ACC FOR LOG CANONICAL THRESHOLDS

CHRISTOPHER D. HACON, JAMES M^{C}KERNAN, AND CHENYANG XU

To Vyacheslav Shokurov on the occasion of his sixtieth birthday

Abstract. We show that log canonical thresholds satisfy the ACC.

Contents

1. Introduction  2
2. Description of the proof  5
2.1. Sketch of the proof  6
3. Preliminaries  12
3.1. Notation and Conventions  12
3.2. The volume  13
3.3. Divisorially log terminal modifications  15
3.4. DCC sets  15
3.5. Bounded pairs  16
4. Adjunction  19
5. Global to local  23
6. Upper bounds on the volume  25
7. Birational boundedness  27
8. Numerically trivial log pairs  32
9. Proofs of Theorems  34
10. Proofs of Corollaries  37
11. Accumulation points  39

Date: August 22, 2013.

The first author was partially supported by NSF research grant by NSF research grants DMS-0757897 and DMS-1300750 and by a Simons Investigator award from the Simons Foundation to Christopher Hacon, the second author was partially supported by NSF research grant no: DMS-0701101, DMS-1200656, and the third author was partially supported by NSF research grant no: 1159175 and by a special research fund in China. We would like to thank Valery Alexeev, János Kollár, and Vyacheslav Shokurov, as well as the referee, for some helpful comments. We would also like to thank Osamu Fujino, Kento Fujita and Hiromu Tanaka for pointing out an error in a previous version of the paper.

References  48

1. Introduction

We work over an algebraically closed field of characteristic zero. ACC stands for the ascending chain condition whilst DCC stands for the descending chain condition.

Suppose that (X, Δ) is a log canonical pair and M ≥ 0 is ℝ-Cartier. The log canonical threshold of M with respect to (X, Δ) is

lct(X, Δ; M) = sup{ t ∈ ℝ | (X, Δ + tM) is log canonical }.

Let 𝔗 = 𝔗_{n}(I) denote the set of log canonical pairs (X, Δ), where X is a variety of dimension n and the coefficients of Δ belong to a set I ⊂ [0, 1]. Set

LCT_{n}(I, J) = { lct(X, Δ; M) | (X, Δ) ∈ 𝔗_{n}(I) },

where the coefficients of M belong to a subset J of the positive real numbers.

Theorem 1.1 (ACC for the log canonical threshold). Fix a positive integer n, I ⊂ [0, 1] and a subset J of the positive real numbers.

If I and J satisfy the DCC then LCT_{n}(I, J) satisfies the ACC.

(1.1) was conjectured by Shokurov [33], see also [22] and [24]. When the dimension is three, [22] proves that 1 is not an accumulation point from below and (1.1) follows from the results of [3]. More recently (1.1) was proved for complete intersections [10] and even when X belongs to a bounded family, [11].

The log canonical threshold is an interesting invariant of the pair (X, Δ) and the divisor M which is a measure of the complexity of the singularities of the triple (X, Δ; M). It has made many appearances in many different forms, especially in the case of hypersurfaces, see [24],

[25] and [34]. The ACC for the log canonical threshold plays a role in inductive approaches to higher dimensional geometry. For example, after [6], we have the following application of (1.1):

Corollary 1.2. Assume termination of flips for ℚ-factorial kawamata log terminal pairs in dimension n − 1.

Let (X, Δ) be a kawamata log terminal pair where X is a ℚ-factorial projective variety of dimension n. If K_{X} + Δ is numerically equivalent to a divisor D ≥ 0 then any sequence of (K_{X} + Δ)-flips terminates.

(1.1) is a consequence of the following theorem, which was conjectured by Alexeev [3] and Kollár [22]:

Theorem 1.3. Fix a positive integer n and a set I ⊂ [0, 1] which satisfies the DCC. Let 𝔇 be the set of log canonical pairs (X, Δ) such that the dimension of X is n and the coefficients of Δ belong to I.

Then there is a constant δ > 0 and a positive integer m with the following properties:

(1) the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇 },

also satisfies the DCC. Further, if (X, Δ) ∈ 𝔇 and K_{X} + Δ is big, then

(2) vol(X, K_{X} + Δ) ≥ δ, and

(3) ϕ_{m(K_{X}+Δ)} is birational.

Note that, by convention, ϕ_{m(K_{X}+Δ)} = ϕ_{⌊m(K_{X}+Δ)⌋}. (1.3) was proved for surfaces in [3]. (1.3) is a generalisation of [15, 1.3], which deals with the case that (X, Δ) is the quotient of a smooth projective variety Y of general type by its automorphism group.

One of the original motivations for (1.3) is to prove the boundedness of the moduli functor for canonically polarised varieties, see [26]. We plan to pursue this application of (1.3) in a forthcoming paper.

To state more results it is convenient to give a simple reformulation of (1.1):

Theorem 1.4. Fix a positive integer n and a set I ⊂ [0, 1], which satisfies the DCC.

Then there is a finite subset I_{0} ⊂ I with the following properties: If (X, Δ) is a log pair such that

(1) X is a variety of dimension n,

(2) (X, Δ) is log canonical,

(3) the coefficients of Δ belong to I, and

(4) there is a non kawamata log terminal centre Z ⊂ X which is

contained in every component of Δ, then the coefficients of Δ belong to I_{0}.

(1.4) follows, cf. [33], [27, §18], almost immediately from the existence of divisorial log terminal modifications and from:

Theorem 1.5. Fix a positive integer n and a set I ⊂ [0, 1], which satisfies the DCC.

Then there is a finite subset I_{0} ⊂ I with the following properties: If (X, Δ) is a log pair such that (1) X is a projective variety of dimension n,

(2) (X, Δ) is log canonical,

(3) the coefficients of Δ belong to I, and

(4) K_{X} + Δ is numerically trivial, then the coefficients of Δ belong to I_{0}.

We use finiteness of log canonical models to prove a boundedness result for log pairs:

Theorem 1.6. Fix a positive integer n and two real numbers δ and ϵ > 0.

Let 𝔇 be a set of log pairs (X, Δ) such that • X is a projective variety of dimension n,

• K_{X} + Δ is ample,

• the coefficients of Δ are at least δ, and

• the total log discrepancy of (X, Δ) is greater than ϵ. If 𝔇 is log birationally bounded then 𝔇 is a bounded family.

Log birationally bounded is defined in (3.5.1). We use (1.5) and (1.6) to prove some boundedness results about Fano varieties.

Corollary 1.7. Fix a positive integer n, a real number ϵ > 0 and a set I ⊂ [0, 1] which satisfies the DCC.

Let 𝔇 be the set of all log pairs (X, Δ), where • X is a projective variety of dimension n,

• the coefficients of Δ belong to I,

• the total log discrepancy of (X, Δ) is greater than ϵ,

• K_{X} + Δ is numerically trivial, and

• −K_{X} is ample. Then 𝔇 forms a bounded family.

As a consequence we are able to prove a result on the boundedness of Fano varieties which was conjectured by Batyrev (cf. [9]):

Corollary 1.8. Fix two positive integers n and r.

Let 𝔇 be the set of all kawamata log terminal pairs (X, Δ), where X is a projective variety of dimension n and −r(K_{X} + Δ) is an ample Cartier divisor.

Then 𝔇 forms a bounded family.

Definition 1.9. Let (X, Δ) be a log canonical pair, where X is projective of dimension n and −(K_{X} + Δ) is ample. The Fano index of (X, Δ) is the largest real number r such that we can write

−(K_{X} + Δ) ∼_{ℝ} rH,

where H is a Cartier divisor.

Fix a set I ⊂ [0, 1] and a positive integer n. Let 𝔇 be the set of log canonical pairs (X, Δ), where X is projective of dimension n, −(K_{X} + Δ) is ample and the coefficients of Δ belong to I.

The set

R = R_{n}(I) = { r ∈ ℝ | r is the Fano index of (X, Δ) ∈ 𝔇 },

is called the Fano spectrum of 𝔇.

Corollary 1.10. Fix a set I ⊂ [0, 1] and a positive integer n. If I satisfies the DCC then the Fano spectrum satisfies the ACC.

(1.10) was proved in dimension 2 in [1] and for R ∩ [n − 2, ∞) in [2]. Now given any set which satisfies the ACC it is natural to try to iden-

tify the accumulation points. (1.1) implies that LCT_{n}(I) = LCT_{n}(I, ℕ) satisfies the ACC. Kollár, cf. [24], [32], [20], conjectured that the accumulation points in dimension n are log canonical thresholds in dimension n − 1:

Theorem 1.11. If 1 is the only accumulation point of I ⊂ [0, 1] and I = I_{+} then the accumulation points of LCT_{n}(I) are LCT_{n−1}(I)−{ 1 }. In particular, if I ⊂ ℚ then the accumulation points of LCT_{n}(I) are rational numbers.

See §3.4 for the definition of I_{+}. (1.11) was proved if X is smooth in [20]. Note that in terms of inductive arguments it is quite useful to identify the accumulation points, especially to know that they are rational.

Finally, recall:

Conjecture 1.12 (Borisov-Alexeev-Borisov). Fix a positive integer n and a positive real number ϵ > 0.

Let 𝔇 be the set of all projective varieties X of dimension n such that there is a divisor Δ where (X, Δ) has log discrepancy at least ϵ and −(K_{X} + Δ) is ample.

Then 𝔇 forms a bounded family.

Note that (1.1), (1.4), (1.5), (1.2) and (1.11) are known to follow from (1.12), (cf. [32]). Instead we use birational boundedness of log pairs of general type cf. (1.3) to prove these results.

2. Description of the proof

Theorem A (ACC for the log canonical threshold). Fix a positive integer n and a set I ⊂ [0, 1], which satisfies the DCC.

Then there is a finite subset I_{0} ⊂ I with the following property: If (X, Δ) is a log pair such that (1) X is a variety of dimension n,

(2) (X, Δ) is log canonical,

(3) the coefficients of Δ belong to I, and

(4) there is a non kawamata log terminal centre Z ⊂ X which is

contained in every component of Δ, then the coefficients of Δ belong to I_{0}.

Theorem B (Upper bounds for the volume). Let n ∈ ℕ and let I ⊂ [0, 1) be a set which satisfies the DCC. Let 𝔇 be the set of kawamata log terminal pairs (X, Δ), where X is projective of dimension n, K_{X} + Δ is numerically trivial and the coefficients of Δ belong to I.

Then the set

{ vol(X, Δ) | (X, Δ) ∈ 𝔇 },

is bounded from above.

Theorem C (Birational boundedness). Fix a positive integer n and a set I ⊂ [0, 1], which satisfies the DCC. Let 𝔅 be the set of log canonical pairs (X, Δ), where X is projective of dimension n, K_{X} + Δ is big and the coefficients of Δ belong to I.

Then there is a positive integer m such that ϕ_{m(K_{X}+Δ)} is birational, for every (X, Δ) ∈ 𝔅.

Theorem D (ACC for numerically trivial pairs). Fix a positive integer n and a set I ⊂ [0, 1], which satisfies the DCC.

Then there is a finite subset I_{0} ⊂ I with the following property: If (X, Δ) is a log pair such that

(1) X is projective of dimension n,

(2) the coefficients of Δ belong to I,

(3) (X, Δ) is log canonical, and

(4) K_{X} + Δ is numerically trivial, then the coefficients of Δ belong to I_{0}.

The proof of Theorem A, Theorem B, Theorem C, and Theorem D proceeds by induction:

• Theorem D_{n−1} implies Theorem A_{n}, cf. (5.3).

• Theorem D_{n−1} and Theorem A_{n−1} imply Theorem B_{n}, cf. (6.2).

• Theorem C_{n−1}, Theorem A_{n−1}, and Theorem B_{n} imply Theorem C_{n}, cf. (7.4).

• Theorem D_{n−1} and Theorem C_{n} implies Theorem D_{n}, cf. (8.1).

2.1. Sketch of the proof. The basic idea of the proof of (1.1) goes back to Shokurov and we start by explaining this.

Consider the following simple family of plane curve singularities,

C = (y^{a} + x^{b} = 0) ⊂ ℂ^{2},

where a and b are two positive integers. A priori, to calculate the log discrepancy c, one should take a log resolution of the pair (X = ℂ^{2}, C), write down the log discrepancy of every exceptional divisor E_{i} with respect to the pair (X, tC) as a function of t and then find out the largest value c of t for which all of these log discrepancies are nonnegative. However there is an easier way. We know that when t = c there is at least one divisor of log discrepancy zero (and every other divisor has non-negative log discrepancy). Let π: Y ⟶ X extract just this divisor. To construct π we simply contract all other divisors on the log resolution.

Almost by definition we can write

K_{Y} + E + cD = π^{∗}(K_{X} + cC),

where E is the exceptional divisor and D is the strict transform of C. Restrict both sides of this equation to E. As the RHS is a pullback, we get a numerically trivial divisor.

To compute the LHS we apply adjunction. E is a copy of ℙ^{1}. One slightly delicate issue is that Y is singular along E and the adjunction formula has to take account of this. In fact Y ⟶ X is precisely the weighted blow up of X = ℂ^{2}, with weights (a, b), in the given coordinates x, y. There are two singular points p and q of Y along C, of index a and b, and D intersects C transversally at another point r. If we apply adjunction we get

(K_{Y} + E + cD)|_{E} = K_{E} + (\frac{a − 1}{a}) p + (\frac{b − 1}{b}) q + cr.

As (K_{Y} +E+cD)|_{E} is numerically trivial we have (K_{Y} +E+cD)·E = 0 so that

−2 + \frac{a − 1}{a} + \frac{b − 1}{b} + c = 0,

and so

c = \frac{1}{a} + \frac{1}{b}.

Now let us consider the general case. As with the example above the first step is to extract divisors of log discrepancy zero, π: Y ⟶ X. To construct π we mimic the argument above; pick a log resolution for the pair (X, Δ + C) and contract every divisor whose log discrepancy is not zero. The fact that we can do this in all dimensions follows from the MMP (minimal model program), see (3.3.1) and π is called a divisorially log terminal modification.

The next step is the same, restrict to the general fibre of some divisor of log discrepancy zero, see (5.1). There are similar formulae for the coefficients of the restricted divisor, see (4.1). In this way, we reduce the problem from a local one in dimension n to a global problem in dimension n − 1, see §5. This explains how to go from Theorem D_{n−1} to Theorem A_{n}, see the proof of (5.3).

The global problem involves log canonical pairs (X, Δ), where X is projective and K_{X} + Δ is numerically trivial. One reason that the dimension one case is easy is that there is only one possibility for X, X must be isomorphic to ℙ^{1}. In higher dimensions it is not hard, running the MMP again, to reduce to the case where X has Picard number one, so that at least X is a Fano variety and Δ is ample. In this case we perturb Δ by increasing one of its coefficients to get a kawamata log terminal pair (X, Λ) such that K_{X} +Λ is ample. We then exploit the fact that some fixed multiple m(K_{X} + Λ) of K_{X} + Λ gives a birational map ϕ_{m(K_{X}+Λ)}. By definition this means that ϕ_{⌊m(K_{X}+Λ)⌋} is a birational map, which in particular means that K_{X} + Λ_{⌊m⌋} (see (3.1) for the definition of Λ_{⌊m⌋}) is big . This forces Δ ≤ Λ_{⌊m⌋} which implies that there are lots of gaps. This explains how to go from Theorem C_{n} to Theorem D_{n}, see the proof of (8.1).

It is clear then that the main thing to prove is that if (X, Δ) is a kawamata log terminal pair, K_{X} + Δ is big and the coefficients of Δ belong to a DCC set then some fixed multiple of K_{X} + Δ, gives a birational map ϕ_{m(K_{X}+Δ)}. Following some ideas of Tsuji, we developed a fairly general method to prove such a result in [15], see (3.5.2) and (3.5.5). We use the technique of cutting non kawamata log terminal centres as developed in [5], see [24]. The main issue is to find a boundary on the non kawamata log terminal centre so that we can run an induction.

There are two key hypotheses to apply (3.5.5). One of them requires that the volume of K_{X} +Δ restricted to appropriate non kawamata log terminal centres is bounded from below. The other places a requirement on the coefficients of Δ which is stronger than the DCC.

The first condition follows by induction on the dimension and a strong version of Kawamata’s subadjunction formula, (4.2), which we now explain. If (X, Λ) is a log pair and V is a non kawamata log terminal centre such that (X, Λ) is log canonical at the generic point of V, then one can write

(K_{X} + Λ)|_{W} = K_{W} + Θ_{b} + J,

where W is the normalisation of V, Θ_{b} is the discriminant divisor and J is the moduli part. Not much is known about the moduli part J beyond the fact that it is pseudo-effective. On the other hand Θ_{b} ≥ 0 behaves very well. If (X, Λ) is log canonical at the generic point of a prime divisor B on W then the coefficient of B in Θ_{b} is at most one. In fact there is a simple way to compute the coefficient of B involving the log canonical threshold. By assumption there is a log canonical place, that is, a valuation with centre V of log discrepancy zero. Then we can find a divisorially log terminal modification g: Y ⟶ X such that the centre of this log canonical place is a divisor S on Y. Note that there is a commutative diagram

[figure: S; Y; W; f; X.; g]

If we pullback K_{X} + Δ to Y and restrict to S we get a divisor Φ′ on S. Let

λ = sup{ t ∈ ℝ | (S, Φ′ + tf^{∗}B) is log canonical over a neighbourhood of the generic point of B },

be the log canonical threshold. Then the coefficient of B in Θ_{b} is 1 − λ.

In practice we start with a divisor Δ whose coefficients belong to I such that (X, Δ) is kawamata log terminal. We then find a divisor Δ_{0}, whose coefficients we have no control on, and V is a non kawamata log terminal centre of (X, Λ = Δ + Δ_{0}). It follows that the coefficients of Φ′ do not behave well and we have no control on the coefficients of Θ_{b}.

To circumvent this we simply mimic the same construction for (X, Δ) rather than (X, Λ). First we construct a divisor Φ on S whose coefficients of Φ belong to D(I), see (4.1). Then we construct a divisor Θ whose coefficients automatically belong to the set

{ a | 1 − a ∈ LCT_{n−1}(D(I)) } ∪ { 1 }.

It is clear from the construction that Θ_{b} ≥ Θ, so that if we bound the volume of K_{W} +Θ from below we bound the volume of (K_{X} +Δ+Δ_{0})|_{W} from below.

On the other hand, as part of the induction we assume that Theorem A_{n−1} holds. Hence LCT_{n−1}(D(I)) satisfies the ACC and the coefficients of Θ belong to a set which satisfies the DCC. The final step is to observe that if we choose V to pass through a general point then it belongs to a family which covers X. If we assume that V is a general member of such a family then we can pullback K_{X} + Δ to this family and restrict to V. It is straightforward to check that the difference between K_{W} + Θ and (K_{X} + Δ)|_{W} on a log resolution of the family is pseudo-effective (for example, if X and V are smooth then this follows from the fact that the first chern class of the normal bundle is pseudo-effective), so that if K_{X} + Δ is big then so is K_{W} + Θ. In this case we know the volume is bounded from below by induction.

We now explain the condition on the coefficients. To apply (3.5.5) we require that either I is a finite set or

I = \{ \frac{r − 1}{r} | r ∈ ℕ \}.

The first lemma, (7.2), simply assumes this condition on I and we deduce the result in this case.

The key is then to reduce to the case when I is finite. Given any positive integer p and a log pair (X, Δ), let Δ_{⌊p⌋} denote the largest divisor less than Δ such that pΔ_{⌊p⌋} is integral. Given I it suffices to find a fixed positive integer p such that if we start with (X, Δ) such that K_{X} + Δ is big and the coefficients belong to I then K_{X} + Δ_{⌊p⌋} is big, since the coefficients of Δ_{⌊p⌋} belong to the finite set

\{ \frac{i}{p} | 1 ≤ i ≤ p \}.

Let

λ = inf{ t ∈ ℝ | K_{X} + tΔ is big },

be the pseudo-effective threshold. A simple computation, (7.4), shows that it suffices to bound λ away from one. Running the MMP we reduce to the case when X has Picard number one. Since K_{X} + λΔ is numerically trivial and kawamata log terminal, Theorem B implies that the volume of Δ is bounded away from one. Passing to a log resolution we may assume that (X, D) has simple normal crossings where D is the sum of the components of Δ. As K_{X} +D is big then so is K_{X} + \frac{r−1}{r}D for any positive integer r which is sufficiently large. It follows that some fixed multiple of K_{X} + \frac{r−1}{r}D gives a birational map, and (3.5.2) implies that (X, D) belongs to log birationally bounded family. In this case, it is easy to bound the pseudo-effective threshold λ away from one, see (7.3). This explains how to go from Theorem B_{n} to Theorem C_{n}, cf. (7.4).

We now explain the last implication. Suppose that (X, Δ) is kawamata log terminal and K_{X} + Δ is numerically trivial. If the volume of Δ is large then we may find a divisor Π numerically equivalent to a small multiple of Δ with large multiplicity at a general point, so that (X, Π) is not kawamata log terminal. In particular we may find Φ arbitrarily close to Δ such that (X, Φ) is not kawamata log terminal. The key lemma is to show that this is impossible, (6.1). By assumption we may extract a divisor S of log discrepancy zero with respect to (X, Φ). After we run the MMP we get down a log pair (Y, S +Γ) where Γ is the strict transform of Δ and both K_{Y} + S + Γ and −(K_{Y} + S + (1 − ϵ)Γ) are ample. Here ϵ > 0 is arbitrarily close to zero. If we restrict to S and apply adjunction, it is easy to see that this contradicts either ACC for the log canonical threshold or ACC for numerically trivial pairs. This explains how to go from Theorem D_{n−1} and Theorem A_{n−1} to Theorem B_{n}, cf. (6.2).

It is interesting to note that if (X, Δ) is log canonical then there is no bound on the volume of Δ:

Example 2.1.1. Let X be the weighted projective surface ℙ(p, q, r), where p, q and r are three positive integers and let Δ be the sum of the three coordinate lines. Then K_{X} + Δ ∼_{ℚ} 0 and

vol(X, Δ) = \frac{(p + q + r)^{2}}{pqr} .

But the set

\{ \frac{(p + q + r)^{2}}{pqr} | (p, q, r) ∈ ℕ^{3} \},

is dense in the positive real numbers, cf. [19, 22.5].

We now explain the proof of (1.11) which mirrors the proof of (1.1). We are given a sequence of log pairs (X, Δ) = (X_{i}, Δ_{i}) and we want to identify the limit points of the log canonical thresholds. The first step is to show that the set of log canonical thresholds is essentially the same as the set of pseudo-effective thresholds. In §5 we showed that every log canonical threshold in dimension n + 1 is a numerically trivial threshold in dimension n. To show the reverse inclusion, one takes the cone (Y, Γ) over a log canonical pair (X, Δ) where K_{X} + Δ is numerically trivial, (11.5).

In this way we are reduced to looking at log canonical pairs (X, Δ) such that K_{X} + Δ is numerically trivial. The basic idea is to generate a component of coefficient one and apply adjunction. To this end, we need to deal with the case where some coefficients of Δ don’t necessarily belong to I but instead they are increasing towards one, (11.7).

Running the MMP we reduce to the case of Picard number one, Case A, Step 1 and Case B, Steps 3 and 5. We may also assume that the non kawamata log terminal locus is a divisor. In particular −K_{X} is ample, any two components of Δ intersect and we may assume that the number of components of Δ is constant, (11.6). If (X, Δ) is not kawamata log terminal then there is a component of coefficient one and we are done, Case B, Step 2.

The argument now splits into two cases. Case A deals with the case that the coefficients of Δ are bounded away from one. In this case if the volume of Δ is arbitrarily large then we can create a component of coefficient one and we reduce to the other case, Case B. Otherwise (1.6) implies that (X, Δ) belongs to a bounded family, which contradicts the fact that the coefficients of Δ are not constant.

So we may assume we are in Case B, namely that some of the coefficients of the components of Δ are approaching one. We decompose Δ as A + B + C where the coefficients of A are approaching one, the coefficients of B are fixed, and we are trying to identify the limit of the coefficients of C. Using the fact that the Picard number of X is one, we may increase the coefficients of A to one and decrease the coefficients of C, without changing the limit of the coefficients of C. At this point we apply adjunction and induction, Case B, Step 6.

3. Preliminaries

3.1. Notation and Conventions. If D = ∑ d_{i}D_{i} is an ℝ-divisor on a normal variety X, then the round down of D is ⌊D⌋ = ∑⌊d_{i}⌋D_{i}, where ⌊d⌋ denotes the largest integer which is at most d, the fractional part of D is {D} = D − ⌊D⌋, and the round up of D is ⌈D⌉ = −⌊−D⌋. If m is a positive integer, then let

D_{⌊m⌋} = \frac{⌊mD⌋}{m}.

Note that D_{⌊m⌋} is the largest divisor less than or equal to D such that mD_{⌊m⌋} is integral.

The sheaf O_{X}(D) is defined by

O_{X}(D)(U) = { f ∈ K(X) | (f)|_{U} + D|_{U} ≥ 0 },

so that O_{X}(D) = O_{X}(⌊D⌋). Similarly we define |D| = |⌊D⌋|. If X is normal, and D is an ℝ-divisor on X, the rational map ϕ_{D} associated to D is the rational map determined by the restriction of ⌊D⌋ to the smooth locus of X.

We say that D is ℝ-Cartier if it is a real linear combination of Cartier divisors. If f: Y ⟶ X is a morphism then D|_{Y} denotes the pullback of D to Y, f^{∗}D. In general D|_{Y} is only well-defined up to ℝ-linear equivalence. However if f(Y) is not contained in the support of D then D|_{Y} is a well-defined ℝ-Cartier divisor. An ℝ-Cartier divisor D on a normal variety X is nef if D · C ≥ 0 for any curve C ⊂ X. We say that two ℝ-divisors D_{1} and D_{2} are ℝ-linearly equivalent, denoted D_{1} ∼_{ℝ} D_{2}, if the difference is an ℝ-linear combination of principal divisors.

A log pair (X, Δ) consists of a normal variety X and a ℝ-Weil divisor Δ ≥ 0 such that K_{X} + Δ is ℝ-Cartier. The support of Δ = ∑_{i∈I} d_{i}D_{i} (where d_{i} ≠ 0) is the sum D = ∑_{i∈I} D_{i}. If (X, Δ) has simple normal crossings, a stratum of (X, Δ) is an irreducible component of the intersection ∩_{j∈J}D_{j}, where J is a non-empty subset of I (in particular, a stratum of (X, Δ) is always a proper closed subset of X). If we are given a morphism X ⟶ T, then we say that (X, Δ) has simple normal crossings over T if (X, Δ) has simple normal crossings and both X and every stratum of (X, D) is smooth over T. We say that the birational morphism f: Y ⟶ X only blows up strata of (X, Δ), if f is the composition of birational morphisms f_{i}: X_{i+1} ⟶ X_{i}, 1 ≤ i ≤ k, with X = X_{0}, Y = X_{k+1}, and f_{i} is the blow up of a stratum of (X_{i}, Δ_{i}), where Δ_{i} is the sum of the strict transform of Δ and the exceptional locus.

A log resolution of the pair (X, Δ) is a projective birational morphism μ: Y ⟶ X such that the exceptional locus is the support of a μ-ample divisor and (Y, G) has simple normal crossings, where G is the support of the strict transform of Δ and the exceptional divisors. If we write

K_{Y} + Γ + ∑ b_{i}E_{i} = μ^{∗}(K_{X} + Δ)

where Γ is the strict transform of Δ, then b_{i} is called the coefficient of E_{i} with respect to (X, Δ). The log discrepancy of E_{i} is a(E_{i}, X, Δ) = 1−b_{i}. The log discrepancy of (X, Δ) is the infimum over all log resolutions of the log discrepancy of any exceptional divisor. The total log discrepancy of (X, Δ) is the minimum of the log discrepancy of (X, Δ) and 1 − a where a ranges over the coefficients of the components of Δ. The pair (X, Δ) is kawamata log terminal (respectively log canonical; purely log terminal; divisorially log terminal) if b_{i} < 1 for all i and ⌊Δ⌋ = 0 (respectively b_{i} ≤ 1 for all i and for all log resolutions; b_{i} < 1 for all i and for all log resolutions; the coefficients of Δ belong to [0, 1] and there exists a log resolution such that b_{i} < 1 for all i). If we drop the condition that Δ ≥ 0 but all of the coefficients of Γ are at most one then we say that (X, Δ) is sub log canonical. A non kawamata log terminal centre is the centre of any valuation associated to a divisor E_{i} with b_{i} ≥ 1. In this paper, we only consider valuations ν of X whose centre on some birational model Y of X is a divisor.

We now introduce some results some of which are well known to experts but which are included for the convenience of the reader.

3.2. The volume.

Definition 3.2.1. Let X be an irreducible projective variety of dimension n and let D be an ℝ-divisor. The volume of D is

vol(X, D) = lim sup_{m→∞} \frac{n!h^{0}(X, O_{X}(mD))}{m^{n}}.

We say that D is big if vol(X, D) > 0.

For more background, see [31].

Lemma 3.2.2. Let X be a projective variety and let (X, Δ) be a log pair.

If D is an ℝ-divisor and vol(X, D) > n^{n} then for every point x ∈ X we may find Π ∼_{ℝ} D such that (X, Δ+Π) is not kawamata log terminal at x ∈ X.

Proof. Arguing as in the proof of [24, 6.7.1] we may assume that x ∈ X is a general point, so that in particular x is a smooth point of X. As the volume is a continuous function of D we may assume that D is a ℚ-divisor, [30, 2.2.44]. The result then follows as in the proof of [24, 6.1]. □

Lemma 3.2.3. Let X be a quasi-projective ℚ-factorial variety and let (X, Δ) be a kawamata log terminal pair.

If (X, Δ + D) is not log canonical, where D ≥ 0 is big, then we may find 0 ≤ D′ ∼_{ℝ} tD, for some 0 < t < 1 such that (X, Δ + D′) has exactly one log canonical place.

Proof. As (X, Δ + D) is not log canonical we may find δ > 0 such that (X, Δ+(1−δ)D) is not log canonical. As D is big we may find divisors A ≥ 0 and B ≥ 0 such that D ∼_{ℝ} A + B and A is ample. Replacing D by (1 − δ)D + δA + δB we may assume that there is an ample divisor A ≥ 0 such that D ≥ A.

Let

π: Y ⟶ X

be a log resolution. We may write

K_{Y} + Γ + ∑ a_{i}E_{i} = π^{∗}(K_{X} + Δ + tD),

where Γ is the strict transform of Δ and a_{i} are linear functions of t. By assumption a_{i} < 1 when t = 0 and there is an index i such that a_{i} > 1 when t = 1. It follows that we may find t ∈ (0, 1) such that a_{i} ≤ 1 for all indices with equality for at least one index i. Possibly using A to tie-break, see [24], we may assume that there is at most one index i such that a_{i} = 1. □

3.3. Divisorially log terminal modifications. If (X, Δ) is not kawamata log terminal then we may find a modification which is divisorially log terminal, so that the non kawamata log terminal locus is a divisor:

Proposition 3.3.1. Let (X, Δ) be a log pair where X is a variety and the coefficients of Δ belong to [0, 1].

Then there is a projective birational morphism π: Y ⟶ X such that (1) Y is ℚ-factorial,

(2) π only extracts divisors of log discrepancy at most zero,

(3) if E = ∑ E_{i} is the sum of the π-exceptional divisors and Γ is the strict transform of Δ, then (Y, Γ + E) is divisorially log terminal and

K_{Y} + E + Γ = π^{∗}(K_{X} + Δ) + ∑_{a(E_{i},X,B)<0} a(E_{i}, X, B)E_{i}.

(4) Further, if (X, Δ) is log canonical and S is a component of Δ then there is a nef divisor of the form −T − F, where T is the strict transform of S and F ≥ 0 is a sum of exceptional divisor whose centres are contained in S.

Any birational morphism π: Y ⟶ X satisfying (1–3) is called a divisorially log terminal modification.

Proof. The proof of (1–3) is due to the first author and can be found in [13], [28, 3.1], and also [4].

Now suppose that (X, Δ) is log canonical and S is a component of Δ. In this case

K_{Y} + E + Γ = π^{∗}(K_{X} + Δ).

Pick ϵ > 0 so that Γ−ϵT ≥ 0. Note that (Y, E+Γ−ϵT) is divisorially log terminal, as Y is ℚ-factorial and (Y, E + Γ) is divisorially log terminal. By Theorem 1.1 of [7] or by Theorem 1.6 of [16], we may replace Y by a log terminal model of (Y, E + Γ − ϵT) over X, gaining the fact that −T is nef over X, at the expense of temporarily losing the property that (Y, Γ + E) is divisorially log terminal, whilst preserving the condition that K_{Y} + E + Γ is log canonical and numerically trivial over X. If g: W ⟶ Y is a divisorially log terminal modification of (Y, Γ + E) and we replace Y by W then g^{∗}(−T) is a nef divisor over X of the correct form. □

3.4. DCC sets. We say that a set I of real numbers satisfies the descending chain condition or DCC, if it does not contain any infinite strictly decreasing sequence. For example,

I = \{ \frac{r − 1}{r} | r ∈ ℕ \},

satisfies the DCC. Let I ⊂ [0, 1]. We define

I_{+} := { 0 } ∪ { j ∈ [0, 1] | j = ∑_{p=1}^{l} i_{p}, for some i_{1}, i_{2}, …, i_{l} ∈ I },

and

D(I) := \{ a ≤ 1 | a = \frac{m − 1 + f}{m} , m ∈ ℕ, f ∈ I_{+} \}.

As usual, \overline{I} denotes the closure of I. Note that the set D(I) appears when we apply adjunction, (4.1).

Proposition 3.4.1. Let I ⊂ [0, 1].

(1) D(D(I)) = D(I) ∪ { 1 }.

(2) I satisfies the DCC if and only if \overline{I} satisfies the DCC.

(3) I satisfies the DCC if and only if D(I) satisfies the DCC.

Proof. Straightforward, see for example [32, 4.4]. □

3.5. Bounded pairs. We recall some results and definitions from [15], stated in a convenient form.

Definition 3.5.1. We say that a set 𝔛 of varieties is birationally bounded if there is a projective morphism Z ⟶ T, where T is of finite type, such that for every X ∈ 𝔛, there is a closed point t ∈ T and a birational map f: Z_{t} −−→ X.

We say that a set 𝔇 of log pairs is log birationally bounded (respectively bounded) if there is a log pair (Z, B), where the coefficients of B are all one, and a projective morphism Z ⟶ T, where T is of finite type, such that for every (X, Δ) ∈ 𝔇, there is a closed point t ∈ T and a birational map f: Z_{t} −−→ X (respectively isomorphism of varieties) such that the support of B_{t} is not the whole of Z_{t} and yet B_{t} contains the support of the strict transform of Δ and any f-exceptional divisor (respectively f(B_{t}) = Δ).

Theorem 3.5.2. Fix a positive integer n and a set I ⊂ [0, 1]∩ℚ, which satisfies the DCC. Let 𝔅_{0} be a set of log canonical pairs (X, Δ), where X is projective of dimension n, K_{X} + Δ is big and the coefficients of Δ belong to I.

Suppose that there is a constant M such that for every (X, Δ) ∈ 𝔅_{0} there is a positive integer k such that ϕ_{k(K_{X}+Δ)} is birational and

vol(X, k(K_{X} + Δ)) ≤ M.

Then the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔅_{0} },

satisfies the DCC.

Proof. Follows from (2.3.4), (3.1) and (1.9) of [15]. □

Recall:

Definition 3.5.3. Let X be a normal projective variety and let D be a big ℚ-Cartier ℚ-divisor on X.

If x and y are two general points of X then, possibly switching x and y, we may find 0 ≤ Δ ∼_{ℚ} (1 − ϵ)D, for some 0 < ϵ < 1, where (X, Δ) is not kawamata log terminal at y, (X, Δ) is log canonical at x and {x} is a non kawamata log terminal centre, then we say that D is potentially birational.

Note that this is a slight variation on the definition which appears in [15], where general is replaced by very general.

Theorem 3.5.4. Let (X, Δ) be a kawamata log terminal pair, where X is projective of dimension n and let H be an ample ℚ-divisor. Suppose there is a constant γ ≥ 1 and a family of subvarieties V ⟶ B with the following property.

If x and y are two general points of X then, possibly switching x and y, we can find b ∈ B and 0 ≤ Δ_{b} ∼_{ℚ} (1 − δ)H, for some δ > 0, such that (X, Δ + Δ_{b}) is not kawamata log terminal at y and there is a unique non kawamata log terminal place of (X, Δ + Δ_{b}) whose centre V_{b} contains x. Further there is a divisor D on W, the normalisation of V_{b}, such that ϕ_{D} is birational and γH|_{W} − D is pseudo-effective.

Then mH is potentially birational, where m = 2p^{2}γ + 1 and p = dim V_{b}.

Proof. Let x and y be two general points of X. Possibly switching x and y, we will prove by descending induction on k that there is a ℚ-divisor Δ_{0} ≥ 0 such that: (♭)_{k} Δ_{0} ∼_{ℚ} λH, for some λ < 2(p − k)pγ + 1, where (X, Δ + Δ_{0}) is log canonical at x, not kawamata log terminal at y and there is a non kawamata log terminal centre Z ⊂ V_{b} of dimension at most k containing x.

Suppose k = p. (X, Δ + Δ_{b}) is not kawamata log terminal but log canonical at x since there is a unique non kawamata log terminal place whose centre contains x. Thus Δ_{0} = Δ_{b} ∼_{ℚ} λH, where λ = 1 − δ < 1, satisfies (♭)_{k} and so this is the start of the induction.

Now suppose that we may find a ℚ-divisor Δ_{0} satisfying (♭)_{k}. We may assume that Z is the minimal non kawamata log terminal centre containing x and that Z has dimension k. Let Y ⊂ W be the inverse image of Z. As x is a general point of X it is also a general point of W, Y and Z. In particular the restriction of γH|_{W} − D to Y is pseudo-effective, Y ⟶ Z is birational, and as ϕ_{D} is birational and x is general, the restriction of ϕ_{D} to Y is birational. Thus

vol(Y, γH|_{Y}) ≥ vol(Y, D|_{Y}) ≥ 1,

where the last inequality is proved, for example, in [14, 2.2]. Note that

vol(Z, γH|_{Z}) = vol(Y, γH|_{Y}),

as H is nef, see for example [23, VI.2.15]. Thus

vol(Z, 2pγH|_{V}) > vol(Z, 2kγH|_{V}) ≥ 2k^{k},

so that by [15, 2.3.5], we may find Δ_{1} ∼_{ℚ} μH, where μ < 2pγ and constants 0 < a_{i} ≤ 1 such that (X, Δ + a_{0}Δ_{0} + a_{1}Δ_{1}) is log canonical at x, not kawamata log terminal at y and there is a non kawamata log terminal centre Z′ containing x, whose dimension is less than k. As

a_{0}Δ_{0} + a_{1}Δ_{1} ∼_{ℚ} (a_{0}λ + a_{1}μ)H,

and

λ′ = a_{0}λ + a_{1}μ < 2(p − k)pγ + 1 + 2pγ = 2(p − (k − 1))pγ + 1,

a_{0}Δ_{0} + a_{1}Δ_{1} satisfies (♭)_{k−1}. This completes the induction and the proof. □

Theorem 3.5.5. Fix a positive integer n. Let 𝔅_{0} be a set of kawamata log terminal pairs (X, Δ), where X is projective of dimension n and K_{X} + Δ is ample.

Suppose that there are positive integers p, k and l such that for every (X, Δ) ∈ 𝔅_{0} we have:

(1) There is a family of subvarieties V ⟶ B such that if x and y are two general points of X then, possibly switching x and y, we can find b ∈ B and 0 ≤ Δ_{b} ∼_{ℚ} (1 − δ)H, for some δ > 0, such that (X, Δ + Δ_{b}) is not kawamata log terminal at y and there is a unique non kawamata log terminal place of (X, Δ + Δ_{b}) whose centre V_{b} contains x, where H = k(K_{X} + Δ). Further there is a divisor D on W, the normalisation of V_{b}, such that ϕ_{D} is birational and lH|_{W} − D is pseudo-effective.

(2) Either pΔ is integral or the coefficients of Δ belong to

\{ \frac{r − 1}{r} | r ∈ ℕ \}.

Then there is a positive integer m such that ϕ_{mk(K_{X}+Δ)} is birational, for every (X, Δ) ∈ 𝔅_{0}.

Proof. Let m_{0} = 2(n − 1)^{2}l + 1. (3.5.4) implies that m_{0}H is potentially birational. But then [15, 2.3.4.1] implies that ϕ_{K_{X}+⌈m_{0}jH⌉} is birational for all positive integers j.

If pΔ is integral then

K_{X} + ⌈m_{0}kp(K_{X} + Δ)⌉ = ⌊(m_{0}kp + 1)(K_{X} + Δ)⌋,

and if the coefficients of Δ belong to

\{ \frac{r − 1}{r} | r ∈ ℕ \},

then

K_{X} + ⌈m_{0}kp(K_{X} + Δ)⌉ = ⌊(m_{0}kp + 1)(K_{X} + Δ)⌋.

Let m = (m_{0} + 1)p. □

4. Adjunction

We will need the following basic result about adjunction (see for example §16 in [27]).

Lemma 4.1. Let (X, Δ = S′ + B) be a log canonical pair, where S has coefficient one in Δ. If S is the normalisation of S′ then there is a divisor Θ = Diff_{S}(B) on S such that

(K_{X} + Δ)|_{S} = K_{S} + Θ.

(1) If (X, Δ) is purely log terminal then (S, Θ) is kawamata log terminal.

(2) If (X, Δ) is divisorially log terminal then (S, Θ) is divisorially log terminal.

(3) If B = ∑ b_{i}B_{i} then the coefficients of Θ belong to the set

D({ b_{1}, b_{2}, …, b_{m} }). In particular, if (X, Δ) is divisorially log terminal and the coefficients of B belong to the set I then the coefficients of Θ belong to the set D(I).

Theorem 4.2. Let I be a subset of [0, 1] which contains 1. Let X be a projective variety of dimension n and let V be an irreducible closed subvariety, with normalisation W. Suppose we are given a log pair (X, Δ) and an ℝ-Cartier divisor Δ′ ≥ 0, with the following properties:

(1) the coefficients of Δ belong to I,

(2) (X, Δ) is kawamata log terminal, and

(3) there is a unique non kawamata log terminal place ν for (X, Δ+ Δ′), with centre V.

Then there is a divisor Θ on W whose coefficients belong to

{ a | 1 − a ∈ LCT_{n−1}(D(I)) } ∪ { 1 },

such that the difference

(K_{X} + Δ + Δ′)|_{W} − (K_{W} + Θ),

is pseudo-effective.

Now suppose that V is the general member of a covering family of subvarieties of X. Let ψ: U ⟶ W be a log resolution of W and let Ψ be the sum of the strict transform of Θ and the exceptional divisors. Then

K_{U} + Ψ ≥ (K_{X} + Δ)|_{U}.

Proof. Since there is a unique non kawamata log terminal place with centre V, it follows that (X, Δ+Δ′) is log canonical but not kawamata log terminal at the generic point of V, see (2.31) of [29]. Let g: Y ⟶ X be a divisorially log terminal modification of (X, Δ + Δ′), (3.3.1), so that the centre of ν is a divisor S on Y and this is the only exceptional divisor with centre V. As (X, Δ + Δ′) is divisorially log terminal, S is normal and so there is a commutative diagram

[figure: S; Y; W; f; X.; g]

We may write

K_{Y} +S+Γ = g^{∗}(K_{X}+Δ)+E and K_{Y} +S+Γ+Γ′ = g^{∗}(K_{X}+Δ+Δ′),

where Γ is the sum of the strict transform of Δ and the exceptional divisors, apart from S. In particular the coefficients of Γ belong to I. As (X, Δ) is kawamata log terminal, E ≥ 0. As g is a divisorially log terminal modification of (X, Δ + Δ′), Γ′ ≥ 0 and (Y, S + Γ) is divisorially log terminal. We may write

(K_{Y} + S + Γ)|_{S} = K_{S} + Φ and (K_{Y} + S + Γ + Γ′)|_{S} = K_{S} + Φ′.

Note that the coefficients of Φ belong to D(I). Let B be a prime divisor on W. Let

μ = sup{ t ∈ ℝ | (S, Φ + tf^{∗}B) is log canonical over a neighbourhood of the generic point of B },

be the log canonical threshold over a neighbourhood of the generic point of B. We define Θ by

mult_{B}(Θ) = 1 − μ.

It is clear that the coefficients of Θ belong to

{ a | 1 − a ∈ LCT_{n−1}(D(I)) } ∪ { 1 }.

Let

λ = sup{ t ∈ ℝ | (S, Φ′ + tf^{∗}B) is log canonical over a neighbourhood of the generic point of B },

be the log canonical threshold over a neighbourhood of the generic point of B. We define a divisor Θ_{b} on W by

mult_{B}(Θ_{b}) = 1 − λ.

As Γ′ ≥ 0 we have Φ ≤ Φ′, so that λ ≤ μ. But then

Θ ≤ Θ_{b}.

Note that Θ_{b} is precisely the divisor defined in Kawamata’s subadjunction formula, see Theorem 1 and 2 of [18] and also (8.5.1) and (8.6.1) of [25]. It follows that the difference

(K_{X} + Δ + Δ′)|_{W} − (K_{W} + Θ_{b}),

is pseudo-effective, so that the difference

(K_{X} + Δ + Δ′)|_{W} − (K_{W} + Θ),

is certainly pseudo-effective.

Now suppose that V is the general member of a covering family of subvarieties of X. We first relate the definition of Θ, which uses the log canonical threshold on S, to a log canonical threshold on X. Let B be a prime divisor on W and let A be its image on V. Pick any ℚ-divisor H ≥ 0 on X which is ℚ-Cartier in a neighbourhood of the generic point of A and which does not contain V such that

mult_{B}(H|_{W}) = 1.

We have

K_{Y} + S + Γ + tg^{∗}H = g^{∗}(K_{X} + Δ + tH) + E,

and so

(K_{Y} + S + Γ + tg^{∗}H)|_{S} = K_{S} + Φ + tf^{∗}B,

over a neighbourhood of the generic point of B. Now if (X, Δ + tH) is not log canonical in a neighbourhood of the generic point of A then K_{Y} + S + Γ + tg^{∗}H is not log canonical over a neighbourhood of the generic point of B. Inversion of adjunction on Y, cf [17], implies that K_{Y} +S +Γ+tg^{∗}H is log canonical over a neighbourhood of the generic point of B if and only if K_{S} + Φ + tf^{∗}B is log canonical over a neighbourhood of the generic point of B. It follows that if

μ = sup{ t ∈ ℝ | (S, Φ + tf^{∗}B) is log canonical over a neighbourhood of the generic point of B },

the log canonical threshold of f^{∗}B over a neighbourhood of the generic point of B, and

ξ = sup{ t ∈ ℝ | (X, Δ + tH) is log canonical at the generic point of A },

the log canonical threshold of H at the generic point of A, then μ ≤ ξ.

By assumption we may pick a component R_{0} of the Hilbert scheme H whose universal family dominates X, such that V is the general member of R_{0}. Let π_{0} : Z_{0} ⟶ R_{0} be the restriction of the normalisation of the universal family. Cutting by hyperplanes in R_{0} we may find R ⊂ R_{0} with V ∈ R such that if π: Z ⟶ R is the restriction of π_{0} then the natural morphism h: Z ⟶ X is generically finite (note that if we take the hyperplanes successively from a fixed sequence of general pencils of hyperplanes then we won’t lose the fact that V is a general element). We may write

K_{Z} + Ξ = h^{∗}(K_{X} + Δ).

Possibly blowing up, we may assume that (Z, Ξ) has simple normal crossings over a dense open subset R_{1} of R. Let U be the fibre of π corresponding to W. As V is a general member of R_{0}, we may assume that r = π(U) ∈ R_{1} and so (U, Ξ|_{U}) has simple normal crossings. As the coefficients of Ξ|_{U} are at most one it follows that (U, Ξ|_{U}) is sub log canonical. Therefore it is enough to check that

K_{U} + Ψ ≥ (K_{X} + Δ)|_{U} = K_{U} + Ξ|_{U},

on the given model and in fact we just have to check that Ψ ≥ Ξ|_{U}.

Let C be a prime divisor on U. If mult_{C} Ξ|_{U} ≤ 0 there is nothing to prove as Ψ ≥ 0. If C is an exceptional divisor of U ⟶ V then mult_{C} Ψ = 1 and there is again nothing to prove as mult_{C} Ξ|_{U} ≤ 1.

Otherwise pick a prime component G of Ξ such that mult_{C}(G|_{U}) = 1. If h(G) is a divisor then let H = h(G)/e where e is the ramification index at G. Note that the pullback of H to W is ℚ-Cartier in a neighbourhood of the generic point of B = ψ(C). Otherwise, pick a ℚ-Cartier divisor H ≥ 0, which does not contain V, such that mult_{G}(h^{∗}H) = 1. Either way, as r ∈ R is general it follows that mult_{C}(h^{∗}H|_{U}) = 1. But then

mult_{B}(H|_{W}) = mult_{C}(h^{∗}H|_{U}) = 1.

We may write

K_{Z} + Ξ + ξh^{∗}H = h^{∗}(K_{X} + Δ + ξH).

As (X, Δ+ξH) is log canonical in a neighbourhood of the generic point of B, K_{Z} + Ξ + ξh^{∗}H is sub log canonical in a neighbourhood of the generic point of C. Note that in a neighbourhood of the generic point of C,

(K_{Z} + Ξ + ξh^{∗}H)|_{U} = K_{U} + Ξ|_{U} + ξC + J,

where J ≥ 0. As r is a general point of R, (U, Ξ|_{U} + ξC + J) is sub log canonical in a neighbourhood of the generic point of C. It follows that

mult_{C} Ξ|_{U} + ξ ≤ 1,

so that

mult_{C} Ψ = mult_{B} Θ = 1 − μ ≥ 1 − ξ ≥ mult_{C} Ξ|_{U}.

Thus Ψ ≥ Ξ|_{U}. □

5. Global to local

Lemma 5.1. Fix a positive integer n and a set 1 ∈ I ⊂ [0, 1].

Suppose (X, Δ) is a log canonical pair where X is a variety of dimension n + 1, the coefficients of Δ belong to I and there is a non kawamata log terminal centre V ⊂ X. Suppose that c ∈ I is the coefficient of some component M of Δ which contains V.

Then we may find a log canonical pair (S, Θ) where S is a projective variety of dimension at most n, the coefficients of Θ belong to D(I), K_{S} + Θ is numerically trivial and some component of Θ has coefficient

\frac{m − 1 + f + kc}{m},

where m, k ∈ ℕ and f ∈ D(I).

Proof. Possibly passing to an open subset of X and replacing V by a maximal (with respect to inclusion) non kawamata log terminal centre, we may assume that X is quasi-projective. If V is a divisor then M = V is a component of Δ with coefficient one so that c = 1. As 1 ∈ I we may take (S, Θ) = (ℙ^{1}, p + q), where p and q are two points of ℙ^{1}.

Otherwise, let π: Y ⟶ X be a divisorially log terminal modification of (X, Δ). Then Y is ℚ-factorial and we may write

K_{Y} + E + Γ = π^{∗}(K_{X} + Δ),

where Γ is the strict transform of Δ, E is the sum of the exceptional divisors and the pair (Y, E + Γ) is divisorially log terminal. By (4) of (3.3.1) we may choose π so that there is a nef divisor of the form −N − F, where N is the strict transform of M and F ≥ 0 is a sum of exceptional divisors whose centres are contained in M.

By assumption π is not an isomorphism over the generic point of V. It follows that N must intersect an exceptional divisor S of π whose centre is V. We may write

(K_{Y} + E + Γ)|_{S} = K_{S} + Θ,

by adjunction, where (S, Θ) is divisorially log terminal, the coefficients of Θ belong to D(I) and some component of Θ has a coefficient of the form

\frac{m − 1 + f + kc}{m},

where m, k ∈ ℕ and f ∈ D(I). Note that N ∩ S dominates V. If v ∈ V is a general point then (S_{v}, Θ_{v}) is divisorially log terminal, S_{v} is projective of dimension at most n, the coefficients of Θ_{v} belong to D(I), some component of Θ_{v} has a coefficient of the form

\frac{m − 1 + f + kc}{m},

and K_{S_{v}} + Θ_{v} is numerically trivial. □

Lemma 5.2. Let I ⊂ [0, 1] be a set which satisfies the DCC. If J_{0} ⊂ [0, 1] is a finite set then

I_{0} = \{ c ∈ I | \frac{m − 1 + f + kc}{m} ∈ J_{0}, for some k, m ∈ ℕ and f ∈ D(I) \}

is a finite set.

Proof. We may assume that c ≠ 0. Suppose that

l = \frac{m − 1 + f + kc}{m} ∈ J_{0}.

Then kc ≤ 1. As I satisfies the DCC, we may find δ > 0 such that c > δ. It follows that k < 1/δ so that k can take on only finitely many values. As J_{0} is finite, we may find ϵ > 0 such that if l < 1 then l < 1 − ϵ. But then m < \frac{1}{ϵ}. If l = 1 then f + kc = 1, in which case we may take m = 1. Either way, we may assume that m takes on only finitely many values.

Fix k, m and l. Then

c = \frac{(ml − m + 1) − f}{k} .

The LHS belongs to I, a set which satisfies the DCC. The RHS belongs to a set which satisfies the ACC. But the only set which satisfies both the DCC and the ACC is a finite set. □

Lemma 5.3. Theorem D_{n−1} implies Theorem A_{n}.

Proof. As I satisfies the DCC so does J = D(I). As we are assuming Theorem D_{n−1}, there is a finite set J_{0} ⊂ J such that if (S, Θ) is a log canonical pair where S is projective of dimension at most n − 1, the coefficients of Θ belong to J and K_{S} + Θ is numerically trivial, then the coefficients of Θ belong to J_{0}. Let

I_{0} = \{ c ∈ I | \frac{m − 1 + f + kc}{m} ∈ J_{0} for some k and m ∈ ℕ and f ∈ I_{+} \}.

As J_{0} is a finite set, (5.2) implies that I_{0} is also a finite set.

Suppose that (X, Δ) is a log canonical pair where X is a quasiprojective variety of dimension n, the coefficients of Δ belong to I, and there is a non kawamata log terminal centre Z ⊂ X which is contained in every component of Δ. (5.1) implies that the coefficients of Δ belong to I_{0}. □

6. Upper bounds on the volume

Lemma 6.1. Using the notation of Theorem B_{n}, Theorem D_{n−1} and Theorem A_{n−1} imply that there is a constant ϵ > 0 with the following property:

If (X, Δ) ∈ 𝔇, where X has dimension n, Δ is big and K_{X} + Φ is numerically trivial, where

Φ ≥ (1 − δ)Δ,

for some δ < ϵ, then (X, Φ) is kawamata log terminal.

Proof. Theorem D_{n−1} and Theorem A_{n−1} imply that we may find ϵ > 0 with the following property: if S is a projective variety of dimension n − 1, (S, Θ) and (S, Θ′) are two log pairs, the coefficients of Θ belong to D(I), and

(1 − ϵ)Θ ≤ Θ′ ≤ Θ,

then (S, Θ) is log canonical if (S, Θ′) is log canonical, and moreover Θ = Θ′ if in addition K_{S} + Θ′ is numerically trivial.

Suppose that (X, Φ) is not kawamata log terminal, where

Φ ≥ (1 − δ)Δ,

for some δ < ϵ and K_{X} + Φ is numerically trivial. As δ < ϵ and Φ is big we may assume that K_{X} + Φ is not log canonical. Pick λ ∈ (0, 1] such that (X, (1 − λ)Δ + λΦ) is log canonical but not kawamata log terminal. As Φ is big, δ < ϵ and (X, Δ) is kawamata log terminal, (3.2.3) implies that, perturbing Φ, we may assume (X, (1 − λ)Δ + λΦ) has only one non kawamata log terminal place.

Replacing Φ by (1 − λ)Δ + λΦ we may assume that (X, Φ) is purely log terminal and the non kawamata log terminal locus is irreducible. Let ϕ: Y ⟶ X be a divisorially log terminal modification of (X, Φ). We may write

K_{Y} + Ψ = ϕ^{∗}(K_{X} + Φ) and K_{Y} + Γ + aS = ϕ^{∗}(K_{X} + Δ),

where S = ⌊Ψ⌋ is a prime divisor, Γ is the strict transform of Δ and a < 1, as (X, Δ) is kawamata log terminal.

As K_{Y} +Ψ is numerically trivial, K_{Y} +Ψ−S is not pseudo-effective. By [8, 1.3.3], we may run f: Y −−→ W the (K_{Y} + Ψ − S)-MMP until we end with a Mori fibre space π: W ⟶ Z. As K_{Y} +Ψ is numerically trivial, every step of this MMP is S-positive, so that the strict transform T of S dominates Z. Let F be the general fibre of π. Replacing Y, Γ and Ψ by F and the restriction of π_{∗}Γ and π_{∗}Ψ to F, we may assume that S, Ψ and Γ are ℚ-linearly equivalent to multiples of the same ample divisor.

In particular K_{Y} + Γ + S is ample. As Ψ ≥ (1 − ϵ)Γ + S, it follows that K_{Y} + (1 − η)Γ + S is numerically trivial, for some 0 < η < ϵ, and K_{Y} + (1 − ϵ)Γ + S is log canonical. We may write

(K_{Y} + (1 − ϵ)Γ + S)|_{S} = K_{S} + Θ_{1},
(K_{Y} + (1 − η)Γ + S)|_{S} = K_{S} + Θ_{2}, and
(K_{Y} + Γ + S)|_{S} = K_{S} + Θ,

where the coefficients of Θ belong to D(I). Note that

(1 − ϵ)Θ ≤ Θ_{1} ≤ Θ_{2} ≤ Θ,

where by (4.1) the first inequality follows from the inequality

t (\frac{m − 1 + f}{m} ) ≤ \frac{m − 1 + tf}{m} for any t ≤ 1.

As (S, Θ_{1}) is log canonical, it follows that (S, Θ) is log canonical. In particular (S, Θ_{2}) is also log canonical. As K_{S} + Θ_{2} is numerically trivial, Θ = Θ_{2}, a contradiction. □

Lemma 6.2. Theorem D_{n−1} and Theorem A_{n−1} imply Theorem B_{n}.

Proof. Let ϵ > 0 be the constant given by (6.1). If (X, Δ) ∈ 𝔇, Δ is big, Π ∼_{ℝ} ηΔ and (X, Π + (1 − η)Δ) is not kawamata log terminal, then (6.1) implies that η ≥ ϵ. But then (3.2.2) implies that

vol(X, Δ) ≤ (\frac{n}{ϵ})^{n} .  □

7. Birational boundedness

Lemma 7.1. Let (X, Δ) be a log pair, where X is a projective variety of dimension n and let D be a big ℝ-divisor.

If vol(X, D) > (2n)^{n} then there is a family V ⟶ B of subvarieties of X such that if x and y are two general points of X then we may find b ∈ B and 0 ≤ Δ_{b} ∼_{ℝ} D such that (X, Δ + Δ_{b}) is not kawamata log terminal at y and there is a unique non kawamata log terminal place of (X, Δ + Δ_{b}) whose centre V_{b} contains x. Further, if B_{1}, B_{2}, …, B_{k} are the irreducible components of B and V_{i} ⟶ B_{i} is the corresponding family then the natural map V_{i} ⟶ X is dominant.

Proof. Let K be the algebraic closure of the function field of X. There is a fibre square

X_{K}  \overline{}X

[figure: Spec K; Spec k.]

Let ξ be the closed point of X_{K} corresponding to the generic point of X, and let Δ_{K} and D_{K} be the pullbacks of Δ and D to X_{K}. (3.2.2) implies that we may find 0 ≤ D_{ξ} ∼_{ℝ} D_{K}/2 such that (X_{K}, Δ_{K} + D_{ξ}) is not log canonical at ξ. By standard arguments we may spread out D_{ξ} to a family of divisors D_{t}, t ∈ T, where there is dominant morphism g: T ⟶ X such that (X, Δ + D_{t}) is not log canonical at x = g(t) and where D_{t} ∼_{ℝ} D/2.

Let y be a general point of X. Pick s such that (X, Δ + D_{s}) is not log canonical at y = g(s), where D_{s} ∼_{ℝ} D/2. Let

β = β_{s,t} = sup{ λ ∈ ℝ | (X, Δ + λ(D_{t} + D_{s})) is log canonical at x },

be the log canonical threshold. Thus (X, Δ + β(D_{s} + D_{t})) is log canonical but not kawamata log terminal at x. Possibly switching s and t, we may assume that (X, Δ+β(D_{s} +D_{t})) is not kawamata log terminal at y. Perturbing, by (3.2.3) we may assume that there is a unique non kawamata log terminal place of (X, Δ + β(D_{t} + D_{s})) whose centre V_{(s,t)} contains x (as y is general, we will not lose the property that (X, Δ + β(D_{t} + D_{s})) is not kawamata log terminal at y). Decomposing B = T × T into finitely many locally closed subsets, we may assume that the log canonical threshold is constant on each irreducible component of B, and moreover that V_{s,t} forms a family V ⟶ B. Possibly discarding components of B, we may assume that every component of V dominates X. Then the image of B in X × X contains an open subset of the form U × U. □

Lemma 7.2. Assume Theorem C_{n−1} and Theorem A_{n−1}. Fix a positive integer p.

Let 𝔅_{1} be the set of kawamata log terminal pairs (X, Δ), where X is projective of dimension n, K_{X} + Δ is big and either pΔ is integral or the coefficients of Δ belong to

\{ \frac{r − 1}{r} | r ∈ ℕ \}.

Then there is a positive integer m such that ϕ_{m(K_{X}+Δ)} is birational, for every (X, Δ) ∈ 𝔅_{1}.

Proof. Passing to a log canonical model of (X, Δ) we may assume that K_{X} + Δ is ample.

Pick a positive integer k such that vol(X, k(K_{X} + Δ)) > (2n)^{n}. We will apply (3.5.5) to k(K_{X} + Δ). (2) holds by hypothesis.

Let

J = { 1 − a | a ∈ LCT_{n−1}(D(I)) } ∪ { 1}.

Theorem A_{n−1} implies that J satisfies the DCC.

Theorem C_{n−1} implies that there is a positive integer l such that if (U, Ψ) is a log canonical pair, where U is projective of dimension at most n − 1, the coefficients of Ψ belong to J and K_{U} + Ψ is big, then ϕ_{l(K_{U} +Ψ)} is birational.

Apply (7.1) to k(K_{X} + Δ) to get a family V ⟶ B. Let b ∈ B be a general point. Let ν: W ⟶ V_{b} be the normalisation of V_{b} and let 0 ≤ Δ_{b} ∼_{ℝ} k(K_{X} + Δ) be the divisor given by (7.1), so that V_{b} is the unique non kawamata log terminal place of (X, Δ + Δ_{b}) containing x. (4.2)_{n} implies that we may find Θ on W such that

(K_{X} + Δ + Δ_{b})|_{W} − (K_{W} + Θ),

is pseudo-effective, where the coefficients of Θ belong to J.

Let ψ: U ⟶ W be a log resolution of (W, Θ) and let Ψ be the sum of the strict transform of Θ and the exceptional divisors. (4.2)_{n} implies that

(K_{U} + Ψ) ≥ (K_{X} + Δ)|_{U},

so that K_{U} + Ψ is big. As the coefficients of Θ belong to J, it follows that the coefficients of Ψ belong to J. But then ϕ_{l(K_{U} +Ψ)} is birational. It is easy to see (1) of (3.5.5) holds.

As the hypotheses of (3.5.5) hold, there is a positive integer m_{0} such that ϕ_{m_{0}k(K_{X}+Δ)} is birational. If vol(X, K_{X} +Δ) ≥ 1 then vol(X, 2(n+ 1)(K_{X} + Δ)) > (2n)^{n} and ϕ_{2m_{0}(n+1)(K_{X}+Δ)} is birational.

Otherwise, if vol(X, K_{X} + Δ) < 1, then we may find k such that

(2n)^{n} < vol(X, k(K_{X} + Δ)) ≤ (4n)^{n}.

It follows that

vol(X, m_{0}k(K_{X} + Δ)) ≤ (4m_{0}n)^{n}.

(3.5.2) implies that there is a constant 0 < δ < 1 such that if (X, Δ) ∈ 𝔅, then

vol(X, K_{X} + Δ) > δ.

In this case,

vol(X, α(K_{X} + Δ)) > (2n)^{n},

where

α = \frac{2n}{δ},

and we may take m = max(m_{0}⌜α⌝, 2m_{0}(n + 1)). □

Lemma 7.3. Using the notation of Theorem C_{n}, assume Theorem C_{n−1}, Theorem A_{n−1}, and Theorem B_{n}.

Then there is a constant β < 1 such that if (X, Δ) ∈ 𝔅 then the pseudo-effective threshold

λ = inf{ t ∈ ℝ | K_{X} + tΔ is big },

is at most β.

Proof. We may assume that 1 ∈ I. Suppose that (X, Δ) ∈ 𝔅. Let π: W ⟶ X be a log resolution of (X, Δ). We may write

K_{W} + Ξ = π^{∗}(K_{X} + Δ) + F,

where Ξ is the strict transform of Δ plus the sum of the exceptional divisors and F ≥ 0 is exceptional as (X, Δ) is log canonical. Let

μ = inf{ t ∈ ℝ | K_{W} + tΞ is big }.

be the pseudo-effective threshold. As π_{∗}(K_{W} + μΞ) = K_{X} + μΔ is pseudo-effective it follows that λ ≤ μ and so it suffices to bound μ away from one. Replacing (X, Δ) by (W, Ξ) we may assume that (X, Δ) has simple normal crossings.

We may assume that λ > 1/2, so that K_{X} is not pseudo-effective. As K_{X} + Δ is big we may find 0 ≤ D ∼_{ℝ} (K_{X} + Δ). If ϵ > 0 then

(1 + ϵ)(K_{X} + λΔ) ∼_{ℝ} K_{X} + μΔ + ϵD,

where μ = (1 + ϵ)λ − ϵ < λ. It follows that if ϵ is sufficiently small then K_{X} + μΔ + ϵD is kawamata log terminal. By [8, 1.4.2], we may run f: X −−→ Y the (K_{X} + λΔ)-MMP with scaling until K_{Y} + Γ is kawamata log terminal and nef, where Γ = f_{∗}(λΔ). Now we may run the (K_{Y} + μf_{∗}Δ)-MMP with scaling of f_{∗}D until we get to a Mori fibre space π: Y ⟶ Z; all steps of this MMP are (K_{Y} + Γ)-trivial, as all steps of this MMP are (K_{Y} + μf_{∗}Δ + ϵf_{∗}D)-trivial, so that (Y, Γ) remains kawamata log terminal and nef. Replacing (X, Δ) by a log resolution, we may assume that f is a morphism. Replacing X by the general fibre of the composition of f and π, we may assume that Z is a point, so that K_{Y} + Γ is numerically trivial.

Suppose that we have a sequence of such log pairs (X_{l}, Δ_{l}) ∈ 𝔅. We may assume that the pseudo-effective threshold is an increasing sequence,

λ_{1} < λ_{2} < λ_{3} < …,

and it suffices to bound this sequence away from one. Let

J = { λ_{l}i | i ∈ I, l ∈ ℕ }.

Then J satisfies the DCC, as λ_{l} are an increasing sequence.

Theorem B_{n} implies that there is a constant C such that vol(Y, Γ) < C for any Γ whose coefficients belong to J. Let α be the smallest nonzero element of J and let G = G_{l} be the sum of the components of Γ = Γ_{l}. Let Y = Y_{l}. Then

vol(Y, K_{Y} + G) = vol(Y, G − Γ)

≤ vol(Y, G)

≤ vol(Y, \frac{1}{α}Γ)

≤ \frac{C}{α^{n}} .

Let D be the sum of the components of Δ. Certainly K_{X} + D is big. We may write

K_{X} + D = f^{∗}(K_{Y} + G) + F,

where F is supported on the exceptional locus. It follows that

vol(X, K_{X} + D) ≤ vol(Y, K_{Y} + G) ≤ \frac{C}{α^{n}} .

Given (X_{l}, D_{l}) we may pick r ∈ ℕ such that

K_{X_{l}} + Θ_{l} = K_{X_{l}} + \frac{r − 1}{r}D_{l}

is big. As the coefficients of Θ_{l} belong to

\{ \frac{r − 1}{r} | r ∈ ℕ \},

(7.2) implies that

{ (X_{l}, Θ_{l}) | l ∈ ℕ },

is log birationally bounded. But then

{ (X_{l}, Δ_{l}) | l ∈ ℕ },

is log birationally bounded. In particular, [15, 1.9] implies that there is a constant δ > 0 such that

vol(X_{l}, K_{X_{l}} + Δ_{l}) ≥ δ,

for every l ∈ ℕ. In this case

δ ≤ vol(X, K_{X}+Δ) ≤ vol(Y, K_{Y} +\frac{1}{λ}Γ) = (\frac{1}{λ}−1)^{n} vol(Y, Γ) ≤ (\frac{1}{λ}−1)^{n}C,

so that we may take

β = \frac{1}{1 + (\frac{δ}{C} )^{1/n}} .  □

Lemma 7.4. Theorem C_{n−1}, Theorem A_{n−1}, and Theorem B_{n} imply Theorem C_{n}.

Proof. Replacing I by

I ∪ \{ \frac{r − 1}{r} | r ∈ ℕ \} ∪ \{1\},

we may assume that 1 is both an accumulation point of I and an element of I. Let α be the smallest non-zero element of I. By (7.3) there is a constant β < 1 such that if (X, Δ) ∈ 𝔅 then the pseudoeffective threshold

λ = inf{ t ∈ ℝ | K_{X} + tΔ is big },

is at most β.

Pick (X, Δ) ∈ 𝔅. Let π: Y ⟶ X be a log resolution of (X, Δ). Then we may write

K_{Y} + Γ = π^{∗}(K_{X} + Δ) + E,

where Γ is the strict transform of Δ plus the sum of the exceptional divisors. Replacing (X, Δ) by (Y, Γ) we may assume that (X, Δ) is log smooth. If S = ⌊Δ⌋, then we may pick r ∈ ℕ such that

K_{X} + Δ′ = K_{X} + \frac{r − 1}{r}S + \{Δ\},

is big. Replacing (X, Δ) by (X, Δ′), we may assume that (X, Δ) is kawamata log terminal.

Pick p such that

p > \frac{2}{α(1 − β)}.

If a is the coefficient of a component of Δ then

\frac{⌊pa⌋}{p} > a − \frac{1}{p}

> a − \frac{α(1 − β)}{2}

≥ a − \frac{a(1 − β)}{2}

= \frac{a(1 + β)}{2}.

It follows that

\frac{β + 1}{2}Δ ≤ Δ_{⌊p⌋} ≤ Δ,

so that K_{X} + Δ_{⌊p⌋} is big. Since the coefficients of Δ_{⌊p⌋} belong to

I_{0} = \{ \frac{i}{p} | 1 ≤ i ≤ p − 1 \},

(7.2) implies that there is a positive integer m such that ϕ_{m(K_{X}+Δ_{⌊p⌋})} is birational. But then ϕ_{m(K_{X}+Δ)} is birational as well. □

8. Numerically trivial log pairs

Lemma 8.1. Theorem D_{n−1} and Theorem C_{n} imply Theorem D_{n}.

Proof. We may assume that 1 ∈ I and n > 1.

As we are assuming Theorem D_{n−1} there is a finite set J_{0} ⊂ J = D(I) with the following property. If (S, Θ) is a log pair such that S is projective of dimension n − 1, the coefficients of Θ belong to J, (S, Θ) is log canonical, and K_{S} +Θ is numerically trivial, then the coefficients of Θ belong to J_{0}. Let I_{1} be the largest subset of I such that D(I_{1}) ⊂ J_{0}. (5.2) implies that I_{1} is finite.

Theorem C_{n} implies that there is a constant m with the following property: if (Y, Γ) is log canonical, Y is a projective variety of dimension n, K_{Y} +Γ is big and the coefficients of Γ belong to I, then ϕ_{m(K_{Y} +Γ)} is birational.

For every 1 ≤ l ≤ m, let

A_{l} = [(l − 1)/m, l/m),

and A_{m+1} = { 1 } so that

[0, 1] = ⋃_{l=1}^{m+1} A_{l}.

Let I_{2} be the union of the largest elements of A_{l} ∩ I (if A_{l} ∩ I does not have a largest element, either because it is empty or because it has infinitely many elements, then we ignore the elements of A_{l} ∩ I). Then I_{2} has at most m + 1 elements, so that I_{2} is certainly finite. Let I_{0} be the union of I_{1} and I_{2}.

Suppose that (X, Δ) satisfies (1–4) of Theorem D_{n}. Let π: Y ⟶ X be a divisorially log terminal modification, so that Y is ℚ-factorial. As (X, Δ) is log canonical if we write

K_{Y} + Γ = π^{∗}(K_{X} + Δ),

then Γ is the strict transform of Δ plus the exceptional divisors, so that (Y, Γ) is numerically trivial and divisorially log terminal. Replacing (X, Δ) by (Y, Γ), we may assume that X is ℚ-factorial. Further (X, Δ) is kawamata log terminal if and only if ⌊Δ⌋ = 0. Suppose that B is a prime component of Δ with coefficient i. It suffices to prove that i ∈ I_{0}. We may assume that i ≠ 1. Suppose that B intersects a component of ⌊Δ⌋. If S is the normalisation of this component then by adjunction we may write

(K_{X} + Δ)|_{S} = K_{S} + Θ,

where the coefficients of Θ belong to J = D(I) by (4.1). As S is projective of dimension n − 1, (S, Θ) is log canonical, and K_{S} + Θ is numerically trivial, the coefficients of Θ belong to J_{0}. But then i ∈ I_{1}.

As K_{X} +Δ is numerically trivial, K_{X} +Δ−iB is not pseudo-effective. By [8, 1.3.3] we may run f: X −−→ Y the (K_{X} +Δ−iB)-MMP until we reach a Mori fibre space. As K_{X} + Δ is numerically trivial, it follows that every step of this MMP is B-positive. If at some step of this MMP we contract a component S of ⌊Δ⌋ then this component intersects B and i ∈ I_{1} by the argument above. Otherwise, it follows that (Y, f_{∗}Δ) is kawamata log terminal if and only if ⌊f_{∗}Δ⌋ = 0. Further B is not contracted and so replacing (X, Δ) by (Y, f_{∗}Δ), we may assume that X is a Mori fibre space π: X ⟶ Z, where B dominates Z.

If Z is not a point, then replacing X by the general fibre of π we are done by induction. So we may assume that X has Picard number one. If ⌊Δ⌋ ≠ 0 then any component S of ⌊Δ⌋ intersects B and so i ∈ I_{1}. Otherwise ⌊Δ⌋ = 0 and we may assume that (X, Δ) is kawamata log terminal.

Suppose that j ∈ I and j > i. Let π: Y ⟶ X be a log resolution of (X, Δ). Let Γ_{0} be the strict transform of Δ, let E by the sum of the exceptional divisors, and let C be the strict transform of B. Set

Γ = Γ_{0} + E + (j − i)C.

Then (Y, Γ) is log canonical and the coefficients of Γ belong to I. We may write

K_{Y} + Γ_{0} + E = π^{∗}(K_{X} + Δ) + F,

where F ≥ 0 contains the full exceptional locus. Pick ϵ > 0 such that F ≥ ϵE. Note that (j − i)C + ϵE > δπ^{∗}B for any δ > 0 sufficiently small, so that

K_{Y} + Γ = (K_{Y} + Γ_{0} + (1 − ϵ)E) + (j − i)C + ϵE,

is big. Hence ϕ_{m(K_{Y} +Γ)} is birational, so that K_{Y} + Γ_{⌊m⌋} is big. But then K_{X} + Λ_{⌊m⌋} is big, where

Λ = π_{∗}Γ = Δ + (j − i)B.

It follows that if i ∈ A_{l}, then j ≥ l/m, so that i is the largest element of the interval A_{l} which also belongs to I. Hence i ∈ I_{2}. □

9. Proofs of Theorems

Proof of (1.5) and (1.4). This is Theorem A and Theorem D. □

Proof of (1.1). Suppose that c_{1}, c_{2}, … ∈ LCT_{n}(I, J), where c_{i} ≤ c_{i+1}. It suffices to show that c_{i} = c_{i+1} for i sufficiently large. By assumption we may find log canonical pairs (X_{i}, Δ_{i}) and ℝ-Cartier divisors M_{i}, where X_{i} is a variety of dimension n, the coefficients of Δ_{i} belong to I, the coefficients of M_{i} belong to J and c_{i} is the log canonical threshold,

c_{i} = sup{ t ∈ ℝ | (X_{i}, Δ_{i} + c_{i}M_{i}) is log canonical }.

Let Θ_{i} = Δ_{i} + c_{i}M_{i} and

K = I ∪ { c_{i}j | i ∈ ℕ, j ∈ J }.

Then (X_{i}, Θ_{i}) is log canonical, X_{i} is a variety of dimension n, the coefficients of Θ_{i} belong to K and there is a non kawamata log terminal centre V contained in the support of M_{i}. Possibly throwing away components of Θ_{i} which don’t contain V and passing to an open subset which contains the generic point of V, we may assume that every component of Θ_{i} contains V.

As K satisfies the DCC, (1.5) implies that the coefficients of Θ_{i} belong to a finite subset K_{0} of K. It follows c_{i} = c_{i+1} for i sufficiently large. □

Proof of (1.3). (3) is Theorem C.

Fix a constant V > 0 and let

𝔇_{V} = { (X, Δ) ∈ 𝔇 | 0 < vol(X, K_{X} + Δ) ≤ V }.

(3) implies that ϕ_{m(K_{X}+Δ)} is birational. (3.5.2) implies that the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇_{V} },

satisfies the DCC, which implies that (1) and (2) of (1.3) hold in dimension n. □

Lemma 9.1. Let Z ⟶ T be a projective morphism to a variety and suppose that (Z, Φ) has simple normal crossings over T. Suppose that the restriction of any irreducible component of Φ to any fibre is irreducible. Suppose that (Z, Φ) is kawamata log terminal and there is a closed point 0 ∈ T such that K_{Z_{0}} + Φ_{0} is big. Let 0 ≤ Θ ≤ Φ be any divisor with the same support as Φ.

Then we may find finitely many birational contractions f_{i}: Z −−→ X_{i} over T such that if f: Z_{t} −−→ Y is the log canonical model of (Z_{t}, Ψ) for some t ∈ T and Θ_{t} ≤ Ψ ≤ Φ_{t} then f = f_{it} for some index i.

Proof. [15, 1.7] implies that K_{Z} + Φ is big over T. Pick

0 ≤ D ∼_{ℝ,T} (K_{Z} + Φ).

Let

B = \frac{ϵ}{1 − ϵ}D.

If we pick ϵ > 0 sufficiently small then K_{Z} + B + Φ is kawamata log terminal and we may find a divisor 0 ≤ Θ′ ≤ Θ with

K_{Z} + Θ = ϵ(K_{Z} + Φ) + (1 − ϵ)(K_{Z} + Θ′).

If Θ ≤ Ξ ≤ Φ then

K_{Z} + Ξ ∼_{ℝ,T} (1 − ϵ)(K_{Z} + B + Ξ′),

where Θ′ ≤ Ξ′ ≤ Ξ. It is proved in [8, 1.1.5] that there are finitely many f_{1}, f_{2}, …, f_{k} birational contractions f_{i}: Z −−→ X_{i} over T such that if g: Z −−→ X is the log canonical model of K_{Z} + Ξ over T then g = f_{i} for some index 1 ≤ i ≤ k.

It suffices to show that if Ξ|_{Z_{t}} = Ψ and g is the log canonical model of K_{Z} + Ξ then f = g_{t}. For this we may assume that T is affine.

In this case the (relative) log canonical model is given by taking Proj

X_{i} = Proj(Z, R(Z, k(K_{Z} + Ξ))),

of the (truncation of the) canonical ring

R(Z, k(K_{Z} + Ξ)) = ⨁_{m∈ℕ} H^{0}(Z, O_{Z}(mk(K_{Z} + Ξ))).

On the other hand [15, 1.7] implies that if k is sufficiently divisible then

R(Z, k(K_{Z} + Ξ)) ⟶ R(Z_{t}, k(K_{Z_{t}} + Ψ)),

is surjective and so f = g_{t}. □

Proof of (1.6). By definition there is a log pair (Z, B) and a projective morphism Z ⟶ T, where T is of finite type with the following property. If (X, Δ) ∈ 𝔇 then there is a closed point t ∈ T and a birational map f: X −−→ Z_{t} such that the support of B_{t} is a divisor on Z_{t} which contains the support of the strict transform of Δ and any f^{−1}-exceptional divisor.

We may assume that T is reduced. Decomposing T into a finite union of locally closed subsets and throwing away some components, we may assume that every fibre Z_{t} is a variety and that B does not contain Z_{t}; blowing up and decomposing T into a finite union of locally closed subsets, we may assume that (Z, B) has simple normal crossings; passing to an open subset of T, we may assume that the fibres of Z ⟶ T are log pairs, so that (Z, B) has simple normal crossings over T; passing to a finite cover of T, we may assume that every stratum of (Z, B) has irreducible fibres over T; decomposing T into a finite union of locally closed subsets, we may assume that T is smooth; finally passing to a connected component of T, we may assume that T is integral.

Let a = 1 − ϵ < 1. By assumption δ ≤ a ≤ 1. Let Φ = aB and Θ = δB, so that Φ, Θ and B have the same support but the coefficients of Φ are all a, the coefficients of Θ are all δ and the coefficients of B are all one. As (Z, Φ) is kawamata log terminal it follows that there are only finitely many valuations of log discrepancy at most one with respect to (Z, Φ). As (Z, Φ) has simple normal crossings there is a sequence of blow ups Y ⟶ Z of strata, which extracts every divisor of log discrepancy at most one. Note that as (Z, Φ) has simple normal crossings over T, it follows that if t ∈ T is a closed point then every valuation of log discrepancy at most one with respect to (Z_{t}, Φ_{t}) has centre a divisor on Y_{t}.

Suppose that (X, Δ) ∈ 𝔇. Then there is a closed point t ∈ T and a birational map f: X −−→ Z_{t} such that the support of B_{t} contains the support of the strict transform of Δ_{t} and any f^{−1}-exceptional divisor. Let p: W ⟶ X and q: W ⟶ Z_{t} resolve f. Let S be the sum of the p-exceptional divisors and let Ξ be the sum of the strict transform of Δ and aS, so that S and Ξ are divisors on W. We may write

K_{W} + Ξ = p^{∗}(K_{X} + Δ) + E,

where E is a sum of p-exceptional divisors and E ≥ 0 as the log discrepancy of (X, Δ) is greater than ϵ.

Let Ψ = q_{∗}Ξ. We may write

p^{∗}(K_{X} + Δ) + E + F = q^{∗}(K_{Z_{t}} + Ψ),

where F is q-exceptional. As p^{∗}(K_{X} + Δ) is nef, it is q-nef so that E + F ≥ 0 by negativity of contraction. If ν is any valuation whose centre is a divisor on X then

a(Z_{t}, Φ_{t}, ν) ≤ a(Z_{t}, Ψ, ν) as Φ_{t} ≥ Ψ
≤ a(X, Δ, ν) as E + F ≥ 0
≤ 1  as the centre of ν is a divisor on X.

Therefore the induced birational map Y_{t} −−→ X is a birational contraction. Thus replacing Z by Y and B by its strict transform union the exceptional divisor, we may assume that g = f^{−1} : Z_{t} −−→ X is a birational contraction. In this case F is p-exceptional and so g is the log canonical model of (Z_{t}, Θ_{t}).

Since there are only finitely integral divisors 0 ≤ B′ ≤ B, replacing B we may assume that Ψ has the same support as B_{t}. K_{Z_{t}} + Φ_{t} is big as K_{Z_{t}} + Ψ is big and Φ_{t} ≥ Ψ. Finally Θ_{t} ≤ Ψ ≤ Φ_{t} and so we are done by (9.1). □

10. Proofs of Corollaries

Proof of (1.2). This follows from (1.1) and the main result of [6]. □

Proof of (1.7). (1.5) implies that there is a finite subset I_{0} ⊂ I such that the coefficients of Δ belong to I_{0}. Thus there is a positive integer r such that rΔ is integral.

On the other hand, Theorem B implies that there is a constant C such that vol(X, Δ) < C. Let D be the sum of the components of Δ. Then K_{X} + D is big and

vol(X, K_{X} + D) = vol(X, D − Δ)
≤ vol(X, D)
≤ vol(X, rΔ)
≤ Cr^{n}.

Let π: Y ⟶ X be a log resolution of (X, Δ). Let G be the sum of the strict transform of the components of Δ and the exceptional divisors. Then (Y, G) has simple normal crossings. Pick η > 0 such that (X, (1 + η)Δ) is kawamata log terminal and the log discrepancy is greater than ϵ. Then K_{X} + (1 + η)Δ is ample and we may write

K_{Y} + Γ = π^{∗}(K_{X} + (1 + η)Δ),

where Γ ≤ G. As K_{Y} + Γ is big it follows that K_{Y} + G is big. (1.3) implies that there is a positive integer m such that ϕ_{m(K_{Y} +G)} is birational, for every (X, Δ) ∈ 𝔇. But then 𝔇 is log birationally bounded by [15, 2.4.2.3-4]. Now apply (1.6). □

Proof of (1.8). Let D = −r(K_{X} + Δ). Then D is an ample Cartier divisor and D − (K_{X} + Δ) is ample. By Kollár’s effective base point free theorem (cf. [21]), there is a fixed positive integer m such that the linear system |mD| is base point free. Pick a general divisor H ∈ |mD|. Then (X, Λ = Δ + \frac{1}{mr}H) is kawamata log terminal and

K_{X} + Λ ∼_{ℚ} 0.

Note the coefficients of Λ belong to the finite set

I = \{ \frac{i}{r} | 1 ≤ i ≤ r − 1 \} ∪ \{ \frac{1}{mr} \}.

There are two ways to proceed. On the one hand we may apply (1.7). Here is a more direct approach. Theorem B implies that

vol(X, Λ),

is bounded from above. But then

vol(X, mD) ≤ (mr)^{n} vol(X, Λ),

is bounded from above. □

Proof of (1.10). Suppose that r_{1} ≤ r_{2} ≤ … is a non-decreasing sequence in R. For each i we may find (X, Δ) = (X_{i}, Δ_{i}) ∈ 𝔇 and a Cartier divisor H such that −(K_{X} + Δ) ∼_{ℝ} rH. By the cone theorem we may find a curve C such that −(K_{X} + Δ) · C ≤ 2n, cf. Theorem 18.2 of [13]. In particular r ≤ 2n, as H · C ≥ 1. By Fujino’s extension, [12], of Kollár’s effective base point free theorem, [21], to the case of log canonical pairs, there is a fixed positive integer m such that the linear system |mH| is base point free. Possibly replacing m by a multiple we may assume that m > 2n. Pick a general divisor D ∈ |mH|.

Then (X, Λ = Δ + \frac{r}{m}D) is log canonical and

K_{X} + Λ ∼_{ℝ} 0.

Then the coefficients of Λ_{i} = Λ belong to the set

I ∪ \{ \frac{r_{i}}{m} | i ∈ ℕ \},

which satisfies the DCC. (1.4) implies that the coefficients of Λ belong to a finite subset. But then r_{i} = r_{i+1} is eventually constant and so R satisfies the ACC. □

11. Accumulation points

Definition 11.1. Given I ⊂ [0, 1] and c ∈ [0, 1] let

D_{c}(I) = \{ a ≤ 1 | a = \frac{m − 1 + f + kc}{m}, k, m ∈ ℕ, f ∈ I_{+} \} ⊂ D(I∪\{ c \}).

Let 𝔑_{n}(I, c) be the set of log canonical pairs (X, Δ) such that X is a projective variety of dimension n, K_{X} + Δ is numerically trivial and we may write Δ = B + C, where the coefficients of B belong to D(I) and the coefficients of C ≠ 0 belong to D_{c}(I).

Let

N_{n}(I) = { c ∈ [0, 1] | 𝔑_{n}(I, c) is non-empty }.

Lemma 11.2. Let n ∈ ℕ and I ⊂ [0, 1].

(1) LCT_{n}(I) ⊂ LCT_{n+1}(I).

(2) N_{n}(I) ⊂ N_{n+1}(I).

(3) If f ∈ I_{+} and k ∈ ℕ then

c = \frac{1 − f}{k} ∈ N_{n}(I).

Proof. Let E be an elliptic curve. If (X, Δ = ∑ d_{i}Δ_{i}) is a log pair then (Y, Γ) is a log pair, where Y = X × E and Γ = ∑ d_{i}(Δ_{i} × E). By construction Γ has the same coefficients as Δ.

Note that (X, Δ) is log canonical if and only if (Y, Γ) is log canonical. This gives (1). Further if c ∈ [0, 1] and (X, Δ) ∈ 𝔑_{n}(I, c) then (Y, Γ) ∈ 𝔑_{n+1}(I, c). This is (2).

Using (2), it suffices to prove (3) when n = 1. Let X = ℙ^{1} and Δ = B + C, where B = fp + fq, C = 2kcr, and p, q and r are three points of ℙ^{1}. Then (X, Δ) ∈ 𝔑_{1}(I, c) (take m = 1) so that c ∈ N_{1}(I). This is (3). □

For technical reasons, it is convenient to introduce a smaller set than 𝔑_{n}(I, c):

Definition 11.3. Given I ⊂ [0, 1] and c ∈ [0, 1] let 𝔎_{n}(I, c) ⊂ 𝔑_{n}(I, c) be the subset consisting of kawamata log terminal pairs (X, Δ), where X is ℚ-factorial of Picard number one.

Let

K_{n}(I) = { c ∈ [0, 1] | 𝔎_{m}(I, c) is non-empty, for some m ≤ n }.

Lemma 11.4. If n ∈ ℕ and I ⊂ [0, 1] then

N_{n}(I ∪ { 1 }) = K_{n}(I).

In particular, N_{n}(I ∪ { 1 }) = N_{n}(I).

Proof. By (2) of (11.2), it suffices to show that

N_{n}(I ∪ { 1 }) ⊂ K_{n}(I).

Suppose that c ∈ N_{n}(I ∪ { 1 }). Then we may find (X, Δ) ∈ 𝔑_{n}(I ∪ { 1 }, c). By assumption we may write Δ = A + B + C, where the coefficients of A are one, the coefficients of B belong to D(I) and the coefficients of C ≠ 0 belong to D_{c}(I).

Let π: X′ ⟶ X be a divisorially log terminal modification of (X, Δ). If we write

K_{X′} + Δ′ = π^{∗}(K_{X} + Δ),

then X′ is projective of dimension n, X′ is ℚ-factorial, (X′, Δ′) is divisorially log terminal and K_{X′} + Δ′ is numerically trivial. Let B′ and C′ be the strict transforms of B and C and let A′ = Δ′ − B′ − C′. Then the coefficients of A′ are one, the coefficients of B′ belong to D(I) and the coefficients of C′ ≠ 0 belong to D_{c}(I). Thus (X′, Δ′) ∈ 𝔑_{n}(I ∪ { 1 }, c). Replacing (X, Δ) by (X′, Δ′) we may assume that X is ℚ-factorial and (X, A + B) is divisorially log terminal. Note that (X, Δ) is kawamata log terminal if and only if A = 0.

Suppose that A and C intersect. Let S be an irreducible component of A which intersects C. Then we may write

(K_{X} + Δ)|_{S} = K_{S} + Θ,

by adjunction, where (S, Θ) is divisorially log terminal and moreover we may write Θ = A′ +B′ +C′, where the coefficients of A′ are one, the coefficients of B′ belong to D(I) and the coefficients of C′ ≠ 0 belong to D_{c}(I). Thus (S, Θ) ∈ 𝔑_{n−1}(I ∪ { 1 }, c). Hence c ∈ N_{n−1}(I ∪ { 1 }) and so c ∈ K_{n−1}(I) ⊂ K_{n}(I), by induction on n.

Let f: X −−→ X′ be a step of the (K_{X} + A + B)-MMP. As K_{X} + Δ is numerically trivial, f is automatically C-positive. Suppose that f is birational. Let A′ = f_{∗}A, B′ = f_{∗}B and C′ = f_{∗}C, so that Δ′ = f_{∗}Δ = A′ + B′ + C′. C′ ≠ 0, as f is C-positive. X′ is a projective variety of dimension n, (X′, Δ′) is log canonical, K_{X′} + Δ′ is numerically trivial, the coefficients of A′ are all one, the coefficients of B′ belong to D(I) and the coefficients of C′ ≠ 0 belong to D_{c}(I). Thus (X′, Δ′) ∈ 𝔑_{n}(I ∪ { 1 }, c). Further X′ is ℚ-factorial and (X′, A′ + B′) is divisorially log terminal. If a component of A is contracted then A and C intersect and we are done. Otherwise (X′, Δ′) is kawamata log terminal if and only if A′ = 0.

If we run the (K_{X} + A + B)-MMP with scaling of an ample divisor then we end with a Mori fibre space. Therefore, replacing (X, Δ) by (X′, Δ′) finitely many times, we may assume that f: X −−→ Z = X′ is a Mori fibre space and C dominates Z. If dim Z > 0 then let z ∈ Z be a general point. Then (X_{z}, Δ_{z}) ∈ 𝔑_{n−k}(I ∪ { 1 }, c), where k = dim Z, and we are done by induction on the dimension.

So we may assume that Z is a point in which case X has Picard number one. If A ≠ 0 then A and C intersect and we are done. If A = 0 then (X, Δ) is kawamata log terminal and so (X, Δ) ∈ 𝔎_{n}(I, c). But then c ∈ K_{n}(I). □

Proposition 11.5. If I ⊂ [0, 1], I = I_{+} and n ∈ ℕ then LCT_{n+1}(I) = N_{n}(I).

Proof. We first show that LCT_{n+1}(I) ⊂ N_{n}(I). Pick 0 ≠ c ∈ LCT_{n+1}(I). By definition we may find a log canonical pair (X, Δ + cM) where X has dimension n + 1, the coefficients of Δ belong to I, M is an integral ℚ-Cartier divisor and there is a non kawamata log terminal centre V contained in the support of M. Possibly passing to an open subset of X and replacing V by a maximal non kawamata log terminal centre, we may assume that V is the only non kawamata log terminal centre of (X, Δ + cM). In particular, (X, Δ) is kawamata log terminal.

If V is a component of M then V has coefficient one in Δ + cM and c = \frac{1−f}{k} ∈ N_{n}(I) by (3) of (11.2). Otherwise let f: Y ⟶ X be a divisorially log terminal modification of (X, Δ + cM). Then Y is ℚ-factorial and we may write

K_{Y} + T + Δ′ + cM′ = f^{∗}(K_{X} + Δ + cM)

where Δ′ and M′ are the strict transforms of Δ and M, T is the sum of the exceptional divisors and the pair (Y, T +Δ′ +cM′) is divisorially log terminal. By (4) of (3.3.1) we may choose f so that T contains the inverse image of V. Let S be an irreducible component of T which intersects M′. Then we may write

(K_{Y} + T + Δ′ + cM′)|_{S} = K_{S} + Θ,

by adjunction, where (S, Θ) is divisorially log terminal and moreover we may write Θ = A + B + C, where the coefficients of A are one, the coefficients of B belong to D(I) and the coefficients of C ≠ 0 belong to D_{c}(I). As S is a non kawamata log terminal centre, the centre of S on X is V so that there is a morphism S ⟶ V. If v ∈ V is a general point then (S_{v}, Θ_{v}) ∈ 𝔑_{k}(I ∪ { 1 }, c), for some k ≤ n. Thus c ∈ N_{k}(I ∪ { 1 }) ⊂ N_{n}(I).

We now show that LCT_{n+1}(I) ⊃ N_{n}(I). Pick 0 ≠ c ∈ N_{n}(I). Then we may find a pair (X, Δ) ∈ 𝔎_{m}(I, c), some m ≤ n. If m < n then we are done by induction on the dimension. Otherwise X has dimension n. As −K_{X} is ample, we may pick d such that −dK_{X} is very ample and embed X into projective space by the linear system | − dK_{X}|.

Let Y be the cone over X and let Γ_{j} be the cone over Δ_{j}. Then Y is a quasi-projective variety of dimension n + 1. Y is ℚ-factorial as X has Picard number one. (Y, Γ = ∑ d_{i}Γ_{i}) is log canonical but not kawamata log terminal at the vertex p of the cone. By assumption we may write

d_{i} = \frac{m_{i} − 1 + f_{i} + k_{i}c}{m_{i}},

for each i, where m_{i} is a positive integer, k_{i} is a non-negative integer (k_{i} = 0 if Γ_{i} is a component of B_{i} and k_{i} > 0 if Γ_{i} is a component of C_{i}) and f_{i} ∈ I_{+}. Since we are working locally around p, the vertex of Y, we may find a cover of π: Ỹ ⟶ Y which ramifies over Γ_{i} to index m_{i} for every i and is otherwise unramified at the generic point of any divisor. We may write

KỸ + Γ̃ = π^{∗}(K_{Y} + Γ),

where the coefficients of Γ̃ belong to the set

{ f_{i} + k_{i}c | i }.

Ỹ is a ℚ-factorial quasi-projective variety of dimension n+1 and (Ỹ , Γ̃) is log canonical but not kawamata log terminal over any point q lying over p. Let

Θ = ∑ f_{i}Γ_{i} and M_{i} = ∑ k_{i}Γ_{i}.

Then the coefficients of Θ belong to I_{+} = I, M_{i} is an integral ℚ-Cartier divisor and

c = sup{ t ∈ ℝ | (X, Θ + tM) is log canonical },

is the log canonical threshold. But then c ∈ LCT_{n+1}(I). □

Lemma 11.6. Let (X, Δ) be a log canonical pair, where X is ℚfactorial of dimension n and Picard number one and K_{X} + Δ is numerically trivial.

If the coefficients of Δ are at least δ then Δ has at most \frac{n+1}{δ} components.

Proof. [27, 18.24] implies that the sum of the coefficients of Δ is at most n + 1. □

Proposition 11.7. Fix a positive integer n and a set I ⊂ [0, 1] whose only accumulation point is one such that I = I_{+}.

Let c_{1}, c_{2}, … ∈ [0, 1] be a strictly decreasing sequence with limit c ≠ 0 with the following property. There is a sequence of log canonical pairs (X_{i}, Δ_{i}) such that X_{i} is a projective variety of dimension n, K_{X_{i}} + Δ_{i} is numerically trivial and we may write Δ_{i} = A_{i} + B_{i} + C_{i}, where the coefficients of A_{i} are approaching one, the coefficients of B_{i} belong to D(I) and the coefficients of C_{i} ≠ 0 belong to D_{c_{i}}(I).

Then c ∈ N_{n−1}(I).

Proof. We may assume that A_{i} and B_{i} + C_{i} have no common components. Replacing B_{i} by B_{i} − ⌊B_{i}⌋ and A_{i} by A_{i} + ⌊B_{i}⌋ we may assume that ⌊Δ_{i}⌋ = ⌊A_{i}⌋. As the coefficients of A_{i} + B_{i} belong to a set which satisfies the DCC, (1.5) implies that not all of the coefficients of C_{i} are increasing. In particular at least one coefficient of C_{i} is bounded away from one.

Let a_{i} be the total log discrepancy of (X_{i}, Δ_{i}).

Case A: lim a_{i} > 0.

In this case, we assume that a_{i} is bounded away from zero.

Case A, Step 1: We reduce to the case X_{i} is ℚ-factorial and the Picard number of X_{i} is one.

As we are assuming that a_{i} is bounded away from zero, A_{i} = 0 and so (X_{i}, Δ_{i}) ∈ 𝔑_{n}(I, c_{i}), so that c_{i} ∈ N_{n}(I) = K_{n}(I), by (11.4). Thus we may assume that (X_{i}, Δ_{i}) ∈ 𝔎_{m}(I, c_{i}), for some m ≤ n. If m < n then we are done by induction. Otherwise we may assume that X_{i} is ℚ-factorial and the Picard number of X_{i} is one.

Possibly passing to a subsequence, (11.6) implies that we may assume that the number of components of B_{i} and C_{i} is fixed. As the only accumulation point of D(I) is one and the coefficients of B_{i} are bounded away from one, possibly passing to a subsequence we may assume that the coefficients of B_{i} are fixed and that the coefficients of C_{i} have the form

\frac{r − 1}{r} + \frac{f}{r} + \frac{kc_{i}}{r} ,

where k, r and f depend on the component but not on i.

Given t ∈ [0, 1], let C_{i}(t) be the divisor with the same components as C_{i} but now with coefficients

\frac{r − 1}{r} + \frac{f}{r} + \frac{kt}{r},

so that C_{i} = C_{i}(c_{i}). Let

h_{i} = sup{ t | (X_{i}, B_{i} + C_{i}(t)) is log canonical },

be the log canonical threshold. Set h = lim h_{i}.

Case A, Step 2: We reduce to the case h > c.

Suppose that h ≤ c. As c_{i} ≤ h_{i}, it follows that h = c. Now

h_{i} ∈ LCT_{n}(D(I)) = N_{n−1}(I),

so that we are done by induction in this case.

Case A, Step 3: We reduce to the case vol(X_{i}, C_{i}) is unbounded. Suppose not, suppose that vol(X_{i}, C_{i}) is bounded from above. Let

d_{i} = \frac{c_{i} + h_{i}}{2} and d = \frac{c + h}{2}.

Then the coefficients of (X_{i}, B_{i} + C_{i}(d)) are fixed. The log discrepancy of (X_{i}, B_{i} + C_{i}(d_{i})) is at least a_{i}/2 so that the log discrepancy of (X_{i}, B_{i} + C_{i}(d)) is bounded away from zero. As h > c, possibly passing to a tail of the sequence, we may assume that d > c_{i} so that K_{X_{i}} + B_{i} + C_{i}(d) is ample. Note that

vol(X_{i}, K_{X_{i}} + B_{i} + C_{i}(d)) = vol(X_{i}, C_{i}(d) − C_{i})

is bounded from above by assumption. (1.3) implies that there is a positive integer m such that ϕ_{m(K_{X_{i}}+B_{i}+C_{i}(d))} is birational. But then { (X_{i}, Δ_{i}) | i ∈ ℕ } is log birationally bounded by [15, 2.4.2.4]. (1.6) implies that (X_{i}, Δ_{i}) belongs to a bounded family. Thus we may find an ample Cartier divisor H_{i} such that the intersection numbers T_{i} · H_{i}^{n−1} and −K_{X_{i}} · H_{i}^{n−1} are bounded, where T_{i} is any component of Δ_{i}. Possibly passing to a subsequence, we may assume that these intersection numbers are constant. But then

(K_{X_{i}} + Δ_{i}) · H_{i}^{n−1} = 0, A_{i} · H_{i}^{n−1} = 0 and B_{i} · H_{i}^{n−1}

is independent of i, whilst C_{i} · H_{i}^{n−1} is not constant, a contradiction. Case A, Step 4: We finish case A. As vol(X_{i}, C_{i}) is unbounded (3.2.2) implies that we may find ϵ_{i} > 0

and divisors 0 ≤ C′_{i} ∼_{ℝ} ϵ_{i}C_{i} such that (X_{i}, Δ_{i}+C′_{i}) is not log canonical. Passing to a subsequence, and using (3.2.3), we may find g_{i} < c_{i} and a divisor

0 ≤ Θ_{i} ∼_{ℝ} C_{i} − C_{i}(g_{i}) with lim g_{i} = c,

such that (X_{i}, Φ_{i} = B_{i} + C_{i}(g_{i}) + Θ_{i}) has a unique non kawamata log terminal place. If ϕ: Y_{i} ⟶ X_{i} is a divisorially log terminal modification then ϕ extracts a unique prime divisor S_{i} of log discrepancy zero with respect to (X_{i}, Φ_{i}). We may write

K_{Y_{i}}+Ψ_{i} = ϕ^{∗}(K_{X_{i}}+Φ_{i}) and K_{Y_{i}}+B′_{i}+C′_{i}+s_{i}S_{i} = ϕ^{∗}(K_{X_{i}}+Δ_{i}),

where S_{i} = ⌊Ψ_{i}⌋, B′_{i} and C′_{i} are the strict transform of B_{i} and C_{i}, and s_{i} < 1, as (X_{i}, Δ_{i}) is kawamata log terminal.

As K_{Y_{i}}+Ψ_{i} is numerically trivial, K_{Y_{i}}+Ψ_{i}−S_{i} is not pseudo-effective. By [8, 1.3.3], we may run f: Y_{i} −−→ W_{i} the (K_{Y_{i}} + Ψ_{i} − S_{i})-MMP until we end with a Mori fibre space π_{i}: W_{i} ⟶ Z_{i}. As K_{Y_{i}} + Ψ_{i} is numerically trivial, every step of this MMP is S_{i}-positive, so that the strict transform T_{i} of S_{i} dominates Z_{i}. Let F_{i} be the general fibre of π_{i}. Replacing Y_{i}, B′_{i}, C′_{i} and Ψ_{i} by F_{i} and the restriction of f_{∗}B′_{i}, f_{∗}C′_{i} and f_{∗}Ψ_{i} to F_{i}, we may assume that S_{i}, Ψ_{i}, B′_{i} and C′_{i} are multiples of the same ample divisor. In particular K_{Y_{i}} + B′_{i} + C′_{i} + S_{i} is ample.

We let C′_{i}(t) denote the strict transform of C_{i}(t). We may write

(K_{Y_{i}} + S_{i} + B′_{i} + C′_{i}(t))|_{S_{i}} = K_{S_{i}} + B′′_{i} + C′′_{i}(t),

where the coefficients of B′′_{i} belong to D(I) and the coefficients of C′′_{i}(t) ≠ 0 belong to D_{t}(I). We let C′′_{i} = C′′_{i}(c_{i}).

There are two cases. Suppose that (S_{i}, B′′_{i} +C′′_{i}) is not log canonical. Let

k_{i} = sup{ t | (S_{i}, B′′_{i} + C′′_{i}(t)) is log canonical },

be the log canonical threshold. Then k_{i} ∈ LCT_{n−1}(D(I)) = N_{n−2}(I). Then k = lim k_{i} ∈ N_{n−2}(I) ⊂ N_{n−1}(I) by induction on n. As (S_{i}, B′′_{i} + C′′_{i}(g_{i})) is kawamata log terminal, k_{i} ∈ (g_{i}, c_{i}). Thus

c = lim c_{i} = lim k_{i} = k ∈ N_{n−1}(I).

Otherwise we may suppose that (S_{i}, B′′_{i} + C′′_{i}) is log canonical. Let

l_{i} = sup{ t | (S_{i}, B′′_{i} + C′′_{i}(t)) is pseudo-effective },

be the pseudo-effective threshold. Then l_{i} ∈ N_{n−1}(I) and l = lim l_{i} ∈ N_{n−1}(I) by induction on n. On the other hand l_{i} ∈ (g_{i}, c_{i}). Thus

c = lim c_{i} = lim l_{i} = l ∈ N_{n−1}(I).

Case B: lim a_{i} = 0.

In this case, we assume that a_{i} approaches 0.

Case B, Step 1: We reduce to the case A_{i} ≠ 0, X_{i} is ℚ-factorial and (X_{i}, Δ_{i}) is kawamata log terminal if and only if ⌊A_{i}⌋ = 0.

Possibly passing to a subsequence we may assume that a_{i} ≥ a_{i+1} and a_{i} ≤ 1. If (X_{i}, Δ_{i}) is not divisorially log terminal or A_{i} ≠ 0 but X_{i} is not ℚ-factorial then let π_{i}: X′_{i} ⟶ X_{i} be a divisorially log terminal modification. If A_{i} = 0 then let π_{i}: X′_{i} ⟶ X_{i} extract a divisor of log discrepancy a_{i}, where X′_{i} is ℚ-factorial. Either way, we may write

K_{X′_{i}} + Δ′_{i} = π_{i}^{∗}(K_{X_{i}} + Δ_{i}),

where Δ′_{i} is a sum of the strict transform of Δ_{i} and a divisor which is exceptional. Let B′_{i} and C′_{i} be the strict transforms of B_{i} and C_{i} and let A′_{i} = Δ′_{i} − B′_{i} − C′_{i} ≠ 0. Then X′_{i} is a ℚ-factorial projective variety of dimension n, (X′_{i}, Δ′_{i}) is a divisorially log terminal pair, K_{X′_{i}} + Δ′_{i} is numerically trivial, the coefficients of A′_{i} ≠ 0 are approaching one, the coefficients of B′_{i} belong to D(I) and the coefficients of C′_{i} ≠ 0 belong to D_{c_{i}}(I). Replacing (X_{i}, Δ_{i}) by (X′_{i}, Δ′_{i}), we may assume that A_{i} ≠ 0 and X_{i} is ℚ-factorial. Moreover (X_{i}, Δ_{i}) is kawamata log terminal if and only if ⌊A_{i}⌋ = 0.

Case B, Step 2: We are done if the support of C_{i} and ⌊A_{i}⌋ intersect. Suppose that a component of C_{i} intersects the normalisation of a

component S_{i} of ⌊A_{i}⌋. Then we may write

(K_{X_{i}} + Δ_{i})|_{S_{i}} = K_{S_{i}} + Θ_{i},

by adjunction. S_{i} is projective of dimension n−1, (S_{i}, Θ_{i}) is log canonical, K_{S_{i}} +Θ_{i} is numerically trivial, and we may write Θ_{i} = A′_{i}+B′_{i}+C′_{i}, where the coefficients of A′_{i} approach one, the coefficients of B′_{i} belong to D(I) and the coefficients of C′_{i} ≠ 0 belong to D_{c_{i}}(I). In this case, the limit c belongs to N_{n−2}(I) ⊂ N_{n−1}(I) by induction.

Case B, Step 3: We are done if f_{i}: X_{i} ⟶ Z_{i} is a Mori fibre space, A_{i} dominates Z_{i} and dim Z_{i} > 0.

Let F_{i} be the general fibre of f_{i}. We may write

(K_{X_{i}} + Δ_{i})|_{F_{i}} = K_{F_{i}} + Θ_{i},

by adjunction. F_{i} is projective of dimension at most n − 1, (F_{i}, Θ_{i}) is log canonical, K_{F_{i}} + Θ_{i} is numerically trivial, and we may write Θ_{i} = A′_{i} + B′_{i} + C′_{i}, where the coefficients of A′_{i} approach one, the coefficients of B′_{i} belong to D(I) and the coefficients of C′_{i} belong to D_{c_{i}}(I).

There are two cases. Suppose that C′_{i} = 0. Then (1.5) implies that the coefficients of A′_{i} are fixed, so that ⌊A′_{i}⌋ = A′_{i}. But then ⌊A_{i}⌋ ≠ 0 dominates Z_{i}. On the other hand, as C′_{i} = 0, C_{i} does not intersect F_{i}, that is, C_{i} does not dominate Z_{i}. But then C_{i} must contain a fibre so that A_{i} and C_{i} intersect and we are done by Case B, Step 2. Otherwise C′_{i} ≠ 0. In this case c_{i} ∈ N_{n−1}(I) so that

c = lim c_{i} ∈ N_{n−2}(I) ⊂ N_{n−1}(I),

by induction.

Case B, Step 4: We reduce to the case (X_{i}, Δ_{i}) is kawamata log terminal.

Suppose not, suppose that (X_{i}, Δ_{i}) is not kawamata log terminal. By Case B, Step 1, this implies that S_{i} = ⌊A_{i}⌋ is not the zero divisor. Let Θ_{i} = Δ_{i} − S_{i}. We run the (K_{X_{i}} + Θ_{i})-MMP with scaling of some ample divisor. Let f_{i}: X_{i} −−→ X′_{i} be a step of the (K_{X_{i}} + Θ_{i})-MMP. As K_{X_{i}} + Δ_{i} is numerically trivial, f_{i} is automatically S_{i}-positive. Let A′_{i} = f_{i∗}A_{i}, B′_{i} = f_{i∗}B_{i} and C′_{i} = f_{i∗}C_{i}. First suppose that f_{i} is birational. If C′_{i} = 0 then (1.5) implies that the coefficients of A′_{i} are all one. As f_{i} contracts C_{i} it does not contract a component of A_{i} and so it follows that the coefficients of A_{i} are all one, that is, S_{i} = A_{i}. As f_{i} contracts C_{i} and f_{i} is S_{i}-positive, C_{i} intersects S_{i} and we are done by Case B, Step 2. Therefore we may assume that C′_{i} ≠ 0 and we may replace (X_{i}, Δ_{i}) by (X′_{i}, Δ′_{i}). As the MMP must terminate with a Mori fibre space, replacing (X_{i}, Δ_{i}) with (X′_{i}, Δ′_{i}) finitely many times, we may assume that f_{i}: X_{i} ⟶ Z_{i} = X′_{i} is a Mori fibre space and S_{i} dominates Z_{i}. By Case B, Step 3, we may assume that Z_{i} is a point. But then the support of S_{i} and C_{i} intersect and we are done by Case B, Step 2.

Case B, Step 5: We reduce to the case X_{i} has Picard number one. We run the (K_{X_{i}} +B_{i}+C_{i})-MMP with scaling of some ample divisor.

Let f_{i}: X_{i} −−→ X′_{i} be a step of the (K_{X_{i}} + B_{i} + C_{i})-MMP. As K_{X_{i}} + Δ_{i} is numerically trivial f_{i} is automatically A_{i}-positive. Let A′_{i} = f_{i∗}A_{i}, B′_{i} = f_{i∗}B_{i} and C′_{i} = f_{i∗}C_{i}. First suppose that f_{i} is birational. Suppose C′_{i} = 0. As f_{i} contracts only one divisor and A_{i} and C_{i} are non-zero by assumption, it follows that A′_{i} ≠ 0. (1.5) implies that the coefficients of A′_{i} are all one, which contradicts the fact that (X_{i}, Δ_{i}) is kawamata log terminal. Therefore we may assume that C′_{i} ≠ 0 and we may replace (X_{i}, Δ_{i}) by (X′_{i}, Δ′_{i}). As the MMP must terminate with a Mori fibre space, replacing (X_{i}, Δ_{i}) with (X′_{i}, Δ′_{i}) finitely many times, we may assume that f_{i}: X_{i} ⟶ Z_{i} = X′_{i} is a Mori fibre space and A_{i} dominates Z_{i}.

By Case B, Step 3 we may assume that Z_{i} is a point, so that X_{i} has Picard number one.

Case B, Step 6: We finish case B and the proof.

Possibly passing to a subsequence, (11.6) implies that we may assume that the number of components of B_{i} and C_{i} is fixed. As the only accumulation point of D(I) is one and the coefficients of B_{i} are bounded away from one, possibly passing to a subsequence we may assume that the coefficients of B_{i} are fixed and that the coefficients of C_{i} have the form

\frac{r − 1}{r} + \frac{f}{r} + \frac{kc_{i}}{r} ,

where k, r and f depend on the component but not on i.

Given t ∈ [0, 1], let C_{i}(t) be the divisor with the same components as C_{i} but now with coefficients

\frac{r − 1}{r} + \frac{f}{r} + \frac{kt}{r},

so that C_{i} = C_{i}(c_{i}).

Let T_{i} be the sum of the components of A_{i}, so that T_{i} has the same components as A_{i} but now every component has coefficient one. Then A_{i} ≤ T_{i} and C_{i}(c) ≤ C_{i}. Note that (X_{i}, A_{i} + B_{i} + C_{i}(c)) is kawamata log terminal as (X_{i}, A_{i} + B_{i} + C_{i}) is kawamata log terminal. Let

s_{i} = sup{ s ∈ [0, 1] | (X_{i}, A_{i} + B_{i} + C_{i}(c) + s(T_{i} − A_{i})) is log canonical } be the log canonical threshold. Then

A_{i} + B_{i} + C_{i}(c) ≤ A_{i} + B_{i} + C_{i}(c) + s_{i}(T_{i} − A_{i}) ≤ T_{i} + B_{i} + C_{i}(c),

As the coefficients of A_{i} + B_{i} + C_{i}(c) belong to a set which satisfies the DCC and the coefficients of T_{i} − A_{i} approach zero, the coefficients of A_{i} + B_{i} + C_{i}(c) + s_{i}(T_{i} − A_{i}) belong to a set which satisfies the DCC. Therefore, possibly passing to a tail of the sequence, (1.4) implies that s_{i} = 1, so that (X_{i}, T_{i} + B_{i} + C_{i}(c)) is log canonical.

Suppose that (X_{i}, T_{i} + B_{i} + C_{i}) is not log canonical. Let

d_{i} = sup{ t ∈ [c, c_{i}) | (X_{i}, T_{i} + B_{i} + C_{i}(t)) is log canonical },

be the log canonical threshold. Then d_{i} ∈ LCT_{n}(D(I)) = N_{n−1}(I) and c = lim d_{i} and so we are done by induction on the dimension.

Thus we may assume that (X_{i}, T_{i} + B_{i} + C_{i}) is log canonical. Let

e_{i} = sup{ t ∈ ℝ | K_{X_{i}} + T_{i} + B_{i} + C_{i}(t)) is pseudo-effective },

be the pseudo-effective threshold. Suppose that e_{i} < c. Let

f_{i} = sup{ t ∈ ℝ | K_{X_{i}} + tT_{i} + B_{i} + C_{i}(c)) is pseudo-effective },

be the pseudo-effective threshold. As e_{i} < c, f_{i} < 1 and lim f_{i} = 1, so that the coefficients of f_{i}T_{i} + B_{i} + C_{i}(c) belong to a set which satisfies the DCC, which contradicts (1.5). Thus e_{i} ≥ c. On the other hand e_{i} < c_{i} as K_{X_{i}} + T_{i} + B_{i} + C_{i} is strictly bigger than K_{X_{i}} + A_{i} + B_{i} + C_{i}, which is numerically trivial. Thus lim e_{i} = c. Possibly passing to a subsequence we may assume that either e_{i} > e_{i+1} for all i or e_{i} = c. In the former case we might as well replace C_{i} = C_{i}(c_{i}) by C_{i}(e_{i}). In this case some component of C_{i} intersects a component S_{i} of T_{i} and we are done by Case B, Step 2. In the latter case we restrict to a component S_{i} of T_{i} and apply adjunction to conclude that c = e_{i} ∈ N_{n−1}(I). □

Proof of (1.11). By (11.5) it suffices to prove that the accumulation points of N_{n}(I) belong to N_{n−1}(I). Suppose that c_{1}, c_{2}, … ∈ [0, 1] is a strictly decreasing sequence of real numbers such that 𝔑(I, c_{i}) is non-empty. Pick (X_{i}, Δ_{i}) ∈ 𝔑(I, c_{i}). By assumption we may write Δ_{i} = B_{i} + C_{i} where the coefficients of B_{i} belong to D(I) and the coefficients of C_{i} ≠ 0 belong to D_{c_{i}}(I), and so (11.7) implies that the limit c belongs to N_{n−1}(I). □

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Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112, USA

E-mail address: hacon@math.utah.edu

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

E-mail address: mckernan@math.mit.edu

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive # 0112, La Jolla, CA 92093-0112, USA

E-mail address: jmckernan@math.ucsd.edu

Beijing International Center of Mathematics Research, 5 Yiheyuan Road, Haidian District, Beijing 100871, China

E-mail address: cyxu@math.pku.edu.cn