ON THE BIRATIONAL AUTOMORPHISMS OF VARIETIES OF GENERAL TYPE

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

In memory of Eckart Viehweg

Abstract. We show that the number of birational automorphisms of a variety of general type X is bounded by c · vol(X, K_{X}), where c is a constant which only depends on the dimension of X.

Contents

1. Introduction  2
1.1. Sketch of the proof of (1.4)  6
2. Preliminaries  11
2.1. Notation and Conventions  11
2.2. Log pairs  12
2.3. The volume  13
2.4. Bounded pairs  15
3. Birationally bounded pairs  17
4. Deformation invariance of log plurigenera  20
5. DCC for the volume of bounded pairs  25
6. Birational geometry of global quotients  32
7. Proof of (1.4) and (1.1)  34
References  35

Date: August 7, 2012.

The first author was partially supported by NSF research grant no: 0757897, the second author was partially supported by NSF research grant no: 0701101, and the third author was partially supported by NSF research grant no: 0969495 and by a special research fund in China. We would like to thank Xinyi Yuan for sparking our initial interest in this problem, János Kollár for many useful conversations about this paper, and Igor Dolgachev and Mihai Păun for some helpful suggestions. We would also like to thank the referee for some helpful comments.

1. Introduction

Throughout this paper, unless otherwise mentioned, the ground field k will be an algebraically closed field of characteristic zero.

Theorem 1.1. If n is a positive integer, then there is a constant c such that the birational automorphism group of any projective variety X of general type of dimension n has at most c · vol(X, K_{X}) elements.

For curves, this is a weak form of the classical Hurwitz Theorem which says that if C is a curve of genus g ≥ 2 with automorphism group G, then |G| ≤ 84(g − 1). Note that vol(C, K_{C}) = 2g − 2 and so this bound may be rephrased as |G| ≤ 42 · vol(C, K_{C}).

This problem has been extensively studied in higher dimensions, see for example, [1], [3], [8], [10], [16], [30], and [31], for surfaces, [9], [26], [28], [32], and [33], in higher dimensions, and [5], for surfaces in characteristic p.

Xiao, [31], proved that if S is a smooth projective surface of general type, with automorphism group G, then |G| ≤ (42)^{2} vol(S, K_{S}) (if S is minimal, then vol(S, K_{S}) = K_{S}^{2}; for the general definition of the volume, see [21, 2.2.31] or (2.3.1)). Xiao shows that we have equality if and only if S is a quotient of C × C, where C is a curve whose automorphism group has cardinality 42(2g −2), by the action of a very special subgroup of the automorphism group of C × C.

Question 1.2. Find an explicit bound for the constant c appearing in (1.1).

If C is a curve with automorphism group of maximal size, that is, | Aut(C)| = 84(g − 1) and

X = C × C × ⋯ × C,

then Aut(X) = n!(42)^{n}(2g −2)^{n} and vol(X, K_{X}) = n!(2g −2)^{n}, so that c ≥ 42^{n}. If we consider the example of the Fermat hypersurface

X = (X_{0}^{m} + X_{1}^{m} + ⋯ + X_{n+1}^{m} = 0) ⊂ ℙ^{n+1},

then Aut(X) ≥ (n + 2)!m^{n+1} and vol(X, K_{X}) = m(m − n − 2)^{n}. If we take m = n + 3 then the ratio

\frac{Aut(X)}{vol(X, K_{X})} ≥ (n + 2)!(n + 3)^{n},

exceeds 42^{n} for n sufficiently large (indeed, n ≥ 5 suffices), so that c is eventually greater than 42^{n}. In fact, c grows faster than n^{n}, so that c grows faster than any exponential function.

It is all too easy to give examples which show that (1.1) fails spectacularly in characteristic p. Consider the finite field 𝔽_{q^{2}} with q^{2} elements, where q = p^{k} is a power of a prime p. Note that the function

𝔽_{q^{2}} ⟶ 𝔽_{q^{2}} given by b ⟶ b̄ = b^{q},

is an involution which plays the role of complex conjugation in characteristic p. Suppose that V = 𝔽_{q^{2}}^{m} is the standard vector space of dimension m over the field 𝔽_{q^{2}}. Then there is a sesquilinear pairing

V × V ⟶ 𝔽_{q^{2}} given by (a, b) ⟶ ∑_{i} a_{i}b̄_{i}.

Let U_{m}(q) denote the group of m×m unitary matrices over the field 𝔽_{q^{2}}, so that U_{m}(q) is the group of linear maps of V preserving the pairing. Recall that U_{m}(q) is a finite simple group of Lie type, see for example [13], whose notation we follow. Note that the Fermat hypersurface

X = (X_{0}^{q+1} + X_{1}^{q+1} + ⋯ + X_{n+1}^{q+1} = 0) ⊂ ℙ^{n+1},

is the projectivisation of the null cone of the pairing, so that Aut(X) ⊃ U_{n+2}(q). We have

|U_{n+2}(q)| = \frac{1}{(n + 2, q + 1)}q^{\binom{n+2}{2}} ∏_{i=2}^{n+2}(q^{i} − (−1)^{i}),

see, for example, the table on page 8 of [13]. Note that both the order of the automorphism group and the volume of the Fermat hypersurface are polynomials f and g in q. f has degree

\binom{n + 2}{2} + \binom{n + 3}{2} − 1,

and g has degree

(n + 1).

If n = 1, the genus is a quadratic polynomial in q and the order of the automorphism group is bounded by a polynomial of degree 4 in g.

Question 1.3. Fix a positive integer n. Can we find positive integers c and d such that if X is any n-dimensional smooth projective variety of general type over an algebraically closed field of arbitrary characteristic, then

| Aut(X)| ≤ c · vol(X, K_{X})^{d}?

It is known that if n = 1 then we may take c = 216 and d = 4 (cf. [25]).

We now explain how to derive (1.1) from a result about the quotient. If Y is a variety of general type, then the automorphism group G = Aut(Y) is known to be finite, see [22]. If f: Y ⟶ X = Y/G is the quotient map, then there is a ℚ-divisor Δ on X such that K_{Y} = f^{∗}(K_{X} + Δ). We call any such log pair (X, Δ) a global quotient, cf. (2.2.1). As

vol(Y, K_{Y}) = |G| · vol(X, K_{X} + Δ),

the main issue is to bound vol(X, K_{X} + Δ) from below:

Theorem 1.4. Fix a positive integer n. Let 𝔇 be the set of log pairs (X, Δ), which are global quotients, where X is projective of dimension n.

(1) the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇 },

satisfies the DCC. Further, there are two constants δ > 0 and M such that 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.

DCC is an abbreviation for the descending chain condition. Note that, by convention, ϕ_{M(K_{X}+Δ)} = ϕ_{⌊M(K_{X}+Δ)⌋} (cf. (2.1)). Note also that the set of volumes of smooth projective varieties of fixed dimension is a discrete set (cf. [29], [15] and [27]). The situation for log pairs is considerably more subtle.

Remark 1.5. [29], [15] and [27] show that if we fix a positive integer n, then the set

{ vol(X, K_{X}) | X is a smooth projective variety of dimension n },

is discrete. However, the corresponding statement fails for kawamata log terminal surfaces, whence also for surface with reduced boundary with simple normal crossings. See [19] for an example.

However we do have:

Conjecture 1.6 (Kollár, cf. [17], [1]). Fix n ∈ ℕ and a set I ⊂ [0, 1] which satisfies the DCC.

If 𝔇 is the set of simple normal crossings pairs (X, Δ), where X is projective of dimension n, and the coefficients of Δ belong to I, then the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇 },

satisfies the DCC.

Alexeev, cf. [1] and [2], proved (1.6) for surfaces. Note that if (X, Δ) is a global quotient, then the coefficients of Δ belong to the set I =

\{ \frac{r − 1}{r} | r ∈ ℕ \}, so that (1.4) is a special case of (1.6).

We hope to give an affirmative answer to (1.6), using some of the techniques developed in this paper. Let

I = \{ \frac{r − 1}{r} | r ∈ ℕ \}.

Assuming an affirmative answer to (1.6) for this particular set, it is interesting to wonder what is the smallest possible volume. Let

(X, Δ) = (ℙ^{n}, \frac{1}{2}H_{0} + \frac{2}{3}H_{1} + \frac{6}{7}H_{2} + ⋯ + \frac{r_{n+1}}{r_{n+1} + 1}H_{n+1}),

where H_{0}, H_{1}, …, H_{n+1} are n+2 general hyperplanes and r_{1}, r_{2}, … are defined recursively by:

r_{0} = 1 and r_{n+1} = r_{n}(r_{n} + 1).

Note that (X, Δ) ∈ 𝔇. It is easy to see that the volume of K_{X} + Δ is

\frac{1}{r_{n+2}^{n}} .

Question 1.7. Find an explicit bound for

δ = min{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇 }.

The most optimistic answer to (1.7) would be

δ = \frac{1}{r_{n+2}^{n}} .

Note that c ≥ \frac{1}{δ}. When n = 1, we have

δ = \frac{1}{r_{3}} = \frac{1}{42},

and the reciprocal is precisely the constant c = 42. On the other hand, one can check that r_{n} grows roughly like

a^{2^{n}},

for some constant a > 1, so that in general there is a huge difference between r_{n} and c_{n}. In fact it is easy to check that

r_{n+1} = ∏_{i=0}^{n}(r_{i} + 1),

see §8 of [18] for more details.

Theorem 1.8 (Deformation invariance of log plurigenera). Let π: X ⟶ T be a projective morphism of smooth varieties. Suppose that (X, Δ) is log canonical and has simple normal crossings over T.

(1) If K_{X} + Δ is kawamata log terminal, and either K_{X} + Δ or Δ is big over T, and m is any positive integer such that mΔ is integral, then h^{0}(X_{t}, O_{X_{t}}(m(K_{X_{t}} + Δ_{t}))) is independent of t ∈ T.

(2) κ_{σ}(X_{t}, K_{X_{t}} + Δ_{t}) is independent of t ∈ T.

(3) vol(X_{t}, K_{X_{t}} + Δ_{t}) is independent of t ∈ T.

For the definitions of κ_{σ} and simple normal crossings over T, see (2.1). We will prove a similar but stronger statement (4.2) which implies (1.8). Obviously, (1.8) is a generalisation of Siu’s theorem on invariance of plurigenera, cf. [24]. We recently learnt that (1) of (1.8) holds even without the assumption that K_{X} + Δ is big, see Theorem 0.2 of [6]. We use (1.8) to prove:

Theorem 1.9. Fix a set I ⊂ [0, 1] which satisfies the DCC. Let 𝔇 be a set of simple normal crossings pairs (X, Δ), which is log birationally bounded (cf. (2.4.1)), such that if (X, Δ) ∈ 𝔇, then the coefficients of Δ belong to I.

Then the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇 },

satisfies the DCC.

1.1. Sketch of the proof of (1.4). The proof of (1.4) is by induction on the dimension n and the proof is divided into two steps. The first step uses some ideas of Tsuji which are used to prove that some fixed multiple of K_{X} defines a birational map, for a variety X of general type, see [29], [15] and [27]. In this step we establish that modified versions of (2) and (3) of (1.4) are equivalent, given that (1.4)_{n−1} holds (that is, (1.4) holds in dimension n − 1). Namely, consider

(2) vol(X, r(K_{X} + Δ)) > δ, and

(3) ϕ_{Mr(K_{X}+Δ)} is birational. We show that if (X, Δ) is a global quotient of dimension n, then there are constants δ > 0 and M such that for every positive integer r, (2) implies (3), see (6.1).

It is clear that if some fixed multiple of r(K_{X} +Δ) defines a birational map, then the volume of r(K_{X} + Δ) is bounded from below, (2.3.2), so that there are constants δ and M such that (3) implies (2). To go the other way, we need to construct a divisor 0 ≤ D ∼_{ℚ} mr(K_{X} + Δ), where m is fixed, which has an isolated non kawamata log terminal centre at a very general point and is not kawamata log terminal at another very general point, (2.3.4). As we know that log canonical models exist by [7], we may assume that K_{X} + Δ is ample, so that lifting divisors from any subvariety is simply a matter of applying Serre vanishing. In this case, it is well known, since the work of Anghern and Siu, [4], that to construct D we need to bound the volume of K_{X} + Δ from below on special subvarieties V of X (specifically, any V which is a non kawamata log terminal centre of (X, Δ + Δ_{0}), where Δ_{0} is proportional to K_{X} + Δ), see (2.3.5) and (2.3.6).

If V = X, there is nothing to prove, as we are assuming that vol(X, r(K_{X} + Δ)) > δ. Otherwise, the dimension of V is less than the dimension of X, and we may proceed by induction on the dimension; as V passes through a very general point of X, V is birational to a global quotient. In fact, we even know that vol(V, (K_{X} + Δ)|_{V}) is bounded from below.

So from now on we assume that (2) implies (3). We may suppose that the constant δ appearing in (2) is at most one. The next step is to prove that (3) holds when r = 1 (that is, (3) of (1.4) holds). There are two cases. If the volume of K_{X} + Δ is at least one, then the volume of K_{X} +Δ is certainly bounded from below, and there is nothing to prove. Otherwise, we may find r > 0 such that δ < 1 ≤ vol(X, r(K_{X} + Δ)) < 2^{n}. But then ϕ_{m(K_{X}+Δ)} is birational, where m = Mr and, at the same time, the volume of m(K_{X} + Δ) is bounded from above. In this case, the degree of the image of ϕ_{m(K_{X}+Δ)} is bounded from above, and so we know that the image belongs to a bounded family. In fact, one can prove that both the degree of the image and the degree of the image of Δ and the exceptional locus have bounded degree, (3.1), so we only need to concern ourselves with a log birationally bounded family of log pairs, (2.4.1). This finishes the first step.

To finish the argument, we need to argue that the volume is bounded from below if we have a log birationally bounded family of log pairs. This is the most delicate part of the argument and is the second step. We use some ideas which go back to Alexeev. Firstly, it is not much harder to prove that the volume of global quotients satisfies the DCC. The first part of the second step is to argue that we only need to worry about log pairs (X, Δ) which are birational to a single pair (Z, B), rather than a bounded family of log pairs. For this we prove a version of deformation invariance of log plurigenera for log pairs, see (1.8). Deformation invariance fails in general (cf. [11, 4.10]); we need to assume that the family has simple normal crossings over the base (which roughly means that every component of Δ is smooth over the base). To reduce to this case involves some straightforward manipulation of a family of log pairs, see the proof of (1.9) in §5. We prove, cf. (4.2), a version of (1.8) which is better suited to induction; to this end, we first show that if we have a family of log pairs over a curve then we can run the MMP in a family, (4.1).

So we are reduced to the most subtle part of the argument. We are given a simple normal crossings pair (Z, B), a set I which satisfies the DCC, and we want to argue that if (X, Δ) is a simple normal crossings pair such that there is a birational morphism f: X ⟶ Z with f_{∗}Δ ≤ B, then the volume of K_{X} + Δ belongs to a set which satisfies the DCC, see (5.1). To fix ideas, let us suppose that Z is a smooth surface (this is the case originally treated by Alexeev). We are given a sequence of simple normal crossings surfaces (X_{i}, Δ_{i}) and birational morphisms f_{i}: X_{i} ⟶ Z.

We have that Φ_{i} = f_{i∗}Δ_{i} ≤ B and the coefficients of Φ_{i} belong to I. Note that we are free to pass to an arbitrary subsequence, so that we may assume that Φ_{i} ≤ Φ_{i+1}. In particular, the volume of K_{Z} + Φ_{i} is not decreasing. The problem is that if we write

K_{X_{i}} + Δ_{i} = f_{i}^{∗}(K_{Z} + Φ_{i}) + E_{i},

then E_{i} might have negative coefficients, so that the volume of K_{X_{i}} +Δ_{i} is less than the volume of K_{Z} + Φ_{i}. Suppose that we write E_{i} = E_{i}^{+} − E_{i}^{−}, where E_{i}^{+} ≥ 0 and E_{i}^{−} ≥ 0 have no common components. Note that E_{i}^{+} has no effect on the volume, as E_{i} is supported on the exceptional locus. What bothers us is the possibility that the E_{i}^{−} involve exceptional divisors that live on higher and higher models.

Clearly, we should consider the limit Φ = lim_{i} Φ_{i}. However, this is not enough, we need to take the limit of the divisors Δ_{i} on various models, and to work with linear systems on these models. It was for just this purpose that the language of b-divisors was introduced by Shokurov. Recall that a b-divisor D is just the choice of a divisor D_{X} on every model X, which is compatible under pushforward.

For us there are three relevant b-divisors. Since we want to work on higher models without changing the volume, or the fact that the coefficients lie in the set I, we introduce, (5.5), the b-divisor M_{Δ_{i}} associated to a log pair (X_{i}, Δ_{i}): given a model Y ⟶ X_{i} we just throw in any exceptional divisor with coefficient one. We next take the limit B of the sequence of b-divisors M_{Δ_{i}}; on Z we just recover the divisor Φ. Finally, we define (5.2), a b-divisor L_{Δ_{i}} that assigns to a model π: Y ⟶ X_{i} the positive part of the log pullback. Actually, this is the most complicated of the three b-divisors, and it is the subtle behaviour of the b-divisors L_{Δ_{i}} which complicates the proof.

If we set Δ′_{i} = Δ_{i} ∧ L_{Φ_{i},X_{i}}, then,

vol(X_{i}, K_{X_{i}} + Δ′_{i}) = vol(X_{i}, K_{X_{i}} + Δ_{i}),

see (2) of (5.3). If we knew that the coefficients of Δ′_{i} belong to a set I ⊂ J that satisfies the DCC, then we would be done. In fact, if L_{Φ} ≤ B, then it is relatively straightforward to conclude that the volume satisfies the DCC, see the proof of (5.1). Unfortunately, since X_{i} ⟶ Z might extract arbitrarily many divisors, it is all too easy to write down examples where the smallest set J which contains the coefficients of every Φ_{i} does not satisfy the DCC (cf. (1.10)).

Therefore, our objective is to find a model Z′ ⟶ Z and suitable modifications Δ′_{i} of Δ_{i} such that L_{Φ′} ≤ B′, where B′ is the limit of M_{Δ′_{i}}. We choose Δ′_{i} so that the difference Δ_{i} − Δ′_{i} is supported only on the strict transform of the exceptional divisors of Z′ ⟶ Z. In this case, it is not hard to arrange for the coefficients of Δ′_{i} to belong to a set I ⊂ J that satisfies the DCC.

We construct Z′ ⟶ Z by induction. For this, we can work locally about a point p in Z. Suppose that p is the intersection of two components B_{1} and B_{2} of Φ. If B_{1} and B_{2} appear with coefficient b_{1} and b_{2} in Φ and π: Z′ ⟶ Z blows up p, then working locally about p, we may write

K_{Z′} + b_{1}B′_{1} + b_{2}B′_{2} + eE = π^{∗}(K_{Z} + b_{1}B_{1} + b_{2}B_{2}),

where e = b_{1}+b_{2}−1. Here primes denote strict transforms and E is the unique exceptional divisor. We suppose that L_{Φ} ≤ B does not hold, so that there is some valuation ν, with centre p, such that L_{Φ}(ν) > B(ν). Since e = b_{1} + b_{2} − 1, the larger b_{1} and b_{2}, the further we expect to be from the inequality L_{Φ} ≤ B. We introduce the weight w, which counts the number of components of Φ of coefficient one, that is, the number of i such that b_{i} = 1. In the case of a surface, the weight is 0, 1 or 2 and it clearly suffices to construct Z′ ⟶ Z so that the weight goes down.

In fact, the extreme cases are relatively easy. If the weight is two, then we just take Z′ ⟶ Z to be the blow up of p. The key point is that then the base locus of the linear system

f′_{i∗}|m(K_{X_{i}} + Δ_{i})| ⊂ |m(K_{Z} + Φ_{i})|,

contains p in its support for all m sufficiently large and divisible and this forces the strict transform of E to be a component of E_{i}^{+} as well. Therefore we are free to decrease the coefficient of E in Φ′ away from 1. At the other extreme, if the weight is zero, then (Z, Φ) is kawamata log terminal and we may find Z′ ⟶ Z which extracts every divisor of coefficient greater than zero. In this case L_{Φ′}(ρ) = 0 for every valuation ρ whose centre on Z′ is not a divisor and the inequality L_{Φ′} ≤ B′ is trivial.

The hard case is when the weight is one, so that one of b_{1} and b_{2} is one. Suppose that b_{2} = 1. If ν is a valuation such that L_{Φ}(ν) > 0 then ν corresponds to a weighted blow up. At this point it is convenient to use the language of toric geometry. A weighted blow up corresponds to a pair of natural numbers (v_{1}, v_{2}) ∈ ℕ^{2}. In fact, if

𝔉 = { v_{1} ∈ ℕ | (1 − b_{1})v_{1} < 1 },

then

{ (v_{1}, v_{2}) ∈ ℕ^{2} | v_{1} ∈ 𝔉 },

is the set of all valuations ν such that L_{Φ}(ν) > 0. The crucial point is that 𝔉 is finite. For every element f_{1} ∈ 𝔉, we pick v_{2} ∈ ℕ, which minimises B(f_{1}, v_{2}) (this makes sense as I satisfies the DCC). We then pick any simple normal crossings model Z′ ⟶ Z on which the centre of every one of these finitely many valuations is a divisor. Using standard toric geometry, one can check that on Z′ every valuation ν′, whose centre belongs to a component of Φ′ of coefficient one, satisfies the property L_{Φ′}(ν′) ≤ B′(ν′). It follows that the weight of (Z′, Φ′) is zero and this completes the induction. The details are contained in the proof of (5.1).

Example 1.10. Consider the behaviour of L_{Φ} in a simple example. We use the notation above. Note that e = b_{1} + b_{2} − 1 is an increasing and affine linear function of b_{1} and b_{2}, whence a continuous function of b_{1} and b_{2}. We are only concerned with the possibility that e > 0, and in this case, L_{Φ,Z′} = b_{1}B′_{1} + b_{2}B′_{2} + eE, by definition. There are two interesting points p_{1} = B′_{1} ∩ E and p_{2} = B′_{2} ∩ E lying over p, and the coefficients of the divisors containing them are b_{1}, e and b_{2}, e. The problem is that we can blow up along either point and keep going.

Suppose that

I = \{ \frac{r − 1}{r} | r ∈ ℕ \}.

Note that if b_{1} = \frac{i}{r} and b_{2} = \frac{r−1}{r} then

e = b_{1} + b_{2} − 1 = \frac{i}{r} + \frac{r − 1}{r} − 1 = \frac{i − 1}{r}.

So, the smallest set J which contains I and which is closed under the operation of picking any two elements b_{1} and b_{2} and replacing them by b_{1} + b_{2} − 1 (provided this sum is non-negative) is ℚ ∩ [0, 1]. Clearly, this set does not satisfy the DCC.

2. Preliminaries

2.1. Notation and Conventions. We will use the notations in [20] and [21].

If D = ∑ d_{i}D_{i} is a ℚ-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 D′ = ∑ d′_{i}D_{i} is another ℚ-divisor, then D ∧ D′ := ∑ min{d_{i}, d′_{i}}D_{i}.

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 a ℚ-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.

If X is a normal projective variety and D is a ℚ-Cartier divisor, κ_{σ}(X, D) denotes the numerical Kodaira dimension, which is defined by Nakayama in [23, V.2.5] as follows: let H be a divisor on X, we define

σ(X, D; H) := max\{ k ∈ ℕ | lim sup_{m→∞} \frac{h^{0}(X, H + mD)}{m^{k}} > 0 \},

and

κ_{σ}(X, D) := max{ σ(X, D; H) | H is a divisor }.

If D is pseudo-effective, we define N_{σ}(X, D) as in [23, III.4] or [7, 3.3.1].

A log pair (X, Δ) consists of a normal variety X and a ℚ-Weil divisor Δ ≥ 0 such that K_{X} + Δ is ℚ-Cartier. The support of Δ = ∑_{i∈I} d_{i}Δ_{i} is the sum D = ∑_{i∈I} Δ_{i}. If (X, Δ) has simple normal crossings then a stratum of (X, Δ) is an irreducible component of the intersection ∩_{j∈J}Δ_{j}, where J is a non-empty subset of I (in particular, a stratum 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, Δ) 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}, D_{i}), where D_{i} is the sum of the strict transform of D 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. Note that the extra assumption that the exceptional locus is the support of a μ-ample divisor is not standard. However it is convenient for our purpose, and it can be always achieved after possibly choosing a higher model. If we write

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

where Γ is the strict transform of Δ, then a_{i} is called the coefficient of E_{i} with respect to (X, Δ). Note that −a_{i} is the discrepancy of the pair (X, Δ), with respect to E_{i}, see [20, 2.25]. A non kawamata log terminal centre is the centre of any valuation ν whose coefficient is at least one.

In this paper, we only consider valuations ν of X whose centre on some birational model Y of X is a divisor. We say that a formal sum B = ∑ a_{ν}ν, where the sum ranges over all valuations of X, is a bdivisor, if the set

F_{X} = { ν | a_{ν} ≠ 0 and the centre of ν on X is a divisor },

is finite. The trace B_{Y} of B is the sum ∑ a_{ν}B_{ν}, where the sum now ranges over the elements of F_{Y}. In fact, to give a b-divisor is the same as to give a collection of divisors on every birational model of X, which are compatible under pushforward.

2.2. Log pairs.

Definition 2.2.1. We say that a log pair (X, Δ) is a global quotient if there is a smooth quasi-projective variety Y and a finite subgroup G ⊂ Aut(Y) such that X = Y/G and if π: Y ⟶ X is the quotient morphism, then K_{Y} = π^{∗}(K_{X} + Δ).

Note that if (X, Δ) is a global quotient, then X is ℚ-factorial, (X, Δ) is kawamata log terminal, and the coefficients of Δ belong to the set

\{ \frac{r − 1}{r} | r ∈ ℕ \},

(cf. [20, 5.15, 5.20]).

Lemma 2.2.2. If (X, Δ) is a log pair and the coefficients of Δ are less than one, then

⌊mΔ⌋ ≤ ⌈(m − 1)Δ⌉,

for every positive integer m, with equality if the coefficients of Δ belong

to the set \{ \frac{r − 1}{r} | r ∈ ℕ \}.

Proof. Easy. □

2.3. The volume.

Definition 2.3.1. Let X be a normal n-dimensional irreducible projective variety and let D be a ℚ-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 [21]. We will need the following simple result:

Lemma 2.3.2. Let X be a projective variety, D a divisor such that the rational map ϕ_{D}: X −−→ ℙ^{n} is birational onto its image Z. Then, the volume of D is greater than or equal to the degree of Z. In particular the volume of D is at least 1.

Proof. This is well known, see for example (2.2) of [15]. □

Definition 2.3.3. 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 very 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.

Lemma 2.3.4. Let X be a normal projective variety and let D be a big ℚ-Cartier divisor on X.

(1) If D is potentially birational, then ϕ_{K_{X}+⌈D⌉} is birational.

(2) If X has dimension n and ϕ_{D} is birational, then (2n + 1)⌊D⌋ is potentially birational. In particular, ϕ_{K_{X}+(2n+1)D} is birational and K_{X} + (2n + 1)D is big.

Proof. Replacing X by a resolution, we may assume that X is smooth. As D is big, we may write D ∼_{ℚ} A + B where A is an ample ℚ-divisor and B ≥ 0. Using A to tie break (cf. [18, 6.9]), we may assume that (X, Δ) is kawamata log terminal in a punctured neighbourhood of x. As

⌈D⌉ − (Δ + ϵB + ⌈D⌉ − D) ∼_{ℚ} ϵA,

is ample, Nadel vanishing implies that

H^{1}(X, O_{X}(K_{X} + ⌈D⌉) ⊗ J (Δ + ϵB + ⌈D⌉ − D)) = 0,

where J (Δ + ϵB + ⌈D⌉ − D) is the multiplier ideal sheaf. But then we may find a section σ ∈ H^{0}(X, O_{X}(K_{X} + ⌈D⌉)) vanishing at y but not at x. In particular, as y is very general, we may also find τ not vanishing at y. But then some linear combination ρ of τ and σ is a section which vanishes at x and not at y. This is (1).

Replacing D by ⌊D⌋, we may assume that D is Cartier. Let X′ be the image of ϕ_{D}. Let x′ = ϕ_{D}(x) and y′ = ϕ_{D}(y) and let Δ′ be the sum of n general hyperplanes through x′ and n general hyperplanes through y′. Let Δ be the strict transform of Δ′. As x and y are general, ϕ_{D} is an isomorphism in a neighbourhood of x and y. It follows that (X, Δ) is not kawamata log terminal at y, (X, Δ) is log canonical at x and if we blow up x then the coefficient of the exceptional divisor is one. It is then easy to see that (2n+1)D is potentially birational and (2) follows from (1). □

We will need the following result from [18]:

Theorem 2.3.5. Let (X, Δ) be a kawamata log terminal pair, where X is projective. Suppose that x and y are two closed points of X. Let Δ_{0} ≥ 0 be a ℚ-Cartier divisor on X such that (X, Δ + Δ_{0}) is log canonical in a neighbourhood of x but not kawamata log terminal at y, and there is a non kawamata log terminal centre V which contains x. Let H be an ample ℚ-divisor on X such that vol(V, H|_{V}) > 2k^{k}, where k = dim V.

Then, possibly switching x and y, there is a ℚ-divisor H ∼_{ℚ} Δ_{1} ≥ 0 and rational numbers 0 < a_{i} ≤ 1 such that (X, Δ + a_{0}Δ_{0} + a_{1}Δ_{1}) is log canonical in a neighbourhood of x but not kawamata log terminal at y, and there is a non kawamata log terminal centre V′ which contains x, such that dim V′ < k.

Proof. By (6.9.1) of [18], we may assume that V is the unique non kawamata log terminal centre which contains x, and we may apply (6.8.1), (6.8.1.3) and (6.5) of [18]. □

Theorem 2.3.6. 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 γ_{0} ≥ 1 is a constant such that vol(X, γ_{0}H) > n^{n}. Suppose ϵ is a constant with the following property:

For very general x in X and every 0 ≤ Δ_{0} ∼_{ℚ} λH such that (X, Δ + Δ_{0}) is log canonical at x, if V is the minimal non kawamata log terminal centre containing x, then vol(V, λH|_{V}) > ϵ^{k}, where k is the dimension of V and λ ≥ 1 is a rational number.

Then mH is potentially birational, where

m = 2γ_{0} (1 + γ)^{n−1}  and  γ = \frac{2n}{ϵ}.

Proof. Let x and y be two very 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 1 ≤ λ < 2γ_{0}(1 + γ)^{n−1−k}, 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 V of dimension at most k containing x.

As

vol(X, 2γ_{0}H) > 2n^{n},

we may find 0 ≤ Φ ∼_{ℚ} 2γ_{0}H, such that (X, Δ + Φ) is not log canonical at either x or y. If

β = sup{ α | K_{X} + Δ + αΦ is log canonical at x },

is the log canonical threshold, then β < 1. Possibly switching x and y, we may assume that (X, Δ + βΦ) is not kawamata log terminal at y. Clearly Δ_{0} = βΦ satisfies (♭)_{n−1}, 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 V is the minimal non kawamata log terminal centre containing x and that V has dimension k. By assumption,

vol(V, λγH|_{V}) > 2k^{k},

so that by (2.3.5), possibly switching x and y, we may find Δ_{1} ∼_{ℚ} μH, where μ < λγ 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 V′ 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 ≤ (1 + γ)λ < 2γ_{0}(1 + γ)^{n−1−(k−1)},

a_{0}Δ_{0} + a_{1}Δ_{1} + max(0, 1 − λ′)B satisfies (♭)_{k−1}, where the support of B ∼_{ℚ} H does not contain either x or y (we only need to add on B in the unlikely event that λ′ < 1). This completes the induction and the proof. □

2.4. Bounded pairs.

Definition 2.4.1. 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 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 such that the support of B_{t} contains the support of the strict transform of Δ and any f-exceptional divisor.

Lemma 2.4.2. Fix a positive integer n.

(1) Let 𝔛 and 𝔜 be two sets of varieties, such that if X ∈ 𝔛, then we may find Y ∈ 𝔜 birational to X. If 𝔜 is birationally bounded, then 𝔛 is birationally bounded.

(2) Let 𝔛 be a set of varieties of dimension n. If there is a constant V such that for every X ∈ 𝔛, we may find a Weil divisor D such that ϕ_{D} is birational and the volume of D is at most V, then 𝔛 is birationally bounded.

(3) Let 𝔇 and 𝔊 be two sets of log pairs, such that if (X, Δ) ∈ 𝔇, then we may find (Y, Γ) ∈ 𝔊, and a birational map f: Y −−→ X, where the support of Γ contains the support of the strict transform of Δ and any f-exceptional divisor. If 𝔊 is log birationally bounded, then 𝔇 is log birationally bounded.

(4) Let 𝔇 be a set of log pairs of dimension n. If there are constants V_{1} and V_{2} such that for every (X, Δ) ∈ 𝔇 we may find a Weil divisor D such that ϕ_{D}: X −−→ Y is birational, the volume of D is at most V_{1}, and if G denotes the sum over the components of the strict transform of Δ and the ϕ^{−1}-exceptional divisors, then G · H^{n−1} ≤ V_{2}, where H is the very ample divisor on Y determined by ϕ_{D}, then 𝔇 is log birationally bounded.

(5) If the set 𝔇 of log pairs is log birationally bounded, then

𝔛 = { X | (X, Δ) ∈ 𝔇 }

is birationally bounded.

Proof. (1), (3) and (5) are clear.

We prove (4). Suppose that Y ⊂ ℙ^{s} is a closed subvariety of dimension n and degree at most V_{1}. Then by the classification of minimal degree subvarieties of projective space, we may assume s ≤ V_{1} + 1 − n. By boundedness of the Chow variety, there are flat morphisms Z ⟶ T and B ⟶ T such that if Y ⊂ ℙ^{s} has dimension n (respectively n − 1) and degree at most V_{1} (respectively V_{2}), then Y is isomorphic to the fibre Z_{t} (respectively B_{t}) over a closed point t ∈ T. Passing to a stratification of T and a log resolution of the generic fibres of Z ⟶ T, we may assume that the fibres of Z ⟶ T are smooth. In particular, (Z, B) is a log pair.

Now suppose that (X, Δ) ∈ 𝔇. By assumption there is a divisor D such that ϕ_{D}: X −−→ Y is birational. The degree of the image is at most the volume of D, that is, at most V_{1}. So there is a closed point t ∈ T such that Y is isomorphic to Z_{t}. By assumption G · H^{n−1} ≤ V_{2} so that we may assume that G corresponds to B_{t}. But then 𝔇 is log birationally bounded. This is (4).

The proof of (2) is similar to and easier than the proof of (4). □

3. Birationally bounded pairs

§3 is devoted to a proof of:

Theorem 3.1. Fix a positive integer n and two constants A and δ > 0. Then the set of log pairs (X, Δ) satisfying

(1) X is projective of dimension n,

(2) (X, Δ) is log canonical,

(3) the coefficients of Δ are at least δ,

(4) there is a positive integer m such that vol(X, m(K_{X} + Δ)) ≤ A, and

(5) ϕ_{K_{X}+m(K_{X}+Δ)} is birational, is log birationally bounded.

The key result is:

Lemma 3.2. Let X be a normal projective variety of dimension n and let M be a base point free Cartier divisor such that ϕ_{M} is birational. Let H = 2(2n + 1)M.

If D is a sum of distinct prime divisors, then

D · H^{n−1} ≤ 2^{n} vol(X, K_{X} + D + H).

Proof. Possibly discarding ϕ_{M}-exceptional components of D, we may assume that no component of D is ϕ_{M}-exceptional. If f: Y ⟶ X is a log resolution of the pair (X, D) and G is the strict transform of D, then

D · H^{n−1} = G · (f^{∗}H)^{n−1},

and

vol(Y, K_{Y} + G + f^{∗}H) ≤ vol(X, K_{X} + D + H).

Replacing (X, D) by (Y, G) and M by f^{∗}M we may assume that (X, D) has simple normal crossings, and possibly blowing up more, that the components of D do not intersect.

Since no component of D is contracted, we may find an ample ℚdivisor A and a ℚ-divisor B ≥ 0, such that

M ∼_{ℚ} A + B,

where B and D have no common components. As K_{X} + D + δB is divisorially log terminal for any δ > 0 sufficiently small, it follows that

H^{i}(X, O_{X}(K_{X} + E + pM)) = 0,

for all positive integers p, i > 0 and any integral Weil divisor 0 ≤ E ≤ D. If we let

A_{m} = K_{X} + D + mH,

then

H^{i}(D, O_{D}(A_{m})) = 0,

for all i > 0 and m > 0 and so there is a polynomial P (m) of degree n − 1, with

P (m) = h^{0}(D, O_{D}(A_{m})),

for m > 0. (2) of (2.3.4) implies that A_{1} = K_{X} + D + H is big and so [7] implies that K_{X} + D + H has a log canonical model. In particular there is a polynomial of Q(m) of degree n, with

Q(m) = h^{0}(X, O_{X}(2mA_{1})),

for any sufficiently divisible positive integer m. Note that the leading coefficients of P (m) and Q(m) are

\frac{D · H^{n−1}}{(n − 1)!}  and  \frac{2^{n} vol(X, K_{X} + D + H)}{n!}.

If D_{i} is a component of D, and M_{i} = (D − D_{i} + (2n + 1)M)|_{D_{i}}, then

H^{0}(X, O_{X}(K_{X} + D + (2n + 1)M)) ⟶ H^{0}(D_{i}, O_{D_{i}}(K_{D_{i}} + M_{i})),

is surjective, and so the general section of H^{0}(X, O_{X}(K_{X} + D + (2n + 1)M)) does not vanish identically on any component of D. Pick sections

s ∈ H^{0}(X, O_{X}(K_{X}+D+(2n+1)M)) and l ∈ H^{0}(X, O_{X}((2n+1)M)),

whose restrictions to each component of D is non-zero. Let

t = s^{⊗2m−1} ⊗ l ∈ H^{0}(X, O_{X}(2mA_{1} − A_{m})).

Consider the following commutative diagram

[figure: 0; O_{X}(A_{m} − D); O_{X}(A_{m}); O_{D}(A_{m}); 0; 0; O_{X}(2mA_{1} − D); O_{X}(2mA_{1}); O_{D}(2mA_{1}); 0,]

where the vertical morphisms are injections induced by multiplying by t.

Note that

H^{0}(X, O_{X}(A_{m})) ⟶ H^{0}(D, O_{D}(A_{m})),

is surjective. Hence every element of H^{0}(D, O_{D}(2mA_{1})) in the image of H^{0}(D, O_{D}(A_{m})) lifts to H^{0}(X, O_{X}(2mA_{1})). Therefore

P (m) ≤ h^{0}(X, O_{X}(2mA_{1})) − h^{0}(X, O_{X}(2mA_{1} − D)).

Note that

Q(m − 1) = h^{0}(X, O_{X}(2(m − 1)A_{1})) ≤ h^{0}(X, O_{X}(2mA_{1} − D)),

as h^{0}(X, O_{X}(2K_{X} + D + 2H)) ≠ 0. It follows that

P (m) ≤ Q(m) − Q(m − 1).

Now compare the leading coefficients of P (m) and Q(m)−Q(m−1). □

Proof of (3.1). Let (X, Δ) be a log pair satisfying the hypotheses of (3.1). If π: Y ⟶ X is a log resolution of (X, Δ) which resolves the indeterminacy of

ϕ = ϕ_{K_{X}+m(K_{X}+Δ)} : X −−→ Z,

and Γ is the strict transform of Δ plus the sum of the exceptional divisors, then (X, Δ) is log birationally bounded if and only if (Y, Γ) is log birationally bounded, by (3) of (2.4.2). On the other hand,

vol(Y, m(K_{Y} + Γ)) ≤ vol(X, m(K_{X} + Δ)) ≤ A,

and ϕ_{K_{Y} +m(K_{Y} +Γ)} is birational.

Replacing (X, Δ) by (Y, Γ), we may assume that

ϕ = ϕ_{K_{X}+m(K_{X}+Δ)} : X ⟶ Z,

is a birational morphism. In particular, if we decompose ⌊K_{X} + m(K_{X} + Δ)⌋ into its mobile part M and its fixed part E, so that

|K_{X} + m(K_{X} + Δ)| = |M| + E,

then M is big and base point free. Let H be a divisor on Z such that M = ϕ^{∗}H, so that H is very ample.

Note that

vol(X, K_{X} + m(K_{X} + Δ)) ≤ vol(X, (m + 1)(K_{X} + Δ))
≤ 2^{n}A.

On the other hand, let G be the sum of the components of the strict transform of Δ on Z. Pick B ∈ |⌊K_{X} + m(K_{X} + Δ)⌋|. Let

α = max(\frac{1}{δ}, 2(2n + 1)).

If D_{0} is the sum of the components of Δ and B which are not contracted by ϕ, then

D_{0} ≤ α(B + Δ).

Note that there is a divisor C ≥ 0 such that

α(B + Δ) + C ∼_{ℚ} α(m + 1)(K_{X} + Δ).

As ϕ is a morphism and M is base point free, (3.2) implies that

G · H^{n−1} ≤ D_{0} · (2(2n + 1)M)^{n−1}
≤ 2^{n} vol(X, K_{X} + D_{0} + 2(2n + 1)M)
≤ 2^{n} vol(X, K_{X} + α(B + Δ) + 2(2n + 1)(M + E + Δ))

≤ 2^{n} vol(X, K_{X} + Δ + α(m + 1)(K_{X} + Δ) + 2(2n + 1)(m + 1)(K_{X} + Δ))

≤ 2^{n}(1 + 2α(m + 1))^{n} vol(X, K_{X} + Δ)
≤ 2^{3n}α^{n} vol(X, (m + 1)(K_{X} + Δ))
≤ 2^{4n}α^{n}A.

Now apply (4) of (2.4.2). □

4. Deformation invariance of log plurigenera

Proposition 4.1. Let (X, Δ) be a ℚ-factorial log pair. Suppose that X ⟶ T is a projective morphism to a smooth curve T whose fibres (X_{t}, Δ_{t}) are terminal, where every component of Δ dominates T. Let 0 ∈ T be a closed point. Suppose that

• either Δ or K_{X} + Δ is big over T, and

• no component of Δ_{0} belongs to the stable base locus of K_{X_{0}} +Δ_{0}. Then we may find a log terminal model f: X −−→ Y of (X, Δ) over T,

such that f is an isomorphism at the generic point of every component of Δ_{0} and f_{0} : X_{0} −−→ Y_{0} is a weak log canonical model of (X_{0}, Δ_{0}).

Proof. We first prove this result under the additional hypothesis that K_{X} + Δ is pseudo-effective over T.

Note that X is smooth in codimension two, as the fibres X_{t} of X ⟶ T are Cartier and smooth in codimension two. Hence

(K_{X} + Δ)|_{X_{0}} = (K_{X} + X_{0} + Δ)|_{X_{0}} = K_{X_{0}} + Δ_{0}.

It follows from [7, 1.4.5] that the coefficient of any valuation μ with respect to (X, X_{0} + Δ) is at most zero, if the centre of μ is neither a component of Δ nor a component of Δ_{0}.

By [7, 1.2], we may run the (K_{X} + Δ)-MMP over T, that is, we may find a sequence g^{1}, g^{2}, …, g^{m−1} of divisorial contractions and flips g^{k}: X^{k} −−→ X^{k+1} starting at X = X^{1} and ending with a log terminal model Y = X^{m} for the pair (X, Δ) over T. Let Δ^{k} denote the pushforward of Δ under the induced birational map f^{k}: X −−→ X^{k}. We will prove by induction on k that (a) g^{k} is an isomorphism at the generic point of any component of Δ_{0}^{k},

and

(b) g_{0}^{k}: X_{0}^{k} −−→ X_{0}^{k+1} is a birational contraction. Suppose that (a–b)_{≤k−1} hold. Then f_{0}^{k}: X_{0} −−→ X_{0}^{k} is a birational contraction which does not contract any components of Δ_{0} and so (X_{0}^{k}, Δ_{0}^{k}) is terminal.

(b)_{≤k−1} implies that no component of Δ_{0}^{k} is a component of the stable base locus of K_{X_{0}^{k}} + Δ_{0}^{k}. Suppose that g^{k} is not an isomorphism at the generic point of a divisor D contained in X_{0}^{k}. Then D is covered by curves C such that

(K_{X_{0}^{k}} + Δ_{0}^{k}) · C = (K_{X^{k}} + Δ^{k}) · C < 0.

It follows that D is a component of the stable base locus of K_{X_{0}^{k}} + Δ_{0}^{k}, so that D is not a component of Δ_{0}^{k}. Thus (a)_{k} holds.

Suppose that G ⊂ X_{0}^{k+1} is a prime divisor, which is not a component of Δ_{0}^{k+1}. By the classification of log canonical surface singularities, we may find a valuation ν with centre G on X^{k+1} whose coefficient d with respect to (X^{k+1}, X_{0}^{k+1}+Δ^{k+1}) is at least zero. As X_{0}^{k} is the pullback of a divisor from T, g^{k} is (K_{X^{k}} +X_{0}^{k}+Δ^{k})-negative and so the coefficient c of ν with respect to (X^{k}, X_{0}^{k}+Δ^{k}) is at least d, with equality if and only if g^{k} is an isomorphism at the generic point of G. By (a)_{k} the centre of ν on X^{k} is not a component of Δ_{0}^{k}. It follows that 0 ≤ d ≤ c ≤ 0, so that c = d = 0 and g^{k} is an isomorphism at the generic point of G. Hence g_{0}^{k} is a birational contraction, that is, (b)_{k} holds. This completes the induction and the proof that (a–b)_{≤m−1} hold.

As g_{0}^{k} is a birational contraction which is (K_{X_{0}^{k}} + Δ_{0}^{k})-negative, for k ≤ m − 1, it follows that f_{0} is a (K_{X_{0}} + Δ_{0})-negative birational contraction. But then f_{0} is a weak log canonical model.

It remains to prove that K_{X} + Δ is pseudo-effective over T. Pick a divisor A which is ample over T and let

λ = inf{ t ∈ ℝ | K_{X} + Δ + tA is π-pseudo-effective },

be the π-pseudo-effective threshold. It is proved in [7] that λ is rational. By what we have already proved we may find a log terminal model f: X −−→ Y of K_{X} + Δ + λA over T such that f_{0} : X_{0} −−→ Y_{0} is a weak log canonical model of K_{X_{0}} + Δ_{0} + λA_{0}. Let G = f_{∗}(K_{X} + Δ + λA). If λ > 0 then K_{X_{0}} + Δ_{0} + λA_{0} is big, so that G_{0} is big and nef. But then G_{0}^{n} > 0 is positive and so G_{t}^{n} = G_{0}^{n} > 0 for every t ∈ T. As G is nef over T, it is big over T, and so K_{X} + Δ + λA is big over T, a contradiction. It follows that λ = 0 so that K_{X} + Δ is pseudo-effective over T. □

Theorem 4.2. Let X ⟶ T be a flat projective morphism of quasiprojective varieties. Let (X, Δ) be a log pair such that the fibres (X_{t}, Δ_{t}) are ℚ-factorial terminal, for every t ∈ T. Assume that every component R of Δ dominates T and that the fibres of the Stein factorisation of R ⟶ T are irreducible. Let m > 1 be any integer such that D = m(K_{X} + Δ) is integral.

If either K_{X} + Δ or Δ is big over T then h^{0}(X_{t}, O_{X_{t}}(D_{t})) is independent of t ∈ T.

Proof. Let R be a component of Δ, let S ⟶ T be the normalisation of the Stein factorisation of R ⟶ T so that S ⟶ T is finite and S is normal, and let

[figure: Y; X; S; T,]

be the fibre square. As S ⟶ T is finite, S is irreducible and Y ⟶ S is flat, Y is a quasi-projective variety. Y is normal by [14, 5.12.7]. Replacing X ⟶ T by Y ⟶ S finitely many times, we may assume that the fibres of R ⟶ T are irreducible, for every component R of Δ. Fix a closed point 0 ∈ T. Replacing T by the intersection of general hyperplane sections containing 0, we may assume that T is a curve. Replacing T by its normalisation and passing to the fibre square, we may assume that T is smooth. As the fibres of X ⟶ T are ℚ-factorial terminal, [11, 3.2] implies that X is ℚ-factorial.

It suffices to show that |D_{0}| = |D|_{X_{0}}. In particular we may suppose that K_{X_{0}} + Δ_{0} is pseudo-effective. Further we are free to work locally about 0. In particular we may assume that T is affine. [7, 1.2] implies that the divisor N_{σ}(X_{0}, K_{X_{0}} + Δ_{0}), defined in (2.1), is a ℚ-divisor. In particular

Θ_{0} = Δ_{0} − Δ_{0} ∧ N_{σ}(X_{0}, K_{X_{0}} + Δ_{0}),

is a ℚ-divisor. By assumption we may find a ℚ-divisor 0 ≤ Θ ≤ Δ whose restriction to X_{0} is Θ_{0}. If we set

μ = \frac{m}{m − 1},

then K_{X} + μΔ is big. Therefore, we may find ℚ-divisors A ≥ 0 and B ≥ 0, where A is ample, the support of A is a prime divisor and X_{0} is not a component of B, such that K_{X} + μΔ ∼_{ℚ} A + B. Possibly passing to an open subset of T we may assume that the components of B dominate T. Pick δ ∈ (0, 1/2) such that (X_{t}, Δ_{t} + δ(A_{t} + B_{t})) is terminal for every t ∈ T. If we let

H = \frac{δ}{m − 1 − δ} A,

then

D − Ξ ∼_{ℚ} K_{X} + Φ + δB + (m − 1 − δ)(K_{X} + Θ + H),

where Φ = (1 − δμ + δ)Δ and Ξ = (m − 1 − δ)(Δ − Θ).

As H_{0} is ample, no component of Θ_{0} + H_{0} belongs to the stable base locus of K_{X_{0}} + Θ_{0} + H_{0}. (4.1) implies that we may find a log terminal model f: X −−→ Y for (X, Θ + H), such that the induced birational map f_{0} : X_{0} −−→ Y_{0} is a weak log canonical model of (X_{0}, Θ_{0} + H_{0}).

Let p: W ⟶ X and q: W ⟶ Y resolve f: X −−→ Y, where p is also a log resolution of (X, Δ + A + B). If we let

G = (m − 1 − δ)f_{∗}(K_{X} + Θ + H),

then G is big and nef and

(m − 1 − δ)p^{∗}(K_{X} + Θ + H) = q^{∗}G + F,

where F ≥ 0 is exceptional for q.

Let W_{0} be the strict transform of X_{0}. As (X_{0}, Φ_{0} +δB_{0}) is kawamata log terminal, inversion of adjunction, [7, 1.4.5], implies that (X, X_{0} + Φ + δB) is purely log terminal. Therefore, if we write

K_{W} + W_{0} = p^{∗}(K_{X} + X_{0} + Φ + δB) + E,

then ⌈E⌉ ≥ 0 is exceptional for p. Let

L = ⌈p^{∗}(D − Ξ) + E − F ⌉.

Possibly passing to an open subset of T, we may assume X_{0} is ℚ-linearly equivalent to zero. In particular,

L − W_{0} ∼_{ℚ} K_{W} + C + q^{∗}G,

where C is the fractional part of −p^{∗}(D − Ξ) − E + F. Hence (W, C) is kawamata log terminal and Kawamata-Viehweg vanishing implies that

H^{1}(W, O_{W} (L − W_{0})) = 0.

Let N = p^{∗}(K_{X} +Θ)−q^{∗}f_{∗}(K_{X} +Θ). As H is ample, p^{∗}H ≤ q^{∗}f_{∗}H, and so

mN = (1 + δ)N + (m − 1 − δ)N ≥ F.

As Ξ ≤ m(Δ − Θ) we have D − Ξ ≥ m(K_{X} + Θ) and so it follows that

M = L − ⌊mq^{∗}f_{∗}(K_{X} + Θ)⌋
= ⌈L − mq^{∗}f_{∗}(K_{X} + Θ)⌉
≥ ⌈mN + E − F ⌉
≥ ⌈E⌉.

Let q_{0} : W_{0} ⟶ X_{0} denote the restriction of q to W_{0} and let L_{0} and M_{0} denote the restriction of L and M to W_{0}. We have

|D_{0}| = |m(K_{X_{0}} + Θ_{0})|  by definition of Θ_{0}
⊂ |mf_{0∗}(K_{X_{0}} + Θ_{0})|  since f_{0} is a birational contraction
= |mq_{0}^{∗}f_{0∗}(K_{X_{0}} + Θ_{0})|
⊂ |L_{0}|  as M_{0} ≥ 0
= |L|_{W_{0}}  since H^{1}(W, O_{W} (L − W_{0})) = 0
⊂ |D|_{X_{0}}  since ⌈E⌉ is exceptional for p.

Thus equality holds, as the reverse inequality holds automatically. □

Proof of (1.8). We first prove (1). Let 0 ∈ T be a closed point. Replacing T by an unramified cover, we may assume that the strata of (X, Δ) intersect X_{0} in strata of (X_{0}, Δ_{0}). Since the only valuations of non-negative coefficient lie over the strata of (X, Δ), replacing (X, Δ) by a blow up, we may assume that (X, Δ) is terminal. Thus (1) follows from (4.2).

Now we prove (2). Pick m_{0} > 0 such that m_{0}Δ is integral and an ample divisor H such that H +m_{0}Δ is very ample. Pick a prime divisor A ∼ H + m_{0}Δ, such that (X, Δ + A) has simple normal crossings over T. If m ≥ m_{0} is any positive integer such that mΔ is integral, then

(X, Δ′ = \frac{m − m_{0}}{m} Δ + \frac{1}{m}A)

is a simple normal crossings pair and it is kawamata log terminal. Further

K_{X} + Δ′ ∼_{ℚ} K_{X} + Δ + H/m,

and Δ′ is big over T. Thus (2) follows from (1).

Note that

vol(X_{t}, K_{X_{t}} + Δ_{t}) = lim_{ϵ→0} vol(X_{t}, K_{X_{t}} + (1 − ϵ)Δ_{t}).

As (X_{t}, (1 − ϵ)Δ_{t}) is kawamata log terminal, (3) follows from (1) and (2). □

5. DCC for the volume of bounded pairs

We prove (1.9) in this section. We first deal with the case that T is a closed point.

Proposition 5.1. Fix a set I ⊂ [0, 1] which satisfies the DCC and a simple normal crossings pair (Z, B), where Z is projective of dimension n. Let 𝔇 be the set of simple normal crossings pairs (X, Δ), where X is projective, the coefficients of Δ belong to I and there is a birational morphism f: X ⟶ Z with Φ = f_{∗}Δ ≤ B.

Then the set

{ vol(X, K_{X} + Δ) | (X, Δ) ∈ 𝔇 },

also satisfies the DCC.

Definition 5.2. Let (X, Δ) be a log pair. If π: Y ⟶ X is a birational morphism, then we may write

K_{Y} + Γ = π^{∗}(K_{X} + Δ) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, π_{∗}Γ = Δ and π_{∗}E = 0.

Define a b-divisor L_{Δ} by setting L_{Δ,Y} = Γ.

Lemma 5.3. Let (X, Δ) be a simple normal crossings pair, where X is a projective variety.

(1) If Y ⟶ X is a birational morphism such that (Y, Θ = L_{Δ,Y}) has simple normal crossings, and Γ−Θ ≥ 0 is exceptional, then

vol(X, K_{X} + Δ) = vol(Y, K_{Y} + Γ).

(2) If f: X ⟶ Z is a birational morphism such that (Z, Φ = L_{Δ,Z}) has simple normal crossings and Θ = Δ ∧ L_{Φ,X}, then

vol(X, K_{X} + Δ) = vol(X, K_{X} + Θ).

Proof. (1) is clear, as

H^{0}(X, O_{X}(m(K_{X} + Δ))) ≃ H^{0}(Y, O_{Y} (m(K_{Y} + Γ))),

for all m.

For (2), suppose m is a sufficiently divisible positive integer. We have

H^{0}(X, O_{X}(m(K_{X} + Δ))) ⊂ H^{0}(Z, O_{Z}(m(K_{Z} + Φ)))
= H^{0}(X, O_{X}(m(K_{X} + L_{Φ,X}))),

and so

H^{0}(X, O_{X}(m(K_{X} + Δ))) = H^{0}(X, O_{X}(m(K_{X} + Θ))).  □

Lemma 5.4. Let (Z, Φ) be a simple normal crossings pair which is log canonical.

If ν is a valuation such that L_{Φ}(ν) > 0, then the centre of ν is a stratum W of (Z, Φ) and there is a birational morphism Y = Y_{ν} ⟶ Z such that ρ(Y/Z) = 1, Y is ℚ-factorial and the centre of ν is a divisor on Y; Y_{ν} is unique with these properties.

Proof. This is a consequence of the existence of log terminal models, which is proved in [7], and uniqueness of log canonical models. □

Definition 5.5. Let (X, Δ) be a log pair. Define a b-divisor M_{Δ} by assigning to any valuation ν,

M_{Δ}(ν) = \begin{cases}mult_{B}(Δ) & if the centre of ν is a divisor B on X, \\ 1 & otherwise. \end{cases}

Definition 5.6. Let B be a b-divisor whose coefficients belong to [0, 1] and let (Z, Φ = B_{Z}) be a model with simple normal crossings. Let Z′ ⟶ Z be a log resolution, and let Σ be a set of valuations σ whose centres are exceptional divisors for Z′ ⟶ Z, such that L_{Φ}(σ) > 0.

For every valuation σ ∈ Σ, let Γ_{σ} = (L_{Φ} ∧ B)_{Y_{σ}}, where Y_{σ} ⟶ Z is defined in (5.4). Let

Θ = ⋀_{σ∈Σ} L_{Γ_{σ},Z′},

the minimum of the divisors L_{Γ_{σ},Z′}.

The cut of (Z, B), associated to Z′ ⟶ Z and Σ, is the pair (Z′, B′), where

B′ = B ∧ M_{Θ},

so that the trace of B′ on Z′ is Θ ∧ B_{Z′} and otherwise B′ is the same b-divisor as B.

We say that the pair (Z′, B′) is a reduction of the pair (Z, B), if they are connected by a sequence of cuts, that is, there are pairs, (Z_{i}, B_{i}), 0 ≤ i ≤ k, starting at (Z_{0}, B_{0}) = (Z, B) and ending at (Z_{k}, B_{k}) = (Z′, B′), such that (Z_{i+1}, B_{i+1}) is a cut of (Z_{i}, B_{i}), for each 0 ≤ i < k.

Lemma 5.7. Let B be a b-divisor whose coefficients belong to a set I ⊂ [0, 1] which satisfies the DCC, and let (Z, Φ = B_{Z}) be a model with simple normal crossings.

Then there is a reduction (Z′, B′) of (Z, B) such that

L_{Φ′} ≤ B′,

where Φ′ = B′_{Z′}.

Proof. If W is a stratum of (Z, Φ), then define the weight w of W as follows: If there is a valuation ν, with centre W, such that B(ν) < L_{Φ}(ν), then let w be the number of components of Φ with coefficient 1 which contain W. Otherwise, if there is no such ν, then let w = −1.

Define the weight of (Z, B) to be the maximum weight of the strata of (Z, Φ).

Suppose the weight of (Z, B) is −1. Then L_{Φ}(ν) ≤ B(ν) for any valuation ν whose centre is a stratum. If ρ is a valuation, whose centre is not a stratum, we have 0 = L_{Φ}(ρ) ≤ B(ρ) (cf. [20, 2.31]). In this case we just take Z′ = Z.

From now on we suppose that the weight w ≥ 0. Suppose that (Z′, B′) is a cut of (Z, B). Then B′ and B have the same coefficients, except for finitely many valuations. In particular, the coefficients of B′ belong to a set I′ ⊃ I which still satisfies the DCC. It suffices therefore to prove that we can find a cut (Z′, B′) of (Z, B) with smaller weight.

Now if (Z′, B′) is a cut of (Z, B), then B′_{Z′} ≤ L_{Φ,Z′}. On the other hand, if ν is any valuation whose centre is not a divisor on Z′, then B(ν) = B′(ν). It follows that the weight of (Z′, B′) is at most the weight of (Z, B). Therefore, as (Z, Φ) has only finitely many strata, we may construct (Z′, B′) étale locally about every stratum. Thus, we may assume that Z = ℂ^{n} and that Φ is supported on the coordinate hyperplanes.

We will use the language of toric geometry, cf. [12]. ℂ^{n} is the toric variety associated to the cone spanned by the standard basis vectors e_{1}, e_{2}, …, e_{n} in ℝ^{n}. If ν is any valuation such that L_{Φ}(ν) > 0, then ν is toric and we will identify ν with an element (v_{1}, v_{2}, …, v_{n}) of ℕ^{n}. Order the components of Φ so that the last w components have coefficient one and let 0 ≤ c_{1}, c_{2}, …, c_{s} < 1 be the initial coefficients, so that n = s + w. With this ordering, we have

L_{Φ}(ν) = 1 − ∑_{i} v_{i}(1 − c_{i}).

(Indeed both sides of this equation are affine linear in v_{1}, v_{2}, …, v_{n} and c_{1}, c_{2}, …, c_{s} and it is easy to check we have equality when either ν is the zero vector or when ν = e_{i}, 1 ≤ i ≤ n.) Consider the finite set

𝔉 = { (v_{1}, v_{2}, …, v_{s}) ∈ ℕ^{s} | ∑_{i} v_{i}(1 − c_{i}) < 1 }.

Given a valuation ν = (v_{1}, v_{2}, …, v_{n}), note that L_{Φ}(ν) > 0 if and only if (v_{1}, v_{2}, …, v_{s}) ∈ 𝔉.

As I satisfies the DCC, for every f = (f_{1}, f_{2}, …, f_{s}) ∈ 𝔉, we may pick a valuation σ = (f_{1}, f_{2}, …, f_{s}, v_{s+1}, v_{s+2}, …, v_{n}), such that

B(σ) = inf{ B(ν) | ν = (f_{1}, f_{2}, …, f_{s}, u_{s+1}, u_{s+2}, …, u_{n}) }.

Let Σ be a set of choices of such valuations σ, so that Σ and 𝔉 have the same cardinality. Let Z′ ⟶ Z be any log resolution of (Z, Φ) such that the centre of every element of Σ is a divisor on Z′. We may assume that the induced birational map Z′ ⟶ Y_{σ} is a morphism, for every σ ∈ Σ. Let (Z′, B′) be the cut of (Z, B) associated to Z′ → Z and Σ.

There are two cases. If w = 0, then Σ = 𝔉 is the set of all valuations of coefficient less than one. It follows that if ν is any valuation whose centre on Z′ is not a divisor, then L_{Φ′}(ν) = 0, so that the weight of (Z′, B′) is −1, which is less than the weight of (Z, B).

Otherwise we may assume that w ≥ 1. Suppose that ν is a valuation whose centre is not a divisor on Z′ such that B′(ν) < 1 and L_{Φ′}(ν) > 0. Then B(ν) = B′(ν) and L_{Φ}(ν) > 0 and so ν = (v_{1}, v_{2}, …, v_{n}) is toric and (v_{1}, v_{2}, …, v_{s}) ∈ 𝔉. By construction, there is an element σ of Σ with the same first s coordinates as ν such that B(σ) ≤ B(ν) < 1. The cone spanned by the standard basis vectors e_{1}, e_{2}, …, e_{n} is divided into m ≤ n subcones by σ (these are the maximal cones of Y_{σ}), where m is the number of non-zero entries of σ, and so ν is a non-negative linear combination of σ and n − 1 vectors taken from e_{1}, e_{2}, …, e_{n}. It follows that

(♯)  ν = ∑_{j≠l} λ_{j}e_{j} + λσ,

for some index 1 ≤ l ≤ n and non-negative real numbers λ_{1}, λ_{2}, …, λ_{n} and λ, where the lth entry of σ is non-zero.

If the centre of ν on Z′ is contained in w components of Φ′ of coefficient one, then the centre of ν on Y_{σ} is also contained in w components of coefficient one of Γ_{σ} = (L_{Φ} ∧ B)_{Y_{σ}}. By assumption, the exceptional divisor of Y_{σ} ⟶ Y has coefficient strictly less than one, and so the centre of ν on Y_{σ} must be contained in the strict transform of the last w coordinate hyperplanes. But then, by standard toric geometry, l ≤ s. Hence the lth entry of ν is non-zero. Comparing the coefficient of e_{l} in (♯), we must have λ = 1, so that ν ≥ σ. In this case,

L_{Φ′}(ν) ≤ L_{Γ_{σ}}(ν)  by definition of B′,
≤ L_{Γ_{σ}}(σ)  as ν ≥ σ,
≤ B(σ)  since Γ_{σ} ≤ B_{Y_{σ}},
≤ B(ν)  by our choice of σ,
= B′(ν)  by definition of B′.

It follows that the weight of (Z′, B′) is indeed smaller than the weight of (Z, B) and this completes the induction and the proof. □

Proof of (5.1). Suppose we have a sequence of log pairs (X_{i}, Δ_{i}) ∈ 𝔇, such that v_{i} ≥ v_{i+1}, where v_{i} := vol(X_{i}, K_{X_{i}} + Δ_{i}). We will show that the sequence v_{1}, v_{2}, … is eventually constant; to this end we are free to pass to a subsequence. Replacing I by Ī∪ {1}, we may assume that I is closed and 1 ∈ I.

By assumption there are projective birational morphisms f_{i}: X_{i} ⟶ Z such that Φ_{i} = f_{i∗}Δ_{i} ≤ B. Note that if ν is a valuation such that M_{Δ_{i}}(ν) ∉ {0, 1}, then the centre of ν is a component of Δ_{i}. On the other hand, if M_{Δ_{i}}(ν) = 1 and M_{Δ_{j}}(ν) = 0 and the centre of ν is not a component of Δ_{i}, then the centre of ν is not a divisor on X_{i} and it is a divisor on X_{j}. Therefore there are only countably many valuations ν such that M_{Δ_{i}}(ν) ≠ M_{Δ_{j}}(ν) for some i and j. Therefore, as I satisfies the DCC, by a standard diagonalisation argument, after passing to a subsequence, we may assume that M_{Δ_{i}}(ν) is eventually a non-decreasing sequence, for all valuations ν. In particular, we may define a b-divisor B by putting

B(ν) = lim_{i→∞} M_{Δ_{i}}(ν).

Note that the coefficients of B belong to I. Let Φ = B_{Z}.

Suppose that (Z′, B′) is a cut of (Z, B) associated to a birational morphism Z′ ⟶ Z and a set of valuations Σ. Let f′_{i}: X_{i} −−→ Z′ be the induced birational map. Note that, if X′_{i} ⟶ X_{i} is a birational morphism and (X′_{i}, Δ′_{i} = M_{Δ_{i},X′_{i}}) has simple normal crossings, then vol(X′_{i}, K_{X′_{i}} + Δ′_{i}) = v_{i} and the coefficients of Δ′_{i} belong to I, so that (X′_{i}, Δ′_{i}) ∈ 𝔇. Therefore, we are free to replace (X_{i}, Δ_{i}) by (X′_{i}, Δ′_{i}). In particular, we may assume that f′_{i} is a birational morphism.

Given σ ∈ Σ, let Γ_{i,σ} = (L_{Φ_{i}} ∧ B)_{Y_{σ}}, where Y_{σ} ⟶ Z is defined in (5.4). Suppose we define a sequence of divisors

Θ_{i} = ⋀_{σ∈Σ} L_{Γ_{i,σ},Z′},

as in (5.6). Suppose that B is a prime divisor on Z′ which is exceptional over Z. Then the coefficient of B in Θ_{i} is the minimum of finitely many affine linear functions of the coefficients of Δ_{i}. It follows that the coefficients of Θ_{1}, Θ_{2}, … belong to a set I′ ⊃ I which satisfies the DCC. Finally, let

Δ′_{i} = Δ_{i} ∧ M_{Θ_{i},X_{i}},

so that we only change the coefficients of divisors which are exceptional for Z′ → Z. In particular, the coefficients of Δ′_{i} belong to I′. On the other hand,

Δ_{i} ∧ L_{Θ_{i},X_{i}} ≤ Δ′_{i} = Δ_{i} ∧ M_{Θ_{i},X_{i}} ≤ Δ_{i},

so that

v_{i} = vol(X_{i}, K_{X_{i}} + Δ′_{i}),

by (2) of (5.3). Finally, note that M_{Δ′_{i}}(ρ) is eventually a non-decreasing sequence for any valuation ρ. In particular, we may define a b-divisor B′ by putting

B′(ρ) = lim_{i→∞} M_{Δ′_{i}}(ρ),

as before.

Hence, (5.7) implies that we may find a reduction (Z′, B′), of (Z, B) and pairs (X′_{i}, Δ′_{i}), whose coefficients belong to a set I′ which satisfies the DCC, such that v_{i} = vol(X′_{i}, K_{X′_{i}} + Δ′_{i}), there is a birational morphism X′_{i} ⟶ Z′, and moreover L_{Φ′} ≤ B′. Replacing (X_{i}, Δ_{i}) by (X′_{i}, Δ′_{i}), I by I′, and Z by Z′, we may therefore assume that L_{Φ} ≤ B.

Note that

v_{i} = vol(X_{i}, K_{X_{i}} + Δ_{i})
≤ vol(Z, K_{Z} + Φ_{i})
≤ lim vol(Z, K_{Z} + Φ_{i})
= vol(Z, K_{Z} + Φ),

as lim Φ_{i} = Φ.

On the other hand, if we fix ϵ > 0, then (Z, (1−ϵ)Φ) is kawamata log terminal. In particular, we may pick a birational morphism f: Y ⟶ Z such that (Y, Ψ = L_{(1−ϵ)Φ,Y}) is terminal. If we let Θ = L_{Φ,Y} and Γ = B_{Y}, then

Ψ ≤ (1 − η)Θ ≤ Θ ≤ Γ

for some η > 0. As Γ is the limit of Γ_{i} = M_{Δ_{i},Y}, it follows that we may find i such that Ψ ≤ Γ_{i}. As (Y, Ψ) is terminal, we have Ψ_{i} = L_{Ψ,X_{i}} ≤ Δ_{i}. But then

vol(Z, K_{Z} + (1 − ϵ)Φ) = vol(Y, K_{Y} + Ψ)
≤ vol(X_{i}, K_{X_{i}} + Ψ_{i})
≤ vol(X_{i}, K_{X_{i}} + Δ_{i}) = v_{i}.

Taking the limit as ϵ goes to zero, we get

vol(Z, K_{Z} + Φ) ≤ v_{i} ≤ vol(Z, K_{Z} + Φ),

so that v_{i} = vol(Z, K_{Z} + Φ) is constant. □

Proof of (1.9). We may assume that 1 ∈ I. By assumption there is a log pair (Z, B) and a projective morphism Z ⟶ T, where T is of finite type, such that 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} contains the support of the strict transform of Δ_{t} and any f^{−1}-exceptional divisor.

Suppose that p: Y ⟶ X is a birational morphism. Then the coefficients of Γ = M_{Δ,Y} belong to I and

vol(X, K_{X} + Δ) = vol(Y, K_{Y} + Γ),

by (1) of (5.3). Replacing (X, Δ) by (Y, Γ), we may assume that f is a morphism and we are free to replace Z and B by higher models.

We may assume that T is reduced. 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 Z_{0} and B_{0} be the fibres over a fixed closed point 0 ∈ T. Let 𝔇_{0} ⊂ 𝔇 be the set of simple normal crossings pairs (Y, Γ), where the coefficients of Γ belong to I, Y is a projective variety of dimension n, and there is a birational morphism g: Y ⟶ Z_{0} with g_{∗}Γ ≤ B_{0}.

Pick (X, Δ) ∈ 𝔇. Let Φ = f_{∗}Δ. Let Σ be the set of all valuations ν whose centre on X is a divisor which is exceptional over Z_{t} such that L_{Φ}(ν) > 0. We may find a birational morphism f′: X′ ⟶ Z_{t}, such that the centre of every element of Σ is a divisor on X′, whilst f′ only blows up strata of (Z_{t}, Φ). Suppose that g: W ⟶ X is a log resolution which resolves the indeterminacy locus of the induced birational map X −−→ X′. If we set Δ′ = M_{Δ,X′}, then the coefficients of Δ′ belong to I and

vol(X, K_{X} + Δ) = vol(W, K_{W} + M_{Δ,W}) ≤ vol(X′, K_{X′} + Δ′),

by (1) of (5.3). If ν is any valuation whose centre is an exceptional divisor for W ⟶ X′ but not for W ⟶ X, then the centre of ν is an exceptional divisor for X ⟶ Z_{t} and so L_{Φ}(ν) = 0, by choice of f′. It follows that

M_{Δ,W} ≥ M_{Δ′,W} ∧ L_{Φ,W} ,

and so (5.3) implies that

vol(W, K_{W} +M_{Δ,W}) ≥ vol(W, K_{W} +M_{Δ′,W} ∧L_{Φ,W}) = vol(X′, K_{X′}+Δ′).

and hence the inequalities above are equalities. In particular

vol(X, K_{X} + Δ) = vol(X′, K_{X′} + Δ′).

Replacing (X, Δ) by (X′, Δ′), we may assume that f only blow ups strata of Φ.

As (Z, B) has simple normal crossings over T and the strata of (Z, B) have irreducible fibres, we may find a sequence of blow ups g: Z′ ⟶ Z of strata of B, which induces the sequence of blow ups determined by f, so that X = Z′_{t}. There is a unique divisor Ψ supported on the strict transform of B and the exceptional locus of g, such that Δ = Ψ_{t}. If Y = Z′_{0} is the fibre over 0 of Z′ ⟶ T and Γ is the restriction of Ψ to Y, then (Y, Γ) ∈ 𝔇_{0}. (1.8) implies that vol(Y, K_{Y} + Γ) = vol(X, K_{X} + Δ).

It follows that

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

Now apply (5.1). □

6. Birational geometry of global quotients

Theorem 6.1 (Tsuji). Assume (1.4)_{n−1}.

Then there is a constant C = C(n) > 2 such that if (X, Δ) is a global quotient, where X is projective of dimension n, and K_{X} + Δ is big, then ϕ_{m(K_{X}+Δ)} is birational for every integer m ≥ C + 1 such that

vol(X, (m − 1)(K_{X} + Δ)) > (Cn)^{n}.

Proof. First note that (2.2.2) implies that

K_{X} + ⌈(m − 1)(K_{X} + Δ)⌉ = ⌊m(K_{X} + Δ)⌋.

As we are assuming (1.4)_{n−1} there is a constant ϵ > 0 such that if (U, Θ) is a global quotient, where K_{U} + Θ is big and U is projective of dimension k at most n − 1, then vol(U, K_{U} + Θ) > ϵ^{k}. Let

C = 2(1 + γ)^{n−1} where γ = \frac{4n}{ϵ}.

By assumption there is a smooth projective variety Y of dimension n and a finite group G ⊂ Aut(Y) such that X = Y/G and if π: Y ⟶ X is the quotient morphism, then K_{Y} = π^{∗}(K_{X} + Δ). As K_{X} + Δ is big, Y is of general type. Replacing (X, Δ) and Y by their log canonical models, which exist by [7], we lose the fact that X and Y are smooth, gain the fact that K_{X} + Δ and K_{Y} are ample, and retain the condition that K_{X} + Δ is kawamata log terminal and K_{Y} is canonical.

We check the hypotheses of (2.3.6), applied to the ample divisor K_{X} + Δ and the constants ϵ/2 and γ_{0} = \frac{m−1}{C} ≥ 1. Clearly

vol(X, γ_{0}(K_{X} + Δ)) > n^{n}.

Suppose that V is a minimal non kawamata log terminal centre of a log pair (X, Δ + Δ_{0}), which is log canonical at the generic point of V. Further suppose that V passes through a very general point of X, and 0 ≤ Δ_{0} ∼_{ℚ} λ(K_{X} + Δ), for some rational number λ ≥ 1.

If Γ_{0} = π^{∗}Δ_{0}, then every irreducible component of π^{−1}(V) is a non kawamata log terminal centre of (Y, Γ_{0}). Let V′ be the normalisation of π^{−1}(V). As H = K_{Y} +Γ_{0} is ample, Kawamata’s subadjunction formula implies that for every η > 0, there is a divisor Φ ≥ 0 on V′ such that

(K_{Y} + Γ_{0} + ηH)|_{V′} = K_{V′} + Φ.

Let W ⟶ V′ be a G-equivariant resolution. As V passes through a very general point of X, W is a union of irreducible varieties of general type. If U = W/G is the quotient, then U is irreducible and we may find a ℚ-divisor Θ such that K_{W} = ψ^{∗}(K_{U} + Θ), where ψ: W ⟶ U is the quotient map.

As (U, Θ) is a global quotient, vol(U, K_{U} + Θ) > ϵ^{k}, where k is the dimension of V. Therefore

|G| vol(V, (K_{X} + Δ + Δ_{0})|_{V}) = vol(V′, (K_{Y} + Γ_{0})|_{V′})
≥ vol(V′, K_{V′})
≥ vol(W, K_{W})
= |G| vol(U, K_{U} + Θ)
≥ |G|ϵ^{k}.

Thus

vol(V, (1 + λ)(K_{X} + Δ)|_{V}) > ϵ^{k},

and so

vol(V, λ(K_{X} + Δ)|_{V}) > (\frac{ϵ}{2})^{k} .

(2.3.6) implies that (m − 1)(K_{X} + Δ) is potentially birational. (1) of (2.3.4) implies that ϕ_{K_{X}+⌈(m−1)(K_{X}+Δ)⌉} is birational. □

7. Proof of (1.4) and (1.1)

Proof of (1.4). By induction on n. Assume (1.4)_{n−1}. By (6.1) there is a constant C = C(n) > 2 depending only on the dimension n such that if (X, Δ) is a global quotient, where X is projective of dimension n and K_{X} + Δ is big, then ϕ_{m(K_{X}+Δ)} is birational, for any m ≥ C + 1 such that

vol(X, (m − 1)(K_{X} + Δ)) > (Cn)^{n}.

Note that the right hand side does not depend on m.

Fix a constant V > n^{n} and let

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

Note that if k is a positive integer such that vol(X, k(K_{X} +Δ)) ≤ C^{n}V, then vol(X, (k + 1)(K_{X} + Δ)) ≤ 2^{n}C^{n}V. It follows that there is a positive integer m ≥ C + 1 such that if (X, Δ) ∈ 𝔇_{V}, then

(Cn)^{n} < vol(X, (m − 1)(K_{X} + Δ)) ≤ 2^{n}C^{n}V,

so that ϕ_{m(K_{X}+Δ)} is birational. (2) of (2.3.4) implies that ϕ_{K_{X}+(2n+1)m(K_{X}+Δ)} is birational. But then (3.1) implies that 𝔇_{V} is log birationally bounded, and so (1.9) implies that the set

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

satisfies the DCC, which implies that (1) and (2) of (1.4) hold in dimension n.

In particular there is a constant δ > 0 such that if (X, Δ) ∈ 𝔇, and K_{X} + Δ is big, then vol(X, K_{X} + Δ) ≥ δ. It follows that ϕ_{M(K_{X}+Δ)} is birational, for any

M > \frac{Cn}{δ} + 1,

and this completes the induction and the proof. □

Proof of (1.1). By (2) of (1.4) there is a constant δ > 0 such that if (X, Δ) is a global quotient, where X is projective of dimension n and K_{X} + Δ is big, then vol(X, K_{X} + Δ) ≥ δ. Let c = \frac{1}{δ}.

Let Y be a projective variety of dimension n of general type. By [7], there is a log canonical model Y −−→ Y′. If G is the birational automorphism group of Y, then G is the automorphism group of Y′. Replacing Y by a G-equivariant resolution of Y′, we may assume that G is the automorphism group of Y. Let π: Y ⟶ X = Y/G be the quotient of Y. Then there is a divisor Δ on X such that K_{Y} = π^{∗}(K_{X} + Δ). By definition, (X, Δ) is a global quotient, X is projective and K_{X} + Δ is big. It follows that vol(X, K_{X} + Δ) ≥ δ. As

vol(Y, K_{Y}) = |G| vol(X, K_{X} + Δ),

it follows that

|G| ≤ c · vol(Y, K_{Y}).  □

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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

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

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

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

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