ON ZARISKI DECOMPOSITION PROBLEM

YURI G. PROKHOROV

Abstract. We discuss different generalizations of Zariski decomposition, relations between them and connections with finite generation of divisorial algebras.

Contents

1. Some preliminary facts 1 2. Classical Zariski decomposition 6 3. On finite generation of divisorial algebras 9 4. s-, σ- and sectional decompositions 12 5. Zariski decomposition in Shokurov’s sense and bss-ampleness 19 6. Zariski decomposition in Fujita’s sense 25 7. Zariski decomposition in CKM’s sense 30 8. Decompositions of b-divisors 32 9. Analytic Zariski decomposition 36 References 38

The purpose of this survey is to clarify some details in §§3-4 of [PLF] related to connections between different generalizations of Zariski decomposition and finite generation of divisorial algebras. The survey almost does not contain new results. All the discussed results were known earlier (see [1], [4], [9], [12], [15], [16], [17], [PLF], [21], [22]).

1. Some preliminary facts

All varieties are assumed to be normal, projective and defined over the field of complex numbers ℂ. However, almost all results can be generalized to the relative situation X/Z. Everywhere, if is not is indicated converse by a divisor we mean an ℝ-divisor, i.e., an ℝ-linear combination of prime Weil divisors. For any divisor D, we put

O_{X}(D) = O_{X}(⌊D⌋), |D| = {F | F ∼ D, f ≥ 0}.

This work was is carried out under the support of grants Leading Scientific Schools 00-15-96085, RFFI 02-01-00441, and INTAS-OPEN 2000-269.

For definitions and main properties of b-divisors we refer to Iskovskikh paper [Isk].

Recall that a divisor D is said to be big, if κ(X, D) = dim X, where

κ(X, D) =^{def} \begin{cases}−∞, & if |nD| = ∅ for all n ∈ ℕ, \\ max_{n∈ℕ}dim Φ_{|nD|}(X), & otherwise, \end{cases}

is the Iitaka D-dimension of (X, D). A divisor D is said to be (semi) ample if D = ∑ α_{i}H_{i}, where H_{i} are integral (semi) ample divisors and α_{i} ∈ ℝ_{>0}. By the Kleiman criterion, the ampleness is equivalent to the positiveness of the divisor on \overline{NE}(X) \ \{0\}. The semiampleness is equivalent to the existence of a contraction α: X → Y and an ample divisor H on Y such that D ∼_{ℝ} α^{∗}H.

Recall that a divisor F on a variety X is said to be b-semiample (bnef ) if there exists a model Y dominating X and an ℝ-Cartier divisor L on Y such that the b-divisor \overline{L} is b-semiample (respectively, b-nef) and F = (\overline{L})_{X}. Equivalent: there exists a birational contraction f: Y → X and a semiample (respectively, nef) ℝ-Cartier divisor L on Y such that f_{∗}L = F. On a nonsingular variety the b-semiampleness and semiampleness (respectively, b-nef and nef properties) are equivalent [23, Th. 6.1]. 1.1. A rational 1-contraction is a dominant rational map α: X −−→ Y with connected fibers such that

dim Center_{Y} G < dim G (= dim X − 1)

for any prime exceptional b-divisor G on X.

According to Hironaka for a rational 1-contraction α: X −−→ Y there exists a hut

(1.2)  [figure: W; g; h; X; α; Y]

satisfying the following conditions:

(i) morphisms g: W → X and h: W → Y are contractions, and the contraction g is birational,

(ii) if a prime divisor A on W is contracted by g, then it also is

contracted by h. Example 1.3. Let Y = ℙ^{1} and let W = ℙ_{Y} (E) be the projectivization of the vector bundle E = \underbrace{O_{ℙ^{1}} ⊕ ⋯ ⊕ O_{ℙ^{1}}}_{n} ⊕O_{ℙ^{1}}(d), d ≥ 1. Let g: W → X ⊂ ℙ^{d+n} be the morphism defined by the linear system |O_{ℙ(E)}(1)|. Then g is a birational contraction and we have diagram (1.2). Here X ⊂ ℙ^{d+n} is a cone over a rational curve C_{d} ⊂ ℙ^{d} with vertex ℙ^{n−1} and the map α is defined by the linear system of planes on X. The base locus of this system is precisely the vertex of the cone. 1.4. For a rational 1-contraction α: X −−→ Y, we may define the pullback of any ℝ-Cartier divisor D as follows: α^{∗}D =^{def} g_{∗}h^{∗}D (it is easy to show that this definition does not depend on the choice of the hut (1.2)). Note however that the map α^{∗} is not functorial: it is possible that (α ∘ β)^{∗} does not coincide with β^{∗}α^{∗}. Similarly we can define α_{∗}. For α_{∗} we always have α_{∗}β_{∗} = (αβ)_{∗} whenever β is a rational 1-contraction.

For a closed subset V ⊂ X, denote by [V]^{r} the subset consisting of all components V_{i} ⊂ V of dimension ≥ r.

A divisor E on X is said to be exceptional with respect to a rational 1contraction α if dim α(E_{i}) < dim E_{i} and α(E_{i}) ≠ Y for all components E_{i} ⊂ Supp(E). In this situation E is said to be very exceptional if for any prime divisor G on Y the divisorial fiber α^{•}(G) over G is not contained in Supp(E) (see [PLF, Def. 3.2] or [9, Prop. 1.10]). Here the divisorial fiber is defined as α^{•}(G) =^{def} [α^{−1}(G)]^{dim X−1}. Remark 1.5. Let α be a morphism. To check that an exceptional divisor E is very exceptional it is sufficient to verify the following property:

for any component E_{i} ⊂ Supp(E) such that dim α(E_{i}) = dim Y − 1 we have Supp(α^{∗}α(E_{i})) ⊄ Supp(E).

Assume that α is a morphism and let d = dim X − dim Y. Define the following closed subset in Y:

𝔈(α, E) =^{def} {y ∈ Y | [α^{−1}(y)]^{d} ⊂ Supp(E)} ,

where α^{−1}(y) = f(h^{−1}(y)). It is clear that 𝔈(α, E) ⊂ α(Supp(E)). Then E is very exceptional if and only if codim 𝔈(α, E) ≥ 2.

The following fact will be frequently used without references: Lemma 1.6 ([19, §1.1], [20, 2.15]). Let f: X → Z be a birational contraction and A be an ℝ-Cartier divisor on X such that

(i) f contracts all components of A with negative coefficients; (ii) for a sufficiently general curve C in a fiber A_{i}/f(A_{i}) in each f-exceptional divisor A_{i} having negative coefficient in A, we have A · C ≤ 0.

Then the divisor A is effective.

We need also the following result:

Lemma 1.7. Let f: X → Z be a contraction (not necessary birational) and let A be a divisor on X. Write A = A^{+} − A^{−}, where A^{+} and A^{−} are effective divisors without common component. Assume that −A is f-nef and A^{−} is very exceptional on Z. Then A^{−} = 0, i.e., A is effective.

Proof. Assume that dim X > dim Z. Let A^{−} ≠ 0. If dim f(Supp(A^{−})) > 0, then we can replace Z with its general hyperplane section Z′ ⊂ Z, X with f^{−1}(Z)′, and A with A|_{f^{−1}(Z)′}. The very exceptionality of A^{−} is preserved. Indeed, it is sufficient to choose a hyperplane section Z′ ⊂ Z so that it does not contain components of the set 𝔈(α, E) of codimension 2. Continuing the process we get the situation when dim f(Supp(A^{−})) = 0. We may also assume that Z is a sufficiently small affine neighborhood of some fixed point o ∈ Z (and f(Supp(A^{−})) = o). Further, all the conditions of lemma are preserved if we replace X with its general hyperplane section X′. If dim Z > 1, then we can reduce our situation to the case dim X = dim Z. Then the statement of the lemma follows by Lemma 1.6 and from the existence of the Stein factorization. Finally, consider the case dim Z = 1 (here we may assume that dim X = 2). In this instance, A · A^{−} = A^{+} · A^{−} − (A^{−})^{2} ≤ −(A^{−})^{2}. By the Zariski lemma the last number is positive, a contradiction. □

Movable and fixed parts of a divisor. For a divisor D, we put

(1.8)  Mov(D) = \begin{cases}− inf_{s∈K(X)^{∗}}\{(s) | D + (s) ≥ 0\} & if |D| ≠ ∅, \\ −∞ & otherwise. \end{cases}

If H^{0}(O(D)) ≠ 0 (that is equivalent to |D| ≠ ∅), we put

Fix(D) = inf{L | L ∼ D, L ≥ 0} = D − Mov(D).

Divisors Mov(D) and Fix(D) are called mobile and fixed parts of a divisor D respectively. If D is effective, then so is Mov(D):

Mov(D) ≥ −(Const) = 0.

Obviously, Mov(D) = Mov(⌊D⌋). Lemma 1.9 (cf. [PLF, Prop. 4.15]). Let D be a divisor such that H^{0}(O(D)) ≠ 0. Then M = Mov(D) satisfies the following properties:

(i) M ≤ D; (ii) the linear system |M| has no fixed components (i.e., the divisor

M is b-free); (iii) if a divisor L ≤ D is b-free, then L ≤ M.

Conversely, if an (integral) divisor M satisfies conditions (i)-(iii), then M = Mov(D).

Proof. It is clear that

D − M = inf_{s∈K(X)^{∗}}{D + (s) | D + (s) ≥ 0} ≥ 0.

Since

0 = inf_{s∈K(X)^{∗}}{M + (s) | D + (s) ≥ 0}

and D ≥ M, we have that the divisor D + (s) is effective if and only if so is M + (s). Hence,

0 = inf_{s∈K(X)^{∗}}{M + (s) | M + (s) ≥ 0}.

This means that the linear system has no fixed components. Finally, let a divisor L ≤ D b-free. Then

L = − inf_{s∈K(X)^{∗}}{(s) | L + (s) ≥ 0} ≤ − inf_{s∈K(X)^{∗}}{(s) | D + (s) ≥ 0} = M.

The last statement follows from the uniqueness of a divisor M satisfying conditions (i)-(iii). □

Lemma 1.10. Let D be a divisor such that H^{0}(O(D)) ≠ 0. Then M = Mov(D) satisfies the following properties:

(i) M ≤ D; (ii) H^{0}(O(M)) = H^{0}(O(D)); (iii) if for a divisor L ≤ D the equality H^{0}(O(L)) = H^{0}(O(D))

holds, then L ≥ M. Conversely, if an (integral) divisor M satisfies conditions (i)-(iii), then M = Mov(D).

Proof. (ii) Since M ≤ D, H^{0}(O(M)) ⊂ H^{0}(O(D)). Let s ∈ H^{0}(O(D)). Then D + (s) ≥ 0 and according to (1.8) we have M + (s) ≥ 0, i.e., s ∈ H^{0}(O(M)).

(iii) Let H^{0}(O(L)) = H^{0}(O(D)). Then L + (s) ≥ 0 if and only if D + (s) ≥ 0. Hence,

M = − inf_{s∈K(X)^{∗}}{(s) | D + (s) ≥ 0} = − inf_{s∈K(X)^{∗}}{(s) | L + (s) ≥ 0}.

Therefore,

L − M = inf_{s∈K(X)^{∗}}{L + (s) | L + (s) ≥ 0} ≥ 0.

□

2. Classical Zariski decomposition

We say that a divisor D is effective modulo ℚ-linear (ℝ-linear) equivalence if D ∼_{ℚ} D′ (respectively, D ∼_{ℝ} D′), where D′ is effective. The effectiveness modulo ℚ-linear equivalence is equivalent to that H^{0}(O(αD)) ≠ 0 for some α ∈ ℕ.

Zariski decomposition and its various generalizations have the form D = P + N, where N is the effective part and P is the “maximal positive” part. In general such a decomposition is defined for effective divisors though many statements work in a more general situation:

Definition 2.1 ([7]). A divisor D on projective variety X is said to be pseudo-effective if there exists an ample divisor H such that D + εH is effective for any ε > 0.

If D is an ℝ-Cartier divisor, then its pseudo-effectiveness is equivalent to any of the following conditions (see [14]):

(i) the class of D is contained in the closure of the cone of effective divisors (in numerical sense), i.e., there exists a sequence of effective divisors D^{(n)} such that lim (D^{(n)} · C) = D · C for any curve C;

(ii) the divisor D + εH is effective (modulo ∼_{ℝ}) for any ample

divisor H and any ε > 0; (iii) D · ℓ ≥ 0 for any nef 1-cycle ℓ on X.

It is easy to see that the property of an ℝ-Cartier divisor to be pseudo-effective is closed under taking pull-backs f^{∗}.

Theorem 2.2 ([23], [7]). Let X be a surface and let D be a pseudoeffective divisor on X. Then there exists an effective divisor N = N(D) = ∑ a_{i}N_{i} on X such that

(i) the divisor P =^{def} D − N is nef;

(ii) either N = 0 or the matrix (N_{i} · N_{j}) is negative definite; (iii) (P · N_{i}) = 0 for all i.

Furthermore, if D is a ℚ-divisor, then so is N. The divisor N(D) is uniquely defined by the class of numerical equivalence of D.

A decomposition D = N + P satisfying conditions (i) - (iii) of Theorem 2.2 is called a Zariski decomposition of a divisor D. The divisor P is its positive and N is its negative (or exceptional [PLF]) parts. Sometimes, by abuse of language, we say that a Zariski decomposition is the divisor N = N(D).

Note that in the two-dimensional case the ℝ- (or ℚ)-Cartier condition is not necessary: the intersection theory is defined for any normal surface (see, e.g.., [18]). We give a sketch of proof with running the “D-MMP” (see [18]):

Sketch of the proof. If D is nef, then we put N(D) = 0. Otherwise there exists an irreducible curve E such that D · E < 0. Since D is pseudo-effective, we have E^{2} < 0 (otherwise E is a nef curve and by definition 2.1, D · E ≥ 0). By the Grauert criterion, the curve E is contractible (at least in the category of normal analytic spaces): f_{1} : X → X_{1}, where the divisor f_{1∗}D is again pseudo-effective. Continuing the process we obtain a model X′ on which the image D′ of D is nef (in other words, we run “D-minimal model program”). Let f: X → X′ be the composition of all contractions. Put P =^{def} f^{∗}D′ and N =^{def} D − P = D − f^{∗}f_{∗}D. Since the divisor −D is nef over X′, we have N ≥ 0 (see Lemma 1.6). The uniqueness follows by Proposition 2.5 below. □

Note however that in the category of projective surfaces contraction X → X′ does not necessarily exist. In other words, the divisor P is not always semiample. Our proof shows that we may guarantee the semiampleness of P for divisors of type D = K_{X} + B whenever the pair (X, B) is log canonical.

In his paper [23] Zariski considered the case of effective and integral divisor D. The generalization to the pseudo-effective case belongs to Fujita [7]. Below, in Example 6.18 we will see that in higher dimensional generalizations one has to consider divisors with irrational coefficients. We discuss the different higher dimensional generalizations of Theorem 2.2 and relations between them.

It is easy to see that Zariski decompositions agree via pull-backs. That is why we have the following.

Proposition 2.3. For any pseudo-effective divisor D on a surface X, there exists a decomposition \overline{D} = 𝒫 + 𝒩 of b-divisors in the group BCDiv_{ℝ}(K(X)) of b-Cartier b-divisors (see[Isk]) that induce Zariski decompositions (\overline{D})_{Y} = 𝒫_{Y} +𝒩_{Y} on all normal projective models Y/X.

Simple examples show also that Zariski decompositions does not agree via f_{∗}. That is why 𝒫_{Y} and 𝒩_{Y} are not elements of BDiv_{ℝ}(K(X)).

Example 2.4. Consider the quadratic birational map ℙ^{2} −−→ ℙ^{2}, (x : y : z) −−→ (yz : xz : xy) and let

X

[figure: σ; f]

ℙ^{2}  ^{}ℙ^{2}

be a resolution of the indeterminacy. Here σ: X → ℙ^{2} and f: X → ℙ^{2} are blowups of triples of different points on ℙ^{2}. Let D =^{def} σ^{∗}H + E_{1} + E_{2} + E_{3}, where H is an ample divisor on ℙ^{2} and let E_{1}, E_{2}, E_{3} be exceptional divisors of σ. Then N(D) = E_{1}+E_{2}+E_{3} and P (D) = σ^{∗}H. But N(f_{∗}D) = 0 ≠ f_{∗}(E_{1} + E_{2} + E_{3}) and P (f_{∗}D) = f_{∗}D ≠ f_{∗}σ^{∗}H. Proposition 2.5. Let X be a surface and let D be a pseudo-effective divisor on X. Let D = N(D)+P (D) be a Zariski decomposition. Then for any nef divisor L such that L ≤ D we have L ≤ P (D).

Proof. Write N = N(D) = ∑ a_{i}N_{i}. Let L ≤ D, where L is nef. For all i we have

N_{i} · (P − L) = −N_{i} · L ≤ 0.

Write P − L = (D − L) − N = F^{♯} − N^{♯}, where F^{♯} and N^{♯} are effective divisors without common components. Since N^{♯} ≤ N,

0 ≥ N^{♯} · (P − L) = N^{♯} · F^{♯} − N^{♯2} ≥ 0.

This gives us N^{♯2} = 0, N^{♯} = 0 and P − L ≥ N. □

Proposition 2.5 follows also by Lemma 1.6 applied to the contraction f from the proof of Theorem 2.2. 2.6. Generalizations. It is clear that Zariski decomposition cannot be generalized in higher dimensions without significant modifications. We formulate general scheme for eventual generalizations. Let D be an ℝ-Cartier divisor on a variety X and let N(D) be the negative part in a (generalized) Zariski decomposition. It is reasonable to claim the following:

(i) the divisor N(D) is effective; (ii) the divisor D − N(D) is “positive” in some sense; (iii) N(D) is “minimal”.

Also it is reasonable to claim that the decompositions agree via pullbacks: N(D) is ℝ-Cartier and if f: Y → X is a birational contraction, then

(2.7)  N(f^{∗}D) = f^{∗}N(D).

Remark 2.8. The last property allow us to introduce Zariski decompositions of b-Cartier b-divisors: for D ∈ BCDiv_{ℝ}(K(X)) there exists 𝒩(D) ∈ BCDiv_{ℝ}(K(X)) and a projective model Y of field K(X) such that 𝒩(D)_{Y′} is a (generalized) Zariski decomposition for any model of Y′ of K(X) dominating Y.

3. On finite generation of divisorial algebras

One of fundamental problems of algebraic geometry is the question about finite generation of algebras

R_{X}D = ⨁_{n≥0} H^{0}(X, O_{X}(nD)).

Here R_{X}D is considered as a subalgebra of the algebra K(X)[t], where each space H^{0}(O_{X}(nD)) is enclosed in the component K(X)t^{n}. Below we present several well known facts. Proposition 3.1 (see, for example, [2, ch. III, §1, 3^{o}], [PLF, Th. 4.6]). Let R = ⨁_{n≥0} R_{n} be a graded algebra over a field k such that R_{0} = k. Then

(i) the algebra R is finitely generated if and only so is the trun-

cated algebra R^{[n_{0}]} =^{def} ⨁_{n≥0} R_{nn_{0}}, n_{0} ∈ ℕ;

(ii) if the algebra R is finitely generated, then there exists n_{0} ∈ ℕ such that the truncated algebra R^{[n_{0}]} is generated by elements degrees 1.

Proposition 3.2. Let D be an integral divisor such that Bs |D| = ∅. Then the algebra R_{X}D is finitely generated.

Proof. First, we prove this in the case, when D is ample. Let X ⊂ ℙ^{N} be an embedding corresponding to a suitable multiplicity n_{0}D of D and let J_{X} ⊂ O_{ℙ^{N}} be the ideal sheaf. From the exact sequence

0 ⟶ J_{X}(n) ⟶ O_{ℙ^{N}}(n) ⟶ O_{X}(nn_{0}D) ⟶ 0

and Serre’s vanishing theorem we obtain that the restrictions H^{0}(O_{ℙ^{N}}(n)) ⟶ H^{0}(O_{X}(n_{0}D)) are surjective for n ≥ n_{1}. Hence there is a surjective map R_{ℙ^{N}}^{[n_{1}]}O(1) → R_{X}^{[n_{0}n_{1}]}D. According to Proposition 3.1 this is proves finite generation of R_{X}D.

Now consider the general case. Let X → X̄ ⊂ ℙ^{N} be a morphism defined by the linear system |n_{0}D|, where n_{0} ≫ 0 and X̄ = φ(X). Consider its Stein factorization X ⟶^{φ} X′ ⟶^{ψ} X̄. Then the divisor H = ψ^{∗}OX̄(1) is ample and Mov(nD) = φ^{∗}H. Since φ_{∗}O_{X} = O_{X′}, R^{[n]}D ≃ R_{X′}H. According to the above the last algebra is finitely generated. □

Zariski decomposition and finite generation of divisorial algebras. 3.3. For a Zariski decomposition D = P + N, there is an isomorphism of graded algebras

(3.4)  R_{X}D ≃ R_{X}P.

Indeed, for each n ∈ ℕ we have

H^{0}(O_{X}(nP)) ⊂ H^{0}(O_{X}(nD)) = H^{0}(O_{X}(Mov(nD))).

On the other hand, the divisor Mov(nD) is nef and by Proposition 2.5 we have Mov(nD) ≤ nP. This gives us the inverse inclusion

H^{0}(O_{X}(Mov(nD))) ⊂ H^{0}(O_{X}(nP)).

Thus the question about finite generation of a divisorial algebra R_{X}D can be reduced to the question about finite generation of the divisorial algebra R_{X}P, where the divisor P is nef. It is well known that the algebra R_{X}D is not always finitely generated: Proposition 3.5 ([23], cf. [PLF, Th. 4.28]). Suppose that an integral effective Cartier divisor D satisfies the following two conditions:

(i) Fix |nD| ≠ ∅ for all n ∈ ℕ; (ii) multiplicities of components Fix |nD| are bounded as n → ∞.

Then the algebra R_{X}D is not finitely generated.

The proposition above enable us to construct a great number of divisors with non-finitely generated algebras R_{X}D (see Example 3.9 below).

Proof. Suppose that the algebra R_{X}D is generated by the finite number of elements u_{1}, …, u_{r}. Let d_{i} = deg u_{i} and d =^{def} max{d_{1}, …, d_{r}}. Then the vector space H^{0}(O_{X}(nD)) is generated by the monomials of the form

u_{1}^{ν_{1}}u_{2}^{ν_{2}} ⋯ u_{r}^{ν_{r}},

where

ν_{i} ∈ ℤ_{≥0},  ∑_{i=1}^{r} d_{i}ν_{i} = n.

Therefore,

Fix |nD| ≥ Min {∑_{i=1}^{r} ν_{i} Fix |d_{i}D| | ∑_{i=1}^{r} d_{i}ν_{i} = n} .

On the other hand,

\frac{d!}{k} Fix |kD| ≥ Fix |d!D| ≠ 0

for all k = 1, …, d. Thus divisors Fix |d_{1}D|,…, Fix |d_{r}D| have at least one common component. This gives us a contradiction. □

The condition (ii) in Proposition 3.5 is automatically satisfied if the divisor D is nef and big: Corollary 3.6. Let D be a nef and big integral Cartier divisor on a surface X such that Fix |nD| ≠ ∅ for all n ∈ ℕ. Then the algebra R_{X}D is not finitely generated.

Proof. Follows by Proposition 3.7 below. □

We say that the base locus Bs |nD| is bounded as n → ∞ if for any birational contraction f: Y → X multiplicities of components of Fix |nf^{∗}D| are bounded as n → ∞. In other words multiplicities of components of the b-divisor Fix(\overline{nD}) are bounded (see 8.1). Proposition 3.7 ([24, Th. 2.2]). Let D be an (integral) Cartier divisor on a variety X. If the base locus Bs |nD| is bounded as n → ∞, then the divisor D is nef. Conversely, if D is nef and big, then Bs |nD| is bounded.

Sketch of the proof. We give an outline of the proof only for dim X = 2. If D · C < 0 for some curve C, then C ⊂ Fix |nD| for any n ∈ ℕ. Furthermore, (nD − mC) · C < 0 whenever m < n\frac{D·C}{C^{2}} , i.e., mC ≤ Fix |nD| for such m. This is means that the multiplicity C in Fix |nD| goes to infinity, a contradiction.

Conversely, assume that D is nef and big. Let E_{i} be fixed components of |nD|. Obviously, we may assume that the surface X and all the E_{i} are nonsingular. Choose a very ample divisor H such that divisors H − K_{X} − E_{i} are ample for all E_{i}. We prove that H^{0}(E_{i}, O_{E_{i}}(mD + H)) ≠ 0. By the Riemann-Roch Theorem this is satisfied if

(mD + H) · E_{i} > p_{a}(E_{i}) − 1 = \frac{1}{2}(K_{X} + E_{i}) · E_{i}.

The last is equivalent to

0 < (2mD+2H −K_{X} −E_{i})·E_{i} = 2mD·E_{i} +H ·E_{i} +(H −K_{X} −E_{i})·E_{i},

that, obviously, is satisfied. Further, H^{1}(O_{X}(mD+H −E_{i})) = 0. From exact sequence

0 ⟶ H^{0}(O_{X}(mD + H − E_{i})) ⟶ H^{0}(O_{X}(mD + H))

⟶ H^{0}(O_{E_{i}}(mD + H)) ⟶ 0

we obtain that E_{i} is not a fixed component of |mD + H|. Therefore the linear system |mD + H| has no fixed component for m ∈ ℤ_{≥0}. Finally, since D is big, for some a ∈ ℕ we have aD ∼ H + F, where F is an effective divisor. Therefore, Fix |(a + m)D| = Fix |F + H + mD| is bounded by the divisor F, a contradiction. □

From Corollary 3.6 (see also Example 4.11 below) taking into account isomorphism (3.4) we obtain the following criterion. Theorem 3.8 ([23], cf. [PLF, Th. 4.28] and Theorem 4.10). Let X be a surface and let D be an effective modulo ℚ-linear equivalence ℚ-Cartier divisor on X. Then the algebra R_{X}D is not finitely generated if and only if κ(X, D) = 2 and the divisor P (D) is not semiample. Example 3.9 (Zariski). Consider a nonsingular cubic curve C ⊂ ℙ^{2}. Pick 12 points P_{1}, …, P_{12} ∈ C so that the divisor O_{C}(4) − ∑ P_{i} is not a torsion in Pic(C). Let σ: X → ℙ^{2} be the blowup of P_{1}, …, P_{12} and let E_{1}, …, E_{12} be the corresponding exceptional divisors. Put D =^{def} σ^{∗}O_{ℙ^{2}}(4) − ∑ E_{i}. It is easy to show that the divisor D is effective modulo linear equivalence, nef and big. Therefore in a Zariski decomposition one has N(D) = 0. Furthermore, D · C̃ = 0 and the birational transform C̃ =^{def} σ^{−1}(C) of the curve C is a fixed component of the linear system |nD| for any n ∈ ℕ (otherwise nD|C̃ = 0). Therefore D satisfies the conditions of Proposition 3.5 and the algebra R_{X}D is not finitely generated.

For a big divisor, there is also the following criterion of finite generation of the algebra R_{X}D: Theorem 3.10 ([24, Th. 1.2], cf. [PLF, Th. 3.18]). Let D be an (integral) big Cartier divisor. Then the following conditions are equivalent:

(i) the algebra R_{X}D is finitely generated; (ii) there exists n ∈ ℕ and a birational contraction f: X̃ → X such that the linear system Mov |nf^{∗}D| defines a birational morphism contracting all the components of Fix |nf^{∗}D|.

4. s-, σ- and sectional decompositions

s-decomposition. Let D be an effective modulo ℚ-linear equivalence divisor. Put

M_{n} =^{def} Mov(nD).

Thus nD = M_{n} + F_{n}, where F_{n} ≥ 0. Put also

(4.1)  P_{s}(D) =^{def} lim sup_{n→∞}(M_{n}/n).

Since M_{n_{1}n_{2}} ≥ n_{2}M_{n_{1}} for all n_{1}, n_{2} ∈ ℕ, we have

nP_{s}(D) ≥ M_{n} for all n ∈ ℕ.

We obtain a decomposition

D = P_{s}(D) + N_{s}(D), N_{s}(D) ≥ 0,

which we call an s-decomposition. The divisor P_{s}(D) is called its positive and N_{s}(D) its negative part. In case, when D is not effective modulo ℚ-linear equivalence, we put P_{s}(D) = −∞. Obviously,

N_{s}(D) = inf_{\begin{matrix}n∈ℕ \\ s∈K(X)^{∗} \end{matrix}} \{D + (s)/n | D + (s)/n ≥ 0\}.

In other words

(4.2)  N_{s}(D) = inf{L | L ∼_{ℚ} D, L ≥ 0}.

Decompositions of such type were used by many authors: [13, §2], [17, §2], [PLF, §§3-4]. Note that P_{s}(D) and N_{s}(D) are not necessarily ℚ-divisors, even if the divisor D is integral (see Example 6.18). Remark 4.3. Let X be a surface and let D be an effective modulo ∼_{ℚ} divisor on X. Then the divisor P_{s} is nef and by Proposition 2.5 we have P ≥ P_{s}, where P = P (D) is the positive part of the classical Zariski decomposition. In general case, this inequality is not an equality (see Example 4.5). If κ(X, D) ≥ 1 and surface ℚ-factorial, then equality P = P_{s} holds, i.e., the s-decomposition coincides with the classical Zariski decomposition.

Indeed, if the divisor D is big, then so is the divisor P and by the Kodaira lemma P = A + F, where A is ample and F is an effective divisor. Since for any 0 < ε < 1, the divisor (1−ε)P +εA is ample and (1 − ε)P + εA ≤ P, we have (1 − ε)P + εA ≤ P_{s}. Therefore, P ≤ P_{s} (see also Proposition 4.21 below).

If κ(X, D) = 1, then κ(X, P) = 1, P^{2} = 0 and for some n ∈ ℕ the linear system Mov(nP) defines a contraction f: X → Z onto a curve. It is easy to see that

P · Mov(nP) = P · Fix(nP) = Mov(nP)^{2} =

Mov(nP) · Fix(nP) = Fix(nP)^{2} = 0.

Hence the divisor Fix(nP) is contained in fibers and is a pull-back of some divisor on Z. Therefore we can write nP ∼ f^{∗}L. Then N_{s}(P) = \frac{1}{n}N_{s}(f^{∗}L) (see 4.6 below). Since Mov(mf^{∗}L) ≥ f^{∗} Mov(mL), P_{s}(f^{∗}L) ≥ f^{∗}P_{s}(L) = f^{∗}L. Thus P_{s}(f^{∗}L) = f^{∗}L, N_{s}(P) = N_{s}(f^{∗}L) = 0, P_{s}(P) = P and P_{s}(D) ≥ P_{s}(P) = P. σ-decomposition. In the work [17] Nakayama defined a similar type of decompositions for any pseudo-effective divisor D, so-called, σdecomposition:

D = P_{σ}(D) + N_{σ}(D).

If D is a big divisor, then

N_{σ}(D) =^{def} N_{s}(D) = inf{L | L ∼_{ℚ} D, L ≥ 0}.

If D is a pseudo-effective, but not big divisor, then we put

N_{σ}(D) =^{def} lim_{ε→0} N_{σ}(D + εA),

where A is an arbitrary ample divisor. (It is easy to show that this definition does not depend on the choice of A).

From now on \overline{Mv}(X) denotes the closed convex cone in N^{1}(X) generated by the classes of mobile Cartier divisors and Mv^{o}(X) denotes the interior of \overline{Mv}(X). By the Kodaira lemma ewe have that if the class of any ℚ-divisor D is contained in Mv^{o}(X), then some multiplicity nD, n ∈ ℕ is an integral mobile divisor [13, §2]. Proposition 4.4 ([17, 2.1. 10]). σ-decomposition of a pseudo-effective divisor D on a nonsingular (enough: on ℚ-factorial) variety satisfies the following properties:

(i) [P_{σ}(D)] ∈ \overline{Mv}(X); (ii) for any divisor L such that L ≤ D and [L] ∈ \overline{Mv}(X) we have

L ≤ P_{σ}(D).

In particular, on a nonsingular surface the σ-decomposition coincides with the classical Zariski decomposition.

Thus P_{σ}(D) ≥ P_{s}(D) for any effective modulo ∼_{ℚ} divisor D. If the divisor D is big, then (by definition) P_{σ}(D) = P_{s}(D). However, this is not true for arbitrary effective divisors even in the two-dimensional case (see Example 4.5 below). Example 4.5. Let C ⊂ ℙ^{2} be a nonsingular cubic curve and let P_{1}, …, P_{9} ∈ C be distinct points such that O_{C}(3) − ∑ P_{i} is not a torsion in Pic(C). Let σ: X → ℙ^{2} be the blowup of points P_{1}, …, P_{9} and let D be the birational transform C. Then dim |nD| = 0 for all n ∈ ℕ. Therefore, P_{s}(D) = 0. On the other hand, D is nef. Hence, P_{σ}(D) = P (D) = D.

Properties of s-decompositions. The following properties are immediate consequences of the definition. 4.6. (i) The negative part of an s-decomposition depends only

on the class of ℚ-linear equivalence of the divisor D. More precisely, if D and D′ are ℚ-linearly equivalent and effective modulo ∼_{ℚ} divisors, then N_{s}(D) = N_{s}(D)′. (ii) If D′ and D′′ are effective modulo ∼_{ℚ} divisors, then

P_{s}(D)′ + P_{s}(D′′) ≤ P_{s}(D′ + D′′), N_{s}(D)′ + N_{s}(D′′) ≥ N_{s}(D′ + D′′).

If additionally D ≥ D′, then P_{s}(D) ≥ P_{s}(D)′. (iii) Let D be an effective modulo ∼_{ℚ} divisor. Then for any α ∈ ℚ_{>0}

we have N_{s}(αD) = αN_{s}(D). (iv) Computing P_{s} we always can replace the limit (4.1) on the

“truncated” limit:

P_{s}(D) = lim sup_{n→∞} \frac{M_{nn_{0}}}{nn_{0}} .

(v) For an ample divisor D, we have P_{s}(D) = D. Proposition 4.7. Let D be an effective modulo ℚ-linear equivalence divisor. If a ℚ-divisor L is b-semiample and L ≤ D, then L ≤ P_{s}(D).

Proof. Write L = ∑ α_{i}H_{i}, where H_{i} are integral b-free divisors and α_{i} ∈ ℝ_{≥0}. Since the coefficients of the divisor L are rational, we can choose numbers α_{i} also to be rational. Therefore nL is an (integral) b-free divisor for some n ∈ ℕ. By Lemma 1.9 we have L ≤ M_{n}/n ≤ P_{s}. □

The following easy statement shows how s-decompositions can be used in the study of divisorial algebras. Proposition 4.8. An s-decomposition of an effective modulo ℚ-linear equivalence divisor satisfies the following properties:

(i) P_{s}(D) ≤ D; (ii) R_{X}D = R_{X}P_{s}(D); (iii) for any divisor L such that L ≤ D and R_{X}D = R_{X}L we have

L ≥ P_{s}(D) (i.e., P_{s} is the smallest divisor satisfying properties (i) and (ii)).

Thus this proposition and [PLF, Remark 3.30] explain and justify introduction of the concept “s-decomposition”.

Proof. Since P_{s} ≥ M_{n}/n we have by Lemma 1.10 that for any n ∈ ℕ we have

H^{0}(O_{X}(M_{n})) = H^{0}(O_{X}(nD)) ⊃ H^{0}(O_{X}(nP_{s})) ⊃ H^{0}(O_{X}(M_{n})).

This proves (ii).

We prove (iii). Since H^{0}(O(nD)) = H^{0}(O(nL)), nL ≥ M_{n} (again by Lemma 1.10). Hence, L ≥ P_{s}(D).

The converse follows from the fact that there exists at most one divisor satisfying properties (i) - (iii). □

Remark 4.9. Proposition 4.8 remains to be true if we replace the condition (iii) with the following:

(iii′) for any divisor L such that L ≤ P_{s}(D) and R_{X}D = R_{X}L we

have L = P_{s}(D). (i.e., P_{s} is the minimal divisor satisfying properties (i) and (ii)). Indeed, by Proposition 4.8 the s-decomposition satisfies properties (i), (ii), (iii)′. Conversely, let P′_{s} satisfies properties (i), (ii), (iii)′. Then from (iii) applied to P_{s} we have P′_{s} ≥ P_{s} and from (iii)′ we obtain P′_{s} = P_{s}. Theorem 4.10 (cf. [PLF, Th. 4.28]). Let D be an effective modulo ℚ-linear equivalence divisor. If the divisorial algebra R_{X}D is finitely generated, then P_{s}(D) = Mov(n_{0}D)/n_{0} for some n_{0} ∈ ℕ (in other words the limit (4.1) stabilizes).

Thus in the case when the algebra R_{X}D is finitely generated, the positive part of s-decomposition is a b-semiample ℚ-divisor. Under additional conditions for the positive part we obtain decompositions discussed in §§5-6. Theorem 4.10 is not a criterion: in this form the converse is not true. In fact, the condition P_{s}(D) = Mov(n_{0}D)/n_{0} is divisorial and is preserved under small birational contractions, while finite generation is essentially more subtle condition. In order to obtain a criterion of finite generation, we must consider the condition for the stabilization of limits on all blowups of the initial variety, i.e., to pass to a b-divisor (see [PLF, Th. 4.28] and Theorem 8.9). We notice that the condition of stabilization does not mean that P_{s}(D) = Mov(nD)/n for all n ≫ 0. However, this condition implies that P_{s}(D) = Mov(nn_{0}D)/(nn_{0}) for all n ∈ ℕ.

Proof. According to Proposition 3.1 there exists n_{0} ∈ ℕ such that the algebra R_{X}(n_{0}D) is generated by the elements u_{1}, …, u_{r} ∈ H^{0}(X, O_{X}(n_{0}D))^{∗} = H^{0}(X, O_{X}(n_{0}D)) \ {0}. For any n ∈ ℕ and for any s ∈ H^{0}(X, O_{X}(nn_{0}D)), we have

s = ∑_{ν_{1}+···+ν_{r}=n} a_{ν_{1},...,ν_{r}}u_{1}^{ν_{1}} ⋯ u_{r}^{ν_{r}}, ν_{1}, …, ν_{r} ∈ ℤ_{>0}, a_{ν_{1},...,ν_{r}} ∈ ℂ.

Hence,

ord_{F} (s) ≥ min_{ν_{1}+···+ν_{r}=n}(ν_{1} ord_{F} (u_{1}) + ⋯ + ν_{r} ord_{F} (u_{r}))

≥ n inf_{u∈H^{0}(O_{X}(n_{0}D))^{∗}} ord_{F} (u).

By definition we obtain Mov(nn_{0}D) ≤ n Mov(n_{0}D). Since the inverse inequality always holds, we have Mov(nn_{0}D) = n Mov(n_{0}D). This proves the theorem. □

Examples 4.11. (i) If κ(X, D) = 0, then M_{n} = 0 or −∞ for all n. Moreover there exists n_{0} ∈ ℕ such that H^{0}(O(n_{0}mD)) = 1 for all m ∈ ℕ. Therefore, M_{n_{0}m} = 0. This gives us P_{s}(D) = 0.

(ii) If the class of the divisor D is contained in the open cone Mv^{o}(X), then D we may be approximated from below by b-semiample ℚ-divisors. Therefore, P_{s}(D) = D.

(iii) Let D be an effective divisor with κ(X, D) = 1. Then there exists n_{0} ∈ ℕ such that dim |nD| > 0 for all n ≥ n_{0}. Thus for every n ≥ n_{0} the linear system |M_{n}| defines a map X −−→ Y_{n} ⊂ ℙ^{N_{n}}. It is clear that K(Y_{n}) is a subfield in K(X) generated by H^{0}(O_{X}(M_{n})). Since M_{n} ≤ M_{n+1}, K(Y_{n}) ⊂ K(Y_{n+1}). All the fields K(Y_{n}) have transcendence degree 1 over ℂ. Therefore the algebraic closure \overline{K(Y_{n_{0}})} of K(Y_{n_{0}}) in K(Y) also has transcendence degree 1. Then there exists n_{1} ≥ n_{0} such that K(Y_{n}) = \overline{K(Y_{n_{0}})} for n ≥ n_{1}. Let Y be a nonsingular curve such that K(Y) = \overline{K(Y_{n_{0}})}. All rational maps X −−→ Y_{n}, n ≥ n_{0} factorise through Y:

X −−→^{g} Y ⟶^{h_{n}} Y_{n},

where h_{n} is a finite morphism. Therefore,

M_{n} = g^{∗}(sup{L ∈ Div(Y) | nD − f^{∗}L ≥ 0})

for n ≫ 0. Hence the divisor

P_{s}(D) = lim_{n→∞} M_{n}/n = g^{∗}(sup{F ∈ Div_{ℚ}(Y) | D − f^{∗}F ≥ 0})

is b-semiample. Obviously, the last divisor is a ℚ-divisor whenever so is D. In particular, we obtain that if D is a ℚ-divisor, then the algebra R_{X}D ≃ R_{X}P_{s} is finitely generated. The constructed map is a particular case of the Iitaka fibration of (X, D).

(iv) (see also Proposition 3.7) Suppose that an ℝ-Cartier divisor D is nef and big. Then by the Kodaira lemma there exists an effective divisor F such that the divisor D − F is ample. Therefore so is D − εF for any 0 < ε < 1. Thus,

P_{s}(D) ≥ P_{s}(D − εF) = D − εF.

Passing to the limit we obtain P_{s}(D) = D (cf. with Example 4.5). Remark 4.12. If N_{s}(D) = 0, then the divisor D is the limit of bsemiample divisors M_{n}/n, i.e., its class is contained in the closure of the cone Mv(X). In general case, this does not imply the b-semiampleness of D (see Example 3.9). Proposition 4.13 (cf. [12]). Let D be a big divisor on a ℚ-factorial variety. Then the negative part of s-decomposition depends only on the class of numerical equivalence of D.

Proof. Let D, D′ be big numerically equivalent divisors and let T =^{def} D′ − D ≡ 0. Since the divisor P_{s}(D) is big, the Kodaira lemma gives us P_{s}(D) = H + F, where F ≥ 0 and H is an ample divisor. Write

D′ = D + T = P_{s}(D) + T + N_{s}(D) = P_{s}(D) − εF + T + εF + N_{s}(D).

For any 0 < ε < 1, the class of P_{s}(D)−εF +T = (1−ε)P_{s}(D)+εH +T is contained in the cone Mv^{o}(X). According to Example 4.11, (ii) we have

P_{s}(D)′ ≥ P_{s}(P_{s}(D) − εF + T) = P_{s}(D) − εF + T.

Passing to the limit as ε → 0, we obtain P_{s}(D) + T ≤ P_{s}(D)′, i.e., N_{s}(D)′ ≤ N_{s}(D). By symmetry we have also the inverse inequality. □

Proposition 4.14. Let f: X → Y be a birational contraction and let D be an effective modulo ∼_{ℚ} divisor on X. Then

(4.15)  f_{∗}P_{s}(D) ≤ P_{s}(f_{∗}D).

Proof. For any n ∈ ℕ, we have f_{∗} Mov(nD) ≤ f_{∗}nD and the divisor f_{∗} Mov(nD) is b-free. Hence, f_{∗} Mov(nD) ≤ Mov(nf_{∗}D). This proves the statement. □

Example 2.4 shows that in general case inequality (4.15) is not an equality.

From Proposition 4.8 we obtain: Corollary 4.16 (cf. [17, Th. 3.5. 3]). Let f: Y → X be a birational contraction of ℚ-factorial varieties and let D be an effective modulo ∼_{ℚ} divisor on X. Then

f^{∗}P_{s}(D) ≥ P_{s}(f^{∗}D).

Note that the s-decomposition does not satisfy condition (2.7): Example 4.17. Let X be a blowup of ℙ^{3} in two distinct points P_{1} and P_{2}, and let D be the birational transform of a plane passing through P_{1} and P_{2}. Since the divisor D is mobile, N_{s}(D) = 0. Now let f: Y → X be the blowup of the birational transform of the line passing through P_{1} and P_{2} and let E be the exceptional divisor. Then N_{s}(f^{∗}D) = E ≠ f^{∗}N_{s}(D).

Sectional decomposition. Definition 4.18 ([13, §2]). A decomposition D = P_{sec}(D)+N_{sec}(D) is called a sectional decomposition if the following conditions are satisfied:

(i) N_{sec}(D) ≥ 0;

(ii) P_{sec}(D) ∈ \overline{Mv}(X) (in particular, P_{sec} is ℝ-Cartier);

(iii) there is an isomorphism of graded algebras

R_{X}D ≃ R_{X}P_{sec}.

From definition and Proposition 4.8 we immediately obtain: Proposition 4.19. Let D be an effective modulo ∼_{ℚ} divisor on a ℚfactorial variety. Then an s-decomposition is also a sectional decomposition. For any sectional decomposition we have P_{sec} ≥ P_{s} Remarks 4.20. (i) Note that unlike s- and the σ-decompositions a

sectional decomposition not necessary unique (in Example 4.5 there exists infinitely many sectional decompositions P_{sec}(D) = tD, 0 ≤ t ≤ 1, while P_{s}(D) = 0).

(ii) The set of all sectional of decompositions is closed: if there exists a sequence D = P_{sec}^{(r)} + N_{sec}^{(r)} of sectional decompositions and the limit P = lim_{r→∞} P_{sec}^{(r)} exists, then and D = P + (D − P) is also a sectional decomposition.

Proposition 4.21 ([13, §2]). Let D be a big divisor on a ℚ-factorial variety. Then a sectional decomposition is unique (and P_{sec} = P_{s} = P_{σ}). Furthermore, by Proposition 4.13 the negative part N_{sec}(D) is uniquely defined by the numerical class of D.

Sketch of the proof. According to Proposition 4.19 it is sufficient to prove the inequality P_{sec} ≤ P_{s}. Note that the divisor P_{sec} is big. As in the proof of Proposition 4.13 by the Kodaira lemma we write P_{sec} = H + F, where H is ample and F is effective. Then for any 0 < ε < 1 the class of P_{sec} − εF = (1 − ε)P_{sec} + εH is contained in the cone Mv^{o}(X). According to Example 4.11, (ii) we have

P_{s}(D) ≥ P_{s}(P_{sec} − εF) = P_{sec} − εF.

Passing to the limit, we obtain P_{sec} ≤ P_{s}. □

5. Zariski decomposition in Shokurov’s sense and bss-ampleness

Bss-ampleness. We need the following, almost obvious lemma. Lemma 5.1. Let D be a b-divisor of the fields K(X) and let L be a divisor on X such that D_{X} ∼_{ℝ} L. Then there exists a unique b-divisor Λ(L, D) such that Λ(L, D)_{X} = L and Λ(L, D) ∼_{ℝ} D. Definition 5.2 ([PLF, Def. 3.2], see also [9, Prop. 1.10]). A divisor D on X is said to be bss-ample if there exists a rational 1-contraction α: X −−→ Y, a (numerically) ample divisor H on Y, and a decomposition

(5.3)  D = D^{m} + E,

satisfying the following conditions:

(i) D^{m} ∼_{ℝ} α^{∗}H and there exists a b-semiample b-divisor H such that H_{X} = α^{∗}H;

(ii) the divisor E is effective and very exceptional with respect to

the map α; (iii) for any b-semiample b-divisor L such that L_{X} ≤ D we have

L ≤ Λ(D^{m}, H). Remark 5.4. (i) According to (iii) the b-divisor Λ(D^{m}, H) is

uniquely defined (i.e., it depends only on D). Therefore divisors D^{m}, E, and the contraction α also are uniquely defined. The b-divisor H is defined up to ℝ-linear equivalence. (ii) In the notation of Definition 5.2 the divisor D is big if and only if the rational 1-contraction α is birational (see [PLF, Prop. 3.20]).

(iii) The condition of (iii) in the definition is automatically satisfied

in cases dim X − dim Y ≤ 1 and dim Y = 0 (see [PLF, 3.4.3]). Examples 5.5. (i) Any semiample divisor D is bss-ample (we can put E = 0 and D^{m} = D). A b-semiample divisor D is bss-ample if and only if there exists the largest b-semiample b-divisor D^{m} = Λ(D^{m}, H) with D_{X}^{m} = D.

(ii) In the case κ(X, D) = 0 the divisor D is bss-ample if and only if D ∼_{ℝ} 0.

(iii) Consider the map α from Example 1.3 and let D be an ndimensional plane on X. Then D is a bss-ample divisor because D = α^{∗}O_{ℙ^{1}}(1).

(iv) A divisor D with κ(X, D) > 0 on a surface is bss-ample if and only if the positive part P (D) of the classical Zariski decomposition is semiample. In this instance decomposition (5.3) coincides with the Zariski decomposition, i.e., N(D) = E.

Indeed, by Zariski’s Main Theorem the map α is a morphism. In the case dim Y = 2 the contraction α is birational and the decomposition D = D^{m} + E coincides with the Zariski decomposition by definition and because the intersection matrix of exceptional divisors is negative definite. If dim Y = 1, then the intersection matrix (E_{i} ·E_{j}) is negative definite by the property of very exceptionality and semi-negativity of the intersection matrix in fibers of X → Y.

Conversely, if P (D) is a semiample divisor, then it defines a contraction α: X → Y, for which the divisor N(D) is exceptional. Since the intersection matrix of components of N(D) is negative definite, in case dim Y = 1, a fiber of the contraction α cannot be contained in Supp(N(D)).

Remark 5.6. Let D be a bss-ample divisor on X and let α: X −−→ Y be the corresponding rational 1-contraction. Then there exists a diagram

(5.7)  [figure: W; g; h; X; α; Y]

and a numerically ample divisor H on Y such that

g^{−1}D + F_{W} ∼_{ℝ} h^{∗}H + E_{W} ,

where divisors F_{W} and E_{W} are exceptional on X and Y, respectively, E_{W} is effective, and the divisor g(E_{W}) is very exceptional on Y. It is clear that H = \overline{h^{∗}H}.

Note however that divisors F_{W} and E_{W} are defined ambiguously. We can take as E_{W} the birational transform of E and in this instance E_{W} will be very exceptional on Y.

Obviously any bss-ample divisor is effective (more precisely, D ∼_{ℝ} D′, where D′ ≥ 0). However, the converse is not always true: the divisor D from Example 3.9 is effective and nef but it is not bss-ample. Indeed, this divisor is not semiample (because its restriction to C̃ is a numerically trivial divisor which is not a torsion).

Proposition 5.8. Let D be a big divisor. Assume that the D-MMP holds (including D-abundance conjecture). Then D is bss-ample.

The conditions of the proposition are satisfied when dim X ≤ 3 (and in any dimension modulo LMMP) in most important cases (see 6.8 below).

Proof. First replace (X, D) with its ℚ-factorialization, and then run the D-MMP. It is clear that bss-ampleness is invariant under birational isomorphisms in codimension 1. Thus it is sufficient to show that bssampleness is preserved under divisorial contractions:

Lemma 5.9. Let φ: X → X′ be a divisorial extremal D-negative contraction and let D′ = φ_{∗}D. Assume that D′ is big and bss-ample. Then so is D.

Proof. Let S be an (irreducible) φ-exceptional divisor. Then D = φ^{∗}D′ + aS, where a ≥ 0 according to Lemma 1.6. Consider Hironaka’s hut

W

[figure: δ; τ]

[figure: X; φ; X′; π; β; Y]

and let α =^{def} β ∘ φ. Then α is a rational 1-contraction. Furthermore, since D′ is big, the map α is birational. We have D′ = D^{′ m} +E′, where D^{′ m} ∼_{ℝ} β^{∗}H and the divisor H is ample. Put

E = φ^{∗}π_{∗}τ^{∗}H − δ_{∗}τ^{∗}H + φ^{∗}E′ + aS, D^{m} = D − E.

Then

D^{m} = φ^{∗}D′ + aS − (φ^{∗}π_{∗}τ^{∗}H − δ_{∗}τ^{∗}H + φ^{∗}E′ + aS) = φ^{∗}(D′ − β ∗ H′ − E′) + α^{∗}H ∼_{ℝ} α^{∗}H.

Further,

φ_{∗}E = π_{∗}τ^{∗}H − φ_{∗}δ_{∗}τ^{∗}H + E′ = E′ ≥ 0.

One can also see that the divisor E is exceptional on Y. Finally, by Lemma 1.6 the divisor φ^{∗}π_{∗}τ^{∗}H + φ^{∗}E′ + aS − δ_{∗}τ^{∗}H is effective. □

After a finite number of contractions and flips X −−→ X_{1} −−→ ⋯ −−→ X_{n} we obtain a model (X_{n}, D_{n}), on which the divisor D_{n} is nef (and big). By our abundance hypothesis the divisor D_{n} is semiample. Hence it is bss-ample. □

Note that it is not possible to omit the bigness condition in Proposition 5.8: Example 5.10. Let X = ℙ^{1} × 𝔽_{1}, let F be a fiber of the projection ℙ^{1} × 𝔽_{1} → ℙ^{1}, and let G = ℙ^{1} × C_{0} ⊂ X, where C_{0} is the negative section of 𝔽_{1}. Consider the divisor D = F + G. Then the contraction φ: X → ℙ^{1} × ℙ^{2} is a (unique) step of D-MMP and the divisor φ_{∗}D is nef. However, D is not bss-ample. Indeed, suppose that there is a decomposition such as in (5.3): D = D^{m} + E. Since the divisor D^{m} is b-semiample, we have that there exist at most a finite number of curves negatively intersecting D^{m}. On the other hand, X is a smooth Fano variety. Hence the divisor D^{m} is nef and the rational 1-contraction α is a morphism. Further, X is a quasihomogeneous variety. We obtain that for E there is only one possibility E = λG, λ ≥ 0. Then D^{m} = F +λG, where 0 ≤ λ ≤ 1. However, G is a fixed component of the linear system |n(F + λG)| for λ > 0. Therefore, λ = 0, D^{m} = F, and the rational 1-contraction α coincides with the projection X → ℙ^{1}. But the divisor G is not exceptional on ℙ^{1}, a contradiction.

Note that (X, Δ) is 0-pair for a suitable boundary Δ.

Zariski decomposition in Shokurov’s sense. Definition 5.11. A decomposition D = D^{m} + D^{e} is called a Zariski decomposition in sense of Shokurov (or simply Shokurov decomposition) if the following conditions are satisfied:

(i) D^{e} ≥ 0;

(ii) there exists a b-semiample b-divisor D^{m} such that D_{X}^{m} = D^{m}; (iii) for any b-semiample b-divisor L such that L_{X} ≤ D we have

L ≤ D^{m}. Thus any bss-ample divisor has a Shokurov decomposition (and it

coincides with (5.3)). In general case, the converse not always true: Example 5.12. Let X and D be such as in Example 5.10. Then we have the following Shokurov decomposition:

D^{m} = P_{s}(D) = F, D^{e} = G.

However, the divisor D is not exceptional for the morphism X → ℙ^{1} defined by the linear system |F |. Therefore D is not bss-ample.

Immediately from definition we obtain the following properties: 5.13. Let D be an effective modulo ℚ-linear equivalence divisor.

(i) If a Shokurov decomposition of D exists, then D^{m} ≥ P_{s}(D). If furthermore D^{m} is a ℚ-divisor, then D^{m} = P_{s}(D) (see Proposition 4.7).

(ii) If a Shokurov decomposition for of D exists, then R_{X}D^{m} =

R_{X}D = R_{X}P_{s}(D) (see Proposition 4.8). Remark 5.14. Let P_{s}(D) be a b-semiample ℚ-divisor. Then for the decomposition D = P_{s} + N_{s} conditions (i) and (ii) of Definition 5.11 are satisfied while (iii) is not necessary satisfied. Instead the following weaker condition holds:

(iii)′ for any b-semiample divisor L such that L ≤ D we have L ≤

D^{m}. Indeed, it is sufficient to show that for any b-semiample divisor L such that L ≤ D the inequality P_{s} ≥ L holds.

Assume that L ≰ P_{s}. Write P_{s} = ∑ α_{i}L_{i} and L = ∑ β_{i}L_{i}, where the L_{i} are integral b-free divisors, α_{i} ∈ ℚ_{>0}, and β_{i} ∈ ℝ_{>0}. Consider the finite dimensional real vector space V generated by the components of divisors P_{s} and L. The conditions Min(P_{s}, L) ≤ x ≤ Max(P_{s}, L) define a parallelepiped 𝔎 (possible not maximal dimension) in V and the bsemiample divisors L_{i} generate a convex rational polyhedron ℜ ⊂ V containing the diagonal [P_{s}, L] of 𝔎. So it is easy to see that there exists a rational divisor L′ into the interior of the set 𝔎 ∩ ℜ. From Proposition 4.7 we obtain L′ ≤ P_{s}. Contradiction with the fact that L ≰ P_{s}.

From Theorem 4.10 and 5.14 we obtain: Corollary 5.15 (cf. [PLF, Remark 3.30, Example 4.30]). Suppose that a divisorial algebra R_{X}D nontrivial and finitely generated. Then there exists a Shokurov decomposition (furthermore, D^{m} = P_{s} is a ℚdivisor).

Remark 5.14 and Corollary 5.15 are particular cases of [PLF, Example 4.30] (see also Proposition 8.12). Proposition 5.16. Let D be a big divisor on a ℚ-factorial variety X. If a Shokurov decomposition for D exists, then D^{m} = P_{s}. In particular, D^{e} depends only on the class of D modulo numerical equivalence.

Proof. Suppose that D^{m} ≩ P_{s}(D). Similar to the proof of Proposition 4.21, we can write D^{m} = H + F, where H is ample and F is an effective divisor. Then for any 0 < ε < 1 the class of the divisor D^{m} − εF = (1 − ε)D^{m} + εH is contained in Mv^{o}(X). According to Example 4.11, (ii) we have

P_{s}(D) ≥ P_{s}(D^{m} − εF) = D^{m} − εF.

Passing to the limit, we obtain D^{m} ≤ P_{s}, a contradiction. □

Example 5.17. The integral divisor D from Example 3.9 has no Shokurov decompositions. In fact, we can assert that if a divisor D on variety of any dimension is nef and for any n ∈ ℕ the linear system |nD| has fixed components of bounded multiplicity, then D has no Shokurov decompositions. Example 5.18. Let D be a divisor from Example 4.5. Since D^{2} = 0 and dim |nD| = 0 for all n ∈ ℕ, we have that D is not semiample. Therefore D has the following Shokurov decomposition D^{m} = 0 and D^{e} = D. It does not coincide with the classical Zariski decomposition N(D) = 0.

In the two-dimensional case there is a criterion of the existence of Shokurov decomposition (see Example 5.5 and Proposition 2.5): Proposition 5.19. Let X be a surface with ℚ-factorial singularities and let D be a divisor on X with the (classical) Zariski decomposition D = P + N. Then the following conditions are equivalent:

(i) the divisor P is semiample; (ii) the divisor D is bss-ample; (iii) there exists a Shokurov decomposition D = D^{m} + D^{e} such that

D^{e} = N and D^{e} = \overline{N}. Example 5.20. If in the conditions of Proposition 5.19 on X there exists a Shokurov decomposition D = D^{m} + D^{e} and κ(X, D) ≥ 1, then D^{e} = N(D).

Indeed, by Proposition 2.5 we always have P (D) ≥ D^{m} ≥ P_{s} and according to Remark 4.3 the equality holds. Theorem 5.21 ([PLF, Th. 3.33], cf. Proposition 5.8, Theorem 7.9). Let (X, B) be a 0-pair, i.e., a Kawamata log terminal pair with K_{X} + B ≡ 0 and let D be an effective modulo ℝ-linear equivalence divisor. Assume the LMMP (including the abundance conjecture). Then there exists a Shokurov decomposition for the divisor D.

In the conditions of this theorem there exist (and coincide between each other) all other Zariski decompositions (see below) for the divisor D.

6. Zariski decomposition in Fujita’s sense

Fujita noticed that property 2.5 characterizes Zariski decomposition: Definition 6.1. A decomposition D = P_{f} + N_{f} is called a Zariski decomposition in Fujita sense (or simply Fujita decomposition) if

(i) N_{f} ≥ 0; (ii) P_{f} is nef; (iii) for any b-nef divisor L such that L ≤ D we have L ≤ P_{f}. It is clear that we may set a question about the existence of such a

decomposition only for pseudo-effective divisors. Remarks 6.2. (i) It follows immediate from the definition that the

negative part of a Fujita decomposition depends only on the class of numerical equivalence of D.

(ii) It is clear that P_{f} ≥ M_{n}/n. Therefore, P_{f} ≥ P_{s}. So,

(6.3)  R_{X}D = R_{X}P_{f} = R_{X}P_{s}.

Example 4.5 shows that the equality P_{f} = P_{s} is not always true (even if the divisor P_{s} is nef). Nevertheless D = P_{f} + N_{f} is a σ-decomposition for any pseudo-effective divisor D (see [17]).

(iii) In dimension 2 the Fujita decomposition coincides with the

classical Zariski decomposition (see Proposition 2.5). Example 6.4. Let D be a big bss-ample ℝ-Cartier divisor. We make use the notation of Remark 5.6. Then in diagram (5.7) there exists a unique decomposition

(6.5)  g^{∗}D = P_{f} + N_{f},

where P_{f} ∼_{ℝ} h^{∗}H and the divisor N_{f} is effective and (very) exceptional on Y. This decomposition is a Fujita decomposition (with semiample positive part).

Indeed, in this instance h is a birational contraction. Put N_{f} = E_{W} −F_{W}. Then the divisor −N_{f}∼_{ℝ}h^{∗}H−g^{∗}D is nef over X. By Lemma 1.6 we have N_{f} ≥ 0. It is clear that N_{f} is exceptional on Y. Assume that a divisor L ≤ D is nef. Then the divisor L − P_{f} ∼_{ℝ} L − h^{∗}H is nef over Y. From Lemma 1.6 we obtain that L ≤ P_{f}. Thus g^{∗}D ∼_{ℝ}P_{f} + N_{f} is a Zariski decomposition in Fujita sense. It is unique.

The statement is no longer true if we omit the condition for D to be big (see Example 1.3) Proposition 6.6. Let D be an effective ℝ-Cartier divisor. Assume that the decomposition D = P_{f} + N_{f} satisfies conditions (i)-(ii) of Definition 6.1. Then (iii) is equivalent to the following

(iii)′ for any birational contraction f: X̂ → X and for any nef divisor F on X̂ such that F ≤ f^{∗}D we have F ≤ f^{∗}P_{f}.

Proof. (iii) ⇒ (iii)′. Since −(f^{∗}P_{f} − F) is nef over X and f_{∗}(f^{∗}P_{f} − F) ≥ 0, we have f^{∗}P_{f} − F ≥ 0. (We used that f_{∗}F is b-nef and f_{∗}F ≤ D, so f_{∗}F ≤ P_{f}).

(iii) ′ ⇒ (iii). Since L is b-nef, there exists a birational contraction f: X̂ → X and a nef divisor L̂ on X̂ such that f_{∗}L̂ = L. Further, L̂ − f^{∗}D is nef over X and f_{∗}(f^{∗}D − L̂) ≥ 0. Hence, f^{∗}D − L̂ ≥ 0 and L ≤ P_{f}. □

However, a Fujita decomposition does not always exists that shows the following simple example: Example 6.7. Consider the linear system |D| on a nonsingular projective variety X such that

(i) |D| has no fixed components; (ii) D is not nef.

For example, similar to Example 4.17 we may take as X a blowup of ℙ^{3} in two distinct points P_{1} and P_{2}, and as D the birational transform of a plane passing through P_{1} and P_{2}. Assume that there exists a decomposition N_{f}(D) in Fujita’s sense. Then P_{f} ≥ M_{1} = D, i.e., P_{f} = D, a contradiction. 6.8. The above shows that it is more naturally to construct a Fujita decomposition for a pull-back f^{∗}D of D under some birational contraction f: Y → X. However even in such a stating, the problem of the existence of a Fujita decomposition fails (see [17]). Nevertheless, its positive decision is expected (for a pseudo-effective divisor D) in the most important cases:

(i) if (X, B) is 0-pair (see Theorem 5.21); (ii) if (X, B) is a Fano log variety; (iii) if D = K_{X} + B, where the pair (X, B) has log canonical sin-

gularities and κ(X, K_{X} + B) ≥ 0 (cf. Theorem 7.9). Argument justifying this hope is, for example, fact proved by Kawamata [12, Prop. 5]: for any effective Cartier divisor D on a toric variety X there exists a toric birational contraction f: Y → X such that the divisor f^{∗}D has a Fujita decomposition and this decomposition coincides with a Shokurov decomposition and with an CKM-decomposition (see below). Remark 6.9. It is clear that the Fujita decomposition satisfies condition (2.7). Therefore (see Remark 2.8) if it exists on Y for the divisor D_{Y}, where D ∈ BCDiv_{ℝ}(K(Y)), then there exists a b-divisor 𝒩(D) ∈ BCDiv_{ℝ}(K(Y)) such that 𝒩(D)_{Y′} is a decomposition in Fujita sense for any model Y′ of the field K(Y) dominating Y. In other words a Fujita decomposition is unique in the birational sense. Definition 6.10. A decomposition D = P_{gf} + N_{gf} is called generalized Fujita decomposition if

(i) N_{gf} ≥ 0; (ii) P_{gf} is b-nef; (iii) for any b-nef divisor L such that L ≤ D we have L ≤ P_{gf}. Another way to generalize Fujita decomposition is to consider the

straightforward analog of Definition 5.11: necessary to claim the existence of a b-nef b-divisor 𝒫 such that 𝒫_{X} = P and satisfying condition 5.11 (iii). Proposition 6.11. Let D be an effective ℝ-Cartier divisor. Assume that there exists a birational contraction f: X̂ → X such that the divisor D^{∗} = f^{∗}D has a Fujita decomposition D^{∗} = P_{f}(D^{∗}) + N_{f}(D^{∗}). Then D = f_{∗}P_{f}(D^{∗}) + f_{∗}N_{f}(D^{∗}) is a generalized Fujita decomposition.

Proof. By construction f_{∗}N_{f}(D^{∗}) is effective and f_{∗}P_{f}(D^{∗}) is b-nef. Let L be a b-nef divisor on X such that L ≤ D. We may assume that there exists a nef divisor L̂ on X̂ such that f_{∗}L̂ = L. Then −(f^{∗}D − L̂) is nef over X and f_{∗}(f^{∗}D − L̂) ≥ 0. By Lemma 1.6 we have f^{∗}D ≥ L̂. But then P_{f}(D^{∗}) ≥ L̂ and f_{∗}P_{f}(D^{∗}) ≥ L. □

Proposition 6.12. Let D be an effective divisor. Assume that there exists a generalized Fujita decomposition D = P_{gf} + F_{gf}. Then

(i) R_{X}D = R_{X}P_{gf}; (ii) P_{gf} ≥ P_{s}(D). If furthermore the divisor D is big and the variety

X is ℚ-factorial, then P_{gf} = P_{s}(D). Conversely, if the divisor D is big, X is ℚ-factorial, and P_{s}(D) is bnef, then there exists the generalized Fujita decomposition for D and P_{gf} = P_{s}(D).

Proof. The proof of (i) is similar to the proof of Proposition 4.8. We prove (ii). Since M_{n} ≤ nP_{gf} for all n ∈ ℕ, we have P_{s}(D) ≤ P_{gf}. Assume now that the divisor D is big. By (i) so is P_{gf}. Fix ε > 0. By the Kodaira lemma P_{gf} = A+F, where A is ample and F is an effective divisor. It is clear that P_{gf} − εF = (1 − ε)P_{gf} + εA is a b-semiample divisor and its class is contained in the open cone Mv^{o}(X). Similar to the proof of Proposition 5.16 we have

P_{s}(D) ≥ P_{s}(P_{gf} − εF) = P_{gf} − εF.

Passing to the limit, we obtain P_{gf} ≤ P_{s}(D).

Finally, let L ≤ D be a b-nef divisor. We prove that L ≤ P_{s}(D). By the Kodaira lemma D = A + F, where the divisor A is ample and F is effective. Take a sufficiently small rational ε > 0. Then (1 + ε)D ≥ L + εA. As above

(1 + ε)P_{s}(D) = P_{s}((1 + ε)D) ≥ P_{s}(L + εA) = L + εA.

Hence, L ≤ P_{s}(D). This proves the proposition. □

6.13. Zariski decomposition for varieties of intermediate Kodaira dimension. Proposition 6.14 ([9, Prop. 1.24, 1.10]). Let f: X → Z be a contraction of nonsingular varieties, let D be a divisor on Z, and let R be a very exceptional divisor on X. Then the pull-back of f^{∗}D + R under some birational contraction g: X′ → X has a Zariski decomposition in Fujita sense if and only if this decomposition exists for the pull-back of D under some birational contraction h: Z′ → Z. Furthermore, the divisor P_{f}(g^{∗}(f^{∗}D + R)) is the pull-back of P_{f}(h^{∗}D) for a suitable choice of g and h: if the diagram

\begin{matrix}X & ⟵^{g} & X′ \\ ^{f}↓ & & ^{f′}↓ \\ Z & ⟵^{h} & Z′ \end{matrix}

is commutative, then P_{f}(g^{∗}(f^{∗}D + R)) = f^{′∗}P_{f}(h^{∗}D).

Using this statement and a formula for canonical divisor of elliptic fibrations Fujita proved the following theorem.

Theorem 6.15 ([9]). Let f: X → Z be a contraction of a threedimensional variety with general fiber being an elliptic curve. Assume that κ(X, K_{X}) ≥ 0. Then there exists a birational contraction g: X′ → X such that g^{∗}K_{X} has a Zariski decomposition in Fujita sense P_{f} +N_{f} with semiample positive part P_{f} (and it coincides with the Shokurov decomposition). In particular, the canonical algebra RK_{X} is finitely generated.

Later the last fact was generalized in [6]: Theorem 6.16 ([6]). Let (X, B) be a Kawamata log terminal pair with κ(X, K_{X} + B) = l ≥ 0, where B is a ℚ-boundary. Then there exists an l-dimensional Kawamata log terminal pair (Z, Δ) with κ(Z, K_{Z} +Δ) = l such that log canonical algebras R(K_{X} + B) and R(K_{Z} + Δ) are quasi-isomorphic [PLF, Def. 4.3]. In particular, questions about finite generation of these algebras are equivalent [PLF, Th. 4.6]. Corollary 6.17. Let (X, B) be a Kawamata log terminal pair with 0 ≤ κ(X, K_{X} + B) ≤ 3. Then the log canonical algebra R(K_{X} + B) is finitely generated. 6.18. An example of a divisor with non-rational Zariski decomposition. Following [4], we present an example of a divisor with a Zariski decomposition in Fujita sense having irrational coefficients.

Let E be an elliptic curve with End(E) ≃ ℤ and let S = E × E. Pick a point P ∈ E and put E_{1} = {P } × E and E_{2} = E × {P }. Then the diagonal Δ and the curves E_{1} and E_{2} form the basis of the space NS_{ℝ}(S). Put Δ′ = Δ − E_{1} − E_{2}. Then Δ′, E_{1}, E_{2} is also a basis. In this basis the quadratic form x^{2} has the following form

(α_{1}E_{1} + α_{2}E_{2} + βΔ′)^{2} = 2α_{1}α_{2} − 2β^{2}.

Therefore the cone of ample divisors in NS_{ℝ}(S) is defined by conditions

(6.19)  α_{1}α_{2} − β^{2} > 0,  α_{1}, α_{2} > 0.

Consider the ℙ^{1}-bundle π: X = ℙ(O_{S}(Δ′) ⊕ O_{S}) → S. Identify S with the zero section. Then O_{S}(S) = O_{S}(Δ′). Put S_{1} = π^{∗}E_{1} and S_{2} = π^{∗}E_{2}. Let H be an integral ample divisor on X and let L =^{def} H|_{S}.

Consider the divisor G(α_{1}, α_{2}) =^{def} L + α_{1}E_{1} + α_{2}E_{2}. For positive α_{1}, α_{2}, define the function

γ(α_{1}, α_{2}) =^{def} sup{β | G(α_{1}, α_{2}) + βΔ′ is nef}

Using (6.19), choose α_{1}, α_{2} ∈ ℚ, α_{1}, α_{2} > 0 so that γ =^{def} γ(α_{1}, α_{2}) is irrational. If δ > γ, then G(α_{1}, α_{2}) + δΔ′ is not nef. Therefore there exists an irreducible curve Γ having negative intersection with G(α_{1}, α_{2}) + δΔ′. Since S is an abelian surface, there exists a family of such curves {Γ_{λ}} on S:

(6.20)  (G(α_{1}, α_{2}) + γΔ′) · Γ_{λ} < 0.

On the other hand, if 0 < δ < γ, then G(α_{1}, α_{2}) + δΔ′ is ample.

Put D =^{def} B + rS, where B =^{def} H + α_{1}S_{1} + α_{2}S_{2} and take r ∈ ℚ so that r > γ. Claim 6.21. Let D be the (effective and big) divisor constructed above. Then N_{f}(D) = (r − γ)S is a Fujita decomposition. In particular, for any birational contraction f: Y → X the pull-back f^{∗}D has no Fujita decompositions with rational coefficients.

Proof. First we show that the divisor P_{f} = D − N_{f}(D) is nef. Indeed, suppose that (B + γS) · C < 0 for some irreducible curve C. Since B is ample, S · C < 0. Therefore, (G(α_{1}, α_{2}) + γΔ′) · C < 0, a contradiction with our choice of γ.

Further, let f: Y → X be a birational contraction and let F be an effective divisor on Y such that f^{∗}D − F = f^{∗}(P_{f} + N_{f}) − F is nef. From (6.20) we have (G(α_{1}, α_{2}) + (γ + ε)Δ′) · Γ_{λ} < 0 for any ε > 0. Hence, (P_{f} + εS) · Γ_{λ} < 0. Let {Γ′_{λ}} be a family of irreducible curves on Y dominating {Γ_{λ}}. Then f^{∗}(P_{f} + εS) · Γ′_{λ} < 0. Since B is ample, f^{∗}S · Γ′_{λ} < 0. Therefore the family {Γ′_{λ}} cover the birational transform S̃ of S. On the other hand,

0 < (f^{∗}D − F) · Γ′_{λ} = (f^{∗}(P_{f} + εS + (r − γ − ε)S) − F) · Γ′_{λ}.

Hence,

−(r − γ − ε)S · Γ′_{λ} < f^{∗}(P_{f} + εS − F) · Γ′_{λ} < −F · Γ′_{λ}.

This means that f_{∗}F > (r − γ − ε)S. Since ε is an arbitrary positive, f_{∗}F ≥ (r − γ)S. Finally, the inequality F ≥ f^{∗}(r − γ)S follows from f_{∗}F ≥ (r − γ)S and the fact that the divisor f^{∗}(r − γ)S − F is nef over X (see Lemma 1.6). □

Nakayama [17] using similar construction, constructed an example of a big (integral) divisor on a nonsingular variety such that its pullback under any blowup has no Fujita decompositions (as well as CKM decompositions, see below).

7. Zariski decomposition in CKM’s sense

Definition 7.1 ([4], [12], [15]). A decomposition D = P_{CKM}(D) + N_{CKM}(D) is called a Zariski decomposition in sense Cutkosky-Kawamata-Moriwaki (or simply CKM-decomposition) if the following conditions are satisfied:

(i) N_{CKM}(D) ≥ 0; (ii) the divisor P_{CKM}(D) is nef; (iii) there is an isomorphism of graded algebras

R_{X}D ≃ R_{X}P_{CKM}(D).

Remark 7.2. Since there is the following embedding of cones

Nef(X) = \overline{Amp}(X) ⊂ \overline{Mv}(X),

we have that an CKM-decomposition is also a sectional decomposition. If D = P_{sec}(D)+N_{sec}(D) is a sectional decomposition (such as in 4.18), then it is an CKM-decomposition if and only if the divisor P_{sec}(D) is nef. Remark 7.3. If D is ℝ-Cartier, then a decomposition N_{CKM}(D) satisfies condition (2.7).

Indeed, let f: Y → X be a birational contraction. It is sufficient to show the existence of isomorphisms

\begin{matrix}H^{0}(O_{X}(nP_{CKM})) ≃ H^{0}(O_{Y} (f^{∗}nP_{CKM})), \\ H^{0}(O_{X}(nD)) ≃ H^{0}(O_{Y} (f^{∗}nD)), \end{matrix}

that follows from the fact that f is a contraction. Proposition 7.4. (i) Let D be a big divisor on a ℚ-factorial va-

riety and let D = P_{f}(D) + N_{f}(D) be its Fujita decomposition. Then it is also an CKM-decomposition.

(ii) Let D be a big ℝ-Cartier divisor. If a decomposition D = P_{CKM}(D) + N_{CKM}(D) exists, then it is a Fujita decomposition.

Proof. The statement of (i) follows by (6.3) and (ii) follows by Propositions 4.21 and 6.12. □

Remark 7.5. In general case (i.e., if D is not big) the CKMdecomposition is not unique and does not coincide with the Fujita decomposition (and even with the classical Zariski decomposition). For example, the divisor D from Example 4.5 has infinitely many CKMdecompositions: N_{CKM}(D) = tD, 0 ≤ t ≤ 1. Example 7.6. An CKM-decomposition on a surface with ℚ-factorial singularities satisfies condition (iii) of Theorem 2.2 and instead of (ii) we have only

(ii)′ the matrix (N_{i} · N_{j}) is seminegative definite. Indeed, we may assume that P 2_{CKM} = 0. If P_{CKM} · N_{i} > 0, then (P_{CKM} + εN_{i})^{2} = ε(2P_{CKM} · N_{i} + εN_{i}^{2}) > 0 for some 0 < ε ≪ 1, i.e., the divisor P_{CKM} + εN_{i} is big, a contradiction. Thus, P_{CKM} · N_{i} = 0 for all N_{i}. Now suppose that N^{′2} > 0, where N′ =^{def} ∑ ε_{i}N_{i}, |ε_{i}| ≪ 1. Then (P_{CKM} + N′)^{2} = N^{′2} > 0 and the divisor P_{CKM} + N′ is big. Again we have a contradiction.

The following theorem is a consequence of the existence of Iitaka fibration and two-dimensional Zariski decomposition. Theorem 7.7 ([4]). Let D be an effective Cartier divisor. Assume that 1 ≤ κ(X, D) ≤ 2. Then there exists a birational contraction f: Y → X such that f^{∗}D has an CKM-Zariski decomposition N_{CKM}(f^{∗}D) satisfying the following condition

(7.8)  κ(X, D) = ν(Y, P_{CKM}(f^{∗}D)).

Furthermore, if D is a ℚ-divisor, then so is N_{CKM}(D). An CKMdecomposition satisfying condition (7.8) is unique.

It is expected that Zariski decomposition exists for the log canonical divisor. In this instance Zariski decomposition must have good properties: Theorem 7.9 ([1], [15], [16], [12]). Let (X, Δ) be a projective Kawamata log terminal pair such that Δ is a ℚ-boundary and the divisor K_{X} + Δ is big. Assume that there exists a Zariski decomposition in CKM sense for D = K_{X} + Δ. Then the positive part P_{CKM}(D) is a semiample ℚ-divisor. In particular, the algebra R_{X}(K_{X} +Δ) is finitely generated.

An CKM-decomposition in this case coincides with Fujita and Shokurov decompositions.

Theorem 7.9 was proved by Moriwaki [15]-[16] and Kawamata [12] in more general, relative situation. Thus the existence of a (relative) Zariski decomposition in CKM sense for the log canonical divisor of small contractions X → Z is a sufficient condition for the existence of log flips.

8. Decompositions of b-divisors

8.1. Let D be a b-divisor. Similar to (1.8) we put

Mov(D) =^{def} \begin{cases}− inf_{s∈K(X)^{∗}}\{(s) | D + (s) ≥ 0\} & if H^{0}(O(D)) ≠ 0, \\ −∞ & otherwise. \end{cases}

In the case Mov(D) ≠ −∞, we may take a section s_{0} ∈ K(X)^{∗} so that D + (s_{0}) ≥ 0. Then −(s_{0}) ≤ Mov(D) ≤ D. Therefore Mov(D) is a b-divisor. If H^{0}(O(D)) ≠ 0, then we also put

Fix(D) =^{def} inf{L | L ∼ D, L ≥ 0} = D − Mov(D).

B-divisors Mov(D) and Fix(D) defined above are called mobile and fixed parts of D respectively. If a b-divisor D is effective, then so is Mov(D): Mov(D) ≥ −(Const) = 0. It is easy to see from the definition that

(8.2)  (Mov(D))_{X} ≤ Mov(D_{X}),  (Fix(D))_{X} ≥ Fix(D_{X}).

Lemma 8.3. Let D be a b-divisor such that H^{0}(O(D)) ≠ 0. Then M = Mov(D) satisfies the following properties:

(i) M ≤ D; (ii) the b-divisor M is b-free (in particular it is b-Cartier); (iii) if a b-divisor L ≤ D is b-free, then L ≤ M.

Conversely, if an (integral) b-divisor M satisfies conditions (i) - (iii), then M = Mov(D).

Proof. Prove (ii). Similar to Lemma 1.9 we have

Fix(M) = inf{L | L ∼ M, L ≥ 0} = 0.

Hence, Fix(M_{X}) = 0, i.e., the linear system |M_{X}| has no fixed components. Let f: Y → X be a resolution of base points of the linear system |M_{X}| and let |M| be the birational transform of |M_{X}| on Y. Using [PLF, Prop. 3.20] we obtain |M_{X}| ≃ |M_{Y} | ≃ |M|. Since |M_{Y} | has no fixed components, |M_{Y} | = |M| and this linear system has no base points. The rest of the proof is completely similar to Lemma 1.9 □

Similar to Lemma 1.10 one can prove the following Lemma 8.4. Let D be a b-divisor such that H^{0}(O(D)) ≠ 0. Then M = Mov(D) satisfies the following properties:

(i) M ≤ D; (ii) H^{0}(O(M)) = H^{0}(O(D)); (iii) if for a b-divisor L ≤ D the equality H^{0}(O(L)) = H^{0}(O(D))

holds, then L ≥ M. Conversely, if an (integral) b-divisor M satisfies conditions (i) - (iii), then M = Mov(D).

Here (as well as everywhere) we assume that sections of the sheaf O(D) are elements of the field K(X):

H^{0}(O(D)) = {s ∈ K(X) | D + (s) ≥ 0}.

Therefore the statement remains to be true for infinitely dimensional spaces H^{0}(O(·)). Definition 8.5. A b-divisor G is said to be pbs-ample if there exists a sequence b-semiample b-divisors G_{i} such that lim_{i→∞} G_{i} = G.

Definition 8.6. A decomposition of a b-divisor D = D^{m} + D^{e} is called a pbs-decomposition (pseudo b-semiample decomposition) if it satisfies the following properties:

(i) D^{e} ≥ 0; (ii) D^{m} is pbs-ample; (iii) for any pbs-ample b-divisor L such that D^{m} ≤ L ≤ D we have

L = D^{m}. Here D^{m} is the maximal pbs-ample part and D^{e} is the fixed part. Pbs-decompositions generalize divisorial Shokurov decomposi-

tions 5.11. Indeed, let D = D^{m} + D^{e} and D^{m} be a decomposition and the corresponding b-divisor from 5.11 and let D^{e} = D^{e} (the last is considered as an equality of b-divisors). Put also D = D^{m} + D^{e}. Then by definition D = D^{m} + D^{e} is a pbs-decomposition and D = D_{X}. 8.7. Similar to the divisorial case we say that a b-divisor D is effective modulo ℚ-linear equivalence if D ∼_{ℚ} D′, where D′ ≥ 0. Equivalent: H^{0}(O(n_{0}D)) ≠ 0 for some n_{0} ∈ ℕ. Let D be an effective modulo ℚ-linear equivalence b-divisor. Denote

M_{n} =^{def} Mov(nD).

Since M_{n}/n ≤ D, there exists the limit

𝒫_{s}(D) =^{def} lim sup_{n→∞}(M_{n}/n).

Since M_{n_{1}n_{2}} ≥ n_{2}M_{n_{1}} for all n_{1}, n_{2} ∈ ℕ, we have

n𝒫_{s}(D) ≥ M_{n} for all n ∈ ℕ.

It is clear that −(s_{0}) ≤ 𝒫_{s}(D) ≤ D, where 0 ≠ s_{0} ∈ H^{0}(O(n_{0}D)). Hence, 𝒫_{s}(D) is also a b-divisor. Thus we obtain a decomposition

D = 𝒫_{s}(D) + 𝒩_{s}(D), 𝒩_{s}(D) =^{def} D − 𝒫_{s}(D) ≥ 0,

which we call an s-decomposition of a b-divisor. Similar to (4.2) we have

(8.8)  𝒩_{s}(D) = inf{L | L ∼_{ℚ} D, sL ≥ 0}.

By construction 𝒫_{s}(D) is a pbs-ample b-divisor.

It is easy to see that properties 4.6 hold also for the b-divisorial s-decomposition (if one replaces divisors with b-divisors).

The concept of s-decomposition of b-divisors allows us to formulate a criterion of finite generation of a b-divisorial algebra R_{X}D =^{def} ⊕_{n≥0}H^{0}(X, O_{X}(nD)) (cf. Theorem 4.10).

Theorem 8.9 (Limiting Criterion [PLF, Th. 4.28]). Let D be an effective modulo ℚ-linear equivalence b-divisor. Then the b-divisorial algebra R_{X}D is finitely generated if and only if 𝒫_{s}(D) = Mov(n_{0}D)/n_{0} for some n_{0} ∈ ℕ. 8.10. Example 2.4 shows that the divisorial s-decomposition D = P_{s}(D) + N_{s}(D) does not agree via pull-backs f_{∗}. Therefore the system P_{s}(D_{X}) does not form a b-divisor. In particular, 𝒫_{s}(D)_{X} ≠ P_{s}(D_{X}). However, from (8.2) we always have

𝒫_{s}(D)_{X} ≤ P_{s}(D_{X}), 𝒩_{s}(D)_{X} ≥ N_{s}(D_{X}).

Where equalities are achieved “birationally asymptotically”: Lemma 8.11.

𝒫_{s}(D)_{X} = inf_{f: Y →X} f_{∗}P_{s}(D_{Y}), 𝒩_{s}(D)_{X} = sup_{f: Y →X} f_{∗}N_{s}(D_{Y}),

where the infimum and supremum are taken over all birational contractions f: Y → X.

Proof. Consider the following sets of divisors on X

𝔖 =^{def} {L_{X} | L ∼_{ℚ} D, L ≥ 0},

𝔖_{Y} =^{def} {L_{X} | L ∼_{ℚ} D, L_{Y} ≥ 0},

where Y is a birational model dominating X. Then 𝔖_{Y′} ⊂ 𝔖_{Y} if Y′ dominates Y and 𝔖 = ⋂_{Y/X} 𝔖_{Y}. According to (4.2) and (8.8) we have

𝒩_{s}(D)_{X} = inf 𝔖 and f_{∗}N_{s}(D_{Y}) = inf 𝔖_{Y}. Hence,

𝒩_{s}(D)_{X} = inf 𝔖 = inf ⋂_{Y/X} 𝔖_{Y} = sup_{Y/X} inf 𝔖_{Y} = sup_{f: Y →X} f_{∗}N_{s}(D_{Y}).

This proves the statement. □

It follows from definitions that 𝒫_{s}(D) ≤ D^{m} for any effective modulo ℚ-linear equivalence b-divisor D (such that a decomposition D = D^{m}+ D^{e} exists).

Completely similar to 5.14 one can prove the following. Proposition 8.12. Suppose that 𝒫_{s}(D) is a rational b-semiample bdivisor. Then for any b-semiample b-divisor L ≤ D the inequality L ≤ 𝒫_{s}(D) holds.

According to Limiting Criterion 8.9 the conditions of proposition are satisfied, for example, in case when the b-divisorial algebra R_{X}D is finitely generated.

In conclusion we mention an interesting result of Fujita:

Theorem 8.13 ([10]). Let D be a big Cartier divisor on a ddimensional variety X and let

v(D) =^{def} lim sup_{t→∞} (\frac{d!}{t^{n}} dim H^{0}(X, O(tD)))

be its volume. Then for any ε > 0, there exists a birational contraction f: Y → X and a decomposition

f^{∗}D = P_{ε} + N_{ε}

in a sum of ℚ-Cartier divisors, where • N_{ε} ≥ 0;

• the divisor P_{ε} is semiample;

• v(D) − ε < v(P_{ε}) = (P_{ε})^{d} < v(D).

9. Analytic Zariski decomposition

In conclusion we mention about complex analytic approach to the constructing of Zariski decompositions [21], [22] (see also [3]). Main definitions and facts on complex currents can be found in [11] or [5].

Let T be a closed positive (1, 1)-current on the open unit ball in ℂ^{n} with center at 0. The Lelong number Θ(T, 0) of T at 0 is the number

Θ(T, 0) =^{def} lim_{r→0} \frac{T (χ(r)ω^{n−1})}{π^{n−1}r^{2(n−1)}}

where ω = \frac{\sqrt{−1}}{2} ∑ dz_{i} ∧ dz̄_{i} and χ(r) is the characteristic function of the open ball in ℂ^{n} with center at 0 and radius r. This number is invariant under the changes of coordinates. Thus it is possible to define the Lelong number Θ(T, x) for a closed positive (1, 1)-current at a point x ∈ X on any complex variety X. If T is given by integration over an analytic subvariety of codimension 1, then Θ(T, x) is the usual multiplicity of this subvariety at x.

Further we suppose that X is a nonsingular projective complex variety. For any analytic subset V ⊂ X, we can define

Θ(T, V) =^{def} inf{Θ(T, x) | x ∈ V }.

If V irreducible, then Θ(T, V) coincides with the Lelong number Θ(T, x) at a very general point x ∈ V. The last enable us to define Siu decomposition (see [5, Ch. III, (8.16)]):

(9.1)  T = T′ + ∑_{V} Θ(T, V)[V],

where the (infinite in general) sum is taken over all prime divisors V and [V] is current given by integration over V. Here T′ is also a closed positive (1, 1)-current such that the set {x ∈ X | Θ(T, x) ≥ ε} has codimension ≥ 2 for any ε > 0.

Let f: Y → X be a surjective morphism and let T be a closed positive (1, 1)-current on X. Locally we may write T = \sqrt{−1}∂ ¯∂φ, where φ is a plurisubharmonic function [5, Ch. III, Prop. 1.19]. Therefore we can define the pull-back of a current T by the formula f^{∗}T = \sqrt{−1}∂ ¯∂f^{∗}φ (this definition does not depend on the choice of the function φ).

It is possible also to define a “b-divisorial” version of the Siu decomposition: for a closed positive (1, 1)-current T we put

(9.2)  S(T) = ∑_{V} Θ(T, V)V,

where the sum is taken over all divisorial discrete valuations V of the field K(X) and Θ(T, V) is computed on a (nonsingular projective) birational model f: Y → X, on which the center of V is a prime divisor: Θ(T, V) = Θ(f^{∗}T, V). Strictly speaking S(T) is not a b-divisor, since S(T)_{X} can be an infinite sum. Nevertheless, S(T) satisfies the condition S(T)_{Y_{1}} = g_{∗}S(T)_{Y_{2}} whenever there is a birational morphism g: Y_{1} → Y_{2}. Theorem 9.3 ([21]). Let D be a big divisor on X. Then there exists a (closed, positive) (1, 1)-current T such that

(i) T represents the cohomology class c_{1}(D) of D in H^{2}(X, ℝ); (ii) for any birational contraction f: Y → X, any nonnegative integer m, and any point y ∈ Y the following approximation formulas for Θ(f^{∗}T, y) hold:

(9.4)  min_{L∈|f^{∗}mD|} (mult_{y} L) ≥ mΘ(f^{∗}T, y)
(9.5)  lim inf_{m→∞} (\frac{1}{m} min_{L∈|f^{∗}mD|} (mult_{y} L)) = Θ(f^{∗}T, y).

Such a current T is called an analytic Zariski decomposition (AZD) of D.

Equality (9.5), in particular, asserts that the Siu decomposition of a current T is a divisor and coincides with the s-decomposition N_{s}(D). Furthermore, the formal sum S(T) is a b-divisor and S(T) = 𝒩_{s}(\overline{D}). Corollary 9.6. Let D be a divisor on X.

(i) If D is nef and big, then c_{1}(D) can be represented by a closed positive (1, 1)-current T on X such that Θ(T) ≡ 0.

(ii) Conversely, if c_{1}(D) can be represented by a closed positive (1, 1)-current T on X such that Θ(T) ≡ 0, then D is nef.

The author would like to thank Professors V. V. Shokurov and V. A. Iskovskikh for useful discussions as well as for that they looked through the preliminary manuscript and gave numerous remarks. A part of this work was carried out at Max-Planck-Institut für Mathematik in 2001 and at the Isaac Newton Institute for Mathematical Sciences in 2002. Author very is grateful to the research staff and administrations of these institutes for hospitality and wonderful working environment.

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Department of Algebra, Faculty of Mathematics, Moscow State Lomonosov University, Leninskie Gory, 117234 Moscow, Russia

E-mail address: prokhoro@mech.math.msu.su