegative integer and Λbe a set of rational numbers satisfying the descending
chain condition. Assume Conjecture 1and Conjecture 2in dimension c. There is a constant 𝐼(Λ,𝑐),
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30 F. Figueroa et al.
only depending on Λand c, satisfying the following. Let (𝑋, 𝐵)be a projective klt log Calabi–Yau pair
with coefficients in Λand dimension c. Then, there is a section
0≠𝑠∈𝐴(𝑋, 𝐼(Λ,𝑐)(𝐾𝑋+𝐵)).
Note that for a klt log Calabi–Yau pair an admissible section is nothing else than a section which is
invariant under the pull-back via crepant birational transformations.
Finally, we prove the following lemma that allows us to produce admissible sections on possibly
disconnected dlt pairs, once we know the existence of admissible sections on connected dlt pairs. The
proof is similar to that of [44, Proposition 3.2.8].
Lemma 4.7. Let d be a positive integer. Let (𝑋, 𝐵)be a possibly disconnected projective dlt log
Calabi–Yau pair. Assume that for every component (𝑋𝑖,𝐵
𝑖)of (𝑋, 𝐵), we have a nontrivial section on
𝐴(𝑋𝑖,𝐼(𝐾𝑋𝑖+𝐵𝑖)). Then, we have that 𝐴(𝑋, 𝐼(𝐾𝑋+𝐵)) admits a nowhere vanishing section.
Proof. Let (𝑋, 𝐵)be a possibly disconnected projective dlt log Calabi–Yau pair. We write (𝑋𝑖,𝐵
𝑖)for
its components for 𝑖∈{1,...,𝑘}. By assumption, for each i,wehave
0≠𝑠𝑖∈𝐴(𝑋𝑖,𝐼(𝐾𝑋𝑖+𝐵𝑖)).
For 𝜆𝑖∈C, we define
𝑠(𝜆1𝑠1,...,𝜆
𝑘𝑠𝑘)∈𝐻0(𝑋, 𝑁(𝐾𝑋+𝐵)).
Let 𝐺=Bir(𝑋, 𝐵). We claim that the image of Gin 𝐺𝐿(𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵)))) is finite. We denote by
𝜌𝐼:Bir(𝑋, 𝐵)→𝐺𝐿(𝐻0(𝑋, O𝑋(𝐼(𝐾𝑋+𝐵))))
the usual map induced by pulling back sections. Thus, we want to show that 𝜌𝐼(𝐺)is finite. By [19], the
finiteness of 𝜌𝐼(𝐺)is known if Xis connected. Thus, we need to reduce the disconnected case to the
connected one. Note that for every 𝑔∈𝐺,wehavethat𝜌𝐼(𝑔)𝑘!has finite order by [19, Theorem 3.15]
and the fact that the order of any permutation in 𝑆𝑘divides 𝑘!. Hence, we conclude that 𝜌𝐼(𝐺)is a finitely
generated subgroup of finite exponent of a general linear group, where the bound is determined by 𝑘!
and the least common multiple of the orders of the pluricanonical representations of each irreducible
component. Indeed, notice that 𝜌𝐼(𝐺)is finitely generated, as it is the extension of two finite groups:
the image via 𝜌𝐼of the subgroup fixing the irreducible components of X, which is isomorphic to the
product of the pluricanonical representations of each irreducible component (hence, a finite group by
[19]), and a subgroup of 𝑆𝑘. By a theorem due to Burnside, known as the bounded Burnside problem
for linear groups [8, Theorem 6.13], we conclude that 𝜌𝐼(𝐺)is finite.
Consider the section
𝑡
𝜎∈𝐺
𝜎(𝑠).
By construction, we have that 𝑡∈𝐴(𝑋, 𝐼(𝐾𝑋+𝐵)). Indeed, tis invariant under the action of any
birational transformation of (𝑋, 𝐵). Thus, the restriction of tto every log canonical center is also
invariant. It suffices to show that tis nontrivial on each component of X. By considering orbits of the
action, we may assume that Bir(𝑋, 𝐵)acts transitively on the components of X. Consider the basis
((0,...,𝑠
𝑖,...,0))1≤𝑖≤𝑘of 𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵))). Since the sections 𝑠𝑖are admissible, in this basis,
the action of 𝜌𝐼(𝑔)is represented by a matrix whose diagonal entries are either 0 (if 𝑔(𝑋𝑖)≠𝑋𝑖)or
1(if𝑔(𝑋𝑖)=𝑋𝑖). Hence, by observing that the matrix associated to the identity element of Gis the
identity matrix, it follows that the action of 𝜎∈𝐺𝜎in this basis is given by a nontrivial matrix whose
diagonal entries are all integers greater than or equal to 1. By the transitivity of the action and the fact
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that the action of 𝜎∈𝐺𝜎is given by a matrix whose diagonal are positive integers, we deduce that we
can find 𝜆𝑖∈Cfor which tis nonzero on all components.
Now, we are ready to prove the main theorem of this section.
Theorem 4.8. Let c be a nonnegative integer and Λbe a set of rational numbers satisfying the descending
chain condition. Assume Conjecture 1and Conjecture 2in dimension c. There is a constant 𝐼(Λ,𝑐), only
depending on Λand c, satisfying the following. Let (𝑋, 𝐵)be a projective dlt pair with coefficients in Λ
and coregularity c. Assume that 𝐼(Λ,𝑐)(𝐾𝑋+𝐵)∼0. Then, there is a nowhere vanishing admissible
section
0≠𝑠∈𝐴(𝑋, 𝐼(Λ,𝑐)(𝐾𝑋+𝐵)).
Proof. Let 𝐼(Λ,𝑐)be the positive integer given by Lemma 4.6. Without loss of generality, we may
assume that 𝐼(Λ,𝑐)is even.
By induction on i, we prove that every i-dimensional log canonical center Vof (𝑋, 𝐵)satisfies that
0≠𝑠𝑉∈𝐴(𝑉, 𝐼(Λ,𝑐)(𝐾𝑉+𝐵𝑉)),(4.1)
where (𝑉, 𝐵𝑉)is the pair obtained by dlt adjunction of (𝑋, 𝐵)to V.If𝑖=𝑐, then the pair is klt and the
statement follows from Lemma 4.6.
Now, assume that the statement holds for every irreducible i-dimensional dlt center of (𝑋, 𝐵).Let
Wbe a log canonical center of (𝑋, 𝐵)of dimension 𝑖+1. The pair (𝑊, 𝐵𝑊)obtained from adjunction
is dlt of dimension 𝑖+1 and it holds that 𝐼(Λ,𝑐)(𝐾𝑊+𝐵𝑊)∼0. Let 𝑊0be the union of all the log
canonical centers of (𝑊, 𝐵𝑊).Let(𝑊0,𝐵
𝑊0)be the pair obtained by performing adjunction of (𝑊, 𝐵𝑊)
to (𝑊0,𝐵
𝑊0). Hence, (𝑊0,𝐵
𝑊0)is an i-dimensional semi-dlt pair with 𝐼(Λ,𝑐)(𝐾𝑊0+𝐵𝑊0)∼0. Observe
that 𝑊0may have multiple irreducible components. Let 𝑛𝑈:𝑈→𝑊0be the normalization of 𝑊0.Let
(𝑈, 𝐵𝑈)be the pair obtained by log pull-back of (𝑊0,𝐵
𝑊0)to U. Then, we have that (𝑈, 𝐵𝑈)is a
possibly disconnected projective dlt pair of dimension iand coregularity c.By[16, Theorem 1.4], we
know that every component has coregularity c. Furthermore, we have that 𝐼(Λ,𝑐)(𝐾𝑈+𝐵𝑈)∼0. By
Equation (4.1) in dimension i, we have that each irreducible component 𝑈𝑗of Usatisfies that
0≠𝑠𝑈,𝑗 ∈𝐴(𝑈𝑗,𝐼(Λ,𝑐)(𝐾𝑈𝑗+𝐵𝑈𝑗)).
By Lemma 4.7, we conclude that there exists a nowhere vanishing section
0≠𝑠𝑈∈𝐴(𝑈, 𝐼(Λ,𝑐)(𝐾𝑈+𝐵𝑈)).
By Lemma 4.3, we conclude that this section descends to 𝑠𝑊0∈𝐻0(𝑊0,O𝑊0(𝐼(Λ,𝑐)(𝐾𝑊0+𝐵𝑊0))).
Note that we have 𝑛∗
𝑈𝑠𝑊0=𝑠𝑈. By Lemma 4.2, we conclude that 𝑠𝑊0∈𝐴(𝑊0,𝐼(Λ,𝑐)(𝐾𝑊0+𝐵𝑊0)).
By Lemma 4.5, we conclude that there exists
0≠𝑡𝑊∈𝑃𝐴(𝑊, 𝐼(Λ,𝑐)(𝐾𝑊+𝐵𝑊)).
Finally, since Wis connected, we conclude by Lemma 4.4, that there is a section
0≠𝑡𝑊∈𝐴(𝑊, 𝐼(Λ,𝑐)(𝐾𝑊+𝐵𝑊)).
This finishes the inductive step.
We conclude, that for every 𝑖∈{𝑐,...,dim 𝑋},everyi-dimensional log canonical center of
(𝑋, 𝐵)admits an admissible section. In particular, we get a nowhere vanishing section 0 ≠𝑠𝑋∈
𝐴(𝑋, 𝐼(Λ,𝑐)(𝐾𝑋+𝐵)) as claimed.
The previous theorem allows controlling the index of semilog canonical pairs once we can control
the index of their normalization.
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32 F. Figueroa et al.
Theorem 4.9. Let c be a nonnegative integer and Λbe a set of rational numbers satisfying the descending
chain condition. Assume Conjecture 1and Conjecture 2in dimension c. There is a constant 𝐼(Λ,𝑐), only
depending on Λand c, satisfying the following. Let (𝑋, 𝐵)be a projective semilog canonical pair with
coefficients in Λand coregularity c. Let (𝑌, 𝐵
𝑌)be a Q-factorial dlt modification of a normalization of
(𝑋, 𝐵). Assume that 𝐼(Λ,𝑐)(𝐾𝑌+𝐵𝑌)∼0. Then, we have that 𝐼(Λ,𝑐)(𝐾𝑋+𝐵)∼0.
Proof. We can consider 𝐼(Λ,𝑐)as in Theorem 4.8. By Theorem 4.8 and Lemma 4.7, we know that
there exists a nowhere vanishing preadmissible section
0≠𝑠𝑌∈𝑃𝐴(𝑌,𝐼(Λ,𝑐)(𝐾𝑌+𝐵𝑌)).
By Lemma 4.3, we conclude that the linear equivalence 𝐼(Λ,𝑐)(𝐾𝑋+𝐵)∼0 holds.
In the case of dimension 0, Conjecture 1and Conjecture 2are trivial. Indeed, the only variety of
interest is Spec(K), no boundary is allowed for dimensional reasons, and Bir(Spec(K)) is trivial. Thus,
we get the following statement.
Theorem 4.10. Let (𝑋, 𝐵)be a projective semilog canonical Calabi–Yau pair of coregularity 0 and
𝜆be its Weil index. Let (𝑌, 𝐵
𝑌)be a Q-factorial dlt modification of a normalization of (𝑋, 𝐵).If
2𝜆(𝐾𝑌+𝐵𝑌)∼0, then 2𝜆(𝐾𝑋+𝐵)∼0.
Finally, Conjecture 1and Conjecture 2are known in the case of klt pairs of dimension 1 or 2 (see,
e.g., [44]). We get the following statement.
Theorem 4.11. Let Λbe a set of rational numbers satisfying the descending chain condition. There
exists a constant 𝐼(Λ), only depending on Λ, satisfying the following. Let (𝑋, 𝐵)be a projective semilog
canonical Calabi–Yau pair of coregularity 1 (resp. 2) such that B has coefficients in Λ.Let(𝑌, 𝐵
𝑌)be a
Q-factorial dlt modification of a normalization of (𝑋, 𝐵).If𝐼(Λ)(𝐾𝑌+𝐵𝑌)∼0, then 𝐼(Λ)(𝐾𝑋+𝐵)∼0.
Let us note that Conjecture 1is known for klt 3-folds (see, e.g., [44]). However, Conjecture 2is still
unknown in the case of klt Calabi–Yau 3-folds.
4.1. Lifting complements from nonnormal divisors in fibrations
In this subsection, we prove a statement about lifting complements from nonnormal divisors in fibrations.
Theorem 4.12. Let 𝜆, d and c be nonnegative integers. Assume that Conjecture 1(𝑐)and Conjecture 2(𝑐)
hold. Let 𝐼𝐼(𝐷𝜆,𝑑−1,𝑐,0)be the integer provided by Theorem 6(𝑑−1,𝑐). Up to replacing I with a
bounded multiple, further assume that I is divisible by the integer provided by Theorem 4.9(𝐷𝜆,𝑐).Let
(𝑋, 𝐵)be a projective d-dimensional log Calabi–Yau pair. Assume that the following conditions hold:
◦XisQ-factorial and klt;
◦there is a fibration 𝑋→𝑊,whichisa(𝐾𝑋+𝐵−𝑆)-Mori fiber space;
◦a component 𝑆⊂𝐵which is ample over the base and (𝑋, 𝐵 −𝑆)is dlt;
◦the morphism 𝑆→𝑊has connected fibers;
◦the coefficients of B belong to 𝐷𝜆; and
◦the pair (𝑋, 𝐵)has coregularity c.
Then, we have that 𝐼(𝐾𝑋+𝐵)∼0.
Proof. The proof is formally identical to the proof of Theorem 2.31, with the only difference that we
need to appeal to the results in §4since Smay not be normal. For completeness, we include a full proof
of the statement.
Let (𝑋, 𝐵),S,𝑓:𝑋→𝑊and 𝐼𝐼(𝐷𝜆,𝑑−1,𝑐,𝜆)be as in the statement. First, we show that we
can apply the inductive hypothesis to S.
By [19, Example 2.6], the pair obtained by adjunction of (𝑋, 𝐵−𝜖𝐵+𝜖𝑆)to Sis semilog canonical.
In particular, Sis 𝑆2. In turn, by letting 𝜖→0, it follows that the pair obtained by adjunction of (𝑋, 𝐵)
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Forum of Mathematics, Sigma 33
to Sis semilog canonical. In particular, let (𝑆, 𝐵𝑆)denote the pair obtained by adjunction from (𝑋, 𝐵),
and let (𝑆𝜈,𝐵
𝑆𝜈)denote its normalization. By [10, Lemma 2.28], (𝑆𝜈,𝐵
𝑆𝜈)has coregularity c. Then,
by Lemma 2.17,(𝑆𝜈,𝐵
𝑆𝜈)satisfies the assumptions of Theorem 6(𝑑−1,𝑐)with constant I. Then, by
Theorem 4.9,wehave
𝐼(𝐾𝑆+𝐵𝑆)∼0.(4.2)
By [29, Proposition 4.32], Sis seminormal. Then, by [12, Lemma 2.3] and the fact that 𝑆→𝑊has
connected fibers, we have 𝑓∗O𝑆=O𝑊. Lastly, we observe that, if dim 𝑋−dim 𝑊=1, since 𝑆→𝑊
has connected fibers, it follows that (𝐵−𝑆)hor ≠0.
Now, consider the short exact sequence
0→O𝑋(𝐼(𝐾𝑋+𝐵)−𝑆)→O𝑋(𝐼(𝐾𝑋+𝐵)) → O𝑆(𝐼(𝐾𝑆+𝐵𝑆)) → 0.(4.3)
The exactness of Equation (4.3) follows verbatim as the exactness of Equation (2.3). Since 𝐼(𝐾𝑋+𝐵)−
𝑆∼Q,𝑓 −𝑆, the divisor −𝑆is f-ample and dim 𝑊<dim 𝑋,wehave
𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵)−𝑆)=0.
Similarly, we write
𝐼(𝐾𝑋+𝐵)−𝑆∼Q,𝑓 −𝑆∼Q,𝑓 𝐾𝑋+(𝐵−𝑆).
First, assume that 𝐵hor ≠𝑆. Note that Xis klt and 𝐵−𝑆is f-ample since fis a Mori fiber space and the
assumption that 𝐵hor ≠𝑆. Thus, by the relative version of Kawamata–Viehweg vanishing, we have
𝑅1𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵)−𝑆)=0.
Now, assume that 𝐵hor =𝑆. By the equality 𝐵hor =𝑆and the fact that fis a Mori fiber space, we have
𝐼(𝐾𝑋+𝐵)−𝑆∼Q,𝑓 −𝑆∼Q,𝑓 𝐾𝑋+(𝐵−𝑆)∼
Q,𝑓 𝐾𝑋+𝐵ver ∼Q,𝑓 𝐾𝑋.
Thus, we obtain
𝐼(𝐾𝑋+𝐵)−𝑆−𝐾𝑋∼Q,𝑓 0.
Since Xis a klt variety, by [28, Theorem 1-2-7], we have that 𝑅1𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵)−𝑆)is torsion
free. To conclude that it vanishes, it suffices to show that it has rank 0. As observed at the end of the
previous paragraph, we have that dim 𝑋≥dim 𝑍+2 under the additional assumption 𝐵hor =𝑆. Then,
by applying Kawamata–Viehweg vanishing to a general fiber [30, Theorem 2.70], we conclude that the
rank of 𝑅1𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵)−𝑆)is 0, thus implying that 𝑅1𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵)−𝑆)=0.
Therefore, by pushing forward Equation (4.3)viaf, we obtain
𝑓∗O𝑋(𝐼(𝐾𝑋+𝐵)) 𝑓∗O𝑆(𝐼(𝐾𝑆+𝐵𝑆)).
Now, taking global sections, we have
𝐻0(𝑋,O𝑋(𝐼(𝐾𝑋+𝐵))) =𝐻0(𝑆, O𝑆(𝐼(𝐾𝑆+𝐵𝑆))) =𝐻0(𝑆, O𝑆)≠0.(4.4)
By [13, Lemma 3.1], (4.4)impliesthat𝐼(𝐾𝑋+𝐵)∼0.
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34 F. Figueroa et al.
5. Relative complements
In this section, we prove an inductive statement regarding the existence of complements for Fano type
morphisms with bounded coregularity.
5.1. Lifting sections from a divisor
In this subsection, we introduce some tools to lift complements from a divisor of a log Fano pair. Let X
be a Fano type variety and (𝑋, 𝐵,M)be a generalized log canonical pair for which −(𝐾𝑋+𝐵+M𝑋)
is nef. The main theorem of this subsection implies that we can lift complements for (𝑋, 𝐵,M)from a
component Sof 𝐵under some suitable conditions explained in the following theorem.
Theorem 5.1. Let d, c and p be nonnegative integers and R⊂Q>0be a finite set. Let 𝑁𝑁(𝐷(R),
𝑑−1,𝑐,𝑝)be the integer provided by Theorem 7(𝑑−1,𝑐). Assume that N is divisible by p and by
𝐼R.Let𝜋:𝑋→𝑍be a Fano type morphism, where X is a d-dimensional variety. Let (𝑋, 𝐵, M)be a
generalized log canonical pair over Z and 𝑧∈𝑍a point satisfying the following conditions:
◦the generalized pair (𝑋, 𝐵, M)has coregularity at most c over z;
◦the divisor B has coefficients in R;
◦𝑝Mis b-Cartier; and
◦the divisor −(𝐾𝑋+𝐵+M𝑋)is nef over Z.
Assume that there exists 𝐵1≤𝐵and 𝛼∈(0,1]for which:
◦the generalized pair (𝑋, 𝐵1,𝛼M)is generalized log canonical but it is not generalized klt over z;
◦the divisor −(𝐾𝑋+𝐵1+𝛼M𝑋)is big and nef over Z.
Then, (𝑋, 𝐵,M)admits an N-complement over z.
In order to prove the main theorem of this section, we take inspiration from [3, §6.6]. In particular,
we will first prove a weaker statement.
Proposition 5.2. Let d, c and p be nonnegative integers and R⊂Q>0be a finite set. Let
𝑁𝑁(𝐷(R),𝑑−1,𝑐,𝑝)be the integer provided by Theorem 7(𝑑−1,𝑐). Assume that N is divis-
ible by p and by 𝐼R.Let𝜋:𝑋→𝑍be a Fano type morphism, where X is a d-dimensional variety.
Let (𝑋, 𝐵,M)beaQ-factorial generalized log canonical pair over Z and 𝑧∈𝑍a point satisfying the
following conditions:
◦the generalized pair (𝑋, 𝐵, M)has coregularity at most c over z;
◦the divisor B has coefficients in R;
◦𝑝Mis b-Cartier; and
◦the divisor −(𝐾𝑋+𝐵+M𝑋)is nef over Z.
Assume there exists a boundary Γon X and 𝛼∈(0,1)for which:
◦the generalized pair (𝑋, Γ,𝛼M)is generalized plt over z;
◦we have that 𝑆=Γ⊂𝐵intersects the fiber over z; and
◦the divisor −(𝐾𝑋+Γ+𝛼M𝑋)is ample over Z.
Then, (𝑋, 𝐵,M)admits an N-complement over z.
Proof. We will proceed by induction on the dimension, keeping the coregularity constant. Over several
steps, we will lift a complement from a divisor. Since the statement is local over 𝑧∈𝑍, in the course
of the proof we are free to shrink Zaround z. In particular, all linear equivalences that are relative to Z
can be assumed to hold globally. We add the fractions with denominator pto the set R. This does not
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Forum of Mathematics, Sigma 35
change the value of N, hence proving the statement for this new finite set is the same as proving it for
the original R.
Step 1. In this step, we reduce to the case where (𝑋, 𝐵, M)is Q-factorial generalized dlt.
Let (𝑋,𝐵
,M)be a Q-factorial dlt modification of (𝑋, 𝐵, M).PickEexceptional such that −𝐸is
ample over X. Notice that the existence of Eis guaranteed by the hypothesis that Xis Q-factorial. Also,
let (𝑋,Γ,𝛼M)be the trace of (𝑋, Γ,M)on 𝑋. We observe that Γmay no longer be effective.
Since (𝑋, Γ,𝛼M)is generalized plt with Γ⊂𝐵and 𝑋→𝑋only extracts divisor that appear
with coeffcient 1 in 𝐵,for0 <𝜆1, the datum of (𝑋,(1−𝜆)𝐵+𝜆Γ,(1−𝜆+𝜆𝛼)M)is actually
a generalized pair (i.e., its boundary is effective) and it is generalized plt with 1 −𝜆+𝜆𝛼 ∈(0,1).
Furthermore,
−(𝐾𝑋+(1−𝜆)𝐵+𝜆Γ+(1−𝜆+𝜆𝛼)M𝑋)
is the pull-back of a divisor on Xthat is relatively ample over Z. Thus, for 𝜀>0 small enough, we have
that
−(𝐾𝑋+(1−𝜆)𝐵+𝜆Γ+𝜀𝐸 +(1−𝜆+𝜆𝛼)M𝑋)
is ample over Z. Hence, up to replacing (𝑋, 𝐵,M)with (𝑋,𝐵
,M)and (𝑋, Γ,𝛼M)with (𝑋,(1−
𝜆)𝐵+𝜆Γ+𝜀𝐸, (1−𝜆+𝜆𝛼)M)we can assume that Xis Q-factorial and (𝑋, 𝐵,M)is generalized dlt.
Step 2. In this step, we prove that 𝑆→𝜋(𝑆)is a contraction.
As 𝛼M𝑋is the push-forward of a divisor that is nef over Z, its diminished base locus does not contain
any divisor. Let 𝜋:𝑋→𝑋be a model where Mdescends. Let 𝐾𝑋+Γ+𝛼M𝑋be the crepant pullback
of 𝐾𝑋+Γ+𝛼M𝑋(this Γis different from the one defined in step 1). For any 0 <𝛿<1, we can write
(1−𝛿)(𝐾𝑋+Γ+𝛼M𝑋)=𝐾𝑋+Γ+(𝛼M𝑋−𝛿(𝐾𝑋+Γ+𝛼M𝑋)).
As 𝛼M𝑋is nef and −(𝐾𝑋+Γ+𝛼M𝑋)is big and nef over Z,by[32, Example 2.2.19] there exists
an effective divisor 𝐸such that 𝛼M𝑋−𝛿(𝐾𝑋+Γ+𝛼M𝑋)∼
Q𝐴
𝑘+1
𝑘𝐸, for all positive integers k,
where each 𝐴
𝑘is ample over Z.
So, we can write (1−𝛿)(𝐾𝑋+Γ+𝛼M𝑋)∼
Q𝐾𝑋+Γ+𝐴
𝑘+1
𝑘𝐸. If we choose klarge enough
and 𝐴
𝑘generically, then the subpair (𝑋,Γ+𝐴
𝑘+1
𝑘𝐸)is sub-plt. With those choices fixed, we define
𝐴=𝜋∗𝐴
𝑘,𝐸=1
𝑘𝜋∗𝐸. Therefore,
(1−𝛿)(𝐾𝑋+Γ+𝛼M𝑋)∼
Q𝐾𝑋+Γ+𝐴+𝐸,
with (𝑋, 𝐴 +𝐸+Γ)being plt. Call 𝐺𝐴+𝐸+Γ.For𝛿small enough, we have that −(𝐾𝑋+𝐺)is
ample over Zand 𝐺=𝑆. From the exact sequence
0→O𝑋(−𝑆)→O𝑋→O𝑆→0,
we get the exact sequence
𝜋∗O𝑋→𝜋∗O𝑆→𝑅1𝜋∗O𝑋(−𝑆).
Since −𝑆=𝐾𝑋+𝐺−𝑆−(𝐾𝑋+𝐺), with (𝑋, 𝐺 −𝑆)being klt and −(𝐾𝑋+𝐺)being ample over
Z, we have that 𝑅1𝜋∗O𝑋(−𝑆)=0 by the relative Kawamata–Viehweg vanishing theorem. Therefore,
𝜋∗O𝑋→𝜋∗O𝑆is surjective.
Let 𝑔◦𝜋:𝑆→𝑍→𝑍be the Stein factorization of 𝜋:𝑆→𝑍. Then O𝑍=𝜋∗(O𝑋)→
𝜋∗O𝑆=𝑔∗O𝑍is surjective. As O𝑍→𝑔∗O𝑍factors as O𝑍→O𝜋(𝑆)→𝑔∗O𝑍, the morphism
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36 F. Figueroa et al.
O𝜋(𝑆)→𝑔∗(O𝑍)is surjective. Then, it is an isomorphism, as the map 𝑍→𝑍is finite. Hence,
𝑍→𝜋(𝑆)is an isomorphism and 𝑆→𝜋(𝑆)is a contraction. Restricting 𝐾𝑋+𝐺to Sshows that Sis
of Fano type over 𝜋(𝑆).
Step 3. In this step, we use adjunction and consider a complement on S.
Consider a log resolution 𝑓:𝑋→𝑋of (𝑋, 𝐵,M)such that Mdescends on 𝑋, and write 𝐾𝑋+𝐵𝑋
𝑓∗(𝐾𝑋+𝐵).Let𝑆be the strict transform of Sand 𝑔:𝑆→𝑆be the induced morphism. Let (𝑆, 𝐵𝑆,N)
be the generalized pair obtained by adjunction of (𝑋, 𝐵,M)to S. By Lemma 2.17, the coefficients of
𝐵𝑆are in 𝐷(R)and the b-divisor 𝑝Nis b-Cartier. By Lemma 2.24, the coregularity of (𝑆, 𝐵𝑆,N)is at
most c. By Theorem 7(𝑑−1,𝑐)if dim 𝜋(𝑆)=0 or by the inductive hypothesis if dim 𝜋(𝑆)>0, the
divisor 𝐾𝑆+𝐵𝑆+M𝑆has an Ncomplement 𝐵+
𝑆over zwith coregularity at most c. In the following
steps, we will lift 𝐵+
𝑆to an N-complement 𝐵+
𝑋of 𝐾𝑋+𝐵+M𝑋over zwith coregularity at most c.
Step 4. In this step, we introduce some divisors and prove some properties of these divisors.
Define Ω𝑋𝐵𝑋−𝐵≥0
𝑋and 𝑇𝑋𝑁Ω𝑋−(𝑁+1)Ω𝑋−𝑁(𝐾𝑋+𝐵𝑋+M𝑋). We write
𝐾𝑋+Γ𝑋𝑓∗(𝐾𝑋+Γ). Now, we define a divisor 𝑃𝑋in the following way. For any prime divisor
𝐷𝑋≠𝑆, we set coeff𝐷𝑋(𝑃𝑋)=−coeff 𝐷𝑋Γ𝑋+𝑁Ω𝑋−(𝑁+1)Ω𝑋 and coeff 𝑆(𝑃𝑋)=0.
Hence, 𝑃𝑋is an integral divisor such that 𝐽𝑋Γ𝑋+𝑁Ω𝑋−(𝑁+1)Ω𝑋+𝑃𝑋is a boundary,
(𝑋,𝐽
𝑋,𝛼M𝑋)is generalized plt and 𝐽𝑋=𝑆.For𝐷𝑋≠𝑆not exceptional over X,as𝑁𝐵 is
integral, we have that coeff𝐷𝑋(𝑁Ω𝑋)is an integer. Thus, coeff𝐷𝑋(𝑁+1)Ω𝑋=coeff𝐷𝑋(𝑁Ω𝑋).
So, coeff𝐷𝑋(𝑃𝑋)=−coeff 𝐷𝑋(Γ𝑋)=0.We conclude that 𝑃𝑋is exceptional over X.
Step 5. In this step, we lift sections from 𝑆to 𝑋using Kawamata–Viehweg vanishing.
Observe that:
𝑇𝑋+𝑃𝑋=𝑁Ω𝑋−(𝑁+1)Ω𝑋−𝑁(𝐾𝑋+𝐵𝑋+M𝑋)+𝑃𝑋
=𝐾𝑋+Γ𝑋−(𝐾𝑋+Γ𝑋)+𝑁Ω𝑋−(𝑁+1)Ω𝑋−𝑁(𝐾𝑋+𝐵𝑋+M𝑋)+𝑃𝑋
=𝐾𝑋+𝐽𝑋−(𝐾𝑋+Γ𝑋)−𝑁(𝐾𝑋+𝐵𝑋+M𝑋).
Then, we have that −(𝐾𝑋+Γ𝑋+𝛼M𝑋)−𝑁(𝐾𝑋+𝐵𝑋+M𝑋)+𝛼M𝑋is big and nef over Zand
(𝑋,𝐽
𝑋−𝑆)is klt. Therefore, up to shrinking Zaround z, the relative Kawamata–Viehweg vanishing
theorem implies that ℎ1(𝑋,O𝑋(𝑇𝑋+𝑃𝑋−𝑆)) =0. So, we obtain
𝐻0(𝑋,O𝑋(𝑇𝑋+𝑃𝑋)) → 𝐻0(𝑋,O𝑋((𝑇𝑋+𝑃𝑋)|
𝑆)) → 𝐻1(𝑋,O𝑋(𝑇𝑋+𝑃𝑋−𝑆)) =0.
This means that we can lift sections of (𝑇𝑋+𝑃𝑋)|
𝑆from 𝑆to 𝑋.
Step 6. In this step, we introduce a divisor 𝐺𝑆which is linearly equivalent to (𝑇𝑋+𝑃𝑋)|𝑆.
We have −𝑁(𝐾𝑆+𝐵𝑆+N𝑆)=−𝑁(𝐾𝑆+𝐵+
𝑆+𝐵𝑆−𝐵+
𝑆+N𝑆)∼−𝑁(𝐵𝑆−𝐵+
𝑆)=𝑁(𝐵+
𝑆−𝐵𝑆)≥0.
Define 𝐾𝑆+𝐵𝑆+N𝑆𝑔∗(𝐾𝑆+𝐵𝑆+N𝑆). Then, we have that −𝑁(𝐾𝑆+𝐵𝑆+N𝑆)∼𝑁𝑔∗(𝐵+
𝑆−𝐵𝑆)≥0.
Then, it follows that −𝑁(𝐾𝑋+𝐵𝑋+M𝑋)|
𝑆=−𝑁(𝐾𝑆+𝐵𝑆+N𝑆)∼𝑁𝑔∗(𝐵+
𝑆−𝐵𝑆). We define 𝐺𝑆
𝑁𝑔∗(𝐵+
𝑆−𝐵𝑆)+𝑁Ω𝑋|𝑆−(𝑁+1)Ω𝑋|𝑆+𝑃𝑋|𝑆. By definition, we have 𝐺𝑆∼(𝑇𝑋+𝑃𝑋)|
𝑆.
Step 7. In this step, we prove that 𝐺𝑆is effective and that it lifts to an effective divisor 𝐺𝑋on 𝑋.
Assume 𝐺𝑆is not effective, then there exists some prime divisor 𝐶𝑆with coeff𝐶𝑆(𝐺𝑆)<0. As
𝑁𝑔∗(𝐵+
𝑆−𝐵𝑆)and 𝑃𝑋are effective, we must have that coeff𝐶𝑆(𝑁Ω𝑋|𝑆−(𝑁+1)Ω𝑋|𝑆) is negative.
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Forum of Mathematics, Sigma 37
Since 𝑋is a log resolution, we have that the restriction to 𝑆commutes with taking the integral (resp.
fractional) part of a divisor whose support is involved in the log resolution. In particular, observe that
coeff𝐶𝑆(𝑁Ω𝑋|𝑆−(𝑁+1)Ω𝑋|𝑆) =coeff 𝐶𝑆(−Ω𝑋|𝑆+{(𝑁+1)Ω𝑋|𝑆}) ≥ − coeff𝐶𝑆(Ω𝑋|𝑆)>−1.
As 𝐺𝑆is integral by the previous inequality its coefficients cannot be negative. Therefore, by Step 4,
we can lift 𝐺𝑆to an effective divisor 𝐺𝑋∼𝑇𝑋+𝑃𝑋with support not containing 𝑆and such that
𝐺𝑋|𝑆=𝐺𝑆.
Step 8. In this step, we introduce a divisor 𝐵+≥𝐵for which 𝑁𝐵+∼−𝑁(𝐾𝑋+M𝑋).
Since 𝑁𝐵 is integral, (𝑁+1)Ω=𝑁Ω, where Ωis the push-forward of Ω𝑋. Similarly, we call T,
Pand Gthe push-forwards of 𝑇𝑋,𝑃𝑋and 𝐺𝑋, respectively. We have that 𝑃=0as𝑃𝑋is exceptional
and therefore 𝑇=𝑇+𝑃∼𝐺. Hence, we have that −𝑁(𝐾𝑋+𝐵+M𝑋)=𝑇=𝑇+𝑃∼𝐺≥0. Therefore,
𝑁(𝐾𝑋+𝐵++M𝑋)∼0, where we define 𝐵+𝐵+1
𝑁𝐺.
Step 9. In this step, we prove that (𝑋, 𝐵+,M)is generalized log canonical over some neighbourhood of
z, thus proving that 𝐵+is an N-complement for (𝑋, 𝐵,M)over z.
We first prove that 1
𝑁𝐺|𝑆=𝐵+
𝑆−𝐵𝑆. Note that we have the following chain of Q-linear equivalences:
𝑅𝑋𝐺𝑋−𝑃𝑋+(𝑁+1)Ω𝑋−𝑁Ω𝑋∼𝑇𝑋+(𝑁+1)Ω𝑋−𝑁Ω𝑋=−𝑁𝑓∗(𝐾𝑋+𝐵+M𝑋)∼
Q0/𝑋.
Since 𝑁Ωis integral, we have that (𝑁+1)Ω=𝑁Ω. Therefore, as 𝑃𝑋is f-exceptional, 𝑓∗(𝑅𝑋)=𝐺
and 𝑅𝑋is the pull-back of G. Observe that
𝑁𝑔∗(𝐵+
𝑆−𝐵𝑆)=𝐺𝑆−𝑃𝑆+(𝑁+1)Ω𝑋|𝑆−𝑁Ω𝑋|𝑆=(𝐺𝑋−𝑃𝑋+(𝑁+1)Ω𝑋−𝑁Ω𝑋)|
𝑆=𝑅𝑋|𝑆.
Therefore, 𝑔∗(𝐵+
𝑆−𝐵𝑆)=1
𝑁𝑅𝑋|𝑆=𝑔∗(1
𝑁𝐺|𝑆), implying that 𝐵+
𝑆−𝐵𝑆=1
𝑁𝐺|𝑆.
We now have that 𝐾𝑆+𝐵+
𝑆+N𝑆=𝐾𝑆+𝐵𝑆+𝐵+
𝑆−𝐵𝑆+N𝑆=(𝐾𝑋+𝐵+1
𝑁𝑅+M𝑋)|
𝑆=(𝐾𝑋+𝐵++M𝑋)|
𝑆.
By inversion of adjunction, (𝑋, 𝐵+,M)is generalized log canonical near S. Moreover, it has coregularity
cby [10, Lemma 2.30].
If (𝑋, 𝐵+,M)is not generalized log canonical near the fiber over z, then (𝑋, 𝑎𝐵++(1−𝑎)Γ),M)is
also not generalized log canonical near the fiber over zfor 𝑎<1 close enough to 1. The generalized pair
(𝑋, 𝐵+,M)is generalized log canonical near S, therefore a component of the generalized non-klt locus
of (𝑋, 𝑎𝐵++(1−𝑎)Γ),M)is not near S.ButSis also a component of the generalized log canonical locus.
Hence, the generalized non-klt locus of (𝑋, 𝑎𝐵++(1−𝑎)Γ),M)is disconnected near the fiber over z.
This is a contradiction as −(𝐾𝑋+𝑎𝐵++(1−𝑎)Γ+M)=−𝑎(𝐾𝑋+𝐵++M)−(1−𝑎)(𝐾𝑋+Γ+M),isbig
and nef over Z, so the connectedness principle can be applied (see, e.g., [3, Lemma 2.14]). Therefore,
(𝑋, 𝐵+,M)is generalized log canonical near the fiber over z.
Proof of Theorem 5.1.We will proceed in several steps to reduce to Proposition 5.2. Without loss of
generality, we may replace Xwith a small Q-factorial modification.
Step 1. In this step, we define a boundary divisor 𝐵2≤𝐵1and reduce to the case where −(𝐾𝑋+𝐵2+
𝛼𝑏M𝑋)is big and nef for some 𝑏∈(0,1).
For any 0 <𝑎<1, we have that
𝑎(𝐾𝑋+𝐵+M𝑋)+(1−𝑎)(𝐾𝑋+𝐵1+𝛼M𝑋)=𝐾𝑋+(𝑎𝐵 +(1−𝑎)𝐵1)+(𝑎+(1−𝑎)𝛼)M𝑋
is antibig and anti-nef, hence we can replace 𝐵1by 𝑎𝐵 +(1−𝑎)𝐵1and 𝛼by 𝑎+(1−𝑎)𝛼to obtain 𝐵1
with coefficients as close as needed to the coefficients of B.
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38 F. Figueroa et al.
Let 𝐵2𝑏𝐵1for some 𝑏<1. Since Xis of Fano type over Z,−(𝐾𝑋+𝐵1+𝛼M𝑋)defines a
contraction 𝑋→𝑉over Z. We run an MMP on −(𝐾𝑋+𝐵2+𝛼𝑏M𝑋)over V. In the resulting model
−(𝐾𝑋+𝐵
2+𝛼𝑏M𝑋)is big and nef over V. By the definition of V,−(𝐾𝑋+𝐵
2+𝛼𝑏M𝑋)(1−𝑡)−(𝐾𝑋+
𝐵
1+𝛼M𝑋)𝑡is nef over Zfor tclose enough to 1, which is equivalent to saying that −(𝐾𝑋+𝐵
2+𝛼𝑏M𝑋)
is nef over Zfor bclose enough to 1.
By taking aclose enough to 1, we have that 𝐾𝑋+𝐵+M𝑋is nonnegative over V, hence the MMP
we ran is (𝐾𝑋+𝐵+M𝑋)-nonnegative. So, by Lemma 2.22 the coregularity of (𝑋, 𝐵, M𝑋)remains
unchanged.
Then, we can replace (𝑋, 𝐵,M)with (𝑋,𝐵
,M),𝐵1with 𝐵
1and 𝐵2with 𝐵
2to have also that
−(𝐾𝑋+𝐵
2+𝛼𝑏M𝑋)is big and nef over Z.As−(𝐾𝑋+𝐵1+𝛼M𝑋)is big and nef over Z, we have that
there is Aample and Eeffective, such that −(𝐾𝑋+𝐵1+𝛼M𝑋)∼
Q,𝑍 𝐴+𝐸.
Step 2. In this step, we separate into cases depending on whether the generalized log canonical centers
of (𝑋, 𝐵1,𝛼M)are contained in the support of E.
We can take a generalized dlt modification of (𝑋, 𝐵1,𝛼M𝑋), so we can assume that (𝑋, 𝐵1,𝛼M𝑋)
is generalized dlt. If Supp 𝐸contains no generalized log canonical center of (𝑋, 𝐵1,𝛼M), then (𝑋, 𝐵1+
𝜀𝐸, 𝛼M)is generalized dlt for 𝜀>0 small enough.
We have that −(𝐾𝑋+𝐵1+𝜀𝐸 +𝛼M𝑋)∼
Q.𝑍 (1−𝜀)( 𝜀
1−𝜀𝐴+−(𝐾𝑋+𝐵1+𝛼M𝑋)) is ample over
Z. Hence, by altering the coefficients of 𝐵1+𝜀𝐸, we can produce a divisor Γthat lets us conclude by
Proposition 5.2.
If Supp 𝐸does contain some generalized log canonical center of (𝑋, 𝐵1,𝛼M), then for 0 <𝑟<1
we define 𝐵𝑟𝑟𝐵1+(1−𝑟)𝐵2and 𝛼𝑟(𝑟+𝑏(1−𝑟))𝛼. Then, we define 𝑡𝑟to be the generalized
log canonical threshold of 𝐸+𝐵1−𝐵𝑟with respect to (𝑋, 𝐵𝑟,𝛼
𝑟M)over z. Since Xis of Fano type,
it is klt. Furthermore, since (𝑋, 𝐵,M)is generalized log canonical and 0 <𝑏,𝑟<1, it follows that
(𝑋, 𝐵𝑟,𝛼
𝑟M)is generalized klt. In particular, we have 𝑡𝑟>0. We have
−(𝐾𝑋+𝐵𝑟+𝑡𝑟(𝐸+𝐵1−𝐵𝑟)+𝛼𝑟M𝑋)=−(𝐾𝑋+𝐵1+𝛼M𝑋)+𝐵1−𝐵𝑟−𝑡(𝐸+𝐵1−𝐵𝑟)+(𝛼−𝛼𝑟)M𝑋
∼R,𝑍 𝐴+𝐸+(1−𝑡𝑟)(𝐵1−𝐵𝑟)−𝑡𝐸 +(𝛼−𝛼𝑟)M𝑋
=𝑡𝑟𝐴+(1−𝑡𝑟)(𝐴+𝐸+(𝐵1−𝐵𝑟)) + (𝛼−𝛼𝑟)M𝑋
∼R,𝑍 𝑡𝑟𝐴−(1−𝑡𝑟)(𝐾𝑋+𝐵𝑟+𝛼M𝑋)+(𝛼−𝛼𝑟)M𝑋
∼R,𝑍 𝑡𝑟𝐴−(1−𝑡𝑟)(𝐾𝑋+𝐵𝑟+𝛼𝑟M𝑋)+𝑡𝑟(𝛼−𝛼𝑟)M𝑋
=𝑡𝑟(𝐴+(𝛼−𝛼𝑟)M𝑋)−(1−𝑡𝑟)(𝐾𝑋+𝐵𝑟+𝛼𝑟M𝑋).
If we pick rclose enough to 1, then 𝛼−𝛼𝑟tends to 0, hence 𝐴−(𝛼−𝛼𝑟)M𝑋is ample over Zfor rclose
enough to 1. Fixing such an r, it follows that −(𝐾𝑋+𝐵𝑟+𝑡𝑟(𝐸+𝐵1−𝐵𝑟)+𝛼𝑟M𝑋)is ample over Z.
Step 3. In this step, we separate into cases according to the round-down of the divisor Θ𝐵𝑟+𝑡𝑟(𝐸+
𝐵1−𝐵𝑟).
If we have Θ=0, then we let (𝑋,Θ,𝛼
𝑟M)be a dlt modification of (𝑋, Θ,𝛼
𝑟M). We can
assume that every component of Θintersects the fiber over z, after shrinking Z. Furthermore,
since (𝑋, 𝐵𝑟,𝛼
𝑟M)is generalized klt, Θis the exceptional divisor of 𝑋→𝑋. An MMP on
𝐾𝑋+Θ+𝛼𝑏M𝑋over Xends with X,asΘis the exceptional divisor of 𝑋→𝑋and Xis klt and
Q-factorial. The last step of this MMP would be a divisorial contraction 𝑋 →𝑋contracting one prime
divisor 𝑆 with (𝑋,𝑆
,𝛼
𝑟M𝑋 )generalized plt and −(𝐾𝑋 +𝑆 +𝛼𝑟M𝑋 )ample over X. Furthermore
𝑆 is a component of both Θ and 𝐵
1, where 𝐾𝑋 +Θ and 𝐾𝑋 +𝐵
1are the pull-backs of 𝐾𝑋+Θ
and 𝐾𝑋+𝐵1, respectively.
As −(𝐾𝑋+Θ+𝛼𝑟M𝑋)is ample over Zand −(𝐾𝑋 +𝑆+𝛼𝑟M𝑋 )is ample over X, a linear combination
of 𝑆 and Θ yields Γ such that −(𝐾𝑋 +Γ +𝛼𝑟M𝑋 )is ample over Zand (𝑋,Γ,𝛼
𝑟M𝑋 )is
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Forum of Mathematics, Sigma 39
plt with Γ=𝑆. We can apply Proposition 5.2 here. As an N-complement on 𝑋 would induce an
N-complement on X, we have reduced to the case where Θ≠0.
It only remains to deal with the case where Θ≠0. In this case, there is a component Sof
Θ≤𝐵1≤𝐵. On a dlt modification of (𝑋, 𝑆, 𝛼𝑟M𝑋), we can perturb the coefficients of Θto get
that Θis irreducible and −(𝐾𝑋+Γ+𝛼𝑟M𝑋)is ample. Therefore, we can obtain the N-complement
by Proposition 5.2, as desired. As an N-complement on 𝑋 induces an N-complement on X,weare
done.
5.2. Relative complements
In this subsection, we study the existence of complements in the relative setting. The main theorem of
this subsection states that we can lift complements from lower coregularity pairs when all the generalized
log canonical centers are horizontal over the base.
Theorem 5.3. Let d, c and p be nonnegative integers and Λ⊂Qbe a closed set satisfying the DCC.
Assume Theorem 7(𝑐−1)holds. There exists a constant 𝑁𝑁(Λ,𝑐, 𝑝)satisfying the following.
Let 𝜋:𝑋→𝑍be a Fano type morphism, where X is a d-dimensional variety, and dim 𝑍>0.
Let (𝑋, 𝐵,M)be a generalized log canonical pair over Z and 𝑧∈𝑍a point satisfying the following
conditions:
◦the generalized pair (𝑋, 𝐵, M)has coregularity at most c over z;
◦the coefficients of B belong to Λ;
◦every generalized log canonical center of (𝑋, 𝐵, M)dominates Z;
◦𝑝Mis b-Cartier; and
◦the divisor −(𝐾𝑋+𝐵+M𝑋)is nef over Z.
Then, there exists an N-complement for (𝑋, 𝐵, M)over z.
Proof. We will proceed in several steps to be able to lift complements using Theorem 5.1.
Step 1. In this step, we reduce to the case in which the boundary coefficients belong to a finite set,
(𝑋, 𝐵,M)has coregularity 𝑐−1 over, and 𝐵has a vertical component intersecting the fiber over 𝑧.
By Theorem 3.5, there exists a finite set Rand a relative pair (𝑋,𝐵
,M)with coeff(𝐵)⊂Rsuch
that if (𝑋,𝐵
,M)is N-complemented, then so is (𝑋, 𝐵, M). So, we can replace our generalized pair
(𝑋, 𝐵,M)with (𝑋,𝐵
,M). Hence, we may assume the coefficients of Bbelong to the finite set R.Up
to taking a Q-factorial dlt modification, we may further assume that Xis Q-factorial.
We pick an effective Cartier divisor Non Zpassing through z.Welettbe the generalized log canonical
threshold of 𝑞∗𝑁with respect to (𝑋, 𝐵, M)over z. By the connectedness principle [16, Theorem 1.7]
and the assumption that all the generalized log canonical centers of (𝑋, 𝐵, M)dominate Z, we know
that the coregularity of (𝑋, 𝐵 +𝑡𝑞∗𝑁, M)is at most 𝑐−1.
Let (𝑋,𝑇,M)be a generalized dlt modification of (𝑋, 𝐵 +𝑡𝑞∗𝑁, M)over z.Let𝐵be the strict
transform of Bon 𝑋.LetΩbe a boundary such that 𝐵≤Ω≤𝑇, coeff(Ω)⊂Rand some
component Sof Ωis ver tical over Zintersecting the fiber 𝜋−1(𝑧).LetΩ=𝜋∗Ω.
WerunanMMPoverZon −(𝐾𝑋+Ω+M𝑋).As−(𝐾𝑋+Ω+M𝑋)=−(𝐾𝑋+𝑇+M𝑋)+(𝑇−Ω),
with −(𝐾𝑋+𝑇+M.)nef over Zand (𝑇−Ω)effective, the MMP ends with a minimal model, which
we denote by 𝑋.
If (𝑋,Ω,M𝑋 )has an N-complement over z, then (𝑋,Ω,M𝑋)has an N-complement over zby
Lemma 2.12.As𝐵≤Ω, we have that also (𝑋, 𝐵,M𝑋)has an N-complement by Lemma 2.13. So, we
can replace (𝑋, 𝐵, M𝑋)with (𝑋,Ω,M𝑋 ), to obtain 𝐵having a component intersecting the fiber
over z, with −(𝐾𝑋+𝐵+M𝑋)nef over Z. Notice that, after this reduction, the coregularity has decreased,
and it is no longer the case that all generalized log canonical centers dominate Z.
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40 F. Figueroa et al.
Step 2. In this step, we define a divisor 𝐵1which satisfies the hypothesis of Theorem 5.1.
For any prime divisor Dthat vertical is over Z, we set coeff𝐷(𝐵1)coeff 𝐷(𝐵). For any prime
divisor Dhorizontal over Z, we set coeff𝐷(𝐵1)coeff𝐷(𝑎𝐵)for 𝑎<1, close enough to 1.
As Xis of Fano type over Z, we have that −𝐾𝑋is big over Z. Therefore, −(𝐾𝑋+𝑎𝐵 +𝑎M𝑋)=
−𝑎(𝐾𝑋+𝐵+M𝑋)−(1−𝑎)𝐾𝑋and −(𝐾𝑋+𝐵1+𝑎M𝑋)are big over Z. The generalized pair (𝑋, 𝐵1,𝑎M)is
generalized log canonical as 𝐵1≤𝐵and 𝐵1contains the same vertical component as 𝐵intersecting
the fiber of z.
Step 3. In this step, we reduce to the case in which −(𝐾𝑋+𝐵1+M𝑋)is big and nef over Z.
Since 𝜋is a Fano type morphism and −(𝐾𝑋+𝐵+M𝑋)is nef over Z,−(𝐾𝑋+𝐵+M𝑋)is semiample
over Z.Let𝑋→𝑉over Zbe the contraction defined by −(𝐾𝑋+𝐵+M𝑋). We run an MMP on
−(𝐾𝑋+𝐵1+𝑎M𝑋)over V. In the resulting model −(𝐾
𝑋+𝐵
1+𝑎M𝑋)is big and nef over V.Bythe
definition of V,−(𝐾𝑋+𝐵
1+𝑎M𝑋)(1−𝑡)−(𝐾𝑋+𝐵+M𝑋)𝑡is nef over Zfor t close enough to 1,
which is equivalent to picking aclose enough to 1. We have that 𝐾𝑋+𝐵+M𝑋is trivial over V, hence
the MMP is (𝐾𝑋+𝐵+M𝑋)-nonnegative. Applying Lemma 2.22, the coregularity of (𝑋, 𝐵, M)remains
unchanged after this MMP. Thus, we can replace (𝑋, 𝐵,M), with (𝑋,𝐵
,M)and 𝐵1with 𝐵
1with a
close enough to 1.
Step 4. In this step, we conclude by applying Theorem 5.1.
Let 𝑁(R,𝑐−1,𝑝)be the positive integer provided by Theorem 7(𝑐−1). Taking Nto be the least
common multiple of 𝑁(R,𝑐 −1,𝑝),𝐼Rand p. Hence, Ndepends only on Λ,𝑐−1 and p. Indeed, R
only depends on Λ,𝑐and p. By Theorem 5.1 the generalized pair (𝑋, 𝐵,M)admits an N-complement,
where Nonly depends on Λ,cand p.
6. Canonical bundle formula
In this section, we prove a special version of the canonical bundle formula. We obtain an effective
canonical bundle formula that is independent of the dimension of the domain. It only depends on the
coregularity of the fibers.
Theorem 6.1. Let d, c be nonnegative integers and Λ⊂Qbe a closed set satisfying the descending
chain condition. Assume Theorem 7(𝑐−1)holds. There exists a set ΩΩ(Λ,𝑐)⊂Qsatisfying the
descending chain condition and a positive integer 𝑞𝑞(Λ,𝑐)satisfying the following. Let 𝜋:𝑋→𝑍
be a fibration from a d-dimensional projective variety X to a projective base Z with dim 𝑍>0.Let
(𝑋, 𝐵)be a log canonical pair satisfying the following conditions:
◦the fibration 𝜋is of Fano type over a nonempty open set U of Z;
◦every log canonical center of (𝑋, 𝐵)is horizontal over Z;
◦the pair (𝑋, 𝐵)is log Calabi–Yau over Z;
◦the coefficients of B are in Λ; and
◦the coregularity of (𝑋, 𝐵)is at most c.
Then, we can write
𝑞(𝐾𝑋+𝐵)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍),
where (𝑍, 𝐵𝑍,N)is a generalized log canonical pair such that
◦𝐵𝑍is the discriminant part of the adjunction for (𝑋, 𝐵)over Z;
◦the coefficients of 𝐵𝑍belong to Ω; and
◦the divisor 𝑞Nis b-nef and b-Cartier.
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Forum of Mathematics, Sigma 41
Proof. The proof is given in several steps. In Step 1, we make a choice of qand N. In Steps 2–4, we find
Ωand show that the coefficients of 𝐵𝑍belong to Ω.Wealsoprovethat𝑞N𝑍is integral. In Step 5, we
show that 𝑞Nis b-Cartier. We observe that, by the assumptions on the log canonical centers of (𝑋, 𝐵)
and on the coregularity of (𝑋, 𝐵), it follows that dim 𝑍≤𝑐.
Step 1. In this step, we find qand make a choice of N.
Let qbe the integer Nin the statement of Theorem 5.3(𝑑, 𝑐 −1). Here, we assumed Theorem 7(𝑐−1).
Fix a general closed point 𝑧∈𝑈.LetHbe a general hyperplane section of Upassing through z. Then
(𝑋, 𝐵 +𝜋∗𝐻)is log Calabi–Yau and satisfies
coreg(𝑋, 𝐵 +𝜋∗𝐻)≤𝑐−1.
This implies that the absolute coregularity of (𝑋, 𝐵)over zis at most 𝑐−1. By Theorem 5.3(𝑑, 𝑐 −1),
there is a q-complement 𝐾𝑋+𝐵+of 𝐾𝑋+𝐵over zwith 𝐵+≥𝐵. Note that qonly depends on Λand
c. Since 𝐾𝑋+𝐵is Q-trivial over Z,𝐵+−𝐵∼Q0 over zand hence 𝐵+=𝐵near the generic fiber of 𝜋.
Therefore, 𝑞(𝐾𝑋+𝐵)∼0 over the generic point of Z. Thus, we can find a rational function son Xsuch
that 𝑞𝐿 𝑞(𝐾𝑋+𝐵)+Div(𝑠)is zero over the generic point of Z. Note that 𝐿∼Q0 over Z,sowecan
write 𝐿=𝜋∗𝐿𝑍for some Q-Cartier Q-divisor 𝐿𝑍on Z. Define
N𝑍𝐿𝑍−(𝐾𝑍+𝐵𝑍),
where 𝐵𝑍is the discriminant part of adjunction for (𝑋, 𝐵)over Z. Similarly, for any birational morphism
𝑔:𝑍→𝑍, we can define N𝑍as follows. Let 𝑓:𝑋→𝑋be a higher birational model of Xsuch that
the rational map 𝑋𝑍is a morphism. Write 𝐾𝑋+𝐵𝑋for the pull-back of 𝐾𝑋+𝐵and 𝐵𝑍be the
discriminant part of adjunction for (𝑋,𝐵
𝑋)over 𝑍. We define
N𝑍𝑔∗𝐿𝑍−(𝐾𝑍+𝐵𝑍).
The data of N𝑍, for all birational models 𝑍→𝑍, determine a b-divisor Non Z.
Step 2. In this step, we reduce to the case when the base Zis a curve.
Assume dim 𝑍≥2. Let Hbe a general hyperplane section of Zand Gbe the pull-back of Hto X.By
adjunction, we can write
(𝐾𝑋+𝐵+𝐺)|𝐺=𝐾𝐺+𝐵𝐺
for some divisor 𝐵𝐺on G. By Lemma 2.17, there exists a set Λsatisfying the DCC, having rational
accumulation points and depending only on Λsuch that the coefficients of 𝐵𝐺belong to Λ.Wemay
replace Λwith Λto assume that the coefficients of 𝐵𝐺belong to Λ. Since dim 𝑍≤𝑐, this replacement
can only happen at most 𝑐−1 times and is hence allowed. By Lemma 2.25, the pair (𝐺, 𝐵𝐺)over H
satisfies the same conditions as (𝑋, 𝐵)over Zin the statement of this theorem. Furthermore, let 𝐵𝐻
denote the discriminant part of the adjunction for (𝐺, 𝐵𝐺)over H.LetDbe any prime divisor on Zand
Ca component of 𝐷∩𝐻. Then
coeff𝐷(𝐵𝑍)=coeff 𝐶(𝐵𝐻).
Pick a general 𝐻∼𝐻, and let 𝐾𝐻=(𝐾𝑍+𝐻)|𝐻, which is properly defined as a Weil divisor. Define
N𝐻=(𝐿𝑍+𝐻)|𝐻−(𝐾𝐻+𝐵𝐻).
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42 F. Figueroa et al.
Similarly to the way we defined the b-divisor Non Z, we can define Nas a b-divisor on H. Then, we have
𝑞(𝐾𝐺+𝐵𝐺)∼𝑞(𝐿+𝐺)|𝐺∼𝑞𝜓∗(𝐿𝑍+𝐻)|𝐻∼𝑞𝜓∗(𝐾𝐻+𝐵𝐻+N𝐻).
Thus, N𝐻is the moduli part of (𝐺, 𝐵𝐺)over H. Moreover, we have 𝐵𝐻+N𝐻=(𝐵𝑍+N𝑍)|𝐻.
This implies that coeff𝐶(𝐵𝐻+N𝐻)=coeff 𝐷(𝐵𝑍+N𝑍)and hence coeff𝐶(N𝐻)=coeff𝐷(N𝑍).In
particular, 𝑞N𝐻is integral if and only if 𝑞N𝐻is integral.
As a result, to prove that coeff(𝐵𝑍)belongs to a fixed set Ωand that 𝑞N𝑍is integral, we may replace
(𝑋, 𝐵)→𝑍with (𝐺, 𝐵𝐺)→𝐻. By repeating this process until the base of the fibration is a curve, we
may assume that dim 𝑍=1.
Step 3. In this step, we show the existence of Ω.
By Step 2, we can assume dim 𝑍=1. By [3, Lemma 2.11], the variety Xis of Fano type over Z.
Pick any closed point 𝑧∈𝑍.Lettbe the log canonical threshold of 𝜋∗𝑧with respect to (𝑋, 𝐵)around
z. Set Γ=𝐵+𝑡𝜋∗𝑧, and let (𝑋,Γ)be a Q-factorial dlt modification of (𝑋, Γ). Then 𝐾𝑋+Γ∼Q0
over Zand Γhas a component with coefficient 1 mapping to z. Pick a boundary 𝐵on 𝑋satisfying the
following conditions:
◦we have ˜
𝐵≤𝐵≤Γ, where ˜
𝐵is the strict transform of Bon 𝑋;
◦the coefficients of 𝐵are in Λ; and
◦the divisors 𝐵and Γhave the same round-down, that is, 𝐵=Γ.
By construction, Supp(Γ−𝐵)is contained in the strict transform of Supp(𝜋∗𝑧).If𝑡<1, 𝐵has a
component Tmapping to zwhich is exceptional over X. Note that 𝑋is of Fano type over Z.Weruna
−(𝐾𝑋+𝐵)-MMP with scaling over Z. Since −(𝐾𝑋+𝐵)∼
QΓ−𝐵is pseudo-effective, this MMP
terminates with a model (𝑋 ,𝐵
)such that −(𝐾𝑋 +𝐵)is semiample over Zby [1, Theorem 1.1]. The
divisor Tis not contracted by this MMP because 𝑇Supp(Γ−𝐵). Furthermore, by Lemma 2.22,we
have
coreg(𝑋,𝐵
)=coreg(𝑋,𝐵
)≤coreg(𝑋,Γ)≤𝑐.
As a result, up to losing the dlt condition, we may replace (𝑋,𝐵
)by (𝑋,𝐵
)to assume that the
following properties hold:
◦𝑋is Fano type over Z;
◦(𝑋,𝐵
)is a log canonical pair over Z;
◦the coefficients of 𝐵are in Λ;
◦coreg(𝑋,𝐵
)≤𝑐;
◦−(𝐾𝑋+𝐵)is semiample over Z; and
◦𝐵has a component mapping to z.
If 𝑡=1, we may simply take 𝐵=Γ
so the above conditions hold as well for (𝑋,𝐵
).
By Theorem 5.1 and our choice of q, the pair (𝑋,𝐵
)has a q-complement (𝑋,𝐵
+)over zwith
𝐵+≥𝐵. Pushing forward 𝐵+to Xgives a q-complement (𝑋, 𝐵+)of (𝑋, 𝐵)over zwith 𝐵+≥𝐵.
Furthermore, 𝐾𝑋+𝐵+has a non-klt center mapping to z, since its pull-back 𝐾𝑋+𝐵+does. Note that
𝐵+−𝐵∼Q0 over z,so𝐵+−𝐵must be a multiple of 𝜋∗𝑧over z. This implies that 𝐵+=𝐵+𝑡𝜋∗𝑧over z
since 𝐾𝑋+𝐵+has a non-klt center mapping to z.
Pick a component Sof 𝜋∗𝑧.Letb,𝑏+and mbe the coefficients of Sin B,𝐵+and 𝜋∗𝑧, respectively.
Then 𝑏+=𝑏+𝑡𝑚 and 𝑡=𝑏+−𝑏
𝑚. By construction, 𝑞𝑏+and mare integers, 𝑏+≤1 and 𝑏∈Λ.By
Lemma 2.18,tbelongs to a fixed set Σ(depending only on 𝐼,Λ) which satisfies ACC and has rational
accumulation points. Since the coefficient of zin 𝐵𝑍is 1 −𝑡, it belongs to a set Ωdepending only on I
and Λsuch that Ωsatisfies DCC and has rational accumulation points.
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Forum of Mathematics, Sigma 43
Step 4. In this step, we show that 𝑞N𝑍is integral.
By Step 2, we can assume that Zis a curve. By Step 1, the equivalence 𝑞(𝐾𝑋+𝐵)∼0 holds over
some nonempty open subset 𝑉⊆𝑍such that Supp 𝐵𝑍⊆𝑍\𝑉.Let
Θ=𝐵+
𝑧∈𝑍\𝑉
𝑡𝑧𝜋∗𝑧,
where 𝑡𝑧is the log canonical threshold of 𝜋∗𝑧with respect to (𝑋, 𝐵)over z. Note that 𝐾𝑋+Θ∼Q0 over
Z.LetΘ𝑍be the discriminant part of adjunction for (𝑋, Θ)over Z. Then
Θ𝑍=𝐵𝑍+
𝑧∈𝑍\𝑉
𝑡𝑧𝑧=
𝑧∈𝑍\𝑉
𝑧
is an integral divisor. Moreover, by Step 3, the pair (𝑋, Θ)is a q-complement of (𝑋, 𝐵)over each
𝑧∈𝑍\𝑉.OverV,wehave𝑞(𝐾𝑋+Θ)∼𝑞(𝐾𝑋+𝐵)∼0. Thus, 𝑞(𝐾𝑋+Θ)∼0 over Zby [3, Lemma
2.4]. Furthermore, from the equalities
𝑞(𝐾𝑋+Θ)=𝑞(𝐾𝑋+𝐵)+𝑞(Θ−𝐵)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍)+𝑞𝜋∗(Θ𝑍−𝐵𝑍)∼𝑞𝜋∗(𝐾𝑍+Θ𝑍+N𝑍),
we obtain that 𝑞(𝐾𝑍+Θ𝑍+N𝑍)is an integral divisor and hence 𝑞N𝑍is integral.
Step 5. In this step, we show that 𝑞N𝑍is nef Cartier on some resolution 𝑍→𝑍.
The nefness follows from [3, Theorem 3.6], so we just need to show that 𝑞N𝑍is integral. Denote
the birational morphism 𝑍→𝑍by g.AsinStep1,let 𝑓:𝑋→(𝑋, 𝐵)be a log resolution such
that the rational map 𝜋:𝑋𝑍is a morphism. Let 𝑈0⊆𝑈be a nonempty open set such that
𝑈
0𝑔−1(𝑈0)→𝑈0is an isomorphism. Let Δbe the sum of the birational transform of Band reduced
exceptional divisors of fbut with all components mapping outside of 𝑈0removed. Then, the generic
point of every log canonical center of (𝑋,Δ)lies inside 𝑈0.
Let Tbe the normalization of the main component of 𝑍×𝑍𝑋. Run an MMP on 𝐾𝑋+Δover T
with scaling of some ample divisor. This MMP terminates with a Q-factorial dlt pair (𝑋,Δ)such
that 𝐾𝑋 +Δ is nef over T.Let𝑋0=𝜋−1(𝑈0).Over𝑈0𝑈
0,wehave𝑍×𝑍𝑋𝑋0, and hence
𝐾𝑋 +Δ is nef over 𝑋0⊆𝑋. By the negativity lemma, over 𝑈
0, the divisor 𝐾𝑋 +Δ is equal to the
log pull-back of 𝐾𝑋+𝐵. This shows that (𝑋,Δ)is a dlt modification of (𝑋, 𝐵)over 𝑈0. In particular,
𝑋 is Fano type over 𝑈
0. Furthermore, every log canonical center of (𝑋,Δ)dominates 𝑈
0. Indeed,
(𝑋, 𝐵)satisfies the same property and the generic point of every log canonical center of (𝑋 ,Δ)lies
inside 𝑈
0. Thus, we have
coreg(𝑋,Δ )=coreg(𝑋, 𝐵)≤𝑐.
By [1, Theorem 1.4], we can run an MMP on 𝐾𝑋 +Δover 𝑍which terminates with a good minimal
model over 𝑍. Since this MMP is trivial over 𝑈
0, the dual complex and hence the coregularity of
(𝑋,Δ)does not change under this MMP. Abusing the notation, we again denote the good minimal
model by 𝑋.Let𝜋 :𝑋 →𝑍 over 𝑍be the morphism induced by the relatively semiample divisor
𝐾𝑋 +Δ. Since 𝐾𝑋 +Δ ∼Q0 over 𝑈
0,𝑍 →𝑍is birational and 𝑈
0, the preimage of 𝑈
0in 𝑍,is
isomorphic to 𝑈
0. Thus, every log canonical center of (𝑋,Δ)also dominates 𝑍.
Let Wbe a common resolution of Xand 𝑋, and set 𝛼:𝑊→𝑋and 𝛽:𝑊→𝑋. By construction,
over the preimage of 𝑈0in W,wehave𝛼∗(𝐾𝑋+𝐵)=𝛽∗(𝐾𝑋 +Δ). Write 𝐾𝑋 +𝐵 =𝛽∗𝛼∗(𝐾𝑋+𝐵)
and 𝐿 =𝛽∗𝛼∗𝐿, where 𝑞𝐿 =𝑞(𝐾𝑋+𝐵)+Div(𝑠)as in Step 1. Let 𝑃 =Δ
−𝐵,whichisQ-trivial
over 𝑍 and supported outside of 𝑈
0. Hence, 𝑃 =𝜋∗𝑃𝑍 for some Q-divisor 𝑃𝑍 on 𝑍.LetΔ𝑍
be the discriminant part of adjunction for (𝑋,Δ)on 𝑍.Let𝐵𝑍 =Δ
𝑍 −𝑃𝑍 , then 𝐵𝑍 is the
discriminant part of adjunction for (𝑋,𝐵
)on 𝑍. By the definition of Nin Step 1, we have
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44 F. Figueroa et al.
𝑞(𝐾𝑋 +Δ)=𝑞(𝐾𝑋 +𝐵 +𝑃)∼𝑞𝜋∗(𝐾𝑍 +𝐵𝑍 +N𝑍 +𝑃𝑍 )∼𝑞𝜋 ∗(𝐾𝑍 +Δ𝑍 +N𝑍 ).
Here, 𝐿𝑍 is the pull-back of 𝐿𝑍in Step 1. This shows that N𝑍 is the moduli part of (𝑋,Δ)over
𝑍. Since the coefficients of Δ are in Λand the coregularity of (𝑋 ,Δ)is at most c, we may apply
Steps 2–4 to show that 𝑞N𝑍 is an integral divisor. Thus, 𝑞N𝑍is also an integral divisor and hence
Cartier as 𝑍is smooth.
We show that Theorem 6.1 also holds for generalized pairs (𝑋, 𝐵,M)in the special case that M𝑋∼Q0
over the base Z.
Theorem 6.2. Let d, c and p be nonnegative integers and Λ⊂Qbe a closed set satisfying the descending
chain condition. Assume Theorem 7(𝑐−1)holds. There exists a set ΩΩ(Λ,𝑐,𝑝)⊂Qsatisfying the
descending chain condition and a positive integer 𝑞𝑞(Λ,𝑐,𝑝),bothΩand q only depending on Λ
and c, and satisfying the following. Let 𝜋:𝑋→𝑍be fibration from a d-dimensional projective variety
X to a projective base Z with dim 𝑍>0.Let(𝑋, 𝐵, M)be a generalized pair for which
◦the generalized pair (𝑋, 𝐵, M)is generalized log canonical;
◦the fibration 𝜋is of Fano type over a nonempty open set U of Z;
◦every generalized log canonical center of (𝑋, 𝐵, M)is horizontal over Z;
◦the divisors 𝐾𝑋+𝐵+M𝑋and M𝑋are Q-trivial over Z;
◦the coefficients of B are in Λ,
◦𝑝Mis b-Cartier; and
◦the coregularity of (𝑋, 𝐵)is at most c.
Then, we can write
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍),
where (𝑍, 𝐵𝑍,N)is a generalized log canonical pair such that
◦𝐵𝑍is the discriminant part of the adjunction for (𝑋, 𝐵,M)over Z;
◦the coefficients of 𝐵𝑍belong to Ω; and
◦the divisor 𝑞Nis b-nef and b-Cartier.
Proof. We proceed in several steps. In the first step, we apply the canonical bundle formula for pairs.
In the rest of the proof, we show that Mis the pull-back of a b-nef divisor on the base and control the
Cartier index of this b-nef divisor where it descends.
Step 1. We show that the pair (𝑋, 𝐵)satisfies the conditions in the statement of Theorem 6.1.
By Lemma 2.5,(𝑋, 𝐵)is log canonical. Furthermore, every log canonical center of (𝑋, 𝐵)is also a
log canonical center of (𝑋, 𝐵, M), and hence it dominates U.
Thus, by Theorem 6.1, there exists Ωand I, depending only on Λand csuch that
𝑞(𝐾𝑋+𝐵)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+P𝑍),
where 𝐵𝑍and Pare the discriminant and moduli part of adjunction for the fibration (𝑋, 𝐵)→𝑍. Since
(𝑋, 𝐵)is log canonical, (𝑍, 𝐵𝑍,P)is generalized log canonical. Furthermore, the coefficients of 𝐵𝑍
belong to Ωand 𝑞Pis nef Cartier on any high resolution of Z. By replacing qwith 𝑝𝑞, we can assume
that qis a multiple of p.
Step 2. In this step, we express Mas a pull-back of some b-divisor from Z.
Let 𝑔:𝑍→𝑍be a log resolution of (𝑍, 𝐵𝑍+P)such that Pdescends on 𝑍and P𝑍is nef. Let
𝑓:𝑋→𝑋be a resolution such that the rational map 𝜋:𝑋𝑍is a morphism and M𝑋is nef. By
the negativity lemma, we can write
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Forum of Mathematics, Sigma 45
𝑓∗M𝑋=M𝑋+𝐸𝑋
for some effective f-exceptional Q-divisor 𝐸𝑋. Since M𝑋∼Q0 over Z,𝐸𝑋is vertical over Z, so there
is a nonempty subset of Uover which 𝐸𝑋=0 and M𝑋∼Q0. Since 𝜋is of Fano type over U,the
general fibers of 𝜋are of Fano type and hence rationally connected. Thus, the general fibers of 𝜋are
also rationally connected. By [3, Lemma 2.44], after replacing 𝑋and 𝑍with possibly higher birational
models, we can write
𝑝M𝑋∼𝑝𝜋∗𝑇𝑍
for some Q-divisor 𝑇𝑍on 𝑍such that 𝑝𝑇𝑍is nef Cartier. Let Tbe the b-divisor on Zwith the data
𝑔:𝑍→𝑍and 𝑇𝑍(i.e., Tdescends on 𝑍as 𝑇𝑍).
Step 3. In this step, we show that 𝑞M𝑋∼𝑞𝜋∗T𝑍.
As in Step 2, write
𝑓∗M𝑋=M𝑋+𝐸𝑋.
Then 𝐸𝑋is vertical and Q-linearly trivial over 𝑍(since M𝑋∼Q0 over 𝑍), so we can write
𝐸𝑋=𝜋∗𝐸𝑍for some effective Q-divisor 𝐸𝑍.If𝐸𝑍has a component 𝐷𝑍which maps onto a
divisor Din Z, then 𝐸𝑋=𝜋∗𝐸𝑍has a component mapping onto D, contradicting the fact that 𝐸𝑋
is f-exceptional. Thus, 𝐸𝑍is g-exceptional. Note that 𝜋∗(T𝑍+𝐸𝑍)=M𝑋+𝐸𝑋∼Q0 over Z,so
T𝑍+𝐸𝑍∼Q0 over Z. This implies that 𝑔∗T𝑍=T𝑍+𝐸𝑍.Now,wehave
𝑞𝑓∗M𝑋=𝑞(M𝑋+𝐸𝑋)∼𝑞𝜋∗(T𝑍+𝐸𝑍)=𝑞𝜋∗𝑔∗T𝑍=𝑞𝑓∗𝜋∗T𝑍
and hence 𝑞M𝑋∼𝑞𝜋∗T𝑍. In particular, 𝑞Tis a b-Cartier divisor. From now on, we consider (𝑍, 𝐵𝑍,P+
T)as a generalized pair, with moduli part P+T.
Step 4. In this step, we show that the generalized pair (𝑍, 𝐵𝑍,P+T)is generalized log canonical.
Write 𝐾𝑋+𝐵𝑋=𝑓∗(𝐾𝑋+𝐵).Let𝐵𝑍be the discriminant part of the adjunction for (𝑋,𝐵
𝑋)over
𝑍. We can assume that (𝑍,Supp 𝐵𝑍+Supp 𝐸𝑍+Supp P𝑍+Supp T𝑍)is log smooth. By construction,
we have
𝐾𝑍+𝐵𝑍+𝐸𝑍+P𝑍+T𝑍=𝑔∗(𝐾𝑍+𝐵𝑍+P𝑍+T𝑍).
Thus, it suffices to show that every coefficient of 𝐵𝑍+𝐸𝑍is at most one. Let Dbe a prime divisor
on 𝑍.Let𝑡𝐷be the log canonical threshold of (𝑋,𝐵
𝑋)with respect to 𝜋∗𝐷over the generic point
of D. Since (𝑋, 𝐵, M)is generalized log canonical, (𝑋,𝐵
𝑋+𝐸𝑋)is sublog canonical. Furthermore,
𝐸𝑋=𝜋∗𝐸𝑍, so the coefficient of Din 𝐸𝑍is at most 𝑡𝐷(otherwise 𝐸𝑋≥(𝑡𝐷+𝜖)𝜋∗𝐷for some
𝜖>0 and this violates the sublog canonical condition). By definition, coeff𝐷(𝐵𝑍)=1−𝑡𝐷. Thus,
coeff𝐷(𝐵𝑍+𝐸𝑍)≤1−𝑡𝐷+𝑡𝐷=1, as desired.
Step 5. In this step, we conclude that the generalized pair (𝑍, 𝐵𝑍,P+T)satisfies the desired properties.
By Step 1, the coefficients of 𝐵𝑍belong to Ωand 𝑞Pis b-nef and b-Cartier. By Steps 2 and 3,
𝑞Tis b-Cartier. By Step 4, (𝑍, 𝐵𝑍,P+T)is generalized log canonical. Finally, by Steps 1 and 3,
we have
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+P𝑍+T𝑍).
Proposition 6.3. Assume that Theorem 7(𝑐−1)holds. Then, Theorem 8(𝑐)holds.
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46 F. Figueroa et al.
Proof. Theorem 7(𝑐−1)implies that Theorem 6.2(𝑐)holds.
Let (𝑋, 𝐵,M)be a generalized pair in the statement of Theorem 8(𝑐). We may replace (𝑋, 𝐵, M)by a
Q-factorial generalized dlt modification and assume that (𝑋, 𝐵,M)is Q-factorial generalized dlt. Since
Xis of Fano type over Z, we can run an MMP on M𝑋over Zto get a model (𝑋,𝐵
,M)such that M𝑋is
semiample over Z. After replacing (𝑋, 𝐵, M)with (𝑋,𝐵
,M), up to losing the generalized dlt property
for (𝑋, 𝐵, M), we may assume that M𝑋is semiample over Z.Let 𝑋→𝑍be the morphism induced by
M. Since Mis trivial on a general fiber of 𝜋, the morphism 𝑍→𝑍is birational. After replacing Zwith
𝑍, we may assume that M∼Q0 over Z. Now, the result follows from Theorem 6.2(𝑐).
7. Proof of the theorems
In this section, we prove the main theorems of this article. In this section, we use the notation from Sec-
tion: ‘Strategy of the Proof’, we write Theorem 𝑋(𝑑, 𝑐)for Theorem Xin dimension dand coregularity
at most c. The following is the boundedness of complements for Fano type pairs of coregularity 0. Note
that the following theorem is an unconditional version of Theorem 7(0).
Theorem 7.1. Let p be a positive integer. Let Λ⊂Qbe a closed set satisfying the descending chain
condition. There exists a constant 𝑁𝑁(Λ,𝑝)satisfying the following. Let X be a Fano type variety
and (𝑋, 𝐵,M)be a generalized pair of absolute coregularity 0. Assume that the following conditions
hold:
◦the coefficients of B belong to Λ;
◦𝑝Mis b-Cartier.
Then, there exists a boundary 𝐵+≥𝐵such that
◦the generalized pair (𝑋, 𝐵+,M)is generalized log canonical;
◦we have that 𝑁(𝐾𝑋+𝐵++M)∼0; and
◦the equality coreg(𝑋, 𝐵+,M)=0holds.
Proof. First, we replace Λwith its derived closure (see Lemma 2.16). We let RR(Λ,0,𝑝)⊂Λbe
the finite subset provided by Theorem 3.5. This finite subset only depends on Λand p.
By [14, Theorem 1.2], there is a constant 𝑁(Λ,𝑑,0,𝑝)such that every generalized pair (𝑋, 𝐵,M)as
in the statement and of dimension at most dadmits an 𝑁(Λ,𝑑,0,𝑝)-complement. We will proceed by
induction on d. We may assume that 𝑁(Λ,𝑑,0,𝑝)is divisible by 𝐼(Λ)and pfor ever y d. Throughout
the proof, we assume that 𝑁(Λ,𝑑,0,𝑝)is minimal with such properties.
By Theorem 3.5, we may assume that the coefficients of Bbelong to R.Let𝐵+Γbe a Q-complement
of (𝑋, 𝐵,M)of coregularity 0. Let (𝑌, 𝐵
𝑌+Γ𝑌,M)be a generalized dlt modification of (𝑋, 𝐵 +Γ,M).
Here, Γ𝑌is the strict transform of the fractional part of Γand 𝐵𝑌is the reduced exceptional plus the strict
transform of 𝐵+Γ. By Lemma 2.13,Yis a Fano type variety. Thus, we may run a −(𝐾𝑌+𝐵𝑌+M𝑌)-
MMP which terminates with a good minimal model Zsince −(𝐾𝑌+𝐵𝑌+M𝑌)∼
QΓ𝑌is effective.
Let 𝐵𝑍be the strict transform of 𝐵𝑌on Zand M𝑍be the trace of Mon Z. Note that (𝑍, 𝐵𝑍,M)
is a generalized pair of coregularity 0 and −(𝐾𝑍+𝐵𝑍+M𝑍)is a semiample divisor. In order to
produce an N-complement for (𝑋, 𝐵,M), it suffices to produce an N-complement for (𝑍, 𝐵𝑍,M)
(see Lemma 2.12 and Lemma 2.13). Hence, we may replace (𝑋, 𝐵,M)with (𝑍, 𝐵𝑍,M)and assume
that −(𝐾𝑋+𝐵+M𝑋)is semiample and that (𝑋, 𝐵, M)has coregularity 0. By [14, Theorem 1.2],
then this reduction shows that Theorem 7(𝑑,0)holds for any d; this will be used to then appeal to
Theorem 5.1.
If 𝐾𝑋+𝐵+M𝑋∼Q0, then the statement follows from [13, Theorem 1]. Hence, we may assume
that the ample model Wof −(𝐾𝑋+𝐵+M𝑋)is positive-dimensional. Since (𝑋, 𝐵,M)has coregularity
0 and dim 𝑊>0, then some generalized log canonical center of (𝑋, 𝐵,M)is vertical over W.Wemay
replace (𝑋, 𝐵,M)with a generalized dlt modification and assume that some 𝑆⊂Supp𝐵is vertical
over W. Write Ξ=𝐵−𝑆.Let𝑋be the ample model of Ξ+M𝑋over W. Notice that, since Xis of Fano
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Forum of Mathematics, Sigma 47
type and Sis ver tical over W,𝑋𝑋is a birational contraction. Let Ξ,𝐵
be the push-forward of Ξ,𝐵
on 𝑋. Note that Sis not contained in Bs(Ξ+M𝑋/𝑊). Hence, Sis a generalized log canonical place
of (𝑋,𝐵
−𝜖Ξ,(1−𝜖)M)for every 𝜖>0 small enough. Let (𝑌, 𝐵
𝑌,M)beaQ-factorial generalized
dlt modification of (𝑋,𝐵
,M).Let𝜋:𝑌→𝑋be the associated projective morphism. We write
Ξ𝑌+M𝑌=𝜋∗(Ξ+M𝑋).
By construction, the following conditions hold:
◦the generalized pair (𝑌, 𝐵
𝑌,M)is generalized dlt and −(𝐾𝑌+𝐵𝑌+M𝑌)is nef;
◦the generalized pair (𝑌, 𝐵
𝑌−𝜖Ξ𝑌,(1−𝜖)M)is generalized dlt, it is not generalized klt, and the
divisor −(𝐾𝑌+𝐵𝑌−𝜖Ξ𝑌+(1−𝜖)M𝑌)is big and nef.
By Lemma 2.12 and Lemma 2.13,anN-complement of (𝑌, 𝐵
𝑌,M)induces an N-complement of
(𝑋, 𝐵,M). By Theorem 5.1, we conclude that (𝑌, 𝐵
𝑌,M)admits an 𝑁(𝐷(R),𝑑−1,0,𝑝)-complement.
Notice that we can rely on Theorem 5.1 in lower dimension, since we are proceeding by induction on
d, and the conjectures to which Theorem 5.1 is conditional are known in coregularity 0 (by the remarks
under the statement of Theorem 7). Hence, (𝑋, 𝐵, M)admits a 𝑁(𝐷(R),𝑑−1,0,𝑝)-complement. Note
that 𝐷(R)⊂Λ.Sothisisalsoa𝑁(Λ,𝑑−1,0,𝑝)-complement. Thus, by the minimality of 𝑁(Λ,𝑑,0,𝑝),
we have that
𝑁(Λ,𝑑,0,𝑝)≤𝑁(Λ,𝑑−1,0,𝑝).
This implies that 𝑁(Λ,𝑑,0,𝑝)is bounded above by 𝑁(Λ,1,0,𝑝). This finishes the proof.
Proof of Theorem 4.We follow the notation of the proof of Theorem 7.1.By[13, Corollary 3], a log
Calabi–Yau pair with standard coefficients and coregularity 0 has coefficients in 1
2,1. In particular,
a generalized log canonical threshold with standard coefficients and coregularity 0 belongs to 1
2,1
(see Definition 2.26 and [10, Theorem 4.3]). On the other hand, by Corollary 3.3, a generalized pseudo-
effective threshold with standard coefficients and coregularity 0 is either 1
2or 1. In the proof of Theorem
3.5,thesetR(S,0,2)only consists of log canonical thresholds of coregularity 0 and pseudo-effective
thresholds of coregularity 0. Hence, by the proof of Theorem 3.5, we may assume that the coefficients
of Bbelong to {1
2,1}, that is, we have that
R0R(S,0,2)=1
2,1.
Note that 𝐼S=1 (see Definition 2.14) and 𝑝=2 in this case. By the proof of Theorem 7.1, we conclude
that
𝑁(S,𝑑,0,2)=𝑁(R0,𝑑,0,2)≤𝑁(R0,𝑑−1,0,2),
for every 𝑑≥2. Then, the proof follows as 𝑁(S,1,0,2)=2.
Proposition 7.2. Assume that Theorem 8(𝑐)holds. Then, Theorem 6(𝑐)holds.
Proof. By Lemma 2.16, we may assume that Λis derived. By [13, Theorem 2], we may assume that
the coefficients of Bbelong to a finite subset Λ0⊂Λ.Let𝜆0be the smallest positive integer such that
𝜆0Λ0⊂Zand pdivides 𝜆0.Let𝐼(Λ0,𝑐,𝑑,𝑝)be the smallest positive integer that is divisible by the
index of all the generalized pairs as in the statement of dimension at most d.Apriori,𝐼(Λ0,𝑐,𝑑,𝑝)
may not exist. However, due to Conjecture 1and Lemma 2.30, we know that 𝐼(Λ0,𝑐,𝑐,𝑝)is finite.
We proceed by induction on the dimension d. Assume that 𝐼(Λ0,𝑑 −1,𝑐,𝑝)is finite. We may assume
that 𝐼(Λ0,𝑑−1,𝑐,𝑝)is divisible by 𝜆0.Let(𝑋, 𝐵, M)be a d-dimensional generalized pair as in the
statement. We write 𝐼(𝑋, 𝐵, M)for the index of this generalized log Calabi–Yau pair.
By Theorem 2.29, we may replace (𝑋, 𝐵,M)with a Kollár–Xu model and assume the following
conditions hold:
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48 F. Figueroa et al.
◦the generalized pair (𝑋, 𝐵,M)is generalized dlt;
◦there is a contraction 𝜋:𝑋→𝑍for which 𝐵fully supports a 𝜋-semiample and 𝜋-big divisor; and
◦every generalized log canonical center of (𝑋, 𝐵,M)dominates Z.
In particular, we know that Zhas dimension at most c.
We will proceed in two different cases, depending on the coefficients of 𝐵+M𝑋and the rational
connectedness of X.
Case 1: We assume that Mis numerically nontrivial and Xis rationally connected.
We will proceed in two different sub-cases, depending on the coefficients of {𝐵}+M𝑋.
Case 1.1: We assume that {𝐵}+M𝑋is Q-trivial on the general fiber of 𝜋.
We observe that, given the running assumption that Mis numerically nontrivial, in this subcase we
have dim 𝑍>0. By Theorem 8(𝑐), we can write
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑍+𝐵𝑍+N𝑍),
where (𝑍, 𝐵𝑍,N)is a generalized klt log Calabi–Yau pair and the positive integer qonly depends on
Λ0,cand p. The coefficients of 𝐵𝑍belong to a set Ω, which satisfies the DCC, and only depends on Λ0,
cand p. Finally, the divisor 𝑞Nis b-Cartier. We conclude that
𝐼(𝑋, 𝐵,M)≤lcm(𝑞, 𝐼 (Ω,𝑐,𝑐,𝑞)).
Note that the value on the r.h.s. only depends on Λ0,𝑐and p.
Case 1.2: We assume that {𝐵}+M𝑋is nontrivial on the general fiber of 𝜋.
We run a (𝐾𝑋+𝐵)-MMP over Z. Since 𝐾𝑋+𝐵is not pseudo-effective over Z, this minimal model
program terminates with a Mori fiber space 𝜋:𝑋→𝑊over Z. We denote the push-forward of Bto 𝑋
by 𝐵. We may replace (𝑋, 𝐵, M)with (𝑋,𝐵
,M)and assume that 𝐾𝑋+𝐵is antiample over W.In
this reduction, we may give up the generalized dlt property for (𝑋, 𝐵, M), while (𝑋, 𝐵,M)remains
generalized dlt. By the reduction to the Kollár–Xu model, the divisor 𝐵contains a prime component
Swhich dominates W. By construction, the general fiber of 𝑆→𝑊is rationally connected. Since Wis
rationally connected, we conclude that Sis rationally connected. Since (𝑋, 𝐵,M)is generalized dlt,
Sis normal. Let (𝑆, 𝐵𝑆,N)be the generalized pair obtained by adjunction. Then, by Lemma 2.17 and
[10, Theorem 2], we know that the coefficients of 𝐵𝑆belong to Λ0and 𝑝Nis b-Cartier. We conclude that
𝐼(Λ0,𝑑−1,𝑐,𝑝)(𝐾𝑆+𝐵𝑆+N𝑆)∼0.
By Theorem 2.31, we conclude that
𝐼(Λ0,𝑑−1,𝑐,𝑝)(𝐾𝑋+𝐵+M𝑋)∼0.
Putting Case 1.1 and Case 1.2 together, we conclude that if Mis numerically nontrivial and Xis
rationally connected, then we have that
𝐼(𝑋, 𝐵,M)≤max{𝐼(Λ0,𝑑−1,𝑐,𝑝),lcm(𝑞, 𝐼 (Ω,𝑐,𝑐,𝑞))}.
Case 2: We assume that either Mis the trivial b-divisor or Mis numerically trivial and Xis rationally
connected.
In the latter case, by [13, Lemma 3.9] we know that 𝑝M𝑋∼0 where 𝑋→𝑋is a resolution on
which Mdescends. Replacing Mwith 𝑝M, we may assume that the first case holds.
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Forum of Mathematics, Sigma 49
Let (𝑋, 𝐵)be a log Calabi–Yau pair as in the statement. We assume that the coefficients of Bbelong
to Λ0. We may assume that 𝑑>𝑐. We replace (𝑋, 𝐵)with a Q-factorial dlt modification. Let 𝑆⊂𝐵be
a prime component. We run a (𝐾𝑋+𝐵−𝑆)-MMP. This terminates with a Mori fiber space 𝜋:𝑋→𝑊.
We denote by 𝐵the push-forward of Bto 𝑋. We replace (𝑋, 𝐵)with (𝑋,𝐵
). Note that Sis ample
over the base Wand (𝑋,𝐵 −𝑆)is dlt. Let (𝑆, 𝐵𝑆)be the pair obtained by adjunction. Then, (𝑆, 𝐵𝑆)is
a semilog canonical log Calabi–Yau pair by [19, Example 2.6]. Furthermore, by Lemma 2.17 and [10,
Theorem 2], the coefficients of 𝐵𝑆belong to Λ0. By Theorem 4.9, up to replacing 𝐼(Λ0,𝑑 −1,𝑐)with
lcm(𝐼(Λ0,𝑑 −1,𝑐),𝐼
𝑎(Λ0,𝑐)), we may assume that
𝐼(Λ0,𝑑−1,𝑐,0)(𝐾𝑆+𝐵𝑆)∼0.
Here, 𝐼𝑎(Λ0,𝑐)is the constant from Theorem 4.9. By construction, we have that (𝑋, 𝐵 −𝑆)is dlt, Xis
Q-factorial and klt, and 𝑆⊂𝐵is ample over W. If the fibers of 𝑆→𝑊are connected, then we can
apply Theorem 4.12, to conclude that
𝐼(Λ0,𝑑−1,𝑐,0)(𝐾𝑋+𝐵)∼0.
Otherwise, we can apply Theorem 2.32, to conclude that
𝐼(Λ0,𝑑−1,𝑐,0)(𝐾𝑋+𝐵)∼0.
Putting Case 1 and Case 2 together, we conclude that every generalized pair (𝑋, 𝐵,M)of dimension
das in the statement satisfies that
𝐼(𝑋, 𝐵,M)≤max{𝐼(Λ0,𝑑−1,𝑐,𝑝),lcm(𝑞, 𝐼 (Ω,𝑐,𝑐,𝑞))}.
Hence, we have that
𝐼(Λ0,𝑑,𝑐,𝑝)≤max{𝐼(Λ0,𝑑 −1,𝑐,𝑝),lcm(𝑞, 𝐼(Ω,𝑐,𝑐,𝑞))}.
Proceeding inductively, we conclude that
𝐼(Λ0,𝑑,𝑐,𝑝)≤max{𝐼(Λ0,𝑐,𝑐,𝑝),𝑞𝐼(Ω,𝑐,𝑐,𝑐)}.
The r.h.s. does not depend on d. This finishes the proof of the proposition.
Lemma 7.3. Let 𝜆be a positive integer. Let (P1,𝐵
P1,MP1)be a generalized log Calabi–Yau pair. Assume
that the coefficients of 𝐵P1belong to 𝐷𝜆and 2𝜆MP1is Weil. Then, 𝐼(𝐾1
P+𝐵P1+MP1)∼0for some 𝐼=𝑚𝜆,
where 𝑚≤120𝜆is an even positive integer. Furthermore, if MP1=0, then 𝐻0(P1,OP1(𝐼(𝐾1
P+𝐵P1)))
admits an admissible section.
Proof. We prove the second statement. Let 𝐺=Aut(P1,𝐵
P1). The group Gis a finite extension of a
torus. First, we assume that Gis finite, which turns to imply that 𝐵P1is supported in at least three points.
Let (P1,𝐵
P1)→(P1,𝐵
P1)be the quotient by Ggiven by the Hurwitz formula. By pulling back to P1,it
suffices to show that
ℎ0(P1,OP1(𝐼(𝐾P1+𝐵
P1))) ≠0
for some 𝐼=𝑚𝜆, where 𝑚≤120𝜆. We write
𝐵
P1=
𝑘
𝑖=11−1
𝑚𝑖
+𝑙𝑘
𝑗=1
𝑝𝑗,𝑘
𝜆
𝑚𝑖{𝑝𝑖},
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50 F. Figueroa et al.
where the 𝑝𝑗,𝑘 ’s are positive integers. We may assume that 𝑚1≥···≥𝑚𝑘.If𝑚1≤5, then for 𝐼=120𝜆,
we have that 𝐼𝐵
P1is Weil. If 𝑚1>5, then 𝑚3=···=𝑚𝑘=1. If 𝑚2>2𝜆, then 𝑚1>2𝜆and
coeff 𝑝1(𝐵
P1)+coeff 𝑝2(𝐵
P1)+coeff 𝑝3(𝐵
P1)>1−1
2𝜆+1−1
2𝜆+1
𝜆=2.
This leads to a contradiction. Hence, we may assume that 𝑚2≤2𝜆. In this case, we have that 𝑚2𝜆𝐵
P1
is Weil. Thus, it suffices to take 𝐼=𝑚2𝜆with 𝑚2≤2𝜆.
Now, we assume that Gis a finite extension of a torus. This implies that 𝐵P1is supported in two
points, so we may assume that 𝐵P1={0}+{∞}and that 𝐺G𝑚Z
2. Note that G𝑚acts trivially on
B-representations as it is connected. Hence, in this case, 2(𝐾P1+𝐵P1)admits an admissible section.
In the first statement, we need to control the index of the generalized pair. The same argument we
used in the previous paragraph to control the index of (P1,𝐵
P1)applies to (P1,𝐵
P1,MP1). The only
difference is that due to the presence of M, it could be that 𝐵P1is supported at only one point or it could
even be empty. In this case, 2𝜆(𝐾P1+𝐵P1+MP1)is integral, and the claims follow.
Proof of Theorem 2.Let (𝑋, 𝐵, M)be a d-dimensional generalized log Calabi–Yau pair as in the state-
ment. We follow the proof of Proposition 7.2. In Case 1.1, we can write
𝑞(𝜆)(𝐾𝑋+𝐵+M𝑋)∼𝑞(𝜆)𝜋∗(𝐾P1+𝐵P1+MP1),
where (P1,𝐵
P1,MP1)is a generalized log Calabi–Yau pair for which the coefficients of 𝐵P1belong
to a DCC set Ωand 2𝜆MP1is Weil. Here, we can take 𝑞(𝜆)=2𝜆. Indeed, the constant 𝑞(𝜆)in the
canonical bundle formula depends on the existence of bounded complements with standard coefficients
and relative absolute coregularity 0. By Theorem 5.3 and Theorem 4such relative complement can be
chosen to be a 2𝜆-complement. On the other hand, we can take Ω=𝐷𝜆. Indeed, in this case, (𝑋, 𝐵,M)
admits a log canonical center which has a finite dominant map to P1. Thus, the coefficients of 𝐵P1
can be computed by the adjunction formula and Riemann–Hurwitz. By Lemma 7.3, we conclude that
𝐼(𝐾P1+𝐵P1+MP1)∼0 for some integer 𝐼=𝑚𝜆 where 𝑚≤120𝜆. We conclude that 𝐼(𝐾𝑋+𝐵+M𝑋)∼0
for the same choice of I.
In Case 1.2 and Case 2, the index of (𝑋, 𝐵,M)divides the index of a possibly nonnormal log Calabi–
Yau pair of coregularity 1 and dimension 𝑑−1. Inductively, we reduce to the one-dimensional case
which follows by Theorem 4.9 and Lemma 7.3.
Proof of Theorem 3.In a similar fashion as the proof of Theorem 2, this follows from the proof of
Proposition 7.2 and the classification of log Calabi–Yau pair structures on P1with standard coefficients.
Proposition 7.4. Assume that Theorem 6(𝑐)holds and Theorem 8(𝑐)holds. Then, Theorem 7(𝑐)holds.
Proof. By Lemma 2.16, we may assume that Λis derived. Let 𝑁(Λ,𝑑,𝑐,𝑝)be the smallest positive
integer for which every generalized pair (𝑋, 𝐵,M)of dimension das in the statement admits an
N-complement. By Theorem 3.5, there exists a finite subset R⊂Λfor which
𝑁(R,𝑑,𝑐,𝑝)=𝑁(Λ,𝑑,𝑐,𝑝)
for every d. We proceed by induction on d. We may assume that every complement throughout the proof
is divisible by pand 𝐼R. We write 𝑁(𝑋, 𝐵,M)for the smallest positive integer for which (𝑋, 𝐵, M)
admits an N-complement of coregularity c.Let𝐵+Γbe a Q-complement of (𝑋, 𝐵, M)that computes
the absolute coregularity. Let (𝑌, 𝐵
𝑌+Γ𝑌+𝐸,M)be a generalized Q-factorial dlt modification of
(𝑋, 𝐵 +Γ𝑌,M), where 𝐵𝑌(resp. Γ𝑌) is the strict transform of the fractional part of B(resp. Γ). Then,
the generalized pair (𝑌, 𝐵
𝑌+𝐸,M)has coregularity c.Weruna−(𝐾𝑌+𝐵𝑌+𝐸)-MMP with scaling,
which terminates with a good minimal model Z. By Lemma 2.22, the generalized pair (𝑍, 𝐵𝑍+𝐸𝑍)
has coregularity c. By Lemma 2.13 and Lemma 2.12, we have that 𝑁(𝑋, 𝐵, M)≤𝑁(𝑍, 𝐵𝑍+𝐸𝑍,M).
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Forum of Mathematics, Sigma 51
In order to give an upper bound for 𝑁(𝑋, 𝐵, M), we may replace (𝑋, 𝐵, M)with (𝑍, 𝐵𝑍+𝐸𝑍,M).
Thus, we may assume that coreg(𝑋, 𝐵,M)=𝑐and −(𝐾𝑋+𝐵+M𝑋)is semiample. Let Wbe the ample
model of −(𝐾𝑋+𝐵+M𝑋). We proceed in three different cases, depending on the dimension of W.
Case 1: In this case, we can assume that dim 𝑊=0.
Then, we have that 𝑁(𝐾𝑋+𝐵+M𝑋)∼0forsomeNthat only depends on Λ,cand pby Theorem 6(𝑐).
Case 2: In this case, we assume that dim 𝑊=dim 𝑋.
In this case, we have that −(𝐾𝑋+𝐵+M𝑋)is a nef and big divisor. We may assume that (𝑋, 𝐵, M)is
Q-factorial and generalized dlt. By Theorem 5.1, we conclude that 𝑁(𝑋,𝐵
,M)≤𝑁(Λ,𝑑−1,𝑐,𝑝)=
𝑁(R,𝑑−1,𝑐,𝑝). In this case, we conclude that
𝑁(𝑋, 𝐵,M)≤𝑁(Λ,𝑑−1,𝑐,𝑝)=𝑁(R,𝑑−1,𝑐,𝑝).
Case 3: In this case, we assume that 0 <dim 𝑊<dim 𝑋.
We run a ({𝐵}+M𝑋)-MMP over Wwhich terminates with a good minimal model 𝑋𝑋→𝑊
over W. By Lemma 2.22, the coregularity of (𝑋, 𝐵, M)is unaffected by this MMP. Let 𝑊→𝑊be the
ample model of {𝐵}+M𝑋over the base. First, assume that dim 𝑊=dim 𝑋. In this case, {𝐵}+M𝑋is
big over W. Hence, for 𝜖>0 small enough, we have that the generalized pair (𝑋,𝐵
−𝜖{𝐵},(1−𝜖)M)
is generalized log canonical but not generalized klt and the divisor
−(𝐾𝑋+𝐵−𝜖{𝐵}+(1−𝜖)M𝑋)
is big and nef. Note that 𝑁(𝑋, 𝐵,M)=𝑁(𝑋,𝐵
,M). By Theorem 5.1, we conclude that
𝑁(𝑋, 𝐵,M)≤𝑁(Λ,𝑑−1,𝑐,𝑝)=𝑁(R,𝑑−1,𝑐,𝑝).
From now on, we assume that 0 <dim 𝑊<dim 𝑋. We separate in two cases, depending on the log
canonical centers of (𝑋,𝐵
,M).
Case 3.1: In this case, we assume that there is a generalized log canonical center of (𝑋,𝐵
,M)that is
vertical over 𝑊.
We may assume that (𝑋,𝐵
,M)is generalized dlt. Let 𝑆⊂𝐵be a prime component that is
vertical over 𝑊. In this case, 𝐵
hor is big over the base. We run a 𝐵
hor-MMP over 𝑊which terminates
with a good minimal model 𝑋
0, and we consider its ample model 𝑋 over 𝑊. We have the following
commutative diagram:
(𝑋, 𝐵,M)//___
(𝑋,𝐵
,M)//___
𝜋
(𝑋
0,𝐵
0,M)𝜓//
wwooooooooooo
(𝑋,𝐵
,M)
ssggggggggggggggggggggggggg
𝑊𝑊
.
oo
Note that all the previous models are crepant. Hence, we have that
𝑁(𝑋, 𝐵,M)=𝑁(𝑋,𝐵
,M)=𝑁(𝑋,𝐵
,M).(7.1)
By construction, the following conditions are satisfied:
◦the variety Wis an ample model for −(𝐾𝑋+𝐵+M𝑋);
◦the variety 𝑊is an ample model for {𝐵}+𝑀𝑋over W;
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52 F. Figueroa et al.
◦the variety 𝑋
0is a good minimal model for 𝐵
0,hor over 𝑊; and
◦the variety 𝑋 is an ample model for 𝐵
hor over 𝑊.
We conclude that the divisor
−(𝐾𝑋 +𝐵 −𝜖{𝐵}−𝛿𝐵
hor +(1−𝜖)M𝑋 )
is ample for 𝜖𝛿>0 small enough. We claim that the generalized pair
(𝑋,𝐵
−𝜖{𝐵}−𝛿𝐵
hor,(1−𝜖)M)
is generalized log canonical but not generalized klt. Note that (𝑋
0,𝐵
0−𝜖{𝐵
0}−𝛿𝐵
0,hor,(1−𝜖)M)is
generalized log canonical and not generalized klt as 𝑆
0is a component of 𝐵
0−𝜖{𝐵
0}−𝛿𝐵
0,hor.The
morphism 𝜓is (𝐾𝑋
0+𝐵
0−𝜖{𝐵
0}−𝛿𝐵
0,hor +(1−𝜖)M𝑋
0)-trivial, so the claim follows. By Theorem
5.1 and the sequence of equalities (7.1), we conclude that
𝑁(𝑋, 𝐵,M)≤𝑁(Λ,𝑑−1,𝑐,𝑝)=𝑁(R,𝑑−1,𝑐𝑝).
Thus, in this case, we have that 𝑁(𝑋, 𝐵, M)≤𝑁(R,𝑑−1,𝑐).
Case 3.2: In this case, we assume that all the generalized log canonical centers of (𝑋,𝐵
,M)are
horizontal over 𝑊.
Let 𝜋:𝑋→𝑊be the projective contraction. We may apply Theorem 8(𝑐)to obtain a linear
equivalence:
𝑞(𝐾𝑋+𝐵+M𝑋)∼𝑞𝜋∗(𝐾𝑊+𝐵𝑊+N𝑊),
where the following conditions are satisfied:
◦the generalized pair (𝑊,𝐵
𝑊,N)is of Fano type, has dimension 𝑑𝑊≤𝑐and is exceptional (i.e., its
absolute coregularity is equal to its dimension 𝑑𝑊);
◦the positive integer qonly depends on Λ,cand p;
◦the coefficients of 𝐵𝑊belong to a DCC set Ωwhich only depends on Λ,cand p; and
◦the b-nef divisor 𝑞Nis b-Cartier.
Indeed, if (𝑊,𝐵
𝑊,N)is not exceptional, by pulling back a non-klt complement of it, we obtain a
complement for (𝑋,𝐵
,M)of coregularity strictly less than c. This leads to a contradiction. By [14,
Theorem 1.2], any generalized pair (𝑊,𝐵
𝑊,N)as above admits an 𝑁(Ω,𝑑
𝑊,𝑑
𝑊,𝑞)-complement.
By pulling back, we obtain an N-complement for (𝑋,𝐵
,M)for some
𝑁≤lcm(𝑞, 𝑁 (Ω,𝑑
𝑊,𝑑
𝑊,𝑞)).
Putting Case 1 through Case 3 together, we conclude that
𝑁(𝑋, 𝐵,M)≤max{𝑁(R,𝑑−1,𝑐,𝑝),lcm(𝑞, 𝑁 (Ω,1,1,𝑞)),...,lcm(𝑞, 𝑁(Ω,𝑐,𝑐,𝑞))}.
for every d-dimensional generalized pair (𝑋, 𝐵,M)as in the statement. Proceeding inductively, we
conclude that every generalized pair (𝑋, 𝐵, M)as in the statement of the theorem satisfies that
𝑁(𝑋, 𝐵,M)≤max{𝑁(R,𝑐,𝑐,𝑝),lcm(𝑞, 𝑁 (Ω,1,1,𝑞)),...,lcm(𝑞, 𝑁 (Ω,𝑐,𝑐,𝑞))}.
Observe that the number on the r.h.s. only depends on Λ,𝑐 and p. This finishes the proof of the
implication.
Lemma 7.5. Let (P1,𝐵
P1,MP1)be a generalized log Calabi–Yau pair for which coeff (𝐵P1)∈𝐷𝑡(Z>0)
and 2MP1is Cartier. If 𝑡>5
6, then 𝑡=1.
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Forum of Mathematics, Sigma 53
Proof. Each coefficient of 𝐵P1is either standard or of the form
1−1
𝑚+𝑡
𝑚>1−1
6𝑚>5/6.(7.2)
The only sets of standard coefficients whose sum is less than 7
6are {1}and {1
2,1
2}. Thus, 𝐵P1is supported
in at most 3 points and 𝑡=1.
Proof of Theorem 5.Let (𝑋, 𝐵, M)be a d-dimensional generalized pair as in the statement. Following
Step 1 of the proof of Theorem 3.5,lettbe a log canonical threshold of coregularity 1 (or a pseudo-
effective threshold) of a prime divisor with respect to (𝑋, 𝐵, M).By[10, Lemma 3.2] (or proof of
Lemma 3.2 in the case where tis a pseudo-effective threshold), we can construct a generalized log
Calabi–Yau pair
(P1,𝐵
P1,MP1)
for which coeff(𝐵P1)∈𝐷𝑡({1}) and 2MP1is Cartier. By Lemma 7.5, we conclude that 𝑡=1 provided
that 𝑡>5/6. Hence, by the proof of Theorem 3.5, we may assume that the coefficients of Bbelong to
R11
2,2
3,3
4,4
5,5
6.
Proceeding as in the proof of Proposition 7.4, we conclude that
𝑁(𝑋, 𝐵,M)≤𝑁(R1,𝑑,1,2)≤lcm(2,𝑁(R1,1,1,2)).
Note that we can take 𝑞=2 due to Theorem 1. We conclude that 𝑁∈{2,4,6}.
We prove the three main theorems of the article. The theorems are proved together inductively.
Proof of Theorems 6,7and 8.Note that Theorem 8(0)is trivial. By [13, Theorem 1], we conclude that
Theorem 6(0)holds. By Theorem 7.1, we know that Theorem 7(0)holds. Assume that Theorem 6(𝑐−1),
Theorem 7(𝑐−1)and Theorem 8(𝑐−1)hold. By Proposition 6.3, we conclude that Theorem 8(𝑐)
holds. By Proposition 7.2, we conclude that Theorem 6(𝑐)holds. By Proposition 7.4, we conclude that
Theorem 7(𝑐)holds. This finishes the proof of the theorems.
Finally, we prove the application to klt singularities.
Proof of Theorem 9.Let (𝑋;𝑥)be a klt singularity of absolute coregularity 0. Let (𝑋, Γ0;𝑥)be a strictly
log canonical pair of coregularity 0 at x.By[43, Lemma 1], there exists a plt blow-up 𝜋:𝑌→𝑋that
extracts a unique exceptional divisor Ethat is a log canonical place of (𝑋, Γ0;𝑥).3In particular, the
pair (𝐸,Diff𝐸(0)) is a Fano pair of absolute coregularity 0 and with standard coefficients (see, e.g.,
[39, Proposition 3.9]). By Theorem 4,(𝐸,Diff𝐸(0)) admits a 1- or 2-complement. By Steps 4–9 of the
proof of Proposition 5.2, we can lift this complement to a 1- or 2-complement of (𝑋;𝑥)at x.
Proof of Theorem 10.The proof is analogous to the one of Theorem 9by replacing Theorem 4with
Theorem 5.
Acknowledgements. This project was initiated in the Minimal Model Program Learning Seminar. The authors would like to
thank Mirko Mauri for many discussions that led to some of the ideas of this article. Lastly, the authors would like to thank the
anonymous referee for useful comments and suggestions that helped the authors improve the clarity of this work.
Funding statement. SF was partially supported by ERC starting grant #804334. FF received partial financial support by the
NSF under János Kollár’s grant number DMS-1901855.
Competing interest. The authors have no competing interest to declare.
3The fact that the divisor computing the plt blow-up of (𝑋;𝑥)can be chosen to be a log canonical place of (𝑋, Γ0;𝑥)is
implicit in the proof of [43, Lemma 1].
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54 F. Figueroa et al.
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