# Boundedness of $n$-complements on surfaces

Xiangze Zeng

###### Abstract.

In light of Shokurov’s result [16, Theorem 3] in 2020, the main purpose of
this paper is to prove the boundedness of $n$-complements for surface pairs
$(X/Z\ni o,B)$ under the same setting, without the asssumptions that $X/Z$ is
of weak Fano type or $(X/Z\ni o,B)$ has a klt $\mathbb{R}$-complement.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Basic Properties of Complements
    2. 2.2 Rational Approximation of Coefficients
    3. 2.3 Connectedness Lemma
    4. 2.4 Inversion of Adjunction
    5. 2.5 Minimal Dlt Modification
    6. 2.6 Zariski Decomposition
    7. 2.7 0-contractions
    8. 2.8 Elliptic Fibrations
    9. 2.9 Known Boundedness Results
  3. 3 Boundedness for Curves
    1. 3.1 Classification of $\mathbb{R}$-complementary curves
    2. 3.2 Boundedness of $n$-complements on curves
  4. 4 A special big case of Theorem 1.4
  5. 5 Proof of the Main Theorem 1.4

##  1\. Introduction

The concept of $n$-complements naturally arises from the study on the
anticanonical linear system $|-nK_{X}|$, and was introduced by V.V.Shokurov in
[14]. Later this concept of $n$-complements was generalized to
$\mathbb{R}$-complements and Shokurov asked: does the existence of an
$\mathbb{R}$-complement necessarily imply the existence of an $n$-complement
for some integer $n$? And if so,what kind of restrictions can we put on this
$n$? The widely expected result is that there exists a finite set
$\mathcal{N}$ of positive integers such that, for a certain class of relative
pairs $(X/Z\ni o,B)$, as long as $(X/Z\ni o,B)$ has an
$\mathbb{R}$-complement, it also has an $n$-complement for some
$n\in\mathcal{N}$. We’ll refer to this statement as ”boundedness of
$n$-complements” throughout this paper. Boundedness of $n$-complements on
surfaces was established by V.V.Shokurov in [15]. In [1] C.Birkar proved
boundedness of $n$-complements for pairs $(X,B)$ of Fano type with hyperstand
multiplicities in arbitrary dimensions, and later Shokurov generalized this
boundedness result to all relative pairs of weak Fano type in [16]. In this
paper, we’ll generalize Shokurov’s boundedness result Theorem 3 in [16] to all
surface pairs $(X/Z\ni o,B)$, without the weak Fano type assumption on $X/Z$
and without the restrictions on the multiplicities of $B$. To better state our
main theorem 1.4, we recall the formal definitions of complements first:

######  Definition 1.1.

Let $(X/Z\ni o,D)$ be a pair with a proper local morphism $X/Z\ni o$ and an
$\mathbb{R}$-divisor $D$. Then a pair $(X/Z\ni o,D^{+})$ with the same local
morphism is called an $\mathbb{R}$-complement of $(X/Z\ni o,D)$ if the
following holds:

  1. (1)

$D^{+}\geq D$;

  2. (2)

$(X,D^{+})$ is lc;

  3. (3)

$K_{X}+D^{+}\sim_{\mathbb{R}}0/Z\ni o$.

######  Definition 1.2.

Let $(X/Z\ni o,D)$ be a pair with a proper local morphism $X/Z\ni o$ and an
$\mathbb{R}$-divisor $D=\sum d_{i}D_{i}$, and $n$ is a positive integer. Then
a pair $(X/Z\ni o,D^{+}=\sum d_{i}^{+}D_{i})$ with the same local morphism is
called an $n$-complement of $(X/Z\ni o,D)$ if the following holds:

  1. (1)

for every prime divisor $D_{i}$,

$d_{i}^{+}\geq\begin{cases}1&\text{if $d_{i}=1$;}\\\
\frac{\lfloor(n+1)d_{i}\rfloor}{n}&\text{otherwise;}\end{cases}$

  2. (2)

$(X,D^{+})$ is lc;

  3. (3)

$K_{X}+D^{+}\sim_{n}0/Z\ni o$.

Now we are ready to state our main result Theorem 1.4, with the complementary
index $n$ under the restrictions given by Definition 1.3 below.

######  Definition 1.3.

Given these data: a positive integer $I$, a nonrational vector
$v\in\mathbb{R}^{l}$, a direction $e$ of the rational affine span $\langle
v\rangle$ and a real number $\epsilon>0$. We say an integer $n$ is under
complementary restrictions with respect to $I,v,e,\epsilon$ if there exists a
vector $v_{n}\in\mathbb{R}^{l}$ satisfying the following:

  1. (1)

Divisibility: $I|n$;

  2. (2)

Denominator with $n$: $nv_{n}\in\mathbb{Z}^{l}$;

  3. (3)

Approximation: $||v_{n}-v||<\epsilon/n$;

  4. (4)

Directional Approximation: $||\frac{v_{n}-v}{||v_{n}-v||}-e||<\epsilon$.

######  Theorem 1.4.

Suppose we have the data $I,v,e,\epsilon$ as in Definition 1.3. Then we can
find a finite set $\mathcal{N}=\mathcal{N}(I,v,e,\epsilon)$ such that:

  1. (1)

Restrictions: $\mathcal{N}$ consists of integers under complementary
restrictions;

  2. (2)

Existence of $n$-complements: suppose we have a surface pair $(X/Z\ni o,B)$
with a proper local morphism $X/Z\ni o$, connected central fiber $X_{o}$ and
an $\mathbb{R}$-boundary $B$. Then if $(X/Z\ni o,B)$ has an
$\mathbb{R}$-complement, it also has an $n$-complement for some
$n\in\mathcal{N}$.

I would like to thank Professor Shokurov for his help and guidance. This work
wouldn’t be completed without his deep insights and infinite patience.

##  2\. Preliminaries

###  2.1. Basic Properties of Complements

In this section we list some basic properties of $n$-complements frequently
used throughout this paper. Lemma 2.1 implies that $n$-complements are
preserved by birational pushforwards, while Lemma 2.2 is a monotonicity result
for $n$-complements.

######  Lemma 2.1.

([12, Proposition 4.3.1]). Let $f:Y/Z\to X/Z$ be a birational contraction, $D$
a subboundary on $Y$. If $(Y/Z,D)$ has an $n$-complement $(Y/Z,D^{+})$, then
$(X/Z,f(D))$ has an $n$-complement $(X/Z,f(D^{+}))$.

######  Remark 1.

If we put certain restrictions on the coefficients of the boundary $D$, we can
also pull back complements along a birational contraction. See [12,
Proposition 4.3.2] for one of such result.

######  Lemma 2.2.

([16, Proposition 1]). Let $(X/Z,D)$ and $(X/Z,D\,^{\prime})$ be two pairs on
$X$ with $D\geq D\,^{\prime}$. If $(X/Z,D)$ has an $n$-complement
$(X/Z,D^{+})$, then $(X/Z,D^{+})$ is also an $n$-complement of
$(X/Z,D\,^{\prime})$.

###  2.2. Rational Approximation of Coefficients

We’ll quote and use certain known boundedness results on $n$-complements for
the main proof. Since these positive results are established under the
assumption that coefficients of $B$ are contained in a hyperstandard set, or
more generally, a dcc set of rational numbers in [0,1], we briefly introduce
the definition of hyperstandard sets and a procedure of lower rational
approximation of coefficients using hyperstandard sets here. For more details
of related constructions, properties and proofs, see Construction 1 from [16].

######  Definition 2.3.

([11, 3.2]). Let $\mathfrak{R}$ be a set of rational numbers in $[0,1]$. Then
a hyperstandard set is

$\Phi=\Phi(\mathfrak{R})=\\{1-\frac{r}{l}|r\in\mathfrak{R},\text{$l$ is a
positive integer}\\}\cap[0,1].$

######  Remark 2.

This definition of hyperstandard sets is a generalization of the set $S$ of
standard multiplicities where
$S=\\{1\\}\cup\\{1-1/l|l\in\mathbb{N}^{+}\\}=\Phi(\\{0,1\\})$. Hence to make
sure that the hyperstandard set $\Phi(\mathfrak{R})$ contains $S$, we’ll
always assume $\\{0,1\\}\subset\mathfrak{R}$.

For a finite set $\mathcal{N}$ of positive integers, and a hyperstandard set
$\Phi=\Phi(\mathfrak{R})$ as in Definition 2.3, we can construct a new set
$\Gamma(\mathcal{N},\Phi)$ of rational numbers in [0,1] as follows:

$\Gamma(\mathcal{N},\Phi)=\\{1-\frac{r}{l}+\frac{1}{l}(\sum\limits_{n\in\mathcal{N}}\frac{m_{n}}{n+1})|r\in\mathfrak{R},l\in\mathbb{N}^{+},m_{n}\in\mathbb{N}\\}\cap[0,1].$

$\Gamma(\mathcal{N},\Phi)$ is again a hyperstandard set by the following
lemma.

######  Lemma 2.4.

([16, Corollary 4]). Let $\mathcal{N}$ be a nonempty finite set of positive
integers and $\mathfrak{R}$ be a finite set of rational numbers in [0,1]. Then
there exists a new finite set $\mathfrak{R}\,^{\prime}$ of rational numbers in
[0,1] such that

$\Gamma(\mathcal{N},\Phi)=\Phi(\mathfrak{R}\,^{\prime}).$

Moreover, $\Gamma(\mathcal{N},\Phi)$ is a dcc set of rational numbers in [0,1]
with only one accumulation point $1\in\Gamma(\mathcal{N},\Phi)$.

######  Construction 1.

Using the fact $\Gamma(\mathcal{N},\Phi)$ is a dcc set with only one
accumulation point 1 from Lemma 2.4, for any $b\in[0,1]$, there exists a
unique largest rational approximation $b_{\mathcal{N},\Phi}\leq b$ in
$\Gamma(\mathcal{N},\Phi)$. For simplicity, in the following text we’ll always
take $\mathfrak{R}=\\{0,1\\}$ and denote this $b_{\mathcal{N},\Phi}$ by
$b_{\mathcal{N}}$. Similarly, for a boundary $B$ on $X$, we can construct its
low approximation $B_{\mathcal{N}}$.

The following lemma shows that this low approximation of the boundary $B$
preserves $n$-complements when $n\in\mathcal{N}$.

######  Lemma 2.5.

([16, Corollary 6]). Let $(X/Z,B)$ be a pair with a boundary $B$. If
$(X/Z,B_{\mathcal{N}})$ has an $n$-complement $(X/Z,B^{+})$ with
$n\in\mathcal{N}$, then $(X/Z,B^{+})$ is also an $n$-complement of $(X/Z,B)$.

###  2.3. Connectedness Lemma

In this section we prove a modified version of the well-known Connectedness
Lemma, which is Lemma 2.7. Let’s first recall the concept of locus of lc
singularities in Definition 2.6.

######  Definition 2.6.

Let $X$ be a normal variety, $(X,D)$ a log pair. Then a subvariety $W\subset
X$ is a center of lc singularities, if there exists a birational contraction
$f:Y\to X$ and a divisor $E$ (not necessarily exceptional) on $Y$, such that
the discrepancy $a(E,D,X)\leq-1$ and $f(E)=W$. The union of all centers of lc
singularities is called the locus of lc singularities of $(X,D)$ and denoted
by $\operatorname{LCS}(X,D)$.

######  Lemma 2.7.

([14, 5.7], [9, Theorem 17.4]). Let $(X/Z\ni o,D)$ be a pair with a proper
local morphism $g:X\to Z\ni o$ and connected central fiber $g^{-1}(o)$,
$D=\sum d_{i}D_{i}$ a divisor on $X$ such that:

  1. (1)

if $d_{i}<0$, then $D_{i}$ is very $g$-exceptional in the sense that
$g_{*}(\mathcal{O}_{X}(D_{i}))=\mathcal{O}_{Z}$;

  2. (2)

$-(K_{X}+D)$ is big and nef over $Z$.

Then $\operatorname{LCS}(X,D)$ is connected in a neighborhood of the fiber
$g^{-1}(o)$.

###### Proof.

Let $X\to Z\,^{\prime}\to Z$ be the Stein factorization of $g$. Since the
central fiber $g^{-1}(o)$ is connected and $X\to Z\,^{\prime}$ has connected
fibers, the fiber of $Z\,^{\prime}\to Z$ over $o$ is a single point
$o\,^{\prime}$. Then we can apply the well-known connectedness lemma to the
contraction $X\to Z\,^{\prime}$, and conclude that $\operatorname{LCS}(X,D)$
is connected near the fiber of $X\to Z\,^{\prime}$ over $o\,^{\prime}$, which
is the same as $g^{-1}(o)$ set-theoretically. ∎

The connectedness principle might not work for pairs $(X/Z\ni o,D)$ without
the big assumption. However, we have a generalization of 2.7 for surface
0-pairs:

######  Lemma 2.8.

([14, Theorem 6.9]). Let $X$ be a surface, $g:X\to Z\ni o$ a proper local
morphism with connected central fiber $g^{-1}(o)$, $D$ an effective divisor on
$X$ such that $K_{X}+D\equiv 0$. If $\operatorname{LCS}(X,D)$ is disconnected
in a neighborhood of $g^{-1}(o)$, then

  1. (1)

$(X,D)$ is plt in a neighborhood of $g^{-1}(o)$, and

  2. (2)

$\operatorname{LCS}(X,D)\cap g^{-1}(o)$ has two connected components.

######  Remark 3.

For a detailed discussion on Lemma 2.8 in higher dimensions and for pairs with
$-(K_{X}+D)$ nef, see [6, Theorem 1.2] or [2, Theorem 1.2].

###  2.4. Inversion of Adjunction

In this section we recall the fact that inversion of adjunction holds for log
canonical surface pairs, which is Lemma 2.9 below.

######  Lemma 2.9.

Let $(X,S+D)$ be a log pair with $X$ a smooth surface, $S\subset X$ a smooth
curve, and $D$ an effective $\mathbb{R}$-divisor on $X$ such that
$S\not\subset\operatorname{Supp}D$. Then $(S,D|_{S})$ is lc if and only if
$(X,S+D)$ is lc in a neighborhood of $S$.

The following is a special generalization of Lemma 2.9 which we’ll use in the
proof of Theorem 4.4 later.

######  Corollary 2.10.

Let $(X,S+D)$ be a log pair with $X$ a smooth surface, $S\subset X$ a smooth
curve, $D=\sum d_{i}D_{i}$ an $\mathbb{R}$-divisor on $X$ such that
$S\not\subset\operatorname{Supp}D$, and $D_{i}.S=0$ for all $i$ with
$d_{i}<0$. Then if $(S,D|_{S})$ is lc, $(X,S+D)$ is lc in a neighborhood of
$S$.

###### Proof.

Since $D_{\geq 0}|_{S}=D|_{S}$ by our assumption where $D_{\geq
0}=\sum_{d_{i}>0}d_{i}D_{i}$, we can apply Lemma 2.9 to the pair $(X,S+D_{\geq
0})$. Hence $(X,S+D_{\geq 0})$ is lc in a neighborhood of $S$. Since $D\leq
D_{\geq 0}$, $(X,S+D)$ is also lc in a neighborhood of $S$. ∎

######  Remark 4.

One can find a more general version of Lemma 2.9 in higher dimensions and for
a wider range of varieties $X$ in [7]. For inversion of adjunction in the log
terminal case, see [10, Theorem 5.50].

###  2.5. Minimal Dlt Modification

In some of our proofs, we need to lift lc pairs to a higher model with better
singularities. That’s where Minimal Dlt Modification of a lc pair as follows
comes in handy.

######  Proposition 2.11.

([12, Proposition 3.1.2]). Let $(X,D)$ be a lc normal surface pair where $D$
is a boundary. Then there exists a blow-up, namely the minimal dlt
modification $f:Y\to X$ such that

  1. (1)

$f^{*}(K_{X}+D)=K_{Y}+D_{Y}+\sum E_{i}$, where $D_{Y}$ is the birational
transform of $D$ and $E_{i}$ are all the exceptional divisors over $X$;

  2. (2)

$(Y,D_{Y}+\sum E_{i})$ is dlt;

  3. (3)

$Y$ is $\mathbb{Q}$-factorial.

###  2.6. Zariski Decomposition

Traditionally Zariski decomposition are defined for effective divisors on
nonsingular surfaces, on which the existence and uniqueness of Zariski
decomposition is purely a linear algebraic result. There are generalizations
of this classical concept to a wider range of algebraic varieties and divisors
in higher dimensions, and here in this section we recall Zariski decomposition
for pseudo-effective divisors on normal $\mathbb{Q}$-factorial surfaces in the
relative setting ([13, Section 2]), where the 2-dimensional intersection
theory works well.

######  Definition 2.12.

Let $X/Z$ be a relative projective normal $\mathbb{Q}$-factorial surface, and
$D$ be any divisor on $X$. Then a decomposition $D=M+F$ is called a Zariski
decomposition of $D$ if the following holds:

  1. (1)

$F=\sum f_{i}D_{i}$, where $D_{i}$ are distinct vertical prime divisors over
$Z$, and $f_{i}>0$;

  2. (2)

$M$ is nef over $Z$;

  3. (3)

the intersection matrix $(d_{ij})_{i\times j}$ is negative definite, where
$d_{ij}=D_{i}.D_{j}$;

  4. (4)

$M.D_{i}=0$ for all $i$.

If such a decomposition exists and is unique, we usually call $M$ the mobile
part and $F$ the fixed part of $D$.

######  Remark 5.

See [13] for a generalization of Zariski decomposition to higher dimensions.

It is well-known that Zariski decomposition exists and is unique for surface
pairs:

######  Theorem 2.13.

([13, Theorem 2.2]). Let $X/Z$ be a projective relative normal
$\mathbb{Q}$-factorial surface and $D$ be a pseudo-effective divisor on $X$.
Then there exists a unique Zariski decomposition $D=M+F$ of $D$.

######  Proposition 2.14.

([13, Proposition 2.5]). Assumptions the same as in Theorem 2.13 and let $L$
be a nef divisor over $Z$. If $L\leq D$, we have $L\leq M$.

###### Proof.

Let $M-L=(D-L)-F=M\,^{\prime}-F\,^{\prime}$, where $M\,^{\prime}$ and
$F\,^{\prime}$ are effective divisors with no common components. Then to show
$L\leq M$, it suffices to show $F\,^{\prime}=0$. On one hand, since $D-L$ is
effective, by our construction $F\,^{\prime}\leq F$. Hence $M.F\,^{\prime}=0$
by Definition 2.12(4), and it follows that
$(M-L).F\,^{\prime}=-L.F\,^{\prime}\leq 0$ since $L$ is nef over $Z$. On the
other hand,
$(M-L).F\,^{\prime}=(M\,^{\prime}-F\,^{\prime}).F\,^{\prime}\geq-(F\,^{\prime})^{2}\geq
0$, where $M\,^{\prime}.F\,^{\prime}\geq 0$ by construction,
$(F\,^{\prime})^{2}\leq 0$ by Definition 2.12(3) and the fact
$F\,^{\prime}\leq F$. Combining these, we have $(F\,^{\prime})^{2}=0$, and
$F\,^{\prime}=0$ follows from Definition 2.12(3). ∎

######  Remark 6.

This property can be rephrased as: $M$ is the biggest nef divisor $L$ such
that $L\leq D$. Another useful result from Proposition 2.14 is: if
$D\,^{\prime}\leq D$ are two pseudo-effective divisors on a surface $X$, then
$M\,^{\prime}\leq M$. Here $M\,^{\prime}$ and $M$ are the mobile parts of
$D\,^{\prime}$ and $D$ respectively.

In general the mobile part $M$ is not necessarily semi-ample. But we can
establish this useful property in the following special case.

######  Lemma 2.15.

Let $(X/Z,B)$ be a lc pair, where $X/Z$ is a projective relative normal
surface, $B$ is effective and $D=K_{X}+B$ is pseudo-effective. Then the mobile
part $M$ of $D$ is semi-ample over $Z$, and there exists a birational
contraction $f:X/Z\to X\,^{\prime}/Z$ such that the fixed part $F$ of $D$ is
$f$-exceptional. Additionally, if $D$ is effective $M$ is also effective.

###### Proof.

Run $(K_{X}+B)$-MMP on the pair $(X/Z,B)$, and suppose at each step we have an
extremal contraction $f_{i}:(X_{i},B_{i})\to(X_{i+1},B_{i+1})$. Set
$E_{i}:=K_{X_{i}}+B_{i}-f_{i}^{*}(K_{X_{i+1}}+B_{i+1})$, then $-E_{i}$ is
$f_{i}$-nef and $f_{i}$-exceptional, hence by Negativity Lemma $E_{i}$ is
effective. Let $X\,^{\prime}/Z$ be the end result of this MMP, and $f:X/Z\to
X\,^{\prime}/Z$ be the composition of all $f_{i}$ with $B\,^{\prime}=f_{*}B$,
$D\,^{\prime}=f_{*}D$, then $E:=K_{X}+B-f^{*}(K_{X\,^{\prime}}+B\,^{\prime})$
is also effective and $f$-exceptional. Note $D=K_{X}+B=f^{*}D\,^{\prime}+E$ is
a Zariski Decomposition of $D$, since $f^{*}D\,^{\prime}$ is nef over $Z$, and
$E\geq 0$ is $f$-exceptional. By uniqueness we then have $M=f^{*}D\,^{\prime}$
and $F=E$. Now $f$ is indeed a birational contraction for which $F$ is
$f$-exceptional. The semi-ampleness of $D\,^{\prime}$ over $Z$ follows from
the relative abundance theorem in dimension 2: $(X\,^{\prime},B\,^{\prime})$
is lc, and $K_{X\,^{\prime}}+B\,^{\prime}=D\,^{\prime}$ is nef over $Z$ by
construction. Hence $M=f^{*}D\,^{\prime}$ is also semi-ample over $Z$.

When $D$ is effective, we can apply Proposition 2.14 to our case with $L=0$,
then we get $M$ is also effective. ∎

###  2.7. 0-contractions

In this section we briefly review the definition of $0$-contractions. For more
detailed definitions and properties, we refer to [16, Section 7].

######  Definition 2.16.

Suppose $f:X\to Z$ is a contraction of normal algebraic varieties, $B$ is an
$\mathbb{R}$-divisor on $X$. Then in the usual sense $f$ is a $0$-contraction
if it satisfies (1) and (2) in the following:

  1. (1)

$(X,B)$ is lc generically over $Z$;

  2. (2)

$K_{X}+B\sim_{\mathbb{R}}0$ over $Z$;

  3. (3)

the horizontal part $B^{h}$ of $B$ with respect to $f$ is generically a
boundary over $Z$.

But in this paper we always assume $0$-contractions also have property (3).

######  Definition 2.17.

Under the same setting as in Definition 2.16, we have two divisors on $Z$
naturally related to $B$. One is called the divisorial part of adjunction
$B_{\operatorname{div}}$ of $f$, which is uniquely determined by this
$0$-contraction; the other is the moduli part of adjunction
$B_{\operatorname{mod}}$ of $f$, which is defined up to $\sim_{\mathbb{R}}$.
For convenience we’ll also use the b-divisor form
$\mathcal{B}_{\operatorname{mod}}$ in this section. For explicit constructions
of these two divisors, we refer to [16, Section 7.2] or [11, 7.2]. The
divisors $B_{\operatorname{div}}$ and $B_{\operatorname{mod}}$ satisfy the
following canonical bundle formula:

$K_{X}+B\sim_{\mathbb{R}}f^{*}(K_{Z}+B_{\operatorname{div}}+B_{\operatorname{mod}}).$

######  Definition 2.18.

For a $0$-contraction $f:(X,B)\to Z$, we call a positive integer $I$ its
adjunction index if

  1. (1)

$\mathbb{K}_{X}+\mathbb{B}\sim_{I}f^{*}(\mathbb{K}_{Z}+\mathcal{B}_{\operatorname{div}}+\mathcal{B}_{\operatorname{mod}})$,
in particular,
$K_{X}+B\sim_{I}f^{*}(K_{Z}+B_{\operatorname{div}}+B_{\operatorname{mod}})$;

  2. (2)

$I\mathcal{B}_{\operatorname{mod}}$ is b-Cartier and
$\mathcal{B}_{\operatorname{mod}}$ is defined modulo $\sim_{I}$, in
particular,
$\mathbb{K}_{X}+\mathbb{B}\sim_{I,Z}f^{*}(\mathbb{K}_{Z}+\mathcal{B}_{\operatorname{div}})$.

Actually, (2) implies (1). See [16, 7.3] for this result and more details.

A well-known result about the moduli part $\mathcal{B}_{\operatorname{mod}}$
is the following theorem.

######  Theorem 2.19.

([16, Theorem 8]). Let $f:(X,B)\to Z$ be a $0$-contraction as in Definition
2.16, and $B$ is an effective $\mathbb{Q}$-divisor generically over $Z$. Then
$\mathcal{B}_{\operatorname{mod}}$ is a b-$\mathbb{Q}$-divisor of $Z$, defined
up to a $\mathbb{Q}$-linear equivalence. Moreover,
$\mathcal{B}_{\operatorname{mod}}$ is b-nef.

Proposition 2.20 below guarantees us that we can pull back $n$-components on
the base $Z$ back to $X$ along a $0$-contraction $f:(X,B)\to Z$ if $n$ is
divisible by the adjunction index $I$.

######  Proposition 2.20.

([16, (10) in Section 7.5]). Suppose we have a $0$-contraction $f:(X,B)\to Z$
with adjunction index $I$, $X$ is a surface and $Z$ is a curve. If $n$ is an
integer with $I|n$, then $(Z,B_{\operatorname{div}}+B_{\operatorname{mod}})$
has an $n$-complement $\Rightarrow$ $(X,B)$ has an $n$-complement.

###  2.8. Elliptic Fibrations

We refer to [12, Section 7.2] for a more complete theory on elliptic
fibrations. Here we only list several definitions and propositions which are
of importance to us later.

######  Definition 2.21.

An elliptic fibration is a contraction $f:X\to Z$ from a surface to a curve
such that its general fiber is a smooth elliptic curve. If we have
$K_{X}\equiv 0$ over $Z$ and $X$ is smooth additionally, we say $f$ is
minimal.

######  Definition 2.22.

Consider a pair $f:(X,B)\rightarrow Z\ni P$ from a surface to a curve. We say
$(X/Z\ni P,B+pf^{*}P)$ is maximally lc if $(X/Z\ni P,B+pf^{*}P)$ is lc and for
any $q>p$, $(X/Z\ni P,B+qf^{*}P)$ is not lc.

Kodaira’s classification of degeneration of elliptic fibers [8] lists all
possible types of these singular fibers. We’ll recall this classical result in
Theorem 2.23 below, and restate two results in the view of canonical bundle
formula and complements respectively.

######  Theorem 2.23.

([12, Theorem 7.2.10]). Let $f:X\to Z\ni P$ be a minimal elliptic fibration
near $P$. By the canonical bundle formula we have
$K_{X}=f^{*}(K_{Z}+D_{\operatorname{div}})$ with effective
$D_{\operatorname{div}}=d_{P}P$. Then the following table lists all possible
values of $d_{P}$ for possible types of fibers $f^{*}P$.

Type |  $m$In | II | III | IV |  I${}^{*}_{b}$ |  II∗ |  III∗ |  IV∗
---|---|---|---|---|---|---|---|---
$d_{P}$ | $1-\frac{1}{m}$ | $\frac{1}{6}$ | $\frac{1}{4}$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $\frac{5}{6}$ | $\frac{3}{4}$ | $\frac{2}{3}$

######  Proposition 2.24.

([15, Theorem 3.1]). Assumptions the same as in Theorem 2.23 and choose $p>0$
such that $(X/Z\ni P,pf^{*}P)$ is maximally lc. Then $K_{X}+pf^{*}P$ has index
1,2,6,4,3 respectively for the types $m$Ib, $m$I${}^{*}_{b}$, II/II∗,
III/III∗, IV/IV∗.

###### Proof.

By [15, Theorem 3.1], $(X/Z\ni P,pf^{*}P)$ has a regular $n$-complement
$(X/Z\ni P,B^{+})$ for $n=1,2,6,4,3$ in the corresponding cases. Since
$K_{X}+pf^{*}P\equiv K_{X}+B^{+}\equiv 0/Z\ni P$, we have $B^{+}=qf^{*}P$ for
some $q\geq p$. Now $(X/Z\ni P,B^{+}=qf^{*}P)$ is lc and $(X/Z\ni P,pf^{*}P)$
is maximally lc, we have $q=p$. Hence $K_{X}+pf^{*}P=K_{X}+B^{+}$ has an index
$n$. ∎

######  Proposition 2.25.

([11, Example 7.16]). Let $f:(X,B)\to Z$ be a $0$-contraction with $X/Z$ an
elliptic fibration. Then $f$ has an adjunction index $12$.

###  2.9. Known Boundedness Results

Now we are ready to restate two known results for a certain class of pairs
$(X/Z,B)$ where $X/Z$ is of Fano type and $B$ has hyperstandard
multiplicities. We’ll use their special versions for surface pairs in the
proof of the main Theorem 1.4 later. To begin with, we state the following
result about the existence of adjunction index for such pairs:

######  Theorem 2.26.

([16, Theorem 9]). Let $\mathfrak{R}$ be a finite set of rational numbers in
$[0,1]$ and $d$ be a positive integer. Then there exists a positive integer
$I=I(d,\mathfrak{R})$ such that, for any contraction $f:(X,B)\to Z$ with

  1. (1)

$f$ is a 0-contraction;

  2. (2)

$X$ is of Fano type over $Z$ with $\dim X=d$;

  3. (3)

the coefficients of $B^{h}$ are in $\Phi(\mathfrak{R})$.

Then $f$ has an adjunction index $I$.

######  Corollary 2.27.

Let $\mathfrak{R}$ be a finite set of rational numbers in $[0,1]$. Then there
exists a natural number $I=I(\mathfrak{R})$ such that, for any 0-contraction
$f:(X,B)\to Z$ from a surface to a curve with coefficients of $B^{h}$ in
$\Phi(\mathfrak{R})$, $f$ has an adjunction index $I$.

###### Proof.

Let $F$ be a general fiber of $f$. By the adjunction formula
$(K_{X}+B+F)|_{F}=K_{F}+B_{F}$. Taking degrees both sides,
$0=(K_{X}+B+F).F\geq\deg K_{F}$, hence $F$ is either rational or elliptic.

Case 1: In the case $f$ is a $\mathbb{P}^{1}$-fibration, we see $B_{F}\geq 0$
and hence $B^{h}\neq 0$. We can then run a $K_{X}$-MMP over $Z$ and get
$g:(X,B)\to(X\,^{\prime},B\,^{\prime}=g_{*}B)$. $(X\,^{\prime},B\,^{\prime})$
can’t be a minimal model, otherwise $-B\,^{\prime}\equiv K_{X\,^{\prime}}$ is
nef over $Z$, which contradicts that $B^{h}\neq 0$. Hence we get a Mori fiber
space $X\,^{\prime}/Z$, and $X\,^{\prime}$ is of Fano type over $Z$. We can
then apply Theorem 2.26 to the 0-contraction $(X\,^{\prime},B\,^{\prime})\to
Z$ ($B\,^{\prime}=g_{*}B$ also has coefficients in $\Phi(\mathfrak{R})$) and
conclude that it has an adjunction index $I_{1}=I(\mathfrak{R})$. Since
$K_{X}+B=g^{*}(K_{X\,^{\prime}}+B\,^{\prime})$ is crepant, $I_{1}$ is also an
adjunction index for $(X,B)\to Z$.

Case 2: In the case $f$ is an elliptic fibration, $B^{h}=0$. Then Proposition
2.25 concludes that $f$ has an adjunction index $I_{2}=12$.

Combining the two cases above, we can take
$I=I(\mathfrak{R})=lcm(I_{1},I_{2})$. ∎

######  Lemma 2.28.

([5, Theorem 5.5], [1, Proposition 6.3]) Assumptions the same as in Corollary
2.27. Then there exists another finite set $\mathfrak{S}\subset[0,1]$ of
rational numbers depending only on $\mathfrak{R}$ such that the discriminant
$B_{\operatorname{div}}$ on $Z$ has coefficients in $\Phi(\mathfrak{S})$.

######  Remark 7.

When the relative dimension of $f$ is 1, [11, Theorem 8.1] generalizes this
Corollary 2.27 to all cases without the assumption $X/Z$ is of Fano type. Also
see [5, Theorem 5.5] for another version of this result.

Now let’s recall the index conjecture for pairs $(X,B)$, which is established
as Theorem 2.30 below in the case $X$ is a surface.

######  Theorem 2.29.

([1, Theorem 1.8]). Let $\mathfrak{R}$ be a finite set of rational numbers in
$[0,1]$ and $d$ be a positive integer. Then there exists a positive integer
$n=n(d,\mathfrak{R})$ such that, for any pair $(X/Z\ni o,B)$ and a contraction
$f:(X,B)\to Z$ with

  1. (1)

$(X,B)$ is lc of dimension $d$;

  2. (2)

$X$ is of Fano type over $Z$;

  3. (3)

the coefficients of $B$ are in $\Phi(\mathfrak{R})$;

  4. (4)

$-(K_{X}+B)$ is nef over $Z$.

There is an $n$-complement $(X/Z\ni o,B^{+})$ of $(X/Z\ni o,B)$ with
$B^{+}\geq B$. Moreover, this complement is also an $mn$-complement for any
positive integer $m$.

######  Theorem 2.30.

([5, Proposition 5.3], [11, Corollary 1.11]). Let $\mathfrak{R}$ be a finite
set of rational numbers in $[0,1]$. Then there exists a positive integer
$I=I(\mathfrak{R})$ with the following. If $(X,B)$ is a surface pair such that

  1. (1)

$(X,B)$ is lc;

  2. (2)

$K_{X}+B\sim_{\mathbb{R}}0$;

  3. (3)

the coefficients of $B$ are in $\Phi(\mathfrak{R})$.

Then $I(K_{X}+B)\sim 0$.

###### Proof.

By taking a dlt modification (Proposition 2.11), we can suppose $(X,B)$ is
dlt. Then we have the following cases:

Case 1: $B=0$ and $X$ has canonical singularities. We take a minimal
resolution $g:Y\to X$. Then $K_{Y}-\sum a_{i}E_{i}=g^{*}K_{X}$ where $E_{i}$
are $g$-exceptional prime divisors. On one hand, $a_{i}\geq 0$ since $X$ has
canonical singularities; on the other hand, since $K_{Y}$ is nef over $X$, by
Negativity Lemma $a_{i}\leq 0$. Hence $a_{i}=0$ and $K_{Y}=g^{*}K_{X}$. We can
then assume $X$ is smooth. In this case we can quote the classical result that
$12K_{X}\sim 0$ if $K_{X}\sim_{\mathbb{Q}}0$ and set $I_{1}=12$.

Case 2: $B=0$ and $X$ has klt singularities. By the classification of log
terminal surface singularities [10, Proposition 4.18], $X$ has quotient
singularities and the existence of an index $n$ follows. A classical result
from [4] has that $nK_{X}\sim 0$ for some $n\leq 21$ in this case. Hence for
our result we can take $I_{2}=21!$.

Case 3: $B\neq 0$. Run $K_{X}$-MMP. Let $h:X\to X\,^{\prime}$ be the composite
of all contractions and $B\,^{\prime}=h_{*}B$. Note $B\,^{\prime}$ also has
coefficients in $\Phi(\mathfrak{R})$. If $X\,^{\prime}$ is a minimal model,
$K_{X\,^{\prime}}$ is nef, $B\,^{\prime}\geq 0$ and
$K_{X\,^{\prime}}+B\,^{\prime}\sim_{\mathbb{R}}0$. Hence $B\,^{\prime}=0$ and
by the previous case, $K_{X\,^{\prime}}+B\,^{\prime}$ has an index $I_{2}$.

If $X\,^{\prime}\to T$ is a Mori fiber space and $T$ is a curve, from our
assumption we see that $f:(X\,^{\prime},B\,^{\prime})\to T$ is a
0-contraction, hence by Corollary 2.27 $f$ has an adjunction index
$J=J(\mathfrak{R})$ and

$K_{X\,^{\prime}}+B\,^{\prime}\sim_{J}f^{*}(K_{T}+B\,^{\prime}_{T}).$

Here
$B\,^{\prime}_{T}=B\,^{\prime}_{\operatorname{div}}+B\,^{\prime}_{\operatorname{mod}}$
is taken effective. By Lemma 2.28, $B\,^{\prime}_{\operatorname{div}}$ has
coefficients in some hyperstandard set $\Phi(\mathfrak{S})$ and since
$JB\,^{\prime}_{\operatorname{mod}}$ is integral,
$B\,^{\prime}_{\operatorname{mod}}$ has coefficients of the type $i/J$ for
$i=0,1,...,J$. Hence by adding $i/J$, $i=1,2,...,J-1$ to $\mathfrak{S}$ we get
$\mathfrak{R}\,^{\prime}$ and the coefficients of $B\,^{\prime}_{T}$ are in
the hyperstandard set $\Phi(\mathfrak{R}\,^{\prime})$, which depends only on
$\mathfrak{R}$. Note $\deg K_{T}\leq 0$, $T$ is of Fano type and
$K_{T}+B\,^{\prime}_{T}\sim_{\mathbb{R}}0$. We can then apply Theorem 2.29 to
$(T,B\,^{\prime}_{T})$ to conclude that it has an index $N$, $N$ depends only
on $\mathfrak{R}$. Then take $I_{3}:=\operatorname{lcm}(N,J)$, we have
$I_{3}(K_{X\,^{\prime}}+B\,^{\prime})\sim
f^{*}(I_{3}(K_{T}+B\,^{\prime}_{T}))\sim 0$, $I_{3}$ is an index for
$K_{X\,^{\prime}}+B\,^{\prime}$, and $I_{3}$ depends only on $\mathfrak{R}$.

If $T$ is a point, $X\,^{\prime}$ is klt log del Pezzo and
$\rho(X\,^{\prime})=1$, in particular $X\,^{\prime}$ has Fano type. Then by
Theorem 2.29, $K_{X\,^{\prime}}+B\,^{\prime}$ has an index
$I_{4}=I_{4}(\mathfrak{R})$.

Finally, since $K_{X}+B=h^{*}(K_{X\,^{\prime}}+B\,^{\prime})$ is crepant, an
index $I$ for $K_{X\,^{\prime}}+B\,^{\prime}$ is also an index for $K_{X}+B$.
We can then take our $I$ in the theorem to be the least common multiple of
$I_{j}$, $j=1,2,3,4$. Notice that all $I_{j}$ depend only on $\mathfrak{R}$,
hence so does $I$. ∎

######  Remark 8.

As in the theorem, we call $I$ an $index$ for the 0-pair $(X/Z\ni o,B)$ if
$I(K_{X}+B)\sim 0$.

######  Remark 9.

It is known that the index conjecture holds true for non-klt 0-pairs $(X,B)$
when $\dim X\leq 3$ ([3, Corollary 1.12]). Using this result, we can avoid the
use of Theorem 2.29 in Case 3 of our proof.

######  Remark 10.

The index conjecture is widely expected to be true in arbitrary dimensions,
especially in the case $(X,B)$ is not klt. The klt case is much more difficult
and remains a mystery even in dimension 3. By Theorem 2.29, the index
conjecture is true in the case $X$ is of Fano type.

##  3\. Boundedness for Curves

###  3.1. Classification of $\mathbb{R}$-complementary curves

To find $n$-complements for a surfaces pair $(X,B)$, a natural thought is to
lift $n$-complements of the restriction of the pair on some curves. Hence we
need a boundedness result of $n$-complements for certain kinds of curves $C$.
In view of Connectedness Lemma 2.7, we focus on the case $C$ is a connected
complete nodal curve. Proposition 3.1 below gives a complete classification of
$\mathbb{R}$-complementary curve pairs $(C,B)$ in this case.

######  Proposition 3.1.

Let $(C,B)$ be a pair, where $C$ is a connected complete nodal curve and $B$
is a boundary. If $(C,B)$ has an $\mathbb{R}$-complement, then one of the
following occurs:

  1. (1)

$C$ is a nonsingular rational curve and $\deg B\leq 2$;

  2. (2)

$C$ is a nonsingular elliptic curve and $B=0$;

  3. (3)

$C$ is an irreducible rational curve with only one nodal singularity and
$B=0$;

  4. (4)

$C=\sum\limits_{i=1}^{n}C_{i}$ is a cycle of smooth rational curves $C_{i}$,
namely $C_{i}$ are the irreducible components and $C_{i}.C_{j}=1$ if $|i-j|=1$
or $\\{i,j\\}=\\{1,n\\}$, $C_{i}.C_{j}=0$ otherwise. And $B=0$;

  5. (5)

$C=\sum\limits_{i=1}^{n}C_{i}$ is a chain of smooth rational curves $C_{i}$,
namely $C_{i}$ are the irreducible components and $C_{i}.C_{j}=1$ if
$|i-j|=1$, $C_{i}.C_{j}=0$ otherwise. If $B_{i}$ is the restriction of $B$ on
$C_{i}$, then $\deg{B_{1}}\leq 1$, $\deg{B_{n}}\leq 1$ and $B_{i}=0$
otherwise.

We need a lemma before the proof.

######  Lemma 3.2.

Suppose $C$ is a connected complete curve with only nodal singularities and
$C_{i}$ are the irreducible components. Consider the composition
$\pi:\tilde{C}_{i}\to C_{i}\to C$ where $\tilde{C}_{i}\to C_{i}$ are the
normalization of $C_{i}$, then we have

$K_{\tilde{C}_{i}}=\pi^{*}K_{C}-\sum P_{j}.$

Here $P_{j}$ are all the points on $\tilde{C}_{i}$ mapped to singularities of
$C$.

###### Proof.

By the existence of an $\mathbb{R}$-complement $(C,B^{+})$ we have
$K_{C}+B^{+}\sim_{\mathbb{R}}0$ and $B^{+}\geq B\geq 0$. Taking degree both
sides, we have $\deg{K_{C}}\leq 0$. Moreover, since $B^{+}$ is supported on
the smooth locus of $C$, $B^{+}$ doesn’t affect singularities and we can
assume $(C,0)$ is lc.

Now assume $C$ is nonsingular. Then $\deg{K_{C}}=2g(C)-2\leq 0$, $g(C)\leq 1$.
If $g(C)=1$, $K_{C}\sim 0$, $B=B^{+}=0$ and we get case 1). If $C$ is
rational, $K_{C}+B^{+}\sim_{\mathbb{R}}0$ is equivalent to $\deg{B^{+}}=2$.
Existence of such $B^{+}$ is equivalent to $\deg B\leq 2$ and we get case 2).

If $C$ is singular and irreducible, consider its normalization $f:\tilde{C}\to
C$. By Lemma 3.2 we have $f^{*}K_{C}=K_{\tilde{C}}+\sum P_{j}$. Since
$\deg{K_{C}}\leq 0$, $\deg{(K_{\tilde{C}}+\sum P_{j})}=2g(\tilde{C})-2+l\leq
0$. Here $l$ is the number of points on $\tilde{C}$ mapped to singularities of
$C$. Suppose $C$ has $r$ nodal singularities, then $l=2r\leq 2$, $r=1$ and
$\tilde{C}$ is rational. As a result $K_{C}\sim 0$, $B=B^{+}=0$ and this is
case 3).

For the rest of cases $C$ is reducible. For $\pi_{i}:\tilde{C}_{i}\to C$
defined in Lemma 3.2, we have $\deg{(K_{\tilde{C}_{i}}+\sum
P_{j})}=2g(\tilde{C}_{i})-2+l_{i}\leq 0$, where $l_{i}$ is the number of
points on $\tilde{C}_{i}$ mapped to singularities of $C$. There are at least
one singularity of $C$ lying on $C_{i}$ by the connectedness assumption, so
$l_{i}\geq 1$. Hence $g(\tilde{C}_{i})=0$ and $l_{i}\leq 2$. If $C_{i}$ has a
nodal singularity itself, we’ll get $l_{i}\geq 3$ and contradiction. Hence
$C_{i}$ are rational and smooth for all $i$. Now $l_{i}\leq 2$ means every
$C_{i}$ has at most 2 intersections with other $C_{j}$. Starting with $C_{1}$,
we can rearrange $C_{i}$ such that $C_{i}$ intersects $C_{i+1}$ for $1\leq
i\leq n-1$. If $C_{n}$ intersects $C_{1}$, then $l_{i}=2$ for all $i$, and
$K_{C_{i}}+\sum P_{j}+B^{+}_{i}\sim 0$ implies $B_{i}=B^{+}_{i}=0$, hence
$B=0$. This is case 4). Otherwise $C_{1}$ doesn’t meet $C_{n}$,
$l_{1}=l_{n}=1$ and $l_{i}=0$ for $2\leq i\leq n-1$. By the same reasoning
$B_{i}=0$ for $2\leq i\leq n-1$. For $i=1$ or $n$, $K_{C_{i}}+\sum
P_{j}+B^{+}_{i}\sim 0$ is equivalent to $\deg{B^{+}_{i}}=1$, so
$\deg{B_{i}}\leq 1$. This is case 5). ∎

###  3.2. Boundedness of $n$-complements on curves

Now we are ready to prove the boundedness of $n$-complements under
complementary restrictions, which is Corollary 3.6. Corollary 3.6 follows from
Theorem 3.5 below about simultaneous construction of $n$-complements for
curves. Before we start, We’ll need some lemmas and notations. All norms
throughout are maximal absolute value norm.

######  Notation 1.

For a real number $a\in[0,1]$ and a positive integer $n$, we introduce a real
number $a^{[n]}$ as in Definition 1.2 of $n$-complements: $a^{[n]}=1$ if
$a=1$, and $a^{[n]}=\frac{\lfloor(n+1)a\rfloor}{n}$ otherwise. Note
$a^{[n]}\leq 1$ always holds.

######  Lemma 3.3.

For arbitrary n real numbers $a_{1},a_{2},...a_{n}$, $\sum\lfloor
a_{i}\rfloor\leq\lfloor\sum a_{i}\rfloor$.

######  Lemma 3.4.

Suppose we have a continuous map $\lambda:W\to V$ between two finite linear
spaces with norms, a vector $e\in W$ such that $\lambda(e)\neq 0$ and a real
number $\epsilon>0$. Then we can find a $\delta>0$ such that for every
$e\,^{\prime}$ with $||e-e\,^{\prime}||<\delta$, we have
$\lambda(e\,^{\prime})\neq 0$ and
$||\frac{\lambda(e)}{||\lambda(e)||}-\frac{\lambda(e\,^{\prime})}{||\lambda(e\,^{\prime})||}||<\epsilon$.

######  Theorem 3.5.

Suppose we have $I,v,e,\epsilon$ as in Definition 1.3, a positive integer $d$
and $m$ positive rational numbers $c_{1},c_{2},...,c_{m}$. Then there exists a
finite set $\mathcal{N}=\mathcal{N}(I,v,e,\epsilon,d,c_{1},...,c_{m})$ such
that the following holds:

  1. (1)

$\mathcal{N}$ consists of positive integers under complementary restrictions
with respect to $I,v,e,\epsilon$;

  2. (2)

For $m$ vectors $B_{i}=(b_{i1},...,b_{id})\in\mathbb{R}^{d}$ satisfying
$\sum_{j=1}^{d}b_{ij}\leq c_{i}$ and $b_{ij}\in[0,1]$ for $i=1,2,...,m$,
$j=1,2,...,d$, there exists an $n\in\mathcal{N}$ such that
$\sum^{d}_{j=1}b_{ij}^{[n]}\leq c_{i}$ for $i=1,2,...,m$.

###### Proof.

Consider the following subset of $\mathbb{R}^{md}$:

$S:=\\{(B_{1},...,B_{m})|\sum^{d}_{j=1}b_{ij}\leq
c_{i},b_{ij}\in[0,1],i=1,2,...,m,j=1,2,...,d\\}.$

$S$ is then compact. For every point $B$ in $S$ we will define an open
neighborhood $U_{B}$ of $B$ and an integer $n_{B}$ satisfying the
complementary restrictions, such that for any point
$(A_{1},A_{2},...,A_{m})\in U_{B}$ and this $n_{B}$,
$\sum^{d}_{j=1}a_{ij}^{[n_{B}]}\leq c_{i}$ for $i=1,2,...,m$. The sets $U_{B}$
form an open cover of $S$, by compactness it has a finite subcover $\cup
U_{B_{q}}$. Then $\mathcal{N}=\\{n_{B_{q}}\\}$ is our desired set.

We fix a point $B\in S$. Without loss of generality we can assume $b_{1j}<1$
if $j\leq r_{1}$ and $b_{1j}=1$ otherwise. Then we set the positive constants
$\delta_{1}=\min_{j\leq r_{1}}\\{1-b_{1j},1/(2r_{1})\\}$ and $\delta_{i}$
similarly; in the special case $r_{i}=0$ let $\delta_{i}=\infty$. Define
$\lambda:\mathbb{R}^{l+md}\to\mathbb{R}^{l}$,
$(b_{1},b_{2},...,b_{l+md})\mapsto(b_{1},b_{2},...,b_{l})$ to be the
projection onto first $l$ entries. Let $u=(v,B)$ and choose some direction
$e\,^{\prime}$ of the rational affine span $\langle u\rangle$ such that
$\lambda(e\,^{\prime})=e$. For this $\lambda$, $e\,^{\prime}$ and $\epsilon$
we have a $\delta$ as in Lemma 3.4. Since we can make this $\delta$
arbitrarily small, in the following we’ll fix a $\delta<\min_{1\leq i\leq
m}\\{\epsilon,\delta_{i}\\}$. Now we can choose some $n$ under complementary
restrictions with respect to $I\,^{\prime},u,e\,^{\prime},\delta$ where
$I\,^{\prime}=lcm(I,q_{i})$ and $q_{i}$ are the denominators of $c_{i}$. Then
we have an approximation $u_{n}=(v_{n},B_{n})$ ($v_{n}=\lambda(u_{n})$) with
denominator $n$ of $u$ such that $||u_{n}-u||<\delta/n$ and
$||\frac{u_{n}-u}{||u_{n}-u||}-e\,^{\prime}||<\delta$.

We claim $n=n_{B}$ is an index satisfying complementary restrictions with
respect to $I,v,e,\epsilon$. Divisibility $I|n$ is clear since
$I|I\,^{\prime}$. Since $\lambda(e\,^{\prime})=e\neq 0$, we can apply Lemma
3.4 to $\lambda$, $e\,^{\prime}$ and $\frac{u_{n}-u}{||u_{n}-u||}$. As a
result $||\frac{v_{n}-v}{||v_{n}-v||}-e||<\epsilon$, which is 4) in Definition
1.3. From $||u_{n}-u||<\delta/n<\epsilon/n$, we also have
$||v_{n}-v||<\epsilon/n$. This is 3) in Definition 1.3. So $n$ satisfies 1) in
the theorem indeed. As an addendum we have an approximation
$||B_{n}-B||<\delta/n$ of $B$ with denominator $n$.

Let $B_{n}=((B_{1})_{n},...,(B_{m})_{n})$ be this approximation and we’ll
construct the neighborhood $U_{B}$. We focus on $B_{1}$ and $(B_{1})_{n}$ for
now. Let $(B_{1})_{n}=(\frac{M_{1}}{n},...,\frac{M_{d}}{n})$ where $M_{j}$ are
integers. Then set

$U_{1}=\\{(a_{1},..,a_{d})\Big{|}|a_{j}-\frac{M_{j}}{n}|<\frac{\delta}{n},a_{j}<1-\delta,j\leq
r_{1};a_{j}>1-\frac{1}{2n(d-r_{1})},j>r_{1};\sum a_{j}\leq c_{1}\\}.$

One can check $B_{1}$ is indeed in $U_{1}$: for $j\leq r_{1}$,
$||(B_{1})_{n}-B_{1}||<\delta/n$ implies
$|b_{1j}-\frac{M_{j}}{n}|<\frac{\delta}{n}$, and $b_{1j}\leq
1-\delta_{1}<1-\delta$ by our choice of $\delta$; for $j>r_{1}$ this is clear
since $b_{1j}=1$.

For any $(a_{1},...,a_{d})$ in $U_{1}$, let $\epsilon_{j}=na_{j}-M_{j}$. For
$j\leq r_{1}$, by construction $|\epsilon_{j}|<\delta$ and $a_{j}<1$, so we
have

$a^{[n_{j}]}=\frac{\lfloor(n+1)a_{j}\rfloor}{n}=\frac{\lfloor
M_{j}+\epsilon_{j}+a_{j}\rfloor}{n}=\frac{\lfloor\epsilon_{j}+a_{j}\rfloor}{n}+a_{j}-\frac{\epsilon_{j}}{n}<a_{j}+\frac{\delta}{n}.$

Note $\epsilon_{j}+a_{j}<\delta+1-\delta=1$, so
$\lfloor\epsilon_{j}+a_{j}\rfloor\leq 0$. Summing these up for $j$ from $1$ to
$r_{1}$ and note $r_{1}\delta<r_{1}\delta_{1}\leq 1/2$ we get

(1) $\sum_{j\leq r_{1}}a^{[n]}_{j}<\sum_{j\leq
r_{1}}a_{j}+\frac{r_{1}\delta}{n}<\sum_{j\leq r_{1}}a_{j}+\frac{1}{2n}$

For $j>r_{1}$ part we have an estimate

(2)
$\sum_{j>r_{1}}a^{[n]}_{j}\leq\sum_{j>r_{1}}1<\sum_{j>r_{1}}(a_{j}+\frac{1}{2n(d-r_{1})})=\sum_{j>r_{1}}a_{j}+\frac{1}{2n}$

Adding (1) and (2) we have $\sum a^{[n]}_{j}<\sum a_{j}+\frac{1}{n}\leq
c_{1}+\frac{1}{n}$; in the special cases $r_{1}=0$ and $r_{1}=d$, we don’t
have (1) or (2), so we have actually a stronger inequality $\sum
a^{[n]}_{j}<c_{1}+\frac{1}{2n}$. This implies $\sum a^{[n]}_{j}\leq c_{1}$
since the denominator of $c_{1}$ divides $n$ and $\sum a^{[n]}_{j}$ has
denominator $n$.

We construct open sets $U_{i}$ similarly. Then the set $U_{B}=(U_{1}\times
U_{2}\times...\times U_{m})\cap S$ is an open neighborhood of $B$ meeting all
of our requirements. ∎

######  Remark 11.

In the proof we only used the assumption $b_{ij}\geq 0$ for the compactness of
$S$. So we can drop this condition if the set $S$ of boundaries is already
compact. For example, this is the case when we consider all subboundaries
$B_{i}$ with $\sum_{j=1}^{d}b_{ij}=c_{i}$ and $b_{ij}\leq 1$.

######  Corollary 3.6.

Suppose we have the data $I,v,e,\epsilon$ as in Definition 1.3. Then we can
find a finite set $\mathcal{N}=\mathcal{N}(I,v,e,\epsilon)$ such that:

  1. (1)

Restrictions: $\mathcal{N}$ consists of integers under complementary
restrictions;

  2. (2)

Existence of $n$-complements: suppose we have a curve pair $(C,B)$ where $C$
is a connected nodal curve, $B$ is a boundary supported on the smooth locus of
$C$, and $-(K+B)$ is nef on every irreducible component of $C$. Then $(C,B)$
has an $n$-complement $(C,B^{+})$ for some $n\in\mathcal{N}$, where $B^{+}$
has the same support as $B$.

###### Proof.

By Proposition 3.1, the only interesting cases are 3) and 5), since in other
cases $B=0$, and $(C,0)$ is an $n$-complement of itself for any $n$. For case
3), we can join small multiplicities of $B$ together by Lemma 3.3 until there
are at most one of them $<1/2$. As a result we can assume there are at most 4
nonzero multiplicities in $B$. Applying Theorem 3.5 to the case $m=1$,
$c_{1}=2$, $d=4$, we get a finite set $\mathcal{N}_{1}$. We can then find
$n\in\mathcal{N}_{1}$ such that $B^{[n]}=\sum b_{i}^{[n]}P_{i}$ has degree
$\leq 2$, and some boundary $B^{+}\geq B^{[n]}$ with degree 2. Then
$(C,B^{+})$ is an $n$-complement of $(C,B)$ by definition.

For case 5), similarly we can assume there are at most 2 nonzero
multiplicities in $B_{1}$ and $B_{h}$. Applying Theorem 3.5 to $m=2$,
$c_{1}=c_{2}=1$ and $d=2$, we get another finite set $\mathcal{N}_{2}$. We can
then find $n\in\mathcal{N}_{2}$ such that $B_{i}^{[n]}=\sum
b_{ij}^{[n]}P_{ij}$ has degree $\leq 1$, and some boundary $B_{i}^{+}\geq
B_{i}^{[n]}$ with degree 1 for $i=1,h$. Then $(C,B^{+}=B^{+}_{1}+B^{+}_{h})$
is an $n$-complement of $(C,B)$.

Now $\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}$ is our desired set. ∎

##  4\. A special big case of Theorem 1.4

In this chapter, using our boundedness result Corollary 3.6 for curves,
Kawamata-Viehweg vanishing and Connectedness Lemma 2.7, we show that Theorem
1.4 holds when $-(K_{X}+B)$ is big and nef, which is Corollary 4.5. This case
is crucial in the proof of Theorem 1.4 later. We’ll need the following
preliminary results.

######  Notation 2.

For any divisor $D$, we have the unique decomposition of $D=D_{\geq 0}-D_{\leq
0}$ into its positive and negative part, where $D_{\geq 0}$ and $D_{\leq 0}$
are two effective divisors with no common components.

######  Lemma 4.1.

Let $X$ be a smooth surface, $S\subset X$ a nonsingular curve and $D$ a
divisor with $S\not\subset\operatorname{Supp}D$. If $-(K+S+D)$ is nef on the
support of $D_{\leq 0}$ and $D_{\leq 0}$ is contractible, then $S$ has no
intersection with components of $D_{\leq 0}$.

###### Proof.

If $D_{\leq 0}=0$, the result is clear. We assume $D_{\leq 0}\neq 0$ and use
induction on the number of components of $D_{\leq 0}$. Since $D_{\leq 0}$ is
contractible, by negative definiteness we have $(D_{\leq 0})^{2}<0$. Hence
there exists an irreducible curve $E\subset\operatorname{Supp}D_{\leq 0}$ such
that $D_{\leq 0}.E<0$. It follows that

$(K+S).E\leq(K+S+D_{\geq 0}).E\leq D_{\leq 0}.E<0.$

Since $E$ is contractible, we also have $E^{2}<0$. Hence $E$ is a $(-1)$-curve
and $S.E=0$.

Now let $g:X\to Y$ be the contraction of $E$ and consider $g(S)$ and $g(D)$ on
$Y$. Let’s check that $K_{Y}+g(S)+g(D)=g_{*}(K+S+D)$ also satisfies all the
assumptions of the lemma on $Y$. For an irreducible curve
$C\subset\operatorname{Supp}(g(D)_{\leq 0})$ we have

$g_{*}(K+S+D).C=(K+S+D).g^{*}C=(K+S+D).(\tilde{C}+aE)\leq 0.$

Here the total transform $g^{*}C$ consists of two parts: $\tilde{C}$, the
birational transform of $C$ on $X$ and the exceptional part $aE$ with $a\geq
0$. They are both in the support of $D_{\leq 0}$, hence $(K+S+D).\tilde{C}\leq
0$ and $(K+S+D).E\leq 0$ by assumption. It’s then established
$-(K_{Y}+g(S)+g(D))$ is nef on $\operatorname{Supp}g(D)_{\leq 0}$. It’s also
clear that $g(D)_{\leq 0}=g(D_{\leq 0})$ is contractible and has strictly
smaller number of components than $D_{\leq 0}$. Hence by induction hypothesis,
$g(S)$ has no intersection with $\operatorname{Supp}g(D)_{\leq 0}$. Since $S$
is disjoint from $E$ and $\operatorname{Supp}D_{\leq
0}=\operatorname{Supp}g(D)_{\leq 0}\cup E$, we get our result. ∎

######  Corollary 4.2.

Notations and assumptions the same as in Lemma 4.1. The result still holds if
$S$ is a sum of nonsingular curves.

######  Lemma 4.3.

Let $(X,B)$ be a lc log surface pair, $D\subset\operatorname{Supp}B$ is a
prime divisor. Then $D$ is $\mathbb{Q}$-Cartier.

###### Proof.

Let $f:Y\to X$ be a log resolution of $(X,B)$, and
$K_{Y}+B_{Y}=f^{*}(K_{X}+B)+\sum a_{i}E_{i}$. Here $B_{Y}$ is the birational
transform of $B$, $E_{i}$ are the exceptional divisors. Choose a
$0<\epsilon\ll 1$, then $f$ is also a log resolution of the pair
$(X,B-\epsilon D)$. Let $K_{Y}+B_{Y}-\epsilon D_{Y}\equiv_{f}\sum
a\,^{\prime}_{i}E_{i}$, where $D_{Y}$ is the birational transform of $D$. Then
$\sum(a_{i}-a\,^{\prime}_{i})E_{i}\equiv_{f}\epsilon D_{Y}$ is $f$-nef. By
negativity lemma, $a\,^{\prime}_{i}\geq a_{i}\geq-1$ since $K_{X}+B$ is lc.
Hence $(X,B-\epsilon D)$ is numerically lc, which is the same as lc on
surfaces. It follows that $K_{X}+B-\epsilon D$ is $\mathbb{Q}$-Cartier and $D$
is also $\mathbb{Q}$-Cartier. ∎

######  Theorem 4.4.

Fix the data $I,v,e,\epsilon$ as in Definition 1.3. Let $(X/Z\ni o,B)$ be a
log surface pair with a proper local morphism $g:X\to Z\ni o$ and a connected
central fiber $X_{o}$. Assume further

  1. (1)

$-(K_{X}+B)$ is big and nef over $Z$;

  2. (2)

$B$ is a boundary, and $(X,B)$ is lc.

Then there exists a finite set $\mathcal{N}=\mathcal{N}(I,v,e,\epsilon)$
consisting of integers under complementary restrictions such that $(X/Z\ni
o,B)$ has an $n$-complement for some $n\in\mathcal{N}$.

###### Proof.

We claim $\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}$ is one desired
finite set. Here we take $\mathcal{N}_{1}$ to be the set $\mathcal{N}$ as in
[16, Theorem 3] for 2-dimensional case. $\mathcal{N}_{2}$ is the set
$\mathcal{N}(I,v,e,\epsilon)$ from Corollary 3.6.

Step 1: In this step we’ll first reduce to the case when there are no
horizontal divisors in the reduced center $\lfloor B\rfloor$. Suppose the
theorem already holds for $\mathcal{N}$ when horizontal parts of $\lfloor
B\rfloor$ are empty. We’ll use induction on the number of horizontal
components in $\lfloor B\rfloor$. If there is any prime horizontal divisor $D$
in $\lfloor B\rfloor$, we decrease its multiplicity by
$\epsilon<\frac{1}{N+1}$ where $N$ is the biggest number in $\mathcal{N}$.
Explicitly, we set $B\,^{\prime}=B-\epsilon D$, then the number of horizontal
components in $\lfloor B\,^{\prime}\rfloor$ drops by 1.

For this new pair $(X/Z\ni o,B\,^{\prime})$, we check it still satisfies the
assumptions in the theorem. By Lemma 4.3 $K_{X}+B\,^{\prime}$ is again
$\mathbb{R}$-Cartier and $B\,^{\prime}$ is also a boundary. For (1), since
$-(K_{X}+B)$ is big and nef over $Z$, and $\epsilon D$ is horizontal, hence
big and nef over $Z$, the sum $-(K_{X}+B\,^{\prime})=-(K_{X}+B)+\epsilon D$ is
also big and nef over $Z$. For (2), it’s clear since $(X,B)$ is lc and
$B-B\,^{\prime}=\epsilon D$ is effective and $\mathbb{Q}$-Cartier.

Now by induction hypothesis $(X/Z\ni o,B\,^{\prime})$ has an $n$-complement
$(X/Z\ni o,B^{+})$ for some $n\in\mathcal{N}$. We claim $(X/Z\ni o,B^{+})$ is
also an $n$-complement for $(X/Z\ni o,B)$. For this, we only need to check the
multiplicities $d$, $d\,^{\prime}$ and $d^{+}$ of $D$ in $B$, $B\,^{\prime}$
and $B^{+}$ respectively. We have $d=1$, $d\,^{\prime}=1-\epsilon$ and
$d^{+}\geq\lfloor(n+1)(1-\epsilon)\rfloor/n$. Note $(n+1)(1-\epsilon)\geq n$
by our choice of $\epsilon$, $d^{+}=1$. Hence by definition $(X/Z\ni o,B^{+})$
is also an $n$-complement of $(X/Z\ni o,B)$.

Step 2: In this step we consider the case when $(X,B)$ is klt. By assumption
(1), $X/Z\ni o$ is then of Fano type, hence $(X/Z\ni o,B)$ has a klt
$\mathbb{R}$-complement. Then by [16, Theorem 3] $(X/Z\ni o,B)$ has an
$n$-complement for some $n\in\mathcal{N}_{1}$.

Step 3: In this step we assume $(X,B)$ is lc but not klt. We’ll construct an
$n$-complement $(X/Z\ni o,B^{+})$ of $(X/Z\ni o,B)$ for some
$n\in\mathcal{N}_{2}$ in this case.

Let $h:Y\stackrel{{\scriptstyle f}}{{\longrightarrow}}X\stackrel{{\scriptstyle
g}}{{\longrightarrow}}Z$ be a log resolution of $(X,B)$, and
$f^{*}(K_{X}+B)=K_{Y}+\sum d_{j}D_{j}$ be the crepant pullback. Define
$S:=\sum_{j:d_{j}=1}D_{j}$ and $A:=\sum_{j:d_{j}<1}d_{j}D_{j}$. Now we check
the conditions from Corollary 3.6 on the pair $(S,A|_{S})$. Firstly,
$S\neq\emptyset$ since $(X,B)$ is not klt, $S$ is connected near the central
fiber $h^{-1}(o)$ by Lemma 2.7, and $S$ is snc because $f$ is a log resolution
of $(X,B)$. Hence $S$ is a nonempty connected nodal curve near $h^{-1}(o)$.
Secondly, $A_{\leq 0}$ only consists of $f$-exceptional divisors since $B$ is
effective, hence $A_{\leq 0}$ is contractible. Moreover,
$-(K_{Y}+S+A)=-f^{*}(K_{X}+B)$ is nef over $Z$ by assumption 1), and in
particular $-(K_{Y}+S+A)$ is nef on the support of $A_{\leq 0}$. Hence by
Corollary 4.2, $S$ has no intersection with components of $A_{\leq 0}$,
therefore $A|_{S}$ is a boundary divisor of $S$. Since $S+A$ is snc, $A|_{S}$
is supported on the smooth locus of $S$. Finally, $-(K_{S}+A|_{S})$ is nef on
every irreducible component of $S$, since $-(K_{Y}+S+A)$ is nef over $Z$ and
$S$ is vertical by our reduction in Step 1. Hence we can apply Corollary 3.6
to the pair $(S,A|_{S})$ and get an $n$-complement $(S,(A|_{S})^{+})$ for some
$n\in\mathcal{N}_{2}$, where $(A|_{S})^{+}$ has the same support as $A|_{S}$.

Now $-(K_{Y}+S+A)$ is big and nef over $Z$, and $S+A$ is snc since $f$ is a
log resolution of $(X,B)$, by Kawamata-Viehweg vanishing we have

$\displaystyle
R^{1}h_{*}(\mathcal{O}_{Y}(-nK_{Y}-(n+1)S-\lfloor(n+1)A\rfloor))=$
$\displaystyle
R^{1}h_{*}(\mathcal{O}_{Y}(K_{Y}+\lceil-(n+1)(K_{Y}+S+A)\rceil))=0.$

This implies the surjection

$\displaystyle
H^{0}(Y,\mathcal{O}_{Y}(-nK_{Y}-nS-\lfloor(n+1)A\rfloor))\twoheadrightarrow$
$\displaystyle H^{0}(S,\mathcal{O}_{S}(-nK_{S}-\lfloor(n+1)A\rfloor|_{S})).$

Since $(S,(A|_{S})^{+})$ is an $n$-complement of $(S,A|_{S})$ and all
multiplicities in $A|_{S}$ are $<1$ by construction,
$n(A|_{S})^{+}-\lfloor(n+1)A|_{S}\rfloor\in|-nK_{S}-\lfloor(n+1)A|_{S}\rfloor|$.
From the surjectivity, there exists
$\overline{A}\in|-nK_{Y}-nS-\lfloor(n+1)A\rfloor|$ which restircts to
$n(A|_{S})^{+}-\lfloor(n+1)A|_{S}\rfloor$. Set
$A^{+}=\frac{1}{n}(\overline{A}+\lfloor(n+1)A\rfloor)$, then
$A^{+}|_{S}=(A|_{S})^{+}$ and $n(K_{Y}+S+A^{+})\sim 0$.

Step 4: In this step we show $(X/Z\ni o,B^{+})$ is an $n$-complement of
$(X/Z\ni o,B)$, where $B^{+}=f_{*}(S+A^{)}$. Since $n(K_{Y}+S+A^{+})\sim 0$,
we have $n(K_{X}+B^{+})\sim 0$. Moreover,
$A^{+}=\frac{1}{n}(\overline{A}+\lfloor(n+1)A\rfloor)\geq\frac{1}{n}\lfloor(n+1)A\rfloor$
since $\overline{A}\geq 0$. Combining this with the fact that coefficients of
$A$ are $<1$, $(X/Z\ni o,B^{+})$ satisfies condition 1) in Definition 1.2. Now
it remains to check that $(X,B^{+})$ is lc near the fiber over $o$.

Firstly, we show that $(Y,S+A^{+})$ is lc near $S$. For this, we apply
inversion of adjunction 2.10 on every irreducible component $S_{i}$ of $S$:
$Y$ and $S_{i}$ are both smooth since $f$ is a log resolution, and it’s also
clear $S_{i}\not\subset\operatorname{Supp}(S-S_{i}+A^{+})$. Moreover, since
$\operatorname{Supp}A^{+}_{\leq 0}\subset\operatorname{Supp}A_{\leq 0}$ and
$S$ has no intersection with components of $A_{\leq 0}$, $S$ also has no
intersection with components of $A^{+}_{\leq 0}$. Hence it remains to check
that the restriction $(S-S_{i}+A^{+})|_{S_{i}}$ is a boundary on $S_{i}$.
Since $A^{+}|_{S}=(A|_{S})^{+}$ and is supported on the smooth locus of $S$,
$A^{+}|_{S}=\sum\limits_{i}A^{+}|_{S_{i}}$ is the sum of its disjoint
restrictions on $S_{i}$. Moreover, $A^{+}|_{S_{i}}$ and $(S-S_{i})|_{S_{i}}$
have no common components. Hence it suffices to show $(S-S_{i})|_{S_{i}}$ and
$A^{+}|_{S_{i}}$ are both boundaries. On one hand, $(S-S_{i})|_{S_{i}}$ is a
boundary since $S=\sum\limits_{i}S_{i}$ is snc. On the other hand,
$A^{+}|_{S}$ is also a boundary since $(S,A^{+}|_{S}=(A|_{S})^{+})$ is an
$n$-complement of $(S,A|_{S})$ with $A|_{S}$ a boundary. In particular
$A^{+}|_{S_{i}}$ are boundaries for every $i$.

Secondly, we prove a stronger result that $(Y,S+A^{+})$ is lc near any fiber
of $f:Y\to X$ that intersects $S$. Since
$A^{+}=\frac{1}{n}(\overline{A}+\lfloor(n+1)A\rfloor)$ and $\overline{A}$ is
effective, components in $A^{+}$ with negative coefficients are contracted,
assumption 1) is satisfied in Corollary 2.7. Assumption 2) also holds since
$K_{Y}+S+A^{+}\sim 0$ and $f$ is birational. Hence we can apply connectedness
lemma 2.7 to $(Y/X,S+A^{+})$, and conclude that
$\operatorname{LCS}(Y,S+A^{+})$ is connected in a neighborhood of any fiber of
$f$ that intersects $S$. Now assume $(Y,S+A^{+})$ is not lc near one of such
fiber $F$. Then from the facts that $(Y,S+A^{+})$ is lc but not klt near $S$,
$F\cap S\neq\emptyset$ and $\operatorname{LCS}(Y,S+A^{+})$ is connected near
$F$, we can find an irreducible curve $T$ in the fiber $F$, such that $T$ is
contained in Supp$(S+A^{+})$ with multiplicity $1$, $T$ passes through a point
where $(Y,S+A^{+})$ is not lc and a point where $(Y,S+A^{+})$ is lc. Consider
the different $A\,^{\prime}=(S+A^{+}-T)|_{T}$, then $A\,^{\prime}$ is
effective by Lemma 4.1. On one hand, $T$ contains a point where $(Y,S+A^{+})$
is not lc. Hence by Lemma 2.9, $(T,A\,^{\prime})$ is not lc, and this leads to
a point on $A\,^{\prime}$ with multiplicity $>1$. On the other hand, the point
where $(Y,S+A^{+})$ is lc leads to a different point on $A\,^{\prime}$ with
multiplicity $\geq 1$. Hence $\deg A\,^{\prime}>2$ and
$\deg(K_{T}+A\,^{\prime})>0$. This contradicts the fact that
$K_{T}+A\,^{\prime}=(K_{Y}+S+A^{+})|_{T}$ is antinef over $X$. Therefore,
$(Y,S+A^{+})$ is lc near any fiber of $f$ that intersects $S$. Moreover,
$f^{*}(K_{X}+B^{+})=K_{Y}+S+A^{+}$ is crepant since $n(K_{Y}+S+A^{+})\sim 0$.
Hence by pushing forward to $X$, we conclude that $(X,B^{+})$ is lc near
$f(S)$.

Finally, if $\operatorname{LCS}(X,B^{+})$ is disconnected in a neighorhood of
the fiber over $o$, we can apply Lemma 2.8 to the pair $(X,B^{+})$ since
$K_{X}+B^{+}\sim 0$. We then conclude that $(X,B^{+})$ is plt in a
neighborhood of $g^{-1}(o)$, hence lc. Otherwise, in a neighborhood of the
fiber over $o$, the non-klt center $\operatorname{LCS}(X,B^{+})$ is connected
and contains $f(S)$. If $(X,B^{+})$ is not lc near the fiber over $o$,
$\operatorname{LCS}(X,B^{+})$ also contains a point $P\notin f(S)$ where
$(X,B^{+})$ is not lc. Now by the same reasoning as in the paragraph above, we
get a contradiction. Hence $(X,B^{+})$ is also lc near the fiber over $o$ in
this case.

∎

Now we can prove the following special case of the main theorem 1.4.

######  Corollary 4.5.

Suppose we have the data $I,v,e,\epsilon$ as in Definition 1.3. Then we can
find a finite set $\mathcal{N}=\mathcal{N}(I,v,e,\epsilon)$ such that:

  1. (1)

Restrictions: $\mathcal{N}$ consists of integers under complementary
restrictions;

  2. (2)

Existence of $n$-complements: suppose we have a log surface pair $(X/Z\ni
o,B)$ with a proper local morphism $X/Z\ni o$, connected central fiber $X_{o}$
and an $\mathbb{R}$-boundary $B$. Then if $-(K_{X}+B)$ is big over $Z$, and
$(X/Z\ni o,B)$ has an $\mathbb{R}$-complement, it also has an $n$-complement
for some $n\in\mathcal{N}$.

###### Proof.

Just take $\mathcal{N}$ to be the same set from Theorem 4.4. By assumption
$(X/Z\ni o,B)$ has an $\mathbb{R}$-complement $(X/Z\ni o,B+B\,^{\prime})$
where $B\,^{\prime}\sim-(K_{X}+B)$ is big over $Z$ and effective. Then we can
take the Zariski decomposition $B\,^{\prime}=M+F$ of $B\,^{\prime}$ by Theorem
2.13. For the pair $(X/Z\ni o,B+F)$, the conditions (1) and (2) in Theorem 4.4
hold: for (1), $-(K_{X}+B+F)\sim_{\mathbb{R}}M/Z\ni o$ is big over $Z$ since
$M$ is the mobile part of the big divisor $B\,^{\prime}$ over $Z$, and $M$ is
nef over $Z$ by (2) in Definition 2.12; for (2), notice that
$(X,B+B\,^{\prime})=(X,B+F+M)$ is lc, $M$ is effective by Lemma 2.15 and
$Qq$-Cartier by Lemma 4.3, hence $(X,B+F)$ is also lc. Moreover, $B+F\leq
B+B\,^{\prime}$ is also a boundary. Hence Theorem 4.4 applies to $(X/Z\ni
o,B+F)$ to generate an $n$-complement of $(X/Z\ni o,B+F)$ for some
$n\in\mathcal{N}$, which is also an $n$-complement of $(X/Z\ni o,B)$ by Lemma
2.2. ∎

##  5\. Proof of the Main Theorem 1.4

Using the techniques and results introduced in the previous chapters, we are
now ready to give the proof of Theorem 1.4.

######  Proposition 5.1.

[12, Theorem 7.2.11]. Let $f:(X,B)\to Z\ni o$ be a contraction from a normal
surface $X$ to a smooth curve $Z$ such that:

  1. (1)

$K_{X}+B\sim_{\mathbb{R}}0/Z\ni o$ and $B$ is a boundary;

  2. (2)

the general fiber of $f$ is a nonsingular rational curve.

Then for any set of data $I,v,e,\epsilon$ as in Definition 1.3, $(X/Z\ni o,B)$
has an $n$-complement for some $n\in\mathcal{N}=\mathcal{N}(I,v,e,\epsilon)$.
Here $\mathcal{N}$ is the same as in Corollary 4.5.

###### Proof.

After a log terminal modification (Proposition 2.11), we can assume that $X$
is smooth, and by adding $pf^{*}o$ where $p$ is the log canonical threshold we
can assume $K_{X}+B$ is maximally lc, and it follows that the reduced part
$C:=\lfloor B\rfloor$ is non-empty. Then choose a sufficiently general point
$P$ such that $P$ is contained in only one component $C_{1}$ of $C$ and
Supp$B$ is nonsingular at $P$. Then we get a higher model $f:X\to Z\ni o$ such
that the fiber $F=f^{-1}o$ contains a (-1)-curve $E$ with multiplicity 1, $E$
is not contained in Supp$B$ and $(F-E).E=1$. It suffices to prove the result
for this new model, since Lemma 2.1 promises that $n$-complements are
compatible with crepant pullbacks. By 2) in our assumption we have
$K_{X}.F=-2$. Hence $K_{X}.(F-E)=-1$ and$(F-E)^{2}=(F-E).(-E)=-1$, we can
contract $\operatorname{Supp}(F-E)$ and get

$f:X\stackrel{{\scriptstyle g}}{{\longrightarrow}}Y\to Z.$

Now that $g$ is birational, by Corollary 4.5 $K_{X}+B$ has an $n$-complement
$K_{X}+B^{+}$ over $o\,^{\prime}=g(F-E)$. Numerically this is just
$n(K_{X}+B^{+})\equiv 0$ over $F-E$. To show that this $n$-complement
$K_{X}+B^{+}$ can be extended to an $n$-complement, it suffices to show
$n(K_{X}+B^{+})\sim 0/Z\ni o$. But in our case linear equivalence is the same
as numerical equivalence, it suffices to show $n(K_{X}+B^{+})\equiv 0$ over
the fiber $F$ over $o$. Hence it remains to show $(K_{X}+B^{+}).E=0$. On one
hand, $K_{X}.E=-1$ since $E$ is a (-1)-curve. On the other hand,
$B^{+}.E=C_{1}.E=1$. The first equality holds since $E$ only intersects the
component $C_{1}$ by our construction, and $C_{1}$ also has coefficient 1 in
$B^{+}$ as in $B$ because $K_{X}+B^{+}$ is an $n$-complement of $K_{X}+B$. The
second equality is again by our construction of $E$. ∎

###### Proof of Theorem 1.4.

We proceed in the following steps.

Step 1: In this step, we’ll construct the finite set
$\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}\cup\mathcal{N}_{3}$ first.

In our definition, there are three sets $\mathcal{N}_{i}$, $i=1,2,3$ involved.
They are defined as follows, which correspond to the three global cases in
Step 4 below. Here by dimension we mean the Iitaka dimension of the divisor
$M$, which will also be defined in Step 4.

  1. (1)

2-dimensional case: $\mathcal{N}_{1}=\mathcal{N}(I,v,e,\epsilon)$ as in
Corollary 4.5.

  2. (2)

1-dimensional case: $\mathcal{N}_{2}=\mathcal{N}(lcm(I,J),v,e,\epsilon)$ as in
Corollary 3.6. Here $J=J(\mathcal{N}_{1})$ is the adjunction index determined
by the hyperstandard set $\Gamma(\mathcal{N}_{1},\Phi)$ as in Corollary 2.27.

  3. (3)

0-dimensional case: $\mathcal{N}_{3}=\\{N\\}$, where
$N\in\mathcal{N}(lcm(I,I\,^{\prime},12),v,e,\epsilon)$ is an integer under
complementary restrictions with respect to given data (as in Definition 1.3).
Here $I\,^{\prime}$ is the index determined by the hyperstandard set
$\Gamma(\mathcal{N}_{1},\Phi)$ as in Theorem 2.30.

Note all $\mathcal{N}_{i}$, $i=1,2,3$ consist of integers under complementary
restrictions with respect to $I,v,e,\epsilon$ from our construction, it
remains to check (2) from Theorem 1.4, namely we need to find an
$n$-complement of $(X/Z\ni o,B)$ for some $n\in\mathcal{N}$.

Step 2: In this step, we assume $X\to Z$ is a surjective contraction without
loss of generality.

We take the Stein factorization $X\to
Z\,^{\prime}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}Z$ of $X\to Z$.
Since $X_{o}$ is connected and $X\to Z\,^{\prime}$ is a contraction,
$o\,^{\prime}=\phi^{-1}o$ is a single point. We claim that an $n$-complement
$K_{X}+B^{+}$ of $(X/Z\,^{\prime}\ni o\,^{\prime},B)$ is also an
$n$-complement of $(X/Z\ni o,B)$ (and the same holds for
$\mathbb{R}$-complements). For this, we just need to check condition 3) in
Definition 1.2, namely $K_{X}+B^{+}\sim_{n}0/Z\ni o$ is equivalent to
$K_{X}+B^{+}\sim_{n}0/Z\,^{\prime}\ni o\,^{\prime}$. We rephrase this linearly
trivial condition: $K_{X}+B^{+}\sim_{n}0/Z\ni o$ means $n(K_{X}+B^{+})\sim 0$
near $X_{o}$, and the claim follows since the central fibers
$X_{o\,^{\prime}}$ and $X_{o}$ only differ in multiplicities. Based on this
claim, we can then replace the pair $(X/Z\ni o,B)$ by $(X/Z\,^{\prime}\ni
o\,^{\prime},B)$.

Step 3: In this step, we assume $(X,B)$ is dlt and
$K_{X}+B\sim_{\mathbb{R}}0/Z\ni o$. It follows that $X$ is
$\mathbb{Q}$-factorial.

Indeed, by our assumption $(X/Z\ni o,B)$ has an $\mathbb{R}$-complement
$(X/Z\ni o,B^{+})$ with $B^{+}\geq B$. Since $(X,B^{+})$ is lc and $B^{+}$ is
a boundary, by Proposition 2.11, we can take the minimal dlt modification
$f:Y\to X$ of $K_{X}+B^{+}$. Let $B_{Y}^{+}$ be the strict transform of
$B^{+}$. Then we have $K_{Y}+B_{Y}^{+}+E=f^{*}(K_{X}+B^{+})$, $E$ is a reduced
exceptional divisor. By Lemma 2.1, an $n$-complement for $(Y/Z\ni
o,B^{+}_{Y}+E)$ is also an $n$-complement for $(X/Z\ni o,B^{+})$. By Lemma
2.2, an $n$-complement for $(X/Z\ni o,B^{+})$ is also an $n$-complement for
$(X/Z\ni o,B)$. Hence to find an $n$-complement for $(X/Z\ni o,B)$, we can
replace $X$ by $Y$, $B$ by $B^{+}_{Y}+E$. The pair $(Y/Z\ni o,B^{+}_{Y}+E)$ is
indeed a dlt 0-pair with a boundary $B^{+}_{Y}+E$, and is itself an
$\mathbb{R}$-complement.

Step 4 (global case): In this step, we show the existence of an $n$-complement
of $(X,B)$ for some $n\in\mathcal{N}$ in the global case $Z=pt$.

Take the low approximation $B_{\mathcal{N}_{1}}$ of $B$ as in Construction 1
and set $D=B-B_{\mathcal{N}_{1}}$. By Construction 1, $D$ is effective and
$\operatorname{Supp}D\subset\operatorname{Supp}(B-\lfloor B\rfloor)$. Hence
for $0<\delta\ll 1$, $(X,B+\delta D)$ is also dlt, hence lc. Applying Lemma
2.15 to the pair $(X,B+\delta D)$, the mobile part $\delta M$ of
$K_{X}+B+\delta D\sim_{\mathbb{R}}\delta D$ is effective and semiample. Hence
a multiple of $M$ induces a morphism $\varphi:X\to T$. Then we distinguish
three cases as follows based on $\iota(X,M)=\dim T$.

Case 1: if $\dim T=2$, $(X,B)$ has an $n$-complement for some
$n\in\mathcal{N}_{1}$.

In this case $M$ is big, hence
$-(K_{X}+B_{\mathcal{N}_{1}})\sim_{\mathbb{R}}D\geq M$ is also big, and the
pair $(X,B_{\mathcal{N}_{1}})$ has an $\mathbb{R}$-complement $(X,B)$. Hence
by Corollary 4.5, $(X,B_{\mathcal{N}_{1}})$ has an $n$-complement for some
$n\in\mathcal{N}_{1}$, which is also an $n$-complement for the pair $(X,B)$ by
Lemma 2.5.

Case 2: if $\dim T=1$, $(X,B)$ has an $n$-complement for some
$n\in\mathcal{N}_{2}$.

Let $\varphi:(X,B)\to T$ be the contraction induced by a multiple of the
effective divisor $M$. In this case $T$ is a smooth projective curve. Then
$\varphi$ is a $0$-contraction of relative dimension 1: $(X,B)$ is generically
lc over $T$, $B$ is generically a boundary over $T$, and
$K_{X}+B\sim_{\mathbb{R}}0$ over $T$. We claim that $\varphi$ contracts all
components of $D=M+F$. Since $\varphi$ is induced by a multiple of the
effective divisor $M$, $M=\varphi^{*}L$ for some ample divisor $L$ on $T$ and
all the components of $M$ are contracted. For a component $C$ of the fixed
part $F$, by Definition 2.12 we have
$0=M.C=\varphi^{*}L.C=L.(\varphi)_{*}C=0$, $C$ is also contracted by
$\varphi$. Then the horizontal part $B^{h}$ of $B=B_{\mathcal{N}_{1}}+D$ over
$T$ is contained in the support of $B_{\mathcal{N}_{1}}$, whose coefficients
belong to the hyperstandard set $\Gamma(\mathcal{N}_{1},\Phi)$. We can then
apply Corollary 2.27 and conclude that $\varphi$ has an adjunction index $J$.

Now we can apply the canonical bundle formula to $\varphi$ and get

$(K_{X}+B)\sim_{J}\varphi^{*}(K_{T}+B_{\operatorname{div}}+B_{\operatorname{mod}}).$

Here $B_{\operatorname{mod}}$ is defined up to $J$-linear equivalence and
$JB_{\operatorname{mod}}$ is integral. Since $B$ is a boundary,
$B_{\operatorname{div}}$ is also a boundary. By Theorem 2.19,
$B_{\operatorname{mod}}$ is nef. Hence $K_{T}+B_{T}\sim_{\mathbb{R}}0$ and
$\deg{B_{T}}\geq 0$, where
$B_{T}=B_{\operatorname{div}}+B_{\operatorname{mod}}$. This implies $T$ is
either elliptic or rational. If $T$ is elliptic, on one hand we have
$B_{T}\sim_{\mathbb{R}}0$ and $B_{\operatorname{div}}=0$ follows. But on the
other hand $B=(B-M)+\varphi^{*}L$ with $B-M=B_{\mathcal{N}_{1}}+F$ effective
and $L$ ample, hence $B_{\operatorname{div}}=(B-M)_{\operatorname{div}}+L$ is
big, contradiction. So $T\simeq\mathbb{P}^{1}$. We can then choose a suitable
representative $B_{\operatorname{mod}}\geq 0$ to make $B_{T}$ a boundary on
$T$: for example, choose $JB_{\operatorname{mod}}=\sum P_{i}$ with $P_{i}$
distinct closed points on $T$ which are not in the support of
$B_{\operatorname{div}}$. Then by Corollary 3.6, $(T,B_{T})$ has an
$n$-complement for some $n\in\mathcal{N}_{2}$, and by our construction of
$\mathcal{N}_{2}$, the adjunction index $J$ of $\varphi:(X,B)\to T$ divides
$n$. Hence by Proposition 2.20, we can pullback the $n$-complement of
$(T,B_{T})$ to an $n$-complement of $(X,B)$.

Case 3: if $\dim T=0$, $(X,B)$ has an $n$-complement for some
$n\in\mathcal{N}_{3}$.

In this case $\iota(X,M)=0$ implies $M=0$. By Lemma 2.15, we have a birational
contraction $g:X\to X\,^{\prime}$ contracting the fixed part $F$. Now the pair
$(X\,^{\prime},B\,^{\prime}=g_{*}B)$ is also lc since
$K_{X}+B=g^{*}(K_{X\,^{\prime}}+B\,^{\prime})$ is crepant and $(X,B)$ is lc,
$K_{X\,^{\prime}}+B\,^{\prime}=g_{*}(K_{X}+B)\sim_{\mathbb{R}}0$ and
$B\,^{\prime}=g_{*}B=g_{*}(B_{\mathcal{N}_{1}}+F)=g_{*}B_{\mathcal{N}_{1}}$
has coefficients in $\Gamma(\mathcal{N}_{1},\Phi)$. We can then apply Theorem
2.30 and conclude that $I\,^{\prime}(K_{X\,^{\prime}}+B\,^{\prime})\sim 0$.
Hence $I\,^{\prime}(K_{X}+B)\sim 0$ and $(X,B)$ is an $n$-complement of itself
for any $n$ divisible by $I\,^{\prime}$, and in particular for some
$n\in\mathcal{N}_{3}$.

Step 5 (local case): In this step, we show the existence of an $n$-complement
of $(X/Z\ni o,B)$ for some $n\in\mathcal{N}$ in the local case $\dim Z\geq 1$.

In the local case, we can always replace $B$ by $B+t\varphi^{*}o$ where
$\varphi:X\to Z$ such that $K_{X}+B+t\varphi^{*}o$ is maximally lc, since
$K_{X}+B+t\varphi^{*}o\sim_{\mathbb{R}}0/Z\ni o$ also holds and by Lemma 2.2
an $n$-complement of $(X/Z\ni o,B+t\varphi^{*}o)$ is also an $n$-complement of
$(X/Z\ni o,B)$. Hence we assume $(X/Z\ni o,B)$ is maximally lc. Then we also
have three cases based on the dimension of $Z$.

Case 1: If $\varphi$ is birational, then $(X/Z\ni o,B)$ has an $n$-complement
for some $n\in\mathcal{N}_{1}$. This follows immediately from Corollary 4.5
since $K_{X}+B\sim_{\mathbb{R}}0/Z\ni o$.

Case 2: If $Z$ is a curve and $B^{h}\neq 0$, then $(X/Z\ni o,B)$ has an
$n$-complement for some $n\in\mathcal{N}_{1}$. This is just Proposition 5.1
since a general fiber of $\varphi$ is a smooth rational curve in this case.

Case 3: If $Z$ is a curve and $B^{h}=0$, then $(X/Z\ni o,B)$ has a
$12$-complement, and in particular an $n$-complement for some
$n\in\mathcal{N}_{3}$.

In this case $\varphi:X\to Z$ is an elliptic fibration. If $K_{X}.C<0$ for
some irreducible curve $C$ in the fiber $\varphi^{-1}o$, $C$ can’t be the full
fiber. Then $K_{X}.C<0$ and $C^{2}<0$, $C$ is a $(-1)$-curve and we can
contract $C$. After contraction of all such $(-1)$-curves, we get $g:X\to
X\,^{\prime}$ where $K_{X\,^{\prime}}$ is nef over $Z$. Since
$K_{X\,^{\prime}}.F=0$ for a general fiber $F$ of $X\,^{\prime}/Z$, we have
$K_{X\,^{\prime}}\equiv 0/Z$. Then $X\,^{\prime}/Z$ is a minimal elliptic
fibration with $K_{X\,^{\prime}}+B\,^{\prime}\sim_{\mathbb{R}}0/Z\ni o$
maximally lc, where $B\,^{\prime}=g_{*}B$. By Proposition 2.24,
$12(K_{X\,^{\prime}}+B\,^{\prime})\sim 0/Z\ni o$ and since
$K_{X}+B=g^{*}(K_{X\,^{\prime}}+B\,^{\prime})$ is crepant, $12(K_{X}+B)\sim
0/Z\ni o$ also holds. It follows that $(X/Z\ni o,B)$ is an $n$-complement of
itself for any $n$ divisible by 12, and in particular for any
$n\in\mathcal{N}_{3}$. ∎

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