BOUNDEDNESS OF PLURICANONICAL MAPS OF VARIETIES OF GENERAL TYPE

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

Abstract. Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r_{n} such that if X is a smooth projective variety of general type and dimension n, then ϕ_{rK_{X}}: X −−→ ℙ(H^{0}(O_{X}(rK_{X}))) is birational for all r ≥ r_{n}.

1. Introduction

One of the main problems of complex projective algebraic geometry is to understand the structure of the pluricanonical maps. If X is of general type, then by definition the pluricanonical map ϕ_{rK_{X}}: X −−→ ℙ(H^{0}(O_{X}(rK_{X}))) is birational for all sufficiently large r. It is a natural question to then ask if there is an r_{n} such that ϕ_{rK_{X}} is birational as soon as r ≥ r_{n}, uniformly for any variety of general type of dimension n. When X is a curve, it is well known that ϕ_{rK_{X}} is birational for r ≥ 3 and when X is a surface, Bombieri proved in [3] that ϕ_{rK_{X}} is birational, for r ≥ 5.

However, starting with threefolds the problem is substantially harder, since there are threefolds of general type for which the minimal model is necessarily singular. In fact it is easy to write down examples of threefolds of general type, for which the index of K_{X} (the smallest multiple of K_{X} which is Cartier) on a minimal model is arbitrarily large (for example, take any hypersurface of sufficiently large degree in any weighted projective space). In this case, a priori, the degree of K_{X} (equivalently the volume of K_{X} on a smooth model) could be arbitrarily small. Even the case when the minimal model is smooth, or at least the index of K_{X} is bounded, has attracted considerable attention, see for example, [7], [2], [19] and [5], and there have only been partial results in the general case, see [11], [16] and [17].

The first author was partially supported by NSA research grant no: MDA904-03-1-0101 and by a grant from the Sloan Foundation. We are in debt to [22] and the beautiful ideas and techniques introduced in that paper. We would like to thank Meng Chen for pointing out that a previous version of (1.6) was too optimistic. We would also like to thank the referee, whose extensive comments were very useful.

Recently, Tsuji settled this problem for 3-folds and assuming the minimal model program, for all higher dimensional varieties, see [22]. Using ideas of Tsuji, we prove the following:

Theorem 1.1. For any positive integer n, there exists an integer r_{n} such that if X is a smooth projective variety of general type and dimension n, then ϕ_{rK_{X}}: X −−→ ℙ(H^{0}(O_{X}(rK_{X}))) is birational for all r ≥ r_{n}.

(1.1) has some interesting consequences:

Corollary 1.2. Let n be a positive integer and M be a positive constant. Then the family of all smooth projective varieties of general type of dimension n and volume of K_{X} at most M, is birationally bounded.

Corollary 1.3. Let n be a positive integer. Then there is a positive constant η_{n} > 0 such that given any smooth projective variety X of general type of dimension n, the volume of K_{X} is at least η_{n}.

As Maehara points out in [18] (see also [22]), one also recovers the Severi-Iitaka conjecture:

Corollary 1.4. For any fixed variety X, there exist only finitely many dominant rational maps π: X −−→ Y (modulo birational equivalence), where Y is a variety of general type.

The proof of the above results closely follows Tsuji’s original strategy; in fact this note grew out of our desire to understand [22]. Even though our proof is based on the ideas of this paper, it only relies on the techniques of algebraic geometry. In particular we do not use the “Analytic Zariski Decomposition” and we do not make use of the results of [23] (which we were unable to follow). The main tools employed in this paper are the algebraic techniques of the minimal model program and in particular the theory of log canonical centres and of multiplier ideals.

We now give a short informal sketch of the proof of (1.1), which closely follows the strategy of Tsuji’s proof, see [22]. As one would expect, the goal is to produce a ℚ-divisor Δ ∼_{ℚ} λK_{X} such that Δ has an isolated log canonical centre at a general point x ∈ X. There is a well established strategy for producing divisors with log canonical centres at a general point x ∈ X, as soon as the volume is sufficiently large, so that the rational number λ depends only on the volume of K_{X}.

The problem is that the smallest log canonical centre V containing x might well be of positive dimension. In order to produce an isolated log canonical centre, one proceeds by inductively cutting down the (dimension of the) log canonical centres. As x is general, a resolution W of V must have general type, and so by induction we can certainly find a ℚ-divisor Θ on W with the required properties.

At this point, the idea is to lift sections from V to the whole of X. In principle this ought to be straightforward; indeed if K_{X} is nef, then we can lift Θ to the whole of X, as an easy application of Serre vanishing. In practice we cannot assume that K_{X} is nef, and this step of the argument forms the technical heart of the paper. In fact we generalise some results and techniques of [20] and [10] for extending pluricanonical forms from a divisor to the whole space. Instead of just lifting pluricanonical forms, we lift certain log pluricanonical forms, from a log canonical centre to the whole of X, using a variant of multiplier ideal sheaves, see §3. It is then not too hard to prove that we can then lift log canonical centres from W to X, see §4.

Then we prove, assuming (1.1) in dimension less than n, that ϕ_{rK_{X}} is birational, where r is a fixed linear function of 1/ vol(X)^{1/n}, which only depends on the dimension. If vol(K_{X}) ≥ 1, the theorem follows easily. If vol(K_{X}) < 1, one then shows that X belongs to a birationally bounded family and the existence of r_{n} is then clear.

As already pointed out, the main technical point is (3.17), the result which shows that certain log forms may be lifted from a divisor to the ambient variety. We hope that it will find other applications in a variety of contexts. For example, in [6], we use this result to prove a conjecture of Shokurov which in particular implies that the fibers of a resolution of a variety with log terminal singularities are rationally chain connected.

We now raise some questions and problems, which are closely related to (1.1):

Problem 1.5. Find explicit (hopefully small) values of r_{n}.

Even determining r_{3} would seem interesting. Indeed in the past twenty years or so, our knowledge of threefolds has increased considerably; in particular we have Reid’s powerful Riemann-Roch formula for terminal threefolds and we are able to write down many explicit examples of threefolds of general type. Thus determining the value of r_{3} would seem to be a reasonable problem. In fact we raise the following:

Question 1.6. Is the rational map ϕ_{27K_{X}} always birational, whenever X is a threefold of general type?

Note that we have little or no evidence for an affirmative answer to (1.6), apart from the examples of Fletcher in [8]. In particular for the threefold X_{46}, a hypersurface in the weighted projective space ℙ(4, 5, 6, 7, 23), see [8], ϕ_{mK_{X}} is birational iff m = 23, or m ≥ 27.

Finally it seems natural to conjecture that we can drop the condition that X is of general type in (1.1):

Conjecture 1.7. There is a positive integer r_{n} with the following property:

Let X be a smooth variety of non-negative Kodaira dimension, of dimension n. Then the rational map ϕ_{rK_{X}} is birationally equivalent to the Iitaka fibration.

Unfortunately the methods of this paper don’t seem to apply directly to (1.7).

2. Preliminaries

2.1. Notation and conventions. We work over the field of complex numbers ℂ. A ℚ-Cartier divisor D on a normal variety X is nef if D ·C ≥ 0 for any curve C ⊂ X. We say that two ℚ-divisors D_{1}, D_{2} are ℚ-linearly equivalent (D_{1} ∼_{ℚ} D_{2}) if there exists an integer m > 0 such that mD_{i} are linearly equivalent. Given a morphism of normal varieties f: X ⟶ Y, we say that two divisors D_{1} and D_{2} on X are f-linearly equivalent (D_{1} ∼_{f} D_{2}) if there is a Cartier divisor B on Y such that D_{1} ∼ D_{2} + f^{∗}B. We say that D_{1} and D_{2} are f-numerically equivalent (D_{1} ≡_{f} D_{2}) if there is a ℚ-Cartier divisor B on Y such that D_{1} ≡ D_{2} + f^{∗}B. A pair (X, Δ) is a normal variety X and a ℚ-Weil divisor Δ such that K_{X} + Δ is ℚ-Cartier. We will say that a pair (X, Δ) is a log pair, if in addition Δ is effective. We say that (X, Δ) is a smooth pair if X is smooth and Δ is a ℚ-divisor with simple normal crossings support. A projective morphism μ: Y ⟶ X is a log resolution of the pair (X, Δ) if Y is smooth and μ^{−1}(Δ) ∪ {exceptional set of μ} is a divisor with simple normal crossings support. We write μ^{∗}(K_{X} +Δ) = K_{Y} +Γ and Γ = ∑ a_{i}Γ_{i} where Γ_{i} are distinct reduced irreducible divisors. The pair (X, Δ) is kawamata log terminal (resp. log canonical) if there is a log resolution μ: Y ⟶ X as above such that the coefficients of Γ are strictly less than one i.e. a_{i} < 1 for all i (resp. a_{i} ≤ 1). The number 1 − a_{i} is the log discrepancy of Γ_{i} with respect to (X, Δ). We say that a subvariety V ⊂ X is a log canonical centre if it is the image of a divisor of log discrepancy at most zero. We will denote by LLC(X, Δ, x) the set of all log canonical centres containing a point x ∈ X. A log canonical place is a valuation corresponding to a divisor of log discrepancy at most zero. A log canonical centre is pure if K_{X} + Δ is log canonical at the generic point of V. If moreover there is a unique log canonical place lying over the generic point of V, we will say that V is an exceptional log canonical centre.

2.2. Volumes.

Definition 2.1. Let X be an irreducible projective variety and let D be a big ℚ-divisor. The volume of D, is

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

When D is very ample, vol(D) is just the degree of the image of X in ℙH^{0}(X, O_{X}(D)). When D is nef, then vol(D) = D^{dim(X)}. It turns out that the volume of D only depends on its numerical class and one can extend the volume function to a continuous function vol: N_{ℝ}^{1}(X) ⟶ ℝ (cf. [14]).

Lemma 2.2. Let X be a projective variety, D a divisor such that ϕ_{D} is birational with image Z. Then, the volume of D is at least the degree of Z and hence at least 1.

Proof. We may replace X by an appropriate birational model and in particular we may assume that ϕ := ϕ_{D} is a morphism. Since ϕ is birational, its image Z is a non-degenerate subvariety of projective space ℙ^{N} and hence its degree is at least 1. From the inclusion

ϕ^{∗}O_{ℙ^{N}}(1)|Z ↪ O_{X}(D),

one sees that vol(D) ≥ vol(O_{ℙ^{N}}(1)|Z) ≥ 1. □

2.3. Log Canonical Centres. Here we recall some well known results regarding log canonical centres. The first Lemma shows that when the volume of a ℚ-divisor L is big, it is possible to produce a numerically equivalent ℚ-divisor Δ_{x} with high multiplicity at a general point x.

Lemma 2.3. Let V be an irreducible projective variety of dimension d, L a big ℚ-Cartier divisor on V, and x ∈ V a smooth point. If for a positive rational number α, one has

vol(L) > α^{d},

then for any sufficiently divisible integer k ≫ 0, there exists a divisor

A = A_{x} ∈ |kL|  with  mult_{x}(A) > kα.

This is easily seen by a parameter count comparing the number of sections in O_{V} (kL) and the number of conditions one needs to impose for a function to vanish at a smooth point to order at least kα. The next lemma shows that if the multiplicity of a ℚ-divisor at a given point is sufficiently big, then the given point belongs to an appropriate log canonical centre.

Lemma 2.4. Let (X, Δ) be a log pair, x a smooth point of X. If mult_{x}(Δ) ≥ dim X, then LLC(X, Δ, x) ≠ ∅. If mult_{x}(Δ) < 1, then LLC(X, Δ, x)=∅.

Proof. See [14] 9.3.2 and 9.5.13. □

Assume that (X, Δ) is log canonical at x. It is often useful to assume that Z = LLC(X, Δ, x) is an irreducible variety. To this end recall (cf.

[9] and [1] (3.4))

Lemma 2.5. Let X be a normal variety and let Δ be an effective ℚ-Cartier divisor such that K_{X} +Δ is ℚ-Cartier. Assume that x ∈ X is a kawamata log terminal point of X and that (X, Δ) is log canonical near x ∈ X. If W_{1}, W_{2} ∈ LLC(X, Δ, x), and W is an irreducible component of W_{1} ∩ W_{2} containing x, then W is in LLC(X, Δ, x). Therefore, if (X, Δ) is not kawamata log terminal at x, then LLC(X, Δ, x) has a unique minimal irreducible element, say V. Moreover, there exists an effective ℚ-divisor E such that

LLC(X, (1 − ϵ)Δ + ϵE, x) = {V }

for all 0 < ϵ ≪ 1. We may also assume that there is a unique log canonical place lying above V. If X is projective, x ∈ X is general and L is a big divisor, then one can take E = aL for some positive number a.

2.4. Non-vanishing and birational maps. Recall the following well known method for producing sections of adjoint line bundles:

Lemma 2.6. Let X be a smooth projective variety and D a big divisor on X. Let 0 < λ < 1 a rational number and assume that for every general point x ∈ X, there exists Δ = Δ_{x} ∼_{ℚ} λD a ℚ-divisor such that x is a maximal (in terms of inclusion) element of LCS(X, Δ, x). Then h^{0}(O_{X}(K_{X} + D)) > 0 and h^{0}(O_{X}(K_{X} + 2D)) ≥ 2.

Proof. For m ≫ 0, let G be a general divisor in |mD|. Then x is not contained in the support of G and so x is the only element of LLC(X, Δ + \frac{1−λ}{m}G, x). Let μ: X′ ⟶ X be a log resolution of (X, Δ + \frac{1−λ}{m}G) and let

I := I(X, Δ + \frac{1 − λ}{m}G) := μ_{∗}O_{X′}(K_{X′/X} − ⌞μ^{∗}(Δ + \frac{1 − λ}{m}G)⌟).

We may assume that μ^{∗}|mD| = F + |M| where F has simple normal crossings and |M| is base point free and hence M is nef and big. Then

μ^{∗}D − ⌞μ^{∗}(Δ + \frac{1 − λ}{m}G)⌟ ∼_{ℚ} μ^{∗}D − ⌞μ^{∗}(Δ + \frac{1 − λ}{m}F)⌟ ∼_{ℚ}

\{μ^{∗}(Δ + \frac{1 − λ}{m}F)\} + \frac{1 − λ}{m}M.

So μ^{∗}D−⌞μ^{∗}(Δ+ \frac{1−λ}{m}G)⌟ is numerically equivalent to the sum of a fractional divisor with simple normal crossings and a nef and big divisor. By Kawamata-Viehweg vanishing, one sees that R^{i}μ_{∗}O_{X′}(K_{X′} +μ^{∗}D − ⌞μ^{∗}(Δ + \frac{1−λ}{m}G)⌟) = 0 for i > 0. So there is a short exact sequence

0 ⟶ O_{X}(K_{X}+D)⊗I ⟶ O_{X}(K_{X}+D) ⟶ \frac{O_{X}(K_{X} + D)}{O_{X}(K_{X} + D) ⊗ I} ⟶ 0.

Again by Kawamata-Viehweg vanishing, one has that

h^{1}(O_{X}(K_{X} +D)⊗I) = h^{1}(O_{X′}(K_{X′} +μ^{∗}D −⌞μ^{∗}(Δ+ \frac{1 − λ}{m}G)⌟)) = 0.

Since x is the only element of LLC(X, Δ + \frac{1−λ}{m}G, x), one has that ℂ_{x} is a direct summand of \frac{O_{X}(K_{X}+D)}{O_{X}(K_{X}+D)⊗I} and so O_{X}(K_{X} + D) is generated at x.

Pick a general point x_{1}. Then we may find D_{1} ∼_{ℚ} λD with an isolated log canonical centre at x_{1}. Now pick a general point x_{2}, not in the support of D_{1}, and pick a general divisor D_{2} ∼_{ℚ} λD with an isolated log canonical centre at x_{2}. As x_{2} and D_{2} are general, D_{2} does not contain x_{1}. Thus x_{1} and x_{2} are isolated log canonical centres of D_{1} + D_{2} ∼_{ℚ} 2λD.

The rest of the proof that h^{0}(O_{X}(K_{X} + 2D)) ≥ 2 now follows the line above. □

Given a line bundle with at least two sections, one has a rational map to ℙ^{1}. The following Lemma gives a way to produce birational maps.

Lemma 2.7. Let X be a smooth projective variety. Suppose that h^{0}(X, O_{X}(mK_{X})) ≥ 2 for all m ≥ m_{0} and let X′ ⟶ ℙ^{1} be any morphism induced by sections of O_{X}(m_{0}K_{X}) on an appropriate birational model X′ of X. If F denotes the general fiber and |sK_{F} | induces a birational morphism, then |tK_{X}| induces a birational morphism for all t ≥ m_{0}(2s + 2) + s.

Proof. Following Theorem 4.6 of [11] and its proof, one sees that |(m_{0}(2s+ 1) + s)K_{X}| gives a birational map. Since mK_{X} is effective for all m ≥ m_{0}, the assertion follows. □

It is also possible to prove a converse to (2.6):

Lemma 2.8. Let (X, Δ) be a log pair of dimension n, and let D be an integral Weil divisor such that the image Y of the rational map ϕ_{D} has dimension n.

Then there is an open set U such that for any point x ∈ U, we may find Δ′ ∼ nD, such that x is the only element of LLC(X, Δ + Δ′).

Proof. Let U be the complement of the singular locus, the support of Δ, and the locus where the map ϕ_{D} is not an étale morphism.

Pick x ∈ X and let y ∈ Y be a point of the image ϕ_{D}: X ⟶ Y. By assumption, Y sits inside projective space ℙ^{N}. Pick n + 1 general hyperplanes which contain y, H_{1}, H_{2}, …, H_{n+1} and n + 1 rational numbers a_{1}, a_{2}, …, a_{n+1}, such that

0 < a_{i} < 1 and  ∑_{i} a_{i} = n.

Set

Γ = ∑ a_{i}H_{i}.

Then y is the only log canonical centre of K_{Y} + Γ. Let W be the normalisation of the graph of ϕ = ϕ_{D}, so that there is a commutative diagram

W

[figure: p; q]

X  ^{ϕ  ^{}}Y.

Let D_{i} = p_{∗}q^{∗}H_{i}, and set

Δ′ = ∑ a_{i}D_{i}.

Then each D_{i} is linearly equivalent to D, so that Δ′ is linearly equivalent to nD. On the other hand, as ϕ is étale over y, it follows that x is an isolated log canonical centre of K_{X} + Δ + Δ′. □

2.5. Log additivity of the Kodaira dimension.

Lemma 2.9. Let X and Y be smooth projective varieties and Δ an effective ℚ-divisor on X with simple normal crossings support. Let π: X ⟶ Y be a morphism and suppose that K_{X} + Δ is log canonical on the general fiber W of π, and κ(W, (K_{X} + Δ)|_{W}) ≥ 0. Then

π_{∗}O_{X}(m(K_{X/Y} + Δ)),

is weakly positive, for any m which is sufficiently divisible. In particular for any ample line bundle H on Y and any rational number ϵ > 0, one has that κ(K_{X/Y} + Δ + ϵπ^{∗}H) ≥ dim Y.

Proof. It suffices to prove that

π_{∗}O_{X}(m(K_{X/Y} + Δ_{h})),

is weakly positive, where Δ_{h} consists of those components of Δ which dominate Y. In fact both [4], (4.13) and [15], (9.8) prove that this sheaf is weakly positive, as soon as mΔ_{h} is integral. □

Remark 2.10. In fact both [4] and [15] prove a much stronger result, by putting an orbifold structure on the base which takes into account the multiple fibers of the morphism π. We will not need this stronger form.

The following consequences of (2.9) are well known:

Corollary 2.11. Let X, Y, Δ, H be as in (2.9).

(1) If K_{Y} is pseudoeffective, then for all rational numbers ϵ > 0

κ(K_{X} + Δ + ϵπ^{∗}H) ≥ dim Y.

(2) If Y is of general type, κ(K_{X} + Δ) ≥ dim Y.

Corollary 2.12. Let X, Y, and Δ be as in (2.9).

Then there exists an ample line bundle A on X such that for all m sufficiently big and divisible, h^{0}(X, O_{X}(m(K_{X/Y} + Δ)) + A) ≥ 0.

Proof. Let n = dim X. Fix a very ample line bundle H on X such that A = K_{X} + (n + 2)H is also ample. By (2.9), for all m sufficiently big and divisible, κ(m(K_{X/Y} + Δ) + H) ≥ 0. By [14] 11.2.13, one sees that h^{0}(X, O_{X}(K_{X} + (n + 2)H + m(K_{X/Y} + Δ))) ≥ 0. □

3. Lifting sections

In this section, we prove that certain log pluricanonical forms extend from a divisor to the ambient space. We follow the exposition of [10] very closely, which proves that pluricanonical forms extend, indicating those places where the definitions and proofs have to be modified.

Definition 3.1. Let (X, Δ) be a smooth log pair, where every component of Δ has coefficient one and let D be a divisor on X. Let μ: W ⟶ X be a birational morphism from a smooth variety W and let Θ be the support of all divisors of log discrepancy zero. Assume that there is a decomposition

μ^{∗}D = P + M,

in Div(W) ⊗ ℝ such that P is μ-nef and M is effective, where Θ ∪ M ∪ {the exceptional set of μ} has normal crossing support, and Θ and M have no components in common. The multiplier ideal sheaf I_{Δ,M} is defined by the following formula

I_{Δ,M} = μ_{∗}O_{W} (K_{W/X} + Θ − μ^{∗}Δ − ⌞M⌟).

Note that any component of the strict transform of Δ is automatically a component of Θ. Equivalently, in the notation analogous to [10], one has that

μ_{∗}O_{W} (⌜P ⌝ + K_{W} + Θ) = I_{Δ,M}(D + K_{X} + Δ).

When Δ = 0, this reduces to the usual definition of I_{M}. The notation ⌜P ⌝ is somewhat misleading as ⌜P ⌝ ≠ ⌜P′⌝ for two ℚ-linearly equivalent divisors. Therefore we replace it by μ^{∗}D − ⌞M⌟.

Remark 3.2. The definition of Θ is motivated by two conflicting requirements:

(1) On the one hand, we require

(K_{W} + Θ) − μ^{∗}(K_{X} + Δ),

to be effective and exceptional. This forces us to include any divisors of log discrepancy zero into Θ.

(2) On the other hand, to be able to apply vanishing, we will need the supports of M and Θ to have no components in common. Thus we cannot allow any divisor in Θ, which belongs to M.

Note that I_{Δ,M} is a coherent sheaf of ideals of O_{X} which is determined only by M, Δ and μ. Since (K_{W} +Θ)−μ^{∗}(K_{X} +Δ) and μ^{∗}Δ−Θ are effective, one has that

(∗)  μ_{∗}O_{W} (−⌞M⌟) ⊂ I_{Δ,M} ⊂ I_{M}.

We will need the following easy fact:

Lemma 3.3. Let ν: W′ ⟶ W be a birational morphism of smooth projective varieties, G a reduced divisor on W, Γ a ℚ-divisor on W with ⌞Γ⌟ ≤ 0 and such that G + Γ has simple normal crossings support and where G, Γ have no components in common. Then (W, G + Γ) is log canonical and the set of divisors in W′ of log discrepancy 0 for (W, G + Γ) and for (W, G) coincide.

Proof. See 2.31 of [13]. □

Lemma 3.4. Notation as in (3.1). If ν: W′ ⟶ W is a birational morphism, such that

ν^{∗}M ∪ (ν ∘ μ)^{∗}Δ ∪ {exceptional locus of (ν ∘ μ)},

has simple normal crossings, then I_{Δ,ν^{∗}M} is also defined and I_{Δ,M} = I_{Δ,ν^{∗}M}.

Proof. Let Θ′ be the sum of all the divisors in W′ of log discrepancy zero for (X, Δ). Then Θ′ is also the sum of all divisors in W′ of log discrepancy zero for (W, μ^{∗}Δ − K_{W/X}) and so for (W, Θ) (as one has μ^{∗}Δ − K_{W/X} = Θ − F where F ≥ 0 has no component in common with Θ and Θ + F has simple normal crossings support, and so (3.3) applies). Since M and Θ have no common components and M + Θ has simple normal crossings, one sees that M contains no log canonical centres of (W, Θ). Thus ν^{∗}M has no components in common with Θ′ and so I_{Δ,ν^{∗}M} is defined. Note that

M = ⌞M⌟ + {M}  so that,
ν^{∗}M = ν^{∗}⌞M⌟ + ν^{∗}{M}  and so,
⌞ν^{∗}M⌟ = ν^{∗}⌞M⌟ + ⌞ν^{∗}{M}⌟.

By definition

I_{Δ,ν^{∗}M} = (ν ∘ μ)_{∗}O_{W′}(K_{W′/X} + Θ′ − (ν ∘ μ)^{∗}Δ − ⌞ν^{∗}M⌟) =

μ_{∗} (O_{W} (K_{W/X} + Θ − μ^{∗}Δ − ⌞M⌟) ⊗ ν_{∗}O_{W′}(K_{W′/W} + Θ′ − ν^{∗}Θ − ⌞ν^{∗}{M}⌟)) .

Suppose that

K_{W′/W} + Θ′ − ν^{∗}Θ − ⌞ν^{∗}{M}⌟,

is effective and exceptional for ν. Then

ν_{∗}O_{W′}(K_{W′/W} + Θ′ − ν^{∗}Θ − ⌞ν^{∗}{M}⌟) = O_{W} ,

and the lemma follows. We may write

K_{W′} + Θ^{∗} = ν^{∗}(K_{W} + Θ + {M}) + E,

where Θ^{∗} is the reduced divisor whose components are the divisors on W′ of log discrepancy zero for (W, Θ + {M}) and hence for (W, Θ). By (3.3), Θ^{∗} = Θ′ and so

K_{W′/W} + Θ′ − ν^{∗}Θ − ⌞ν^{∗}{M}⌟ = ⌜E⌝,

is effective and exceptional as desired. □

It is important to be able to compare multiplier ideal sheaves. Here is the first basic result along these lines:

Lemma 3.5. Let (X, Δ) be a smooth log pair, where every component of Δ has coefficient one and let D be a divisor on X. Assume that there is a birational map μ: W ⟶ X with the properties of (3.1). Let Δ′ ≤ Δ be another divisor, where every component of Δ′ has coefficient one.

Then

I_{Δ,M} ⊂ I_{Δ′,M}.

Proof. Let F be an exceptional divisor extracted by μ. Then the coefficient of F in

(K_{W} + Θ) − μ^{∗}(K_{X} + Δ),

is either the log discrepancy minus one, whenever the log discrepancy is at least one, or it is zero. In particular the coefficient of F is an increasing function of the log discrepancy of F. It follows easily that

(K_{W} + Θ) − μ^{∗}(K_{X} + Δ) ≤ (K_{W} + Θ′) − μ^{∗}(K_{X} + Δ′),

which in turn gives the inclusion of ideal sheaves. □

Theorem 3.6. Let (X, Δ) be a smooth log pair, where every component of Δ has coefficient one and let π: X ⟶ S be a projective morphism of algebraic varieties. Let M be an integral divisor such that M ≡_{π} L + G where

(1) L is π-nef and there is an effective ℚ-divisor B such that A = L − B is π-ample,

(2) Δ and G ∪ B have no components in common, ⌞G⌟ = 0 and (3) the support of Δ ∪ B ∪ G has simple normal crossings. Then R^{p}π_{∗}O_{X}(K_{X} + Δ + M) = 0, for all p > 0.

Proof. As L is π-nef, L−δB is π-ample, for any δ > 0. Pick ϵ > 0 such that ϵΔ + L − δB is π-ample. As Δ and G ∪ B have no components in common and every component of Δ has coefficient one, we see that (X, (1 − ϵ)Δ + G + δB) is kawamata log terminal for any δ sufficiently small. But then, since

Δ + M ≡_{π} (ϵΔ + L − δB) + (1 − ϵ)Δ + G + δB,

we may apply Kawamata-Viehweg vanishing. □

We fix some notation:

(1) (Y, Γ) is a smooth log pair,

(2) the divisor X ⊂ Y is a component of Γ of coefficient one,

(3) (Y, Γ) is log canonical,

(4) S is the germ of an algebraic variety,

(5) π: Y ⟶ S is a projective morphism,

(6) Δ is the restriction of Γ − X to X.

Note that we make no requirement about how X sits in Y, with respect to π. We will also assume that every coefficient of Γ has coefficient one, until (3.16).

Definition 3.7. A divisor D on Y will be called π-effective for the pair (Y, X) if the natural homomorphism

π_{∗}O_{Y} (D) ⟶ π_{∗}O_{X}(D),

is not zero. It is called π-ℚ-effective for the pair (Y, X) if mD is π-effective for the pair (Y, X) for some positive integer m. D is said to be π-big for the pair (Y, X) if we can write mD = A + B for a positive integer m, a π-ample divisor A and a π-effective divisor B for the pair (Y, X).

A divisor D on X will be called π-transverse for (X, Δ) if the sheaf O_{X}(D) is π-generated at the generic point of every log canonical centre of K_{X} + Δ. D will be called π-ℚ-transverse for (X, Δ) if mD is π-transverse for some m > 0.

Now we turn to the definition of sheaves J_{Δ,D}^{0}, I_{Δ,D}^{0}, J_{Δ,D}^{1}, I_{Δ,D}^{1}, which generalize the sheaves J_{D}^{0}, I_{D}^{0}, J_{D}^{1}, I_{D}^{1} of Definition 2.10 of [10].

We will need a:

Definition 3.8. Let (X, Δ) be a log pair and let μ: W ⟶ X be a birational morphism. We say that μ is canonical if every exceptional divisor extracted by μ has log discrepancy at least one.

If (X, Δ) is a pair consisting of a smooth variety and a divisor with simple normal crossings support, then note that μ is canonical provided it is an isomorphism in a neighbourhood of the generic point of any irreducible component of the intersection of the components of Δ, that is provided it is an isomorphism in a neighbourhood of any log canonical centre. Note also that if μ is canonical, then Θ is the strict transform of Δ.

Definition 3.9. Let D be a π-ℚ-transverse divisor for (X, Δ). For each positive integer m, such that mD is π-transverse, pick a proper birational canonical morphism μ_{m}: W_{m} ⟶ X from a smooth variety W_{m}, such that

(1) there is a decomposition μ_{m}^{∗}(mD) = P_{m} +M_{m} in Div(W_{m}) such that the natural homomorphism

(π ∘ μ_{m})_{∗}O_{W_{m}}(P_{m}) ⟶ π_{∗}O_{X}(mD),

is an isomorphism,

(2) P_{m} is π ∘ μ_{m}-free, and (3) M_{m} ∪Θ_{m} ∪{the exceptional set of μ_{m}} is effective, with normal

crossing support, where Θ_{m} is defined as in (3.1). We define

J_{Δ,D}^{0} = ⋃ I_{Δ, \frac{1}{m} M_{m}},

where the union is taken over all positive integers m such that mD is π-transverse. If D itself is π-transverse then we set I_{Δ,D}^{0} = I_{Δ,M_{1}}.

Now suppose that D is a π-ℚ-transverse divisor on Y for (Y, Γ). For every positive integer m, such that mD is a π-transverse divisor for (Y, Γ), pick a birational morphism μ_{m}: W_{m} ⟶ Y from a smooth variety W_{m}, such that

(1) there is a decomposition μ_{m}^{∗}(mD) = Q_{m} +N_{m} in Div(W_{m}) such that the natural homomorphism

(π ∘ μ_{m})_{∗}O_{W_{m}}(Q_{m}) ⟶ π_{∗}O_{Y} (mD),

is an isomorphism,

(2) Q_{m} is π ∘ μ_{m}-free, and

(3) N_{m} is effective, and X_{m}+N_{m}+Θ_{m}+{the exceptional set of μ_{m}} has normal crossing support, where X_{m} is the strict transform of X and Θ_{m} is defined as in (3.1).

By assumption X_{m} is not contained in the support of N_{m}. We define

J_{Δ,D}^{1} = ⋃ I_{Δ, \frac{1}{m} N_{m}|_{Xm}},

where the union is taken over all positive integers m such that mD is π-transverse for the pair (Y, Γ). If D itself is π-transverse for (Y, Γ), then we set I_{Δ,D}^{1} = I_{Δ,N_{1}|_{X_{1}}}.

Remark 3.10. We will not make use of the sheaves I_{Δ,D}^{i}, for i = 0 and 1. We have included their definition for the sake of completeness.

Lemma 3.11. Let D be a π-ℚ-transverse divisor for (Y, Γ) (respectively for (X, Δ)). Then the sheaf J_{Δ,D}^{i} is defined, it does not depend on the resolution μ_{m} and we may choose μ_{m} to be canonical for any m sufficiently divisible, where i = 1 (respectively i = 0). Further, we may assume that there is a μ_{m}-ample divisor whose support is exceptional.

Proof. This is an easy consequence of (3.4) and of the Resolution Lemma of [21]. □

Lemma 3.12. Let D be a π-ℚ-transverse divisor for (Y, Γ). (1) J_{Δ,D}^{1} ⊂ J_{Δ,D|_{X}}^{0} (2) If B is an effective integral divisor, which does not contain X

and D + B is π-ℚ-transverse for (Y, Γ), then

J_{Δ,D}^{i}(−B) ⊂ J_{Δ,D+B}^{i}, i = 0, 1.

(3) There is a positive integer m such that

J_{Δ,D}^{0} = I_{Δ, \frac{1}{m} M_{m}}  and  J_{Δ,D}^{1} = I_{Δ, \frac{1}{m} N_{m}|_{Xm}}.

Proof. (1) follows, since N_{m}|_{X_{m}} ≥ M_{m} as divisors on X_{m}. (2) is clear.

(3) follows as X is Noetherian. □

Lemma 3.13. Let D be a π-ℚ-transverse divisor for (Y, X, Γ). Then (1) J_{Δ,αD}^{i} ⊂ J_{Δ,D}^{i}, for any rational number α ≥ 1 and i = 0, 1.

(2) J_{Δ,D}^{i} ⊂ J_{Δ,D+L}^{i}, for any π-free divisor L, and i = 0, 1.

(3) Im(π_{∗}O_{Y} (D) ⟶ π_{∗}O_{X}(D)) ⊂ π_{∗}J_{Δ,D}^{1}(D).

Proof. First of all, notice that Definition (3.9) also makes sense for ℚdivisors. (1) and (2) are clear. We may assume that μ_{m} is chosen so that μ_{m}^{∗}D = P + M in Div(W_{m}) and

(π ∘ μ_{m})_{∗}O_{W_{m}}(P) ⟶ π_{∗}O_{Y} (D),

is an isomorphism. Since M > M_{m}/m one sees that there are inclusions

O_{W_{m}}(μ_{m}^{∗}D − M) ⊂ O_{W_{m}}(μ_{m}^{∗}D − ⌞M_{m}/m⌟) ⊂ O_{W_{m}}(μ_{m}^{∗}D).

We now push forward via (π ∘ μ_{m}). Since P = μ_{m}^{∗}D − M, both the left and right hand sides push forward to π_{∗}O_{Y} (D). So the image of π_{∗}O_{Y} (D) is contained in the image of (π ∘ μ_{m})_{∗}O_{W_{m}}(μ_{m}^{∗}D − ⌞M_{m}/m⌟) and hence in π_{∗}J_{Δ,D}^{1}(D) (cf. equation (∗), after (3.2)). □

The following lemma is the key result that will allow us to generalize the results of Siu and Kawamata to the setting of log-forms:

Lemma 3.14. Suppose that D is a divisor on Y and suppose that there is an effective divisor B on Y such that

(1) A = D − B is π-ample, and

(2) B is π-ℚ-transverse for (Y, Γ). Then

π_{∗}J_{Δ,D}^{1}(D+K_{X}+Δ) ⊂ Im(π_{∗}O_{Y} (D + K_{Y} + Γ) ⟶ π_{∗}O_{X}(D + K_{X} + Δ)).

Proof. Notation as in (3.9). There is an exact sequence

0 ⟶ O_{W_{m}}(K_{W_{m}}+Θ_{m}−X_{m}+μ_{m}^{∗}D−⌞\frac{N_{m}}{m} ⌟) ⟶ O_{W_{m}}(K_{W_{m}}+Θ_{m}+μ_{m}^{∗}D−⌞\frac{N_{m}}{m} ⌟)

⟶ O_{X_{m}}(K_{X_{m}} + (Θ_{m} − X_{m})|_{X_{m}} + (μ_{m}^{∗}D − ⌞\frac{N_{m}}{m} ⌟)|_{X_{m}}) ⟶ 0.

Claim 3.15. If m is sufficiently large then

R^{1}(π ∘ μ_{m})_{∗}(O_{W_{m}}(K_{W_{m}} + Θ_{m} − X_{m} + μ_{m}^{∗}D − ⌞\frac{N_{m}}{m} ⌟)) = 0.

Proof of Claim. By (3.11), we may assume that μ_{m} is canonical and there is an effective divisor E supported on the exceptional locus such that μ_{m}^{∗}A − E is π ∘ μ_{m}-ample. As μ_{m} is canonical, E + {N_{m}/m} and Θ_{m} − X_{m} have no common component. Since B is π-ℚ-transverse, we may assume that μ_{m}^{∗}(mB) ∼_{π,ℚ} N′ where N′ and Θ_{m} − X_{m} have no common component and N_{m} + N′ + Θ_{m} + {exceptional locus of μ_{m}} has simple normal crossings support. Notice that since D − B is πample, we have that D is π-ℚ-transverse and that N_{m} ≤ N′. We now apply (3.6) to Δ = Θ_{m} − X_{m}, M = μ_{m}^{∗}D − ⌞N_{m}/m⌟, G = {N_{m}/m} and L = Q_{m}/m. Notice that L − (\frac{N′−N_{m}}{m} + E) ∼_{π∘μ_{m}} μ_{m}^{∗}A − E is π ∘ μ_{m}-ample and \frac{N′−N_{m}}{m} + E is effective. The hypothesis of (3.6) are now verified and so

R^{p}(π ∘ μ_{m})_{∗}O_{W_{m}}(K_{W_{m}} + Θ_{m} − X_{m} + μ_{m}^{∗}D − ⌞\frac{N_{m}}{m} ⌟)

vanishes for p > 0. □

Thus there is a surjective homomorphism

π_{∗}O_{Y} (D + K_{Y} + Γ) ⊃ (π ∘ μ_{m})_{∗}O_{W_{m}}(K_{W_{m}} + Θ_{m} + μ_{m}^{∗}D − ⌞\frac{N_{m}}{m} ⌟)

⟶ (π∘μ_{m})_{∗}O_{X_{m}}(K_{X_{m}}+Θ_{m}−X_{m}+μ_{m}^{∗}D−⌞\frac{N_{m}}{m} ⌟) = J_{Δ,D}^{1}(D+K_{X}+Δ),

hence the assertion. □

Theorem 3.16. Let X ⊂ Y be a smooth divisor in a smooth variety. Fix a positive integer e. Let H be a sufficiently π-very ample divisor and set A = (dim X + 1)H. Assume that

(1) Γ is a ℚ-divisor with simple normal crossings support such that Γ contains X with coefficient one and (Y, Γ) is log canonical;

(2) k is a positive integer such that kΓ is integral;

(3) C ≥ 0 is an integral divisor not containing X;

(4) Given D = k(K_{Y} +Γ), D′ = D|_{X} = k(K_{X} +Δ) and G = D+C, then D′ is π-ℚ-pseudo-effective (i.e. D′ + ϵH′ is π-ℚ-effective for any ϵ > 0) and that G is π-ℚ-transverse for (Y, ⌜Γ⌝).

Then

J_{mD′+eH′}^{0}(−mC) ⊂ J_{mG+eH+A}^{1}, ∀m > 0,

where H′ := H|_{X}.

Proof. It suffices to prove the stronger inclusion

J_{mD′+eH′}^{0}(−mC) ⊂ J_{⌜Δ⌝,mG+eH+A}^{1}.

This is a modification of the proof of Lemma 3.7 of [10], which we follow quite closely. We prove the inclusion by induction on m, in the case e = 1, the general case being left to the reader.

By assumption mD′ + H′ is π-big, C does not contain X and mG + H + A is π-ℚ-transverse for (Y, ⌜Γ⌝) for any non-negative integer m. In particular, both sides of the proposed inclusion are well-defined.

If m = 0 the result is clear. Assume the result for m. Suppose that

Γ = ∑_{j} γ_{j}Γ(j), where 0 < γ_{j} := \frac{g_{j}}{k} ≤ 1.

Then set

Γ_{i} = ∑_{j} γ_{i}(j)Γ(j) where γ_{i}(j) = \begin{cases}0 & if 1 ≤ i ≤ k − g_{j}, \\ 1 & if k − g_{j} < i ≤ k + 1. \end{cases}

With this choice of Γ_{i}, we have

X ≤ Γ_{1} ≤ Γ_{2} ≤ Γ_{3} ≤ ⋯ ≤ Γ_{k} = Γ_{k+1} = ⌜Γ⌝,

and each Γ_{i} is integral. Let Δ_{i} be the restriction of Γ_{i} − X to X. Set

D_{i} = K_{Y} + Γ_{i} and D_{≤i} = ∑_{j≤i} D_{j}.

With this choice of D_{i}, we have D = D_{≤k}. Possibly replacing H by a multiple, we may assume that H_{i} = D_{≤i−1} + Γ_{i} − X + H and A_{i} = H + D_{≤i} + A are π-ample, for 1 ≤ i ≤ k. We are now going to prove, by induction on i, that

J_{mD′+H′}^{0}(−mC) ⊂ J_{Δ_{i+1},mG+D_{≤i}+H+A}^{1} where 0 ≤ i ≤ k,

where we adopt the convention that D_{≤0} = 0.

Since Δ_{1} ≤ ⌜Δ⌝, it follows by (3.5)

J_{⌜Δ⌝,mG+H+A}^{1} ⊂ J_{Δ_{1},mG+H+A}^{1},

and so the case i = 0 follows from the inclusion

J_{mD′+H′}^{0}(−mC) ⊂ J_{⌜Δ⌝,mG+H+A}^{1},

which we are assuming by induction on m. Assume the result up to i − 1. Then

π_{∗}J_{mD′+H′}^{0}(mD + D_{≤i} + H + A)

= π_{∗}J_{mD′+H′}^{0}(−mC)(mG + D_{≤i} + H + A)

⊂ π_{∗}J_{Δ_{i},mG+D_{≤i−1}+H+A}^{1}(mG + D_{≤i} + H + A)

= π_{∗}J_{Δ_{i},mG+D_{≤i−1}+H+A}^{1}(mG + D_{≤i−1} + H + A + K_{Y} + Γ_{i})

⊂ Im(π_{∗}O_{Y} (mG + D_{≤i} + H + A) ⟶ π_{∗}O_{X}(mG + D_{≤i} + H + A))

⊂ π_{∗}J_{Δ_{i+1},mG+D_{≤i}+H+A}^{1}(mG + D_{≤i} + H + A),

where we use induction to get the inclusion of the second line in the third line, (3.14) applied to the basic identity

mG + D_{≤i−1} + H + A = A_{i−1} + B,

to get the inclusion of the fourth line in the fifth line, and (3) of (3.13) to get the inclusion of the fifth line in the sixth line. As

J_{mD′+H′}^{0}(mD+D_{≤i}+H+A) = J_{mD′+H′}^{0}((mD+H)+H_{i}+(dim X)H+K_{X}),

is π-generated by (a trivial generalization of) (3.2) of [10], it follows that

J_{mD′+H′}^{0}(−mC) ⊂ J_{Δ_{i+1},mG+D_{≤i}+H+A}^{1}.

This completes the induction on i. It follows that

J_{mD′+H′}^{0}(−mC) ⊂ J_{Δ_{k+1},mG+D_{≤k}+H+A}^{1},

so that, by (2) of (3.12),

J_{mD′+H′}^{0}(−(m + 1)C) ⊂ J_{⌜Δ⌝,(m+1)G+H+A}^{1}.

But by (1) and (2) of (3.13),

J_{(m+1)D′+H′}^{0} ⊂ J_{mD′+H′}^{0}.

Thus

J_{(m+1)D′+H′}^{0}(−(m + 1)C) ⊂ J_{⌜Δ⌝,(m+1)G+H+A}^{1},

and this completes the induction on m and the proof. □

Corollary 3.17. Let X ⊂ Y be a smooth divisor in a smooth variety. Let H be a sufficiently π-very ample divisor and set A = (dim X +1)H. Assume that

(1) Γ is a ℚ-divisor with simple normal crossings support such that Γ contains X with coefficient one and (Y, Γ) is log canonical; (2) C ≥ 0 is a ℚ-divisor not containing X;

(3) K_{X} + Δ is π-ℚ-pseudo-effective, where Δ = (Γ − X)|_{X} and

K_{Y} + Γ + C is π-ℚ-transverse for (Y, ⌜Γ⌝). For any positive integer m, such that m(K_{Y} + Γ) is integral, the image of the natural homomorphism

π_{∗}O_{Y} (m(K_{Y} + Γ + C) + H + A) ⟶ π_{∗}O_{X}(m(K_{X} + Δ + C) + H + A),

contains the image of the sheaf π_{∗}O_{X}(m(K_{X} + Δ) + H) considered as a subsheaf of π_{∗}O_{X}(m(K_{X} + Δ + C) + H + A) by the inclusion induced by any divisor in mC + |A| not containing X.

Proof. Let H′ = H|_{X}. Fixing a section σ ∈ H^{0}(Y, O_{Y} (A)) not vanishing on X, we can view

J_{m(K_{X}+Δ)+H′}^{0}(m(K_{Y} + Γ) + H),

as a subsheaf of

J_{m(K_{X}+Δ)+H′}^{0}(m(K_{Y} + Γ) + H + A),

via the map induced by multiplication by σ. As we have seen in the proof of (3.16),

π_{∗}J_{m(K_{X}+Δ)+H′}^{0}(m(K_{Y} + Γ) + H + A) =

π_{∗}J_{m(K_{X}+Δ)+H′}^{0}(−mC)(m(K_{Y} + Γ + C) + H + A),

is contained in the image of π_{∗}O_{Y} (m(K_{Y} + Γ + C) + H + A). Since

π_{∗}O_{X}(m(K_{X} + Δ) + H) = π_{∗}J_{m(K_{X}+Δ)+H′}^{0}(m(K_{Y} + Γ) + H),

the assertion now follows easily. □

4. Lifting log canonical centres

We fix some notation for this section. Let (X, Δ) be a log pair, where X is ℚ-factorial, and Δ is an effective ℚ-divisor. Let V be an exceptional log canonical centre of K_{X} + Δ. Let f: W ⟶ V be a resolution of V. Let Θ be an effective ℚ-divisor on W. Suppose that there are positive rational numbers λ and μ such that Δ ∼ λK_{X} and Θ ∼ μK_{W}. Let ν = (λ + 1)(μ + 1) − 1. Suppose that W is of general type.

The main result of this section is:

Theorem 4.1. There is a very general subset U of V with the following property:

Suppose that W′ ⊂ W is a pure log canonical centre of K_{W} + Θ, whose image V′ ⊂ V intersects U.

Then, for every positive rational number δ, we may find an effective divisor Δ′ on X such that V′ is a pure log canonical centre of K_{X} +Δ′, where Δ′ ∼ (ν + δ)K_{X}. Now suppose that we may write K_{X} ∼ A + E, where A is ample, E is effective and V is not contained in E. Then we may assume that V′ is an exceptional log canonical centre of K_{X} + Δ′.

(4.1) is the main step in the proof of (1.1). In fact (1.1) is a standard consequences of (4.1), see §6.

The idea is to lift Θ as a ℚ-divisor, using (3.17). However this is more delicate than it might first appear, as we cannot lift Θ directly to X. Instead, we are able to find a sequence of successively better approximations Θ_{m}. In particular we have to pass to a resolution of X, and to this end, we introduce some more notation.

Let π: Y ⟶ X be a log resolution of (X, Δ). Let E be the unique exceptional divisor of log discrepancy zero with centre V. Possibly blowing up further, we may assume that the restriction p: E ⟶ V of π to E factors g: E ⟶ W through f. Thus we have a commutative diagram,

E_{ξ}  ⊂  E ^{⊂} Y

[figure: g]

W

[figure: f]

[figure: ξ; ∈; V; p; ⊂; X.; π]

Let ξ be the generic point of V and let E_{ξ} be the generic fibre of p.

Lemma 4.2. We may find a positive integer m_{0} and a positive integer s, with the following properties:

For every sufficiently large and divisible integer m, there is an effective divisor G_{m} on Y such that

(1) G_{m} does not contain E,

(2) G_{m} + sE ∼ (m_{0} + m(1 + λ))π^{∗}K_{X},

(3) G_{m}|_{E} = mg^{∗}Θ + B_{m}, where the restriction of B_{m} to the inverse image of a fixed open subset of V is effective, and

(4) B_{m}|_{E_{ξ}} belongs to a fixed linear system.

Proof. We may assume that

K_{Y} + Γ = π^{∗}(K_{X} + Δ) + F,

where Γ and F are effective, with no common components and F is exceptional. We may write Γ = E + Γ^{h} + Γ^{v}, F = F^{h} + F^{v} where the components of Γ^{h} and F^{h} are the components of Γ − E and F whose images contain V. As V is an exceptional log canonical centre for (X, Δ), {Γ^{h}} = Γ^{h} and Γ^{h}|_{E_{ξ}} = (Γ − E)|_{E_{ξ}} and so,

(K_{E} + Γ^{h})|_{E_{ξ}} = (K_{Y} + Γ)|_{E_{ξ}} = F |_{E_{ξ}} ≥ 0.

By Corollary (2.12), we may assume that

h^{0}(E, O_{E}(m(K_{E/W} + Γ^{h}) + H)) > 0,

where H is a sufficiently ample divisor on Y. Therefore, there is an injection

g^{∗}|mK_{W} | ⟶ |m(K_{E} + Γ^{h}) + H|,

induced by a choice of a divisor in |m(K_{E/W} + Γ^{h}) + H|.

We now apply (3.17). Possibly blowing up further, we may assume that the image of every element of LLC(Y, E+⌜Γ^{h}⌝) contains V. As the divisors E + ⌜Γ^{h}⌝ and F have no common components, it follows that K_{Y} + Γ = π^{∗}(K_{X} + Δ) + F is π-ℚ-transverse for (Y, E + ⌜Γ^{h}⌝). Notice also that as W is of general type, (K_{Y} + E + Γ^{h})|_{E} = (K_{E} + Γ^{h}|_{E}) is ℚ-effective, by (2.11). By Corollary (3.17), with S = Spec(ℂ), it follows that elements of

|m(K_{E} + Γ^{h}|_{E}) + H|_{E}| ↪ |m(K_{E} + (Γ − E)|_{E}) + H|_{E} + A|_{E}|,

can be lifted to elements of

|m(K_{Y} + Γ) + H + A| = |m(π^{∗}(K_{X} + Δ) + F) + H + A|.

Denote by S_{m} the lift of mg^{∗}Θ to |mμ(π^{∗}(K_{X} +Δ)+F)+H +A|. As K_{X} is big, there is a fixed positive integer m_{0} and an effective divisor

S_{0} ∈ |m_{0}π^{∗}K_{X} − H − A|.

Then

S_{m}+S_{0} ∈ |(m_{0}+m(1+λ)μ)π^{∗}K_{X}+mμF | = |(m_{0}+m(1+λ)μ)π^{∗}K_{X}|+mμF

so that we may write S_{m} + S_{0} = G′_{m} + mμF for some

G′_{m} ∈ |(m_{0} + m(1 + λ)μ)π^{∗}K_{X}|.

We may write

G′_{m} = G_{m} + sE,

where G_{m} does not contain E and s is a positive integer. Since S_{m} does not contain E and S_{0} is a fixed divisor, it is clear that s is fixed. Since G_{m} + mμF contains S_{m}, it follows that we may write

G_{m}|_{E} = mg^{∗}Θ + B_{m},

where the only components of B_{m} with negative coefficients, are supported on g^{∗}Θ. Thus (3) holds, and when we restrict to E_{ξ} we may ignore g^{∗}Θ. Thus

B_{m}|_{E_{ξ}} ∼ ((m_{0} + m(1 + λ)μ)π^{∗}K_{X} − sE)|_{E_{ξ}} = (−sE)|_{E_{ξ}}.
□

Proof of (4.1). Let U be any very general subset of V, such that (1) the morphism p restricted to p^{−1}(U) is smooth,

(2) every exceptional divisor whose centre intersects U, contains V,

(3) every component of Δ which intersects U, contains V, and (4) every component of B_{m} whose image intersects U, dominates V,

for every sufficiently large and divisible integer m, as in (4.2). Let G_{m} be the sequence of divisors whose existence is guaranteed by

(4.2). Set

Γ_{m} = G′_{m}/m, Θ_{m} = π_{∗}Γ_{m} and Δ_{m} = a_{m}Δ + Θ_{m},

where a_{m} ≤ 1 is chosen so that the log discrepancy of E with respect to K_{X} + Δ_{m} is zero.

Claim 4.3. V′ is a pure log canonical centre of K_{X} + Δ_{m}.

Proof of Claim. Since this result is local about the generic point of V, passing to an open subset of X, we may assume that properties (1-4) hold, and that B_{m} is effective. In this case

K_{Y} + E + F_{m} + G_{m}/m = π^{∗}(K_{X} + Δ_{m}) where
K_{Y} + c_{m}E + F_{m} = π^{∗}(K_{X} + a_{m}Δ),

F_{m} is a divisor, which does not contain E and c_{m} = 1 − s/m < 1 is a positive number. Restricting to E, we get

K_{E} + F_{m}|_{E} + g^{∗}Θ + B_{m}/m.

Let E′ be the inverse image of W′. Since the morphism p is smooth, E′ is a pure log canonical centre of K_{E} + g^{∗}Θ. Since V is an exceptional log canonical centre, we have ⌞Γ^{h}⌟ = 0. Thus K_{E} + Γ^{h} is kawamata log terminal. Since B_{m}|_{E_{ξ}} belongs to a fixed linear system and p is smooth over U, it follows that the restriction of B_{m} to any fibre of p over U, belongs to a fixed linear system. Thus K_{E} + Γ^{h}|_{E} + B_{m}/m is kawamata log terminal, for m large enough. As F_{m} ≤ Γ^{h}, it follows that K_{E} + F_{m}|_{E} + B_{m}/m is also kawamata log terminal, for m large enough. It follows that E′ is a pure log canonical centre of K_{E} +F_{m}|_{E} + g^{∗}Θ + B_{m}/m.

We now apply inversion of adjunction, see (17.7) of [12]. (17.7) is only stated for effective divisors. However it is easy to see that (17.6) and (17.7) of [12], extend to the case when the components of negative coefficient are contracted by a birational map, since this is the case for (17.4) of [12]. □

Clearly 0 ≤ a_{m} ≤ 1. Moreover, a_{m} approaches 1 as m approaches infinity, as the coefficient s/m of E in Γ_{m} approaches zero. Pick m so that m_{0}/m < δ/2. Then, using the decomposition K_{X} ∼ A + E, there is a perturbation Δ′ ∼ (ν +δ)K_{X} of Δ_{m}, such that V′ is an exceptional log canonical centre of K_{X} + Δ′. □

5. Effective birational freeness

In this section we give an effective bound for the rth pluricanonical map of a variety of general type, which only depends on the volume of K_{X} and the value of this constant for general subvarieties of X:

Proposition 5.1. Let X be a smooth variety of general type of dimension n. Let s′ be a positive integer such that for every subvariety V of X which contains a very general point of X, the map ϕ_{s′K_{V}} is birational. Set

s = (⌞\frac{n}{vol(K_{X})^{1/n}} ⌟ + 2) (ns′ + 1) .

Then ϕ_{rK_{X}} is birational, for all r ≥ 4s(s′ + 1) + s′.

Proof. By (2.3) and (2.4), for every very general point x ∈ X, we may find a ℚ-divisor Δ ∼_{ℚ} λK_{X} with

λ < ⌞\frac{n}{vol(K_{X})^{1/n}} ⌟ + 1,

such that LLC(X, Δ, x) ≠ ∅. By (2.5), we may assume that (X, Δ) is log canonical near x, that LLC(X, Δ, x) contains a unique element V and that there is a unique place of log canonical singularities dominating V. Let f: W ⟶ V be a resolution of V. Then, by (2.8), for a general point x′ ∈ V, we may find a divisor Θ on W, such that x′ is an isolated log canonical centre of K_{W} + Θ, where Θ ∼_{ℚ} ns′K_{W}. By (4.1) it follows that we may find Δ′ ∼ (s − 1)K_{X} such that x is an isolated point of K_{X} + Δ′. By (2.6) it follows that h^{0}(2sK_{X}) ≥ 2, and so the result follows from (2.7). □

Corollary 5.2. Let X be a smooth variety of general type of dimension n. Let s′ be a positive integer such that for every subvariety V of X which contains a very general point of X, the map ϕ_{s′K_{V}} is birational.

If the volume of K_{X} is at most v, then X is birational to a subvariety of degree at most

(8(ns′ + 1)(s′ + 1))^{n} max (n^{n}, 3^{n}v) .

Proof. By (5.1), ϕ_{rK_{X}} is birational, where

r = 4s(s′ + 1) + s′ and s = (⌞\frac{n}{vol(K_{X})^{1/n}} ⌟ + 2) (ns′ + 1) .

Now

vol(rK_{X}) = r^{n} vol(K_{X})

= (4s(s′ + 1) + s′)^{n} vol(K_{X}) ≤ ((4s + 1)(s′ + 1))^{n} vol(K_{X})

≤ (⌞\frac{n}{vol(K_{X})^{1/n}} ⌟ + 3)^{n} (4(ns′ + 1)(s′ + 1))^{n} vol(K_{X})

≤ max (\frac{n^{n}}{vol(K_{X})}, 3^{n}) (8(ns′ + 1)(s′ + 1))^{n} vol(K_{X})

≤ (8(ns′ + 1)(s′ + 1))^{n} max (n^{n}, 3^{n}v) ,

and so we are done by (2.2). □

6. Proof of (1.1)

Lemma 6.1. Let π: X ⟶ B be a bounded family of projective varieties of general type. Then there is a positive integer R such that if Y is a resolution of any fiber of π, then ϕ_{rK_{Y}} is birational for all r ≥ R.

Proof. Let Y be a fiber over the generic point ξ of a component of B. If we pick a projective resolution of Y, then we may extend this to an open neighbourhood of ξ. Thus, by Noetherian induction, possibly replacing B by a union of locally closed subsets, we may assume that π is projective and smooth. If ϕ_{rK_{Y}} is birational, then this map remains birational over an open neighbourhood of ξ, and we are done by Noetherian induction. □

Proof of (1.1). We proceed by induction on the dimension. Assume that (1.1) holds up to dimension n−1. By assumption there is a positive integer r_{n−1} such that ϕ_{rK_{W}} is birational, for any smooth variety W of general type of dimension at most n − 1, and for any r ≥ r_{n−1}.

Now if x is a very general point of X, and V is any subvariety containing x, then any resolution W of V is of general type. Hence if vol(K_{X}) ≥ 1, then we are done by (5.1). Thus we may assume that vol(K_{X}) < 1. By (5.2), it follows that X is birational to a subvariety of projective space of bounded degree. But then the set of all varieties of general type such that vol(K_{X}) < 1 is birationally bounded, and we are done by (6.1). □

Proof of (1.2). By (1.1) and (2.2) it follows that any smooth variety of general type of dimension n, where vol(K_{X}) ≤ M, is birational to a subvariety of projective space of degree at most (r_{n})^{n}M, and the result follows. □

Proof of (1.3). If vol(K_{X}) ≥ 1 there is nothing to prove, otherwise we apply (1.2) and (6.1). □

Proof of (1.4). This is an immediate consequence of the main result of [18]. □

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

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

Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA

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