Anti-pluricanonical systems on Fano varieties

Caucher Birkar

Abstract. In this paper, we study the linear systems |−mKX | on Fano varieties X withklt singularities. In a given dimension d, we prove |−mKX | is non-empty and contains anelement with ”good singularities” for some number m depending only on d; if in additionX is ε-lc for some ε > 0, then we show that we can choose m depending only on d and εso that | −mKX | defines a birational map. Further, we prove Shokurov’s conjecture onboundedness of complements, and show that certain classes of Fano varieties form boundedfamilies.

Contents

1. Introduction 22. Preliminaries 62.1. Hyperstandard sets 62.2. Divisors and morphisms 62.3. Pairs 72.4. Fano pairs 72.6. Generalised polarised pairs 82.8. Exceptional and non-exceptional pairs 92.9. Complements 102.10. Bounded families of pairs 102.12. Families of subvarieties 112.15. Potentially birational divisors 122.16. Non-klt centres 132.18. Numerical Kodaira dimension 132.20. Pseudo-effective threshold 142.22. Volume of divisors 142.25. The restriction exact sequence 152.27. Descent of nef divisors 152.29. Pairs with large boundaries 152.31. Divisors with log discrepancy close to 0 162.33. Boundary coefficients close to 1 163. Adjunction 173.1. Divisorial adjunction 173.4. Adjunction for fibre spaces 193.8. Adjunction on non-klt centres 213.12. Lifting sections from non-klt centres 244. Effective birationality 264.1. Singularities in bounded families 264.3. Effective birationality for Fano varieties with good Q-complements 27

Date: March 22, 2016.2010 MSC: 14J45, 14E30, 14C20, 14E05.

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4.6. Effective birationality for nearly canonical Fano varieties 325. Proof of Theorem 1.4 346. Boundedness of complements 376.1. General remarks 376.2. Hyperstandard coefficients under adjunction for fibre spaces 386.4. Pulling back complements from the base of a fibration 406.6. Lifting complements from a non-klt centre 416.11. Boundedness of complements 487. Boundedness of exceptional pairs 497.1. A directed MMP 497.2. Bounds on singularities 507.4. From complements to Theorem 1.3 507.6. Bound on exceptional thresholds 517.8. Bound on anti-canonical volumes 537.10. Bound on lc thresholds 537.12. From bound on lc thresholds to boundedness of varieties 547.14. From complements to Theorem 1.10 558. Boundedness of relative complements 579. Anti-canonical volume 5910. Proofs of main results 61References 62

1. Introduction

We work over an algebraically closed field of characteristic zero. Given a smooth projec-tive variety W , the minimal model program predicts that W is birational to a projectivevariety Y with canonical singularities such that either KY is ample, or Y admits a fibrationwhose general fibres X are Calabi-Yau varieties or Fano varieties. In other words, onemay say that, birationally, every variety is in some sense constructed from varieties X withgood singularities such that either KX is ample or numerically trivial or anti-ample. Soit is quite natural to study such special varieties with the hope of obtaining some sort ofclassification theory. They are also very interesting in moduli theory, arithmetic geometry,and mathematical physics.

When X is one-dimensional the linear system |KX | determines its geometry to a largeextent. However, in higher dimension, one needs to study |mKX | or |−mKX | for all m ∈ N(depending on the type of X) in order to investigate the geometry of X. If KX is ample,then there is m depending only on the dimension such that |mKX | defines a birational em-bedding into some projective space, by Hacon-McKernan [12] and Takayama [39]. If KX isnumerically trivial, there is m such that |mKX | is non-empty but it is not clear whether wecan choose m depending only on the dimension. When KX is anti-ample, that is when X isFano, in this paper we study boundedness and singularity properties of the linear systems|−mKX | in a quite general setting in conjunction with Shokurov’s theory of complements.

Effective non-vanishing. Our first result is a consequence of boundedness of complements(1.7 below). We state it separately because it involves little technicalities.

Theorem 1.1. Let d be a natural number. Then there is a natural number m dependingonly on d such that if X is any Fano variety of dimension d with klt singularities, then the

Anti-pluricanonical systems on Fano varieties 3

linear system |−mKX | is non-empty, that is, h0(−mKX) 6= 0. Moreover, the linear systemcontains a divisor M such that (X, 1

mM) has lc singularities.

Obviously the statement also holds if we replace Fano with the more general notion ofweak Fano, that is, if −KX is nef and big. The theorem was proved by Shokurov in dimen-sion two [37].

Effective birationality for ε-lc Fano varieties. If we bound the singularities of X, wethen have a much stronger statement than the non-vanishing of 1.1.

Theorem 1.2. Let d be a natural number and ε > 0 a real number. Then there is anatural number m depending only on d and ε such that if X is any ε-lc weak Fano varietyof dimension d, then | −mKX | defines a birational map.

Note that m indeed depends on d as well as ε because the theorem implies the volumevol(−KX) is bounded from below by 1

md . Without the ε-lc assumption, vol(−KX) can getarbitrarily small or large [14, Example 2.1.1]. In dimension 2, the theorem is a consequenceof Alexeev [3], and in dimension 3, special cases are proved by Jiang [19] using differentmethods.

Boundedness of certain classes of Fano varieties. Fano varieties come in two flavours:non-exceptional and exceptional. A Fano variety X is non-exceptional if there is 0 ≤ P ∼Q−KX such that (X,P ) is not klt. Otherwise we say X is exceptional. In the non-exceptionalcase we can create non-klt centres which sometimes can be used to do induction, i.e. liftsections and complements from such centres (eg, see 6.8 below). We do not have that luxuryin the exceptional case. Instead we show that there is a ”limited number” of them, that is:

Theorem 1.3. Let d be a natural number. Then the set of exceptional weak Fano varietiesof dimension d forms a bounded family.

Exceptional pairs and generalised polarised pairs can be defined similarly. We will extend1.3 to such pairs (see 1.10 below) which is important for our proofs. In a different directionwe have:

Theorem 1.4. Let d be a natural number, and ε and δ be positive real numbers. Considerprojective varieties X satisfying:• (X,B) is ε-lc of dimension d for some boundary B,• B is big and KX +B ∼R 0, and• the coefficients of B are more than or equal to δ.

Then the set of such X forms a bounded family.

Hacon and Xu proved the theorem assuming the coefficients of B belong to a fixed DCCset of rational numbers [16, Theorem 1.3] relying on the special case when −KX is ample[14, Corollary 1.7]. The theorem can be viewed as a special case of the following conjecturedue to Alexeev and Borisov brothers.

Conjecture 1.5 (BAB). Let d be a natural number and ε a positive real number. Thenthe set of ε-lc Fano varieties X of dimension d forms a bounded family.

The results and ideas developed in this paper will hopefully lead to an eventual proof ofthis conjecture. The conjecture is often stated in the log case for pairs (X,B) but it is nothard to reduce it to the above version. The conjecture has been proved in dimension two[3], for smooth X [29], for toric X [9], for threefolds of Picard number one and terminal

4 Caucher Birkar

singularities [24], for threefolds with canonical singularities [30], in dimension three withKX having bounded Cartier index [8], and in any dimension with KX having boundedCartier index [14, Corollary 1.8 ].

Next we show 1.5 in lower dimension implies a weak form of 1.5, more precisely:

Theorem 1.6. Let d be a natural number and ε a positive real number. Assume Conjecture1.5 holds in dimension d − 1. Then there is a number v depending only on d and ε suchthat if X is an ε-lc weak Fano variety of dimension d, then vol(−KX) ≤ v. In particular,such X are birationally bounded.

In dimension 3, the boundedness of vol(−KX) was proved by Lai [32] for X of Picardnumber one and by Jiang [20] in general who also proves the birational boundedness of X in[19] using different methods. Theorem 1.6 gives new proofs of their results since Conjecture1.5 is known in dimension 2.

Boundedness of complements. Shokurov introduced the theory of complements whileinvestigating threefold log flips [38, §5]. The notion of complement involves both bound-edness and singularities of the linear systems | −mKX |. It is actually defined in the moregeneral setting of pairs. See 2.9 for relevant definitions.

The following theorem was conjectured by Shokurov [37, Conjecture 1.3] who proved itin dimension 2 [37, Theorem 1.4] (see also [34, Corollary 1.8], and [38] for some cases).

Theorem 1.7. Let d be a natural number and R ⊂ [0, 1] be a finite set of rational numbers.Then there exists a natural number n divisible by I(R) and depending only on d and Rsatisfying the following. Assume (X,B) is a projective pair such that• (X,B) is lc of dimension d,• B ∈ Φ(R), that is, the coefficients of B are in Φ(R),• X is Fano type, and• −(KX +B) is nef.

Then there is an n-complement KX + B+ of KX + B such that B+ ≥ B. Moreover, thecomplement is also an mn-complement for any m ∈ N. In particular, if we let B = T + ∆where T = bBc, then the linear system

| − nKX − nT − b(n+ 1)∆c |is not empty.

Here I(R) denotes the smallest natural number so that Ir ∈ Z for every r ∈ R, andΦ(R) is the set of hyperstandard multiplicities associated to R (see 2.1). Fano type means(X,G) is klt and −(KX +G) is ample for some boundary G.

Prokhorov and Shokurov [35][34] prove various inductive statements regarding comple-ments. They [34, Theorem 1.4] also show that 1.7 follows from two conjectures in the samedimension : the BAB conjecture (1.5 above) and the adjunction conjecture for fibre spaces[34, Conjectue 7.13]. In dimension 3, one only needs the BAB [34, Corollary 1.7], and thisalso can be dropped if in addition we assume (X,B) is non-exceptional.

Both the BAB and the adjunction conjectures are very difficult problems. In this paperwe replace the BAB with its special cases 1.3 and 1.4, and we replace the adjunction conjec-ture with the theory of generalised polarised pairs developed in [7]. See [36] for Shokurov’swork on adjunction.

Boundedness of complements in the relative setting. Complements are also defined inthe relative setting for a given contraction X → Z (see 2.9). In particular, when X → Z

Anti-pluricanonical systems on Fano varieties 5

is the identity morphism, boundedness of complements is simply a local statement aboutsingularities near a point on X.

Theorem 1.8. Let d be a natural number and R ⊂ [0, 1] be a finite set of rational numbers.Then there exists a natural number n divisible by I(R) and depending only on d and Rsatisfying the following. Assume (X,B) is a pair and X → Z a contraction such that• (X,B) is lc of dimension d and dimZ > 0,• B ∈ Φ(R),• X is Fano type over Z, and• −(KX +B) is nef over Z.

Then for any point z ∈ Z, there is an n-complement KX +B+ of KX +B over z such thatB+ ≥ B. Moreover, the complement is also an mn-complement for any m ∈ N.

The theorem was proved by Shokurov [37] in dimension 2, and by Prokhorov andShokurov [35] in dimension 3. They also essentially show that 1.8 in dimension d fol-lows from 1.7 in dimension d− 1 [35, Theorem 3.1].

Boundedness of complements for generalised polarised pairs. We prove boundednessof complements in the even more general setting of generalised polarised pairs. This is ofindependent interest but also fundamental to our proofs. It gives enough flexibility for ourinductive arguments to go through unlike Theorem 1.7. For the relevant definitions, see 2.6and 2.9.

Theorem 1.9. Let d and p be natural numbers and R ⊂ [0, 1] be a finite set of rationalnumbers. Then there exists a natural number n divisible by pI(R) and depending onlyon d, p, and R satisfying the following. Assume (X ′, B′ + M ′) is a projective generalisedpolarised pair with data φ : X → X ′ and M such that• (X ′, B′ +M ′) is generalised lc of dimension d,• B′ ∈ Φ(R) and pM is Cartier,• X ′ is Fano type, and• −(KX′ +B′ +M ′) is nef.

Then there is an n-complement KX′ + B′+ + M ′ of KX′ + B′ + M ′ such that B′+ ≥ B′.Moreover, the complement is also an mn-complement for any m ∈ N.

Generalised polarised pairs behave similar to usual pairs in many ways. For example alsosee the effective birationality results for polarised pairs of general type established in [7].

Boundedness of exceptional pairs. As mentioned earlier, in order to carry out ourinductive arguments we need boundedness of exceptional pairs as in the next result.

Theorem 1.10. Let d and p be natural numbers and R ⊂ [0, 1] be a finite set of rationalnumbers. Let (X ′, B′+M ′) be as in Theorem 1.9 and assume (X ′, B′+M ′) is exceptional.Then (X ′, B′) is log bounded.

About this paper. The main tools used in this paper are the minimal model program[31][6], the theory of complements [35][34][37], creating families of non-klt centres usingvolumes [14][15][26, §6], and the theory of generalised polarised pairs [7].

We briefly outline the structure of the paper. In Sections 2, we gather some of the toolsused in the paper, and prove certain basic results. In Section 3, we discuss various types of

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adjunction, recall some known results, and prove some new results (eg, 3.11, 3.13) crucialfor later sections. In Section 4, we prove 1.2 under some additional assumptions (4.4, 4.5,4.7). In Section 5, we prove 1.4. In Section 6, we develop the theory of complements forgeneralised polarised pairs, and prove various inductive statements (eg, 6.5, 6.8), and discussbehaviour of boundary coefficients under adjunction for fibre spaces (6.3). In Section 7,we study exceptional pairs and treat 1.3 and 1.10 inductively (eg, 7.3, 7.5, 7.9, 7.15). InSection 8, we discuss complements in the relative setting (8.2). In Section 9, we prove 1.6.Finally, in Section 10, we give the proofs of all the main results except those proved earlier.

It is worth mentioning that some of the results stated for varieties only can be easilyextended to the log case (eg, 1.2, 1.6) but for simplicity we treat the non-log case only.

Acknowledgements. This work was partially supported by a grant of the LeverhulmeTrust. Part of this work was done while visiting National Taiwan University in May andAugust 2015 with the support of the Mathematics Division (Taipei Office) of the NationalCenter for Theoretical Sciences, and arranged by Jungkai Chen. I would like to thank themfor their hospitality. I am indebted to Vyacheslav V. Shokurov for teaching me the theoryof complements. I am grateful to Jinsong Xu and Yifei Chen for their comments.

2. Preliminaries

All the varieties in this paper are defined over an algebraically closed field of characteristiczero unless stated otherwise.

2.1. Hyperstandard sets. For a subset V ⊆ R and a number a ∈ R, we define V ≥a =v ∈ V | v ≥ a. We similarly define V ≤a, V <a, and V >a.

Let R be a subset of [0, 1]. Following [34, 3.2] we define

Φ(R) = 1− r

m| r ∈ R,m ∈ N

to be the set of hyperstandard multiplicities associated to R. We usually assume 0, 1 ∈ Rwithout mention, so Φ(R) includes Φ(0, 1), the set of usual standard multiplicities. Notethat if we add 1− r to R for each r ∈ R, then we get R ⊂ Φ(R).

Now assume R ⊂ [0, 1] is a finite set of rational numbers. Then Φ(R) is a DCC set ofrational numbers whose only accumulation point is 1. Let I = I(R) be the smallest naturalnumber so that Ir ∈ Z for every r ∈ R. If n ∈ N is divisible by I(R), then nb ≤ b(n+ 1)bcfor every b ∈ Φ(R) [34, Lemma 3.5].

2.2. Divisors and morphisms. Let X be a normal variety, and let M be an R-divisoron X. We often write H i(M) instead of H i(X,OX(bMc)) and write |M | instead of | bMc |.We denote the coefficient of a prime divisor D in M by µDM . If every non-zero coefficientof M belongs to a set Φ ⊆ R, we write M ∈ Φ. Writing M =

∑miMi where Mi are the

distinct irreducible components, the notation M≥a means∑

mi≥amiMi, that is, we ignore

the components with coefficient < a. One similarly defines M≤a,M>a, and M<a.Now let f : X → Z be a morphism to a normal variety. We say M is horizontal over Z

if the induced map SuppM → Z is dominant, otherwise we say M is vertical over Z. If Nis an R-Cartier divisor on Z, we often denote f∗N by N |X .

Again let f : X → Z be a morphism to a normal variety, and let M and L to be R-Cartierdivisors on X, and z ∈ Z. We say M ∼ L over z (resp. M ∼Q L over z)(resp. M ∼R Lover z) if there is a Cartier (resp. Q-Cartier)(resp. R-Cartier) divisor N on Z such thatM − L ∼ f∗N (resp. M − L ∼Q f∗N)(resp. M − L ∼R f∗N) holds over a neighbourhoodof z.

Anti-pluricanonical systems on Fano varieties 7

For a birational map X 99K X ′ (resp. X 99K X ′′)(resp. X 99K X ′′′)(resp. X 99K Y )whose inverse does not contract divisors, and for an R-divisor M on X we denote thepushdown of M to X ′ (resp. X ′′)(resp. X ′′′)(resp. Y ) by M ′ (resp. M ′′)(resp. M ′′′)(resp.MY ).

2.3. Pairs. In this paper a sub-pair (X,B) consists of a normal quasi-projective variety Xand an R-divisor B such that KX + B is R-Cartier. If the coefficients of B are at most 1we say B is a sub-boundary, and if in addition B ≥ 0, we say B is a boundary. A sub-pair(X,B) is called a pair if B ≥ 0 (we allow coefficients of B to be larger than 1 for practicalreasons).

Let φ : W → X be a log resolution of a sub-pair (X,B). Let KW + BW be the pulbackof KX + B. The log discrepancy of a prime divisor D on W with respect to (X,B) is1 − µDBW and it is denoted by a(D,X,B). We say (X,B) is sub-lc (resp. sub-klt)(resp.sub-ε-lc) if every coefficient of BW is ≤ 1 (resp. < 1)(resp. ≤ 1 − ε). When (X,B) is apair we remove the sub and say the pair is lc, etc. Note that if (X,B) is a lc pair, then thecoefficients of B necessarily belong to [0, 1].

Let (X,B) be a sub-pair. A non-klt place of (X,B) is a prime divisor D on birationalmodels of X such that a(D,X,B) ≤ 0. A non-klt centre is the image on X of a non-kltplace. When (X,B) is sub-lc, a non-klt centre is also called a lc centre.

2.4. Fano pairs. Let (X,B) be a pair and X → Z a contraction. We say (X,B) is logFano over Z if it is lc and −(KX +B) is ample over Z; if B = 0 we just say X is Fano overZ. The pair is called weak log Fano over Z if it is lc and −(KX + B) is nef and big overZ; if B = 0 we say X is weak Fano over Z. We say X is Fano type over Z if (X,B) is kltweak log Fano over Z for some choice of B; it is easy to see this is equivalent to existenceof a big/Z Q-boundary Γ so that (X,Γ) is klt and KX + Γ ∼Q 0/Z.

Assume X is Fano type over Z. Then we can run the MMP over Z on any R-CartierR-divisor D on X which ends with some model Y [6]. If DY is nef over Z, we call Y aminimal model over Z for D, but if there is a DY -negative extremal contraction Y → T/Zwith dimY > dimT , we call Y a Mori fibre space over Z for D.

Lemma 2.5. Let (X,B) be a projective lc pair and X → Z a contraction onto a smoothcurve. Assume X is Fano type over some non-empty open set U ⊆ Z. Further assume B isa Q-boundary, KX +B ∼Q 0/Z, and that the generic point of each non-klt centre of (X,B)maps into U . Then X is Fano type over Z.

Proof. There is a big Q-boundary Γ such that over U we have: (X,Γ) is klt and KX +Γ ∼Q0. Let (Y,BY ) be a Q-factorial dlt model of (X,B). There is a big Q-divisor ΓY withcoefficients ≤ 1 such that its pushdown to X is Γ, and over U , KY + ΓY is the pullback ofKX + Γ. Replacing Γ with tΓ + (1− t)B and replacing ΓY with tΓY + (1− t)BY for somesufficiently small rational number t > 0, we can assume ΓY is a boundary. In particular, Yis Fano type over U . Now replacing (X,B) with (Y,BY ) and replacing (X,Γ) with (Y,ΓY ),we can assume X is Q-factorial.

Since (X,Γ) is klt over U and since the generic point of every non-klt centre of (X,B)maps into U , replacing Γ with tΓ + (1− t)B as above, we can assume (X,Γ) is klt, globally.Running an MMP/Z on KX+Γ terminates with a minimal model X ′ on which KX′+Γ′ ∼Q0/Z because Z is a curve. Let KX + Γ be the crepant pullback of KX′ + Γ′. Then (X,Γ)is sub-klt and KX + Γ ∼Q 0/Z. Note that each component of Γ with negative coefficientmaps outside U . On the other hand, since the generic point of each non-klt centre of (X,B)maps into U , adding to B a small multiple of the fibre over each point in Z \ U , we can

8 Caucher Birkar

assume SuppB contains the union of the support of all such fibres. Thus replacing Γ withtΓ + (1− t)B again, we can assume Γ ≥ 0. Therefore, X is Fano type over Z.

2.6. Generalised polarised pairs. For the basic theory of generalised polarised pairs see[7, Section 4]. We recall some of the main notions.

(1) A generalised polarised pair consists of a normal variety X ′ equipped with projective

morphisms Xφ→ X ′ → Z where φ is birational and X is normal, an R-divisor B′ ≥ 0, and

an R-Cartier divisor M on X which is nef/Z such that KX′ +B′ +M ′ is R-Cartier, whereM ′ := φ∗M . We usually refer to the pair by saying (X ′, B′+M ′) is a generalised pair with

data Xφ→ X ′ → Z and M . Note that if Y → X is a projective birational morphism from

a normal variety, then there is no harm in replacing X with Y and replacing M with itspullback to Y . On the other hand, when Z is not relevant we usually drop it and do notmention it: in this case one can just assume X ′ → Z is the identity. When Z is a point wealso drop it but say the pair is projective.

Replacing X we can assume φ is a log resolution of (X ′, B′). We can write

KX +B +M = φ∗(KX′ +B′ +M ′)

for some uniquely determined B. For a prime divisor D on X the generalised log discrepancya(D,X ′, B′ +M ′) is 1− µDB. A generalised non-klt centre is the image of a component ofB with coefficient ≥ 1, and the generalised non-klt locus is the union of all the generalisednon-klt centres.

We say (X ′, B′ + M ′) is generalised lc (resp. generalised klt)(resp. generalised ε-lc) ifevery coefficient of B is ≤ 1 (resp. < 1)(resp. ≤ 1− ε). We say (X ′, B′+M ′) is generaliseddlt if (X ′, B′) is dlt and if every generalised non-klt centre of (X ′, B′ + M ′) is a non-kltcentre of (X ′, B′). If in addition the connected components of bB′c are irreducible, we saythe pair is generalised plt.

(2) Let (X ′, B′+M ′) be a generalised pair as in (1) and let ψ : X ′′ → X ′ be a projectivebirational morphism from a normal variety. Replacing φ we can assume φ factors throughψ. We then let B′′ and M ′′ be the pushdowns of B and M on X ′′ respectively. In particular,

KX′′ +B′′ +M ′′ = ψ∗(KX′ +B′ +M ′).

If B′′ ≥ 0, then (X ′′, B′′ +M ′′) is also a generalised pair with data X → X ′′ → Z and M .If (X ′′, B′′ +M ′′) is Q-factorial generalised dlt and if every exceptional prime divisor of ψappears in B′′ with coefficients one, we then say (X ′′, B′′+M ′′) is a Q-factorial generaliseddlt model of (X ′, B′ + M ′). Such models exist if (X ′, B′ + M ′) is generalised lc, by [7,Remark 4.5 (1)].

(3) Let (X ′, B′+M ′) be a generalised pair as in (1). Assume that D′ on X ′ is an effectiveR-divisor and that N on X is an R-divisor which is nef/Z such that D′ + N ′ is R-Cartierwhere N ′ = φ∗N . The generalised lc threshold of D′ +N ′ with respect to (X ′, B′ +M ′) isdefined as

sups | (X ′, B′ + sD′ +M ′ + sN ′) is generalised lc

where the pair in the definition comes with data Xφ→ X ′ → Z and M + sN .

(4) We prove a connectedness principle similar to the usual one.

Lemma 2.7 (Connectedness principle). Let (X ′, B′ +M ′) be a generalised pair with data

Xφ→ X ′ → Z and M where X ′ → Z is a contraction. Assume −(KX′ + B′ + M ′) is nef

and big over Z. Then the generalised non-klt locus of (X ′, B′+M ′) is connected near eachfibre of X ′ → Z.

Anti-pluricanonical systems on Fano varieties 9

Proof. We can assume φ is a log resolution. Write

KX +B +M = φ∗(KX′ +B′ +M ′).

Let X ′′ be a minimal model of KX +M over X ′ which exists by [7, Lemma 4.4]. We writeKX′′ +B′′ +M ′′ for the pushdown of KX +B +M . Then B′′ is anti-nef/X ′. Thus by thenegativity lemma, B′′ ≥ 0. The generalised non-klt locus of (X ′′, B′′ +M ′′) maps onto thegeneralised non-klt locus of (X ′, B′ +M ′), hence replacing the latter pair with the formerwe can assume X ′ is Q-factorial.

With the above notation, the generalised non-klt locus of (X ′, B′+M ′) is just φ(SuppB≥1).We can write

−(KX +B +M) ∼R A+ C/Z

where A is ample and C ≥ 0. Replacing φ we can assume φ is a log resolution of (X ′, B′+C ′).Pick a sufficiently small ε > 0, let G ∼R M+εA/Z be general, and let ∆ = B+εC+G. ThenKX +∆ ∼R 0/X ′, so KX +∆ = φ∗(KX′+∆′). Moreover, SuppB≥1 = Supp ∆≥1. Thus thenon-klt locus of the pair (X ′,∆′) is equal to the generalised non-klt locus of (X ′, B′+M ′).Therefore, the result follows from the usual connected principle [28, Theorem 17.4] because

−(KX′ + ∆′) ∼R −(1− ε)(KX′ +B′ +M ′)/Z

is nef and big over Z.

(5) Let (X ′, B′+M ′) be a projective generalised klt pair. Assume A′ := −(KX′+B′+M ′)

is nef and big. We show X ′ is Fano type. Using the notation of (1), let A = φ∗A′. ThenA ∼R H+G where H is ample and G ≥ 0. Take a small ε > 0 and a general C ∼R εH+M .Then

KX +B + εG+ C ∼R KX +B +M + εA = φ∗(KX′ +B′ +M ′ + εA′),

hence if we let ∆ = B+ εG+C, then KX + ∆ = φ∗(KX′ + ∆′) which shows (X ′,∆′) is klt.Since −(KX′ + ∆′) is nef and big, X ′ is Fano type.

(6) Let (X ′, B′ + M ′) be a projective generalised lc pair where X ′ is Fano type and−(KX′ + B′ + M ′) is nef. Assume X ′′ → X ′ is a birational morphism from a normalprojective variety. Let KX′′ +B′′ +M ′′ be the pullback of KX′ +B′ +M ′ where B′′ is thepushdown of B and M ′′ is the pushdown of M . We show X ′′ is Fano type too, assumingevery exceptional/X ′ component of B′′ has positive coefficient. There is a Q-boundary Γ′

such that (X ′,Γ′) is klt and −(KX′ + Γ′) is nef and big. Let KX′′ + Γ′′ be the pullbackof KX′ + Γ′. Let ∆′′ = (1 − t)Γ′′ + tB′′ for some t ∈ (0, 1) sufficiently close to 1. Then(X ′′,∆′′ + tM ′′) is generalised klt and −(KX′′ + ∆′′ + tM ′) is nef and big. Now apply (5).

2.8. Exceptional and non-exceptional pairs. (1) Let (X,B) be a projective pair suchthat KX +B+P ∼R 0 for some R divisor P ≥ 0. We say the pair is non-exceptional (resp.strongly non-exceptional) if we can choose P so that (X,B + P ) is not klt (resp. not lc).We say the pair is exceptional if (X,B + P ) is klt for every choice of P .

(2) Now let (X ′, B′ + M ′) be a projective generalised pair with data φ : X → X ′ andM . Assume KX′ +B′ +M ′ + P ′ ∼R 0 for some R-divisor P ′ ≥ 0. We say the pair is non-exceptional (resp. strongly non-exceptional) if we can choose P ′ so that (X ′, B′+P ′+M ′)is not generalised klt (resp. not generalised lc). We say the pair is exceptional if (X ′, B′ +P ′ +M ′) is generalised klt for every choice of P ′. Here we consider (X ′, B′ + P ′ +M ′) asa generalised pair with data φ : X → X ′ and M .

10 Caucher Birkar

2.9. Complements. (1) We first recall the definition for usual pairs. Let (X,B) be a pairwhere B is a boundary and let X → Z be a contraction. Let T = bBc and ∆ = B−T . Ann-complement of KX +B over a point z ∈ Z is of the form KX +B+ where• (X,B+) is lc,• n(KX +B+) ∼ 0 holds over some neighbourhood of z, and• nB+ ≥ nT + b(n+ 1)∆c holds over some neighbourhood of z.

From the definition one sees that

−nKX − nT − b(n+ 1)∆c ∼ nB+ − nT − b(n+ 1)∆c ≥ 0

over some neighbourhood of z which in particular means the linear system

| − nKX − nT − b(n+ 1)∆c |

is not empty over z.(2) Now let (X ′, B′ +M ′) be a projective generalised pair with data φ : X → X ′ and M

where B′ ∈ [0, 1]. Let T ′ = bB′c and ∆′ = B′ − T ′. An n-complement of KX′ +B′ +M ′ is

of the form KX′ +B′+ +M ′ where• (X ′, B′+ +M ′) is generalised lc,

• n(KX′ +B′+ +M ′) ∼ 0 and nM is Cartier, and

• nB′+ ≥ nT ′ + b(n+ 1)∆′c.

From the definition one sees that

−nKX′ − nT ′ −⌊(n+ 1)∆′

⌋− nM ′ ∼ nB′+ − nT ′ −

⌊(n+ 1)∆′

⌋≥ 0

which in particular means the linear system

| − nKX′ − nT ′ −⌊(n+ 1)∆′

⌋− nM ′|

is not empty. We can also define complements for generalised pairs in the relative settingbut for simplicity we will not deal with those.

2.10. Bounded families of pairs. A couple (X,D) consists of a normal projective varietyX and a divisor D on X whose coefficients are all equal to 1, i.e. D is a reduced divisor.The reason we call (X,D) a couple rather than a pair is that we are concerned with Drather than KX +D and we do not want to assume KX +D to be Q-Cartier or with nicesingularities.

We say that a set P of couples is birationally bounded if there exist finitely many pro-jective morphisms V i → T i of varieties and reduced divisors Ci on V i such that for each(X,D) ∈ P there exist an i, a closed point t ∈ T i, and a birational isomorphism φ : V i

t 99K Xsuch that (V i

t , Cit) is a couple and E ≤ Cit where V i

t and Cit are the fibres over t of themorphisms V i → T i and Ci → T i respectively, and E is the sum of the birational transformof D and the reduced exceptional divisor of φ. We say P is bounded if we can choose φ tobe an isomorphism.

A set R of projective pairs (X,B) is said to be log birationally bounded (respectivelylog bounded) if the set of the corresponding couples (X,SuppB) is birationally bounded(respectively bounded). Note that this does not put any condition on the coefficients ofB, eg we are not requiring the coefficients of B to be in a finite set. If B = 0 for all the(X,B) ∈ R we usually remove the log and just say the set is birationally bounded (resp.bounded).

Anti-pluricanonical systems on Fano varieties 11

Lemma 2.11. Let P be a bounded set of couples and e ∈ R>0. Then there is a finite setI ⊂ R depending only on P and e satisfying the following. Let (X,D) ∈ P and assumeR ≥ 0 is a non-zero integral divisor on X such that KX +D+rR ≡ 0 for some real numberr ≥ e. Then r ∈ I.

Proof. We can choose an effective very ample Cartier divisor A on X such that (X,A+D)belongs to a bounded set of couples Q depending only on P. Moreover, realising Ad−1 as a1-cycle inside the smooth locus of X, the intersection number L ·Ad−1 makes sense and isan integer for any integral divisor L. On the other hand, by the boundedness assumption,(KX +D) ·Ad−1 belongs to some finite set depending only on P. Thus from

(KX +D + rR) ·Ad−1 = 0

and the assumption r ≥ e we deduce that R ·Ad−1 is bounded from above, hence r belongsto some finite set I depending only on P and e.

2.12. Families of subvarieties. Let X be a normal projective variety. A bounded familyG of subvarieties of X is a family of (closed) subvarieties such that there are finitely manymorphisms V i → T i of projective varieties together with morphisms V i → X such thatV i → X embeds in X the fibres of V i → T i over closed points, and each member of thefamily G is isomorphic to a fibre of some V i → T i over some closed point. Note that wecan replace the V i → T i so that we can assume the set of points of T i corresponding tomembers of G is dense in T i. We say the family G is a covering family of subvarieties of Xif the union of its members contains some non-empty open subset of X. In particular, thismeans V i → X is surjective for at least one i. When we say G is a general member of Gwe mean there is i such that V i → X is surjective, the set A of points of T i correspondingto members of G is dense in T i, and G is the fibre of V i → T i over a general point of A (inparticular, G is among the general fibres of V i → T i).

Note that our definition of a bounded family here is compatible with 2.10. Indeed assumeG is a family of subvarieties of X which is bounded according to the definition in 2.10. Thenthere are finitely many possible Hilbert polynomials (with respect to a fixed ample divisoron X) of the members of the family. Consider the Hilbert scheme H of X and take theuniversal family H → H. There are closed subvarieties T i of H and irreducible componentsV i of the reduction of V i, where V i = T i ×H H → T i is the induced family, so that eachG ∈ G is isomorphic to a fibre of V i → T i over some closed point. By choosing the T i

carefully, we can assume that, for each i, the members of G correspond to a dense set offibres of V i → T i. Since we obtained V i → T i from the Hilbert scheme, we have an inducedmorphism V i → X which embeds in X the fibres of V i → T i over closed points. Therefore,G is a bounded family of subvarieties according to the definition in the last paragraph.

The next lemma is useful in applications when we want to replace V i → T i so thatV i → X becomes generically finite (eg, see the proof of 3.11).

Lemma 2.13. Let f : V → T be a contraction between smooth projective varieties andg : V → X a surjective morphism to a normal projective variety. Let t be a closed point ofT and F the fibre of f over t. Further assume

(1) the induced map F → X is birational onto its image,(2) f is smooth over t,(3) g is smooth over g(ηF ), and g(ηF ) is a smooth point of X where ηF is the generic

point of F .

12 Caucher Birkar

Let S be a general hypersurface section of sufficiently large degree passing through t, letU = f∗S, and assume U → X is surjective. Then U and S are smooth, U → S is smoothover t, and U → X is smooth over g(ηF ).

Proof. Let v be a general closed point of F , and let G be the fibre of g over x := g(v). Thescheme-theoretic intersection F ∩ G is the fibre of F → X over x, hence it is the reducedpoint v, by (1). In particular, TF,v ∩ TG,v = v where TF,v and TG,v are the tangentspaces to F and G at v. On the other hand, F ∩G is also the fibre of the induced morphismG → T over t, so the fibre is again just v. Now since f is smooth over t, we have anexact sequence of tangent spaces

0→ TF,v → TV,v → TT,t → 0.

Thus the kernel of the map TG,v → TT,t is TF,v ∩ TG,v, hence TG,v → TT,t is injective.Therefore, G→ T is a closed immersion near v, by Lemma 2.14 below.

Since S is a general hypersurface section of sufficiently large degree passing through t andsince T is smooth, S is smooth too. Moreover, since G is smooth by (3), S|G is smooth aswell: indeed S|G is smooth outside v by [17, Chapter III, Remark 10.9.2] and also smoothat v as G is smooth and G→ T is a closed immersion near v.

By construction, U → S is smooth over t, and U is a smooth variety outside F . Moreover,U is smooth at every point of F as F is the fibre of U → S over t which is smooth andt ∈ S is smooth. Therefore, U is smooth.

Since g is smooth near G, dimG = dimV − dimX. So

dimU ∩G = dimG− 1 = dimV − dimX − 1 = dimU − dimX

where we think of the scheme-theoretic intersection U ∩ G as the fibre of U → X overx. Thus U → X is flat over x by [17, Chapter III, Exercise 10.9] which in turn impliesU → X is smooth over x [17, Chapter III, Exercise 10.2] as U ∩G = U |G = S|G is smooth.Therefore, U → X is smooth over g(ηF ).

Lemma 2.14. Let h : X → Z be a projective morphism of normal varieties, x ∈ X be aclosed point, and z = h(x). Assume h−1z = x, and assume the map on tangent spacesTX,x → TZ,z is injective. Then h is a closed immersion near x.

Proof. Since h−1z = x, by considering the Stein factorisation of h, we can see thath is a finite morphism over z. Thus shrinking X,Z we can assume h is finite. Usingh−1z = x once more, we deduce that every open neighbourhood of x in X containsthe inverse image under h of some open neighbourhood of z in Z. This implies the map onstalks (h∗OX)z → OX,x is an isomorphism. Thus OX,x is a finitely generated OZ,z-moduleas h∗OX is coherent.

On the other hand, since the map TX,x → TZ,z is injective, the dual map mz/m2z →

mx/m2x is surjective where mz and mx are the maximal ideals of OZ,z and OX,x. Now

apply [17, II, Lemma 7.4] to show the homomorphism OZ,z → OX,x is surjective whichimplies OZ → h∗OX is surjective near z. Therefore, h is a closed immersion near x since his finite.

2.15. Potentially birational divisors. Let X be a normal projective variety, and let Dbe a big Q-Cartier Q-divisor on X. We say that D is potentially birational [14, Definition3.5.3] if for any pair x and y of general closed points of X, possibly switching x and y, wecan find 0 ≤ ∆ ∼Q (1− ε)D for some 0 < ε < 1 such that (X,∆) is not klt at y but (X,∆)is lc at x and x is a non-klt centre.

Anti-pluricanonical systems on Fano varieties 13

If D is potentially birational, then |KX + dDe| defines a birational map [15, Lemma2.3.4].

2.16. Non-klt centres. (1) The next statement is a variant of [14, Lemma 3.2.3].

Lemma 2.17. Let (X,B) be a projective pair where B is a Q-boundary, and let D ≥ 0 bea big Q-Cartier Q-divisor. Let x, y ∈ X be closed points, and assume (X,B) is klt near x,(X,B+D) is lc near x with a unique non-klt centre G containing x, and (X,B+D) is notlc near y. Then there exist rational numbers 0 ≤ t s ≤ 1 and a Q-divisor 0 ≤ D′ ∼Q Dsuch that (X,B + sD + tD′) is not lc near y but it is lc near x with a unique non-klt placewhose centre contains x, and the centre of this non-klt place is G.

Proof. Let φ : W → X be a log resolution. Then φ∗D ∼Q A + C where A ≥ 0 is ampleand general, and C ≥ 0. Let C ′ = φ∗C and D′ = φ∗(A + C). We can assume φ is a logresolution of (X,B +D +D′). Write

KW + Γs,t = φ∗(KX +B + sD + tD′).

Let T be the sum of the components of⌊Γ≥01,0

⌋whose image contains x. By assumption,

φ(S) = G for every component S of T . Now pick t > 0 sufficiently small and let s be thelc threshold of D with respect to (X,B + tD′) near x. Then s is sufficiently close to 1.

Moreover,⌊Γ≥0s,t

⌋⊆⌊Γ≥01,t

⌋=⌊Γ≥01,0

⌋, so any component of

⌊Γ≥0s,t

⌋whose image contains x, is

a component of T . Now possibly after perturbing the coefficients of C and replacing A, we

can assume⌊Γ≥0s,t

⌋has only one component S such that x ∈ φ(S). Since S is a component

of T , we have φ(S) = G.

(2) Let X be a normal projective variety of dimension d and D an ample Q-divisor.Assume vol(D) > (2d)d. Then there is a bounded family of subvarieties of X such that foreach pair x, y ∈ X of general closed points, there is a member G of the family and thereis 0 ≤ ∆ ∼Q D such that (X,∆) is lc near x with a unique non-klt place whose centrecontains x, that centre is G, and (X,∆) is not lc at y [14, Lemma 7.1].

Now assume A is an ample and effective Q-divisor. Pick a pair x, y ∈ X of general closedpoints and let ∆ and G be as above chosen for x, y. If dimG = 0, or if dimG > 0 andvol(A|G) ≤ dd, then we let G′ := G and let ∆′ := ∆ +A. On the other hand, if dimG > 0and vol(A|G) > dd, then there is 0 ≤ ∆′ ∼Q ∆ +A and there is a proper subvariety G′ ⊂ Gsuch that (X,∆′) is lc near x with a unique non-klt place whose centre contains x, thatcentre is G′, and (X,∆′) is not lc at y, by [26, Theorem 6.8.1 and 6.8.1.3] and by Lemma

2.17. Repeating this process d − 1 times, we find 0 ≤ ∆(d−1) ∼Q D + (d − 1)A and a

proper subvariety G(d−1) ⊂ G such that (X,∆(d−1)) is lc near x with a unique non-klt

place whose centre contains x, that centre is G(d−1), (X,∆(d−1)) is not lc at y, and either

dimG(d−1) = 0 or vol(A|G(d−1)) ≤ dd. In particular, all such centres G(d−1) form a boundedfamily of subvarieties of X.

2.18. Numerical Kodaira dimension. For an R-divisor D on a normal projective varietyX, we denote the numerical Kodaira dimension of D by κσ(D), as defined in [33]. If D ispseudo-effective, a decomposition D = Pσ(D) +Nσ(D) is defined in [33] where Nσ(D) ≥ 0.

Lemma 2.19. Let P be a bounded set of smooth projective varieties X with κσ(KX) = 0.Then there is a number l ∈ N such that h0(lKX) 6= 0 for every X ∈ P.

Proof. Perhaps after replacing P, we can assume there is a smooth projective morphismf : V → T of smooth varieties such that every X ∈ P appears as a fibre of f over some

14 Caucher Birkar

closed point. Then by [15, Theorem 1.8 (2)], κσ(KF ) = 0 for every fibre F of f . Applying[11] to the geometric generic fibre and applying semi-continuity of cohomology shows thatthere is l ∈ N such that h0(lKF ) 6= 0 for every fibre F .

2.20. Pseudo-effective threshold.

Lemma 2.21. Let P be a log bounded set of log smooth projective pairs (X,B). Then thereis a number λ > 0 such that if (X,B) ∈ P and if KX is not pseudo-effective, then KX +λBis not pseudo-effective.

Proof. Perhaps after replacing P, we can assume there exist a smooth projective morphismf : V → T of smooth varieties and a reduced divisor S on V , which is simple normal crossingover T , such that if (X,B) ∈ P, then X is a fibre of f over some closed point and SuppBis inside the restriction of S to X. Replacing B, we can assume B = S|X . Now apply [15,Theorem 1.8 (2)].

2.22. Volume of divisors. Recall that the volume of an R-divisor D on a normal projec-tive variety X of dimension d is defined as

vol(D) = lim supm→∞

h0(bmDc)md/d!

.

Lemma 2.23. Let X be a normal projective variety of dimension d, D an R-Cartier R-divisor with κσ(D) > 0, and A an ample Q-divisor. Then limm→∞ vol(mD +A) =∞.

Proof. Replacing X with a resolution and replacing A appropriately, we can assume X issmooth. Replacing D with Pσ(D), we can assume Nσ(D) = 0. Let C be a curve cut out bygeneral members of |A|. By [33, Chapter V, Theorem 1.3], perhaps after replacing A with amultiple, we can assume C does not intersect Bs | bmDc+A| for any m > 0. In particular,for each m, there is a resolution φ : W → X such that φ∗(bmDc + A) decomposes as thesum of a free divisor F and fixed part G such that SuppG does not intersect φ−1C. Then

vol(mD + 2A) ≥ vol(bmDc+ 2A) = vol(φ∗(bmDc+ 2A)) ≥

vol(F + φ∗A)) ≥ F · (φ∗A)d−1 = φ∗(bmDc+A) · (φ∗A)d−1 = (bmDc+A) · C.Since (bmDc+A) ·C is not a bounded function of m, vol(mD+ 2A) is not bounded, hencelimm→∞ vol(2mD + 2A) =∞ which implies the lemma.

Lemma 2.24. Let d ∈ N and let P be a set of pairs (X,A) where X is smooth projective ofdimension ≤ d with κσ(KX) > 0, and A is big and base point free. Then for each numberq, there is a number p such that for every (X,A) ∈ P we have vol(pKX +A) > q.

Proof. If the statement is not true, then there exist a number q ∈ N, a strictly increasingsequence of numbers pi ∈ N, and a sequence of pairs (Xi, Ai) ∈ P such that vol(piKXi +Ai) ≤ q for every i. Since A is base point free, we may assume A is irreducible. Let Xi → Zibe the contraction defined by |Ai| and let Ci be the pushdown of Ai. Since vol(Ai) ≤ q, thecouples (Zi, Ci) form a log bounded family. Replacing Xi with an appropriate resolutionof Zi and replacing Ai accordingly, we can assume (Xi, Ai) is log smooth and log bounded.Replacing the sequence, we can assume there is a projective morphism f : V → T of smoothvarieties and a reduced divisor A ≥ 0 on V which is simple normal crossing over T andsuch that for each i, Xi appears as a general fibre of f over some closed point and thatAi = A|Xi .

Anti-pluricanonical systems on Fano varieties 15

Fix a general fibre F of f and let AF = A|F . Then by [15, Theorem 1.8 (3)], for each i,

vol(KF +1

piAF ) = vol(KXi +

1

piAi),

hencevol(piKF +AF ) = vol(piKXi +Ai) ≤ q.

Pick l so that lAF ∼ HF+LF whereHF is ample and LF is big. Then vol(lpiKF+HF ) ≤ ldq.This contradicts Lemma 2.23.

2.25. The restriction exact sequence. Let X be a normal projective variety, S a normalprime divisor and L an integral Q-Cartier divisor on X. We have an exact sequence

0→ OX(L− S)→ OX(L)→ F → 0

where F is an OS-module.

Lemma 2.26. Assume (X,B) is dlt for some boundary B. Let U be the largest open subsetof X on which L is Cartier. If the codimension of the complement of S ∩U in S is at leasttwo, then F ' OS(L|S).

Proof. Note that L|S is well-defined up to linear equivalence. Let A be an ample Cartierdivisor on X. Since (X,B) is dlt, by duality [31, Corollary 5.27] and Serre vanishing,hi(L − nA) = 0 and hi(L − S − nA) = 0 for any i < d = dimX and any n 0. Thushi(F(−nA)) = 0 for i < d− 1 and any n 0. This implies F is a Cohen-Macaulay sheafon S [31, Corollary 5.72]. In particular, F is S2, hence F is determined by F|U∩S . On theother hand, OS(L|S) is reflexive, hence it is determined by its restriction to S ∩ U . Nowthe result follows from the fact that on S ∩ U , F and OS(L|S) are isomorphic.

2.27. Descent of nef divisors.

Lemma 2.28. Let f : X → Z be a contraction from a smooth projective variety to a normalprojective variety with rationally connected general fibres. Assume M is a nef Cartierdivisor on X such that M ∼Q 0 on the generic fibre of f . Then there exist resolutionsφ : W → X and ψ : V → Z such that the induced map W 99K V is a morphism andφ∗M ∼ 0/V .

Proof. Since M is nef and M ∼Q 0 on the generic fibre of f , we can find φ and ψ so thatφ∗M ∼Q 0/V [25, Proposition 2.1]. Replacing X with W and replacing Z with V , wecan assume M ∼Q 0/Z. Since the general fibres of f are rationally connected, KX is notpseudo-effective over Z. Running an MMP/Z on KX with scaling of some ample divisoras in [6], we get a Mori fibre space X ′ → T ′/Z. Since M is Cartier and M ∼Q 0/Z, M ′ isCartier, by the cone theorem [31, Theorem 3.7], where M ′ is the pushdown of M . Moreover,M ′ ∼ 0/T ′ again by [31, Theorem 3.7], hence M ′ is the pullback of some nef Cartier divisorN ′ on T ′. Let T be a resolution of T ′. Then the general fibres of T → Z are rationallyconnected as they are dominated by the general fibres of f . Now replace X with T andreplace M with the pullback of N ′ to T . Then apply induction on dimension.

2.29. Pairs with large boundaries.

Lemma 2.30. Let (X,B) be a projective klt pair of dimension d, and let M be a nef Cartierdivisor. Let a > 2d be a real number. Then any MMP on KX +B + aM is M -trivial, i.e.the extremal rays in the process intersect M trivially. If M is big, then KX + B + aM isalso big.

16 Caucher Birkar

Proof. The fact that any MMP on KX + B + aM is M -trivial follows from boundednessof extremal rays [23]. The Cartier condition of M is preserved in the process by the conetheorem. Now assume M is big. Then we can run an MMP which terminates with somemodel [6]. Since M is big, the MMP ends with a minimal model otherwise we get aMori fibre space Y → T and by boundedness of extremal rays, MY ≡ 0/T which is notpossible. Replacing X with the minimal model, we can assume KX +B+ aM is nef. ThenKX + B + a′M is nef for any a′ < a sufficiently close to a otherwise there is an extremalray R such that (KX +B + aM) ·R = 0 and M ·R > 0 which contradicts boundedness ofextremal rays. Therefore, KX +B + aM is big.

2.31. Divisors with log discrepancy close to 0.

Lemma 2.32. Let d ∈ N and Φ ⊂ [0, 1] be a DCC set. Then there is ε > 0 depending onlyon d and Φ such that if (X,B) is a projective pair and D is a prime divisor on birationalmodels of X satisfying• (X,B) is lc of dimension d and (X, 0) is klt,• KX +B ∼R 0 and B ∈ Φ, and• a(D,X,B) < ε,

then a(D,X,B) = 0.

Proof. If the lemma does not hold, then there exist a decreasing sequence εi of numbersapproaching 0 and a sequence (Xi, Bi), Di of pairs and divisors as in the statement suchthat 0 < a(Di, Xi, Bi) < εi. Since (Xi, 0) is klt, there is a birational morphism X ′i → Xi

extracting only Di. Let KX′i+B′i be the pullback of KXi +Bi, and let bi = 1−a(Di, Xi, Bi)

which is the coefficient of Di in B′i. Note that B′i ∈ Φ′ := Φ∪bi. Replacing the sequence,we can assume Φ′ is a DCC set. Now we get a contradiction, by [14, Theorem 1.5], becausebi is not finite.

2.33. Boundary coefficients close to 1. Let d, p ∈ N, ε ∈ R>0, and let Φ ⊂ [0, 1] bea DCC set. Let (X ′, B′ + M ′) be a projective generalised pair of dimension d with dataφ : X → X ′ and M such that B′ ∈ Φ∪(1−ε, 1], pM is Cartier, −(KX′+B

′+M ′) is a limit ofmovable R-divisors, and there is 0 ≤ P ′ ∼R −(KX′ +B′+M ′) such that (X ′, B′+P ′+M ′)is generalised lc. Assume X ′ is Q-factorial Fano type. Let Θ′ be the boundary whosecoefficients are the same as B′ except that we replace each coefficient in (1 − ε, 1) with 1.That is, Θ′ = (B′)≤1−ε + d(B′)>1−εe.

Assuming ε is sufficiently small, depending only on d, p,Φ, we will prove the following.Run an MMP on −(KX′ + Θ′ +M ′) and let X ′′ be the resulting model. Then:

(1) (X ′,Θ′ +M ′) is generalised lc,(2) the MMP does not contract any component of bΘ′c,(3) −(KX′′ + Θ′′ +M ′′) is nef, and(4) (X ′′,Θ′′ +M ′′) is generalised lc.

Proof of (1). Assume it is not true. Then there exist a decreasing sequence εi of numbersapproaching 0, and a sequence (X ′i, B

′i + M ′i) of generalised pairs as above such that if Θ′i

is the divisor derived from B′i using εi, then (X ′i,Θ′i + M ′i) is not generalised lc. There

exist boundaries B′i ≤ Γ′i ≤ Θ′i with Γ′i ∈ Φ ∪ (1 − εi, 1] and a component D′i of Γ′i withcoefficient ti ∈ (1 − εi, 1) such that ti is the generalised lc threshold of D′i with respect to(X ′i,Γ

′i − tiD′i + M ′i). Replacing the sequence we can assume the union of the coefficients

of all the Γ′i is a DCC set. Then we get a contradiction by the ACC for generalised lc

Anti-pluricanonical systems on Fano varieties 17

thresholds [7, Theorem 1.4].

Proof of (2). If this is not true, then there exist a decreasing sequence εi of numbersapproaching 0, and a sequence (X ′i, B

′i + M ′i) of generalised pairs as above such that if Θ′i

is the divisor derived from B′i using εi, then the MMP contracts some component S′i ofbΘ′ic. Since all our assumptions are preserved under the MMP, we may assume the first

step of the MMP is a birational contraction X ′i → X ′i which contracts S′i. Let R′i be thecorresponding extremal ray.

As −(KX′ + B′ + M ′) is a limit of movable R-divisors, −(KX′i+ B′i + M ′i) · R′i ≥ 0.

Moreover, S′i · R′i < 0 and if B′i is the same as B′i except that we increase the coefficientof S′i to 1, then B′i ≤ B′i ≤ Θ′i and −(KX′i

+ B′i + M ′i) · R′i ≥ 0. Therefore, there exist

boundaries B′i ≤ Γ′i Θ′i with Γ′i ∈ Φ ∪ (1 − εi, 1] such that −(KX′i+ Γ′i + M ′i) · R′i = 0

and S′i is a component of bΓ′ic. Moreover, there is a component D′i of Γ′i with coefficient

ti ∈ (1 − εi, 1) such that D′i · R′i > 0. Now ti is the generalised lc threshold of D′i with

respect to (X ′i, Γ′i − tiD′i + M ′i) where Γ′i, D

′i, M

′i are the pushdowns of Γ′i, D

′i,M

′i . Again

this contradicts [7, Theorem 1.4].

Proof of (3). Assume this is not true. Then there exist a decreasing sequence εi ofnumbers approaching 0, and a sequence (X ′i, B

′i + M ′i) of generalised pairs as above such

that if Θ′i is the divisor derived from B′i using εi, then the MMP gives an extremal non-birational contraction X ′′i → T ′′i which is −(KX′′i

+ Θ′′i +M ′′i )-negative. Under our assump-

tions −(KX′i+ B′i + M ′i) is nef over T ′′i . So there exist boundaries B′i ≤ Γ′i Θ′i with

Γ′i ∈ Φ ∪ (1− εi, 1] such that −(KX′′i+ Γ′′i +M ′′i ) ≡ 0/T ′′i . Moreover, there is a component

D′i of Γ′i with coefficient ti ∈ (1− εi, 1) such that D′i is ample over T ′′i . By restricting to thegeneral fibres of X ′′i → T ′′i we get a contradiction in view of the global ACC for generalisedpairs [7, Theorem 1.5].

Proof of (4). This can be proved as in (1).

3. Adjunction

In this section we discuss various kinds of adjunction that would be needed in subsequentsections.

3.1. Divisorial adjunction. Let (X ′, B′ + M ′) be a generalised pair with data Xφ→ X ′

and M . Assume that S′ is the normalisation of a component of B′ with coefficient 1, andthat S is its birational transform on X. Replacing X we may assume φ is a log resolutionof (X ′, B′). Write

KX +B +M = φ∗(KX′ +B′ +M ′)

and let

KS +BS +MS := (KX +B +M)|Swhere BS = (B − S)|S and MS = M |S . Let ψ be the induced morphism S → S′ and letBS′ = ψ∗BS and MS′ = ψ∗MS . Then we get the equality

KS′ +BS′ +MS′ = (KX′ +B′ +M ′)|S′

which we refer to as generalised (divisorial) adjunction. Note that

KS +BS +MS = ψ∗(KS′ +BS′ +MS′)

18 Caucher Birkar

It is obvious that BS′ depends on both B′ and M . When M = 0, we recover the usualdivisorial adjunction.

By [7, Remark 4.7], if (X ′, B′ +M ′) is generalised lc, then BS′ is a boundary divisor onS′, i.e. its coefficients belong to [0, 1]. We consider (S′, BS′ + MS′) as a generalised pair

which comes with data Sψ→ S′ and MS . It is also clear that (S′, BS′ +MS′) is generalised

lc if (X ′, B′ +M ′) is generalised lc.

Lemma 3.2 (Generalised inversion of adjunction). Let (X ′, B′ + M ′) be a Q-factorial

generalised pair with data Xφ→ X ′ and M . Assume S′ is a component of B′ with coefficient

1, and that (X ′, S′) is plt. Let

KS′ +BS′ +MS′ = (KX′ +B′ +M ′)|S′

be given by generalised adjunction. If (S′, BS′ +MS′) is generalised lc, then (X ′, B′ +M ′)is generalised lc near S′.

Proof. Assume (X ′, B′ + M ′) is not generalised lc near S′. We can assume φ is a logresolution. Write

KX +B +M = φ∗(KX′ +B′ +M ′).

By assumption φ(B>1) intersects S′. Pick α ∈ (0, 1) sufficiently close to 1, let Γ′ =(1− α)S′ + αB′, and write

KX + Γ + αM = φ∗(KX′ + Γ′ + αM ′).

Then (X ′,Γ′ + αM ′) is not generalised lc near S′. On the other hand, since (X ′, S′) is pltand (S′, BS′ +MS′) is generalised lc, (S′,ΓS′ + αMS′) is generalised klt where

KS′ + ΓS′ + αMS′ = (KX′ + Γ′ + αM ′)|S′

is generalised adjunction. Thus replacing B′ with Γ′ and M with αM , we can assume(S′, BS′ +MS′) is generalised klt.

Pick an ample divisor A ≥ 0 and an effective divisor C ≥ 0 such that A+C ∼ 0/X ′ andthat S is not a component of C. Let ε > 0 be small and pick a general 0 ≤ G ∼R εA+M/X ′.Let ∆ := B+G+εC. Then KX+∆ ∼R 0/X ′, hence KX+∆ = φ∗(KX′+∆′). In particular,(X ′,∆′) is not lc near S′ as ∆ ≥ B. On the other hand, by assumption, (S,BS) is sub-kltwhere KS + BS = (KX + B)|S . This implies (S,∆S) is also sub-klt by construction of ∆where KS + ∆S = (KX + ∆)|S . Therefore, letting KS′ + ∆S′ = (KX′ + ∆′)|S′ we see that(S′,∆S′) is klt because KS + ∆S is the pullback of KS′ + ∆S′ . This contradicts the usualinversion of adjunction [21].

The next result is similar to [34, Proposition 3.9].

Lemma 3.3. Let p ∈ N and R ⊂ [0, 1] be a finite set of rational numbers. Then there is afinite set of rational numbers S ⊂ [0, 1] depending only on p and R satisfying the following.

Assume (X ′, B′ +M ′) is a generalised lc pair of dimension d with data Xφ→ X ′ → Z and

M , assume S′ is the normalisation of a component of bB′c, and assume• B′ ∈ Φ(R) and pM is Cartier, and• BS′ is given by the generalised adjunction

KS′ +BS′ +MS′ = (KX′ +B′ +M ′)|S′ .

Then BS′ ∈ Φ(S).

Anti-pluricanonical systems on Fano varieties 19

Proof. As in [7, Remark 4.8], by taking hypersurface sections, we can reduce the lemma tothe case dimX ′ = 2. Let V be a prime divisor on S′ and assume µVBS′ < 1. Then as inthe proof of [7, Proposition 4.9], (X ′, B′) is plt near the image of the generic point of Vand the coefficient of V in BS′ is of the form

µVBS′ = 1− 1

l+

n∑i=1

biαil

+β

pl

where l ∈ N is divisible by the Cartier index along V of any Weil divisor, bi are thecoefficients of B′, and αi, β ∈ Z≥0.

Expanding R we can assume p−1p ∈ R, hence 1

p ∈ Φ(R). Put αn+1 := β and bn+1 := 1p .

So we can write µVBS′ = 1 − 1l +

∑n+1i=1

biαil . For each i there is ri ∈ R and mi ∈ N such

that bi = 1 − rimi

. Let s := 1 −∑n+1

i=1 biαi. Then µVBS′ = 1 − sl and s > 0. Removing

the zero terms and re-arranging the indexes we can assume s = 1 −∑t

i=1 biαi, and that

biαi > 0 for every i otherwise µVBS′ = 1− 1l is just a standard coefficient. Note that since

s > 0, we have∑t

i=1 biαi < 1, hence αi are bounded. Moreover, bi ≥ 1− 1mi

which means

bi ≥ 12 if mi > 1.

Now assume t = 1. Then either m1 ≤ 2, or m1 > 2 and α1 = 1, hence in any case s = s′

m1

where there are only finitely many possibilities for s′. Thus µVBS′ = 1− sl = 1− s′

m1l∈ Φ(S)

if we choose S so that it contains all such s′. We can then assume t > 1. If mi = 1 forevery i, then there are finitely many possibilities for s = 1 −

∑ti=1(1 − ri)αi and we let

s′ = s. But if some mi > 1, say for simplicity m1 > 1, then α1 = 1 and mi = 1 for i > 1.Then s = r1

m1−∑t

i=2(1− ri)αi, hence m1 is bounded, so s = s′

m1for some s′ for which there

are only finitely many possibilities. As before taking S so that it contains all such s′, weget µVBS′ ∈ Φ(S).

3.4. Adjunction for fibre spaces. (1) Let (X,B) be a projective sub-pair and let f : X →Z be a contraction with dimZ > 0 such that (X,B) is sub-lc near the generic fibre of f , andKX + B ∼R 0/Z. Below we recall a construction based on [22] giving a kind of canonicalbundle formula which we refer to as adjunction for fibre spaces [1][34, §7].

For each prime divisor D on Z we let tD be the lc threshold of f∗D with respect to (X,B)over the generic point of D, that is, tD is the largest number so that (X,B + tDf

∗D) issub-lc over the generic point of D. Of course f∗D may not be well-defined everywhere butat least it is defined over the smooth locus of Z, in particular, near the generic point ofD, and that is all we need. Next let bD = 1 − tD, and then define BZ =

∑bDD where

the sum runs over all the prime divisors on Z. By assumption, KX +B ∼R f∗LZ for someR-Cartier R-divisor LZ on Z. Letting MZ = LZ − (KZ +BZ) we get

KX +B ∼R f∗(KZ +BZ +MZ).

We call BZ the discriminant part and MZ the moduli part of adjunction. Obviously BZis uniquely determined but MZ depends on the choice of LZ . However, MZ is determineduniquely up to R-linear equivalence.

Now let φ : X ′ → X and ψ : Z ′ → Z be birational morphisms from normal projectivevarieties and assume the induced map f ′ : X ′ 99K Z ′ is a morphism. LetKX′+B

′ = φ∗(KX+B). Similar to above we can define a discriminant divisor BZ′ , and taking LZ′ = ψ∗LZgives a moduli divisor MZ′ so that

KX′ +B′ ∼R f ′∗(KZ′ +BZ′ +MZ′)

and BZ = ψ∗BZ′ and MZ = ψ∗MZ′ .

20 Caucher Birkar

(2) We want to show MZ depends only on (X,B) near the generic fibre of f , evenbirationally. We make this more precise. In addition to the assumptions of (1), suppose weare given another projective sub-pair (X,B) and a contraction X → Z such that KX+B ∼R0/Z. Moreover, suppose we have a birational map X 99K X/Z, and suppose there is acommon resolution of X and X on which the pullback of KX + B and KX + B are equal

near the generic fibre over Z. Let BZ′ and MZ′ be the discriminant and moduli parts ofadjunction on Z ′ defined for (X,B) and the contraction X → Z.

Lemma 3.5. Under the above notation and assumptions, MZ′ ∼R MZ′.

Proof. Replacing both X and X with a common resolution over Z ′, replacing KX +B andKX + B with their crepant pullbacks, and replacing Z with Z ′ we can assume X = X,

Z ′ = Z, and that B = B near the generic fibre. Then B − B is vertical over Z andB − B ∼R 0/Z. So B − B = f∗P for some P on Z. Therefore, by definition of thediscriminant part, BZ = BZ + P from which we get MZ ∼R MZ because

KZ +BZ +MZ + P ∼R KZ +BZ +MZ .

(3) When (X,B) is lc over the generic point of Z we have:

Theorem 3.6. Under the notation and assumptions of (1), suppose (X,B) is lc near thegeneric fibre of f . Let Z ′ be a sufficiently high resolution of Z. Then

(i) MZ′ is pseudo-effective, and(ii) if B is a Q-divisor, then MZ′ is nef and for any birational morphism Z ′′ → Z ′ from

a normal projective variety, MZ′′ is the pullback of MZ′.

Proof. Let Γ be the boundary divisor obtained from B by removing components that havenegative coefficients or are vertical/Z. Note that over the generic point of Z we have Γ = B.Let φ : W → X be a log resolution and let ΓW be the sum of the horizontal/Z part of thereduced exceptional divisor of φ and the birational transform of Γ. We can choose W sothat every lc centre of (W,ΓW ) is horizontal/Z. Running an MMP on KW + ΓW over Xwe reach a model X ′ so that (X ′,Γ′) is a Q-factorial dlt model of (X,B) over the genericpoint of Z. Let KX′ +B′ be the pullback of KX +B. Replacing (X,B) with (X ′, B′) andreplacing (X,Γ) with (X ′,Γ′) we can assume (X,Γ) is Q-factorial dlt with every lc centrehorizontal/Z, and that every component of Γ is horizontal/Z.

We prove (ii) first. By [5, Theorem 1.4] we can replace X with a model on which KX +Γis semi-ample over Z, hence defining a contraction X → Y/Z. Since KX + Γ ∼Q 0/Y ,by Lemma 3.5, we can replace Z with Y and replace B with Γ, hence assume (X,B) isQ-factorial dlt. The claim then follows from Fujino-Gongyo [10] (this relies on Ambro [2]in the klt case).

Now we prove (i). By standard arguments we can find Q-divisors Bi and numbersαi ∈ [0, 1] with

∑αi = 1 such that B =

∑αiB

i, (X,Bi) is lc over the generic point ofZ, and KX + Bi ∼Q 0/Z. Let Bi

Z′ and M iZ′ be the discriminant and moduli parts defined

for (X,Bi) over Z. Let D be a prime divisor on Z and let ti be the lc threshold of f∗Dwith respect to (X,Bi) over the generic point of D. Then (X,B + (

∑αiti)f

∗D) is lc overthe generic point of D, hence t ≥

∑αiti where t is the lc threshold of f∗D with respect

to (X,B) over the generic point of D. Then 1 − t ≤ αi(1 − ti) from which we deduceBZ ≤

∑αiB

iZ . Similarly we can prove BZ′ ≤

∑αiB

iZ′ . This shows MZ′ ∼R

∑αiM

iZ′ +P

for some P ≥ 0. Therefore, MZ′ is pseudo-effective as each M iZ′ is nef.

Anti-pluricanonical systems on Fano varieties 21

(4) We relate the singularities of (X,B) and (Z,BZ +MZ) though in a rough sense.

Lemma 3.7. Let ε ∈ R. Under the notation and assumptions of (1), suppose there isa prime divisor S on some birational model of X such that a(S,X,B) ≤ ε and that Sis vertical over Z. Then there is a resolution Z ′ → Z and a component T of BZ′ withcoefficient ≥ 1− ε.

Proof. Pick resolutions X ′ → X and Z ′ → Z so that the induced map f ′ : X ′ 99K Z ′ is amorphism and so that S is a prime divisor on X ′ and its image on Z ′ is a prime divisor T .Let KX′ +B′ be the pullback of KX +B. Since a(S,X,B) ≤ ε, the coefficient of S in B′ isat least 1− ε. Thus the lc threshold of f ′∗T with respect to (X ′, B′) over the generic pointof T is at most ε. Therefore, the coefficient of T in BZ′ is at least 1− ε.

3.8. Adjunction on non-klt centres.

Theorem 3.9 ([14, Theorem 4.2]). Let Φ be a subset of [0, 1] which contains 1. Let X be aprojective variety of dimension d, and let G be a subvariety, with normalisation F . Supposewe are given a boundary B and an R-Cartier divisor ∆ ≥ 0, with the following properties:• (X,B) is klt and B ∈ Φ, and• there is a unique non-klt place of (X,B + ∆) whose centre is G.

Then there is a boundary ΘF on F whose coefficients belong to

a | 1− a ∈ LCTd−1(D(Φ)) ∪ 1such that

KF + ΘF + PF ∼R (KX +B + ∆)|Fwhere PF is pseudo-effective.

Now suppose that G is a general member of a covering family of subvarieties of X. Letψ : F ′ → F be a log resolution of (F,ΘF ), and let ΘF ′ be the sum of the birational transformof ΘF and the reduced exceptional divisor of ψ. Then

KF ′ + ΘF ′ ≥ (KX +B)|F ′ .

In the theorem D(Φ) is a set associated to Φ with the property: D(Φ) is DCC iff Φ isDCC [14, 3.4]. The set LCTd−1(D(Φ)) stands for the set of all lc thresholds of integraleffective divisors with respect to pairs (S,Γ) of dimension d− 1 such that Γ ∈ D(Φ).

In the proof of the next two lemmas we recall the construction of ΘF and PF and theproof of 3.9 following [14] but with more details.

Lemma 3.10. Let (X,B),∆, G, F, and ΘF , be as in Theorem 3.9 where G is not necessarilya general member of a covering family. Let M ≥ 0 be a Q-Cartier Q-divisor on X withcoefficients ≥ 1 and such that G * SuppM . Then for every component D of MF := M |Fwe have µD(ΘF +MF ) ≥ 1.

Proof. Let Γ be the sum of (B + ∆)<1 and the support of (B + ∆)≥1. Put N = B + ∆−Γwhich is supported in bΓc. Let φ : W → X be a log resolution and let ΓW be the sum ofthe reduced exceptional divisor of φ and the birational transform of Γ. Let

NW = φ∗(KX +B + ∆)− (KW + ΓW ).

Then φ∗NW = N ≥ 0 and NW is supported in bΓW c. Now run an MMP/X on KW + ΓWwith scaling of some ample divisor. We reach a model Y on which KY + ΓY is a limitof movable/X divisors. Applying the general negativity lemma (cf. [5, Lemma 3.3]), wededuce NY ≥ 0. In particular, if U ⊆ X is the set of points where (X,B + ∆) is lc, then

22 Caucher Birkar

NY = 0 over U and (Y,ΓY ) is a Q-factorial dlt model of (X,B + ∆) over U . Since thereis a unique non-klt place of (X,B + ∆) with centre G, (X,B + ∆) is lc but not klt at thegeneric point of G. In particular, we can assume there is a unique component S of bΓY cmapping onto G. Moreover, G ∩ U 6= ∅, so G is not inside the image of NY .

Let h : S → F be the contraction induced by S → G. By divisorial adjunction we canwrite

KS + ΓS +NS = (KY + ΓY +NY )|S ∼R 0/F

where NS = NY |S is vertical over F . If S is exceptional over X, then let ΣY be the sumof the exceptional/X divisors on Y plus the birational transform of B. Otherwise let ΣY

be the sum of the exceptional/X divisors on Y plus the birational transform of B plus(1 − µGB)S. In any case, S is a component of bΣY c and ΣY ≤ ΓY . Applying adjunctionagain we get KS + ΣS = (KY + ΣY )|S . Obviously ΣS ≤ ΓS .

Now ΘF is defined as follows. For each prime divisor D on F , let t be the lc threshold ofh∗D with respect to (S,ΣS) over the generic point of D, and then let µDΘF := 1 − t [14,proof of Theorem 4.2]. One then chooses PF to satisfy

KS + ΓS +NS ∼R h∗(KF + ΘF + PF ).

Let ∆F and RF be the discriminant and the moduli parts of adjunction for (S,ΓS + NS)over F . Since ΣS ≤ ΓS + NS , we have ΘF ≤ ∆F , hence PF − RF ∼R ∆F − ΘF ≥ 0. ByTheorem 3.6, RF is pseudo-effective which implies PF is pseudo-effective too.

Let Σ′Y := ΣY +MY and Σ′S = ΣS +MS where MY = M |Y and MS = MY |S . We defineΘ′F similar to ΘF . That is, for each prime divisor D on F , let t′ be the lc threshold ofh∗D with respect to (S,Σ′S) over the generic point of D, and then let µDΘ′F := 1 − t′. Itis easy to see t′ + µDMF = t. This means Θ′F = ΘF + MF . On the other hand, since thecoefficients of M are at least 1 and since S is not a component of MY ,

SuppMY ⊆ Supp bΣY +MY − Sc = Supp⌊Σ′Y − S

⌋.

Therefore, SuppMS ⊂ bΣ′Sc. In particular, as h∗MF = MS , if D is a component of MF ,then every component of h∗D mapping onto D is a component of bΣ′Sc. Thus t′ ≤ 0 whereas above t′ is the lc threshold of h∗D with respect to (S,Σ′S) over the generic point of D.Therefore, µD(ΘF +MF ) = µDΘ′F = 1− t′ ≥ 1.

Lemma 3.11. Let (X,B),∆, G, F, and ΘF be as in Theorem 3.9 where G is a generalmember of a covering family of subvarieties. Assume (X,B) is ε-lc for some ε > 0. Thenthere is a sub-boundary BF on F such that KF +BF = (KX+B)|F , that (F,BF ) is sub-ε-lc,and BF ≤ ΘF .

Proof. Step 1. There is a contraction f : V → T of normal projective varieties such thatG is isomorphic to a general fibre of f , and there is a surjective morphism V → X whoserestriction to each fibre of f over a closed point is a closed immersion (see 2.12). In par-ticular, identifying G with the fibre mentioned, the induced morphism G → X maps Gonto itself. Taking normalisations of V and T , we get a contraction f : V → T of normalprojective varieties such that the fibre of f corresponding to G is just F the normalisationof G. Taking resolutions of V and T we get a contraction f ′ : V ′ → T ′ of smooth projectivevarieties. Letting F ′ be the fibre of f ′ corresponding to F , we see that the induced mor-phism F ′ → X is birational onto its image. Moreover, we can assume there is a Cartierdivisor P ≥ 0 on X whose support contains SuppB and the singular locus of X and suchthat Q′ := Supp g′∗P is relatively simple normal crossing over some open subset of T ′ where

Anti-pluricanonical systems on Fano varieties 23

g′ is the induced map V ′ → X.

Step 2. Fixing G, by construction, f ′ is smooth over t′ = f ′(F ′), g′ is smooth overg′(ηF ′), and g′(ηF ′) is smooth on X. If g′ is generically finite let W ′ := V ′ and R′ := T ′. Ifnot, applying Lemma 2.13, there is a general smooth hypersurface section H ′ of T ′ passingthrough t′ so that if U ′ = f ′∗H ′, then U ′ and H ′ are smooth, U ′ → H ′ is smooth overt′, and U ′ → X is surjective and smooth over g′(ηF ′). Repeating this we get a smoothsubvariety R′ of T ′ passing through t′ so that the induced family W ′ → R′ is smoothover t′, W ′ is smooth, W ′ → X is surjective and generically finite and etale over g′(ηF ′).Let QW ′ = Q′|W ′ . Then by construction, QW ′ |F ′ = Q′|F ′ is reduced and simple normalcrossing. Therefore, near F ′ the divisor QW ′ is reduced and any prime divisor C ′ on F ′ iscontained in at most one component of QW ′ .

Step 3. Let KW ′ + BW ′ be the pullback of KX + B to W ′. Here KW ′ and BW ′ areuniquely determined as Weil divisors (we assume we have already fixed a choice of KX).Let W ′ → W → X be the Stein factorisation of W ′ → X. By the Riemann-Hurwitz for-mula, each coefficient of BW is ≤ 1− ε where KW +BW is the pullback of KX +B. Thus(W,BW ) is sub-ε-lc by [31, Proposition 5.20], hence (W ′, BW ′) is also sub-ε-lc. On the otherhand, by our choice of P , any component of BW ′ with positive coefficient is mapped intoP , hence it is a component of QW ′ . Moreover, since G was chosen general, it is not insideSuppKX ∪ SuppP . Thus since W ′ → X is etale over g′(ηF ′), F

′ is not inside SuppBW ′nor inside SuppKW ′ . Defining BF ′ = BW ′ |F ′ we get KF ′ + BF ′ = (KW ′ + BW ′)|F ′ whereKF ′ = KW ′ |F ′ follows from the fact that W ′ → R′ is smooth near F ′ (note that KF ′ isdetermined as a Weil divisor). Let KF +BF be the pushdown of KF ′ +BF ′ which satisfiesKF +BF = (KX +B)|F .

Step 4. To show (F,BF ) is sub-ε-lc, it is enough to show (F ′, BF ′) is sub-ε-lc. This in

turn follows if show (F ′, AF ′) is ε-lc where AW ′ := B≥0W ′ and AF ′ = AW ′ |F ′ . By the lastparagraph, SuppAW ′ ⊆ SuppQW ′ , hence

SuppAF ′ = SuppAW ′ |F ′ ⊆ SuppQW ′ |F ′ = SuppQ′|F ′ .

So SuppAF ′ is simple normal crossing as SuppQ′|F ′ is simple normal crossing. Therefore,it is enough to show each coefficient of AF ′ is ≤ 1− ε. Let C ′ be a component of AF ′ withpositive coefficient. Then there is a component D′ of AW ′ with positive coefficient so that C ′

is a component of D′|F ′ . Since D′ is a component of QW ′ , by the second paragraph of thisproof, D′ is uniquely determined and D′|F ′ is reduced. Then µD′BW ′ = µD′AW ′ = µC′AF ′ .But µD′BW ′ ≤ 1− ε because (W ′, BW ′) is sub-ε-lc, hence µC′AF ′ ≤ 1− ε.

Step 5. It remains to show BF ≤ ΘF . Assume C is a prime divisor on F with µCBF > 0.Let C ′ on F ′ be the birational transform of C. Then µC′BF ′ > 0. Thus there is a uniquecomponent D′ of B>0

W ′ with positive coefficient such that C ′ is a component of D′|F ′ . Inparticular, µC′BF ′ ≤ µD′BW ′ as D′|F ′ is reduced. Let L = cP and let LW ′ = L|W ′ andLF ′ = L|F ′ where c is a number. Choose c so that µD′LW ′ = 1. Then µC′LF ′ = µD′LW ′ =1.

We use the notation of the proof of Lemma 3.10. Recall that we constructed a birationalmorphism Y → X which we denote by π, a boundary ΣY , and a component S of bΣY cmapping onto G. Let LS = L|S and let t be the lc threshold of LS with respect to (S,ΣS)over the generic point ηC of C. By construction, µC(L|F ) = µC′LF ′ = 1, so under h : S → Fthe divisor LS is equal to h∗C over ηC . Therefore, µCΘF = 1− t.

24 Caucher Birkar

Let s be the lc threshold of L with respect to (X,B) near ν(ηC) where ν denotes F → G.Let I be the minimal non-klt centre of (X,B+sL) which contains ν(ηC). Since G * SuppL,G * I. Let LY = π∗L and write KY + BY = π∗(KX + B). Let IY be a non-klt centreof (Y,BY + sLY ) which maps onto I. Since ΣY ≥ BY , IY is also a non-klt centre of(Y,ΣY + sLY ). Note that IY 6= S.

Step 6. In this step we assume X is Q-factorial. We will show t ≤ s. This follows if weshow that some non-klt centre of (S,ΣS + sLS) maps onto C. Let Π be the fibre of π overa general closed point of ν(C) and let H be the corresponding fibre of S → G. Then Π isconnected, and since X is Q-factorial, Π is inside the union of the exceptional divisors ofπ, hence Π ⊂ bΣY c. Moreover, IY intersects Π.

Let E be the connected component ofH which maps into C under S → F . If a componentR of bΣY c − S intersects E, then R ∩ S gives a non-klt centre of (S,ΣS + sLS) mappingonto C (note that (bΣY c − S) ∩ S is vertical over F ). Assume there is no such R. ThenE = H = Π otherwise since Π is connected, there is a connected chain of curves Z1, . . . , Zrin Π such that Zi * E, Z1 intersects E, and the chain connects E to some point in Π \ E:this is not possible because Z1 cannot be inside bΣY c − S, so it is inside S, hence it isinside E as it intersects E, a contradiction. Therefore, IY intersects E, so by inversion ofadjunction IY ∩S produces a non-klt centre of (S,ΣS +sLS) intersecting E, and the centremaps onto C as required. Thus we have proved t ≤ s.

Now since (X,B + sL) is lc near ν(ηC), (W ′, BW ′ + sLW ′) is sub-lc over ν(ηC) whichimplies

µD′BW ′ + s = µD′(BW ′ + sLW ′) ≤ 1

where D′ is as in Step 5. Therefore,

µCBF + t ≤ µC′BF ′ + s ≤ µD′BW ′ + s ≤ 1

which in turn gives µCBF ≤ 1− t = µCΘF .

Step 7. Finally assume X is not Q-factorial and let X → X be a small Q-factorialisation.Let B,∆, G, etc, be the birational transforms of B,∆, G, etc. We can assume Y → X andW ′ → X both factor through X → X. Let F be the normalisation of G. Then we haveinduced morphisms S → F → F , and we can define ΘF whose pushdown to F is just ΘF .Also we can write KF +BF = (KX +B)|F where the pushdown of BF is just BF . Thus it

is enough to show BF ≤ ΘF . Now apply the arguments of Step 6 on X.

3.12. Lifting sections from non-klt centres. We now show that under suitable assump-tions we can lift sections from a lc centre. This is a key ingredient of the proof of 4.7 innext section.

Proposition 3.13. Let d, r ∈ N and ε ∈ R>0. Then there is l ∈ N depending only on d, r, εsatisfying the following. Let (X,B),∆, G, F,ΘF , and PF be as in Theorem 3.9 where G isa general member of a covering family of subvarieties. Assume in addition that• X is Fano of dimension d, and B = 0,• ∆ ∼Q −(n+ 1)KX for some n ∈ N,• h0(−rnKX |F ) 6= 0, and• PF is big and for any choice of PF ≥ 0 in its Q-linear equivalence class the pair

(F,ΘF + PF ) is ε-lc.

Then h0(−lnrKX) 6= 0.

Anti-pluricanonical systems on Fano varieties 25

Proof. Step 1. We will use the notation of the proof of Lemma 3.10. First note thatsince B = 0 and ∆ is a Q-divisor, ΘF is a Q-boundary, so we can assume PF is a Q-divisor. Remember that (X,∆) is lc near the generic point of G. We will show thatunder our assumptions G is actually an isolated non-klt centre of (X,∆). Recall thatbΓY c has a unique component S mapping onto G. Letting ∆Y = ΓY + NY we haveKY +∆Y = π∗(KX +∆) where π denotes Y → X. Let KS +∆S = (KY +∆Y )|S . Applyingadjunction for fibre spaces we get

KS + ∆S ∼Q f∗(KF + ∆F +RF )

where ∆F is the discriminant part and RF is the moduli part. Note that ∆F + RF ∼QΘF + PF .

Assume G is not an isolated non-klt centre. Then some non-klt centre Z 6= S of (Y,∆Y )intersects S by the connectedness principle, hence some component of b∆Y − Sc intersectsS as the non-klt locus of (Y,∆Y ) is equal to b∆Y c. This in turn gives a component ofb∆Sc which is vertical over F as there is a unique non-klt place of (Y,∆Y ) with centre G.Applying Lemma 3.7, there is a high resolution F ′ → F so that some component T of ∆F ′

has coefficient at least 1.

Step 2. Pick a sufficiently small rational number t > 0. Since PF is big, we can assumePF = AF +CF where AF is ample and CF ≥ 0 both being Q-divisors. Let ΩF = ΘF +CFand let KF ′+ΩF ′ be the pullback of KF +ΩF . Then the coefficient of T in tΩF ′+(1−t)∆F ′

is more than 1− ε. As RF ′ is nef (3.6), we can find

0 ≤ JF ′ ∼Q tAF ′ + (1− t)RF ′where AF ′ is the pullback of AF . Therefore,

(F ′, tΩF ′ + (1− t)∆F ′ + JF ′)

is not sub-ε-lc. Moreover, as KF ′ + ∆F ′ + RF ′ is the pullback of KF + ∆F + RF and asKF ′ + ΩF ′ +AF ′ is the pullback of KF + ΩF +AF , we see that

KF ′ + tΩF ′ + (1− t)∆F ′ + JF ′

is Q-linearly trivial over F , hence it is the pullback of

KF + tΩF + (1− t)∆F + JF

where JF is the pushdown of JF ′ . Therefore,

(F, tΩF + (1− t)∆F + JF )

is not ε-lc.On the other hand, we have

tΩF + (1− t)∆F + JF ∼Q tΘF + tCF + (1− t)∆F + tAF + (1− t)RF= tΘF + tPF + (1− t)∆F + (1− t)RF ∼Q ΘF + PF .

Moreover, by construction, ∆F ≥ ΘF , hence

tΩF + (1− t)∆F ≥ tΘF + (1− t)∆F ≥ ΘF .

This is a contradiction. So we have proved that G is an isolated non-klt centre of (X,∆)and that (Y,∆Y ) is plt near S.

Step 3. Let E = π∗(−rnKX). Then E is integral near S because KX is Cartier near thegeneric point of G and because S does not intersect any exceptional divisor of π except Sitself. Let V be a prime divisor on S. If V is horizontal over G, then E is Cartier near the

26 Caucher Birkar

generic point of V . Assume V is vertical over G. We want to show the Cartier index ofE near the generic point of V is bounded. Let p be the Cartier index of KY + S near thegeneric point of V . Then µV ∆S ≥ 1− 1

p and the Cartier index of E near the generic point

of V divides p [38, Proposition 3.9]. Therefore, 1p ≥ ε otherwise we can apply Lemma 3.7

and apply arguments similar to step 1 to choose PF so that (F,ΘF + PF ) is not ε-lc whichwould give a contradiction. This means that p is bounded depending only on ε, hence theCartier index of E near the generic point of V is also bounded.

Step 4. Using the notation of the proof of 3.10, ∆Y = ΓY + NY . Pick 2 ≤ l ∈ N anddefine L by the relation

dlE − bΓY c −NY e = lE − bΓY c −NY + L.

By construction, NY + L is supported in bΓY c − S which is disjoint from S by Step 2. As

KY + ∆Y = π∗(KX + ∆) ∼Q −nπ∗KX

the Q-divisor I = lE − (KY + ∆Y ) is nef and big. We can write

dlE − bΓY c −NY e = lE − bΓY c −NY + L

∼Q KY + ∆Y + I − bΓY c −NY + L = KY + ΓY − bΓY c+ L+ I.

Since (Y,ΓY − bΓY c+ L) is klt, by Kawamata-Viehweg vanishing

h1(dlE − bΓY c −NY e) = 0.

By the previous step, we can choose a bounded l as above such that if U is the largestopen set on which dlE − bΓY c + S −NY e is Cartier, then the codimension of S \ (S ∩ U)in S is at least 2 where we use the fact that bΓY c − S +NY does not intersect S and thatE is integral near S. Thus by Lemma 2.26, we have the exact sequence

0→ OY (dlE − bΓY c −NY e)→ OY (dlE − bΓY c+ S −NY e)→ OS(dlEe|S)→ 0.

NowdlEe|S = lE|S = f∗(−lnrKX |F )

and by assumption, h0(−lnrKX |F ) 6= 0, hence h0(dlEe|S) 6= 0. Therefore,

h0(dlE − bΓY c+ S −NY e) 6= 0

hence h0(dlEe) 6= 0 which implies h0(−lnrKX) 6= 0.

4. Effective birationality

In this section we prove Theorem 1.2 under certain extra assumptions. These specialcases are crucial for the proof of all the main results of this paper.

4.1. Singularities in bounded families.

Proposition 4.2. Let ε ∈ R>0 and let P be a bounded set of couples. Then there is δ ∈ R>0

depending only on ε,P satisfying the following. Let (X,B) be a projective pair and let T bea reduced divisor on X. Assume• (X,B) is ε-lc and (X,SuppB + T ) ∈ P,• L ≥ 0 is an R-Cartier R-divisor on X,• L ∼R N for some R-divisor N supported in T , and• the absolute value of each coefficient of N is at most δ.

Then (X,B + L) is klt.

Anti-pluricanonical systems on Fano varieties 27

Proof. We may assume all the members of P have the same dimension, say d. We provethe proposition by induction on d. Let (X,B), T, L,N be as in the statement for some δ,and assume (X,B + L) is not klt. First assume d = 1. Then degL ≥ ε, hence degN ≥ ε.This is not possible if δ is small enough because degN ≤ δ deg T and deg T is bounded. Sowe can assume d ≥ 2.

We can find a log resolution φ : W → X of (X,B + T ) such that if we write

KW +BW = φ∗(KX +B) + E

where BW ≥ 0 and E ≥ 0 have no common components, and let TW be the sum ofthe birational transform of T and the reduced exceptional divisor φ, then (W,BW ) is ε-lc and (W, SuppBW + TW ) belongs to a bounded set of couples depending only on P.Let LW = φ∗L and NW = φ∗N . Then there is m ∈ N depending only on P so thatthe absolute value of each coefficient of NW is ≤ mδ. Therefore, we can replace P andreplace (X,B), T, L,N, δ with (W,BW ), TW , LW , NW ,mδ, hence assume from now on that(X,SuppB + T ) is log smooth. Moreover, we may replace T so that B and T have nocommon components and T is very ample.

Now assume δ is sufficiently small. Let D be a prime divisor on birational models of Xsuch that a(D,X,B+L) ≤ 0. We show D is not a divisor on X. Since T is very ample andT d is bounded, from T d−1 · L = T d−1 · N ≤ δT d we deduce if l is a coefficient of L, thenl ≤ δT d. This shows that D cannot be a divisor on X as we can assume µD(B + L) < 1.Therefore, D is exceptional/X.

Let V be the centre of D on X. First assume V is not inside SuppB. In this case wecan remove B and assume B = 0. Let H be a general member of |rT | intersecting V wherer is sufficiently large depending on P. Then H is irreducible and smooth, (X,H) is pltbut (X,H + L) is not plt near any component of V ∩H. This implies (H,LH) is not kltnear any component of V ∩H where LH = L|H [31, Theorem 5.50]. Letting NH = N |H ,we see that NH is supported in TH = T |H with absolute value of coefficients ≤ δ. Byconstruction, (H,TH) belongs to a bounded set of couples Q. So applying induction we geta contradiction as δ can be chosen according to Q depending only on P.

Now assume V is inside some component S of B. Let ∆ = B+ (1− b)S where b = µSB.Then (X,∆) is plt and b∆c = S. Moreover, by the arguments of the third paragraph of thisproof we can assume l = µSL < ε ≤ 1−b, hence B+L ≤ ∆+L−lS. Thus S and V are bothnon-klt centres of (X,∆ +L− lS), so it cannot be plt near V . Let KS + ∆S = (KS + ∆)|Sand LS = (L − lS)|S . Then (S,∆S) is ε-lc but (S,∆S + LS) is not klt. Perhaps afteradding some components to T , we can assume S ∼ S′ where S′ is supported in T and withbounded coefficients. By construction, LS ∼R NS := (N − lS′)|S is supported on TS := T |Sand the absolute value of each coefficient of NS is at most nδ where n ∈ N depends onlyon P. Moreover, (S,Supp ∆S + TS) belongs to a bounded set of couples R depending onlyon P. Now applying induction we again get a contradiction.

4.3. Effective birationality for Fano varieties with good Q-complements.

Proposition 4.4. Let d ∈ N and ε, δ ∈ R>0. Then there exists a number v depending onlyon d, ε, and δ satisfying the following. Assume• X is an ε-lc Fano variety of dimension d,• m ∈ N is the smallest number such that | −mKX | defines a birational map,• n ∈ N is a number such that vol(−nKX) > (2d)d, and

28 Caucher Birkar

• nKX +N ∼Q 0 for some Q-divisor N ≥ 0 with coefficients ≥ δ.

Then mn < v.

Proof. Step 1. If the proposition does not hold, then there is a sequence Xi,mi, ni, Ni ofFano varieties, numbers, and divisors as in the statement such that the numbers mi

niform

a strictly increasing sequence approaching ∞. Let φi : Wi → Xi be a log resolution so thatthe movable part of φ∗i (−miKXi) is base point free. Let AWi be a general divisor linearlyequivalent to the movable part and let RWi be the fixed part. So φ∗i (−miKXi) ∼ AWi +RWi .We denote the pushdown of AWi , RWi to Xi by Ai, Ri respectively. Note that RWi is onlya Q-divisor but Ri is integral.

Step 2. In this step i is fixed. Applying 2.16 (2), there is a bounded covering family ofsubvarieties of Xi (as in 2.12) such that for any two general closed points xi, yi ∈ Xi wecan choose a member Gi of the family and choose a Q-divisor 0 ≤ ∆i ∼Q −(ni + 1)KXi

so that (Xi,∆i) is lc near xi with a unique non-klt place whose centre contains xi, thatcentre is Gi, and (Xi,∆i) is not lc near yi. Note that since xi, yi are general, we canassume Gi is a general member of the family. Recall from 2.12 that this means the familyis given by finitely many morphisms V j → T j of projective varieties with accompanyingsurjective morphisms V j → X and that each Gi is a general fibre of one of these morphisms.Moreover, we can assume the points of T j corresponding to such Gi are dense in T j . Letdi := maxdimV j − dimT j.

If di = 0, that is, if dimGi = 0 for all the Gi, then −2(ni + 1)KXi is potentiallybirational, hence |KXi − 2(ni + 1)KXi | defines a birational map [15, Lemma 2.3.4] whichmeans mi ≤ 2ni + 1 giving a contradiction as we can assume mi/ni 0. Thus we canassume di > 0, so there is j such that dimGi > 0 for all the Gi appearing as general fibresof V j → T j .

Define li ∈ N to be the smallest number so that vol(−liKXi |Gi) > dd for all the Giwith positive dimension. Then we can assume there is j so that if Gi is a general fibre ofV j → T j , then Gi is positive dimensional and vol(−(li − 1)KXi |Gi) ≤ dd.

Step 3. Assume lini

is bounded from above by some natural number a. Then after

replacing ni with dani and applying the second paragraph of 2.16 (2), for each i, we canreplace the positive dimensional Gi with new ones of smaller dimension, and replace thefamily accordingly, hence decrease the number di. Repeating the process we get to thesituation in which we can assume li

niis an increasing sequence approaching∞ otherwise we

get the case di = 0 which yields a contradiction as in Step 2. On the other hand, if mili

is

not bounded from above, then we can assume mili

is an increasing sequence approaching∞,

hence we can replace ni with li in which case lini

is bounded and we can argue as before.

So we can assume mili

is bounded from above.In order to get a contradiction in the following steps it suffices to consider only those

Gi which are positive dimensional and vol(−(li − 1)KXi |Gi) ≤ dd. By Step 2, there is asub-family of such Gi appearing as general fibres of some V j → T j . From now on when wemention Gi we assume it is positive dimensional and it satisfies the inequality just stated.In particular,

vol(−miKXi |Gi) = (mi

li − 1)d vol(−(li − 1)KXi |Gi) ≤ (

mi

li − 1)ddd

Anti-pluricanonical systems on Fano varieties 29

is bounded from above, so vol(Ai|Gi) is bounded from above where Ai is as in Step 1.

Step 4. For each i pick a general Gi as in the last paragraph and let Fi be its nor-malisation. By Theorem 3.9 and the ACC for lc thresholds [14, Theorem 1.1], there is aQ-boundary ΘFi with coefficients in a fixed DCC set Φ depending only on d such that wecan write

(KXi + ∆i)|Fi ∼Q KFi + ∆Fi := KFi + ΘFi + PFi

where PFi is pseudo-effective. By replacing ni and adding to ∆i we can assume PFi ≥ 1miMFi

where MFi = Mi|Fi and Mi := Ai +Ri.As Gi is among the general members of a covering family of subvarieties, by Lemma 3.11,

we can write KXi |Fi = KFi + ΛFi for some sub-boundary ΛFi such that (Fi,ΛFi) is sub-ε-lcand ΛFi ≤ ΘFi ≤ ∆Fi .

Step 5. Fix a rational number ε′ ∈ (0, ε) such that ε′ < min Φ>0. Let fi : F′i → Fi be a log

resolution of (Fi,∆Fi) so that the induced map F ′i 99KWi is a morphism. Let AF ′i = AWi |F ′iwhich is big and base point free, and let HF ′i

∈ |6dAF ′i | be general. We define a boundary

ΩF ′ias follows. Let D be a prime divisor on F ′i . Then let the coefficient of D be

µDΩF ′i:=

1− ε′ if D is exceptional/Fi,1− ε′ if D is a component of MF ′i

:= Mi|F ′i ,ε′ if D is a component of Θ∼Fi

but not of MF ′i,

12 if D = HF ′i

,

0 otherwise

where Θ∼Fiis the birational transform of ΘFi . The pair (F ′i ,ΩF ′i

) is log smooth, and by

taking ε′ < 12 , we can assume it is ε′-lc. By Lemma 2.30, KF ′i

+ ΩF ′iis big.

Step 6. The aim of this step is show that vol(KF ′i+ ΩF ′i

) is bounded from above. Thevolume

vol(KFi + ∆Fi + 5dMFi)

is bounded from above because of the relations

KFi + ∆Fi ∼Q −niKXi |Fi , 5dMFi ∼Q −5dmiKXi |Fi

and the fact that vol(−(ni + 5dmi)KXi |Fi) is bounded from above, by the end of Step 3.Next we claim

vol(KFi + ΩFi) ≤ vol(KFi + ∆Fi + 5dMFi)

where ΩFi is the pushdown of ΩF ′i. This follows if we show ∆Fi + 5dMFi − ΩFi is big.

As PFi is big, it is enough to show ΘFi + 5dMFi − ΩFi is big. Note that by Lemma 3.10,µD(ΘFi +MFi) ≥ 1 for any component D of MFi .

Let D be a component of ΩFi . Then either D is a component of MFi or a component ofΘFi or D = HFi the pushdown of HF ′i

. In the first case,

µDΩFi = 1− ε′ < 1 ≤ µD(ΘFi +MFi).

If D is as in the second case but not the first case, then

ε′ = µDΩFi ≤ µDΘFi ≤ µD(ΘFi +MFi).

So we get ΩFi − 12HFi ≤ ΘFi +MFi . On the other hand,

4dMFi −1

2HFi ∼Q 4dAi|Fi + 4dRi|Fi − 3dAFi

30 Caucher Birkar

is big because 4dAi|Fi ≥ 4dAFi where AFi is the pushdown of AF ′i . This implies the bignessof ∆Fi + 5dMFi − ΩFi .

Finally since

vol(KF ′i+ ΩF ′i

) ≤ vol(KFi + ΩFi)

we get the required boundedness of vol(KF ′i+ ΩF ′i

).

Step 7. We want to show (F ′i ,Ω′Fi

) is log birationally bounded. Let ΣF ′i:= Supp ΩF ′i

.First we show

vol(KF ′i+ ΣF ′i

+ 2(2d+ 1)AF ′i )

is bounded from above. Since KF ′i+ ΩF ′i

is big and since the coefficients of ΩF ′ibelong to

ε′, 12 , 1− ε′, there is α ∈ (0, 1) depending only on d and ε′ such that KF ′i

+αΩF ′iis big [14,

Lemma 7.3]. Since ΩF ′i≥ 1

2HF ′iand since each coefficient of (1−α)ΩF ′i

is at least (1−α)ε′,

taking a large number p, say p = 3(2d+1)(1−α)ε′ , we get

vol(KF ′i+ ΣF ′i

+ 2(2d+ 1)AF ′i ) ≤ vol(KF ′i+ ΩF ′i

+ p(1− α)ΩF ′i)

≤ vol(KF ′i+ ΩF ′i

+ p(KF ′i+ αΩF ′i

) + p(1− α)ΩF ′i)

≤ vol((1 + p)(KF ′i+ ΩF ′i

))

which shows the left hand side volume is bounded from above. By construction, AF ′i is base

point free and |AF ′i | defines a birational map. Thus by [15, Lemma 3.2], ΣF ′i· AdimF ′i−1

F ′iis bounded from above. Therefore, (F ′i ,Ω

′Fi

) is log birationally bounded by [15, Lemma2.4.2(4)] as the volume of AF ′i is bounded.

If F ′i → Fi is the contraction defined by AF ′i , then (Fi,ΣFi) is log bounded where ΣFi

is

the pushdown of ΣF ′i. Thus there is a log resolution F i → Fi of (Fi,ΣFi

) such that if ΣF i

is the sum of the reduced exceptional divisor of F i → Fi and the birational transform ofΣFi

, then (F i,ΣF i) is log smooth and log bounded. Now we define a new boundary ΓF ′i by

letting its coefficients to be the same as those of ΩF ′iexcept if D is a component of Θ∼Fi

,

then we let µDΓF ′i = 1− ε′. So ΓF ′i −ΩF ′i≥ 0 and Supp ΓF ′i = Supp ΩF ′i

. Replacing F ′i with

a higher resolution, we can assume the induced map F ′i 99K F i is a morphism. Let ΓF ibe

the pushdown of ΓF ′i . Then ΓF i≤ ΣF i

, hence (F i,ΓF i) log smooth, log bounded, and ε′-lc.

Step 8. Let MF ibe the pushdown of MF ′i

. The pair (F i,SuppMF i) is log bounded

because SuppMF i⊆ Supp ΓF i

as SuppMF ′i⊆ Supp ΓF ′i . We show that the coefficients of

MF iare bounded from above. There is a number b independent of i such that for each i we

can pick an ample Cartier divisor BF iso that bAF i

−BF iis big where AF i

is the pushdownof AF ′i . Let r = dimFi which we can assume to be independent of i, and let BF ′i be thepullback of BF i

. Then

MF i·Br−1

F i= MF ′i

·Br−1F ′i≤ vol(MF ′i

+BF ′i ) ≤ vol(MF ′i+ bAF ′i ) ≤ vol((1 + b)MF ′i

)

because AF ′i ≤MF ′i. Therefore, MF i

·Br−1F i

is bounded from above which implies the coef-

ficients of MF iare bounded from above.

Step 9. Let KF ′i+ ΛF ′i be the pullback of KFi + ΛFi . Since ΛFi ≤ ΘFi , every component

D of ΛF ′i with positive coefficient is either exceptional/Fi or a component of Θ∼Fi. Since

Anti-pluricanonical systems on Fano varieties 31

(F ′i ,ΛF ′i ) is sub-ε-lc, for such D we have

µDΛF ′i ≤ 1− ε < 1− ε′ = µDΓF ′ihence we get ΛF ′i ≤ ΓF ′i .

Let KF ′i+ ∆F ′i

be the pullback of KFi + ∆Fi . Let ∆F ibe the pushdown of ∆F ′i

and letΛF i

be the pushdown of ΛF ′i . Define IF ′i := ∆F ′i− ΛF ′i and let IF i

= ∆F i− ΛF i

be itspushdown. Then

∆F i= ΛF i

+ IF i≤ ΓF i

+ IF i.

Also note that if IFi is the pushdown of IF ′i , then

IFi := ∆Fi − ΛFi ≥ ΘFi − ΛFi ≥ 0

which means IF ′i ≥ 0 and IF i≥ 0.

Step 10. Note that so far we have not used δ and the divisor Ni. Let NFi := Ni|Fi andlet JFi := 1

δNFi . Let D be a component of JFi . Since the coefficients of 1δNi are at least 1,

by Lemma 3.10,µD(∆Fi + JFi) ≥ µD(ΘFi + JFi) ≥ 1.

Thus (Fi,∆Fi + JFi) is not klt, so (F ′i ,∆F ′i+ JF ′i ) is not sub-klt where JF ′i = JFi |F ′i . Let

JF ibe the pushdown of JF ′i . Since KF ′i

+ ∆F ′i+ JF ′i is nef,

KF ′i+ ∆F ′i

+ JF ′i ≤ g∗i (KF i

+ ∆F i+ JF i

)

where gi is the morphism F ′i → F i. Thus (F i,∆F i+JF i

) is not sub-klt. Let LF i:= IF i

+JF i.

Then (F i,ΓF i+ LF i

) is not klt because ΓF i+ LF i

≥ ∆F i+ JF i

.On the other hand, since

IF ′i = ∆F ′i− ΛF ′i ∼Q (KXi + ∆i)|F ′i −KXi |F ′i ∼Q −(ni + 1)KXi |F ′i ∼Q

ni + 1

miMF ′i

and since JF ′i ∼Rnimiδ

MF ′i, we get

LF i∼R (

ni + 1

mi+

nimiδ

)MF i.

Therefore, the coefficients of the right hand side get arbitrarily small when mi grows large.This contradicts Proposition 4.2.

Proposition 4.5. Let d ∈ N and ε, δ ∈ R>0. Then there exists a number m ∈ N dependingonly on d, ε, and δ satisfying the following. Assume X is an ε-lc Fano variety of dimensiond such that KX +B ∼Q 0 for some Q-divisor B ≥ 0 with coefficients ≥ δ. Then | −mKX |defines a birational map.

Proof. If the proposition is not true, then there is a sequence of Fano varieties Xi andQ-divisors Bi satisfying the assumptions of the proposition but such that if mi ∈ N isthe smallest number so that | − miKXi | defines a birational map, then the mi form astrictly increasing sequence approaching ∞. Let ni ∈ N be the smallest number so thatvol(−niKXi) > (2d)d. By Proposition 4.4, there is a number v independent of i such thatmini

< v. Therefore, vol(−miKXi) is bounded from above because we can assume ni > 1hence

vol(−miKXi) = (mi

ni − 1)d vol(−(ni − 1)KXi) ≤ (

mi

ni − 1)d(2d)d.

Let φi : Wi → Xi be a resolution so that φ∗i (−miKXi) ∼ AWi +RWi with AWi base pointfree and RWi the fixed part of φ∗i (−miKXi). We can assume AWi is general in its linear

32 Caucher Birkar

system. We then get miKXi + Ai + Ri ∼ 0 where Ai, Ri will denote the pushdowns ofAWi , RWi . Let EWi be the sum of the reduced exceptional divisor of φi and the support ofthe birational transform of Ri. Define ΣWi = EWi +AWi . Then, by Lemma 2.30,

KWi + ΣWi + 2(2d+ 1)AWi

is big. Moreover its volume is bounded from above because

vol(KXi + Σi + 2(2d+ 1)Ai) ≤ vol(KXi +Ri + (4d+ 3)Ai)

≤ vol((4d+ 3)Ri + (4d+ 3)Ai) ≤ vol(−(4d+ 3)miKXi)

where Σi is the pushdown of ΣWi . Therefore, (Wi,ΣWi) is log birationally bounded by [15,

Lemmas 3.2 and 2.4.2(4)]. More precisely, if Wi → Wi is the contraction defined by AWi ,

then (Wi,ΣWi) is log bounded. Thus perhaps after replacing Wi there is a log resolution

W i → Wi of (Wi,ΣWi) such that the induced map Wi →W i is a morphism and (W i,ΣW i

)is log smooth and log bounded where ΣW i

is the pushdown of ΣWi .

Let KWi + ΛWi be the pullback of KXi and let KW i+ ΛW i

be its pushdown on W i. Thecrepant pullback of

KXi + ∆i := KXi +1

miAi +

1

miRi ∼Q 0

to W i is

KW i+ ∆W i

:= KW i+ ΛW i

+1

miAW i

+1

miRW i

∼Q 0

where AW iand RW i

are pushdowns of AWi and RWi . Note that the coefficients of ΛW iare

at most 1−ε and the support of ∆W iis inside ΣW i

. On the other hand, since vol(AWi+RWi)is bounded from above, one can show the coefficients of AW i

+RW iare bounded from above

by calculating intersection numbers as in Step 8 of the proof of Proposition 4.4. So if mi issufficiently large, then the coefficients of 1

miAW i

+ 1miRW i

are sufficiently small. Therefore,

if we pick ε′ ∈ (0, ε), then there is a boundary ΓW isupported on ΣW i

such that (W i,ΓW i)

is ε′-lc and such that ∆W i≤ ΓW i

for i 0.

Now let Li := 1δBi which has coefficients ≥ 1. Then (Xi,∆i + Li) is not klt and KXi +

∆i +Li is ample. Therefore, if LW iis the pushdown of Li|Wi , then (W i,∆W i

+LW i) is not

sub-klt which in turn implies (W i,ΓW i+LW i

) is not klt. This contradicts Proposition 4.2

because LW i∼R 1

δmiAW i

+ 1δmi

RW i.

4.6. Effective birationality for nearly canonical Fano varieties.

Proposition 4.7. Let d ∈ N. Then there exist numbers τ ∈ (0, 1) and m ∈ N dependingonly on d satisfying the following. If X is a τ -lc Fano variety of dimension d, then |−mKX |defines a birational map.

Proof. Step 1. If the proposition does not hold, then there is a sequence Xi of Fano varietiesof dimension d and an increasing sequence εi of numbers in (0, 1) approaching 1 such that Xi

is εi-lc and if mi ∈ N is the smallest number such that | −miKXi | defines a birational map,then the mi form an increasing sequence approaching∞. Let ni ∈ N be the smallest numberso that vol(−niKX) > (2d)d. First we want to show mi

niis bounded from above. Assume

this is not the case , so we can assume the mini

form an increasing sequence approaching ∞.We will derive a contradiction by the end of Step 4.

Anti-pluricanonical systems on Fano varieties 33

We apply Steps 1-9 of the proof of Proposition 4.4 in our situation. Recall that thosesteps did not use δ and the Ni. We use the notation and assumptions made in those steps.

Step 2. By Lemma 3.11, (Fi,ΛFi) is sub-εi-lc. Pick ε ∈ (0, 1) so that ε < εi for every i.Assume (Fi,∆Fi) is not ε-lc for any i. Then for each i, there is a prime divisor Di on somebirational model of Fi such that a(Di, Fi,∆Fi) < ε. Replacing F ′i we can assume Di is adivisor on F ′i . By assumption, µDi∆F ′i

> 1− ε, and since (Fi,ΛFi) is εi-lc, µDiΛF ′i ≤ 1− εi.So

µDiIF ′i = µDi(∆F ′i− ΛF ′i ) > εi − ε

which is bounded from below. Therefore, if c is a sufficiently large number, then (Fi,∆Fi +cIFi) is not klt for every i as we can assume µDi(∆F ′i

+cIF ′i ) > 1. Moreover, KFi +∆Fi +cIFi

is ample. Thus (F i,∆F i+ cIF i

) is not sub-klt for every i. Letting LFi := (c + 1)IFi , the

pair (F i,ΓF i+ LF i

) is not klt for every i because

∆F i+ cIF i

≤ ΓF i+ LF i

where LF iis the pushdown of LF ′i = LFi |F ′i . We get a contradiction by Proposition 4.2 as

LF i∼R (c+1)(ni+1)

miMF i

and as the coefficients of the right hand side approach 0 when i goes

to ∞. Therefore, from now on we can assume (Fi,∆Fi) is ε-lc for every i. Note that theabove arguments show (Fi,∆Fi) is ε-lc for any choice of PFi ≥ 0 in its R-linear equivalenceclass.

Now since the εi approach 1, we can choose ε close enough to 1 so that 1− ε < s for any0 < s ∈ Φ where Φ is the DCC set to which the coefficients of ΘFi belong. In particular,ΛFi ≤ 0 for every i otherwise if Di is a component of ΛFi with positive coefficient, thenfrom ΛFi ≤ ΘFi ≤ ∆Fi we deduce µDi∆Fi > 1−ε contradicting the ε-lc property of (Fi,∆Fi).

Step 3. By construction,

KF i+ ΛF i

+1

miMF i

∼Q 0

Moreover, any component of ΛF iwith positive coefficient is exceptional/Fi, hence a compo-

nent of ΓF i, and its coefficient in ΛF i

is at most 1−εi. So the coefficients of (ΛF i+ 1miMF i

)≥0

get arbitrarily small as i gets large. As (F i, Supp ΓF i) is log bounded and

Supp(ΛF i+

1

miMF i

)≥0 ⊆ Supp ΓF i,

we can assume KF iis pseudo-effective for every i, by Lemma 2.21.

First assume κσ(KF i) > 0 for every i. Then for each number q there is a number p such

that vol(pKF i+AF i

) > q for every i, by Lemma 2.24. This in turn gives vol(pKFi+AFi) > q.Therefore, both sides of the inequality

vol(mi

ni(KFi + ∆Fi) +AFi) ≥ vol(

mi

niKFi +AFi)

go to ∞ as i goes to ∞. But

vol(mi

ni(KFi + ∆Fi) +AFi) = vol(

mi

ni(−niKXi)|Fi +AFi)

≤ vol((−miKXi +Ai)|Fi) ≤ vol((−2miKXi)|Fi)

and the right hand side is bounded from above, a contradiction. Thus from now on we canassume κσ(KF i

) = 0 for every i.

34 Caucher Birkar

Step 4. Since κσ(KF i) = 0, there is r ∈ N such that h0(rKF i

) 6= 0 for every i, by Lemma

2.19. Then h0(rKFi) 6= 0 for every i, hence rKFi ∼ TFi for some integral divisor TFi ≥ 0.First assume TFi 6= 0 for every i. Then r(KFi + ∆Fi) ∼Q QFi := r∆Fi + TFi . Thus

(r + 1)(KFi + ∆Fi) ∼Q KFi + ∆Fi +QFi

and (Fi,∆Fi + QFi) is not klt as QFi ≥ TFi . Therefore, letting LFi := IFi + QFi , notingLF i∼R (ni+1

mi+ rni

mi)MF i

, and arguing similar to Step 2 we get a contradiction.Now we can assume TFi = 0 for every i. Then

h0(−rKXi |Fi) = h0(−r(KFi + ΛFi)) = h0(−rΛFi) 6= 0

for every i because ΛFi ≤ 0. Thus by Step 2 and Proposition 3.13, perhaps after replacingr with a multiple, h0(−nirKXi) 6= 0 for every i. Then niKXi + Ni ∼Q 0 for some Ni ≥ 0with coefficients ≥ 1

r . We can then apply Proposition 4.4 to deduce that mini

is boundedfrom above, a contradiction.

Step 5. The volume vol(−miKXi) is bounded from above because for each i either ni = 1in which case mi is bounded, or ni > 1 in which case

vol(−miKXi) = (mi

ni − 1)d vol(−(ni − 1)KXi) ≤ (

mi

ni − 1)d(2d)d

is bounded from above. Recall that φi : Wi → Xi is a resolution so that φ∗i (−miKXi) ∼AWi +RWi with AWi base point free and RWi the fixed part of φ∗i (−miKXi). We can assumeAWi is general in its linear system. Since vol(−miKXi) is bounded from above, as in theproof Proposition 4.5, we can assume there exist log bounded log smooth pairs (W i,ΣW i

)

and birational morphisms Wi →W i so that Supp ΣW icontains the exceptional divisors of

Wi 99K Xi and the support of AW i+ RW i

, the pushdown of AWi + RWi . Moreover, thecoefficients of AW i

+RW iare bounded from above.

Let KWi + ΛWi be the pullback of KXi and let KW i+ ΛW i

be its pushdown on W i. The

crepant pullback of KXi + 1miAi + 1

miRi to W i is

KW i+ ΛW i

+1

miAW i

+1

miRW i

∼Q 0.

Note that the coefficients of ΛW iare at most 1− εi which are either negative or approach 0

as i goes to∞. So if i is sufficiently large, then the coefficients of (ΛW i+ 1miAW i

+ 1miRW i

)≥0

are sufficiently small. Thus KW iis pseudo-effective for every i 0, by Lemma 2.21. This

is a contradiction because KWi is not pseudo-effective hence KW iis not pseudo-effective

for every i. Therefore, mi is bounded as required.

5. Proof of Theorem 1.4

Lemma 5.1. Let (X,B),∆, G, F,ΘF , and PF be as in Theorem 3.9 where G is not nec-essarily a general member of a covering family. Assume X is Q-factorial near the genericpoint of G, PF is big, and (X,B + ∆) has a non-klt centre intersecting G. Then for anyε ∈ (0, 1) we can choose PF ≥ 0 so that (F,ΘF + PF ) is not ε-lc.

Proof. Recall (Y,ΓY +NY ) and (S,ΓS +NS) of the proof of Lemma 3.10. Let I 6= G be amaximal non-klt centre intersecting G. We can assume some component of bΓY c−S mapsonto I. Let x ∈ I ∩G be a closed point. Since bΓY c is the non-klt locus of (Y,ΓY + NY ),by the connectedness principle, bΓY c is connected near the fibre of Y → X over x. Thus

Anti-pluricanonical systems on Fano varieties 35

some component of bΓY c − S intersects the fibre of S → G over x, hence (bΓY c − S)|S 6= 0which is vertical over F which in turn implies (S,ΓS +NS) has a divisorial non-klt centrevertical over F . Therefore, by Lemma 3.7, there is a resolution F ′ → F such that if ΩF ′

and RF ′ are the discriminant and moduli parts of adjuction on F ′ for (S,ΓS +NS) over F ,then ΩF ′ has a coefficient ≥ 1 and RF ′ is pseudo-effective. Since ΩF ≥ ΘF and since PF isbig, taking an average (1− t)(ΩF +RF ) + t(ΘF +PF ) for some small t ∈ (0, 1) we can findPF ≥ 0 in its R-linear equivalence class so that (F,ΘF + PF ) is not ε-lc.

Proposition 5.2. Let d ∈ N and ε, δ ∈ R>0. Then there is a number v depending only ond, ε, and δ such that for any X as in Theorem 1.4 we have vol(−KX) < v.

Proof. Step 1. If the statement is not true, then there is a sequence of pairs (Xi, Bi) sat-isfying the properties listed in Theorem 1.4 such that vol(−KXi) is an increasing sequenceapproaching ∞. Taking a Q-factorialisation we can assume Xi is Q-factorial. Note that Xi

is Fano type. Run an MMP on −KXi ∼R Bi and let X ′i be the resulting model. Since Biis big, −KX′i

is nef and big, and since KXi + Bi ∼R 0, (X ′i, B′i) is ε-lc. Thus if X ′i → X ′′i

is the contraction defined by −KX′i, then X ′′i is a Fano. Replacing (Xi, Bi) with (X ′′i , B

′′i ),

we can assume Xi is Fano. Moreover, modifying Bi, we can assume it is a Q-boundary. ByProposition 4.5, there is m ∈ N such that |−mKXi | defines a birational map for every i. Letφi : Wi → Xi be a resolution so that φ∗(−mKXi) decomposes as the sum of a base point freepart AWi and fixed part RWi . Since vol(−KXi) is an increasing sequence approaching ∞,there is a strictly decreasing sequence ai ∈ Q>0 approaching 0 so that vol(−aiKXi) > (2d)d

for each i.

Step 2. In this paragraph, fix i. Applying 2.16 (2), there exists a covering family ofsubvarieties of Xi such that for each pair of general closed points xi, yi ∈ Xi there exist ageneral member Gi of the family and a Q-divisor 0 ≤ ∆i ∼Q −aiKXi such that (Xi,∆i) islc at xi with a unique non-klt place whose centre contains xi, that centre is Gi, and (Xi,∆i)is not lc at yi. Recall from 2.12 that such Gi are among the general fibres of finitely manymorphisms V j → T j , and we can assume for each j the points on T j corresponding to theGi are dense. Since KXi + ∆i is anti-ample, by the connectedness principle, dimGi > 0.Let bi ∈ Q be the smallest number so that vol(−biKXi |Gi) ≥ dd + 1 for all the Gi; we canassume equality is obtained on a subfamily of the Gi which are general fibres of one of themorphisms V j → T j .

Assume bi is not bounded from below, so we can assume bi is a strictly decreasing sequenceapproaching 0. Applying the second paragraph of 2.16 (2), replacing ai with ai+(d−1)bi andreplacing the ∆i, we can replace each Gi with one having smaller dimension. Introducingnew bi as above and repeating the process leads us to the case when bi is bounded frombelow otherwise we get the case dimGi = 0 which is not possible as mentioned above.

In order to get a contradiction, for each i, it is enough to consider a sub-family of theGi satisfying vol(−biKXi |Gi) = dd + 1 and which are general fibres of one of the mor-phisms V j → T j . From now on when we mention Gi we mean one of those. In particular,vol(−KXi |Gi) is bounded from above as bi is bounded from below.

Step 3. For each i pick a general Gi as in the last paragraph, and let Fi be its normali-sation. Let F ′i → Fi be a resolution so that F ′i 99K Wi is a morphism. Since vol(−KXi |Gi)is bounded from above, the volume of AF ′i := AWi |F ′i is also bounded from above. ByTheorem 3.9, there is a Q-boundary ΘFi with coefficients in a fixed DCC set Φ depending

36 Caucher Birkar

only on d such that we can write

(KXi + ∆i)|Fi ∼Q KFi + ∆Fi := KFi + ΘFi + PFi

where we can assume PFi ≥ is big. LetMi := 1δBi and pick ε′ ∈ (0, ε) such that ε′ < min Φ>0.

Define a boundary ΩF ′ias in Step 5 of the proof of Proposition 4.4 using the same notation.

Then KF ′i+ ΩF ′i

is big and the coefficients of ΩFi belong to ε′, 12 , 1− ε′.

We can show vol(KF ′i+ ΩF ′i

) is bounded from above similar to Step 6 of the proof of 4.4by noting that

vol(KFi + ∆Fi + 5dmMFi) = vol(−(ai − 1 +5dm

δ)KXi |Fi)

is bounded from above, and by showing ∆Fi + 5dmMFi − ΩFi is big. Moreover, as in Step7 of the proof of 4.4, letting ΣF ′i

= Supp ΩF ′i, we can show vol(KF ′i

+ ΣF ′i+ 2(2d+ 1)AF ′i )

is bounded from above and deduce (F ′i ,Ω′Fi

) is log birationally bounded as vol(AF ′i ) is

bounded. Therefore, we can assume we have birational morphisms F ′i → F i so that if

ΩF iis the pushdown of ΩF ′i

, then (F i,ΩF i) is log smooth and log bounded. We define a

new boundary ΓF ′i by letting its coefficients to be the same as those of ΩF ′iexcept if D is a

component of Θ∼Fiwe set the coefficient µDΓF ′i to be 1−ε′. Let ΓF i

be the pushdown of ΓF ′i .

Step 4. By the connectedness principle, the non-klt locus of (Xi,∆i) is connected. Since(Xi,∆i) is not lc at some point, the pair has a non-klt centre intersecting Gi but not equalto Gi. By Lemma 5.1, we can choose PFi ≥ 0 so that (Fi,∆Fi) is not ε′-lc, hence we canassume ∆F ′i

has a component Di with coefficient > 1− ε′. On the other hand, by Lemma

3.11, (F ′i ,ΛF ′i ) is sub-ε-lc where KF ′i+ ΛF ′i = KXi |F ′i . Thus letting IF ′i := ∆F ′i

−ΛF ′i we see

that µDiIF ′i > ε− ε′. Therefore, there is c such that (Fi,∆Fi + cIFi) is not klt for every i.Note that by Lemma 3.11, ΛFi ≤ ΘFi ≤ ∆Fi , so IFi ≥ 0.

Now since (Xi, Bi + ∆i) has a unique non-klt place with centre Gi, by Lemma 3.11,(Fi, BFi) is sub-ε-lc where KFi + BFi = (KXi + Bi)|Fi . Note that BFi = ΛFi + Bi|Fi .Moreover,

BFi + (1 + c)IFi = ΛFi +Bi|Fi + ∆Fi − ΛFi + cIFi = ∆Fi + cIFi +Bi|Fi

hence (Fi, BFi + (1 + c)IFi) is not klt. In addition, KFi +BFi + (1 + c)IFi is ample becauseKFi +BFi ∼Q 0 and

IFi = KFi + ∆Fi − (KFi + ΛFi) ∼Q (KXi + ∆i −KXi)|Fi ∼Q −aiKXi |Fi

is ample.On the other hand, since ΛFi ≤ ΘFi , we have

Supp(BFi)≥0 ⊆ Supp(ΘFi +Bi|Fi).

Thus if KF ′i+BF ′i and KF i

+BF iare the crepant pullbacks of KFi +BFi to F ′i and F i, then

by the sub-ε-lc property of (F ′i , BF ′i ) and the construction of ΓF ′i , we get BF ′i ≤ ΓF ′i which

in turn gives BF i≤ ΓF i

. Therefore, (F i,ΓF i+(1+c)IF i

) is not klt as (F i, BF i+(1+c)IF i

)is not sub-klt where IF i

is the pushdown of IF ′i .

Finally, IF ′i ∼Q aiBi|F ′i and the pushdown of Bi|F ′i on F i is supported on ΓF iand has

bounded coefficients by noting Bi|F ′i ∼Q1m(AWi + RWi)|F ′i and calculating intersection

numbers as in Step 8 of the proof of Proposition 4.4. Therefore, we can apply Proposition4.2 to get a contradiction.

Anti-pluricanonical systems on Fano varieties 37

Proof. (of Theorem 1.4) Taking a Q-factorialisation we can assume X is Q-factorial. Runan MMP on −KX and let X ′ be the resulting model. Since B is big, −KX′ is nef and big,that is, X ′ is a weak Fano. It is enough to show such X ′ are bounded because then thereis a klt n-complement of KX′ which in turn gives a klt n-complement of KX for some nindependent of X, hence we can apply [16, Theorem 1.3]. So replacing X with X ′ we canassume X is weak Fano. Now −KX defines a birational contraction X → X ′′ and replacingX with X ′′ we can assume X is Fano. Pick ε′ ∈ (0, ε). Let ∆ = (1 + t)B for some t > 0 sothat (X,∆) is ε′-lc. By [14, Theorem 1.6], it is enough to show (X,∆) is log birationallybounded which is equivalent to showing (X,B) is log birationally bounded.

By Proposition 4.5, there is m ∈ N depending only on d, ε, δ such that | −mKX | definesa birational map. Let φ : W → X be a log resolution so that φ∗(−mKX) decomposes asthe sum of a base point free AW and fixed part RW . By Proposition 5.2, vol(−KX) isbounded from above, hence vol(AW ) is bounded from above. Let ΣW be the sum of thereduced exceptional divisor of φ and the support of the birational transform of B. Thenvol(KW + ΣW + 2(2d+ 1)AW ) is bounded from above because

vol(KX + Σ + 2(2d+ 1)A) ≤ vol(KX +1

δB + 2(2d+ 1)A) ≤ vol(−rKX)

for some r depending only on d, δ,m where Σ and A are the pushdowns of ΣW and AW .Therefore, (W,ΣW ) is log birationally bounded by [15, Lemmas 3.2 and 2.4.2(4)] whichimplies (X,B) is also log birationally bounded as required.

6. Boundedness of complements

In this section we develop the theory of complements for generalised pairs following[37][35][34]. We prove various inductive statements before we come to the main result ofthis section (6.13).

6.1. General remarks. Let (X ′, B′ + M ′) be a projective generalised pair with dataφ : X → X ′ and M .

(1) Assume there is B′+ ≥ B′ such that (X ′, B′+ +M ′) is generalised lc, nM is Cartier,

and n(KX′+B′+ +M ′) ∼ 0 for some n ∈ N. We show KX′+B′+ +M ′ is an n-complement

of KX′ + B′ + M ′. Writing B′ = T ′ + ∆′ where T ′ = bB′c, we need to show nB′+ ≥nT ′ + b(n+ 1)∆′c. Let D′ be a prime divisor and b′ and b′+ be its coefficients in B′ and

B′+. If b′+ = 1, then either b′ = 1 in which case nb′+ = nb′, or b′ < 1 in which casenb′+ = n ≥ b(n+ 1)b′c. So assume b′+ < 1, say b′+ = i

n . Then

nb′+ = i =⌊(n+ 1)b′+

⌋≥⌊(n+ 1)b′

⌋.

(2) Assume X ′ 99K X ′′ is a birational map to a normal projective variety. Replacing Xwe can assume the induced map ψ : X 99K X ′′ is a morphism. Let M ′′ = ψ∗M and assume

φ∗(KX′ +B′ +M ′) + P = ψ∗(KX′′ +B′′ +M ′′)

for some P ≥ 0 and B′′ ≥ 0. Suppose KX′′+B′′+M ′′ has an n-complement KX′′+B

′′++M ′′

with B′′+ ≥ B′′. We claim KX′ + B′ + M ′ has an n-complement KX′ + B′+ + M ′ withB′+ ≥ B′. Let C ′′ = B′′+ − B′′, let C ′ = φ∗(P + ψ∗C ′′), and let B′+ = B′ + C ′. Then

n(KX′ +B′+ + C ′) ∼ 0 and

φ∗(KX′ +B′+ +M ′) = ψ∗(KX′′ +B′′+ +M ′′)

hence (X ′, B′+ + C ′) is generalised lc. Now apply (1).

38 Caucher Birkar

(3) Assume X ′ 99K X ′′ is a partial MMP on −(KX′ + B′ + M ′) and B′′,M ′′ are thepushdowns of B′,M ′. Then there is P ≥ 0 as in (2). Thus if KX′′ + B′′ + M ′′ has an n-

complement KX′′ +B′′+ +M ′′ with B′′+ ≥ B′′, then KX′ +B′+M ′ has an n-complementKX′ +B′+ +M ′ with B′+ ≥ B′.

6.2. Hyperstandard coefficients under adjunction for fibre spaces. The existenceof S in the next result is similar to [34, Lemma 9.3(i)].

Proposition 6.3. Let d ∈ N and R ⊂ [0, 1] be a finite set of rational numbers. AssumeTheorem 1.8 holds in dimension d. Then there exist q ∈ N and a finite set of rationalnumbers S ⊂ [0, 1] depending only on d,R satisfying the following. Assume (X,B) is apair and f : X → Z a contraction such that• (X,B) is projective lc of dimension d, and dimZ > 0,• KX +B ∼Q 0/Z and B ∈ Φ(R),• X is Fano type over some non-empty open subset U ⊆ Z, and• the generic point of each non-klt centre of (X,B) maps into U .

Then we can write

q(KX +B) ∼ qf∗(KZ +BZ +MZ)

where BZ and MZ are the discriminant and moduli parts of adjunction (as in 3.4), BZ ∈Φ(S), and for any high resolution Z ′ → Z the moduli divisor qMZ′ is nef Cartier.

Proof. Step 1. Let q = n be the number given by Theorem 1.8 which depends only on d,R.Then there is a q-complement KX +B+ of KX +B over some point z ∈ U with B+ ≥ B.Since over z we have KX + B ∼Q 0 and q(KX + B+) ∼ 0, and since B+ ≥ B, we haveB+ = B near the generic fibre of f . Therefore, q(KX + B) ∼ 0 over the generic point ofZ, hence there is a rational function α on X such that qL := q(KX + B) + Div(α) is zeroover the generic point of Z. In particular, q(KX + B) ∼ qL and L is vertical/Z. SinceL ∼Q 0/Z, L = f∗LZ for some LZ on Z. Let MZ := LZ − (KZ + BZ) where BZ is thediscriminant part of adjunction for (X,B) over Z. Thus

q(KX +B) ∼ qL = qf∗LZ = qf∗(KZ +BZ +MZ)

and MZ is the moduli part of adjunction for (X,B) over Z. Note that MZ is not unique:it depends on the choice of α and KZ .

Step 2. Our aim until the end of Step 3 is to show the existence of S and to show qMZ isintegral. First we reduce these claims to the case dimZ = 1. Assume dimZ > 1. Let D bea prime divisor on Z. Let H be a general hyperplane section of Z and G its pullback to X.Let KG+BG = (KX +B+G)|G. Since G is a general member of a free linear system, eachnon-klt centre of (G,BG) is a component of the intersection of a non-klt centre of (X,B)with G, hence its generic point maps into U ∩H. Moreover, G is Fano type over U ∩H.Let BH be the discriminant part of adjunction for (G,BG) over H. Let C be a componentof D∩H. Using inversion of adjunction one can show that µDBZ = µCBH (cf. [4, proof ofLemma 3.2]). By Lemma 3.3, there is a finite set of rational numbers T ⊂ [0, 1] dependingonly on R such that BG ∈ Φ(T). So applying induction on dimension, there is a finite setof rational numbers S ⊂ [0, 1] depending only on d − 1,T hence depending only on d,Rsuch that BH ∈ Φ(S). Therefore, BZ ∈ Φ(S).

Let g be the induced map G → H. Pick a general H ′ ∼ H and let KH = (KZ +H ′)|H where the restriction is well-defined as H is a general hyperplane section, and KH is

Anti-pluricanonical systems on Fano varieties 39

determined as a Weil divisor. Letting MH := (LZ +H ′)|H − (KH +BH), we have

q(KG +BG) ∼ qg∗(KH +BH +MH)

hence MH is the moduli part of (G,BG) over H. Moreover, BH + MH = (BZ + MZ)|H ,hence µC(BH +MH) = µD(BZ +MZ) which implies µCMH = µDMZ as µCBH = µDBZ .Therefore, qµDMZ is integral iff qµCMH is integral. So repeating the process we reducethe problem to the case dimZ = 1.

Step 3. In this step we assume Z is a curve. By Lemma 2.5, X is Fano type over Z.Pick a closed point z ∈ Z. Let t be the lc threshold of f∗z with respect to (X,B). LetΓ = B + tf∗z and let (X ′,Γ′) be a Q-factorial dlt model of (X,Γ) so that bΓ′c has acomponent mapping to z. Note that KX′ + Γ′ ∼Q 0/Z. Then there is a boundary B′ ≤ Γ′

such that B′ ∈ Φ(R), bB′c has a component mapping to z, and B∼ ≤ B′ where B∼ is thebirational transform of B. Now X ′ is Fano type over Z and −(KX′ + B′) ∼Q Γ′ − B′/Z.Run an MMP on −(KX′ +B′) over Z and let X ′′ be the resulting model. Then X ′′ is Fanotype over Z, B′′ ∈ Φ(R), and −(KX′′ + B′′) is nef over Z. Moreover, (X ′′, B′′) is lc, as(X ′′,Γ′′) is lc and B′′ ≤ Γ′′. By our choice of q which comes from Theorem 1.8, KX′′ +B′′

has a q-complement KX′′ + B′′+ over z with B′′+ ≥ B′′. Thus by the arguments of 6.1 inthe relative setting, there is a q-complement KX′ +B′+ of KX′ +B′ over z with B′+ ≥ B′.Pushing KX′ + B′+ down to X gives a q-complement KX + B+ of KX + B over z withB+ ≥ B such that (X,B+) has a non-klt centre mapping to z. Now B+ −B ∼Q 0 over z,hence B+ −B is vertical over Z. Thus over z, the divisor B+ −B is just a multiple of thefibre f∗z. Therefore, B+ = B + tf∗z over z as (X,B+) has a non-klt centre mapping to z.

Recall that the coefficient of z in BZ is 1− t. Pick a component S of f∗z and let b andb+ be its coefficients in B and B+. If m is its coefficient in f∗z, then b+ = b + tm, hence

t = b+−bm . Now b = 1− r

l for some r ∈ R and l ∈ N, so t = sm where s = b+−1+ r

l . If b+ = 1,

then t = rlm . If b+ < 1, then as s ≥ 0 and as qb+ is integral we get 1− 1

l ≤ 1− rl ≤ b

+ ≤ q−1q ,

so l ≤ q, hence there are finitely many possibilities for s. This proves the existence of S.Now we show qMZ is integral. By Step 1, q(KX + B) ∼ 0 over some non-empty open

set V ⊆ Z such that SuppBZ ⊆ Z \ V . Let Θ = B +∑

z∈Z\V tzf∗z where tz is the lc

threshold of f∗z with respect to (X,B). If ΘZ is the discriminant part of adjunction for(X,Θ) over Z, then ΘZ = BZ +

∑z∈Z\V tzz, hence ΘZ is a reduced divisor. Moreover, by

the above arguments, KX + Θ is a q-complement of KX + B over each z ∈ Z \ V , henceq(KX + Θ) ∼ 0/Z. Therefore, since

q(KX + Θ) = q(KX +B) + q(Θ−B)

∼ qf∗(KZ +BZ +MZ) + qf∗(ΘZ −BZ) = qf∗(KZ + ΘZ +MZ)

we deduce q(KZ+ΘZ+MZ) is Cartier. This implies qMZ is integral as KZ+ΘZ is integral.

Step 4. It remains to show qMZ′ is nef Cartier where Z ′ → Z is a high resolution anddimZ is arbitrary. The nefness follows from Theorem 3.6, so we just need to show qMZ′

is integral. Let X ′ → X be a log resolution of (X,B) so that X ′ 99K Z ′ is a morphism.Let U0 ⊆ U be an open set over which Z ′ → Z is an isomorphism. Let ∆′ be the sumof the birational transform of B and the reduced exceptional divisor of X ′ → X but withall the components mapping outside U0 removed. We can assume the generic point of anynon-klt centre of (X ′,∆′) maps into U0. Run an MMP on KX′ + ∆′ over Z ′ ×Z X withscaling of some ample divisor. By [5, Theorem 1.9], the MMP terminates over U ′0 ⊂ Z ′, theinverse image of U0. In fact we reach a model X ′′ such that over U ′0 the pair (X ′′,∆′′) is a

40 Caucher Birkar

Q-factorial dlt model of (X,B), hence KX′′ + ∆′′ ∼Q 0 over U ′0 and X ′′ is Fano type overU ′0. Now by [5, Theorem 1.4][14], we can run an MMP/Z ′ on KX′′ + ∆′′ which terminateswith a good minimal model over Z ′ because the generic point of every non-klt centre of(X ′′,∆′′) is mapped into U ′0. Abusing notation, we denote the minimal model again by X ′′

which is Fano type over U ′0.Let f ′′ : X ′′ → Z ′′/Z ′ be the contraction defined by KX′′ + ∆′′. By construction, on a

common resolution W of X and X ′′, the pullbacks of KX + B and KX′′ + ∆′′ are equalover U ′′0 ⊂ Z ′′, the inverse image of U0. Let KX′′ + B′′ and L′′ be the pushdown toX ′′ of the pullback of KX + B and L to W , respectively. Let P ′′ = ∆′′ − B′′ which isvertical and ∼Q 0 over Z ′′, hence it is the pullback of some Q-divisor PZ′′ on Z ′′. Denoteby ∆Z′′ the discriminant part of adjunction on Z ′′ defined for (X ′′,∆′′) over Z ′′. Then∆Z′′ = BZ′′ + PZ′′ where BZ′′ is the discriminant part of adjunction on Z ′′ defined for(X,B) over Z. Moreover,

q(KX′′ + ∆′′) = q(KX′′ +B′′ + P ′′) ∼ q(L′′ + P ′′) = qf ′′∗(LZ′′ + PZ′′)

= qf ′′∗(KZ′′ + ∆Z′′ +MZ′′)

where LZ′′ is the pullback of LZ in Step 1, and MZ′′ = LZ′′ − (KZ′′ + BZ′′) is the modulipart of both (X ′′,∆′′) over Z ′′ and (X,B) over Z. Now by Steps 2-3, qMZ′′ is an integraldivisor, hence qMZ′ is integral as well which means it is Cartier as Z ′ is smooth.

6.4. Pulling back complements from the base of a fibration.

Proposition 6.5. Assume Theorem 1.9 holds in dimension ≤ d−1 and Theorem 1.8 holdsin dimension d. Then Theorem 1.9 holds in dimension d for those (X ′, B′ +M ′) such thatthere is a contraction X ′ → V ′ so that KX′ +B′+M ′ ∼Q 0/V ′, dimV ′ > 0, and M ′ is notbig/V ′.

Proof. Step 1. Replacing (X ′, B′+M ′) with a Q-factorial generalised dlt model as in 2.6(2)and applying 6.1(2), we can assume X ′ is Q-factorial. Since M ′ is not big/V ′, X ′ → V ′ isnot birational. After running an MMP/V ′ on M ′ and applying 6.1(2) once more, we canassume M ′ is semi-ample/V ′. So X ′ → V ′ factors through a contraction f ′ : X ′ → T ′ suchthat dimX ′ > dimT ′ and M ′ ∼Q 0/T ′.

By construction, KX′ + B′ ∼Q 0/T ′, and by assumption X ′ is Fano type. Thus byProposition 6.3 (which needs Theorem 1.8 in dimension d) there exist q ∈ N and a finiteset of rational numbers S ⊂ [0, 1] depending only on d,R such that

q(KX′ +B′) ∼ qf ′∗(KT ′ +BT ′ + PT ′)

where BT ′ and PT ′ are the discriminant and moduli divisors of adjunction for fibre spacesapplied to (X ′, B′) over T ′, and such that BT ′ ∈ Φ(S) and qPT is nef Cartier for any highresolution T → T ′. We can assume q is divisible by p.

Step 2. Pick a sufficiently high resolution ψ : T → T ′ so that the moduli part PT isnef and it satisfies the pullback property of Theorem 3.6 (ii). We consider (T ′, BT ′ + PT ′)as a generalised pair with data ψ : T → T ′ and PT . It is generalised lc because (X ′, B′)is lc. Replace X so that the induced map f : X 99K T is a morphism. Since M is nef,φ∗M ′ = M + E for some exceptional/X ′ and effective Q-divisor E. Since M ′ ∼Q 0/T ′, Eis vertical/T ′, so there is a non-empty open subset of T ′ over which E = 0 and M ∼Q 0.Since X ′ is Fano type, the general fibres of f ′ are also Fano type, hence they are rationallyconnected which in turn implies the general fibres of f are rationally connected [13][40].Thus perhaps after replacing X and T and applying Lemma 2.28, pM ∼ pf∗MT for a Q-divisor MT on T so that pMT is nef Cartier. Moreover, since E is vertical and ∼Q 0 over T ,

Anti-pluricanonical systems on Fano varieties 41

E = f∗ET for some effective Q-divisor ET . On the other hand, since E is exceptional/X ′,applying [5, Lemma 3.2] to E over T ′ shows that E is very exceptional over T ′ which inturn implies ET is exceptional over T ′. In particular, this means MT ′ := ψ∗MT is Q-Cartieras MT + ET ∼Q 0/T ′. By construction, q(PT +MT ) is nef Cartier and qM ′ ∼ qf ′∗MT ′ .

Step 3. Now we consider (T ′, BT ′+PT ′+MT ′) as a generalised pair with data ψ : T → T ′

and PT +MT . We show it is is generalised lc. We can assume (T,BT + ET ) is log smoothwhere BT is the discriminant divisor on T . By construction

KT +BT + ET + PT +MT = ψ∗(KT ′ +BT ′ + PT ′ +MT ′).

So it is enough to show (T,BT + ET ) is sub-lc which in turn is equivalent to saying thatevery coefficient of BT + ET is ≤ 1. Let KX + B be the pullback of KX′ + B′. Let Dbe a prime divisor on T . By definition of the discriminant divisor, µDBT = 1 − tD wheretD is the largest number so that (X,B + tDf

∗D) is sub-lc over the generic point of D.Since (X ′, B′ + M ′) is generalised lc, (X,B + E) is sub-lc, and since E = f∗ET , we haveµDET ≤ tD which implies µDBT + µDET ≤ 1.

Step 4. To summarise we have proved: (T ′, BT ′+PT ′+MT ′) is generalised lc, BT ′ ∈ Φ(S),q(PT +MT ) is Cartier, and

q(KX′ +B′ +M ′) ∼ qf ′∗(KT ′ +BT ′ + PT ′ +MT ′)

is anti-nef. Moreover, T ′ is Fano type as X ′ is Fano type (this follows from [2]). Now byinduction on dimension, KT ′+BT ′+PT ′+MT ′ has an n-complementKT ′+B

+T ′+PT ′+MT ′ for

some n divisible by qI(S) and depending only on dimT ′, q,S such that GT ′ := B+T ′−BT ′ ≥

0. So n depends only on d, p,R. Denote the pullback of GT ′ to T,X,X ′ by GT , G,G′

respectively. Let B′+ = B′ +G′. Then

n(KX′ +B′+

+M ′) = n(KX′ +B′ +M ′ +G′)

∼ nf ′∗(KT ′ +BT ′ + PT ′ +MT ′ +GT ′) = nf ′∗(KT ′ +B+T ′ + PT ′ +MT ′) ∼ 0.

Thus by 6.1(1), KX′+B′++M ′ is an n-complement of KX′+B

′+M ′ if we show (X ′, B′++M ′) is generalised lc.

Let C be a prime divisor on some birational model of X ′. Replacing X,T we canassume C is a divisor on X and that its image on T is a divisor, say D. The pullback ofKX′ +B′+ +M ′ to X is KX +B +E +G+M . It is enough to show µC(B +E +G) ≤ 1.Since (T ′, B+

T ′ + PT ′ +MT ′) is generalised lc, and since

KT +BT + ET +GT + PT +MT = ψ∗(KT ′ +B+T ′ + PT ′ +MT ′)

we have µD(BT +ET +GT ) ≤ 1. Letting tD be as in Step 3 we get µD(ET +GT ) ≤ tD whichimplies (X,B+E+G) is sub-lc over the generic point of D. Therefore, µC(B+E+G) ≤ 1as required.

6.6. Lifting complements from a non-klt centre.

Proposition 6.7. Assume Theorem 1.9 holds in dimension d−1. Then Theorem 1.9 holdsin dimension d for those (X ′, B′ +M ′) such that• B′ ∈ R,• (X ′,Γ′ + αM ′) is Q-factorial generalised plt for some Γ′ and α ∈ (0, 1),• −(KX′ + Γ′ + αM ′) is ample, and• S′ = bΓ′c is irreducible and it is a component of bB′c.

42 Caucher Birkar

Proof. Step 1. The idea is to construct a complement on S′ and then lift it to X ′. We canassume the given map φ : X → X ′ is a log resolution and that the induced map ψ : S 99K S′

is a morphism where S is the birational transform of S′. By generalised adjunction we canwrite

KS′ +BS′ +MS′ = (KX′ +B′ +M ′)|S′ .By Lemma 3.3, BS′ ∈ Φ(S) for some finite set of rational numbers S ⊂ [0, 1] which onlydepends on p,R. Restricting KX′ + Γ′ + αM ′ to S′ shows that S′ is Fano type, by 2.6(5).On the other hand, pMS = pM |S is Cartier. Thus by Theorem 1.9 in dimension d−1, thereis n ∈ N divisible by pI(S) which depends only on d−1, p,S such that KS′+BS′+MS′ hasan n-complement KS′ + B+

S′ + MS′ with B+S′ ≥ BS′ . Then n depends only on d, p,R and

replacing it with nI(R) we can assume it is divisible by pI(R). In particular, nB′ is integralas B′ ∈ R. We will show there is an n-complement KX′ +B′+ +M ′ of KX′ +B′+M ′ withB′+ ≥ B′.

Step 2. Write

N := −(KX +B +M) := −φ∗(KX′ +B′ +M ′)

and let T =⌊B≥0

⌋and ∆ = B − T . Define

L := −nKX − nT − b(n+ 1)∆c − nM

which is an integral divisor. Note that

L = n∆− b(n+ 1)∆c+ nN.

Now write

KX + Γ + αM = φ∗(KX′ + Γ′ + αM ′).

Replacing Γ′ with (1−a)Γ′+aB′ and replacing αM with ((1−a)α+a)M for some a ∈ (0, 1)sufficiently close to 1, we can assume α is sufficiently close to 1 and B − Γ has sufficientlysmall (positive or negative) coefficients.

Step 3. Let P be the unique integral divisor so that

Λ := Γ + n∆− b(n+ 1)∆c+ P

is a boundary, (X,Λ) is plt, and bΛc = S (in particular, we are assuming Λ ≥ 0). Moreprecisely, we let µSP = 0 and for each prime divisor D 6= S, we let

µDP = −µD bΓ + n∆− b(n+ 1)∆cc

which satisfies

µDP = −µD bΓ−∆ + 〈(n+ 1)∆〉cwhere 〈(n + 1)∆〉 is the fractional part of (n + 1)∆. This implies 0 ≤ µDP ≤ 1 for anyprime divisor D: indeed we can assume D 6= S; if D is a component of T , then D is nota component of ∆ but µDΓ ∈ (0, 1), hence µDP = 0; if D is not a component of T , thenµD(Γ−∆) = µD(Γ−B) is sufficiently small, hence 0 ≤ µDP ≤ 1.

We show P is exceptional/X ′. AssumeD is a component of P which is not exceptional/X ′.Then D 6= S, and since nB′ is integral, µDn∆ is integral, hence µD b(n+ 1)∆c = µDn∆which implies µDP = −µD bΓc = 0, a contradiction.

Step 4. Let

A′ = −(KX′ + Γ′ + αM ′)

Anti-pluricanonical systems on Fano varieties 43

and let A = φ∗A′. Then

L+ P = n∆− b(n+ 1)∆c+ nN + P

= KX + Γ + αM +A+ n∆− b(n+ 1)∆c+ nN + P

= KX + Λ +A+ αM + nN

Since A+αM +nN is nef and big, h1(L+P −S) = 0 by the Kawamata-Viehweg vanishingtheorem, hence

H0(L+ P )→ H0((L+ P )|S)

is surjective.

Step 5. Let RS′ := B+S′ −BS′ which satisfies

−n(KS′ +BS′ +MS′) ∼ nRS′ ≥ 0.

Letting RS be the pullback of RS′ we get

nNS := nN |S = −n(KS +BS +MS) = −nψ∗(KS′ +BS′ +MS′) ∼ nRS ≥ 0.

By construction,

(L+ P )|S = (n∆− b(n+ 1)∆c+ nN + P )|S∼ GS := nRS + n∆S − b(n+ 1)∆Sc+ PS

where ∆S = ∆|S and PS = P |S .We show GS ≥ 0. Assume C is a component of GS with negative coefficient. Then there

is a component D of n∆−b(n+ 1)∆c with negative coefficient such that C is a componentof D|S . But

µC(n∆S − b(n+ 1)∆Sc) = µC(−∆S + 〈(n+ 1)∆S〉) ≥ −µC∆S = −µD∆ > −1

which gives µCGS > −1 and this in turn implies µCGS ≥ 0 because GS is integral, acontradiction. Therefore GS ≥ 0, and by Step 4, L + P ∼ G for some effective divisor Gwhose support does not contain S and G|S = GS .

Step 6. By the previous step,

−nKX′ − nT ′ −⌊(n+ 1)∆′

⌋− nM ′ = L′ = L′ + P ′ ∼ G′ ≥ 0

where L′ is the pushdown of L, etc. Since nB′ is integral, b(n+ 1)∆′c = n∆′, so

nN ′ = −n(KX′ +B′ +M ′) = −nKX′ − nT ′ − n∆′ − nM ′ = L′ ∼ nR′ := G′ ≥ 0.

Let B′+ = B′ +R′. Then n(KX′ +B′+ +M ′) ∼ 0.

By 6.1 (1), it remains to show (X ′, B′+ +M ′) is generalised lc. First we show

KS′ +B+S′ +MS′ = (KX′ +B′

++M ′)|S′

which is equivalent to showing R′|S′ = RS′ . Since

nR := G− P + b(n+ 1)∆c − n∆ ∼ L+ b(n+ 1)∆c − n∆ = nN ∼Q 0/X ′

and since b(n+ 1)∆′c − n∆′ = 0 as n∆′ is integral, we get φ∗nR = G′ = nR′ and that Ris the pullback of R′. Now

nRS = GS − PS + b(n+ 1)∆Sc − n∆S

= (G− P + b(n+ 1)∆c − n∆)|S = nR|Swhich means RS = R|S , hence RS′ = R′|S′ .

44 Caucher Birkar

By generalised inversion of adjunction (3.2), (X ′, B′+ +M ′) is generalised lc near S′. Let

Ω′ := aB′+

+ (1− a)Γ′ and F = (a+ (1− a)α)M

for some a ∈ (0, 1) close to 1. If (X ′, B′+ + M ′) is not generalised lc away from S′, then(X ′,Ω′+F ′) is also not generalised lc away from S′. But then KX′ + Ω′+F ′ is anti-ampleand its generalised non-klt locus has at least two disjoint components one of which is S′.This contradicts the connectedness principle (2.7). Thus (X ′, B′+ +M ′) is generalised lc.

Proposition 6.8. Assume Theorem 1.9 holds in dimension ≤ d−1 and Theorem 1.8 holdsin dimension d. Then Theorem 1.9 holds in dimension d for those (X ′, B′ +M ′) such that• B′ ∈ R,• (X ′, B′ +M ′) is not generalised klt, and• either KX′ +B′ +M ′ 6∼Q 0 or M ′ 6∼Q 0.

Proof. Step 1. Taking a Q-factorial generalised dlt model of (X ′, B′ +M ′) we can assumeX ′ is Q-factorial and that (X ′, B′) is not klt. Let X ′ → Z ′ be the contraction defined by−(KX′ + B′ +M ′). Running an MMP on M ′ over Z ′ and replacing X ′ with the resultingmodel we can assume M ′ is nef/Z ′. Note that since the MMP is an MMP on −(KX′ +B′),the non-klt property of (X ′, B′) is preserved. Let X ′ → V ′/Z ′ be the contraction definedby M ′. If dimZ ′ > 0, then dimV ′ > 0. If dimZ ′ = 0, then again dimV ′ > 0 since in thiscase M ′ 6∼Q 0. In particular, if M ′ is not big over Z ′, then we can apply Proposition 6.5.

From now on we can assume M ′ is nef and big over Z ′. Then

−(KX′ +B′ + αM ′) = −(KX′ +B′ +M ′) + (1− α)M ′

is globally nef and big for some rational number α < 1 close to 1 which will be fixedthroughout the proof. The contraction defined by −(KX′ + B′ + αM ′) is nothing butX ′ → V ′. After running an MMP on B′ over V ′ we can assume B′ is nef over V ′, hence

−(KX′ + βB′ + αM ′) = −(KX′ +B′ + αM ′) + (1− β)B′

is also globally nef and big for any rational number β satisfying α β < 1. Note thatsince the latter MMP is KX′+B

′-trivial, the non-klt property of (X ′, B′) is again preserved.

Step 2. Since X ′ is Fano type and Q-factorial, (X ′, 0) is klt. Thus since (X ′, B′ + M ′)is generalised lc, (X ′, βB′ + αM ′) is generalised klt. Let (X ′′, B′′) be a Q-factorial dltmodel of (X ′, B′), and let M ′′ be the pullback of M ′. Writing the pullback of KX′ + βB′

as KX′′ + B′′, perhaps after increasing β, we can assume the coefficients of B′′ − B′′ aresufficiently small. Replacing X ′, B′,M ′ with X ′′, B′′,M ′′ and renaming B′′ to B′, we have:(X ′, B′+αM ′) is generalised klt, −(KX′ + B′+αM ′) is nef and big, and the coefficients of

B′− B′ are sufficiently small. Moreover, every generalised non-klt centre of (X ′, B′+αM ′)is a non-klt centre of (X ′, B′): if D is a prime divisor on birational models of X ′ such thata(D,X ′, B′ + αM ′) = 0, then a(D,X ′, B′) = 0 because

a(D,X ′, B′ + αM ′) = αa(D,X ′, B′ +M ′) + (1− α)a(D,X ′, B′).

Step 3. Write

−(KX′ +B′ + αM ′) ∼Q A′ +G′

Anti-pluricanonical systems on Fano varieties 45

where A′, G′ ≥ 0 are Q-divisors and A′ is ample. First assume that SuppG′ does notcontain any generalised non-klt centre of (X ′, B′ + αM ′). Then, for some small δ > 0,

−(KX′ +B′ + αM ′ + δG′) ∼Q (1− δ)( δ

1− δA′ +A′ +G′)

is ample and (X ′, B′+ δG′+αM ′) is generalised lc whose generalised non-klt locus is equalto the generalised non-klt locus of (X ′, B′ + αM ′) which is in turn equal to the non-kltlocus of (X ′, B′). Pick a component S′ of bB′c and let Γ′ = S′ + a(B′ − S′ + δG′) for somea < 1 close to 1. Then (X ′,Γ′+αM ′) is generalised plt, bΓ′c = S′, and −(KX′ + Γ′+αM ′)is ample. Now apply Proposition 6.7.

Step 4. From now on we assume that SuppG′ contains some generalised non-klt centreof (X ′, B′ + αM ′). Let t be the generalised lc threshold of G′ + B′ − B′ with respect to

(X ′, B′ + αM ′) and let Ω′ := B′ + t(G′ + B′ − B′). As (X ′, B′ + αM ′) is generalised klt,t > 0. We can assume the given morphism φ : X → X ′ is a log resolution of (X ′, B′ +G′).Write

KX +Bα + αM = φ∗(KX′ +B′ + αM ′)

andKX + Bα + αM = φ∗(KX′ + B′ + αM ′)

from which we get Bα− Bα = φ∗(B′− B′). Perhaps after replacing B′ with bB′+ (1− b)B′for some small b > 0, we can assume the coefficients of Bα − Bα are sufficiently small.

Let G = φ∗G′. Since SuppG′ contains some generalised non-klt centre of (X ′, B′+αM ′),

we can assume G and bBαc≥0 have a common component, say T . Now

KX + Bα + t(Bα − Bα +G) + αM = φ∗(KX′ + Ω′ + αM ′).

Since µT Bα is sufficiently close to µTBα = 1, we deduce t is sufficiently small. Moreover,letting Ω = Bα + t(Bα − Bα +G) we have

Ω ≤ Bα + tG and bΩc≥0 ⊆ bBα + tGc≥0 = bBαc≥0 .We will use these observations in Step 6.

Step 5. We show that −(KX′ + Ω′ + αM ′) is ample. By construction

−(KX′ + B′ + αM ′) = −(KX′ +B′ + αM ′) +B′ − B′ ∼Q A′ +G′ +B′ − B′.Thus

−(KX′ + Ω′ + αM ′) = −(KX′ + B′ + t(G′ +B′ − B′) + αM ′)

= −(KX′ + B′ + αM ′)− t(G′ +B′ − B′)∼Q A′ +G′ +B′ − B′ − t(G′ +B′ − B′)

= (1− t)( t

1− tA′ +A′ +G′ +B′ − B′)

which is ample.

Step 6. Assume bΩ′c 6= 0 and pick a component S′ of bΩ′c. By Step 4, S′ is a componentof bB′c. We then define Γ′ similar to Step 3 by perturbing the coefficients of Ω′, say byletting Γ′ = S′ + a(Ω′ − S′) for some a < 1 close to 1, so that bΓ′c = S′, (X ′,Γ′ + αM ′) isgeneralised plt, and −(KX′ + Γ′ + αM ′) is ample. Then we apply Proposition 6.7.

Now assume bΩ′c = 0. Let Ω be the sum of the birational transform of Ω′ and thereduced exceptional divisor of X → X ′. So Ω−Ω is effective and exceptional/X ′. Running

46 Caucher Birkar

an MMP/X ′ on KX +Ω+αM contracts all the components of Ω−Ω as KX +Ω+αM ≡Ω−Ω/X ′, hence we reach a model X ′′/X ′ such that if Ω′′ and M ′′ are the pushdowns of Ω

and M , then (X ′′,Ω′′+αM ′′) is a Q-factorial generalised dlt model of (X ′,Ω′+αM ′). Theexceptional prime divisors of X ′′ → X ′ all have coefficient 1 in Ω′′. Moreover, any primeexceptional divisor D of X → X ′ not contracted over X ′′ is a component of bΩc≥0, hence a

component of bBαc≥0, by Step 4. Thus if B′′ is the pushdown of Bα, then KX′′+B′′+αM ′′

is the pullback of KX′ +B′+αM ′ to X ′′, and B′′ is the sum of the birational transform ofB′ and the reduced exceptional divisor of X ′′ → X ′. Since (X ′, B′ +M ′) is generalised lc,M ′′ is the pullback of M ′ and KX′′ +B′′ +M ′′ is the pullback of KX′ +B′ +M ′.

Let ∆′′ be the sum of the birational transform of B′ and the reduced exceptional divisorof X ′′ → X ′. Note that ∆′′ ≤ Ω′′ and (X ′′, ∆′′+αM ′′) is generalised dlt. Run an MMP/X ′

on KX′′ + ∆′′ + αM ′′. The MMP ends with X ′ because (X ′, B′ + αM ′) is Q-factorialgeneralised klt. The last step of the MMP is a divisorial contraction which contracts acomponent S′′ of bΩ′′c. Abuse notation and replace X ′′ → X ′ with that last contraction.

By construction, (X ′′, ∆′′+αM ′′) is generalised plt and −(KX′′+∆′′+αM ′′) is ample overX ′. Defining

Γ′′ = a∆′′ + (1− a)Ω′′

for a sufficiently small a > 0 we can check that (X ′′,Γ′′+αM ′′) is generalised plt, S′′ = bΓ′′c,and −(KX′′ + Γ′′ + αM ′′) is globally ample because

−(KX′′ + Γ′′ + αM ′′) = −a(KX′′ + ∆′′ + αM ′′)− (1− a)(KX′′ + Ω′′ + αM ′′)

and because −(KX′′ + Ω′′+αM ′′) is the pullback of the ample divisor −(KX′ + Ω′+αM ′).Now apply Proposition 6.7 to KX′′ +B′′ +M ′′.

Lemma 6.9. Assume Theorem 1.9 holds in dimension ≤ d− 1 and Theorem 1.8 holds indimension d. Then Theorem 1.9 holds in dimension d for those (X ′, B′ +M ′) such that• B′ ∈ R, and• (X ′, B′ +M ′) is strongly non-exceptional.

Proof. By assumption there is P ′ ≥ 0 such that KX′ + B′ + M ′ + P ′ ∼R 0 and (X ′, B′ +P ′ + M ′) is not generalised lc. In particular, P ′ 6= 0. Modifying P ′ we can replace ∼Rwith ∼Q. Let t be the generalised lc threshold of P ′ with respect to (X ′, B′ + M ′). Thent < 1. Let Ω′ = B′ + tP ′ and let (X ′′,Ω′′ + M ′′) be a Q-factorial generalised dlt modelof (X ′,Ω′ + M ′). Let Θ′′ = Ω′′ − tP ′∼ where P ′∼ is the birational transform of P ′. Let πdenote X ′′ → X ′ and let P ′′ be the pullback of P ′. Then Θ′′ ∈ R, X ′′ is Fano type, and

−(KX′′ + Θ′′ +M ′′) = −(KX′′ + Ω′′ − tP ′∼ +M ′′)

= tP ′∼ − π∗(KX′ + Ω′ +M ′) = tP ′∼ − tP ′′ − π∗(KX′ +B′ +M ′)

∼Q tP ′∼ + (1− t)P ′′.Run an MMP on −(KX′′ + Θ′′ + M ′′) and let X ′′′ be the resulting model. Since P ′′ ≥ 0is nef and non-zero, its pushdown P ′′′ 6= 0, hence −(KX′′′ + Θ′′′ + M ′′′) is nef but 6∼Q 0.Moreover, since P ′′ is semi-ample, there is Q′′ ≥ 0 such that KX′′ + Ω′′ + Q′′ + M ′′ ∼Q 0and (X ′′,Ω′′+Q′′+M ′′) is generalised lc. Therefore, (X ′′′,Θ′′′+M ′′′) is generalised lc butnot generalised klt.

By 6.1 (3), if KX′′′+Θ′′′+M ′′′ has an n-complement KX′′′+Θ′′′++M ′′′ with Θ′′′+ ≥ Θ′′′,

then KX′′ + Θ′′ + M ′′ has an n-complement KX′′ + Θ′′+ + M ′′ with Θ′′+ ≥ Θ′′ which inturn gives an n-complement KX′ +B′+ +M ′ of KX′ +B′+M ′ with B′+ ≥ B′. Now applyProposition 6.8 to KX′′′ + Θ′′′ +M ′′′.

Anti-pluricanonical systems on Fano varieties 47

Lemma 6.10. Assume Theorem 1.9 holds in dimension ≤ d− 1 and Theorem 1.8 holds indimension d. Then Theorem 1.9 holds in dimension d for those (X ′, B′ +M ′) such that• B′ ∈ R, and• (X ′, B′ +M ′) is non-exceptional.

Proof. By assumption there is P ′ ≥ 0 such that KX′ + B′ + M ′ + P ′ ∼R 0 and (X ′, B′ +P ′ +M ′) is not generalised klt. Modifying P ′ we can replace ∼R with ∼Q. We can assume(X ′, B′ +P ′ +M ′) is generalised lc otherwise (X ′, B′ +M ′) is strongly non-exceptional, sowe can apply Lemma 6.9. Applying the construction in the proof of Lemma 6.9, by takingt = 1, and replacing (X ′, B′+M ′) with (X ′′′,Θ′′′+M ′′′), we can assume (X ′, B′+M ′) is notgeneralised klt and that X ′ is Q-factorial. Applying Proposition 6.8, we can further assumeM ′ ∼Q 0 and that KX′ + B′ + M ′ ∼Q 0. In particular, KX′ + B′ ∼Q 0, M = φ∗M ′ ∼Q 0,and (X ′, B′) is not klt.

Since X ′ is Fano type, Pic(X ′) and Pic(X) are torsion-free (cf. [18, Proposition 2.1.2]).Therefore, if r is the Cartier index of KX′ + B′ + M ′, then r(KX′ + B′ + M ′) ∼ 0, henceKX′ + B′ + M ′ is an rp-complement of itself. Thus it is enough to show r is bounded.Moreover, pM ∼ 0, so pM ′ ∼ 0, hence we only need to show the Cartier index of KX′ +B′

is bounded.By Lemma 2.32, there is ε ∈ (0, 1) depending only on d,R such that if D is any prime

divisor on birational models ofX ′ with a(D,X ′, 0) < ε, then a(D,X ′, B′) = 0. LetX ′′ → X ′

be the birational contraction which extracts exactly those D with a(D,X ′, 0) < ε. ThenX ′′ is Fano type and ε-lc. Moreover, if KX′′ +B′′ is the pullback of KX′ +B′, then all theexceptional divisors of X ′′ → X ′ appear in B′′ with coefficient 1. Replacing (X ′, B′) with(X ′′, B′′) we can assume X ′ is ε-lc. After running an MMP on KX′ we can assume we havea Mori fibre structure X ′ → T ′. Applying Proposition 6.5, we can assume dimT ′ = 0, soX ′ is an ε-lc Fano.

By Lemma 6.9, there is a number n depending only on d such that if Y ′ is any stronglynon-exceptional Fano variety of dimension d with klt singularities, then KY ′ has an n-complement. We can assume pI(R)|n. On the other hand, by Proposition 4.5, there ism ∈ N depending only on d, ε,R such that | −mKX′ | defines a birational map. Replacingn once more we can assume m|n. So | − nKX′ | also defines a birational map. Replacingφ : X → X ′ we can assume φ∗(−nKX′) is linearly equivalent to the sum of a generalbase point free divisor A and fixed part R. Then n(KX′ + 1

nR′ + 1

nA′) ∼ 0. We claim

(X ′, 1nR′ + 1

nA′) is lc. If not, then (X ′, 0) is strongly exceptional, hence by our choice of

n we have an n-complement KX′ + C ′+ of KX′ . Since nC ′+ ∈ | − nKX′ | and (X ′, C ′+) islc, we deduce (X ′, 1nR

′ + 1nA′) is also lc because A′ +R′ ∈ | − nKX′ | is a general member.

This is a contradiction.Now let ∆′ = 1

2B′ + 1

2nR′ and N ′ = 1

2nA′. Since

2n(KX′ + ∆′ +N ′) ∼ n(KX′ +B′)

it is enough to show the Cartier index of KX′ + ∆′+N ′ is bounded. Assume (X ′,∆′+N ′)is klt. Let ε′ = min ε2 ,

12n. We claim (X ′,∆′ + N ′) is ε′-lc. If not, then there is some

prime divisor D with 0 < a(D,X ′,∆′ + N ′) < ε′. Then either 0 < a(D,X ′, B′) whichimplies ε ≤ a(D,X ′, B′) by Lemma 2.32, or 0 < a(D,X ′, 1nR

′ + 1nA′) which implies 1

n ≤a(D,X ′, 1nR

′ + 1nA′). In either case we get

ε′ ≤ a(D,X ′,∆′ +N ′) =1

2a(D,X ′, B′) +

1

2a(D,X ′,

1

nR′ +

1

nA′)

48 Caucher Birkar

a contradiction. So (X ′,∆′+N ′) is ε′-lc. Therefore, X ′ belongs to a bounded family by [14,Corollary 1.7] as the coefficients of ∆′ + N ′ belong to a fixed finite set, hence the Cartierindex of KX′ + ∆′ +N ′ is bounded as required.

Now we can assume (X ′,∆′ + N ′) is not klt. Consider this pair as a generalised pairwith data X → X ′ and N = 1

2nA. It is not generalised klt and obviously N ′ 6∼Q 0, henceby Proposition 6.8, KX′ + ∆′ + N ′ is an l-complement of itself for some bounded l whichimplies the Cartier index of KX′ + ∆′ +N ′ is at most l.

6.11. Boundedness of complements.

Construction 6.12 Let (X ′, B′+M ′) be as in Theorem 1.9 and assume X ′ is Q-factorial.Let ε ∈ (0, 1). Since X ′ is Fano type, −KX′ is big, hence running an MMP on −KX′ ends

with a model X ′ such that −KX′ is nef and big. Take a general Q-divisor C ′ ∼Q −KX′

with coefficient ≤ 1−ε. Let KX′+C ′ be the crepant pullback of KX′+ C ′. Since X ′ 99K X ′

is also the process of an MMP on C ′ and since C ′ ≥ 0, we get C ′ ≥ 0.Let D be a prime divisor on birational models of X ′ such that a(D,X ′, C ′) < ε. We

claim a(D,X ′, B′ +M ′) < ε. Indeed if P ′ ∼Q −(KX′ +B′ +M ′) is general, then

a(D,X ′, B′ +M ′) = a(D,X ′, B′ + P ′ +M ′) = a(D, X ′, B′ + P ′ + M ′)

≤ a(D, X ′, 0) = a(D, X ′, C ′) = a(D,X ′, C ′) < ε.

Let X ′′ → X ′ be the extraction of all the prime divisors D such that a(D,X ′, C ′) < ε. LetKX′′ + B′′ + M ′′ be the pullback of KX′ + B′ + M ′. Then any component of B′′ whichis exceptional/X ′ has coefficient > 1 − ε. Also note that if KX′′ + C ′′ is the pullback ofKX′+C

′, then (X ′′, C ′′) is ε-lc in codimension at least two, i.e. any D with a(D,X ′′, C ′′) < εis already a divisor on X ′′.

Now assume ε is sufficiently small, depending only on d, p,R. Let Θ′′ be the boundarywhose coefficients are the same as B′′ except that we replace each coefficient in (1 − ε, 1)with 1 (similar to 2.33). That is, Θ′′ = (B′′)≤1−ε+ d(B′′)>1−εe. Run an MMP on −(KX′′ +Θ′′ +M ′′) and let X ′′′ be the resulting model. Then we argue that• (X ′′,Θ′′ +M ′′) is generalised lc,• the MMP does not contract any component of bΘ′′c,• −(KX′′′ + Θ′′′ +M ′′′) is nef,• (X ′′′,Θ′′′ +M ′′′) is generalised lc, and• X ′′′ is ε-lc.The first four claims follow from 2.33. We prove the last claim. Any component of C ′′

with coefficient in (1 − ε, 1) is a component of B′′ with coefficient in (1 − ε, 1], hence acomponent of bΘ′′c, so it is not contracted by the MMP. Thus (X ′′′, C ′′′) is ε-lc in codimen-sion at least two, hence X ′′′ is ε-lc. Note that if (X ′′′,Θ′′′ + M ′′′) is generalised klt, thenbΘ′′′c = 0, so C ′′′ has no component with coefficient in (1− ε, 1) which implies (X ′′′, C ′′′) isε-lc in any codimension.

Finally, since B′′ ∈ Φ(R)∪ (1− ε, 1] and since 1 is the only accumulation point of Φ(R),there is a finite set T ⊂ [0, 1] of rational numbers which includes R and which depends onlyon ε,R such that Θ′′ ∈ T (note that by our choice of ε, I depends only on d, p,R).

Proposition 6.13. Assume Theorems 1.8 and 1.10 hold in dimension ≤ d. Then Theorem1.9 holds in dimension d.

Proof. By induction on d we can assume Theorem 1.9 holds in dimension ≤ d − 1. Let(X ′, B′ + M ′) be as in Theorem 1.9 in dimension d. Replacing (X ′, B′ + M ′) with a Q-factorial generalised dlt model we can assume X ′ is Q-factorial. Pick a sufficiently small

Anti-pluricanonical systems on Fano varieties 49

number ε ∈ (0, 1) as in 6.12. We use the notation and constructions of 6.12. If KX′′′ +

Θ′′′ + M ′′′ has an n-complement KX′′′ + Θ′′′+ + M ′′′ with Θ′′′+ ≥ Θ′′′, then we get ann-complement KX′′ + Θ′′+ +M ′′ of KX′′ + Θ′′+M ′′ with Θ′′+ ≥ Θ′′ which in turn gives ann-complement KX′ + Θ′+ +M ′ of KX′ + Θ′+M ′ with Θ′+ ≥ Θ′ where Θ′ is the pushdownof Θ′′. Since Θ′ ≥ B′, KX′ + Θ′+ +M ′ is an n-complement of KX′ +B′ +M ′. Therefore,replacing X ′ with X ′′′, B′ with Θ′′′, M ′ with M ′′′, and R with T, we can assume B′ ∈ R(where T is as in the end of 6.12).

By Lemma 6.10, we can assume (X ′, B′ + M ′) is exceptional. Since we are assumingTheorem 1.10 in dimension d, X ′ is bounded. So there is a bounded number n divisible bypI(R) such that −n(KX′ +B′ +M ′) is nef and Cartier. Since X ′ is Fano type, we can usethe effective base point free theorem [27], so we can assume −n(KX′ + B′ + M ′) is basepoint free. Now let

G′ ∈ | − n(KX′ +B′ +M ′)|be a general member and let B′+ = B′ + 1

nG′. Then KX′ +B′+ +M ′ is an n-complement

of KX′ +B′ +M ′.

7. Boundedness of exceptional pairs

The aim of this section is to treat Theorems 1.3 and 1.10 inductively.

7.1. A directed MMP. Let (X ′, B′ + M ′) be a projective generalised lc pair with data

Xφ→ X ′ and M . Assume X ′ is Fano type and −(KX′ + B′ + M ′) is nef. Let D′ ≥ 0 be

an R-divisor on X ′ and N a nef R-divisor on X such that D′ + N ′ is R-Cartier and notnumerically trivial. We run an MMP as follows. Let t be the largest number such that

−(KX′ +B′ + tD′ + tN ′ +M ′)

is nef and

(X ′, B′ + tD′ + tN ′ +M ′)

is generalised lc. Then either t is the generalised lc threshold of D′ + N ′ with respect to(X ′, B′ +M ′) or there is an extremal ray R such that

(KX′ +B′ + tD′ + tN ′ +M ′) ·R = 0 and (D′ +N ′) ·R > 0.

In the former case we stop. In the latter case, if R defines a Mori fibre structure, we againstop. Otherwise we perform the flip or divisorial contraction associated to R and continuethe process. That is, in each step we try to increase t keeping the mentioned nefness andgeneralised lc properties and we continue until we hit either a generalised lc threshold or aMori fibre structure. Note that in each step we have (KX′ +B′+M ′) ·R ≤ 0. The processgives an MMP which we refer to as running an MMP for KX′+B

′+M ′ by adding multiplesof D′ +N ′. It terminates with some model X ′′. Let t be the maximum of the t appearingin the process.

We can see from the definition that the intersection of

KX′ +B′ + tD′ + tN ′ +M ′

with each extremal ray in the process is nonnegative. Therefore, replacing φ we can assumethe induced map ψ : X 99K X ′′ is a morphism, and that

ψ∗(KX′′ +B′′ + tD′′ + tN ′′ +M ′′) ≥ φ∗(KX′ +B′ + tD′ + tN ′ +M ′).

50 Caucher Birkar

7.2. Bounds on singularities.

Lemma 7.3. Let d, p ∈ N and Φ ⊂ [0, 1] be a DCC set. Then there is a number ε >0 depending only on d, p,Φ satisfying the following. Let (X ′, B′ + M ′) be a projectivegeneralised pair with data φ : X → X ′ and M such that• (X ′, B′ +M ′) is exceptional of dimension d,• B′ ∈ Φ and pM is Cartier, and• X ′ is Fano type.

Then for any 0 ≤ P ′ ∼R −(KX′ +B′+M ′), the pair (X ′, B′+P ′+M ′) is generalised ε-lc.

Proof. Let (X ′, B′+M ′) and P ′ be as in the statement. Since (X ′, B′+M ′) is exceptional,(X ′, B′ +P ′ +M ′) is klt. Taking a Q-factorialisation we can assume X ′ is Q-factorial. LetD′′ be a prime divisor on birational models of X such that a := a(D′′, X ′, B′ + P ′ +M ′) isminimal. We can assume a < 1. Let X ′′ → X ′ be the birational contraction which extractsexactly D′′; it is the identity morphism if D′′ is already a divisor on X ′. Let KX′′+B

′′+M ′′

be the pullback of KX′ + B′ + M ′, and let P ′′ be the pullback of P ′. Let e and c be thecoefficients of D′′ in B′′ and P ′′ respectively (note that it is possible to have e < 0). Byassumption, e+ c = 1− a.

By 2.6 (6), X ′′ is Fano type. Running an MMP on

−(KX′′ +B′′ + cD′′ +M ′′) ∼Q P ′′ − cD′′ ≥ 0

we get a model X ′′′ on which −(KX′′′ + B′′′ + cD′′′ + M ′′′) is nef. We can assume theinduced maps ψ : X 99K X ′′ and π : X 99K X ′′′ are morphisms. Then

π∗(KX′′′ +B′′′ + cD′′′ +M ′′′) ≥ ψ∗(KX′′ +B′′ + cD′′ +M ′′).

Next run an MMP for KX′′′ + B′′′ + cD′′′ + M ′′′ by adding multiples of D′′′, as in 7.1.Let X be the resulting model on which −(KX + B + cD + M) is nef for some c ≥ c. We

can assume the induced map ρ : X 99K X is a morphism. By construction,

ρ∗(KX +B + cD +M) ≥ π∗(KX′′′ +B′′′ + cD′′′ +M ′′′)

≥ ψ∗(KX′′ +B′′ + cD′′ +M ′′) ≥ φ∗(KX′ +B′ +M ′).

This implies (X,B+ cD+M) is klt as (X ′, B′+M ′) is exceptional. So by definition of theMMP in 7.1, there is a Mori fibre structure X → T so that

−(KX +B + cD +M) ≡ 0/T

and D is ample over T . By construction, the coefficients of B − eD belong to Φ. Nowrestricting KX +B+ cD+M to the general fibres of X → T and applying [7, Theorem 1.5]shows that e+ c is bounded away from 1. This implies e+ c is also bounded away from 1.That is, there is ε > 0 depending only on d, p,Φ such that e + c ≤ 1 − ε. So ε ≤ a, hence(X ′, B′ + P ′ +M ′) is generalised ε-lc.

7.4. From complements to Theorem 1.3.

Lemma 7.5. Assume Theorem 1.9 holds in dimension ≤ d− 1 and Theorem 1.8 holds indimension d. Then Theorem 1.3 holds in dimension d.

Proof. Assume the statement is not true. Then there is a sequence Xi of exceptional weakFano varieties of dimension d such that no infinite subsequence forms a bounded family.Replacing Xi we can assume they are Fano varieties. Let εi be the minimal log discrepancyof Xi. Let ε = lim sup εi.

Anti-pluricanonical systems on Fano varieties 51

First assume ε < 1. Replacing the sequence we can assume εi ≤ ε for every i. There is abirational contraction X ′′i → Xi from a Q-factorial variety which contracts only one primedivisor D′′i , and a(D′′i , Xi, 0) = εi. Let KX′′i

+eiD′′i be the pullback of KXi . Then ei ≥ 1−ε.

Run an MMP, as in 7.1, for KX′′i+ eiD

′′i by adding multiples of D′′i , and let (X ′′′i , eiD

′′′i ) be

the resulting pair. Take common resolutions ψi : Wi → X ′′i and πi : Wi → X ′′′i . Then

π∗i (KX′′′i+ eiD

′′′i ) ≥ ψ∗i (KX′′i

+ eiD′′i ) ≥ ψ∗i (KX′′i

+ eiD′′i ).

This implies (X ′′′i , eiD′′′i ) is exceptional as (X ′′i , eiD

′′i ) is exceptional. In particular, we do

not get a lc threshold in the process, hence there is a Mori fibre structure X ′′′i → T ′′′i suchthat −(KX′′′i

+ eiD′′′i ) is nef and numerically trivial over T ′′′i .

Pick 0 ≤ P ′′′i ∼Q −(KX′′′i+ eiD

′′′i ) and let the crepant pullback of KX′′′i

+ eiD′′′i + P ′′′i

to Xi be KXi + Pi. Then Pi ≥ 0, and applying Lemma 7.3, (Xi, Pi) is ε-lc for some ε > 0independent of i. Thus (X ′′′i , eiD

′′′i ) is also ε-lc. Applying Theorem 1.4 to the restriction

of KX′′′i+ eiD

′′′i to the general fibres of X ′′′i → T ′′′i shows that these fibres are bounded.

Moreover, by Lemma 2.11, ei belongs to a finite set independent of i. Now using Proposition6.5 if dimT ′′′i > 0, or using Theorem 1.4 if dimT ′′′i = 0, we deduce that KX′′′i

+ eiD′′′i has an

n-complement KX′′′i+ B′′′i for some n independent of i such that eiD

′′′i ≤ B′′′i . Therefore,

KXi has an n-complement KXi +Bi, by 6.1 (2). Since Xi is exceptional, (Xi, Bi) is klt. Sothe Xi form a bounded family by Theorem 1.4, a contradiction.

Now we can assume ε = 1 and that the εi form an increasing sequence approaching 1.By Proposition 4.7, there exists m ∈ N such that | −mKXi | defines a birational map forevery i. Pick 0 ≤ Ci ∼ −miKXi . Since Xi is exceptional, (Xi, Bi := 1

mCi) is klt, hence it

is 1m -lc. Therefore, Xi are bounded by Theorem 1.4, a contradiction.

7.6. Bound on exceptional thresholds.

Lemma 7.7. Let d, p ∈ N and let Φ ⊂ [0, 1] be a DCC set. Then there is β ∈ (0, 1)depending only on d, p,Φ satisfying the following. Assume (X ′, B′ + M ′) is a projectivegeneralised pair with data φ : X → X ′ and M such that• (X ′, B′ +M ′) is exceptional of dimension d,• B′ ∈ Φ, and pM is Cartier,• −(KX′ +B′ +M ′) is nef, and• X ′ is Fano type and Q-factorial.

Then (X ′, B′ + αM ′) is exceptional for every α ∈ [β, 1].

Proof. Step 1. Suppose the lemma is not true. Then there exist a strictly increasingsequence of numbers αi approaching 1 and a sequence (X ′i, B

′i + M ′i) of generalised pairs

as in the statement such that (X ′i, B′i + αiM

′i) is non-exceptional. In particular, M ′i is not

numerically trivial. For each i there exists

0 ≤ P ′i ∼R −(KX′i+B′i + αiM

′i) = −(KX′i

+B′i +M ′i) + (1− αi)M ′isuch that (X ′i, B

′i + P ′i + αiM

′i) is not generalised klt. Run an MMP on P ′i and replace

X ′i with the resulting model so that we can assume P ′i is nef. The MMP does not con-tract any divisor because Nσ(P ′i ) = 0, hence all our assumptions are preserved except−(KX′i

+B′i +M ′i) may not be nef any more (but we will not need it).

Step 2. Let ti be the generalised lc threshold of P ′i with respect to (X ′i, B′i + αiM

′i). By

Step 1, ti ≤ 1. Let Ω′i = B′i + tiP′i and let (X ′′i ,Ω

′′i + αiM

′′i ) be a Q-factorial generalised

52 Caucher Birkar

dlt model of (X ′i,Ω′i + αiM

′i). Let Γ′′i be the sum of the birational transform of B′i and the

reduced exceptional of X ′′i → X ′i. Adding 1 to Φ, then Γ′′i ∈ Φ, Γ′′i ≤ Ω′′i , and bΓ′′i c 6= 0.Let G′′i = Ω′′i − Γ′′i . Then from

−(KX′i+ Ω′i + αiM

′i) ∼R (1− ti)P ′i

we get

−(KX′′i+ Γ′′i + αiM

′′i ) = −(KX′′i

+ Ω′′i + αiM′′i ) +G′′i ∼R (1− ti)P ′′i +G′′i ≥ 0

where P ′′i is the pullback of P ′i .

Step 3. By 2.6 (6), X ′′i is Fano type. Run an MMP on −(KX′′i+ Γ′′i + αiM

′′i ) and let

X ′′′i be the resulting model. Since (X ′′i ,Ω′′i +αiM

′′i ) is generalised lc and P ′′i is semi-ample,

(X ′′′i ,Ω′′′i +αiM

′′′i ) is generalised lc which implies (X ′′′i ,Γ

′′′i +αiM

′′′i ) is generalised lc too as

Γ′′′i ≤ Ω′′′i . Moreover, the MMP produces a minimal model, that is, −(KX′′′i+ Γ′′′i +αiM

′′′i )

is nef. By construction, (X ′′′i ,Γ′′′i +αiM

′′′i ) is not generalised klt, and assuming the induced

maps ψi : Xi 99K X ′′i and πi : Xi 99K X ′′′i are morphisms, we have

π∗i (KX′′′i+ Γ′′′i + αiM

′′′i ) ≥ ψ∗i (KX′′i

+ Γ′′i + αiM′′i ).

Step 4. Now run a partial MMP for KX′′′i+ Γ′′′i + αiM

′′′i , as in 7.1, by adding multiples

of M ′′′i , that is, by increasing αi but not exceeding 1. Denote the resulting model by Xi,and assume the process increases αi to αi (so by assumption, αi ≤ 1). If αi = 1 for everyi, then go to Step 5. Otherwise we can assume αi < 1 for every i. By definition of theMMP, either αi is the generalised lc threshold of M i with respect to (Xi,Γi), or there isan extremal ray Ri defining a Mori fibre structure Xi → T i such that

−(KXi+ Γi + αiM i) ·Ri = 0

and M i · Ri > 0. Since lim αi = 1, we can assume the former case holds for every i, by [7,Theorem 1.4], which leads to a contradiction by [7, Theorem 1.5] applied to the restrictionof KXi

+ Γi + αiM i to the general fibres of Xi → T i.

Step 5. Now we assume αi = 1 for every i which in particular means −(KXi+ Γi +M i)

is nef. Assuming the induced map ρi : Xi 99K Xi is a morphism, by 7.1, we have

ρ∗i (KXi+ Γi +M i) ≥ π∗i (KX′′′i

+ Γ′′′i +M ′′′i )

= π∗i (KX′′′i+ Γ′′′i + αiM

′′′i ) + π∗i (1− αi)M ′′′i

≥ ψ∗i (KX′′i+ Γ′′i + αiM

′′i ) + π∗i (1− αi)M ′′′i

from which we get

ψi∗ρ∗i (KXi

+ Γi +M i) ≥ KX′′i+ Γ′′i + αiM

′′i + ψi∗π

∗i (1− αi)M ′′′i .

Now since Mi is nef we get

ψi∗π∗i (1− αi)M ′′′i ≥ ψi∗(1− αi)Mi = (1− αi)M ′′i

which in turn implies

ψi∗ρ∗i (KXi

+ Γi +M i) ≥ KX′′i+ Γ′′i +M ′′i .

Step 6. Pick

0 ≤ Qi ∼R −(KXi+ Γi +M i).

Anti-pluricanonical systems on Fano varieties 53

By Step 5, we can write the crepant pullback of KXi+ Γi +Qi +M i to X ′′i as KX′′i

+ Γ′′i +

Q′′i +M ′′i for some Q′′i ≥ 0. Therefore, (X ′i, B′i +Q′i +M ′i) is not generalised klt and

KX′i+B′i +Q′i +M ′i ∼R 0

where Q′i is the pushdown of Q′i. This is a contradiction as (X ′i, B′i +M ′i) is exceptional.

7.8. Bound on anti-canonical volumes.

Lemma 7.9. Let d, p ∈ N and let Φ ⊂ [0, 1] be a DCC set. Then there is v depending onlyon d, p,Φ satisfying the following. Let (X ′, B′ + M ′) be a projective generalised pair withdata φ : X → X ′ and M such that• (X ′, B′ +M ′) is generalised klt of dimension d,• B′ ∈ Φ and pM is Cartier and big,• KX′ +B′ +M ′ ∼R 0, and• X ′ is Fano type.

Then vol(−KX′) ≤ v.

Proof. If the statement does not hold, then there is a sequence of generalised pairs (X ′i, B′i+

M ′i) as in the statement such that the volumes vol(−KX′i) form a strictly increasing sequence

approaching ∞. After taking a Q-factorialisation we can assume X ′i is Q-factorial. SincepMi is nef and big and Cartier, by Lemma 2.30, KXi + 3dpMi is big, hence KX′i

+ 3dpM ′iis big too. Thus vol(−KX′i

) < vol(3dpM ′i), hence it is enough to show vol(M ′i) is bounded

from above. We can assume the volumes vol(M ′i) form a strictly increasing sequence ofnumbers approaching ∞.

There is a strictly decreasing sequence of numbers δi approaching zero such that vol(δiM′i) >

dd. Thus letting αi = 1− δi,

vol(−(KX′i+B′i + αiM

′i))) = vol(δiM

′i) > dd,

hence there is some

0 ≤ P ′i ∼R −(KX′i+B′i + αiM

′i)

such that (X ′i, B′i+P ′i +αiM

′i)) is not generalised klt. In particular, (X ′i, B

′i+αiM

′i) is non-

exceptional. This contradicts Lemma 7.7 as limαi = 1 and (X ′i, B′i +M ′i) are exceptional.

7.10. Bound on lc thresholds.

Lemma 7.11. Let d, p, l ∈ N and let Φ ⊂ [0, 1] be a DCC set. Then there is a positivereal number t depending only on d, p, l,Φ satisfying the following. Let (X ′, B′ + M ′) be aprojective generalised pair with data φ : X → X ′ and M such that• (X ′, B′ +M ′) is exceptional of dimension d,• B′ ∈ Φ and pM is Cartier and big,• −(KX′ +B′ +M ′) is nef, and• X ′ is Fano type and Q-factorial.

Then for any L′ ∈ | − lKX′ |, the pair (X ′, tL′) is klt.

Proof. If the statement does not hold, then there exist a decreasing sequence of numbersti approaching zero and a sequence (X ′i, B

′i + M ′i) of generalised pairs as in the statement

54 Caucher Birkar

such that (X ′i, tiL′i) is not klt for some L′i ∈ |− lKX′i

|. Since Mi is nef and big and Cartier,by Lemma 2.30,

KX′i+ 3dpM ′i ∼R 3dpM ′i −

1

lL′i

is big.By Lemma 7.7, there is a rational number β ∈ (0, 1) such that (X ′i, B

′i + βM ′i) is excep-

tional for every i. Let si be the generalised lc threshold of L′i with respect to (X ′i, B′i+βM

′i).

Then si ≤ ti. We can assume si <1−β3dpl for every i. Thus

−(KX′i+B′i + siL

′i + βM ′i) = −(KX′i

+B′i +M ′i) + (1− β)M ′i − siL′iis big, hence there is

0 ≤ P ′i ∼R −(KX′i+B′i + siL

′i + βM ′i).

Now (X ′i, B′i+siL

′i+P ′i +βM ′i) is not generalised klt, so (X ′i, B

′i+βM ′i) is non-exceptional.

This is a contradiction.

7.12. From bound on lc thresholds to boundedness of varieties.

Proposition 7.13. Let d,m, v ∈ N and let tl be a sequence of positive real numbers. AssumeTheorem 1.9 holds in dimension ≤ d− 1 and Theorem 1.8 holds in dimension d. Let P bethe set of projective varieties X such that• X is a klt weak Fano variety of dimension d,• KX has an m-complement,• | −mKX | defines a birational map,• vol(−KX) ≤ v, and• for any l ∈ N and any L ∈ | − lKX |, the pair (X, tlL) is klt.

Then P is a bounded family.

Proof. Step 1. Let X ∈ P. Taking a Q-factorialisation we can assume X is Q-factorial.Let φ : W → X be a resolution so that φ∗(−mKX) ∼ AW + RW where AW is base pointfree and RW is the fixed part. We can assume AW is a general member of |AW |. Let A,Rbe the pushdowns of AW , RW . Since KX has an m-complement and since AW is general,(X,B+ := 1

mA + 1mR) is lc, hence KX + B+ is also an m-complement. By Theorem 1.4,

we can assume (X,B+) is not klt. The idea is to construct another complement which isklt. Replacingm we can assumem > 1 so that we can assume A is not a component of bB+c.

Step 2. Let ΣW be the sum of the exceptional divisors of φ and the support of thebirational transform of A+R. Then by Lemma 2.30,

KW + ΣW + 2(2d+ 1)AW

is big. Moreover its volume is bounded from above because the right hand side of

vol(KX + Σ + 2(2d+ 1)A) ≤ vol(KX +A+R+ 2(2d+ 1)A) ≤ vol(−(4d+ 3)mKX)

is bounded from above where Σ is the pushdown of ΣW . Therefore, (W,ΣW ) is log bira-tionally bounded by [15, Lemmas 3.2 and 2.4.2(4)] as vol(AW ) is bounded. Thus as in Step7 of the proof of Proposition 4.4, there is a log bounded family Q of log smooth pairs de-pending only on d,m, v such that, perhaps after replacing W with a higher resolution, thereis (W,ΣW ) ∈ Q and a birational morphism ψ : W → W with ΣW = ψ∗ΣW . Moreover,

letting AW = ψ∗AW we can assume AW = ψ∗AW . Also note that AW ≤ ΣW , so (W,AW )

Anti-pluricanonical systems on Fano varieties 55

is log bounded. Thus since AW is big, there is l ∈ N depending only on the family Q suchthat lAW ∼ GW for some GW ≥ 0 whose support contains ΣW .

Step 3. Let KW + B+W

be the crepant pullback of KX + B+ to W . Then (W,B+W

) is

sub-lc and SuppB+W⊆ ΣW ⊆ SuppGW . Let G = φ∗ψ

∗GW . By construction, lA ∼ G,

so G + lR ∈ | − lmKX |. Thus, by assumption, (X, t(G + lR)) is klt where t := tlm. Inparticular, this means the coefficients of t(G + lR) belong to a fixed finite set dependingonly on t. Decreasing t we can assume it is rational and that t < 1

lm .

Now if (X, 1lm(G+ lR)) is lc, then we let Ω = 1

lm(G+ lR) and n = lm. But if it is not lc,then the pair (X, t(G + lR)) is strongly non-exceptional, hence by Lemma 6.9, there is ndepending only on d, t such that there is Ω ≥ t(G+lR) with (X,Ω) is lc and n(KX+Ω) ∼ 0.

Step 4. Let

∆W := B+W

+t

mAW −

t

lmGW

which satisfies KW + ∆W ∼Q 0. Since AW is not a component of⌊B+W

⌋and since

SuppB+W⊆ SuppGW , there is ε > 0 depending only on t, l,m such that (W,∆W ) is

sub-ε-lc.Let KX + ∆ be the crepant pullback of KW + ∆W to X. Then KX + ∆ ∼Q 0 and (X,∆)

is sub-ε-lc. However,

∆ = B+ +t

mA− t

lmG

has negative coefficients, so we cannot apply Theorem 1.4 at this point.

Step 5. Let Θ = 12∆ + 1

2Ω. Then

Θ =1

2B+ +

t

2mA− t

2lmG+

1

2Ω ≥

1

2B+ +

t

2mA− t

2lmG+

t

2(G+ lR) ≥ 0.

Moreover, (X,Θ) is ε2 -lc, KX + Θ ∼Q 0, and the coefficients of Θ belong to a fixed finite

set depending only on t, l,m, n. Now apply Theorem 1.4.

7.14. From complements to Theorem 1.10.

Proposition 7.15. Assume Theorem 1.9 holds in dimension ≤ d − 1 and Theorem 1.8holds in dimension d. Then Theorem 1.10 holds in dimension d.

Proof. Step 1. Let (X ′, B′ + M ′) be as in Theorem 1.10 in dimension d. It is enough toshow X ′ is bounded because then we can find a very ample Cartier divisor H ′ so that−KX′ ·H ′d−1 is bounded, hence B′ ·H ′d−1 is bounded too as B′ ·H ′d−1 ≤ −KX′ ·H ′d−1,and this implies (X ′, B′) is log bounded. Taking a Q-factorialisation we can assume X ′

is Q-factorial. By Lemma 7.3, (X ′, B′ + M ′) is generalised ε-lc for some ε > 0 dependingonly on d, p,R. In particular, the coefficients of B′ belong to a finite set depending only ond, p,R because 1 is the only accumulation point of R. Extending R we can assume B′ ∈ R.

Run an MMP on −KX′ and let X ′ be the resulting model which is a klt weak Fano.Then KX′ has an n-complement for some n depending only on d by applying Lemma 7.5 if

X ′ is exceptional or by applying Lemma 6.10 otherwise. This implies KX′ also has an n-complement KX′+C ′, by 6.1 (3). On the other hand, since (X ′, B′+M ′) is generalised ε-lc

56 Caucher Birkar

and −(KX′+B′+M ′) is semi-ample, X ′ is ε-lc. Thus by Proposition 4.5, |−mKX′ | definesa birational map for some m depending only on d, ε, n which in turn implies |−mKX′ | alsodefines a birational map. Replacing both m and n by pmn, we can assume m = n andthat p divides m,n. Moreover, replacing φ and C ′ we can assume C ′ = 1

mA′ + 1

mR′ where

φ∗(−mKX′) ∼ A+R with A base point free and R the fixed part.Let ∆′ := 1

2B′ + 1

2mR′ and let N := 1

2M + 12mA. Then (X ′,∆′ + N ′) is generalised lc

and −(KX′ + ∆′ + N ′) is nef. Note that the coefficients of ∆′ belong to a fixed finite setand 2pmN is Cartier.

Step 2. Assume (X ′,∆′ +N ′) is non-exceptional. Then by Lemma 6.10, KX′ + ∆′ +N ′

has an l-complement KX′ + ∆′+ + N ′ for some l depending only on d, p,m,R such thatG′ := ∆′+ −∆′ ≥ 0. Then

lm(KX′ +B′ + 2G′ +M ′) ∼ lm(KX′ +B′ + 2G′ +M ′) + lm(KX′ + C ′)

= lm(2KX′ +B′ +M ′ +1

mR′ +

1

mA′ + 2G′)

= 2lm(KX′ + ∆′ +N ′ +G′) = 2lm(KX′ + ∆′+

+N ′) ∼ 0.

Let B′+ = B′ + 2G′. Since (X ′, B′ +M ′) is exceptional, (X ′, B′+ +M ′) is generalised klt.Thus

(X ′,1

2B′

++

1

2∆′

++

1

2M ′ +

1

2N ′)

is generalised klt, hence exceptional. Now replace B′ with 12B′+ + 1

2∆′+ and replace M

with 12M + 1

2N . Replacing p,R accordingly, we can then assume KX′ +B′ +M ′ ∼Q 0 andthat M is big.

Step 3. Now assume (X ′,∆′ + N ′) is exceptional. By Lemma 7.7, there is a rationalnumber β ∈ (0, 1) depending only on d, p,m,R such that (X ′,∆′ + βN ′) is exceptional.Since N = 1

2M + 12mA and A is base point free and big, there is r ∈ N such that

−r(KX′ + ∆′ + βN ′) = −r(KX′ + ∆′ +N ′) + r(1− β)N ′

is integral and potentially birational where r depends only on d, p,m, β,R. Then

|KX′ − r(KX′ + ∆′ + βN ′)|defines a birational map by [15, Lemma 2.3.4], hence

|mKX′ − rm(KX′ + ∆′ + βN ′)|also defines a birational map which implies

| − rm(KX′ + ∆′ + βN ′)| = |mKX′ +mC ′ − rm(KX′ + ∆′ + βN ′)|defines a birational map as well. In particular, there is ∆′+ ≥ ∆′ such that

rm(KX′ + ∆′+

+ βN ′) ∼ 0.

Since (X ′,∆′+βN ′) is exceptional, (X ′,∆′+ +βN ′) is generalised klt. Now replace B′ and

M with ∆′+ and βN , respectively. Replacing p,R accordingly, from now on we can thenassume KX′ +B′ +M ′ ∼Q 0 and that M is big.

Step 4. Let X ′ be as in Step 1 which is the result of an MMP on −KX′ . Since KX′ +B′ + M ′ ∼Q 0, we can replace X ′ with X ′, hence assume X ′ is a weak Fano. By Step 1,KX′ has an m-complement and | −mKX′ | defines a birational map. Moreover, by Lemma7.9, vol(−KX′) ≤ v for some number v depending only on d, p,R. On the other hand, by

Anti-pluricanonical systems on Fano varieties 57

Lemma 7.11, for each l ∈ N there is a positive real number tl depending only on d, p, l,Rsuch that for any L′ ∈ | − lKX′ |, the pair (X ′, tlL

′) is klt. Therefore, X ′ belongs to abounded family by Proposition 7.13.

8. Boundedness of relative complements

In this section we treat Theorem 1.8 inductively. The results of Sections 6 and 7 rely onthis theorem.

Lemma 8.1. Assume Theorems 1.7 and 1.8 hold in dimension d − 1. Then Theorem 1.8holds in dimension d for those (X,B) and X → Z such that• B ∈ R,• (X,Γ) is Q-factorial plt for some Γ,• −(KX + Γ) is ample over Z,• S := bΓc is irreducible and it is a component of bBc, and• S intersects the fibre over z.

Proof. First we claim that if T is the image of S in Z, then the induced morphism S → Tis a contraction. Let f denote X → Z and let π : V → Z denote the finite part of the Steinfactorisation of S → Z. Then by the properties of Γ and Kawamata-Viehweg vanishing wehave R1f∗OX(−S) = 0 which implies f∗OX → f∗OS is surjective, and this in turn impliesOZ → π∗OV is surjective because OZ = f∗OX and f∗OS = π∗OV . Therefore, V → T is anisomorphism and S → T is a contraction.

Let KS + BS = (KX + B)|S which is anti-nef over T . By Lemma 3.3, BS ∈ Φ(S) forsome finite set S ⊂ [0, 1] of rational numbers depending only on R. Moreover, S is Fanotype over T by restricting KX +Γ to S. Therefore, KS+BS has an n-complement KS+B+

S

over z with B+S ≥ BS , for some n divisible by I(S) and depending only on d,S: this follows

from Theorem 1.7 in dimension d− 1 if dimT = 0 or from Theorem 1.8 in dimension d− 1if dimT > 0. Replacing n we can assume it is divisible by I(R).

The idea is to lift the complement KS+B+S to an n-complement KX+B+ of KX+B over

z with B+ ≥ B. This can be proved by closely following the proof of Proposition 6.7 usingthe right notation and by working over some open affine neighbourhood of z (and shrinkingit if necessary). The inversion of adjunction and the connectedness principle required atthe end of the proof are well-known in this setting [21][28, Theorem 17.4].

Proposition 8.2. Assume Theorems 1.7 and 1.8 hold in dimension d− 1. Then Theorem1.8 holds in dimension d.

Proof. When (X,B) is klt and −(KX +B) is nef and big over Z, the theorem is essentially[35, Theorem 3.1].

Step 1. Pick an effective Cartier divisor N on Z passing through z. Let t be the lcthreshold of f∗N with respect to (X,B) over z. Let Ω = B + tf∗N and let (X ′,Ω′) be aQ-factorial dlt model of (X,Ω). Then X ′ is Fano type over Z. There is ∆′ ≤ Ω′ such that∆′ ∈ Φ(R), some component of b∆′c is vertical over Z intersecting the fibre over z, andB ≤ ∆ where ∆ is the pushdown of ∆′. Run an MMP/Z on −(KX′ + ∆′) and let X ′′ bethe resulting model. Since −(KX′ + ∆′) = −(KX′ + Ω′) + (Ω′−∆′) is pseudo-effective overZ, the MMP ends with a minimal model, that is, −(KX′′ + ∆′′) is nef over Z. Moreover,

if KX′′ + ∆′′ has an n-complement KX′′ + ∆′′+ over z with ∆′′+ ≥ ∆′′, then KX′ + ∆′ hasan n-complement KX′ + ∆′+ over z with ∆′+ ≥ ∆′ which in turn implies KX +B also has

58 Caucher Birkar

an n-complement KX +B+ over z with B+ ≥ B. Since −(KX′ + Ω′) is semi-ample over Z,(X ′′,Ω′′) is lc, hence (X ′′,∆′′) is lc. Replacing (X,B) with (X ′′,∆′′) we can assume bBchas a component intersecting the fibre over z.

Step 2. Let ε > 0 be a sufficiently small number. Let Θ be the boundary whose coeffi-cients are the same as B except that we replace each coefficient in (1 − ε, 1) with 1. Runan MMP/Z on −(KX + Θ) and let X ′ be the resulting model. Arguing as in 2.33, one canshow we can choose ε depending only on d,R so that no component of bΘc is contracted bythe MMP, (X ′,Θ′) is lc, and that −(KX′ + Θ′) is nef over Z. Moreover, the coefficients ofΘ′ belong to some fixed finite set depending only on R, ε. If KX′+Θ′ has an n-complementKX′ + Θ′+ over z with Θ′+ ≥ Θ′, then KX + Θ has an n-complement KX + Θ+ over z withΘ+ ≥ Θ which in turn implies KX + B also has an n-complement KX + B+ over z withB+ ≥ B. Replacing (X,B) with (X ′,Θ′) and extending R, from now on we can assumeB ∈ R. In the following steps we try to mimic the arguments of the proof of Proposition 6.8.

Step 3. Since X is Fano type over Z, −KX is big over Z. So since −(KX + B) is nefover Z,

−(KX + αB) = −α(KX +B)− (1− α)KX

is big over Z for any α ∈ (0, 1). We will assume α is sufficiently close to 1. Define aboundary ∆ as follows. Let D be a prime divisor. If D is vertical over Z, let µD∆ = µDBbut if D is horizontal over Z, let µD∆ = µDαB. Then (X,∆) is lc, αB ≤ ∆ ≤ B, b∆c hasa component intersecting the fibre over z, and −(KX + ∆) is big over Z as ∆ = αB nearthe generic fibre.

Let X → V/Z be the contraction defined by −(KX +B). Run an MMP on −(KX + ∆)over V and let X ′ be the resulting model. Perhaps after replacing α with a larger number,we can assume −(KX′+∆′) is nef and big over Z. Replace (X,B) with (X ′, B′) and replace∆ with ∆′ so that we can assume −(KX + ∆) is nef and big over Z. Let X → T/Z be thecontraction defined by −(KX + ∆).

Let ∆ = β∆ for some β < 1. After running an MMP on −(KX + ∆) over T we can

assume −(KX + ∆) is nef and big over T , hence also nef and big over Z if we replace βwith a number sufficiently close to 1. Taking a Q-factorial dlt model of (X,B), increasing

α, β if necessary, and replacing KX + ∆ and KX + ∆ with their pullbacks we can assume(X,B) is Q-factorial dlt and that there are boundaries ∆ ≤ ∆ ≤ B so that −(KX + ∆)

and −(KX + ∆) are nef and big over Z, some component of b∆c intersects the fibre over

z, and that (X, ∆) is klt.

Step 4. We can write −(KX +∆) ∼R A+G/Z where A ≥ 0 is ample and G ≥ 0. AssumeSuppG does not contain any non-klt centre of (X,∆). Let δ > 0 be sufficiently small. Since

−(KX + ∆ + δG) ∼R (1− δ)( δ

1− δA+A+G)/Z

is ample over Z, by perturbing the coefficients of ∆ + δG we can find a boundary Γ suchthat (X,Γ) is plt, S := bΓc ⊆ bBc is irreducible intersecting the fibre over z, and −(KX+Γ)is ample over Z. So we can apply Lemma 8.1.

Now assume SuppG contains some non-klt centre of (X,∆). Let t be the lc threshold of

G+ ∆− ∆ with respect to (X, ∆) over z. Replacing ∆ we can assume ∆− ∆ is sufficiently

small, hence t is sufficiently small too. Then letting Ω = ∆ + t(G + ∆ − ∆), any non-klt

Anti-pluricanonical systems on Fano varieties 59

centre of (X,Ω) is a non-klt centre of (X,∆). By construction,

−(KX + Ω) = −(KX + ∆ + t(G+ ∆− ∆))

= −(KX + ∆) + ∆− ∆− t(G+ ∆− ∆))

∼R A+G− tG+ (1− t)(∆− ∆)

= (1− t)( t

1− tA+A+G+ ∆− ∆)/Z

which implies −(KX + Ω) is ample over Z because

−(KX + ∆) = −(KX + ∆) + ∆− ∆ ∼R A+G+ ∆− ∆/Z

is nef and big over Z.

Step 5. If bΩc 6= 0, then there is a component S of bΩc ⊆ bBc and there is a boundaryΓ so that (X,Γ) is plt, S = bΓc intersects the fibre over z, and −(KX + Γ) is ample overZ. So we can apply Lemma 8.1.

Now assume bΩc = 0. As in Step 6 of the proof of 6.8 we can assume there is a projectivebirational contraction X ′ → X from a Q-factorial variety such that it contracts only onedivisor S′ which intersects the fibre over z, (X ′, S′) is plt, −(KX′ + S′) is ample over X,and if we denote the pullbacks of KX + Ω and KX + ∆ by KX′ + Ω′ and KX′ + ∆′, then S′

is a component of both bΩ′c and b∆′c. Now since −(KX + Ω) is ample over Z, we can finda boundary Γ′ so that (X ′,Γ′) is plt, S′ = bΓ′c intersects the fibre over z, and −(KX′ + Γ′)is ample over Z. In addition, if KX′+B′ is the pullback of KX +B, then S′ is a componentof bB′c. Now apply Lemma 8.1 to .

9. Anti-canonical volume

In this section we prove Theorem 1.6 which claims that the anti-canonical volumes of ε-lcFano varieties of a given dimension are bounded. Recall that we proved this boundedness forexceptional Fano varieties in 7.9. To deal with the non-exceptional case we need Conjecture1.5 in lower dimension.

Proof. (of Theorem 1.6) Step 1. The birational boundedness claim follows from existence ofv and Theorem 1.2, eg as in Step 7 of the proof of 4.4. If there is no v as in the statement,then there is a sequence Xi of ε-lc Fano varieties of dimension d such that vol(−KXi) is anincreasing sequence approaching ∞. We will derive a contradiction. Fix ε′ ∈ (0, ε). Thenthere exist a decreasing sequence of rational numbers ai approaching 0, and Q-boundariesBi ∼Q −aiKXi such that (2d)d < vol(Bi) and (Xi, Bi) is ε′-lc but not ε′′-lc for any ε′′ > ε′.We can assume ai < 1, hence −(KXi + Bi) is ample. Thus for each i, there is a primedivisor D′i on birational models of Xi such that a(D′i, Xi, Bi) = ε′. If D′i is a divisor on Xi,then we let φi : X

′i → Xi to be a small Q-factorialisation, otherwise we let it be a birational

contraction which extracts only D′i with X ′i being Q-factorial. Let KX′i+ eiD

′i = φ∗iKXi

and let KX′i+ B′i = φ∗i (KXi + Bi). Then ei ≤ 1 − ε but µDiB

′i = 1 − ε′. Therefore, the

coefficient of D′i in P ′i := φ∗iBi = B′i − eiD′i is at least ε− ε′.

Step 2. Let Hi be a general ample Q-divisor so that KXi +Bi+Hi ∼Q 0 and (Xi, Bi+Hi)is ε′-lc, and let H ′i be its pullback to X ′i. Run an MMP on −D′i which ends with a Mori

60 Caucher Birkar

fibre space X ′′i with Mori fibre structure X ′′i → Zi. Letting bi = 1ai− 1 we get biBi ∼Q Hi

and the bi form an increasing sequence approaching ∞. Now

KX′′i+B′′i + biP

′′i ∼Q KX′′i

+B′′i +H ′′i ∼Q 0

and µD′′i biP′′i ≥ bi(ε− ε′). So there is a number si ≥ bi(ε− ε′) so that KX′′i

+ siD′′i ∼Q 0/Zi.

In particular, lim si =∞.Assume dimZi > 0 for every i and let Vi be a general fibre of X ′′i → Zi. Since we are

assuming Conjecture 1.5 in dimension ≤ d − 1, Vi belongs to a bounded family as it is anε′-lc Fano variety. Restricting to Vi we get KVi + siDVi ∼Q 0 where DVi = D′′i |Vi . Thiscontradicts Lemma 2.11.

From now on we can assume dimZi = 0 for every i. By construction,

vol(−KX′′i) ≥ vol(biP

′′i ) ≥ vol(biP

′i ) = vol(biBi) > (2bid)d.

Replacing ε with ε′ and replacing Xi with X ′i we can assume there is a prime divisor Di onXi such that KXi + siDi ∼Q 0 and that the si form an increasing sequence approaching∞.

Step 3. In this step we fix i. By 2.16 (2), there is a bounded covering family of subvarietiesof Xi such that for any pair of general closed points xi, yi ∈ Xi there exist a member Giof the family and a Q-divisor 0 ≤ ∆i ∼Q −aiKXi so that (Xi,∆i) is lc at xi with a uniquenon-klt place whose centre contains xi, that centre is Gi, and (Xi,∆i) is not lc at yi. As−(KXi + ∆i) is ample, dimGi 6= 0 by the connectedness principle.

Let Fi be the normalisation of Gi, and let F ′i → Fi be a resolution so that F ′i 99K Wi isa morphism. By Theorem 3.9, there is a Q-boundary ΘFi with coefficients in a fixed DCCset Φ depending only on d such that we can write

(KXi + ∆i)|Fi ∼Q KFi + ∆Fi := KFi + ΘFi + PFi

where PFi is pseudo-effective. Increasing ai and adding to ∆i we can assume Pi ≥ 0.Let DFi := Di|Fi . By Lemma 3.10, each component of DFi has coefficient at least 1 in

ΘFi +DFi . Replacing ∆i with ∆i+Di and replacing PFi with PFi +DFi , we can assume eachcomponent of DFi := Di|Fi has coefficient at least 1 in ∆Fi . Note that we also need to re-place ai with ai+

1si

which we still can assume to form a decreasing sequence approaching 0.

Step 4. Pick a rational number ε′ ∈ (0, ε). By construction, (Fi,∆Fi) is not ε′-lc. Definea boundary ΣF ′i

on F ′i as follows. Let Si be a prime divisor and let vi be its coefficient in∆F ′i

where KF ′i+ ∆F ′i

is the pullback of KFi + ∆Fi . If vi ≤ 0, then let the coefficient of Siin ΣF ′i

be zero. But if vi > 0, then let the coefficient of Si in ΣF ′ibe the minimum of vi and

1− ε′. Then we can write

ΣF ′i= ∆F ′i

+ EF ′i −NF ′i

where EF ′i , NF ′iare effective with no common components, EF ′i is exceptional/Fi, each

component of NF ′ihas coefficient > 1− ε′ in ∆F ′i

, and NF ′i6= 0. Note that (F ′i ,ΣF ′i

) is ε′-lc.

Let (F ′′i ,ΣF ′′i) be a log minimal model of (F ′i ,ΣF ′i

) over Fi. By construction,

KF ′′i+ ΣF ′′i

= KF ′′i+ ∆F ′′i

+ EF ′′i −NF ′′i∼Q EF ′′i −NF ′′i

/Fi.

So by the negativity lemma, EF ′′i = 0, hence ∆F ′′i= ΣF ′′i

+ NF ′′i≥ 0. Moreover, NF ′′i

6= 0because the birational transform of each component of DFi is a component of NF ′′i

.

Let HF ′′ibe a general Q-divisor so that KF ′′i

+ ∆F ′′i+HF ′′i

∼Q 0 and (F ′′i ,ΣF ′′i+HF ′′i

) is

ε′-lc. Then

KF ′′i+ ΣF ′′i

+HF ′′i+NF ′′i

∼Q 0

Anti-pluricanonical systems on Fano varieties 61

and running an MMP on KF ′′i+ ΣF ′′i

+ HF ′′iwe get a Mori fibre structure Fi → Ti. As

we are assuming Conjecture 1.5 in dimension ≤ d − 1, the general fibres of Fi → Ti arebounded because KFi

is ε′-lc and anti-ample over Ti.

Step 5. By Lemma 3.11, we can write KFi +ΛFi = KXi |Fi where (Fi,ΛFi) is sub-ε-lc and∆Fi − ΛFi ≥ 0. Let KF ′i

+ ΛF ′i be the pullback of KFi + ΛFi . Then

∆F ′i− ΛF ′i ∼Q ∆i|F ′i ∼Q aisiDF ′i

where DF ′i:= Di|F ′i . On the other hand, by construction, NFi

is ample over Ti. Let Cibe one of its components that is ample over Ti. Let C ′i on F ′i be the birational transform

of Ci. Since C ′i is a component of NF ′i, it is a component of ∆F ′i

with coefficient > 1 − ε′which in turn implies it is a component of ∆F ′i

− ΛF ′i with coefficient > ε− ε′.Let ri = si(1− ai). Then KXi + ∆i + riDi ∼Q 0. Thus KF ′i

+ ∆F ′i+ riDF ′i

∼Q 0 whichin turn gives

KF ′i+ ∆F ′i

+risiai

(∆F ′i− ΛF ′i ) ∼Q 0

and then

KFi+ ∆Fi

+risiai

(∆Fi− ΛFi

) ∼Q 0.

But now Ci is a component of risiai

(∆Fi− ΛFi

) whose coefficient is at least ri(ε−ε′)siai

which

approaches ∞ as i grows large. Restricting to the general fibres of Fi → Ti and applyingLemma 2.11 gives a contradiction.

10. Proofs of main results

Recall that we proved Theorem 1.4 in Section 5 and proved Theorem 1.6 in Section9. We prove the other main results by induction so lets assume all the theorems in theintroduction hold in dimension d− 1. They can be verified easily in dimension 1.

Proof. (of Theorem 1.8) This follows from Theorems 1.7 and 1.8 in dimension d − 1, andProposition 8.2.

Proof. (of Theorem 1.3) This follows from Theorem 1.9 in dimension ≤ d− 1, Theorem 1.8in dimension d, and Lemma 7.5.

Proof. (of Theorem 1.10) This follows from Theorem 1.9 in dimension ≤ d − 1, Theorem1.8 in dimension d, and Proposition 7.15.

Proof. (of Theorem 1.9) This follows from Theorems 1.8 and 1.10 in dimension d, andProposition 6.13.

Proof. (of Theorem 1.7) This is a special case of Theorem 1.9.

Proof. (of Theorem 1.1) This is a consequence of Theorem 1.7.

62 Caucher Birkar

Proof. (of Theorem 1.2) This follows from Theorem 1.1 and Proposition 4.5.

References

[1] F. Ambro; The Adjunction Conjecture and its applications. arXiv:math/9903060v3.[2] F. Ambro; The moduli b-divisor of an lc-trivial fibration. Compos. Math. 141 (2005), no. 2, 385–403.[3] V. Alexeev; Boundedness and K2 for log surfaces. Internat. J. Math. 5 (1994), no. 6, 779–810.[4] C. Birkar; Singularities on the base of a Fano type fibration. J. Reine Angew Math., to appear in print.

Published online: DOI: 10.1515/crelle-2014-0033 (2014), 18 pages.[5] C. Birkar, Existence of log canonical flips and a special LMMP, Pub. Math. IHES., 115 (2012), 325-368.[6] C. Birkar, P. Cascini, C. Hacon and J. McKernan, Existence of minimal models for varieties of log

general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405-468.[7] C. Birkar and D-Q. Zhang; Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs,

to appear in Pub. Math. IHES., arXiv:1410.0938v2.[8] A. Borisov; Boundedness of Fano threefolds with log-terminal singularities of given index. J. Math. Sci.

Univ. Tokyo 8 (2001), no. 2, 329–342.[9] L. Borisov; Singular toric Fano varieties. Acad. Sci. USSR Sb. Math. 75 (1993), no. 1, 277283.

[10] O. Fujino, Y. Gongyo; On the moduli b-divisors of lc-trivial fibrations. To appear in Annales de l’InstitutFourier, arXiv:1210.5052v2.

[11] Y. Gongyo; On the minimal model theory for dlt pairs of numerical log Kodaira dimension zero. Math.Res. Lett. 18 (2011), no. 5, 991–1000

[12] C. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent.Math. 166 (2006), 1-25.

[13] C. Hacon and J. McKernan, On Shokurov’s rational connectedness conjecture, Duke Math. J. Volume138, Number 1 (2007), 119-136.

[14] C. D. Hacon, J. McKernan and C. Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014),no. 2, 523-571.

[15] C. D. Hacon, J. McKernan and C. Xu, On the birational automorphisms of varieties of general type,Ann. of Math. (2) 177 (2013), no. 3, 1077-1111.

[16] C. D. Hacon and C. Xu; Boundedness of log Calabi-Yau pairs of Fano type. arXiv:1410.8187v1.[17] R. Hartshorne; Algebraic geometry. Springer (1977).[18] V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties. Algebraic geometry. V., Encyclopaedia Math.

Sci., vol. 47, Springer, Berlin, 1999.[19] C. Jiang, On birational boundedness of Fano fibrations, arXiv:1509.08722v1.[20] C. Jiang, Boundedness of anti-canonical volumes of singular log Fano threefolds, arXiv:1411.6728v2.[21] M. Kawakita; Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), 129-133.[22] Y. Kawamata; Subadjunction of log canonical divisors, II, Amer. J. Math. 120 (1998), 893–899.[23] Y. Kawamata, On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609-611.[24] Y. Kawamata; Boundedness of Q-Fano threefolds. Proceedings of the International Conference on Alge-

bra, Part 3 (Novosibirsk, 1989), 439–445, Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence,RI, 1992.

[25] Y. Kawamata; Pluricanonical systems on minimal algebraic varieties. Invent. Math.79 (1985), 567-588.[26] J. Kollar; Singularities of pairs, in Algebraic Geometry, Santa Cruz 1995, volume 62 of Proc. Symp.

Pure Math Amer. Math. Soc. 1997, 221-286.[27] J. Kollar; Effective base point freeness. Math. Annalen (1993), Volume 296, Issue 1, pp 595-605.[28] J. Kollar et al., Flips and abundance for algebraic threefolds, Asterisque No. 211 (1992).[29] J. Kollar; Y. Miyaoka; S. Mori; Rationally connectedness and boundedness of Fano manifolds. J. Di.

Geom. 36 (1992), 765-769.[30] J. Kollar; Y. Miyaoka; S. Mori; H. Takagi; Boundedness of canonical Q-Fano 3-folds. Proc. Japan Acad.

Ser. A Math. Sci. 76 (2000), no. 5, 73–77.[31] J. Kollar and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134,

Cambridge Univ. Press, 1998.[32] C-J. Lai; Bounding the volumes of singular Fano threefolds. To appear in Nagoya Math. Journal,

arXiv:1204.2593v1.[33] N. Nakayama; Zariski decomposition and abundance. MSJ Memoirs 14, Tokyo (2004).

Anti-pluricanonical systems on Fano varieties 63

[34] Yu. Prokhorov, V.V. Shokurov; Towards the second main theorem on complements. J. Algebraic Ge-ometry, 18 (2009) 151-199.

[35] Yu. Prokhorov; V.V. Shokurov; The first fundamental Theorem on complements: from global to local.(Russian) Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 6, 99–128; translation in Izv. Math. 65 (2001),no. 6, 1169–1196.

[36] V.V. Shokurov, Log Adjunction: effectiveness and positivity. arXiv:1308.5160.[37] V.V. Shokurov; Complements on surfaces. Algebraic geometry, 10. J. Math. Sci. (New York) 102 (2000),

no. 2, 3876–3932.[38] V.V. Shokurov, 3-fold log flips, With an appendix by Yujiro Kawamata. Russian Acad. Sci. Izv. Math.

40 (1993), no. 1, 95–202.[39] S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006),

no. 3, 551-587.[40] Q. Zhang, Rational connectedness of log Q-Fano varieties, J. reine angew. Math. (2006), Issue 590,

Pages 131-142.

DPMMS, Centre for Mathematical SciencesUniversity of Cambridge,Wilberforce Road, Cambridge CB3 0WB, UKc.birkar@dpmms.cam.ac.uk