EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE II

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

Abstract. Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

Contents

1. Introduction  1
2. Notation and conventions  6
3. Preliminary results  9
4. Multiplier ideal sheaves  11
5. Asymptotic multiplier ideal sheaves  15
6. Lifting sections  20
7. Rationality of the restricted algebra  23
8. Proof of (1.3)  25
References  26

1. Introduction

This is the second of two papers whose purpose is to establish:

Theorem 1.1. The canonical ring

R(X, K_{X}) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(mK_{X})),

is finitely generated for every smooth projective variety X.

Date: July 21, 2009.

The first author was partially supported by NSF research grant no: 0456363 and an AMS Centennial fellowship and the second author was partially supported by NSA grant no: H98230-06-1-0059 and NSF grant no: 0701101. We would like to thank F. Ambro, C. Birkar, P. Cascini, J. A. Chen, A. Corti, O. Fujino, S. Keel, J.

Kollár and the referee for valuable suggestions.

Note that Siu has announced a proof of finite generation for varieties of general type, using analytic methods, see [13].

Our proof relies on the ideas and techniques of the minimal model program and roughly speaking in this paper we will show that finite generation in dimension n−1 implies the existence of flips in dimension n. To be more precise, we first recall some definitions:

Definition 1.2. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, and let V be a finite dimensional affine subspace of the real vector space WDiv_{ℝ}(X) of Weil divisors on X. Fix an ℝ-divisor A ≥ 0 and define

V_{A} = { Δ | Δ = A + B, B ∈ V },
L_{A}(V) = { Δ = A + B ∈ V_{A} | K_{X} + Δ is log canonical and B ≥ 0 }.

Then, assuming the following:

Theorem F. Let π: X ⟶ Z be a projective morphism to a normal affine variety. Let (X, Δ = A + B) be a ℚ-factorial kawamata log terminal pair of dimension n, where A ≥ 0 is an ample ℚ-divisor and B ≥ 0. If K_{X} + Δ is pseudo-effective, then

(1) The pair (X, Δ) has a log terminal model μ: X −−→ Y. In particular if K_{X} + Δ is ℚ-Cartier then the log canonical ring

R(X, K_{X} + Δ) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(⌞m(K_{X} + Δ)⌟)),

is finitely generated.

(2) Let V ⊂ WDiv_{ℝ}(X) be the vector space spanned by the components of Δ. Then there is a constant δ > 0 such that if G is a prime divisor contained in the stable base locus of K_{X} + Δ and Ξ ∈ L_{A}(V) such that ∥Ξ − Δ∥ < δ, then G is contained in the stable base locus of K_{X} + Ξ.

(3) Let W ⊂ V be the smallest affine subspace of WDiv_{ℝ}(X) containing Δ, which is defined over the rationals. Then there is a constant η > 0 and a positive integer r > 0 such that if Ξ ∈ W is any divisor and k is any positive integer such that ∥Ξ − Δ∥ < η and k(K_{X} + Ξ)/r is Cartier, then every component of Fix(k(K_{X} + Ξ)) is a component of the stable base locus of K_{X} + Δ.

we prove the existence of pl-flips:

Theorem A. Pl-flips exist in dimension n.

Theorem F and Theorem A are taken directly from [2]. Theorem A_{n} refers to Theorem A in the case when the dimension of X is n.

Theorem 1.3. Theorem F_{n−1} implies Theorem A_{n}.

It is proved in [2] that (1.3) completes the proof of (1.1).

The main ideas used in this paper have their origins in the work of Shokurov on the existence of flips [12] together with the use of the extension theorem of [5] which in turn was inspired by the work of Kawamata, Siu and Tsuji (cf. [7], [14] and [17]). For further history about the details of this problem see [3, §2.1].

In this paper, however we do not make use of the concept of “asymptotic saturation” introduced by Shokurov, and in fact we prove a more general result which does not require the relative weak log Fano condition (see also [1]).

Further treatments of the results of this paper may be found in [1] and [6] (which follows Shokurov’s approach more explicitly).

We now turn to a more detailed description of the results and techniques used in this paper. Recall the following:

Definition 1.4. Let (X, Δ) be a purely log terminal pair and f: X ⟶ Z be a projective morphism of normal varieties. Then f is a plflipping contraction if Δ is a ℚ-divisor and

(1) f is small, of relative Picard number one,

(2) −(K_{X} + Δ) is f-ample,

(3) X is ℚ-factorial,

(4) S = ⌞Δ⌟ is irreducible and −S is f-ample.

The flip of a pl-flipping contraction f: X ⟶ Z is a small projective morphism g: Y ⟶ Z of relative Picard number one, such that K_{Y} +Γ is g-ample, where Γ is the strict transform of Δ.

The flip g is unique, if it exists at all, and it is given by

Y = Proj_{Z} ℜ where ℜ = ⨁_{m∈ℕ: k|m}f_{∗}O_{X}(m(K_{X} + Δ)),

and k is any positive integer such that k(K_{X} +Δ) is integral. Therefore, in order to prove the existence of pl-flips, it suffices to show that ℜ is a finitely generated O_{Z}-algebra. Since this problem is local over Z, we may assume that Z = Spec A is affine and it suffices to prove that

R(X, k(K_{X} + Δ)) = ⨁_{m∈ℕ: k|m}H^{0}(X, O_{X}(m(K_{X} + Δ))),

is a finitely generated A-algebra. It is then natural to consider the restricted algebra

R_{S}(X, k(K_{X} + Δ)) = Im (R(X, k(K_{X} + Δ)) ⟶ R(S, k(K_{S} + Ω))) , whose graded pieces correspond to the images of the restriction homomorphisms

H^{0}(X, O_{X}(m(K_{X} + Δ))) ⟶ H^{0}(S, O_{S}(m(K_{S} + Ω))),

where m = kl is divisible by k and Ω is defined by the adjunction formula

(K_{X} + Δ)|_{S} = K_{S} + Ω,

and k(K_{X} + Δ) is Cartier. Shokurov has shown, cf. (3.2), that the algebra R(X, k(K_{X} +Δ)) is finitely generated if and only if the restricted algebra is finitely generated.

Now, if the natural inclusion

R_{S}(X, k(K_{X} + Δ)) ⊂ R(S, k(K_{S} + Ω)),

were an isomorphism, then (1.3) would follow from (1) of Theorem F_{n−1}. In fact the pair (S, Ω) is kawamata log terminal, dim S = dim X − 1 = n − 1 and since f|_{S} is birational, Ω is automatically big so that, by a standard argument, (1) of Theorem F_{n−1} applies and R(S, k(K_{S} + Ω)) is finitely generated. (3.2) also implies ℜ is finitely generated.

Unluckily this is too much to hope for. However it does suggest that one should concentrate on the problem of lifting sections and the main focus of this paper is to prove the extension result (6.3). In fact (1.3) is a straightforward consequence of (6.3).

To fix ideas, let us start with an example where we cannot lift sections. Let X be the blow up of ℙ^{2} at a point o, with exceptional divisor E. Let S be the strict transform of a line through o, let L_{1}, L_{2} and L_{3} be the strict transforms of general lines in ℙ^{2}, let p = E ∩ S and let p_{i} = L_{i} ∩ S. Then the pair

(X, Δ = S + (2/3)(E + L_{1} + L_{2} + L_{3})),

is purely log terminal but the homomorphism

H^{0}(O_{X}(3l(K_{X} + Δ))) ⟶ H^{0}(O_{S}(3l(K_{S} + Ω))) ≃ H^{0}(O_{ℙ^{1}}(2l)),

is never surjective, where Ω = (Δ−S)|_{S} = 2/3(p+p_{1} +p_{2} +p_{3}) and l is a positive integer. The problem is that the stable base locus of K_{X} +Δ contains E and yet |3(K_{S} + Ω)| is base point free. Notice, however, that

|3l(K_{X} + Δ)|_{S} = |3l(K_{S} + Θ)| + 3l(Ω − Θ),

where Θ = (2/3)(p_{1} + p_{2} + p_{3}) is obtained from Ω by throwing away p. In other words, Θ is obtained from Ω by removing some part of each component contained in the stable base locus of K_{X} + Δ.

Returning to the general setting, one may then hope that the restricted algebra R_{S}(S, l(K_{X} + Δ)) is given by an algebra of the form R(S, l(K_{S} + Θ)) for some kawamata log terminal pair (S, Θ) where 0 ≤ Θ ≤ Ω is a ℚ-divisor obtained from Ω by subtracting components of Ω contained in the stable base locus of K_{X} +Δ. We will now explain how this may be achieved. The tricky thing is to determine exactly how much of the stable base locus to throw away.

It is not hard to reduce to the following situation: π: X ⟶ Z is a projective morphism to a normal affine variety Z, where (X, Δ = S + A + B) is a purely log terminal pair of dimension n, S = ⌞Δ⌟ is irreducible, X and S are smooth, A ≥ 0 is an ample ℚ-divisor, B ≥ 0, (S, Ω = (Δ − S)|_{S}) is canonical and the stable base locus of K_{X} + Δ does not contain S.

Let

Θ_{m} = Ω − Ω ∧ F_{m} where F_{m} = Fix(|m(K_{X} + Δ)|_{S})/m,

and m(K_{X} + Δ) is Cartier. Ω ∧ F_{m} is the minimum of Ω and F_{m}, where the minimum is taken component by component. In particular m(Ω − Θ_{m}) is the biggest divisor contained in Fix(|m(K_{X} + Δ)|_{S}) such that 0 ≤ Θ_{m} ≤ Ω. It follows that

(⊃)  |m(K_{S} + Θ_{m})| + m(Ω − Θ_{m}) ⊃ |m(K_{X} + Δ)|_{S}.

A simple consequence of the main lifting result (6.3) of this paper implies that this tautological inclusion (⊃) is actually an equality,

(=)  |m(K_{S} + Θ_{m})| + m(Ω − Θ_{m}) = |m(K_{X} + Δ)|_{S}.

A technical, but significant, improvement on the proof of the existence of flips which appears in [6] is that the statement of (=) and of (6.3) involves only linear systems and divisors on X and S, even though the proof of (6.3) involves passing to a higher model. The key point is that since (S, Ω) is canonical, it suffices to keep track only of the fixed divisor on S and not of the whole base locus.

To prove (=) we use the method of multiplier ideal sheaves. In fact the main point is to establish an inclusion of multiplier ideal sheaves, (5.3). A proof of (5.3) appeared originally in [5]. We chose to include a proof of this result for the convenience of the reader and we decided to use notation closer to the well established notation used in [10]. Note however that the multiplier ideal sheaves we use, see (4.2), must take into account the divisor Δ (for example consider the case worked out above) and the fact that (S, Ω) is canonical.

In fact, if one assumes the MMP then one should expect (=) to hold. Indeed, if one runs f: X −−→ Y the (K_{X} + Δ)-MMP, almost by definition this will not change the linear systems |m(K_{X} + Δ)|. Since K_{Y} + Γ = K_{X} + f_{∗}Δ is nef, one can lift sections on Y from the strict transform T of S, by an easy application of Kawamata-Viehweg vanishing. In general, however, the linear systems |m(K_{T} + g_{∗}Θ)| are bigger than the linear systems |m(K_{S}+Θ)|, since the induced birational map g: S −−→ T might extract some divisors. However any such divisor must have log discrepancy at most one, so this cannot happen, almost by definition, if K_{S} + Θ is canonical.

In order to establish that R_{S}(X, k(K_{X} +Δ)) is finitely generated, cf. (7.1), and thereby to finish the proof of (1.3), it is necessary and sufficient to show that Θ = lim(Θ_{m!}/m!) is rational (the seemingly strange use of factorials is so that we can use limits rather than limsups). At this point we play off two facts. The first is that since we are assuming that Theorem F holds on S, if m > 0 is sufficiently divisible and Φ is an appropriately chosen ℚ-divisor sufficiently close to Θ, then the base locus of |m(K_{S} +Φ)| and the stable base locus of K_{S} +Θ are essentially the same (basically because K_{S} + Θ and K_{S} + Φ share a log terminal model μ: S −−→ S′ and these two sets of divisors are precisely the divisors contracted by μ). The second is that using (4.1), (6.3) is slightly stronger than (=); one is allowed to overshoot Θ_{m} by an amount ϵ/m, where ϵ > 0 is fixed. (It seems worth pointing out that (4.1) seems to us a little mysterious. In particular, unlike (6.3), we were unable to show that this result follows from the MMP.)

More precisely, since the base locus of |m(K_{S} + Θ_{m})| contains no components of Θ_{m}, by (2) of Theorem F it follows that the stable base locus of K_{S} + Θ contains no components of Θ. If Θ is not rational, then by Diophantine approximation there is a ℚ-divisor 0 ≤ Φ ≤ Ω very close to Θ and an integer k > 0 such that kΦ is integral and mult_{G} Φ > mult_{G} Θ, for some prime divisor G. By (6.3), it actually follows that

|k(K_{S} + Φ)| + k(Ω − Φ) = |k(K_{X} + Δ)|_{S}.

The condition mult_{G} Φ > mult_{G} Θ ensures that G is a component of Fix(k(K_{S} + Φ)), and hence of the stable base locus of K_{S} + Φ. But then G is a component of Θ and of the stable base locus of K_{S} + Θ. This is the required contradiction.

2. Notation and conventions

We work over the field of complex numbers ℂ. Let X be a normal variety. A

\begin{matrix}(integral) divisor \\ ℚ-divisor \\ ℝ-divisor \end{matrix} \} is a \{ \begin{matrix}ℤ-linear \\ ℚ-linear \\ ℝ-linear, \end{matrix}

combination of prime divisors. Given an integral Weil divisor D, we let

R(X, D) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(mD)).

Set

WDiv_{ℚ}(X) = WDiv(X) ⊗_{ℤ} ℚ
WDiv_{ℝ}(X) = WDiv(X) ⊗_{ℤ} ℝ,

where WDiv(X) is the group of Weil divisors on X. The definitions below for ℝ-divisors reduce to the usual definitions for ℚ-divisors and integral divisors, see [2]. Note that the group of ℝ-divisors forms a vector space, with a canonical basis given by the prime divisors. If C = ∑ c_{i}B_{i} and D = ∑ d_{i}B_{i}, where B_{i} are distinct prime divisors, then we write D ≥ 0 if d_{i} ≥ 0 and we will denote by

∥C∥ = max_{i} c_{i}  C ∧ D = ∑_{i} min{c_{i}, d_{i}}B_{i}
⌞C⌟ = ∑_{i}⌞c_{i}⌟B_{i}  {C} = C − ⌞C⌟.

Two ℝ-divisors C and D are

\begin{matrix}linearly equivalent, & C ∼ D \\ ℚ-linearly equivalent, & C ∼_{ℚ} D \\ ℝ-linearly equivalent, & C ∼_{ℝ} D \end{matrix} \} if C − D is a \{ \begin{matrix}ℤ-linear \\ ℚ-linear \\ ℝ-linear, \end{matrix}

combination of principal divisors. Note that if C ∼_{ℚ} D then mC ∼ mD for some positive integer m, but this fails in general for ℝ-linear equivalence. Note also that if two ℚ-divisors are ℝ-linearly equivalent then they are in fact ℚ-linearly equivalent, but that two integral divisors might be ℚ-linearly equivalent without being linearly equivalent. Let

|D| = { C ∈ WDiv(X) | C ≥ 0 , C ∼ D }
|D|_{ℚ} = { C ∈ WDiv_{ℚ}(X) | C ≥ 0 , C ∼_{ℚ} D }
|D|_{ℝ} = { C ∈ WDiv_{ℝ}(X) | C ≥ 0 , C ∼_{ℝ} D }.

If T is a subvariety of X, not contained in the base locus of |D|, then |D|_{T} denotes the image of the linear system |D| under restriction to T. If D is an integral divisor, Fix(D) denotes the fixed divisor of D so that |D| = |D −Fix(D)|+Fix(D) where the base locus of |D −Fix(D)| contains no divisors. More generally Fix(V) denotes the fixed divisor of the linear system V.

The stable base locus of D, denoted by B(D), is the intersection of the support of the elements of |D|_{ℝ} (if |D|_{ℝ} is empty then by convention the stable base locus is the whole of X). The stable fixed divisor is the divisorial support of the stable base locus. The augmented stable base locus of D, denoted by B_{+}(D), is given by the stable base locus of D − ϵA for some ample divisor A and any rational number 0 < ϵ ≪ 1. The diminished stable base locus is defined by

B_{−}(D) = ⋃_{ϵ>0} B(D + ϵA).

In particular we have

B_{−}(D) ⊂ B(D) ⊂ B_{+}(D).

This notation was established in [4] for projective varieties but we will use it in a slightly more general setting, see for example [11].

An ℝ-Cartier divisor D is an ℝ-linear combination of Cartier divisors. An ℝ-Cartier divisor D is nef if D · Σ ≥ 0 for any curve Σ ⊂ X. An ℝ-Cartier divisor D is ample if it is ℝ-linearly equivalent to a positive linear combination of ample divisors (in the usual sense). An ℝ-Cartier divisor D is big if D ∼_{ℝ} A+B, where A is ample and B ≥ 0. A ℚ-Cartier divisor D is a general ample ℚ-divisor if there is an integer m > 0 such that mD is very ample and mD ∈ |mD| is very general.

A log pair (X, Δ) is a normal variety X and an ℝ-Weil divisor Δ ≥ 0 such that K_{X} + Δ is ℝ-Cartier. We say that a log pair (X, Δ) is log smooth, if X is smooth and the support of Δ is a divisor with global normal crossings. A projective birational morphism g: Y ⟶ X is a log resolution of the pair (X, Δ) if X is smooth and the inverse image of Δ union the exceptional locus is a divisor with global normal crossings. Note that in the definition of log resolution we place no requirement that the indeterminacy locus of g is contained in the locus where the pair (X, Δ) is not log smooth. If V is a linear system on X, a log resolution of V and (X, Δ) is a log resolution of the pair (X, Δ) such that if |M| + F is the decomposition of g^{∗}V into its mobile and fixed parts, then |M| is base point free and F union the exceptional locus union the strict transform of Δ is a divisor with simple normal crossings support. If g is a log resolution, then we may write

K_{Y} + Γ = g^{∗}(K_{X} + Δ) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, g_{∗}Γ = Δ and E is g-exceptional. Note that this decomposition is unique. The log discrepancy of a divisor F over X

a(X, Δ, F) = 1 + mult_{F} (E − Γ).

Note that with this definition, a component F of Δ with coefficient b has log discrepancy 1 − b. The log discrepancy does not depend on the choice of model Y, so that the log discrepancy is also a function defined on valuations. A non kawamata log terminal place is any valuation of log discrepancy at most zero and the centre of a non kawamata log terminal place is called a non kawamata log terminal centre. A non terminal centre is any centre of log discrepancy at most one. Note that every divisor on X is by definition a non terminal centre, so the only interesting non terminal centres are of codimension at least two.

The pair (X, Δ) is kawamata log terminal if there are no non kawamata log terminal centres. We say that the pair (X, Δ) is purely log terminal (respectively canonical or terminal) if the log discrepancy of any exceptional divisor is greater than zero (respectively at least one or greater than one). We say that the pair is divisorially log terminal if there is a log resolution g: Y ⟶ X such that all exceptional divisors E ⊂ Y have log discrepancy greater than zero.

3. Preliminary results

In this section we recall several results about finitely generated algebras and in particular we will give a proof of Shokurov’s result that the pl-flip exists if and only if the restricted algebra is finitely generated.

Definition 3.1. Let X be a normal variety, S be a prime divisor and B an integral Weil divisor which is ℚ-Cartier and whose support does not contain S. The restricted algebra R_{S}(X, B) is the image of the homomorphism R(X, B) ⟶ R(S, B|_{S}).

We remark that as B is ℚ-Cartier then B|_{S} is a well defined ℚ-Cartier divisor on S.

Theorem 3.2. Let f: X ⟶ Z be a pl-flipping contraction with respect to (X, Δ). Pick an integer k such that k(K_{X} + Δ) is Cartier.

Then

(1) The flip of f exists if and only if the flip of f exists locally over Z.

(2) If Z = Spec A is affine then the flip f^{+} : X^{+} ⟶ Z exists if and only if the restricted algebra R_{S}(X, k(K_{X} +Δ)) is a finitely generated A-algebra.

Definition 3.3. Let A be a ring, and let R be any graded A-algebra. A truncation of R is any A-algebra of the form

R_{(d)} = ⨁_{m∈ℕ}R_{md},

for a positive integer d.

We start with the following well known result:

Lemma 3.4. Let R be a graded algebra which is an integral domain and let d be a positive integer.

Then R is finitely generated if and only if R_{(d)} is finitely generated.

Proof. Suppose that R is finitely generated. It is easy to write down an action of the cyclic group ℤ_{d} on R so that the invariant ring is R_{(d)}. Thus R_{(d)} is finitely generated by the Theorem of E. Noether which states that the ring of invariants of a finitely generated ring under the action of a finite group is finitely generated.

Suppose now that R_{(d)} is finitely generated. Let f ∈ R_{i}. Then f is a root of the monic polynomial x^{d} − f^{d} ∈ R_{(d)}[x]. It follows that R is integral over R_{(d)} and the result follows by another Theorem of E. Noether on finiteness of integral closures. □

Lemma 3.5. Let S be a normal prime divisor on X and let B an integral Weil divisor which is ℚ-Cartier and whose support does not contain S.

• If R(X, B) is finitely generated then R_{S}(X, B) is finitely generated.

• If S ∼ B and R_{S}(X, B) is finitely generated then R(X, B) is finitely generated.

Proof. Since there is a surjective homomorphism ϕ: R(X, B) ⟶ R_{S}(X, B), it is clear that if R(X, B) is finitely generated then R_{S}(X, B) is finitely generated.

Suppose now that R_{S}(X, B) is finitely generated and S ∼ B. Then there is a rational function g_{1} such that (g_{1}) = S−B. If we consider the elements of R(X, B)_{m} as rational functions, then a rational function g belongs to R(X, B)_{m} if and only if (g) + mB ≥ 0. But if g is in the kernel of ϕ, then there is a divisor S′ ≥ 0 such that (g) + mB = S + S′. It follows that (g/g_{1}) + (m − 1)B = S′ so that g/g_{1} = h ∈ R(X, B)_{m−1}. But then the kernel of ϕ is the principal ideal generated by g_{1}. □

Proof of (3.2). It is well known that the flip f^{+} : X^{+} ⟶ Z exists if and only if the sheaf of graded O_{Z}-algebras

⨁_{m∈ℕ: k|m}f_{∗}O_{X}(m(K_{X} + Δ)),

is finitely generated, cf. [8, 6.4]. Since this can be checked locally, this gives (1).

If Z = Spec A is affine it suffices to check that R(X, k(K_{X} + Δ)) is a finitely generated A-algebra. Since the relative Picard number is one, there are real numbers a and b such that a(K_{X} + Δ) and bS are numerically equivalent over Z. As both −(K_{X} + Δ) and −S are ample ℚ-divisors we may assume that a and b are both positive integers. Moreover, as a(K_{X} + Δ) − bS is numerically trivial over Z, it is semiample over Z by the base point free theorem. In particular, we may replace numerical equivalence by linear equivalence,

a(K_{X} + Δ) ∼_{Z} bS.

But then there is a rational function g and a divisor D on Z such that

a(K_{X} + Δ) = bS + f^{∗}D + (g).

As any line bundle on a quasi-projective variety is locally trivial, possibly passing to an open subset of Z, and using (1), we may assume that D ∼ 0, so that

a(K_{X} + Δ) ∼ bS.

By (3.4) it follows that R(X, k(K_{X} + Δ)) is finitely generated if and only if R(X, S) is finitely generated. Since Z is affine and f is small, S is mobile (that is the fixed divisor is empty) so that S ∼ S′ where S′ ≥ 0 is a divisor whose support does not contain S. By (3.5), R(X, S) is finitely generated if and only if R_{S}(X, S′) is finitely generated. Since a(K_{X} + Δ)|_{S} ∼ bS′|_{S} the result follows by (3.4). □

4. Multiplier ideal sheaves

The main result of this section is:

Theorem 4.1. Let π: X ⟶ Z be a projective morphism to a normal affine variety Z, where (X, Δ = S + A + B) is a log pair, S = ⌞Δ⌟ is irreducible, (X, S) is log smooth, and both A ≥ 0 and B ≥ 0 are ℚ-divisors. Let k be any positive integer and 0 ≤ Φ ≤ Ω = (Δ − S)|_{S} be any divisor such that both k(K_{S} + Φ) and k(K_{X} + Δ) are Cartier. Let C = A/k.

If there is an integer l > 1 and an integral divisor P ≥ 0 such that lA is Cartier, C − \frac{(k−1)}{m}P is ample, (X, Δ + \frac{k−1}{m}P) is purely log terminal and

l|k(K_{S} + Φ)| + m(Ω − Φ) + (mC + P)|_{S} ⊂ |m(K_{X} + Δ + C) + P |_{S},

where m = kl, then

|k(K_{S} + Φ)| + k(Ω − Φ) ⊂ |k(K_{X} + Δ)|_{S}.

See also [14] and [16] for related statements. To prove (4.1), we need a variant of multiplier ideal sheaves:

Definition-Lemma 4.2. Let (X, Δ) be a log smooth pair where Δ is a reduced divisor and let V be a linear system whose base locus contains no non kawamata log terminal centres of (X, Δ). Let μ: Y ⟶ X be a log resolution of V and (X, Δ) and let F be the fixed divisor of the linear system μ^{∗}V. Let K_{Y} + Γ = μ^{∗}(K_{X} + Δ) + E where Γ = ∑ P_{i} is the sum of the divisors on Y of log discrepancy zero.

Then for any real number c ≥ 0, define the multiplier ideal sheaf

J_{Δ,c·V} := μ_{∗}O_{Y} (E − ⌞cF ⌟).

If Δ = 0 we will write J_{c·V} and if D = cG, where G > 0 is a Cartier divisor, we define

J_{Δ,D} := J_{Δ,c·V} ,

where V = {G}.

Proof. We have to show that the definition of the multiplier ideal sheaf is independent of the choice of log resolution. Let μ: Y ⟶ X and μ′: Y′ ⟶ X be two log resolutions of (X, Δ) and V. We may assume that μ′ factors through μ via a morphism ν: Y′ ⟶ Y. Then F′ = ν^{∗}F as μ^{∗}V − F is free, and

E′ − cF′ = K_{Y′} + Γ′ − μ^{′∗}(K_{X} + Δ) − cF′
= K_{Y′} + Γ′ − ν^{∗}(K_{Y} + Γ − E + cF)
= ν^{∗}(E − ⌞cF ⌟) + K_{Y′} + Γ′ − ν^{∗}(K_{Y} + Γ + {cF })
= ν^{∗}(E − ⌞cF ⌟) + G.

Since (Y, Γ + E + F) is log smooth, it follows that (Y, Γ + {cF }) is log canonical and has the same non kawamata log terminal places as (Y, Γ) and hence as (X, Δ). Thus ⌜G⌝ ≥ 0 and since ν_{∗}(K_{Y′} + Γ′) = K_{Y} + Γ, ⌜G⌝ is ν-exceptional. Then

μ′_{∗}O_{Y′}(E′ − ⌞cF′⌟) = μ_{∗}(ν_{∗}O_{Y′}(E′ − ⌞cF′⌟))
= μ_{∗}(ν_{∗}O_{Y′}(ν^{∗}(E − ⌞cF ⌟) + ⌜G⌝))
= μ_{∗}O_{Y} (E − ⌞cF ⌟).  □

We need to develop a little of the theory of multiplier ideal sheaves.

Lemma 4.3. Let (X, Δ) be a log smooth pair where Δ is reduced, let V be a linear system whose base locus contains no non kawamata log terminal centres of (X, Δ) and let G ≥ 0 and D ≥ 0 be ℚ-Cartier divisors whose supports contain no non kawamata log terminal centres of (X, Δ).

Then

(1) J_{Δ,D} = O_{X} if and only if (X, Δ+D) is divisorially log terminal and ⌞D⌟ = 0.

(2) If 0 ≤ Δ′ ≤ Δ then J_{Δ,c·V} ⊂ J_{Δ′,c·V}. In particular, J_{Δ,c·V} ⊂ J_{c·V} ⊂ O_{X}.

(3) If Σ ≥ 0 is a Cartier divisor, D − Σ ≤ G and J_{Δ,G} = O_{X} then I_{Σ} ⊂ J_{Δ,D}.

Proof. (1) follows easily from the definitions.

(2) follows from the fact that a(P, X, Δ′) ≥ a(P, X, Δ) for all divisors P on Y.

To see (3), note that we are free to replace Σ by a smaller Cartier divisor Σ′ such that D − Σ′ ≤ G. As D contains no non kawamata log terminal centres of (X, Δ) we may therefore assume that Σ contains no non kawamata log terminal centres of (X, Δ) as well. Notice that as Σ is Cartier and J_{Δ,G} = O_{X}, we have

J_{Δ,G}(−Σ) = O_{X}(−Σ) = I_{Σ}.

But since D ≤ G + Σ and Σ contains no non kawamata log terminal centres of (X, Δ) we also have

J_{Δ,G}(−Σ) = J_{Δ,G+Σ} ⊂ J_{Δ,D}.  □

We have the following extension of (9.5.1) of [9] or (2.4.2) of [16]:

Lemma 4.4. Let π: X ⟶ Z be a projective morphism to a normal affine variety Z. Let (X, Δ) be a log smooth pair where Δ is reduced, let S be a component of Δ, let D ≥ 0 be a ℚ-Cartier divisor whose support does not contain any non kawamata log terminal centres of (X, Δ) and let Θ = (Δ − S)|_{S}. Let N be a Cartier divisor.

(1) There is a short exact sequence

0 ⟶ J_{Δ−S,D+S} ⟶ J_{Δ,D} ⟶ J_{Θ,D|_{S}} ⟶ 0.

(2) (Nadel Vanishing) If N − D is ample then

H^{i}(X, J_{Δ,D}(K_{X} + Δ + N)) = 0,

for i > 0.

(3) If N − D is ample then

H^{0}(X, J_{Δ,D}(K_{X} + Δ + N)) ⟶ H^{0}(S, J_{Θ,D|_{S}}(K_{X} + Δ + N)),

is surjective.

Proof. By the resolution lemma of [15], we may find a log resolution μ: Y ⟶ X of (X, Δ + D) which is an isomorphism over the generic point of each non kawamata log terminal centre of (X, Δ). If T is the strict transform of S then we have a short exact sequence

0 ⟶ O_{Y} (E−⌞μ^{∗}D⌟−T) ⟶ O_{Y} (E−⌞μ^{∗}D⌟) ⟶ O_{T} (E−⌞μ^{∗}D⌟) ⟶ 0, where E is defined in (4.2). Now μ_{∗}O_{Y} (E − ⌞μ^{∗}D⌟) = J_{Δ,D}. If Γ is the sum of the divisors of log discrepancy zero then

E − μ^{∗}D = (K_{Y} + Γ) − μ^{∗}(K_{X} + Δ + D).

But then

E − μ^{∗}D − T = (K_{Y} + Γ − T) − μ^{∗}(K_{X} + Δ − S + (D + S)),

so that

μ_{∗}O_{Y} (E − ⌞μ^{∗}D⌟ − T) = J_{Δ−S,D+S},

and

(E − μ^{∗}D)|_{T} = K_{T} + (Γ − T)|_{T} − μ^{∗}(K_{S} + Θ + D|_{S}),

so that

μ_{∗}O_{T} (E − ⌞μ^{∗}D⌟) = J_{Θ,D|_{S}}.

Since (Y, Γ + μ^{∗}D) is log smooth and Γ and μ^{∗}D have no common components, (Y, Γ + {μ^{∗}D}) is divisorially log terminal. Choose F ≥ 0 such that −F is μ-ample whose coefficients are sufficiently small so that K_{Y} + Γ + {μ^{∗}D} + F is divisorially log terminal. As

E − ⌞μ^{∗}D⌟ − T − (K_{Y} + Γ − T + {μ^{∗}D} + F) = −μ^{∗}(K_{X} + Δ + D) − F,

is μ-ample, Kawamata-Viehweg vanishing implies that

R^{1}μ_{∗}O_{Y} (E − ⌞μ^{∗}D⌟ − T) = 0,

and this gives (1).

Similarly, Kawamata-Viehweg vanishing implies that

R^{i}μ_{∗}O_{Y} (μ^{∗}(K_{X} + Δ + N) + E − ⌞μ^{∗}D⌟) = 0,

for i > 0. As N − D is ample then, possibly replacing F by a small multiple, we may assume that μ^{∗}(N − D) − F is ample. As

μ^{∗}(K_{X} +Δ+N)+E −⌞μ^{∗}D⌟−(K_{Y} +Γ+{μ^{∗}D}+F) = μ^{∗}(N −D)−F,

is ample, Kawamata-Viehweg vanishing implies that

H^{i}(Y, O_{Y} (μ^{∗}(K_{X} + Δ + N) + E − ⌞μ^{∗}D⌟)) = 0,

for i > 0. Since the Leray-Serre spectral sequence degenerates, this gives (2), and (3) follows from (2). □

Proof of (4.1). Since (X, Δ + \frac{k−1}{m}P) is purely log terminal, (S, Ω + \frac{k−1}{m}P |_{S}) is kawamata log terminal and S is not contained in the support of A or P. If Σ ∈ |k(K_{S} + Φ)| then we may pick a divisor

G ∈ |m(K_{X}+Δ+C)+P | such that G|_{S} = lΣ+m(Ω−Φ+C|_{S})+P |_{S}.

Let

Λ = \frac{k − 1}{m}G + B and N = k(K_{X} + Δ) − K_{X} − S.

As the support of the ℚ-divisor Λ ≥ 0 does not contain S and by assumption

N − Λ ∼_{ℚ} C − \frac{k − 1}{m}P,

is ample, (4.4) implies that sections of H^{0}(S, J_{Λ|_{S}}(k(K_{S} + Ω))) extend to sections of H^{0}(X, O_{X}(k(K_{X} + Δ))). Now

Λ|_{S} − (Σ + k(Ω − Φ))

= \frac{k − 1}{m}(lΣ + m(Ω − Φ + C|_{S}) + P |_{S}) + B|_{S} − (Σ + k(Ω − Φ))

≤ Ω + \frac{k − 1}{m}P |_{S}.

As (S, Ω + \frac{k−1}{m}P |_{S}) is kawamata log terminal, J_{Ω+ \frac{k−1}{m} P |_{S}} = O_{S} and we are done by (3) of (4.3). □

5. Asymptotic multiplier ideal sheaves

Definition 5.1. Let X be a normal variety and let D be a divisor. An additive sequence of linear systems associated to D is a sequence V_{•}, such that V_{m} ⊂ ℙ(H^{0}(X, O_{X}(mD))) and

V_{i} + V_{j} ⊂ V_{i+j}.

Definition-Lemma 5.2. Suppose that (X, Δ) is log smooth, where Δ is reduced and let V_{•} be an additive sequence of linear systems associated to a divisor D. Assume that there is an integer k > 0 such that no non kawamata log terminal centre of (X, Δ) is contained in the base locus of V_{k}.

If c is a positive real number and p and q are positive integers divisible by k then

J_{Δ, \frac{c}{p} ·V_{p}} ⊂ J_{Δ, \frac{c}{q} ·V_{q}} ∀q divisible by p.

In particular the asymptotic multiplier ideal sheaf of V_{•}

J_{Δ,c·V_{•}} = ⋃_{p>0} J_{Δ, \frac{c}{p} ·V_{p}},

is given by J_{Δ,c·V_{•}} = J_{Δ, \frac{c}{p} ·V_{p}}, for p sufficiently large and divisible. If we take V_{m} = |mD| the complete linear system, then define

J_{Δ,c·∥D∥} = J_{Δ,c·V_{•}},

and if S is a component of Δ and we take W_{m} = |mD|_{S}, then define

J_{Θ,c·∥D∥_{S}} = J_{Θ,c·W_{•}},

where Θ = (Δ − S)|_{S}.

Proof. If p divides q then pick a common log resolution μ: Y ⟶ X of V_{p}, V_{q} and (X, Δ) and note that

\frac{1}{q}F_{q} ≤ \frac{1}{p}F_{p},

where F_{p} is the fixed locus of μ^{∗}V_{p} and F_{q} is the fixed locus of μ^{∗}V_{q}. Therefore J_{Δ, \frac{c}{p} ·V_{p}} ⊂ J_{Δ, \frac{c}{q} ·V_{q}}. The equality J_{Δ,c·V_{•}} = J_{Δ, \frac{c}{p} ·V_{p}}, now follows as X is Noetherian. □

We are now ready to state the main result of this section:

Theorem 5.3. Let π: X ⟶ Z be a projective morphism to a normal affine variety Z. Suppose that (X, Δ = S +B) is log smooth and purely log terminal of dimension n, where S = ⌞Δ⌟ is irreducible and let k be a positive integer such that D = k(K_{X} + Δ) is integral. Let A be any ample ℚ-divisor on X. Let q and r be any positive integers such that Q = qA is very ample, rA is Cartier and (j − 1)K_{X} + Ξ + rA is ample for every Cartier divisor 0 ≤ Ξ ≤ j⌜Δ⌝ and every integer 1 ≤ j ≤ k + 1.

If the stable base locus of D does not contain any non kawamata log terminal centres of (X, ⌜Δ⌝), then

J_{∥mD|_{S}∥} ⊂ J_{Θ,∥mD+P ∥_{S}}  for all  m ∈ ℕ,

where Θ = ⌜B⌝|_{S}, p = qn + r and P = pA. Moreover, all sections of the linear system determined by J_{∥mD|_{S}∥}(mD + P) lift, that is we have

π_{∗}J_{∥mD|_{S}∥}(mD + P) ⊂ Im (π_{∗}O_{X}(mD + P) → π_{∗}O_{S}(mD + P)) ,

for all m ∈ ℕ.

We will need some results about the sheaves J_{Δ,c·V_{•}}, most of which are easy generalisations of the corresponding facts for the usual asymptotic multiplier ideal sheaves.

Lemma 5.4. Let π: X ⟶ Z be a projective morphism to a normal affine variety Z and let D be a ℚ-Cartier divisor. Suppose that (X, Δ) is log smooth, Δ is reduced and the stable base locus of D contains no non kawamata log terminal centre of (X, Δ). Then

(1) for any real numbers 0 < c_{1} ≤ c_{2} there is a natural inclusion

J_{Δ,c_{2}·∥D∥} ⊂ J_{Δ,c_{1}·∥D∥},

and

(2) if D is Cartier and S is a component of Δ, then the image of

the map

π_{∗}O_{X}(D) ⟶ π_{∗}O_{S}(D),

is contained in π_{∗}J_{Θ,∥D∥_{S}}(D) where Θ = (Δ − S)|_{S}.

Proof. (1) is immediate from the definitions.

Suppose that D is Cartier. Pick an integer p such that

J_{Θ,∥D∥_{S}} = J_{Θ, \frac{1}{p} ·|pD|_{S}},

and a log resolution μ: Y ⟶ X of |D|, |pD| and (X, Δ). Let T be the strict transform of S, let F_{1} be the fixed locus of μ^{∗}|D| and let F_{p} be the fixed locus of μ^{∗}|pD|. We have

(π ∘ μ)_{∗}O_{Y} (μ^{∗}D − F_{1}) = π_{∗}O_{X}(D) = (π ∘ μ)_{∗}O_{Y} (E + μ^{∗}D).

The first equality follows by definition of F_{1} and the second follows as E ≥ 0 is exceptional. As there are inequalities

μ^{∗}D − F_{1} ≤ μ^{∗}D − ⌞F_{p}/p⌟ ≤ E + μ^{∗}D − ⌞F_{p}/p⌟ ≤ E + μ^{∗}D,

the image of π_{∗}O_{X}(D) is equal to the image of

(π ∘ μ)_{∗}O_{Y} (E + μ^{∗}D − ⌞F_{p}/p⌟).

Thus the image of π_{∗}O_{X}(D) is contained in

(π ∘ μ)_{∗}O_{T} (E + μ^{∗}D − ⌞F_{p}/p⌟) = π_{∗}J_{Θ,∥D∥_{S}}(D).  □

Lemma 5.5. Let π: X ⟶ Z be a projective morphism to a normal affine variety Z and let D be a Cartier divisor. Suppose that (X, Δ) is log smooth and Δ is reduced. Let S be a component of Δ and Θ = (Δ − S)|_{S}.

If B_{+}(D) contains no non kawamata log terminal centres of (X, Δ) then the image of the map

π_{∗}O_{X}(K_{X} + Δ + D) ⟶ π_{∗}O_{S}(K_{S} + Θ + D),

contains

π_{∗}J_{Θ,∥D∥_{S}}(K_{S} + Θ + D).

Proof. Pick an integer p > 1 such that

J_{Θ,∥D∥_{S}} = J_{Θ, \frac{1}{p} ·|pD|_{S}},

and there is a divisor A+B ∈ |pD| where A ≥ 0 is a general very ample divisor and B ≥ 0 contains no non kawamata log terminal centres of (X, Δ). By the resolution lemma of [15], we may find a log resolution μ: Y ⟶ X of |pD| and of (X, Δ) which is an isomorphism over every non kawamata log terminal centre of (X, Δ). Let F_{p} be the fixed divisor of μ^{∗}|pD|, M_{p} = pμ^{∗}D − F_{p} and let Γ and T be the strict transforms of Δ and S. We have a short exact sequence

0 ⟶ O_{Y} (G − T) ⟶ O_{Y} (G) ⟶ O_{T} (G) ⟶ 0,

where G = K_{Y} + Γ + μ^{∗}D − ⌞F_{p}/p⌟. As μ^{∗}A is base point free and μ^{∗}(A + B) ∈ μ^{∗}|pD|, the divisor C = μ^{∗}B − F_{p} ≥ 0. Note that M_{p} − C ∼ μ^{∗}A. As no component of C is a component of Γ, we may pick 0 < δ ≤ 1/p and an exceptional ℚ-divisor F ≥ 0 such that (Y, Γ−T +{F_{p}/p}+δ(C +F)) is divisorially log terminal and μ^{∗}A−F is ample. As |M_{p}| is free, M_{p}/p is nef and so

G − T − (K_{Y} + Γ − T + \{\frac{1}{p}F_{p}\} + δ(C + F)) = \frac{1}{p}M_{p} − δ(C + F)

∼_{ℚ} (\frac{1}{p} − δ)M_{p} + δ(μ^{∗}A − F),

is ample. In particular Kawamata-Viehweg vanishing implies that R^{1}ϕ_{∗}O_{Y} (G − T) = 0 where ϕ = π ∘ μ. Therefore the homomorphism

π_{∗}O_{X}(K_{X}+Δ+D) ⊃ ϕ_{∗}O_{Y} (G) ⟶ ϕ_{∗}O_{T} (G) = π_{∗}J_{Θ,∥D∥_{S}}(K_{S}+Θ+D),

is surjective. □

Theorem 5.6. Let π: X ⟶ Z be a projective morphism, where Z is affine and X is a smooth variety of dimension n.

If D is a Cartier divisor whose stable base locus is a proper subset of X, A is an ample Cartier divisor and H is a very ample divisor then J_{∥D∥}(D + K_{X} + A + nH) is globally generated.

Proof. Pick an integer p > 0 such that if pB ∈ |pD| is a general element, then

J_{∥D∥} = J_{\frac{1}{p} ·|pD|} = J_{B}.

Then by (2) of (4.4), H^{i}(X, J_{∥D∥}(D + K_{X} + A + mH)) = 0 for all i > 0 and m ≥ 0 and we may apply (5.7). □

The following result is well known to experts. We include a proof for the benefit of the reader:

Lemma 5.7. Let π: X ⟶ Z be a projective morphism where X is smooth of dimension n, Z is affine and let H be a very ample divisor.

If F is any coherent sheaf such that H^{i}(X, F(mH)) = 0, for i > 0 and for all m ≥ −n then F is globally generated.

Proof. Pick x ∈ X. Let T ⊂ F be the torsion subsheaf supported at x, and let G = F/T . Then H^{i}(X, G(mH)) = 0 for i > 0 and for all m ≥ −n and F is globally generated if and only if G is globally generated. Replacing F by G we may therefore assume that T = 0.

Pick a general element Y ∈ |H| containing x. As T = 0 there is an exact sequence

0 ⟶ F(−Y) ⟶ F ⟶ G ⟶ 0,

where G = F ⊗ O_{Y}. As H^{i}(Y, G(mH)) = 0, for i > 0 and for all m ≥ −(n − 1), G is globally generated by induction on the dimension. As H^{1}(X, F(−Y)) = 0 it follows that F is globally generated. □

Proof of (5.3). We follow the argument of [5] which in turn is based on the ideas of [7], [14] and [17].

We proceed by induction on m. The statement is clear for m = 0, and so it suffices to show that

J_{∥(m+1)D|_{S}∥} ⊂ J_{Θ,∥(m+1)D+P ∥_{S}},

assuming that

J_{∥tD|_{S}∥} ⊂ J_{Θ,∥tD+P ∥_{S}} for all t ≤ m.

If Δ = ∑ δ_{i}Δ_{i}, where each Δ_{i} is a prime divisor, then for any 1 ≤ s ≤ k, put

Δ^{s} = ∑_{i: δ_{i}>(k−s)/k} Δ_{i}.

We have • each Δ^{s} is integral,

• S = Δ^{1} ≤ Δ^{2} ≤ ⋯ ≤ Δ^{k} = ⌜Δ⌝, and

• Δ = \frac{1}{k} ∑_{s=1}^{k} Δ^{s},

and these properties uniquely determine the divisors Δ^{s}. We let Δ^{k+1} = ⌜Δ⌝. We recursively define integral divisors D_{≤s} by the rule

D_{≤s} = \begin{cases}0 & if s = 0 \\ K_{X} + Δ^{s} + D_{≤s−1} & 1 ≤ s ≤ k. \end{cases}

Note that D_{≤k} = D. By (1) of (5.4) there is an inclusion

J_{∥(m+1)D|_{S}∥} ⊂ J_{∥mD|_{S}∥},

and so it suffices to prove that there are inclusions

(⋆)  J_{∥mD|_{S}∥} ⊂ J_{Θ^{s+1},∥mD+D_{≤s}+P ∥_{S}},

for 0 ≤ s ≤ k, where Θ^{i} = (Δ^{i} − S)|_{S} for 1 ≤ i ≤ k + 1. Thus Θ^{k} = Θ^{k+1} = Θ and Θ^{1} = 0.

We proceed by induction on s. Now

J_{∥mD|_{S}∥} ⊂ J_{Θ,∥mD+P ∥_{S}} ⊂ J_{Θ^{1},∥mD+P ∥_{S}}.

The first inclusion holds by assumption and since Θ^{1} ≤ Θ, (2) of (4.3) implies the second inclusion. Thus (⋆) holds when s = 0.

Now suppose that (⋆) holds for s ≤ t − 1. Note that

mD + D_{≤t} + P = K_{X} + Δ^{t} + (D_{≤t−1} + P) + mD

= mD + K_{X} + (Δ^{t} + D_{≤t−1} + rA) + nQ, (†)

where, by assumption, both D_{≤t−1} + P and Δ^{t} + D_{≤t−1} + rA are ample for any 1 ≤ t ≤ k + 1. In particular B_{+}(mD + D_{≤t−1} + P) contains no log non terminal centres of (X, ⌜Δ⌝). Then

π_{∗}J_{∥mD|_{S}∥}(mD + D_{≤t} + P) ⊂ π_{∗}J_{Θ^{t},∥mD+D_{≤t−1}+P ∥_{S}}(mD + D_{≤t} + P)

⊂ Im (π_{∗}O_{X}(mD + D_{≤t} + P) ⟶ π_{∗}O_{S}(mD + D_{≤t} + P)) ⊂ π_{∗}J_{Θ^{t+1},∥mD+D_{≤t}+P ∥_{S}}(mD + D_{≤t} + P).

The first inclusion holds as we are assuming (⋆) for s = t−1, the second inclusion holds by (†) and (5.5) and the last inclusion follows from (2) of (5.4). But (†) and (5.6) imply that

J_{∥mD|_{S}∥}(mD + D_{≤t} + P),

is generated by global sections and so

J_{∥mD|_{S}∥} ⊂ J_{Θ^{t+1},∥mD+D_{≤t}+P ∥_{S}}.

The inclusion

π_{∗}J_{∥mD|_{S}∥}(mD + P) ⊂ Im (π_{∗}O_{X}(mD + P) → π_{∗}O_{S}(mD + P)) ,

is part of the inclusions proved above when s = k. □

6. Lifting sections

Lemma 6.1. Let D ≥ 0 be a Cartier divisor on a normal variety X, and let Z ⊂ X be an irreducible subvariety.

Then

lim inf \frac{mult_{Z}(|mD|)}{m} = lim \frac{mult_{Z}(|m!D|)}{m!}.

Proof. Note that if a divides b then

\frac{mult_{Z}(|aD|)}{a} ≥ \frac{mult_{Z}(|bD|)}{b},

whence the result. □

The following is essentially proved in [4]; we include a proof for the benefit of the reader:

Lemma 6.2. Let D ⊂ X be a divisor on a smooth variety and Z a closed subvariety.

If lim mult_{Z}(|m!D|)/m! = 0 then Z is not contained in B_{−}(D).

Proof. Let A be any ample divisor. Pick l > 0 such that lA − K_{X} is ample. If m > l is sufficiently divisible then J_{∥mD∥}(m(D + A)) is globally generated by (5.6). But if p > 0 is sufficiently large and divisible and D_{mp} ∈ |mpD| is general, then mult_{Z} D_{mp} = mult_{Z} |mpD| < p and

J_{∥mD∥} = J_{(1/p)D_{mp}}.

But since mult_{Z} D_{mp}/p < 1 it follows that (X, D_{mp}/p) is kawamata log terminal, in a neighbourhood of the generic point of Z. Thus Z is not contained in the co-support of J_{∥mD∥} and so Z is not contained in the base locus of m(D + A). □

Theorem 6.3. Let π: X ⟶ Z be a projective morphism to a normal affine variety Z, where (X, Δ = S + A + B) is a purely log terminal pair, S = ⌞Δ⌟ is irreducible, (X, S) is log smooth, A ≥ 0 is a general ample ℚ-divisor, B ≥ 0 is a ℚ-divisor and (S, Ω + A|_{S}) is canonical, where Ω = (Δ−S)|_{S}. Assume that the stable base locus of K_{X} +Δ does not contain S. Let F = lim F_{l!}, where, for any positive and sufficiently divisible integer m, we let

F_{m} = Fix(|m(K_{X} + Δ)|_{S})/m.

If ϵ > 0 is any rational number such that ϵ(K_{X} + Δ) + A is ample and if Φ is any ℚ-divisor on S and k > 0 is any integer such that

(1) both kΔ and kΦ are Cartier, and

(2) Ω ∧ λF ≤ Φ ≤ Ω, where λ = 1 − ϵ/k, then

|k(K_{S} + Ω − Φ)| + kΦ ⊂ |k(K_{X} + Δ)|_{S}.

Proof. By assumption A = H/m, where H is very ample and a very general element of |H| and m ≥ 2 is an integer. If C = A/k, then

A + (k − 1)C = \frac{2k − 1}{km}H,

and so

(X, Δ + (k − 1)C = S + \frac{2k − 1}{km}H + B)

is purely log terminal, as

\frac{2k − 1}{km} < 1.

On the other hand,

(S, Ω + C|_{S}),

is canonical as we are even assuming that (S, Ω + A|_{S}) is canonical. Pick η > ϵ/k rational so that η(K_{X} + Δ) + C is ample and let μ = 1 − η < λ = 1 − ϵ/k. If l > 0 is any sufficiently divisible integer so that O = l(η(K_{X} + Δ) + C) is very ample, then

G_{l} = Fix(|l(K_{X} + Δ + C)|_{S})/l
= Fix(|lμ(K_{X} + Δ) + O|_{S})/l
≤ Fix(|lμ(K_{X} + Δ)|_{S})/l
= μF_{μl}.

Thus

lim G_{l!} ≤ μ lim F_{l!} = μF.

On the other hand (6.2) implies that there is a positive integer l such that every prime divisor on S which does not belong to the support of F does not belong to the base locus of |l(K_{X} + Δ + C)|. Thus we may pick a positive integer l such that

• k divides l,

• lC is Cartier, and

• G_{l} ≤ λF. Let f: Y ⟶ X be a log resolution of the linear system |l(K_{X} +Δ+

C)| and of (X, Δ + C). We may write

K_{Y} + Γ = f^{∗}(K_{X} + Δ + C) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, f_{∗}Γ = Δ + C and f_{∗}E = 0. Then

H_{l} = Fix(l(K_{Y} + Γ))/l = Fix(lf^{∗}(K_{X} + Δ + C))/l + E.

If Ξ = Γ − Γ ∧ H_{l} then l(K_{Y} + Ξ) is Cartier and Fix(l(K_{Y} + Ξ)) and Ξ share no common components. Since the mobile part of |l(K_{Y} + Ξ)| is free and the support of Fix(l(K_{Y} + Ξ)) + Ξ has normal crossings it follows that the stable base locus of K_{Y} + Ξ contains no non kawamata log terminal centres of (Y, ⌜Ξ⌝) (which are nothing but the strata of ⌜Ξ⌝).

Let H ≥ 0 be any ample divisor on Y. Pick positive integers m and q such that l divides m and Q = qH is very ample. Let T be the strict transform of S, let Γ_{T} = (Γ − T)|_{T} and let Ξ_{T} = (Ξ − T)|_{T}. If

τ ∈ H^{0}(T, O_{T} (m(K_{T} + Ξ_{T}))) = H^{0}(T, J_{∥m(K_{T} +Ξ_{T})∥}(m(K_{T} + Ξ_{T}))),

and σ ∈ H^{0}(T, O_{T} (Q)) then

σ · τ ∈ H^{0}(T, J_{∥m(K_{T} +Ξ_{T})∥}(m(K_{T} + Ξ_{T}) + Q)).

On the other hand, if q is sufficiently large and divisible then by (5.3) H^{0}(T, J_{∥m(K_{T} +Ξ_{T})∥}(m(K_{T} + Ξ_{T}) + Q)) is contained in the image of

H^{0}(Y, O_{Y} (m(K_{Y} + Ξ) + Q)) ⟶ H^{0}(T, O_{T} (m(K_{T} + Ξ_{T}) + Q)).

Hence there is a fixed q such that whenever l divides m, we have

|m(K_{T} + Ξ_{T})| + m(Γ_{T} − Ξ_{T}) + |Q|_{T} | ⊂ |m(K_{Y} + Γ) + Q|_{T} .

If g = f|_{T} : T ⟶ S then g_{∗}Γ_{T} = Ω+C|_{S} and since g_{∗}Ξ_{T} ≤ Ω+C|_{S} and (S, Ω + C|_{S}) is canonical, we have |m(K_{S} + g_{∗}Ξ_{T})| = g_{∗}|m(K_{T} + Ξ_{T})|. Therefore, applying g_{∗}, we obtain

|m(K_{S} + g_{∗}Ξ_{T})| + m(Ω + C|_{S} − g_{∗}Ξ_{T}) + P |_{S} ⊂ |m(K_{X} + Δ + C) + P |_{S},

where P = f_{∗}Q.

Since for every prime divisor L on S we have

mult_{L} G_{l} = mult_{L′} Fix(|l(K_{Y} + Γ)|_{T})/l = mult_{L′} H_{l}|_{T} ,

where L′ is the strict transform of L on T, it follows that

g_{∗}Ξ_{T} − C|_{S} = Ω − Ω ∧ G_{l} ≥ Ω − Ω ∧ λF ≥ Ω − Φ ≥ 0.

Therefore

|m(K_{S} + Ω − Φ)| + mΦ + (mC + P)|_{S} ⊂ |m(K_{X} + Δ + C) + P |_{S},

for any m divisible by l. In particular if we pick m so that C − \frac{k−1}{m}P is ample and (X, Δ + \frac{k−1}{m}P) is purely log terminal then the result follows by (4.1). □

7. Rationality of the restricted algebra

In this section we will prove:

Theorem 7.1. Assume Theorem F_{n−1}.

Let π: X ⟶ Z be a projective morphism to a normal affine variety Z, where (X, Δ = S +A+B) is a purely log terminal pair of dimension n, S = ⌞Δ⌟ is irreducible, (X, S) is log smooth, A ≥ 0 is a general ample ℚ-divisor, B ≥ 0 is a ℚ-divisor and (S, Ω + A|_{S}) is canonical, where Ω = (Δ−S)|_{S}. Assume that the stable base locus of K_{X} +Δ does not contain S. Let F = lim F_{l!} where, for any positive and sufficiently divisible integer m, we let

F_{m} = Fix(|m(K_{X} + Δ)|_{S})/m.

Then Θ = Ω − Ω ∧ F is rational. Moreover if both kΔ and kΘ are Cartier then

|k(K_{S} + Θ)| + k(Ω − Θ) = |k(K_{X} + Δ)|_{S},

and

R_{S}(X, k(K_{X} + Δ)) ≃ R(S, k(K_{S} + Θ)).

Proof. Suppose that Θ is not rational. Let V ⊂ WDiv_{ℝ}(S) be the vector space spanned by the components of Θ. Then there is a constant δ > 0 such that if Φ ∈ V and ∥Φ − Θ∥ < δ then Φ ≥ 0 has the same support as Θ and moreover, by (2) of Theorem F_{n−1}, if G is a prime divisor contained in the stable base locus of K_{S} + Θ then it is also contained in the stable base locus of K_{S} + Φ.

If l(K_{X} + Δ) is Cartier and Θ_{l} = Ω − Ω ∧ F_{l} then

|l(K_{X} + Δ)|_{S} ⊂ |l(K_{S} + Θ_{l})| + l(Ω ∧ F_{l}).

Hence Fix(l(K_{S} + Θ_{l})) does not contain any components of Θ_{l}. In particular the stable base locus of K_{S} + Θ_{l} does not contain any components of Θ_{l}. But we may pick l > 0 so that Θ_{l} ∈ V and ∥Θ_{l}−Θ|| < δ. It follows that no component of Θ is in the stable base locus of K_{S} +Θ.

Let W ⊂ V be the smallest rational affine space which contains Θ.

(3) of Theorem F_{n−1} implies that there is a positive integer r > 0 and a positive constant η > 0 such that if Φ ∈ W, kΦ/r is Cartier and ∥Φ − Θ∥ < η then every component of Fix(k(K_{S} + Φ)) is in fact a component of the stable base locus of K_{S} + Θ.

Pick a rational number ϵ > 0 such that ϵ(K_{X} + Δ) + A is ample. By Diophantine approximation, see (3.7.7) of [2], we may find a positive integer k, a divisor Φ on S and a prime divisor G (necessarily a component of Θ whose coefficient is irrational) such that

(1) 0 ≤ Φ ∈ W,

(2) both kΦ/r and kΔ/r are Cartier,

(3) ∥Φ − Θ∥ < min(δ, η, fϵ/k) where f is the smallest non-zero coefficient of F ≠ 0, and

(4) mult_{G} Φ > mult_{G} Θ.

Claim 7.2. Ω ∧ λF ≤ Ω − Φ ≤ Ω, where λ = 1 − ϵ/k.

Proof of (7.2). Let P be a prime divisor on S and let ω, f, ϕ and θ be the multiplicities of Ω, F, Φ and Θ along P. We just need to check that

(∗)  min(ω, λf) ≤ ω − ϕ.

There are two cases. If ω ≤ f, then θ = 0 so that ϕ = 0 and (∗) holds. If ω ≥ f, then θ = ω − f and since ∥Φ − Θ∥ < fϵ/k,

min(ω, λf) = (1 − \frac{ϵ}{k} ) f ≤ f − (ϕ − θ) = ω − ϕ.  □

(7.2), (2) and (6.3) imply that

|k(K_{S} + Φ)| + k(Ω − Φ) ⊂ |k(K_{X} + Δ)|_{S}.

(4) implies that G is a component of Fix(k(K_{S}+Φ)). (2) and ∥Φ−Θ∥ < η imply that G is a component of the stable base locus of K_{S} + Θ, a contradiction.

Thus Θ is rational. Hence Ω ∧ F is rational, and we are done by (6.3). □

8. Proof of (1.3)

Theorem 8.1. Assume Theorem F_{n−1}.

Let π: X ⟶ Z be a projective morphism to a normal affine variety Z. Suppose that (X, Δ = S + A + B) is a purely log terminal pair of dimension n, S = ⌞Δ⌟ is irreducible and not contained in the stable base locus of K_{X} + Δ, A ≥ 0 is a general ample ℚ-divisor and B ≥ 0 is a ℚ-divisor.

Then there is a birational morphism g: T ⟶ S, a positive integer l and a kawamata log terminal pair (T, Θ) such that K_{T} +Θ is ℚ-Cartier and

R_{S}(X, l(K_{X} + Δ)) ≅ R(T, l(K_{T} + Θ)).

Proof. If f: Y ⟶ X is a log resolution of (X, Δ) then we may write

K_{Y} + Γ′ = f^{∗}(K_{X} + Δ) + E,

where Γ′ ≥ 0 and E ≥ 0 have no common components, f_{∗}Γ′ = Δ and f_{∗}E = 0. If T is the strict transform of S then we may choose f so that (T, Ψ′ = (Γ′ − T)|_{T}) is terminal. Note that T is not contained in the stable base locus of K_{Y} +Γ′ as S is not contained in the stable base locus of K_{X} + Δ.

Pick a ℚ-divisor F such that f^{∗}A − F is ample and (Y, Γ′ + F) is purely log terminal. Pick m > 1 so that m(f^{∗}A − F) is very ample and pick mC ∈ |m(f^{∗}A − F)| very general. Then

(Y, Γ = Γ′ − f^{∗}A + F + C ∼_{ℚ} Γ′),

is purely log terminal (note that since A is general, f^{∗}A is equal to the strict transform of A) and if m is sufficiently large (T, Ψ + C|_{T}) is terminal, where Ψ = (Γ − T)|_{T}.

On the other hand

R(X, k(K_{X} + Δ)) ≅ R(Y, k(K_{Y} + Γ)) and
R_{S}(X, k(K_{X} + Δ)) ≅ R_{T} (Y, k(K_{Y} + Γ)),

for any k sufficiently divisible. Now apply (7.1) to (Y, Γ). □

Proof of (1.3). By (3.2) we may assume that Z is affine and by (3.5), it suffices to prove that the restricted algebra is finitely generated. As Z is affine, S is mobile and as f is birational, the divisor Δ − S is big. But then

Δ − S ∼_{ℚ} A + B,

where A is a general ample ℚ-divisor and B ≥ 0. As S is mobile, we may assume that the support of B does not contain S. Now

K_{X} + Δ′ = K_{X} + S + (1 − ϵ)(Δ − S) + ϵA + ϵB ∼_{ℚ} K_{X} + Δ,

is purely log terminal, where ϵ is any sufficiently small positive rational number. By (3.4), we may replace Δ by Δ′. We may therefore assume that Δ = S + A + B, where A is a general ample ℚ-divisor and B ≥ 0. Since we are assuming Theorem F_{n−1}, (8.1) implies that the restricted algebra is finitely generated. □

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

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

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

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