Numerical analogues of the Kodaira dimension and the Abundance Conjecture

We add further notions to Lehmann’s list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to fill in a gap in Lehmann’s arguments, thus proving that most of these notions are equal. Finally, we show that the Abundance Conjecture, as formulated in the context of the Minimal Model Program, and the Generalized Abundance Conjecture using these numerical analogues to the Kodaira dimension, are equivalent for non-uniruled complex projective varieties.


Introduction
During the last decade a plethora of numerical analogues to the Kodaira dimension for pseudoeffective divisors on (smooth) complex projective varieties was introduced, by Nakayama [18], Boucksom et al. [2], Siu [19] and Lehmann [16]. Lehmann furthermore clarified lots of relations between these numerical dimensions, adding some new notions, ordering them by the way how they are constructed and showing that most of them are at least related by an inequality. However, his results contain a gap leaving the equality of most of these notions unproven, see the discussion in Sect. 2.9. This note fills in the gap in Chapter 3 extending results on the derivatives of the volume function in [2,3]. Furthermore, we slightly extend Lehmann's list and prove some more relations. Finally we show that the Abundance Conjecture as formulated in the context of the Minimal Model Program (see e.g. [17,) is equivalent to a Generalised Abundance Conjecture introduced in [2]. On the way, we prove the birational equivalence of most of these notions of numerical dimension.
In more detail, we will discuss the following notions of numerical dimension, ordered according to their construction method as suggested by Lehmann, and postponing some technical definitions to Sect. 1: This theorem is a consequence of the following net of equalities and inequalities: In Sects. 2 and 3 we will prove the equality ν alg (D) = ν Käh (D) and the inequalities κ ν (D) ≤ ν alg (D), ν Vol,Zar (D) ≤ κ num (D), κ num (D) ≤ κ ν (D) and κ ν,Leh (D) ≤ κ ν (D), and we locate the proofs of the other inequalities in the works of Lehmann [16] and Nakayama [18]. Our proofs of ν Vol,Zar (D) ≤ κ num (D) and κ num (D) ≤ κ ν (D) also rely on Lehmann's ideas. But for the proof of κ ν (D) ≤ ν alg (D) we need a new ingredient: the derivative of the restricted volume, generalizing Thm.A in [3] (see Theorem 3.1). Note that Lehmann [16,Thm.5.3] already shows κ ν,Leh (D) ≤ ν alg (D) but his claim of equality κ ν,Leh (D) = κ ν (D) fails. Since the proof of κ σ (D) ≤ κ ν (D) only works for Nakayama's definition of κ ν (D), we need the new ingredient to close the gap.
The theorem shows that most of the notions in Definition 0.1 are equal. Therefore the following definition is justified: Definition 0.3. Let X be a smooth complex projective variety and D a pseudoeffective R-divisor on X . Then the numerical dimension ν X (D) of D is defined as one of the equal numbers and ν X (D) only depends on the numerical class of D.
In Sect. 4 we show that the numerical dimension of a pseudoeffective divisor behaves well under birational morphisms, following the ideas of Nakayama but explicitly using Theorem 0.2: Proposition 0.4. (= Proposition 4.1) Let f : X → X be a birational morphism between smooth complex projective varieties, let D be a pseudoeffective divisor on X and D a pseudoeffective divisor on X such that D − f * D is an f -exceptional divisor. Then: In a celebrated theorem Boucksom et al. [2,Cor.0.3] show that the canonical divisor K X of a non-uniruled smooth complex variety X is pseudoeffective. Consequently, the numerical dimension of the canonical divisor can be used to state the Abundance Conjecture: Conjecture 0.5. (Abundance Conjecture) Let X be a non-uniruled smooth complex projective variety. Then: Here κ(X ) = κ X (K X ) denotes the Kodaira dimension of the canonical divisor K X , defined e.g. as log m .
In the context of the Minimal Model Program the Abundance Conjecture is formulated under the assumption that minimal models of smooth complex projective varieties exist (see Sect. 4 for definitions): Conjecture 0.6. (Abundance Conjecture, MMP version [17,) Let S be a minimal model of a non-uniruled smooth projective complex variety X . Then |m K S | is base point free for sufficiently divisible and large m ∈ N (that is, K S is semi-ample).
We use the birational invariance of the numerical dimension to show in Sect. 4 that the two Abundance Conjectures as stated above are equivalent: Note that this equivalence is asserted in passing in [6] and proven in all detail in [11,Thm.4.3], using results of [14]. Relying on [1], Gongyo and Lehmann were even able to show that ν X (K X ) = κ X (K X ) already implies the existence of a minimal model.
However, the author still thinks that it is worth presenting the argument for Theorem 4.5, emphasizing in particular that not all the possible definitions of numerical dimension are easily shown to be birationally invariant.

Notions of numerical dimension
In the following X is always a smooth n-dimensional complex projective variety and D a pseudoeffective R-divisor on X .
where the limits are taken in H k,k (X ). Note that other choices of μ m , δ m , β i,m , E i,m satisfying the properties above will yield "smaller" (k, k)-classes α, that is, Käh − α is represented by a positive current.

ν alg (D)
This notion appears first in [16] where it is nevertheless attributed to [2]. In fact, Lehmann uses the algebraic analogue of the moving intersection product [D] k Käh as defined in [3]: To calculate [D] k alg Boucksom, Favre and Jonsson replace the (k, k)-cohomology class β 1,m ∧ . . . ∧ β k,m ∈ H k,k (X m ) by the intersection kcycle class [β 1,m ] · · · [β k,m ] ∈ N k (X m ) and take the limit in N k (X ). In

ν res (D)
This notion is defined in [16]. The diminished or restricted base locus of an Rdivisor The restricted moving intersection D k X |W is constructed in [3] for divisors W and generalized to arbitrary subvarieties where the B i,m are suitably chosen big and nef divisors on the smooth variety X m such that μ m : X m → X is a birational morphism whose center does not contain W (a so-called W -birational model of X ), the Q-divisors μ * m (D + δ m A) − B i,m are effective and μ m -exceptional for a fixed ample divisor A on X and real numbers δ m ↓ 0, and W is the strict μ m -transform of W .

D k
X |W only depends on the numerical class of D. On the cone spanned by classes of big divisors B such that W ⊂ B + (B) (then B is called a W -big divisor), the restricted product is continuous and homogeneous (see [3,Prop.2.9&Prop.4.6] resp. [16,Prop.4.7]). This implies furthermore that for arbitrary W -big divisors B (δ) converging to 0 when δ ↓ 0. Note also that by setting W := X the moving intersection cycle class D k alg can be obtained as a special case of the restricted moving intersection product. Finally, in the calculation of D k X |W one can choose B 1,m = · · · = B k,m (see the proof of [3, Lem.2.6]).

ν Vol (D)
This notion is defined in [16]. Note that the volume of the big R-divisor D + t A can be defined as

ν Vol,res (D)
This notion is introduced in [16] and uses the restricted volume investigated in [? ] (see also [16,Def.2.12] for the definition): Consequently, the restricted volume only depends on the numerical class of D and is continuous and homogeneous on the cone spanned by the classes of W -big divisors B.

ν Vol,Zar (D)
Again this notion is introduced in [16]. Note that morphisms φ : into a positive part P σ and a negative part N σ is constructed by Nakayama [ (1)] showed that the numerical class of P σ (φ * D) lies in the closure of the movable cone Mov(X ) spanned by fixed-part free divisors. For later purposes we need more details of Nakayama's construction of the σ -decomposition: Definition 1.1. Let X be a smooth projective complex variety, B a big R-divsor and a prime divisor on X . We set If D is a pseudoeffective R-divisor and A an ample divisor on X we set is a finite sum.

κ ν (D)
This notion is defined in [ is the locally free sheaf of an effective divisor E W on X , and for every real number b > 0 there exist Note that the condition above is satisfied for any birational morphism ψ : for an effective divisor F W and ample divisor B once it is satisfied for φ and A.

κ σ (D) ≤ κ ν (D)
This inequality is proven in [ Taking the logarithm, dividing by log m and letting m tend to ∞ shows the desired inequality.

κ ν,Leh (D) ≤ κ ν (D)
Let W ⊂ X be an irreducible subvariety, φ : X → X a birational morphism of smooth varieties such that Note that the argument for equality in the proof of [16,Prop.5.3] does not work because projections of finite-dimensional vector spaces are not closed maps. In particular equality could fail if φ * D sits on a non-polyhedral part of the boundary of the big cone Big( X ), as illustrated in the following diagram of a cut through the big cone by the affine plane in NS( X ) R passing through E W , φ * D and E W − 1 b+1 A, for arbitrary b > 0: Note also that Nakayama's proof of κ σ (D) ≤ κ ν (D) does not work if we replace κ ν (D) with κ ν,Leh (D): The definition of κ σ (D) only allows one to find sections of O X ( m D + A), with the ample divisor A on X added.

κ num (D) ≤ κ ν (D)
We adapt [16, Thm.6.7 (7)] and its proof to Nakayama's definition of κ ν (D): Let A be a sufficiently ample divisor on X , and let W ⊂ X be a subvariety such that dim W = κ ν (D) and D W . In particular, for a resolution φ : X → X of W and an ample divisor H on the smooth projective variety X , there exists b > 0 such that xφ * D − y E W + H is not pseudoeffective for any choice of x, y > b.
Choose q ∈ N large enough so that q H − φ * A is pseudoeffective, and consider any sufficiently large m ∈ N. Then the R-divisor mφ * D − q b + 1 E W + q H and hence Since m D + A ≤ m D + A the rate of growth for the right hand side is bounded by a multiple of m dim W q = m κ ν (D) . In particular, there exists a constant C > 0 such that Taking the logarithm, dividing by log m and letting m tend to ∞ shows the desired inequality.

Proof of κ ν ( D) ≤ ν alg ( D)
To show this inequality we cannot just adapt the proof of [16, Thm.6.2(6)≤(1)] to Nakayama's definition of κ ν (D) but need a new ingredient: the derivative of the restricted volume function. The following statement generalizes Thm.A in [3]. Theorem 3.1. Let X be a n-dimensional smooth projective complex variety and V = H 1 ∩ . . . ∩ H n−k a k-dimensional complete intersection variety cut out by very general very ample linearly equivalent divisors H i . If α is a V -big and γ an arbitrary divisor class then To prove this theorem and the inequality we first need further facts on the restricted moving intersection product and volume.

Lemma 3.2. Let X be a smooth projective complex variety, V ⊂ X a subvariety and D a V -pseudoeffective divisor on X . Furthermore, let F ⊂ X be a very general element of a free family of subvarieties, that is, a general element of the family intersects any given algebraic subset of X in the expected codimension. Then for
Proof. This is a generalisation of [16, Lem for big and nef divisors B i,m on X m . Choose F sufficiently general so that it does not contain any of the φ m -exceptional centers and intersects V generically transversally. Then the strict transform V ∩ F of V ∩ F on X m will be a cycle representing the class [φ * m F] · [ V m ]. Thus we can identify the classes and that implies the claimed equality.

Lemma 3.3. Let X be a smooth projective complex variety, V ⊂ X a subvariety of dimension d and D a V -pseudoeffective divisor on X . If k ≤ d and A is an ample divisor on X then
Proof. The Chern character isomorphism K (X ) Q → A(X ) Q shows that D k X |V is numerically trivial if and only if D k X |V · α = 0 for all (n − d + k)-cycles α ∈ A n−d+k (X ) (see [10,Ex.19.1.5]).
Since A n−d+k (X ) is generated by subvarieties of codimension d −k the pseudoeffective class D k X |V ≡ 0 if and only if there exists a subvariety Y ⊂ X of The opposite direction is obvious.

a k-dimensional complete intersection subvariety cut out by very general free big and nef divisors H i linearly equivalent to H and A, B
V -big and nef R-divisors. Then: Proof. This is a generalisation of [ Dividing by m k k! and going to the limit m → ∞ we obtain using the Generalised Fujita Theorem (see 1.5), Lemma 3.2 and the ampleness resp. freeness of A, H and the B i .
In the following, D 1 ≤ V D 2 means that the difference D 2 − D 1 of the two R-divisors D 1 , D 2 on X is effective and the support of D 2 − D 1 does not contain the subvariey V ⊂ X .

a k-dimensional complete intersection subvariety cut out by very general free big and nef divisors H i linearly equivalent to H and B a big and nef R-divisor such that B ≤ V H. If γ is an arbitrary divisor class such that H ± γ is still nef then
for every 0 ≤ t ≤ 1 and some constant c > 0 only depending on H n .
Proof. This is a generalisation of [3,Cor.3.4]. As in [3,Cor.2.4] we can use the assumption that H ± γ is nef to conclude that for 0 ≤ t ≤ 1 and some constant c > 0 only depending on H n , by replacing γ with (H + γ ) − H and using that H + γ ≤ 2H . If we also write B + tγ as the difference of the two nef classes C := B + t (γ + H ) and D := t H then we have furthermore where c once again only depends on H n : Indeed, c is controlled by C i · H n−i , 0 ≤ i ≤ k − 2, and we have C ≤ 3H . Thus we have The result follows by applying Proposition 3.4 to B + tγ = C − D.
Proof of Theorem 3.1. Let H be a very general divisor linearly equivalent to the H i , and assume that α is represented by the R-divisor A ≤ V H and that H ± γ is nef. If this is not the case replace α, γ by multiples sα, sγ with s > 0 sufficiently small. The claim for α, γ still follows, by homogeneity of restricted volumes and moving intersection numbers.
Let β be a nef divisor class on a V -birational model φ : Since V is cut out by the big and nef divisors φ * H i Proposition 3.5 shows for every 0 ≤ t ≤ 1 and some constant c > 0 only depending on H n . Taking the supremum over all nef classes β ≤ V φ * α yields using Lemma 3.2 and the Generalised Fujita Theorem. This holds for every 0 ≤ t ≤ 1, and in fact also for every −1 ≤ t ≤ 0, by replacing γ with −γ .
Exchanging the roles of α + tγ represented by an R-divisor A ≤ 2H and for a constant c possibly larger than c but still only depending on H n . Combining the two inequalities shows that To prove κ ν (D) ≤ ν alg (D) we finally need to connect divisorial Zariski decomposition and algebraic moving intersection product. For the Kähler intersection product this was done in [2, Thm.3.5]. So from now on we assume 1 ≤ k := ν alg (D) < κ ν (D) ≤ n := dim X and derive a contradiction: Fix an ample divisor H on X and > 0. Choose birational models π i : X i → X and big and nef divisors B i ≤ π * i (D + H ) on X i such that the limit of the push forwards π i * (B k i ) calculates the product (D + H ) k alg . [16,Prop.3.5&3.7] tell us that for a suitable effective divisor G on X we can further assume that the big and nef divisors B i satisfy Let W be a smooth k-dimensional intersection of very general very ample divisors such that W ⊂ B − (D) and for all i, the centers of π i intersect W transversally. Set φ : Y → X to be the blow up of X along W , with exceptional divisor E. Fix a very ample divisor H Y on Y . By Nakayama [18, V.2.21] k < κ ν (D) implies that D W , that is, for each sufficiently small > 0 there exists a τ > 0 such that φ * D − τ E + H Y is pseudoeffective and τ → ∞ when → 0.
Choose smooth birational models ψ i : can be computed. In particular there exist birational maps φ i : , and as in the proof of [3, Lem.2.6] we can achieve that H )). By Nakayama [18,III.5.16] H )). Since E i does not appear on the right-hand side, we have Again by Nakayama [18,III.5.16], In particular, we conclude that By definition, So taking the limit over all models Y i we obtain Y is a (k + 1)-dimensional complete intersection subvariety cut out by n − k − 1 very general very ample divisors H i ∈ |H Y |, Theorem 3.1 and Lemma 3.2 imply that where the first and the last equality follow from Lemma 3.2 and the middle equality from [16,Prop.4.20]. By assumption and Lemma 3.3 this last product is positive, contradicting the unboundedness of τ for → 0.

Birational Invariance and Abundance Conjecture
To prove that the Abundance Conjecture 0.5 is equivalent to the MMP-version of the Abundance Conjecture 0.6 we need the birational invariance of the numerical dimension of the canonical bundle: Proof of Claim. If mult E j < mult E j (E) for a j ∈ {1, . . . , k} we subtract a multiple of E j from and E to obtain Pushing forward we have f * ≡ D. Hence for real numbers y i such that y j ≥ 0, Thus, k i=1 y i E i + E is a non-trivial linear combination of the E i numerically equivalent to 0. But this is impossible as numerical classes of f -exceptional prime divisors are always linearly independent: On X , sufficiently general complete intersection curves C avoid all centers f (E i ) but one, hence the strict transform C ⊂ Y intersects the corresponding prime divisor on Y but none else.
The claim implies that σ E i ( f * D + E) ≥ mult E i (E). Taking the limit this also holds when D is only pseudoeffective.
The same holds when φ : Y → Y is a further birational morphism between smooth projective varieties: Using that the numerical dimension can be defined by ν Vol,Zar (see 1.6 and Defini- . Defining the numerical dimension via positive intersection products as ν alg shows that ν Y ( f * D) = ν X (D), together with the projection formula and the fact that f * defines a homomorphism on the intersection rings.
Remark 4.2. The proof above also shows that ν Y ( f * D) = ν Y ( f * D + E) for a pseudoeffective Q-divisor D on X and an effective f -exceptional divisor E on Y even when X is not smooth but only Q-factorial.
Note that for κ σ , this invariance was shown in [18, Prop.2.7(4)&(7)], hence Proposition 4.1 already follows from these results and Theorem 0.2. However, our proof explicitly uses two different ways of defining the numerical dimension (as Nakayama implicitly does, too), and thus demonstrates much better the usefulness of Theorem 0.2. Corollary 4.3. Let X be a non-uniruled smooth projective complex variety and f : Y → X a birational morphism between smooth projective varieties. Then: Proof. By [2, Cor.0.3] the canonical divisors K X and K Y are pseudoeffective on the non-uniruled varieties X and Y . Hence it is possible to calculate their numerical dimension. Since the pullback of canonical forms through a birational morphism is again a canonical form, there exists an effective f -exceptional divisor E such that K Y = f * K X + E. The corollary follows from Proposition 4.1.
A minimal model of a non-uniruled smooth projective complex variety X is a normal variety S such that there exists a sequence of divisorial contractions and flips X = X 0 X 1 · · · X n = S and K S is nef (see e.g. [17, Def.3-3-1andpassim] for further definitions). In particular, S is Q-factorial and has only terminal singularities, that is, every Weil divisor on S is a Q-Cartier divisor and if f : Y → S is a birational morphism from a smooth projective variety Y then in the ramification formula the coefficients a i of all the f -exceptional prime divisors E i are > 0. Note that on a minimal model S it is possible to construct intersection products of (Q-)Cartier divisors and to define the numerical triviality of the resulting (rational) cycles (see [10, 19.1]). Hence it makes sense to set the numerical dimension of a nef (Q-)Cartier divisor D on S equal to ν S (D) := max{k : D k ≡ 0}.
If S is smooth this numerical dimension coincides with the one defined in Definition 0.3, by construction of positive intersection products (see 1.2).
The following result of Kawamata [12] (see also [9,13]) sits at the core of the proof that the two versions of the Abundance Conjecture are equivalent: Proof. By Kawamata's Theorem we only need to prove that κ X (K X ) = κ S (K S ) and ν X (K X ) = ν S (K S ).
The first equality follows from using a common resolution of X and S such that K Y = f * K S + E S = φ * K X + E X where E S and E X are fresp. φ-exceptional effective divisors. Since the sections in H 0 (Y, O Y (m K Y )) = H 0 (Y, O Y (mφ * K X +m E X )) can be interpreted both as rational functions on X and Y , we have H 0 (Y, O Y (m K Y )) ⊂ H 0 (X, O X (m K X )), and since E X is effective the inverse inclusion also holds. Similarly on S, and the equality follows.
For ν X (K X ) = ν S (K S ) we use Corollary 4.3 and Remark 4.2 to deduce the chain of equalities where the last equality follows from the projection formula and the fact that f * defines a homomorphism on the intersection rings.