On connectedness of non-klt loci of singularities of pairs ...

On connectedness of non-klt loci of singularities of pairs

Caucher Birkar

Abstract. We study the non-klt locus of singularities of pairs. We show thatgiven a pair (X,B) and a projective morphism X → Z with connected fibres suchthat −(KX+B) is nef over Z, the non-klt locus of (X,B) has at most two connectedcomponents near each fibre of X → Z. This was conjectured by Hacon and Han.

In a different direction we answer a question of Mark Gross on connectednessof the non-klt loci of certain pairs. This is motivated by constructions in MirrorSymmetry.

1. Introduction

We work over an algebraically closed field of characteristic zero.Pairs and their singularities play a fundamental role in higher dimensional algebraic

geometry. Let’s consider the simplest kind of pair, that is, a projective log smooth pair(X,B) whereX is a smooth projective variety and B =

∑biBi is a divisor with simple

normal crossing singularities and real coefficients bi ≥ 0; here Bi are distinct primedivisors. In this setting, the non-klt locus Nklt(X,B) of (X,B) is the union of the Biwith bi ≥ 1. In general, Nklt(X,B) can have any number of connected componentsas a topological space with the Zariski topology. But in special situations the non-klt locus exhibits interesting behaviour. For example, Shokurov [23, 5.7] proved inthe early 1990’s that if X is a surface and −(KX + B) is ample, then Nklt(X,B) isconnected. This was generalised to higher dimensions by Kollar [17, Theorem 17.4]which is known as the connectedness lemma or connectedness principle. The sameholds if we only assume −(KX +B) to be nef and big, and it also holds in the relativesetting when X is defined over a base Z in which case connectedness holds near eachfibre of X → Z assuming X → Z has connected fibres.

The connectedness principle plays an important role in higher dimensional alge-braic geometry. For example, it is used in the proof of existence of 3-fold flips [23], inthe proof of inversion of adjunction [17, Theorem 17.6], in the proofs of boundednessof complements and boundedness of Fano varieties [4, Proposition 5.1, Proposition6.7], in birational rigidity of Fano varieties [22], etc.

A natural question is what happens if we only assume that −(KX +B) is nef? Inthis case, Nklt(X,B) may not be connected. The easiest example is to take X = P1,B = B1 +B2 where Bi are distinct points. Kollar and Kovacs [18][16, Theorem 4.40]showed that in case KX + B ≡ 0 and the coefficients of B are ≤ 1, if connectednessfails, then this simple example is in a sense the reason, more precisely, X is birationalto a (possibly singular) model (X ′, B′) admitting a contraction X ′ → Y ′ where thegeneral fibres are P1 and bB′c has exactly two disjoint components both horizontalover Y ′ (X ′ is obtained by running a minimal model program on −bBc).

Date: October 16, 2020.2010 MSC: 14J17, 14J32, 14J45, 14E30.

1

2 Caucher Birkar

Hacon and Han [13] investigated the above phenomenon more generally. Theyshowed that if dimX ≤ 4 and if −(KX + B) is nef, then Nklt(X,B) has at mosttwo connected components. They conjectured that this holds in every dimension andthen showed that it follows from the termination of klt flips conjecture. One of ourmain results is to verify their conjecture without assuming termination of klt flips(see Theorem 1.2).

Another main result of this paper (see Theorem 1.4 and Corollaries 1.5, 1.6), ina different but somewhat related direction, is an answer to the following question ofMark Gross. The question is motivated by constructions in Mirror Symmetry, see[12] for related background. In fact this work started in response to this questionwhich was communicated privately.

Question 1.1. Consider the following setup:

• (X,B) is a projective log smooth pair where B =∑Bi is reduced,

• KX +B ≡∑aiBi with ai ≥ 0 real numbers (we say Bi is good if ai = 0),

• (X,C) has a zero-dimensional stratum x where C is the sum of the gooddivisors,• for each stratum V of (X,C), define KV +CV = (KX +C)|V by adjunction.

Then under what conditions CV is connected for every stratum V of dimension ≥ 2?

Here by a stratum of (X,C) we mean X itself or an irreducible component of theintersection of any subset of the irreducible components of C.

Gross pointed out that if (X,B) has a good log minimal model which is log Calabi-Yau, that is, a log minimal model (X ′, B′) withKX′+B′ ∼Q 0, then the connectednessin the question holds by [16, Theorem 4.40] mentioned above. If (X,B) has Kodairadimension zero, then standard conjectures of the minimal model program, includingthe abundance conjecture, imply that (X,B) has such a log minimal model but thecurrent technology of the minimal model program is not enough to guarantee existenceof good log minimal models in general in dimension ≥ 4. The problem is then to findother reasonable assumptions, instead of Kodaira dimension zero, so that a good logminimal model exists or at least that the desired connectedness holds.

It turns out that in the setting of the question it is enough to assume the followingproperty which is of a numerical nature so probably easy to check in explicit examples.Let φ : Y → X be the blowup of X at x and let E be the exceptional divisor. Assumethat φ∗(KX + B) − tE is not pseudo-effective for any real number t > 0 (a divisoris pseudo-effective if it is numerically the limit of effective divisors). Then CV isconnected for every stratum V of dimension ≥ 2. See 1.5 for a statement that worksin a much more general setting.

The non-pseudo-effectivity assumption of the previous paragraph is not as restric-tive as it may seem. In fact, it is conjecturally equivalent to (X,B) having Kodairadimension zero: indeed, assuming standard conjectures of the minimal model pro-gram, (X,B) has a log minimal model (X ′, B′) where KX′ +B′ is semi-ample, that is,|m(KX′ +B′)| is base point free for some sufficiently divisible m ∈ N. Thus there is acontraction g : X ′ → Z ′ such that KX′ +B′ ∼Q g

∗H where H is an ample Q-divisor.Now X 99K X ′ is an isomorphism near x because x does not belong to Supp

∑aiBi.

If dimZ ′ > 0, then we can find 0 ≤ P ′ ∼Q KX′ +B′ so that x′ ∈ SuppP ′ where x′ isthe image of x. This gives 0 ≤ P ∼Q KX +B so that x ∈ SuppP , so in this case

φ∗(KX +B)− tE ∼Q φ∗P − tE

On connectedness of non-klt loci of singularities of pairs 3

is pseudo-effective for some t > 0, a contradiction. Therefore, dimZ ′ = 0 whichexactly means that (X,B) has Kodaira dimension zero.

Since conditions similar to the above non-pseudo-effectivity condition appear againin this text we make a definition to ease notation. Given a projective variety X, apseudo-effective R-Cartier R-divisor L on X, and a prime divisor S over X (that ison birational models of X) we define the threshold

τS(L) = sup{t ∈ R≥0 | φ∗L− tS is pseudo-effective}where φ : W → X is any resolution on which S is a divisor. This is independent ofthe choice of the resolution, see Lemma 2.3. A relative version of the threshold cansimilarly be defined when X is projective over a base Z in which case we denote itby τS(L/Z).

Non-klt loci of anti-nef pairs. Our first main result concerns the non-klt lociof pairs (X,B) with −(KX +B) nef over some base.

Theorem 1.2. Let (X,B) be a pair and f : X → Z be a contraction. Assume−(KX +B) is nef over Z. Assume that the fibre of Nklt(X,B)→ Z over some pointz ∈ Z is not connected. Then we have:

(1) Nklt(X,B) → Z is surjective and its fibre over z has exactly two connectedcomponents;

(2) the pair (X,B) is lc and after base change to an etale neighbourhood of z,there exist a resolution φ : X ′ → X and a contraction X ′ → Y ′/Z such that if

KX′ +B′ := φ∗(KX +B)

and if F ′ is a general fibre of X ′ → Y ′, then (F ′, B′|F ′) is isomorphic to(P1, p1 + p2) for distinct points p1, p2.

Moreover, bB′c has two disjoint irreducible components S′, T ′, both horizon-tal over Y ′, and the images of S′, T ′ on X are the two connected componentsof Nklt(X,B).

As mentioned above, the theorem implies [13, Conjecture 1.1]. After completionof this work we learnt that Filipazzi and Svaldi [10] have also proved this result usingdifferent arguments.

Here is an example of (X,B) as in the theorem on which the non-klt locus hastwo components, one zero-dimensional and the other one-dimensional. Let X = P2,B1 be a line, x a closed point not contained in B1, and B2, . . . , B5 be distinct linespassing through x. Letting

B = B1 +1

2(B2 + · · ·+B5)

we can see that KX + B ≡ 0 and (X,B) is lc and Nklt(X,B) has two components,one is x and the other is B1.

The theorem was already known (in this or other forms) in dimension two [21], incase (X,B) is dlt and KX +B ≡ 0 assuming termination of klt flips [11] and withoutassuming this termination [18][16], and in any dimension assuming termination of kltflips and in dimension ≤ 4 without assuming termination [13].

In the opposite direction we have the following result.

Theorem 1.3. Let (X,B) be a pair and f : X → Z be a contraction. Assume−(KX + B) is nef over Z. Then the fibres of Nklt(X,B) → Z are connected if anyof the following conditions holds:

4 Caucher Birkar

(1) −(KX +B) is big over Z; or(2) Nklt(X,B)→ Z is not surjective; or(3) τS(−(KX +B)/Z) > 0 for every non-klt place S of (X,B).

Here by a non-klt place we mean a prime divisor S over X (that is, on birationalmodels of X) such that the log discrepancy a(S,X,B) = 0.

Case (1) is essentially the connectedness principle mentioned above. Case (3)implies cases (1) and (2) but in practice we first prove cases (1),(2) and then derivecase (3) from Theorem 1.2. There are situations where one can apply (3) but not(1) and (2). For example, consider the following: assume −(KX + B) is semi-ampleover Z defining a non-birational contraction X → T/Z; assume that Nklt(X,B)→ Tis not surjective but Nklt(X,B) → Z is surjective; then τS(−(KX + B)/Z) > 0 forevery non-klt place S of (X,B), so we can apply (3).

See also [13, Corollary 1.3] for some special situations in dimension ≤ 4 or anydimension assuming termination of klt flips.

Non-klt loci for Mirror Symmetry. The following result is the main steptowards answering Question 1.1 which works in a much more general setting.

Theorem 1.4. Assume that

(1) (X,B) is a projective Q-factorial dlt pair where B is a Q-divisor,(2) KX +B is pseudo-effective,(3) x ∈ X is a zero-dimensional non-klt centre of (X,B),(4) x is not contained in the restricted base locus B−(KX +B),(5) if φ : Y → X is the blowup at x with exceptional divisor E, then τE(KX+B) =

0, i.e. φ∗(KX +B)− tE is not pseudo-effective for any real number t > 0.

Then (X,B) has a good log minimal model which is log Calabi-Yau. More precisely,we can run a minimal model program on KX + B ending with a log minimal model(X ′, B′) with KX′ +B′ ∼Q 0.

Recall that for a Q-divisor L on a normal projective variety, the stable base locusis defined as

B(L) :=⋂m

Bs |mL|

where m runs over the natural numbers such that mL is an integral divisor. Therestricted base locus of L is defined as

B−(L) :=⋃

ε∈Q>0

B(L+ εA)

where A is any fixed ample divisor (the locus is independent of the choice of A).

Corollary 1.5. Under the assumptions of Theorem 1.4, suppose that no non-kltcentre of (X,B) is contained in B−(KX + B). For V = X or V a non-klt centreof (X,B), define KV + BV = (KX + B)|V . Then Nklt(V,BV ) is connected whendimV ≥ 2.

In the setting of Question 1.1, conditions (1)-(4) of the theorem are automaticallysatisfied: indeed, (X,B) is log smooth so we have (1); KX +B ≡

∑aiBi with ai ≥ 0,

so KX +B is pseudo-effective which is (2); x is a zero-dimensional stratum of (X,C),so it is a non-klt centre of both (X,C) and (X,B), so we have (3); x is containedonly in the good components of B, so x is not contained in Supp

∑aiBi, hence x is

not contained in B−(KX +B) ⊆ Supp∑aiBi so we have (4).


Applying 1.4 and 1.5, we will prove the following answer to 1.1.

Corollary 1.6. Assume that

(1) (X,B) is a projective log smooth pair where B =∑Bi is reduced,

(2) KX +B ≡∑aiBi with ai ≥ 0 real numbers (we say Bi is good if ai = 0),

(3) (X,C) has a zero-dimensional stratum x where C is the sum of the gooddivisors,

(4) for each stratum V of (X,C), define KV +CV = (KX +C)|V by adjunction,(5) if φ : Y → X is the blowup at x with exceptional divisor E, then τE(KX+B) =

0.

Then (X,B) has a good log minimal model which is log Calabi-Yau, and CV is con-nected for every stratum V of dimension ≥ 2.

Plan of the paper. We will prove 1.2 and 1.3 in Section 3 and 1.4 and 1.6 inSection 4. We will actually prove more general forms of these results in the settingof generalised pairs. Generalised pairs play a key role in the proofs.

Acknowledgements. This work was supported by a grant of the Royal Society.Thanks to Mark Gross for posing Question 1.1 which triggered the start of this workand for related discussions. Thanks to Christopher Hacon and Yifei Chen for helpfulcomments and corrections.

2. Preliminaries

All varieties in this paper are quasi-projective over an algebraically closed field ofcharacteristic zero unless otherwise stated.

2.1. Contractions. A contraction is a projective morphism f : X → Y of varietiessuch that f∗OX = OY (f is not necessarily birational). In particular, f has con-nected fibres and if X → Z → Y is the Stein factorisation of f , then Z → Y is anisomorphism.

2.2. Pseudo-effective thresholds. Given a projective morphism X → Z of vari-eties, a pseudo-effective R-Cartier R-divisor L on X, and a prime divisor S over X(that is on birational models of S) we define the pseudo-effective threshold τS(L/Z)as follows. Pick a birational contraction φ : Y → X from a normal variety so that Sis a Q-Cartier divisor on Y , e.g. a resolution of singularities of X. Then define

τS(L/Z) = sup{t ∈ R≥0 | φ∗L− tS is pseudo-effective/Z}.

The next lemma shows that this is well-defined. When Z is a point, we will simplydenote the threshold by τS(L).

Lemma 2.3. The threshold τS(L/Z) is independent of the choice of φ : Y → X.

Proof. Let φ : Y → X and φ′ : Y ′ → X be birational morphisms from normal vari-eties so that S, S′ are Q-Cartier divisors on Y, Y ′, respectively, where S, S′ representthe same divisor over X, that is, S′ is the birational transform of S. We denote

the corresponding thresholds by τφS (L/Z) and τφ′

S′ (L/Z). It is enough to show that

τφS (L/Z) = τφ′

S′ (L/Z) when the induced map ψ : Y ′ 99K Y is a morphism because inthe general case we can use a common resolution of Y ′, Y .

6 Caucher Birkar

If φ′∗L− tS′ is pseudo-effective/Z, then obviously the pushdown

ψ∗(φ′∗L− tS′) = φ∗L− tS

is pseudo-effective/Z. Thus τφ′

S′ (L/Z) ≤ τφS (L/Z). Conversely, if φ∗L− tS is pseudo-effective/Z, then ψ∗(φ∗L− tS) is pseudo-effective/Z. But

ψ∗(φ∗L− tS) = φ′∗L− tψ∗S ≤ φ′∗L− tS′,

as ψ∗S ≥ S′, hence φ′∗L−tS′ is pseudo-effective/Z (this is where we use the Q-Cartier

condition of S so that we can take the pullback ψ∗S). Thus τφ′

S′ (L/Z) ≥ τφS (L/Z)

which in turn implies the equality τφ′

S′ (L/Z) = τφS (L/Z).�

2.4. Pairs. A sub-pair (X,B) consists of a normal quasi-projective variety X andan R-divisor B such that KX + B is R-Cartier. If the coefficients of B are at most1 we say B is a sub-boundary, and if in addition B ≥ 0, we say B is a boundary. Asub-pair (X,B) is called a pair if B ≥ 0.

Let φ : W → X be a log resolution of a sub-pair (X,B). Let KW + BW be thepulback of KX +B. The log discrepancy of a prime divisor D on W with respect to(X,B) is 1− 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 a(D,X,B) is ≥ 0 (resp. > 0)(resp. ≥ ε) for every D. When(X,B) is a pair we remove the sub and say the pair is lc, etc. Note that if (X,B) isa lc pair, then the coefficients 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 onbirational models of X such that a(D,X,B) ≤ 0. A non-klt centre is the image onX of a non-klt place. When (X,B) is lc, a non-klt centre is also called an lc centre.The non-klt locus Nklt(X,B) of a sub-pair (X,B) is the union of the non-klt centreswith reduced structure.

2.5. Minimal model program (MMP). We will use standard results of the min-imal model program (cf. [19][7]). Assume (X,B) is a pair, X → Z is a projectivemorphism, H is an ample/Z R-divisor, and that KX + B + H is nef/Z. Suppose(X,B) is klt or that it is Q-factorial dlt. Then we can run an MMP/Z on KX + Bwith scaling of H. If (X,B) is klt and if either KX + B or B is big/Z, then theMMP terminates [7]. If (X,B) is Q-factorial dlt, then in general we do not knowwhether the MMP terminates but we know that in some step of the MMP we reacha model Y on which KY + BY , the pushdown of KX + B, is a limit of movable/ZR-divisors: indeed, if the MMP terminates, then the claim is obvious; otherwise theMMP produces an infinite sequence Xi 99K Xi+1 of flips and a decreasing sequenceλi of numbers in (0, 1] such that KXi + Bi + λiHi is nef/Z; by [7][6, Theorem 1.9],limλi = 0; in particular, if Y := X1, then KY + BY is the limit of the movable/ZR-divisors KY +BY + λiHY .

Similar remarks apply to generalised pairs defined below.

2.6. Fano pairs. Let (X,B) be a pair and X → Z a contraction. We say (X,B)is log Fano over Z if it is lc and −(KX + B) is ample over Z; if B = 0 we just sayX is Fano over Z. We say X is of Fano type over Z if (X,B) is klt log Fano overZ for some choice of B; it is easy to see this is equivalent to existence of a big/ZQ-boundary (resp. R-boundary) Γ so that (X,Γ) is klt and KX + Γ ∼Q 0/Z (resp.∼R instead of ∼Q).


Assume X is of Fano type over Z. Then we can run the MMP over Z on anyR-Cartier R-divisor D on X which ends with some model Y [7]. If DY is nef overZ, we call Y a minimal model over Z for D. If DY is not nef/Z, then there is aDY -negative extremal contraction Y → T/Z with dimY > dimT and we call Y aMori fibre space over Z for D.

2.7. Generalised pairs. For the basic theory of generalised polarised pairs see [8,Section 4]. Below we recall some of the main notions and discuss some basic proper-ties.

(1) A generalised pair consists of

• a normal variety X equipped with a projective morphism X → Z,• an R-divisor B ≥ 0 on X, and• a b-R-Cartier b-divisor over X represented by some projective birational mor-

phism X ′φ→ X and R-Cartier divisor M ′ on X ′

such that M ′ is nef/Z and KX +B +M is R-Cartier, where M := φ∗M′.

We usually refer to the pair by saying (X,B +M) is a generalised pair with data

X ′φ→ X → Z and M ′. Since a b-R-Cartier b-divisor is defined birationally, in

practice we will often replace X ′ with a resolution and replace M ′ with its pullback.When Z is not relevant we usually drop it and do not mention it: in this case onecan just assume X → Z is the identity. When Z is a point we also drop it but saythe pair is projective.

Now we define generalised singularities. Replacing X ′ we can assume φ is a logresolution 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 logdiscrepancy a(D,X,B +M) is defined to be 1− µDB′.

We say (X,B+M) is generalised lc (resp. generalised klt)(resp. generalised ε-lc) iffor each D the generalised log discrepancy a(D,X,B+M) is ≥ 0 (resp. > 0)(resp. ≥ε). A generalised non-klt place of (X,B+M) is a prime divisor D on birational modelsof X with a(D,X,B+M) ≤ 0, and a generalised non-klt centre of (X,B+M) is theimage of a generalised non-klt place. The generalised non-klt locus Nklt(X,B + M)of the generalised pair is the union of all the generalised non-klt centres.

We will also use similar definitions when B is not necessarily effective in whichcase we have a generalised sub-pair.

(2) Let (X,B+M) be a generalised pair as in (1). We say (X,B+M) is generaliseddlt if it is generalised lc and if η is the generic point of any generalised non-klt centreof (X,B + M), then (X,B) is log smooth near η and M ′ = φ∗M holds over aneighbourhood of η. Note that when M ′ = 0, then (X,B) is generalised dlt iff it isdlt in the usual sense.

The generalised dlt property is preserved under the MMP. Indeed, assume (X,B+M) is generalised dlt and that X 99K X ′′/Z is a divisorial contraction or a flip withrespect to KX + B + M . Replacing φ we can assume X ′ 99K X ′′ is a morphism.Let B′′,M ′′ be the pushdowns of B,M and consider (X ′′, B′′ +M ′′) as a generalisedpair with data X ′ → X ′′ → Z and M ′. Then (X ′′, B′′ + M ′′) is also generalised dltbecause it is generalised lc and because X 99K X ′′ is an isomorphism over the genericpoint of any generalised non-klt center of (X ′′, B′′ +M ′′).

8 Caucher Birkar

(3) Let (X,B + M) be a generalised pair as in (1) and let ψ : X ′′ → X be aprojective birational morphism from a normal variety. Replacing φ we can assume φfactors through ψ. 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 ′′ → Zand M ′.

Assume that we can write B′′ = ∆′′ + G′′ where (X ′′,∆′′ + M ′′) is Q-factorialgeneralised dlt, G′′ ≥ 0 is supported in b∆′′c, and every exceptional prime divisor ofψ is a component of b∆′′c. Then we say (X ′′,∆′′+M ′′) is a Q-factorial generalised dltmodel of (X,B+M). Such models exist by the next lemma (also see [14, Proposition3.3.1] and [8, Lemma 4.5]). If (X,B +M) is generalised lc, then G′′ = 0.

Lemma 2.8. Let (X,B + M) be a generalised pair with data φ : X ′ → X and M ′.Then the pair has a Q-factorial generalised dlt model.

Proof. Replacing φ we can assume it is a log resolution. Write

KX′ +B′ +M ′ = φ∗(KX +B +M).

Write B = ∆ +G where ∆ is obtained from B by replacing each coefficient > 1 with1. So G ≥ 0 is supported in b∆c. Let ∆′ be the sum of the birational transform of∆ and the reduced exceptional divisor of φ. Let G′ := B′ −∆′.

Run an MMP on KX′ + ∆′ + M ′ over X with scaling of an ample divisor. Wereach a model X ′′ on which KX′′ + ∆′′ +M ′′ is a limit of movable/X R-divisors. Byconstruction,

KX′′ + ∆′′ +M ′′ +G′′ ≡ 0/X.

So for any exceptional/X prime divisor S′′ on X ′′, −G′′|S′′ is pseudo-effective overthe image of S′′ in X. Therefore, by the general negativity lemma [6, Lemma 3.3],G′′ ≥ 0 (note that [6, Lemma 3.3] implicitly assumes the base field is uncountable; ifin our case the base field is countable, then we do a base change and then apply thelemma as in that case for the very general curves C ′′ of S′′ contracted over X, wehave −G′′ · C ≥ 0).

By definition of ∆′, each exceptional prime divisor of X ′′ → X is a componentof b∆′′c. Moreover, each component of G′′ is either exceptional in which case it is acomponent of b∆′′c, or non-exceptional in which case it is the birational transform ofa component of G hence again a component of b∆′′c. Therefore, (X ′′,∆′′ +M ′′) is aQ-factorial generalised dlt model of (X,B +M).

�

Lemma 2.9. Let d, p be natural numbers and Φ be a DCC set of non-negative realnumbers. Then there is a positive real number t > 0 depending only on d, p,Φ satis-fying the following. Assume that (X,B + M) is a Q-factorial generalised pair withdata X ′ → X → Z and M ′ and D ≥ 0 is an R-divisor such that the coefficients ofB,D are in Φ and pM ′ is a Cartier divisor. Then we have:

(1) if (X,B+ (1− t)D+M) is generalised lc, then (X,B+D+M) is generalisedlc;

(2) if (X,B + (1 − t)D + M) is generalised lc, X → Z is a Mori fibre space forthe pair, and KX +B +D +M is nef over Z, then

KX +B +D +M ≡ 0/Z.


Proof. (1) If this is not true, then there exist a strictly decreasing sequence ti of realnumbers approaching 0 and a sequence (Xi, Bi+Mi), Di of pairs and divisors as in thelemma such that (Xi, Bi + (1− ti)Di +Mi) is generalised lc but (Xi, Bi +Di +Mi)is not generalised lc. Let ui be the generalised lc threshold of Di with respect to(Xi, Bi + Mi). Then ui belongs to an ACC set depending only on d, p,Φ, by [8,Theorem 1.5]. On the other hand, 1 > ui ≥ 1 − ti, so the ui form a sequenceapproaching 1. This contradicts the ACC property.

(2) Now we prove the second claim. Again if it is not true, then there exist a strictlydecreasing sequence ti of real numbers approaching 0 and a sequence (Xi, Bi+Mi), Di

of pairs and divisors as in the lemma such that

(Xi, Bi + (1− ti)Di +Mi)

is generalised lc, Xi → Zi a Mori fibre space structure, and with KXi +Bi +Di +Mi

nef over Zi but such that

KXi +Bi +Di +Mi 6≡ 0/Zi.

Let 1− ti < vi < 1 be the number such that

KXi +Bi + viDi +Mi ≡ 0/Zi.

By (1), (Xi, Bi +Di +Mi) is generalised lc. Let Fi be a general fibre of Xi → Zi.Restricting to Fi, we get

KFi +BFi + viDFi +MFi ≡ 0

where (Fi, BFi + viDFi +MFi) naturally inherits the structure of a generalised pair,induced by (Xi, Bi + viDi +Mi), with nef part MF ′

i= M ′i |F ′

iwhere F ′i is the fibre of

X ′i → Zi corresponding to Fi. Now the coefficients of BFi , DFi belong to Φ and pMF ′i

is Cartier. Then we get a contradiction, by the global ACC [8,Theorem 1.6] as thecoefficients of BFi + viDFi are in a DCC but not finite set.

�

Lemma 2.10. Assume that (X,B + M) is a generalised lc pair with data X ′ → Xand M ′. Assume that (X,C+N) is generalised klt with data X ′ → X and N ′. If S isa prime divisor over X with a(S,X,B+M) < 1, then there is a birational contractionY → X from a normal variety such that S is a divisor on Y and −S is ample overX.

Proof. Take a small rational number u > 0 and consider the generalised pair

(X,uC + (1− u)B + uN + (1− u)M)

with nef part uN ′ + (1− u)M ′. The pair is generalised klt and

a(S,X, uC + (1− u)B + uN + (1− u)M) < 1.

Thus replacing (X,B +M) with the pair above, we can assume that (X,B +M) isgeneralised klt.

There exist an ample divisor A′ and an effective divisor G′ on X ′ such that A′ +G′ ∼Q 0/X. Take a small rational number t > 0 and general element L′ ∼Q M

′+ tA′.Letting L be the pushdown of L′ we see that (X,B + tG+ L) is klt and

a(S,X,B + tG+ L) < 1.

Thus replacing (X,B + M) with (X,B + tG + L) we can assume M ′ = 0 and that(X,B) is klt.

10 Caucher Birkar

Now by [7], there is a crepant terminal model (V,BV ) of (X,B). Then S is adivisor on V and the coefficient of S in BV is positive. Since

KV +BV − cS ∼R −cS/X,where c > 0 is sufficiently small, we see that −S has an ample model Y over X. Bythe negativity lemma, S is not contracted over Y . Abusing notation we denote thebirational transform of S on Y again by S. Then on Y , −S is ample over X, so weget the desired model.

�

2.11. Generalised adjunction for fibrations. Consider the following set-up. As-sume that

• (X,B +M) is a generalised sub-pair with data X ′ → X → Z and M ′,• f : X → Z is a contraction with dimZ > 0,• (X,B +M) is generalised sub-lc over the generic point of Z, and• KX +B +M ∼R 0/Z.

We define the discriminant divisor BZ for the above setting. Let D be a prime divisoron Z. Let t be the largest real number such that (X,B+tf∗D+M) is generalised sub-lc over the generic point of D. This makes sense even if D is not Q-Cartier because weonly need the pullback f∗D over the generic point of D where Z is smooth. We thenput the coefficient of D in BZ to be 1− t. Note that since (X,B +M) is generalisedsub-lc over the generic point of Z, t is a real number, that is, it is not −∞ or +∞.Having defined BZ , we can find MZ giving

KX +B +M ∼R f∗(KZ +BZ +MZ)

where MZ is determined up to R-linear equivalence. We call BZ the discriminantdivisor of adjunction for (X,B +M) over Z. If B,M ′ are Q-divisors and KX +B +M ∼Q 0/Z, then BZ is a Q-divisor and we can choose MZ also to be a Q-divisor.

For any birational morphism Z ′ → Z from a normal variety, we can similarly defineBZ′ and MZ′ .

For more details about adjunction for generalised fibrations, we refer to [9] and§6.1 of [2].

Theorem 2.12. Under the above notation, assume that X is projective, (X,B+M)is generalised lc over the generic point of Z, and B,M ′ are Q-divisors, and M ′ isglobally nef. Assume that Z ′ → Z is a high resolution. Then MZ′ is nef, and for anybirational morphism Z ′′ → Z ′, MZ′′ is the pullback of MZ′.

In particular, we can regard (Z,BZ +MZ) as a generalised sub-pair with nef partMZ′ . The theorem is proved in [1] (based on [15]) when M ′ = 0, and in [9] in general.We will use the theorem in the proof of 1.4.

2.13. Connected components and etale neighbourhoods. Let Z be a varietyand g : N → Z be a projective morphism where N is a scheme. In the discussionbelow keep in mind that the topology of the fibre of g over a point z ∈ Z is the sameas the subset topology on g−1{z} induced by the topology on N where g−1{z} meansthe set-theoretic inverse image (a similar fact holds in general for any morphism ofschemes).

(1) The fibres of g are connected iff its fibres over closed points are connected:indeed, assume the latter holds but that the fibre of g over some point z ∈ Z isnot connected; then in the Stein factorisation N → S → Z, the fibre of S → Z


over z is not connected; but then the fibres of S → Z over closed points in someneighbourhood of z are also not connected, hence the fibres of g over the closedpoints in this neighbourhood are not connected, a contradiction.

(2) For each closed point z ∈ Z, there is an etale neighbourhood Z → Z witha closed point z mapping to z such that there is a 1-1 correspondence between theconnected components of N := N ×Z Z, and the connected components of the fibreof g over z [16, 4.38.1]. If the ground field is C, then this essentially says that theconnected components of N over a small analytic neighbourhood of z correspond tothe connected components of the fibre N → Z over Z.

3. Non-klt loci of anti-nef pairs

In this section we prove Theorems 1.2 and 1.3 in the more general framework ofgeneralised pairs. Using generalised pairs is important for the proofs even if one isonly interested in usual pairs.

Theorem 3.1. Let (X,B + M) be a generalised pair with data X ′ → X → Z andM ′ where f : X → Z is a contraction. Assume −(KX +B+M) is nef over Z. Thenthe fibres of

Nklt(X,B +M)→ Z

are connected if any of the following conditions holds:

(1) −(KX +B +M) is big over Z; or(2) Nklt(X,B +M)→ Z is not surjective; or(3) τS(−(KX+B+M)/Z) > 0 for every generalised non-klt place S of (X,B+M).

We prove cases (1) and (2) first and then prove case (3) towards the end of thissection.

Remark 3.2. In view of 2.13, to prove the theorem, it is enough to prove theweaker statement that Nklt(X,B +M) is connected near each fibre of X → Z as allthe conditions (1)-(3) are preserved after etale base change. We explain this point indetail. By 2.13(1), it is enough to show that the fibres of

N := Nklt(X,B +M)→ Z

over closed points are connected. And by 2.13(2), for each closed point z ∈ Z, there

is an etale neighbourhood Z → Z with a closed point z mapping to z such that thereis a 1-1 correspondence between the connected components of N := N ×Z Z, andthe connected components of the fibre of g over z. Fibre product with Z induces ageneralised pair (X, B+M) with data X ′ → X → Z and M ′ where N = Nklt(X, B+

M). It is enough to show that Nklt(X, B+ M) is connected near the fibre of X → Z

over z, hence replacing (X,B + M), X ′ → X → Z and M ′ with (X, B + M),

X ′ → X → Z and M ′, respectively, it is enough to show that Nklt(X,B + M) isconnected near each fibre of X → Z.

Lemma 3.3. Theorem 3.1 (1) holds.

Proof. As pointed out above it is enough to show that Nklt(X,B +M) is connectednear each fibre of X → Z. We can then use [4, Lemma 2.14] which reduces thestatement to the connectedness principle for usual pairs.

�

Lemma 3.4. Theorem 3.1 (2) holds.

12 Caucher Birkar

Proof. Step 1. It is enough to show that Nklt(X,B+M) is connected near each fibreof X → Z over closed points. Assume that Nklt(X,B + M) is not connected nearthe fibre of X → Z over some closed point z. Shrinking Z around z, we can assumethat Nklt(X,B + M) is not connected globally. Extending the ground field we canassume it is not countable.

Let L := −(KX + B + M) and let L′ be the pullback of L on X ′. Since L is nefover Z, L′ is nef over Z, hence M ′ + L′ is nef over Z. Consider the generalised pair(X,B+M +L) with data X ′ → X → Z and M ′+L′. The generalised non-klt locusof (X,B + M + L) coincides with that of (X,B + M) because L being nef over Zmeans that for any prime divisor S over X we have

a(S,X,B +M + L) = a(S,X,B +M).

Thus replacing M ′ with M ′ + L′ we can assume that KX +B +M ≡ 0/Z.

Step 2. Let (X ′′,∆′′ + M ′′) be a Q-factorial generalised dlt model of (X,B + M)which exists by Lemma 2.8. Denoting X ′′ → X by ψ, we have

KX′′ + ∆′′ +G′′ +M ′′ = ψ∗(KX +B +M)

where G′′ ≥ 0 is supported in b∆′′c. We can assume X ′ 99K X ′′ is a morphism, so wecan consider (X ′′,∆′′ +G′′ +M ′′) as a generalised pair with nef part M ′. Since

Nklt(X,B +M) = ψ(Nklt(X ′′,∆′′ +G′′ +M ′′)),

we deduce thatNklt(X ′′,∆′′ +G′′ +M ′′)

is not connected over z. Thus we can replace (X,B +M) with (X ′′,∆′′+G′′+M ′′),hence we can assume that the following condition holds:

(∗) B = ∆ + G where G ≥ 0 is supported in b∆c and (X,∆ − t b∆c + M) isQ-factorial generalised klt for some t ∈ (0, 1).

The condition implies that any generalised non-klt centre of (X,B+M) is containedin the support of t b∆c+G, so we have

b∆c ⊆ Nklt(X,B +M) ⊆ Supp(t b∆c+G) = b∆c ,giving

Nklt(X,B +M) = b∆c .By assumption, Nklt(X,B + M) is vertical over Z, so b∆c is vertical over Z. Thust b∆c+G is also vertical over Z, so

KX + ∆− t b∆c+M ≡ 0

over some non-empty open subset of Z, hence KX +∆− t b∆c+M is pseudo-effectiveover Z.

Step 3. We can run an MMP on KX + ∆ − t b∆c + M over Z with scaling ofsome ample divisor H [8, Lemma 4.4] but we do not claim that the MMP terminates.However, if λi are the numbers that appear in the MMP, then limλi = 0, by [8,Lemma 4.4]. Since

KX + ∆ +G+M ≡ 0/Z,

the divisor t b∆c+G is numerically positive on the extremal ray of each step of theMMP. Clearly the condition (∗) is preserved by the MMP, so the property

Nklt(X,B +M) = b∆c


is also preserved. Also note that since KX + ∆− t b∆c+M is pseudo-effective overZ, every step of the MMP is a divisorial contraction or a flip.

Step 4. We claim that Nklt(X,B +M) remains disconnected near the fibre over zduring the MMP. More precisely, we show that there is a 1-1 correspondence (givenby divisorial pushdown) between the connected components of Nklt(X,B+M) and ofNklt(X ′′, B′′+M ′′) for any model X ′′ appearing in the MMP. Indeed, say X 99K X ′′

is the first step of the MMP which is either a divisorial contraction or a flip. Firstassume X 99K X ′′ is a flip and let X → V be the corresponding flipping contraction.Since t b∆c + G is ample over V , some connected component of Supp(t b∆c + G),say C, intersects every positive-dimensional fibre of X → V . Recall that C is alsoa connected component of Nklt(X,B + M). But by Lemma 3.3, Nklt(X,B + M)is connected near any fibre of X → V as −(KX + B + M) is nef and big over V .Therefore, no connected component of Nklt(X,B + M) other than C intersects theexceptional locus of X → V . Thus each connected component of Nklt(X ′′, B′′+M ′′)is just the birational transform of a connected component of Nklt(X,B + M). Theclaim is then proved in the flip case.

A similar argument shows that if X 99K X ′′ is a divisorial contraction, then exactlyone connected component C of Supp(t b∆c+G), hence of Nklt(X,B+M), intersectsthe exceptional divisor. Moreover, in this case C is not contracted by X 99K X ′′,that is, C is not equal to the exceptional divisor, by the negativity lemma, becauset b∆c+G is ample over X ′′. Thus each connected component of Nklt(X ′′, B′′ +M ′′)is just the divisorial pushdown of a connected component of Nklt(X,B + M). Thisproves the claim in the divisorial contraction case.

Since the claim holds in each step of the MMP, it holds on any model appearingin the MMP.

Step 5. Let F1, . . . , Fr be the irreducible components of the fibre of X → Z overz. We claim that there is i such that Fi is not contained in Nklt(X,B+M) and thatthis is preserved by the MMP, that is, the birational transform of Fi remains outsidethe non-klt locus during the MMP.

By assumption, X → Z is a contraction, so its fibres over z is connected whichmeans the set-theoretic inverse image f−1{z} is connected. On the other hand, weassumed that Nklt(X,B +M) is not connected near the fibre of X → Z over z, so

f−1{z} 6⊆ Nklt(X,B +M) = Supp(t b∆c+G).

Thus some of the Fi are not contained in Nklt(X,B + M); rearranging the indiceswe can assume that F1, . . . , Fs are not contained in Nklt(X,B+M) but Fs+1, . . . , Frare contained.

Let X 99K X ′′ be the first step of the MMP. Recall that t b∆c + G is positive onthe extremal ray of each step of the MMP. Thus if X 99K X ′′ is a divisorial con-traction, then Nklt(X ′′, B′′ +M ′′) contains the image of the exceptional divisor, andif X 99K X ′′ is a flip, then similarly Nklt(X ′′, B′′ + M ′′) contains the flipped locus,that is, the exceptional locus of X ′′ → V where X → V is the corresponding flippingcontraction. This implies that any irreducible component of the fibre of X ′′ → Zover z which is not contained in Nklt(X ′′, B′′+M ′′) is the birational transform of oneof the F1, . . . , Fs. Therefore, after finitely many steps, the irreducible components ofthe fibre over z not contained in the non-klt locus stabilise, that is, replacing X wecan assume that on each model appearing in the MMP, the birational transforms of

14 Caucher Birkar

the F1, . . . , Fs are exactly the irreducible components of the fibre over z which arenot contained in the non-klt locus.

Step 6. In this step Fj denotes one of the components F1, . . . , Fs introduced in theprevious step. Since Fj remains outside Supp(t b∆c+G) in the course of the MMP,the map X 99K X ′′ is an isomorphism near the generic point of Fj for any model X ′′

that appears in a step of the MMP. Since the MMP is an MMP with scaling andsince limλi = 0 for the number λi in the MMP, we deduce that Fj is not containedin the stable base locus

B(KX + ∆− t b∆c+M + λiH)

for any i. In particular, this implies that

(KX + ∆− t b∆c+M + λiH)|Fj

is pseudo-effective for every i. But then

(KX + ∆− t b∆c+M)|Fj

is pseudo-effective. Therefore,

(KX + ∆− t b∆c+M) · C ≥ 0

for any curve C ⊂ Fj outside some countable union of subvarieties of Fj . Since theground field is not countable, this countable union is not equal to Fj .

Step 7. Since the fibre of X → Z over z is connected, there is 1 ≤ j ≤ s such thatFj intersects Nklt(X,B + M). By the previous step, we can find a curve C ⊂ Fjintersecting Nklt(X,B +M) but satisfying

(KX + ∆− t b∆c+M) · C ≥ 0.

Then C intersects Supp(t b∆c+G) but is not contained in it. Therefore,

(t b∆c+G) · C > 0

which contradicts

(KX + ∆− t b∆c+M + t b∆c+G) · C = 0.

�

Next we treat a generalised version of Theorem 1.2.

Theorem 3.5. Let (X,B +M) be a generalised pair with data X ′ → X → Z wheref : X → Z is a contraction. Assume −(KX +B+M) is nef over Z and that the fibreof

g : Nklt(X,B +M)→ Z

over some point z ∈ Z is not connected. Then we have:

(1) g is surjective and its fibre over z has exactly two connected components;(2) the pair (X,B+M) is generalised lc and after base change to an etale neigh-

bourhood of z and replacing X ′ with a high resolution, there exist a contractionX ′ → Y ′/Z such that if

KX′ +B′ +M ′ := φ∗(KX +B +M)

and if F ′ is a general fibre of X ′ → Y ′, then M ′|F ′ ≡ 0 and (F ′, B′|F ′) isisomorphic to (P1, p1 + p2) for distinct points p1, p2.


Moreover, bB′c has two disjoint irreducible components S′, T ′, both horizon-tal over Y ′, and the images of S′, T ′ on X are the two connected componentsof Nklt(X,B +M).

Proof. The proof is similar to the proof of [16, Proposition 4.37] but with some crucialdifferences.

Step 1. By Lemma 3.4, Theorem 3.1(2) holds, so g is surjective as we are assum-ing that the fibre of g over z is not connected (the corresponding argument in [16,Proposition 4.37] instead uses torsion freeness of certain higher direct image sheaveswhen M ′ = 0 and KX +B ∼Q 0/Z).

LetL := −(KX +B +M)

and let L′ be the pullback of L on X ′. Consider the generalised pair (X,B+M +L)with nef part M ′ + L′. Writing

KX′ +B′ +M ′ := φ∗(KX +B +M)

we getKX′ +B′ +M ′ + L′ := φ∗(KX +B +M + L)

meaning B′ is unchanged after adding L. The generalised log discrepancies of the twopairs (X,B + M + L) and (X,B + M) are equal, in particular, the non-klt locus of(X,B+M +L) coincides with that of (X,B+M). Thus replacing M ′ with M ′+L′

we can assume that KX +B +M ≡ 0/Z.

Step 2. Let Z be an etale neighbourhood of z with a point z mapping to z. Fibreproduct with Z gives a generalised pair (X, B+M) with data X ′ → X → Z and M ′,

and Nklt(X, B + M) is the inverse image of Nklt(X,B + M) under the morphism

X → X. If the fibre ofNklt(X, B + M)→ Z

over z has exactly two connected components, then the fibre of g over z also hasexactly two connected components because the former fibre maps surjectively ontothe latter fibre (note we already know the latter fibre is not connected). Therefore,to prove (1) and (2) we are free to replace Z with an etale neighbourhood of z.

Step 3. Let R be the closure of z in Z. Since the fibre of g over z is not connected,shrinking Z around z, we can assume that g−1R is not connected and that everyirreducible component of g−1R maps onto R. In particular, the fibre of g over anypoint ofR is not connected, and we can choose a closed point v ofR so that the numberof connected components of g−1{v} is at least the number of connected components ofg−1{z}. It is then enough to show that g−1{v} has exactly two connected components.

By 2.13, after base change to an etale neighbourhood of v, we can assume thatNklt(X,B +M) is not connected and that distinct connected components of g−1{v}are contained in distinct connected components of Nklt(X,B + M) (note that thenumber of connected components of g−1{v} is unchanged by the base change). It isthen enough to prove (2) without taking further etale base change.

Step 4. After taking a Q-factorial generalised dlt model, as in Step 2 of the proofof Lemma 3.4, we can replace X so that the following holds:

(∗) B = ∆ + G where G ≥ 0 is supported in b∆c and (X,∆ − t b∆c + M) isQ-factorial generalised klt for some real number t > 0.

16 Caucher Birkar

In particular, we have

Nklt(X,B +M) = Supp(t b∆c+G) = b∆c .Moreover, since Nklt(X,B + M) is horizontal over Z, t b∆c + G is also horizontalover Z. Therefore, we see that

KX + ∆− t b∆c+M ≡ −(t b∆c+G)/Z

is not pseudo-effective over Z.

Step 5. We can run an MMP on KX + ∆− t b∆c+M over Z ending with a Morifibre space X ′′ → Y ′/Z [8, Lemma 4.4]. Since t b∆c+ G is positive on the extremalray in each step of the MMP, arguing as in Step 4 of the proof of Lemma 3.4, wesee that the number of the connected components of Nklt(X,B + M) remains thesame throughout the MMP. Moreover, t b∆′′c+G′′ is ample over Y ′. So at least oneirreducible component S′′ of b∆′′c is ample over Y ′ which implies that S′′ intersectsevery fibre of X ′′ → Y ′. In particular, if T ′′ is any vertical/Y ′ component of b∆′′c,then S′′ intersects T ′′. Since (X ′′,∆′′ − t b∆′′c+M ′′) is generalised klt,

Nklt(X ′′, B′′ +M ′′) = Supp(t⌊∆′′⌋

+G′′) =⌊∆′′⌋.

Thus b∆′′c has at least two connected components, hence every irreducible compo-nent of any connected component of b∆′′c is horizontal over Y ′.

Step 6. Let C′′1 , C′′2 be two connected components of Nklt(X ′′, B′′ +M ′′) where S′′

is an irreducible component of C′′1 . Pick an irreducible component T ′′ of C′′2 . Let F ′′

be a general fibre of X ′′ → Y ′. If dimT ′′ ∩ F ′′ > 0, then S′′ intersects T ′′ ∩ F ′′ as S′′

is ample over Y ′, which is not possible as S′′ ∩ T ′′ = ∅. Therefore, dimF ′′ = 1, henceF ′′ ' P1. Since

(KX′′ + ∆′′ +G′′ +M ′′)|F ′′ ≡ 0

and since at least two irreducible components S′′, T ′′ of b∆′′c intersect F ′′ we see thatnear F ′′ we have B′′ = ∆′′ +G′′ = S′′ + T ′′, and that M ′′|F ′′ ≡ 0. Therefore,

(F ′′, B′′|F ′′) = (F ′′, (S′′ + T ′′)|F ′′) ' (P1, p1 + p2)

where p1, p2 are two distinct points.

Step 7. We claim that (X ′′, B′′ + M ′′) is generalised plt whose only generalisednon-klt places are S′′, T ′′. By the previous two steps, C′′1 = S′′, C′′2 = T ′′ are the onlyconnected components of Nklt(X ′′, B′′ + M ′′) otherwise we would find a horizontalover Y ′ component of b∆′′c other than S′′, T ′′ which is not possible. Assume that(X ′′, B′′ +M ′′) has another generalised non-klt place, say R′′′. Consider (X ′′, sB′′ +sM ′′) where s is the smallest number so that the pair is generalised lc. Replacing R′′′

we can assume R′′′ is a generalised non-klt place of (X ′′, sB′′ + sM ′′). Then we canextract R′′′ say via an extremal contraction X ′′′ → X ′′. Note that R′′′ is vertical overY ′. Also R′′′ maps into one of S′′, T ′′, say S′′, so R′′′ does not intersect the birationaltransform T ′′′ of T ′′.

Since X ′′ is of Fano type over Y ′, so is X ′′′. Then we can run an MMP on −R′′′over Y ′ ending with a good minimal model X ′′′′. Since R′′′ is vertical over Y ′, −R′′′′defines a contraction X ′′′′ → V ′/Y ′ where V ′ → Y ′ is birational. Moreover, R′′′′ andT ′′′′ are disjoint: indeed, if KX′′′ +B′′′+M ′′′ is the pullback of KX′′ +B′′+M ′′, thenapplying Theorem 3.5 (1) (through Lemma 3.3), we see that at most one connectedcomponent of Nklt(X ′′′, B′′′+M ′′′) intersects the exceptional locus of each step of the


MMP and that component is the one containing R′′′. However, T ′′′′ is horizontal overV ′ as X ′′′′ → V ′ and X ′′ → Y ′ are the same over the generic point of Y ′. But thenT ′′′′ intersects every fibre of X ′′′′ → V ′, hence it also intersects R′′′′, a contradiction.Therefore, (X ′′, B′′ +M ′′) is generalised plt.

Step 8. Now replacing the given morphism φ : X ′ → X we can assume X ′ is acommon resolution of X,X ′′. Recall

KX′ +B′ +M ′ := φ∗(KX +B +M)

and let F ′ be a general fibre of X ′ → Y ′. Since X ′′ → Y ′ has relative dimensionone and since the exceptional locus of X ′ → X ′′ maps onto a closed subset of X ′′

of codimension ≥ 2, we see that X ′ → X ′′ is an isomorphism over the generic pointof Y ′. Thus if F ′′ is the fibre of X ′′ → Y ′ corresponding to F ′, then (F ′, B′|F ′) isisomorphic to (F ′′, B′′|F ′′) which is in turn isomorphic to (P1, p1 + p2) for distinctpoints p1, p2. Moreover, if S′, T ′ on X ′ are the birational transforms of S′′, T ′′, thenbB′c = S′ + T ′ as we showed that S′′, T ′′ are the only generalised non-klt places of(X ′′, B′′ +M ′′). In addition, M ′|F ′ ≡ 0 as we already showed that M ′′|F ′′ ≡ 0.

It is then clear that the images of S′, T ′ on X are the only generalised non-kltcentres of (X,B +M). Since Nklt(X,B +M) is not connected, the images of S′, T ′

are disjoint and each gives a connected component of Nklt(X,B +M).�

Proof. (of Theorem 3.1) Cases (1) and (2) were proved in Lemmas 3.3, 3.4, respec-tively. We treat case (3). Assume that the fibre of

Nklt(X,B +M)→ Z

over z is not connected. Then by Theorem 3.5, after base change to an etale neigh-bourhood of z, we can assume that (X,B + M) is generalised lc and that replacingX ′ with a high resolution, there is a contraction X ′ → Y ′/Z such that if

KX′ +B′ +M ′ := φ∗(KX +B +M)

and if F ′ is a general fibre of X ′ → Y ′, then M ′|F ′ ≡ 0 and (F ′, B′|F ′) is isomorphicto (P1, p1 + p2) for distinct points p1, p2. Moreover, bB′c has exactly two disjointcomponents S′, T ′, both horizontal over Y ′.

But then −(KX′ +B′+M ′)−tS′ is not pseudo-effective over Y ′ for any real numbert > 0 because

−(KX′ +B′ +M ′) · F ′ = 0

while S′ · F ′ > 0. Therefore, −(KX′ + B′ +M ′)− tS′ is not pseudo-effective over Zfor any t > 0, contradicting the assumption that

τS′(−(KX +B +M)/Z) > 0.

�

Proof. (of Theorem 1.2) This is a special case of Theorem 3.5.�


18 Caucher Birkar

4. Non-klt loci for mirror symmetry

In this section we prove 1.4 and 1.6 but first we prove a generalised version of 1.4which occupies much of the section. Similar to the previous section, using generalisedpairs is crucial for the proofs.

Theorem 4.1. Assume that

(1) (X,B +M) is a projective Q-factorial generalised dlt pair with data X ′ → Xand M ′ where B,M ′ are Q-divisors,

(2) KX +B +M is pseudo-effective,(3) x ∈ X is a zero-dimensional generalised non-klt centre of (X,B +M),(4) x is not contained in the restricted base locus B−(KX +B +M),(5) if ψ : Y → X is the blowup at x with exceptional divisor E, then we have

τE(KX +B +M) = 0, that is,

ψ∗(KX +B +M)− tE

is not pseudo-effective for any real number t > 0.

Then (X,B + M) has a good minimal model which is generalised log Calabi-Yau.More precisely, we can run a minimal model program on KX +B+M ending with aminimal model (X ′′, B′′ +M ′′) with KX′′ +B′′ +M ′′ ∼Q 0.

Before giving the proof of the theorem we prove several lemmas. We start withsome basic properties of the pseudo-effective threshold τ .

4.2. Pseudo-effective thresholds.

Lemma 4.3. Assume that

• X is a normal projective variety,• L is an R-Cartier pseudo-effective R-divisor on X,• φ : Y → X is a birational contraction from a normal variety and S is a prime

divisor on Y such that −S is ample over X, and• T is a prime divisor over X whose centre on X is contained in the centre ofS.

If τS(L) > 0, then τT (L) > 0.

Proof. Take a resolution α : V → X on which T is a divisor and such that theinduced map β : V 99K Y is a morphism. Assume τS(L) > 0. Then φ∗L − tS ispseudo-effective for some real number t > 0. On the other hand, −S is ample overX, hence φ−1{x} ⊆ S for every x ∈ φ(S), so S = φ−1(φ(S)). Since the centre of Ton X is contained in the centre of S, we see that the centre of T on Y is containedin S. Then the coefficient of T in β∗S, say e, is positive. But then

α∗L− teT = β∗(φ∗L− tS) + β∗tS − teT

is pseudo-effective as β∗(φ∗L− tS) is pseudo-effective and tβ∗S− teT ≥ 0. Therefore,τT (L) > 0.

�


• (X,B +M) is a projective generalised lc pair with data φ : X ′ → X and M ′,• (X,C) is klt for some boundary C,• KX +B +M is pseudo-effective,


• S is a prime divisor over X with

a(S,X,B +M) < 1,

and• T is a prime divisor over X whose centre on X is contained in the centre ofS.

If

τS(KX +B +M) > 0,

then

τT (KX +B +M) > 0.

Proof. By Lemma 2.10, there is a birational contraction ψ : Y → X from a normalvariety such that S is a divisor on Y and −S is ample over X. Thus by Lemma 4.3,τS(KX +B +M) > 0 implies τT (KX +B +M) > 0.

�


• (X,B +M) is a projective generalised lc pair with data φ : X ′ → X and M ′,• (X,C) is klt for some boundary C,• KX +B +M is pseudo-effective,• S, T are prime divisors over X with equal centre on X, and• the generalised log discrepancies satisfy

a(S,X,B +M) < 1 and a(T,X,B +M) < 1.

Then

τS(KX +B +M) > 0 iff τT (KX +B +M) > 0.

Proof. This follows from Lemma 4.4.�

Next we look at pseudo-effective thresholds when we modify our pair birationallyin certain situations.


• (X,B +M) is a projective generalised lc pair with data φ : X ′ → X and M ′,• (X,C) is klt for some boundary C,• S is a prime divisor over X with

a(S,X,B +M) = 0,

• writing

KX′ +B′ +M ′ = φ∗(KX +B +M)

we have ∆′ = B′ +R′ ≥ 0 where R′ ≥ 0 is exceptional over X.

Then, if

τS(KX +B +M) = 0,

then

τS(KX′ + ∆′ +M ′) = 0.

20 Caucher Birkar

Proof. By Lemma 2.10, there is a birational contraction ψ : Y → X from a normalvariety such that S is a divisor on Y and −S is ample over X. In particular, S containsthe exceptional locus of ψ, so ψ does not contract any divisor except possibly S.

Assume

τS(KX +B +M) = 0,

but

τS(KX′ + ∆′ +M ′) > 0.

Let ρ : W ′ → X ′ be a resolution so that the induced map α : W ′ 99K Y is a morphism.Let S′ on W ′ be the birational transform of S. Then

ρ∗(KX′ + ∆′ +M ′)− tS′

is pseudo-effective for some real number t > 0. By assumption, S′ is a generalisednon-klt place of (X,B + M), so it is a generalised non-klt place of (X ′, B′ + M ′),hence S′ is not a component of ρ∗R′ otherwise (X ′,∆′+M ′) would not be generalisedlc.

By assumption, R′ is exceptional over X, hence ρ∗R′ is exceptional over X. Thensince S′ is not a component of ρ∗R′ and since the only possible exceptional divisor ofY → X is S, we deduce that ρ∗R′ is exceptional over Y . Thus

ψ∗(KX +B +M)− tS = α∗(ρ∗(φ∗(KX +B +M)) + ρ∗R′ − tS′)

= α∗(ρ∗(KX′ +B′ +M ′) + ρ∗(∆′ −B′)− tS′)= α∗(ρ

∗(KX′ + ∆′ +M ′)− tS′)is pseudo-effective. Therefore, we get

τS(KX +B +M) > 0,

a contradiction.�

4.7. Lifting zero-dimensional non-klt centres.

Lemma 4.8. Assume that (X,B+M) is a generalised dlt pair with data φ : X ′ → Xand M ′, and x ∈ X is a zero-dimensional generalised non-klt centre of (X,B +M).Write

KX′ +B′ +M ′ = φ∗(KX +B +M)

and assume (X ′, B′) is log smooth. Then (X ′, B′ +M ′) has a zero-dimensional gen-eralised non-klt centre x′ mapping to x.

Proof. Note that here we are considering (X ′, B′+M ′) as a generalised sub-pair withdata X ′ → X ′ and M ′. The generalised non-klt centres of (X ′, B′ +M ′) are just thenon-klt centres of (X ′, B′). So we want to show that (X ′, B′) has a zero-dimensionalnon-klt centre x′ mapping to x.

Since (X,B +M) is generalised dlt and x ∈ X is a generalised non-klt centre, bydefinition, (X,B) is log smooth near x and M ′ = φ∗M over a neighbourhood of x.Shrinking X we can assume (X,B) is log smooth and that M ′ = φ∗M . Writing

KX′ + C ′ := φ∗(KX +B),

we see that C ′ = B′ because M ′ = φ∗M , hence x is a non-klt centre of (X,B).Thus in this situation M ′ is not relevant, so removing it from now on we can assumeM ′ = 0.


We can assume d := dimX > 1 as the lemma is obvious in dimension one. Weuse induction on dimension. Since (X,B) is log smooth and x is a zero-dimensionalnon-klt centre, x is an intersection point of d components of bBc passing throughx. Let S be one such component and let S′ be its birational transform on X ′. Byadjunction, define KS + BS = (KX + B)|S and KS′ + BS′ = (KX′ + B′)|S′ . Since(X ′, B′) is log smooth and S′ is a component of B′ with coefficient 1, (S′, BS′) is logsmooth. Denoting the induced morphism S′ → S by π, we have

KS′ +BS′ = π∗(KS +BS).

Moreover, (S,BS) is log smooth and x is a non-klt centre of (S,BS). Thus by induc-tion, there is a zero-dimensional non-klt centre x′ of (S′, BS′) mapping onto x. Butthen x′ is also a non-klt centre of (X ′, B′).

�

4.9. Ample models for certain generalised pairs. In this subsection we showthat certain generalised pairs have ample models in the relative birational setting.

Lemma 4.10. Let (X,B + M) be an lc generalised pair with data X ′φ→ X

g→ Yand M ′ where B,M ′ are Q-divisors and g is a birational contraction. Assume that(Y,BY +MY ) is generalised lc with nef part M ′, where KY +BY +MY is the pushdownof KX +B +M , and that (Y,C) is klt for some boundary C. Then (X,B +M) hasan ample model over Y , i.e.⊕

m≥0g∗OX(m(KX +B +M))

is a finitely generated OY -module.

Proof. We can assume φ is a log resolution of (X,B) and that X ′ → Y is a logresolution of (Y,BY ). Write

KX′ +B′ +M ′ = φ∗(KX +B +M)

and let ∆′ be obtained from B′ by increasing the coefficient of every exceptional/Xprime divisor to 1. Replacing (X,B +M) with (X ′,∆′ +M ′) we can assume that

(∗) (X,B+M) is Q-factorial generalised dlt and any prime exceptional divisor ofX → Y containing a generalised non-klt centre of (X,B+M) is a componentof bBc.

Run an MMP on KX + B + M over Y with scaling of some ample divisor. Wereach a model V on which the pushdown KV + BV + MV is numerically a limit ofmovable/Y R-divisors. Note that property (∗) is preserved because if E is the sumof the exceptional divisors of X → Y which are not components of bBc and t is asufficiently small number, then (X,B+tE+M) is generalised dlt and X 99K V is alsoan MMP on KX +B+tE+M . Replacing X with V we can assume that KX +B+Mis numerically a limit of movable/Y R-divisors.

Since KY +BY +MY is R-Cartier, we can write

KX +B +M +R = g∗(KY +BY +MY )

where R is exceptional over Y . Then −R is numerically a limit of movable/Y R-divisors, so for any exceptional/Y prime divisor S ⊂ X, the divisor −R|S is pseudo-effective over g(S). Therefore, by the general negativity lemma (cf. [6, Lemma 3.3]),we have R ≥ 0 (as in the proof of 2.8, to apply the lemma we can first do a basechange to an uncountable ground field if necessary).

22 Caucher Birkar

We apply induction on the number of the exceptional divisors of g. If there is nosuch divisor, then X is of Fano type over Y because (Y,C) is klt for some C, hence(X,B+M) has an ample model over Y . Now assume there is a prime divisor S thatis exceptional over Y .

First assume that g(S) 6⊂ g(SuppR). Then KX + B + M ∼Q 0 over the genericpoint of g(S). Let t be a sufficiently small rational number. Thus running an MMPon KX + B + tS + M over Y with scaling of some divisor contracts S because overthe generic point of g(S) the MMP is an MMP on S. Let W → U be the step of theMMP where S is contracted. Then X 99K W is an isomorphism in codimension onebecause the MMP cannot contract any divisor other than S. Moreover,

KW +BW +MW ∼Q 0/U

because W → U is extremal and KW + BW + MW ∼Q 0 over the generic point ofg(S). Therefore, if KU +BU +MU has an ample model over Y , then KW +BW +MW

has an ample model over Y which in turn implies that KX + B + M has an amplemodel over Y . Note that (∗) holds on U because X 99K U is also an MMP on

KX +B + tS + uE +M

where E is the sum of the exceptional divisors of X → Y which are not components ofbBc and u is a sufficiently small number. We are then done in this case by induction.

Now we can assume that g(S) ⊂ g(SuppR) for every prime exceptional divisorS that is not a component of bBc. Then all such S are components of R. If not,then since g has connected fibres we can choose one that intersects R but is not acomponent of R so that −R|S is not pseudo-effective over g(S). This contradictsthe third paragraph of this proof. From now on then we can assume that everyexceptional divisor of g is a component of B +R.

Write KX + Γ = g∗(KY + C). Then

(X, (1− α)Γ + αB + αR+ αM)

is generalised klt for some rational number α < 1 sufficiently close to 1, and

KX + (1− α)Γ + αB + αR+ αM ∼Q 0/Y.

From this we can find Θ such that (X,Θ) is klt and KX + Θ ∼Q 0/Y , hence X is ofFano type over Y , so every divisor on X has an ample model over Y .

�

4.11. Fibrations. In this subsection we prove a few results in order to treat Theorem4.1 when the underlying space admits a suitable fibration.

Lemma 4.12. Let (X,B +M) be as in Theorem 4.1. Assume that

• f : X → Z is a contraction where dimZ > 0,• KX +B +M ∼Q 0/Z, and• X is of Fano type over Z.

Then there is a high resolution Z ′ → Z and a closed point z′ ∈ Z ′ such that

• z′ maps to f(x),• z′ is a generalised non-klt centre of (Z ′, BZ′ +MZ′) where the latter is given

by adjunction as in 2.11,• and we have

τG′(KZ +BZ +MZ) = 0

where G′ is the exceptional divisor of the blowup of Z ′ at z′.


Proof. By adjunction, for each birational contraction Z ′ → Z where Z ′ is normal, weget (Z ′, BZ′ + MZ′) as defined in 2.11. Assuming Z ′ → Z is a high log resolutionof (Z,BZ), MZ′ is nef. In particular, (Z ′, BZ′ + MZ′) is a generalised sub-pair and(Z,BZ + MZ) is a generalised pair with nef part MZ′ . Replace the given morphismφ : X ′ → X with a high resolution so that the induced map f ′ : X ′ 99K Z ′ is amorphism. Since KX +B +M is pseudo-effective and

KX +B +M ∼Q f∗(KZ +BZ +MZ),

KZ +BZ +MZ is pseudo-effective.Write

KX′ +B′ +M ′ = φ∗(KX +B +M)

where M ′ is the nef part of (X,B + M). By assumption, (X,B + M) is Q-factorialgeneralised dlt and x is a zero-dimensional generalised non-klt centre. ApplyingLemma 4.8, there is a zero-dimensional generalised non-klt centre x′ of (X ′, B′+M ′)mapping to x.

Let z′ = f ′(x′). We will show z′ is a generalised non-klt centre of (Z ′, BZ′ +MZ′).Let E′ be the exceptional divisor of the blowup of X ′ at x′. Since x′ is a generalisednon-klt centre of (X ′, B′ + M ′), it is a non-klt centre of (X ′, B′), so there are d :=dimX components of bB′c intersecting transversally at x′. So

a(E′, X ′, B′ +M ′) = 0,

that is, E′ is a non-klt place of (X ′, B′ +M ′).Now pick resolutions Z ′′ → Z ′ and X ′′ → X ′ such that f ′′ : X ′′ 99K Z ′′ is a

morphism and if E′′ ⊂ X ′′ is the birational transform of E′, then H ′′ := f ′′(E′′) is adivisor on Z ′′. Write KX′′ +B′′+M ′′ for the pullback of KX′ +B′+M ′. Then E′′ isa component of bB′′c, hence H ′′ is a component of bBZ′′c because the generalised lcthreshold of f ′′∗H ′′ with respect to (X ′′, B′′+M ′′) over the generic point of H ′′ is zeroas E′′ ≤ f ′′∗H ′′. This shows that z′ is a generalised non-klt centre of (Z ′, BZ′ +MZ′)as claimed since H ′′ maps to z′ as E′′ maps to x′.

LetG′ be the exceptional divisor of the blowup of Z ′ at z′. By assumption, (Z ′, BZ′)is log smooth and MZ′ is the nef part of (Z ′, BZ′ + MZ′). Since z′ is a generalisednon-klt centre of (Z ′, BZ′ +MZ′),

a(G′, Z,BZ +MZ) = a(G′, Z ′, BZ′ +MZ′) = 0.

On the other hand, by the previous paragraph,

a(H ′′, Z,BZ +MZ) = a(H ′′, Z ′, BZ′ +MZ′) = 0.

Now since X is of Fano type over Z, there is a boundary D on X such that(X,D) is klt and KX +D ∼Q 0/Z. Thus applying adjunction, we get (Z,DZ +NZ)which is generalised klt. From this we get C so that (Z,C) is klt (as in the proof of2.10). Moreover, G′, H ′′ both map to f(x). Thus, applying Lemma 4.5 to (Z,BZ +MZ), G′, H ′′, we see that to prove that

τG′(KZ +BZ +MZ) = 0

it is enough to prove that

τH′′(KZ +BZ +MZ) = 0.

By assumption,

τE(KX′ +B′ +M ′) = τE(KX +B +M) = 0

24 Caucher Birkar

where E is the exceptional divisor of the blowup of X at x. Also

a(E,X,B +M) = 0

and

a(E′, X,B +M) = a(E′, X ′, B′ +M ′) = 0

where E′ is the exceptional divisor of the blowup of X ′ at x′. Then by Lemma 4.5,

τE′′(KX +B +M) = τE′(KX +B +M) = 0

where recall that E′′ ⊂ X ′′ is the birational transform of E′′. Thus

KX′′ +B′′ +M ′′ − tE′′

is not pseudo-effective for any real number t > 0. This in turn implies that

KX′′ +B′′ +M ′′ − tf ′′∗H ′′

is not pseudo-effective for any real number t > 0 because E′′ ≤ f ′′∗H ′′. But then

KZ′′ +BZ′′ +MZ′′ − tH ′′

is not pseudo-effective as

KX′′ +B′′ +M ′′ − tf ′′∗H ′′ ∼Q f′′∗(KZ′′ +BZ′′ +MZ′′ − tH ′′).

Therefore,

ψ∗(KZ +BZ +MZ)− tH ′′

is not pseudo-effective for any t > 0 where ψ denotes Z ′′ → Z, hence

τH′′(KZ +BZ +MZ) = 0

as required.�


• (X,B +M) is a projective Q-factorial generalised lc pair,• KX +B +M is pseudo-effective,• x ∈ X is a point not contained in B−(KX +B +M),• f : X → Z is a contraction with dimZ > 0 and KX + B + M ∼R f∗L for

some R-Cartier R-divisor L, and• X is of Fano type over Z.

Then

f(x) /∈ B−(L).

Proof. Pick an ample Cartier divisor AZ on Z and let A = f∗AZ . We want to showthat

z := f(x) /∈ B(L+ tAZ)

for any t ∈ R>0. Assume otherwise, that is, assume that

z ∈ B(L+ tAZ)

for some t ∈ R>0. Thus z belongs to the support of every divisor 0 ≤ RZ ∼R L+tAZ .Then f−1{z} ⊂ SuppR for every divisor

0 ≤ R ∼R KX +B +M + tA

because any such R is the pullback of an RZ as above. Therefore,

x ∈ B(KX +B +M + tA).


SinceX is of Fano type over Z, B+M is big over Z. Moreover, B+M has a minimalmodel Y over Z on which BY +MY is semi-ample over Z. Then s(BY +MY ) + tAYis semi-ample for every 0 < s� t (this can be seen by considering the ample modelof BY +MY over Z). Pick one such number s and pick a general

0 ≤ DY ∼R s(BY +MY ) + tAY .

Since X is of Fano type over Z, Y is of Fano type over Z, hence Y has klt singu-larities. Thus

(Y, (1− s)BY + (1− s)MY ))

has generalised klt singularities with nef part (1 − s)M ′ where replacing X ′ we areassuming the induced map X ′ 99K Y is a morphism. Since DY is general, we deducethat

(Y, (1− s)BY +DY + (1− s)MY ))

is generalised klt with nef part (1− s)M ′.On the other hand, since X 99K Y is an MMP on B+M over Z, it is also an MMP

on s(B +M) + tA. Thus DY determines a unique divisor

0 ≤ D ∼R s(B +M) + tA

whose pushdown to Y is DY . Then the pair

(X, (1− s)B +D + (1− s)M))

is generalised klt with nef part (1− s)M ′ because

KX + (1− s)B +D + (1− s)M ∼R KX + (1− s)B + sB + sM + tA+ (1− s)M= KX +B +M + tA ∼R 0/Z

and because(Y, (1− s)BY +DY + (1− s)MY ))

is generalised klt with nef part (1− s)M ′. Moreover, D is big.Now, by [8, Lemma 4.4], we can run an MMP on

KX + (1− s)B +D + (1− s)Mwhich ends with a good minimal model, say V . By the first paragraph,

x ∈ B(KX +B +M + tA)

which means that X 99K V is not an isomorphism near x. That is, x belongs to theexceptional locus of some step of the MMP. But then

x ∈ B(KX +B +M + tA+ uH)

where H is an ample divisor and u > 0 is a sufficiently small real number. Therefore,

x ∈ B(KX +B +M + uH)

as A is semi-ample, hencex ∈ B−(KX +B +M),

a contradiction.�

Lemma 4.14. Assume that Theorem 4.1 holds in dimension ≤ d−1. Let (X,B+M)be as in Theorem 4.1 in dimension d with data X ′ → X and M ′. Suppose that thereexist a birational contraction X → Y and a non-birational contraction Y → Z suchthat

26 Caucher Birkar

• (Y,BY +MY ) is generalised lc with nef part M ′, where KY +BY +MY denotesthe pushdown of KX +B +M ,• Y is of Fano type over Z, and• KY +BY +MY ∼Q 0/Z.

Thenκσ(KX +B +M) = 0

andKX +B +M ∼Q Nσ(KX +B +M) ≥ 0.

Proof. Step 1. Note that κσ and Nσ are defined as in [20]. Since Y is of Fano type overZ, (Y,C) is klt and KY +C ∼Q 0/Z for some C. Thus by Lemma 4.10, (X,B +M)has an ample model over Y , say U . Denoting the morphism U → Y by π, we canwrite

KU +BU +MU +RU = π∗(KY +BY +MY )

for some RU exceptional over Y . Then −RU is ample over Y , so RU ≥ 0 by the nega-tivity lemma. Moreover, SuppRU contains every exceptional divisor of π. Therefore,U is of Fano type over Z as Y is of Fano type over Z by the relative version of [4,2.13(7)], or arguing as in the end of the proof of 4.10. In particular, (U,BU +MU ) hasa minimal model over Z, say V . Note that since KU +BU +MU is pseudo-effective,RU is vertical over Z, so KU +BU +MU ∼Q 0 over the generic point of Z as

KU +BU +MU +RU ∼Q 0/Z.

Step 2. Replacing X ′ we can assume that it is a log resolution of (X,B) and thatthe induced map ρ : X ′ 99K V is a morphism and a log resolution of (V,BV ). Write

KX′ +B′ +M ′ = φ∗(KX +B +M).

Let ∆′ be the sum of the birational transform of BV and the reduced exceptionaldivisor of ρ. Then R′ := ∆′ − B′ is exceptional over V as ρ∗∆

′ = BV = ρ∗B′.

Moreover, R′ ≥ 0: indeed, for any prime divisor D′ exceptional over V , we have

µD′(∆′ −B′) = 1− µD′B′ ≥ 0.

By Lemma 4.8, there is a zero-dimensional generalised non-klt centre x′ of (X ′, B′+M ′) mapping to the given point x. Then x′ is also a generalised non-klt centre of(X ′,∆′ + M ′), and x′ /∈ SuppR′ as (X ′,∆′ + M ′) is generalised lc. On the otherhand, we claim that

x′ /∈ B−(KX′ + ∆′ +M ′).

This follows from

B−(KX′ + ∆′ +M ′) ⊆ B−(KX′ +B′ +M ′) ∪ SuppR′

⊆ φ−1B−(KX +B +M) ∪ SuppR′

and the assumptionx /∈ B−(KX +B +M)

and the fact x′ /∈ SuppR′.

Step 3. By construction,

KX′ + ∆′ +M ′ = ρ∗(KV +BV +MV ) + P ′

where P ′ ≥ 0 is exceptional over V . Thus running an MMP on KX′ + ∆′ +M ′ overV with scaling of some ample divisor contracts P ′ and ends with a minimal model


W/V . In fact, (W,∆W +MW ) is a Q-factorial generalised dlt model of (V,BV +MV )where ∆W +MW is the pushdown of ∆′+M ′. The map X ′ 99KW is an isomorphismnear x′ because x′ is not contained in B−(KX′ + ∆′ +M ′).

Let E (resp. E′) be the exceptional divisor of the blowup of X at x (resp. of X ′

at x′). Then

a(E,X,B +M) = 0 = a(E′, X ′, B′ +M ′) = a(E′, X,B +M).

This implies

a(E′, V, BV +MV ) = a(E′,W,∆W +MW ) = a(E′, X ′,∆′ +M ′) = 0

by the previous paragraph.Now from the assumption

τE(KX +B +M) = 0

we deduce thatτE′(KX +B +M) = 0,

by Lemma 4.5. This in turn implies

τE′(KV +BV +MV ) = 0,

ThusτE′(KW + ∆W +MW ) = 0.

Step 4. Let V → T/Z be the contraction defined by KV + BV + MV . SinceKV +BV +MV ∼Q 0 over the generic point of Z, T → Z is birational but V → T isnot birational. By construction,

κσ(KX +B +M) = κσ(KV +BV +MV ) = κσ(KW + ∆W +MW ).

Moreover, W is of Fano type over Z as V is of Fano type over Z, so W is also of Fanotype over T . Therefore, replacing Z with T , replacing (X,B+M) with (W,∆W+MW )and replacing Y with V , from now on we can assume that KX +B+M ∼Q 0/Z andthat X is of Fano type over Z. In particular, (Z,CZ) is klt for some CZ .

Step 5. Denote X → Z by f and let z := f(x). For each birational contractionZ ′ → Z from a normal variety, consider (Z ′, BZ′ + MZ′) given by adjunction as in2.11. Then by Lemma 4.12, there exist a high resolution ψ : Z ′ → Z and a closedpoint z′ ∈ Z ′ mapping to z such that

• z′ is a generalised non-klt centre of (Z ′, BZ′ +MZ′), so

a(E′, Z,BZ +MZ) = a(E′, Z ′, BZ′ +MZ′) = 0,

• andτG′(KZ +BZ +MZ) = 0

where G′ is the exceptional divisor of the blowup of Z ′ at z′.

Step 6. Let ∆Z′ := B≥0Z′ and PZ′ = ∆Z′ − BZ′ . Then z′ is a generalised non-kltcentre of (Z ′,∆Z′ + MZ′) and z′ /∈ SuppPZ′ . Moreover, by Lemma 4.6 applied to(Z,BZ +MZ) we have

τG′(KZ′ + ∆Z′ +MZ′) = 0.

Moreover, by Lemma 4.13,

z /∈ B−(KZ +BZ +MZ),

28 Caucher Birkar

hence

z′ /∈ B−(KZ′ + ∆Z′ +MZ′) ⊆ ψ−1B−(KZ +BZ +MZ) ∪ SuppPZ′ .

Thus (Z ′,∆Z′ + MZ′) satisfies the same kind of properties listed in Theorem 4.1.Since we are assuming the theorem in dimension ≤ d − 1, (Z ′,∆Z′ + MZ′) has aminimal model which is generalised log Calabi-Yau. Therefore,

κσ(KZ′ + ∆Z′ +MZ′) = 0

and

KZ′ + ∆Z′ +MZ′ ∼Q Nσ(KZ′ + ∆Z′ +MZ′) ≥ 0,

so we have

κσ(KZ +BZ +MZ) = 0

and

KZ +BZ +MZ ∼Q Nσ(KZ +BZ +MZ) ≥ 0

which in turn implies

κσ(KX +B +M) = 0

and that

KX +B +M ∼Q Nσ(KX +B +M) ≥ 0

as desired, by [20, Proposition 6.2.8].�

4.15. Proofs of 1.4 and 4.1.

Proof. (of Theorem 4.1) Step 1. Suppose that

KX +B +M ∼Q Nσ(KX +B +M) ≥ 0.

Run an MMP on KX + B + M with scaling of some ample divisor. After finitelymany steps, no divisor is contracted, so the rest of the MMP can consist of only flips.In particular, we reach a model X ′′ on which KX′′ +B′′+M ′′ is numerically a limit ofmovable R-divisors. This is possible only if every component of Nσ(KX +B +M) iscontracted over X ′′. Therefore, KX′′ +B′′ +M ′′ ∼Q 0 and X ′′ is the minimal modelwe are looking for. In the rest of the proof we show that (X,B + M) satisfies theproperties of the first sentence of this proof.

Step 2. Let Y → X be the blowup of X at x with exceptional divisor E. We canassume that the induced map X ′ 99K Y is a morphism. Writing

KX′ +B′ +M ′ = φ∗(KX +B +M),

let KY +BY +MY be the pushdown of KX′ +B′+M ′. We consider (Y,BY +MY ) asa generalised pair with data X ′ → Y and M ′. There is a zero-dimensional generalisednon-klt centre y of (Y,BY +MY ) mapping to x. Let G be the exceptional divisor ofthe blowup of Y at y. Then

a(E,X,B +M) = 0 = a(G,X,B +M).

Also by assumption,

τE(KY +BY +MY ) = τE(KX +B +M) = 0.

Applying Lemma 4.5, we see that

τG(KY +BY +MY ) = τG(KX +B +M) = 0.


Now (Y,BY +MY ), y satisfies properties similar to (1)-(3), (5) listed in 4.1. It alsosatisfies (4), that is,

y /∈ B−(KY +BY +MY )

as

B−(KY +BY +MY ) ⊆ α−1B−(KX +B +M)

where α denotes Y → X. Therefore, replacing (X,B +M), x with (Y,BY +MY ), y,we can assume that τS(KX +B +M) = 0 for some component S of bBc.

Step 3. Pick a small rational number t > 0. By the last sentence of the previousstep,

KX +B +M − t bBcis not pseudo-effective. Thus we can run an MMP on

KX +B +M − t bBc

ending with a Mori fibre space V → Z. Since KX + B + M is pseudo-effective,KV +BV +MV is nef over Z.

Note that the coefficients of B,M ′ are in a fixed finite set Φ (for the fixed pair(X,B +M)) independent of t. Writing

BV − t bBV c+MV = BV − bBV c+ (1− t) bBV c+MV ,

and applying Lemma 2.9, we deduce that assuming t is sufficiently small, dependingonly on (X,B +M), the pair (V,BV +MV ) is generalised lc and

KV +BV +MV ≡ 0/Z.

Step 4. Replacing X ′ we can assume that φ is a log resolution of (X,B) and thatthe induced map X ′ 99K V is a morphism. Recall

KX′ +B′ +M ′ = φ∗(KX +B +M)

from Step 2. Let ∆′ := B′≥0. Applying Lemma 4.8, we see that there is a zero-dimensional generalised non-klt centre x′ of (X ′, B′+M ′) mapping to x. Note that x′

is also a generalised non-klt centre of (X ′,∆′+M ′). Moreover, if G′ is the exceptionaldivisor of the blowup of X ′ at x′, then by Lemma 4.5,

τG′(KX +B +M) = 0,

so

τG′(KX′ + ∆′ +M ′) = 0

by Lemma 4.6. In addition,

x′ /∈ B−(KX′ + ∆′ +M ′)

because x′ /∈ Supp(∆′ −B′).Therefore, replacing (X,B+M), x with (X ′,∆′+M ′), x′, we can assume that there

is a birational contraction X → V and a non-birational contraction V → Z such that

• (V,BV +MV ) is generalised lc with nef part M ′, where KV +BV +MV denotesthe pushdown of KX +B +M ,• V is of Fano type over Z, and• KV +BV +MV ∼Q 0/Z.

30 Caucher Birkar

Now applying Lemma 4.14, we get

KX +B +M ∼Q Nσ(KX +B +M) ≥ 0

as desired.�


4.16. Proofs of 1.5 and 1.6.

Lemma 4.17. Let (X,B) be a projective dlt pair with KX + B ≡ 0 having a zero-dimensional non-klt centre x. Assume that either V = X or V is a non-klt centreof (X,B), and that dimV ≥ 2. Let KV + BV = (KX + B)|V be given by adjunction(BV = B when V = X). Then the non-klt locus Nklt(V,BV ) = bBV c is connected.

Proof. The equality Nklt(V,BV ) = bBV c follows from the assumption that (X,B) isdlt which implies that (V,BV ) is dlt. Assume that Nklt(V,BV ) is not connected. By[16, Theorem 4.40], all the minimal non-klt centres of (X,B) are birational to eachother, in particular, they have the same dimension. Since we already have a zero-dimensional non-klt centre x, all the minimal non-klt centres are zero-dimensional(we recall the proof of this fact below). Therefore, any minimal non-klt centre of(V,BV ) is also zero-dimensional because any minimal non-klt centre of (V,BV ) is aminimal non-klt centre of (X,B).

Since we assumed that Nklt(V,BV ) = bBV c is not connected, by Theorem 1.2,(V,BV ) has exactly two disjoint non-klt centres. As (V,BV ) is dlt, these centres aretwo disjoint irreducible components of bBV c. This contradicts the fact that (V,BV )has a zero-dimensional non-klt centre.

�

The lemma does not hold if we relax the dlt assumption to lc; see the examplegiven after Theorem 1.2.

In the proof of the lemma we used the fact that all the minimal non-klt centresof (X,B) are zero-dimensional. For convenience, we give the proof here essentiallyfollowing [16, Theorem 4.40]. First applying 1.2, we see that bBc is connected because(X,B) has a zero-dimensional non-klt centre x. Now pick any minimal non-klt centreW of (X,B). Then there exist components D1, . . . , Dn of bBc such that x ∈ D1,W ⊂ Dn, Di intersects Di+1 for each 1 ≤ i < n, and that n is minimal with theseproperties. Then by induction on dimension, any minimal non-klt centre of (D1, BD1)is zero-dimensional where KDi +BDi = (KX +B)|Di . In particular, D1∩D2 containsa zero-dimensional non-klt centre x2 of (D1, BD1) which is in turn a non-klt centre of(X,B). Repeating this argument we find a zero-dimensional non-klt centre of (X,B)in each Di, in particular, in Dn. But W ⊂ Dn, so applying induction to (Dn, BDn)implies that dimW = 0.

Proof. (of Corollary 1.5) By Theorem 1.4, we can run an MMP on KX + B endingwith a minimal model X ′ with KX′ + B′ ∼Q 0. By assumption, no non-klt centreof (X,B) is contained in B−(KX + B). Thus X 99K X ′ is an isomorphism near thegeneric point of each non-klt centre. So each non-klt centre has a birational transformon X ′. On the other hand, if W ′ is any non-klt centre of (X ′, B′), then X 99K X ′

is an isomorphism over the generic point of W ′ because X 99K X ′ is an MMP on


KX + B. Thus there is a 1-1 correspondence between the non-klt centres of (X,B)and (X ′, B′) given by birational transform. In particular, the image x′ on X ′ of thegiven non-klt centre x is a zero-dimensional non-klt centre of (X ′, B′).

Assume that either V = X or that V is a non-klt centre of (X,B). Define KV +BV = (KX +B)|V by adjunction (BV = B if V = X). Since (X,B) is dlt, (V,BV ) isdlt, so

Nklt(V,BV ) = bBV c .Similarly, define KV ′ +BV ′ = (KX′ +B′)|V ′ by adjunction. Then

Nklt(V ′, BV ′) = bBV ′c .Each non-klt centre of (V,BV ) is a non-klt centre of (X,B), and similarly, each non-klt centre of (V ′, BV ′) is a non-klt centre of (X ′, B′). Thus by the previous paragraph,bBV ′c is the birational transform of bBV c.

Now assume that dimV ≥ 2. Since (X ′, B′) is dlt with KX′ + B′ ∼Q 0 having azero-dimensional non-klt centre x′, applying Lemma 4.17 shows that Nklt(V ′, BV ′) =bBV ′c is connected.

Assume that Nklt(V,BV ) = bBV c is not connected. We will derive a contradiction.Since bBV c is not connected but bBV ′c is connected, we can find disjoint componentsS, T of bBV c such that their birational transforms S′, T ′ on X ′ intersect. Let W ′ bean irreducible component of S′ ∩ T ′. Then S′, T ′ are components of bBV ′c, henceW ′ is a non-klt centre of (V ′, BV ′), so W ′ is also a non-klt centre of (X ′, B′). ThusX 99K X ′ is an isomorphism over the generic point of W ′, hence W ′ is the birationaltransform of a non-klt centre W of (X,B), and S, T both contain W , a contradiction.Therefore, bBV c is connected.

�

Proof. (of Corollary 1.6) For each i, let 0 ≤ ui ≤ ai be a rational number such thatui ≤ 1. Let ∆ = B −

∑uiBi. Then

KX + ∆ = KX +B −∑

uiBi ≡∑

(ai − ui)Bi ≥ 0.

In particular, KX + ∆ is pseudo-effective and

B−(KX + ∆) ⊆ Supp∑

(ai − ui)Bi ⊆ B − C.

On the other hand, by assumption, C and B−C have no common components, whereC is the sum of the good components of B, and x is a zero-dimensional non-klt centreof (X,C). Thus x is not contained in B − C which implies that x /∈ B−(KX + ∆).

Moreover, ∆ ≥ C, so x is a zero-dimensional non-klt centre of (X,∆). Also

0 ≤ τE(KX + ∆) ≤ τE(KX +B) = 0

where E is the exceptional divisor of the blow up of X at x. Therefore, (X,∆) satisfiesall the assumptions of Theorem 1.4, so we can run an MMP on KX + ∆ which endswith a log minimal model (X ′,∆′) with KX′ + ∆′ ∼Q 0. In particular, taking ui = 0for every i, we have ∆ = B, so we get the first claim of the corollary.

Assume that V is a stratum of (X,C), that is, either V = X or that V is a non-kltcentre of (X,C). Assume dimV ≥ 2. We want to show that CV is connected whereKV + CV = (KX + C)|V .

From now on we assume that ui > 0 when ai > 0. Then b∆c = C, so the non-klt centres of (X,∆) and (X,C) are the same. So CV = b∆V c where KV + ∆V =(KX + ∆)|V . Moreover, no non-klt centre of (X,C) is contained in B−C, so no non-klt centre of (X,∆) if contained in B−C, hence by the first paragraph of this proof,

32 Caucher Birkar

no non-klt centre of (X,∆) is contained in B−(KX + ∆). Therefore, by Corollary1.5, CV = b∆V c is connected as desired.

�

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DPMMS, Centre for Mathematical SciencesUniversity of Cambridge,Wilberforce Road, Cambridge CB3 0WB, [email protected]

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