The pseudo-effective cone of a compact Kähler manifold and

varieties of negative Kodaira dimension

Sébastien Boucksom^{1} Jean-Pierre Demailly^{2} Mihai Păun^{3} Thomas Peternell^{4} 1Université de Paris VII ^{2}Université de Grenoble I, BP 74 Institut de Mathématiques Institut Fourier 175 rue du Chevaleret UMR 5582 du CNRS 75013 Paris, France 38402 Saint-Martin d’Hères, France

3Université de Strasbourg ^{4}Universität Bayreuth Département de Mathématiques Mathematisches Institut 67084 Strasbourg, France D-95440 Bayreuth, Deutschland

Abstract. We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of “movable curves”, which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non negative Kodaira dimension.^{(1)}

§0 Introduction

One of the major open problems in the classification theory of projective or compact Kähler manifolds is the following geometric description of varieties of negative Kodaira dimension.

0.1 Conjecture. A projective (or compact Kähler) manifold X has Kodaira dimension κ(X) = −∞ if and only if X is uniruled.

One direction is trivial, namely X uniruled implies κ(X) = −∞. Also, the conjecture is known to be true for projective threefolds by [Mo88] and for non-algebraic Kähler threefolds by [Pe01], with the possible exception of simple threefolds (recall that a variety is said to be simple if there is no compact positive dimensional subvariety through a very general point of X). In the case of projective manifolds, the problem can be split into more tractable parts :

(1) The original version of the present paper has been been forwarded very long ago to arXiv (in May 2004), and has been revised several times since then. It had also been submitted to a journal in 2004, but then the submission was cancelled after the referee expressed concerns about certain parts of the paper, as they were written at that time. Although the results of sections 1-5 have been reproduced several times, e.g. in lecture notes of the second named author or in Rob Lazarsfeld’s book [Laz04], a complete version never appeared in a refereed journal. We thank the Journal of Algebraic Geometry for suggesting to repair this omission.

A. If the canonical bundle K_{X} is not pseudo-effective, i.e. not contained in the closure of the cone spanned by classes of effective divisors, then X is uniruled.

B. If K_{X} is pseudo-effective, then κ(X) ≥ 0.

In the Kähler case, the statements should be essentially the same, except that effective divisors have to be replaced by closed positive (1, 1)-currents.

Part B again splits into two pieces:

B1. If K_{X} is pseudo-effective but not big, i.e. on the boundary of the pseudo-effective cone, then there exists a covering family of curves (C_{t}) such that K_{X} · C_{t} = 0.

B2. If K_{X} is pseudo-effective and there exists a covering family (C_{t}) of curves with K_{X} · C_{t} = 0, then κ(X) ≥ 0.

In this paper we give a positive answer to (A) for projective manifolds of any dimension, and deal with (B2), mostly in dimension 4. Part (A) follows in fact from a much more general fact which describes the geometry of the pseudo-effective cone.

0.2 Theorem. A line bundle L on a projective manifold X is pseudo-effective if and only if L · C ≥ 0 for all irreducible curves C which move in a family covering X.

In other words, the dual cone to the pseudo-effective cone is the closure of the cone of “movable” curves. This should be compared with the duality between the nef cone and the cone of effective curves.

0.3 Corollary (Solution of (A)). Let X be a projective manifold. If K_{X} is not pseudoeffective, then X is covered by rational curves.

In fact, if K_{X} is not pseudo-effective, then by (0.2) there exists a covering family (C_{t}) of curves with K_{X} · C_{t} < 0, so that (0.3) follows by a well-known characteristic p argument of Miyaoka and Mori [MM86] (the so called bend-and-break lemma essentially amounts to deform the C_{t} so that they break into pieces, one of which is a rational curve).

In the Kähler case both a suitable analogue to (0.2) and the theorem of Miyaoka-Mori are unknown. It should also be mentioned that the duality statement following (0.2) is actually (0.2) for ℝ-divisors. The proof is based on a use of “approximate Zariski decompositions” and an estimate for an intersection number related to this decomposition. A major tool is the volume of an ℝ-divisor which distinguishes big divisors (positive volume) from divisors on the boundary of the pseudo-effective cone (volume 0).

§1 Positive cones in the spaces of divisors and of curves 3

Concerning (B2) we need to distinguish between covering but not connecting families (C_{t}) on one side, and connecting families on the other side. This latter term “connecting” means that any two point can be joined by a chain of curves C_{t}. For technical purposes it is however better to consider strongly connecting families, i.e., any two sufficiently general points can be joined by a chain of irreducible C′_{t}s. If X has a good minimal model via contractions and flips, then X clearly admits a covering nonconnecting or a strongly connecting family (C_{t}) such that K_{X} · C_{t} = 0; moreover if X simply has a good minimal model, then at least after blowing up this will be the case. Let us say that (C_{t}) is a good covering family, if (C_{t}) is a covering, non-connecting family or a strongly connecting family. Then our remarks justify the division of Problem (B) into the two parts (B1) and (B2), possibly by replacing “covering families” by “good covering families”.

0.4 Theorem. Let X be a smooth projective 4-fold. Assume that K_{X} is pseudoeffective and there is a good covering family (C_{t}) of curves such that K_{X} · C_{t} = 0. Then κ(X) ≥ 0.

One important ingredient of the proof of (0.4) is the quotient defined by the family (C_{t}). The reason for the restriction to dimension 4 is that we use C_{n,m} and the log minimal model program on the base of the quotient of the family (C_{t}). In one circumstance however we have a general result:

0.5 Theorem. Let X be a projective manifold and (C_{t}) a strongly connecting family of curves. Let L be a pseudo-effective ℝ−divisor with L · C_{t} = 0. Then the numerical dimension ν(L) = 0. If L is Cartier, then L is numerically equivalent to a line bundle L′ with κ(L′) = 0.

If L = K_{X}, then in connection with [CP09] we obtain κ(X) = 0. In order to obtain the answer to Problem (B1) (e.g. in dimension 4), we would still need to prove that K_{X} is effective if K_{X} is positive on all good covering families of curves. In fact, in that case, K_{X} should be big, i.e. of maximal Kodaira dimension.

§1 Positive cones in the spaces of divisors and of curves

In this section we introduce the relevant cones, both in the projective and Kähler contexts – in the latter case, divisors and curves should simply be replaced by positive currents of bidimension (n−1, n−1) and (1, 1), respectively. We implicitly use that all (De Rham, resp. Dolbeault) cohomology groups under consideration can be computed in terms of smooth forms or currents, since in both cases we get resolutions of the same sheaf of locally constant functions (resp. of holomorphic sections).

1.1 Definition. Let X be a compact Kähler manifold.

(i) The Kähler cone is the set 𝒦 ⊂ H_{ℝ}^{1,1}(X) of classes {ω} of Kähler forms (this is an open convex cone).

(ii) The pseudo-effective cone is the set ℰ ⊂ H_{ℝ}^{1,1}(X) of classes {T } of closed positive currents of type (1, 1) (this is a closed convex cone). Clearly ℰ ⊃ \overline{𝒦}.

(iii) The Neron-Severi space is defined by

NS_{ℝ}(X) := (H_{ℝ}^{1,1}(X) ∩ H^{2}(X, ℤ)/tors) ⊗_{ℤ} ℝ.

(iv) We set

𝒦_{NS} = 𝒦 ∩ NS_{ℝ}(X), ℰ_{NS} = ℰ ∩ NS_{ℝ}(X).

Algebraic geometers tend to restrict themselves to the algebraic cones generated by ample divisors and effective divisors, respectively. Using L^{2} estimates for \overline{∂}, one can show the following expected relations between the algebraic and transcendental cones (see [Dem90], [Dem92]).

1.2 Proposition. In a projective manifold X, ℰ_{NS} is the closure of the convex cone generated by effective divisors, and \overline{𝒦_{NS}} is the closure of the cone generated by nef ℝ-divisors.

By extension, we will say that \overline{𝒦} is the cone of nef (1, 1)-cohomology classes (even though they are not necessarily integral). We now turn ourselves to cones in cohomology of bidegree (n − 1, n − 1).

1.3 Definition. Let X be a compact Kähler manifold.

(i) We define 𝒩 to be the (closed) convex cone in H_{ℝ}^{n−1,n−1}(X) generated by classes of positive currents T of type (n − 1, n − 1) (i.e., of bidimension (1, 1)).

(ii) We define the cone ℳ ⊂ H_{ℝ}^{n−1,n−1}(X) of movable classes to be the closure of the convex cone generated by classes of currents of the form

μ_{⋆}(ω̃_{1} ∧ … ∧ ω̃_{n−1})

where μ : X̃ → X is an arbitrary modification (one could just restrict oneself to compositions of blow-ups with smooth centers), and the ω̃_{j} are Kähler forms on X̃. Clearly ℳ ⊂ 𝒩.

(iii) Correspondingly, we introduce the intersections

𝒩_{NS} = 𝒩 ∩ N_{1}(X), ℳ_{NS} = ℳ ∩ N_{1}(X),

in the space of integral bidimension (1, 1)-classes

N_{1}(X) := (H_{ℝ}^{n−1,n−1}(X) ∩ H^{2n−2}(X, ℤ)/tors) ⊗_{ℤ} ℝ.

(iv) If X is projective, we define NE(X) to be the convex cone generated by all effective

curves. Clearly \overline{NE(X)} ⊂ 𝒩_{NS}.

(v) If X is projective, we say that C is a strongly movable curve if

C = μ_{⋆}(Ã_{1} ∩ … ∩ Ã_{n−1})

for suitable very ample divisors Ã_{j} on X̃, where μ : X̃ → X is a modification. We let SME(X) to be the convex cone generated by all strongly movable (effective) curves. Clearly \overline{SME(X)} ⊂ ℳ_{NS}.

(vi) We say that C is a movable curve if C = C_{t_{0}} is a member of an analytic family (C_{t})_{t∈S} such that ⋃_{t∈S} C_{t} = X and, as such, is a reduced irreducible 1-cycle. We let ME(X) to be the convex cone generated by all movable (effective) curves.

The upshot of this definition lies in the following easy observation.

1.4 Proposition. Let X be a compact Kähler manifold. Consider the Poincaré duality pairing

H_{ℝ}^{1,1}(X) × H_{ℝ}^{n−1,n−1}(X) ⟶ ℝ, (α, β) ⟼ ∫_{X} α ∧ β.

Then the duality pairing takes nonnegative values

(i) for all pairs (α, β) ∈ \overline{𝒦} × 𝒩; (ii) for all pairs (α, β) ∈ ℰ × ℳ. (iii) for all pairs (α, β) where α ∈ ℰ and β = [C_{t}] ∈ ME(X) is the class of a movable

curve.

Proof. (i) is obvious. In order to prove (ii), we may assume that β = μ_{⋆}(ω̃_{1} ∧…∧ω̃_{n−1}) for some modification μ : X̃ → X, where α = {T } is the class of a positive (1, 1)-current on X and ω̃_{j} are Kähler forms on X̃. Then

∫_{X} α ∧ β = ∫_{X} T ∧ μ_{⋆}(ω̃_{1} ∧ … ∧ ω̃_{n−1}) = ∫_{X} μ^{∗}T ∧ ω̃_{1} ∧ … ∧ ω̃_{n−1} ⩾ 0.

Here, we have used the fact that a closed positive (1, 1)-current T always has a pull-back μ^{⋆}T, which follows from the fact that if T = i∂\overline{∂}φ locally for some plurisubharmonic function in X, we can set μ^{⋆}T = i∂\overline{∂}(φ ∘ μ). For (iii), we suppose α = \{T \} and β = \{[C_{t}]\}. Then we take an open covering (U_{j}) on X such that T = i∂\overline{∂}φ_{j} with suitable plurisubharmonic functions φ_{j} on U_{j}. If we select a smooth partition of unity ∑ θ_{j} = 1 subordinate to (U_{j}), we then get

∫_{X} α ∧ β = ∫_{C_{t}} T_{|C_{t}} = ∑_{j} ∫_{C_{t}∩U_{j}} θ_{j}i∂\overline{∂}φ_{j|C_{t}} ⩾ 0.

For this to make sense, it should be noticed that T_{|C_{t}} is a well defined closed positive (1, 1)-current (i.e. measure) on C_{t} for almost every t ∈ S, in the sense of Lebesgue measure. This is true only because (C_{t}) covers X, thus φ_{j|C_{t}} is not identically −∞ for almost every t ∈ S. The equality in the last formula is then shown by a regularization argument for T, writing T = lim T_{k} with T_{k} = α + i∂\overline{∂}ψ_{k} and a decreasing sequence of smooth almost plurisubharmonic potentials ψ_{k} ↓ ψ such that the Levi forms have a uniform lower bound i∂\overline{∂}ψ_{k} ⩾ −Cω (such a sequence exists by [Dem92]). Then, writing α = i∂\overline{∂}v_{j} for some smooth potential v_{j} on U_{j}, we have T = i∂\overline{∂}φ_{j} on U_{j} with φ_{j} = v_{j}+ψ, and this is the decreasing limit of the smooth approximations φ_{j,k} = v_{j}+ψ_{k} on U_{j}. Hence T_{k|C_{t}} → T_{|C_{t}} for the weak topology of measures on C_{t}.

If 𝒞 is a convex cone in a finite dimensional vector space E, we denote by 𝒞^{∨} the dual cone, i.e. the set of linear forms u ∈ E^{⋆} which take nonnegative values on all elements of 𝒞. By the Hahn-Banach theorem, we always have 𝒞^{^{∨}^{∨}} = \overline{𝒞}.

Proposition 1.4 leads to the natural question whether the cones (𝒦, 𝒩) and (ℰ, ℳ) are dual under Poincaré duality. This question is addressed in the next section. Before doing so, we observe that the algebraic and transcendental cones of (n − 1, n − 1) cohomology classes are related by the following equalities (similar to what we already noticed for (1, 1)-classes, see Prop. 1.2).

1.5 Theorem. Let X be a projective manifold. Then

(i) \overline{NE(X)} = 𝒩_{NS}.

(ii) \overline{SME(X)} = \overline{ME(X)} = ℳ_{NS}.

Proof. (i) It is a standard result of algebraic geometry (see e.g. [Har70]), that the cone of effective cone NE(X) is dual to the cone \overline{𝒦_{NS}} of nef divisors, hence

𝒩_{NS} ⊃ \overline{NE(X)} = 𝒦^{∨}.

On the other hand, (1.4) (i) implies that 𝒩_{NS} ⊂ 𝒦^{∨}, so we must have equality and (i) follows.

Similarly, (ii) requires a duality statement which will be established only in the next sections, so we postpone the proof.

§2 Main results and conjectures

First, the already mentioned duality between nef divisors and effective curves extends to the Kähler case and to transcendental classes. More precisely, [DPa04] gives

2.1 Theorem (Demailly-Păun, 2001). If X is Kähler, then the cones \overline{𝒦} ⊂ H_{ℝ}^{1,1}(X) and 𝒩 ⊂ H_{ℝ}^{n−1,n−1}(X) are dual by Poincaré duality, and 𝒩 is the closed convex cone generated by classes [Y] ∧ ω^{p−1} where Y ⊂ X ranges over p-dimensional analytic subsets, p = 1, 2, …, n, and ω ranges over Kähler forms.

Proof. Indeed, Prop. 1.4 shows that the dual cone 𝒦^{∨} contains 𝒩 which itself contains the cone 𝒩′ of all classes of the form {[Y]∧ω^{p−1}}. The main result of [DPa04] conversely shows that the dual of (𝒩′)^{∨} is equal to \overline{𝒦}, so we must have

𝒦^{∨} = \overline{𝒩′} = 𝒩.

The main new result of this paper is the following characterization of pseudoeffective classes (in which the “only if” part already follows from 1.4 (iii)).

2.2 Theorem. If X is projective, then a class α ∈ NS_{ℝ}(X) is pseudo-effective if (and only if ) it is in the dual cone of the cone SME(X) of strongly movable curves.

In other words, a line bundle L is pseudo-effective if (and only if) L · C ⩾ 0 for all movable curves, i.e., L · C ⩾ 0 for every very generic curve C (not contained in a countable union of algebraic subvarieties). In fact, by definition of SME(X), it is enough to consider only those curves C which are images of generic complete intersection of very ample divisors on some variety X̃, under a modification μ : X̃ → X.

By a standard blowing-up argument, it also follows that a line bundle L on a normal Moishezon variety is pseudo-effective if and only if L·C ≥ 0 for every movable curve C.

The Kähler analogue should be :

2.3 Conjecture. For an arbitrary compact Kähler manifold X, the cones ℰ and ℳ are dual.

The relation between the various cones of movable curves and currents in (1.5) is now a rather direct consequence of Theorem 2.2. In fact, using ideas hinted in [DPS96], we can say a little bit more. Given an irreducible curve C ⊂ X, we consider its normal “bundle” N_{C} = Hom(ℐ/ℐ^{2}, 𝒪_{C}), where ℐ is the ideal sheaf of C. If C is a general

§2 Main results and conjectures 7

member of a covering family (C_{t}), then N_{C} is nef. Now [DPS96] says that the dual cone of the pseudo-effective cone of X contains the closed cone spanned by curves with nef normal bundle, which in turn contains the cone of movable curves. In this way we get :

2.4 Theorem. Let X be a projective manifold. Then the following cones coincide.

(i) the cone ℳ_{NS} = ℳ ∩ N_{1}(X) ;

(ii) the closed cone \overline{SME(X)} of strongly movable curves ;

(iii) the closed cone \overline{ME(X)} of movable curves ;

(iv) the closed cone \overline{ME_{nef}(X)} of curves with nef normal bundle.

Proof. We have already seen that

SME(X) ⊂ ME(X) ⊂ ME_{nef}(X) ⊂ (ℰ_{NS})^{∨}

and

SME(X) ⊂ ME(X) ⊂ ℳ_{NS} ⊂ (ℰ_{NS})^{∨}

by 1.4 (iii). Now Theorem 2.2 implies (ℳ_{NS})^{∨} = \overline{SME(X)}, and 2.4 follows.

2.5 Corollary. Let X be a projective manifold and L a line bundle on X.

(i) L is pseudo-effective if and only if L·C ≥ 0 for all curves C with nef normal sheaf N_{C}.

(ii) If L is big, then L · C > 0 for all curves C with nef normal sheaf N_{C}.

2.5 (i) strenghtens results from [PSS99]. It is however not yet clear whether ℳ_{NS} = ℳ ∩ N_{1}(X) is equal to the closed cone of curves with ample normal bundle (although we certainly expect this to be true).

The most important special case of Theorem 2.2 is

2.6 Theorem. If X is a projective manifold and is not uniruled, then K_{X} is pseudoeffective, i.e. K_{X} ∈ ℰ_{NS}.

Proof. This is merely a restatement of Corollary 0.3, which was proved in the introduction (as a consequence of the results of [MM86]).

Theorem 2.6 can be generalized as follows.

2.7 Theorem. Let X be a projective manifold (or a normal projective variety ). Let ℱ ⊂ T_{X} be a coherent subsheaf. If det ℱ^{∗} is not pseudo-effective, then X is uniruled. In other words, if X is not uniruled and Ω_{X}^{1} → 𝒢 is generically surjective, then det 𝒢 is pseudo-effective.

Proof. In fact, since det ℱ^{∗} is not pseudo-effective, there exists by (2.2) a covering family (C_{t}) such that c_{1}(ℱ) · C_{t} > 0. Hence X is uniruled by [Miy87], [SB92].

2.8. Remark.

(1) In [CP09] Theorem 2.7 is generalized to subsheaves ℱ ⊂ T_{X}^{⊗m}.

(2) Suppose in 2.7 that only κ(det ℱ^{∗}) = −∞. Is X still uniruled ? What can be said if c_{1}(ℱ^{∗}) is on the boundary of the pseudo-effective cone?

Turning to varieties with pseudo-effective canonical bundles, we have the

2.9 Conjecture (part of the “abundance conjecture”). If K_{X} is pseudo-effective, then κ(X) ⩾ 0.

This problem splits into two parts:

(1) If K_{X} is pseudo-effective but not big, i.e. on the boundary of the pseudo-effective cone, then there exists a (good) covering family of curve (C_{t}) such that K_{X} ·C_{t} = 0.

(2) If K_{X} is pseudo-effective and there exists a good covering family (C_{t}) of curves with K_{X} · C_{t} = 0, then κ(X) ≥ 0.

In the last section we will prove (2) in dimension 4, and even parts of it in any dimension.

§3 Zariski decomposition and movable intersections

Let X be compact Kähler and let α ∈ ℰ^{∘} be in the interior of the pseudo–effective cone. In analogy with the algebraic context such a class α is called “big”, and it can then be represented by a Kähler current T, i.e. a closed positive (1, 1)-current T such that T ⩾ δω for some smooth hermitian metric ω and a constant δ ≪ 1.

3.1 Theorem (Demailly [Dem92], [Bou02b, 3.1.24]. If T is a Kähler current, then one can write T = lim T_{m} for a sequence of Kähler currents T_{m} which have logarithmic poles with coefficients in \frac{1}{m}ℤ, i.e. there are modifications μ_{m} : X_{m} → X such that

μ_{m}^{⋆}T_{m} = [E_{m}] + β_{m}

where E_{m} is an effective ℚ-divisor on X_{m} with coefficients in \frac{1}{m}ℤ (the “fixed part”) and β_{m} is a closed semi-positive form (the “movable part”).

Proof. Since this result has already been studied extensively, we just recall the main idea. Locally we can write T = i∂\overline{∂}φ for some strictly plurisubharmonic potential φ. By a Bergman kernel trick and the Ohsawa-Takegoshi L^{2} extension theorem, we get local approximations

φ = lim φ_{m}, φ_{m}(z) = \frac{1}{2m} log ∑_{ℓ} |g_{ℓ,m}(z)|^{2}

where (g_{ℓ,m}) is a Hilbert basis of the space of holomorphic functions which are L^{2} with respect to the weight e^{−2mφ}. This Hilbert basis is also a family of local generators of the globally defined multiplier ideal sheaf ℐ(mT) = ℐ(mφ). Then μ_{m} : X_{m} → X is obtained by blowing-up this ideal sheaf, so that

μ_{m}^{⋆}ℐ(mT) = 𝒪(−mE_{m}).

§3 Zariski decomposition and movable intersections 9

We should notice that by approximating T − \frac{1}{m}ω instead of T, we can replace β_{m} by β_{m} + \frac{1}{m}μ^{⋆}ω which is a big class on X_{m}; by playing with the multiplicities of the components of the exceptional divisor, we could even achieve that β_{m} is a Kähler class on X_{m}, but this will not be needed here.

The more familiar algebraic analogue would be to take α = c_{1}(L) with a big line bundle L and to blow-up the base locus of |mL|, m ≫ 1, to get a ℚ-divisor decomposition

μ_{m}^{⋆}L ∼ E_{m} + D_{m}, E_{m} effective, D_{m} free.

Such a blow-up is usually referred to as a “log resolution” of the linear system |mL|, and we say that E_{m} + D_{m} is an approximate Zariski decomposition of L. We will also use this terminology for Kähler currents with logarithmic poles.

3.2 Definition. We define the volume, or movable self-intersection of a big class α ∈ ℰ^{∘} to be

Vol(α) = sup_{T ∈α} ∫_{X} β^{n} > 0

where the supremum is taken over all Kähler currents T ∈ α with logarithmic poles, and μ^{⋆}T = [E] + β with respect to some modification μ : X̃ → X.

By Fujita [Fuj94] and Demailly-Ein-Lazarsfeld [DEL00], if L is a big line bundle, we have

Vol(c_{1}(L)) = lim_{m→+∞} D_{m}^{n} = lim_{m→+∞} \frac{n!}{m^{n}} h^{0}(X, mL),

and in these terms, we get the following statement.

3.3 Proposition. Let L be a big line bundle on the projective manifold X. Let ϵ > 0. Then there exists a modification μ : X_{ϵ} → X and a decomposition μ^{∗}(L) = E + β with E an effective ℚ-divisor and β a big and nef ℚ-divisor such that

Vol(L) − ε ⩽ Vol(β) ⩽ Vol(L).

It is very useful to observe that the supremum in Definition 3.2 can actually be computed by a collection of currents whose singularities satisfy a filtering property. Namely, if T_{1} = α+i∂\overline{∂}φ_{1} and T_{2} = α+i∂\overline{∂}φ_{2} are two Kähler currents with logarithmic poles in the class of α, then

(3.4)  T = α + i∂\overline{∂}φ,  φ = max(φ_{1}, φ_{2})

is again a Kähler current with weaker singularities than T_{1} and T_{2}. One could define as well

(3.4′)  T = α + i∂\overline{∂}φ,  φ = \frac{1}{2m} log(e^{2mφ_{1}} + e^{2mφ_{2}}),

where m = lcm(m_{1}, m_{2}) is the lowest common multiple of the denominators occuring in T_{1}, T_{2}. Now, take a simultaneous log-resolution μ_{m} : X_{m} → X for which the singularities of T_{1} and T_{2} are resolved as ℚ-divisors E_{1} and E_{2}. Then clearly the associated divisor in the decomposition μ_{m}^{⋆}T = [E] + β is given by E = min(E_{1}, E_{2}). By doing so, the volume ∫_{X_{m}} β^{n} gets increased, as we shall see in the proof of Theorem 3.5 below.

3.5 Theorem (Boucksom [Bou02b]). Let X be a compact Kähler manifold. We denote here by H_{⩾0}^{k,k}(X) the cone of cohomology classes of type (k, k) which have non-negative intersection with all closed semi-positive smooth forms of bidegree (n − k, n − k).

(i) For each integer k = 1, 2, …, n, there exists a canonical “movable intersection product”

ℰ × ⋯ × ℰ → H_{⩾0}^{k,k}(X), (α_{1}, …, α_{k}) ↦ ⟨α_{1} · α_{2} ⋯ α_{k−1} · α_{k}⟩

such that Vol(α) = ⟨α^{n}⟩ whenever α is a big class.

(ii) The product is increasing, homogeneous of degree 1 and superadditive in each argument, i.e.

⟨α_{1} ⋯ (α′_{j} + α′′_{j}) ⋯ α_{k}⟩ ⩾ ⟨α_{1} ⋯ α′_{j} ⋯ α_{k}⟩ + ⟨α_{1} ⋯ α′′_{j} ⋯ α_{k}⟩.

It coincides with the ordinary intersection product when the α_{j} ∈ \overline{𝒦} are nef classes.

(iii) The movable intersection product satisfies the Teissier-Hovanskii inequalities

⟨α_{1} · α_{2} ⋯ α_{n}⟩ ⩾ (⟨α_{1}^{n}⟩)^{1/n} … (⟨α_{n}^{n}⟩)^{1/n} (with ⟨α_{j}^{n}⟩ = Vol(α_{j}) ).

(iv) For k = 1, the above “product” reduces to a (non linear) projection operator

ℰ → ℰ_{1}, α → ⟨α⟩

onto a certain convex subcone ℰ_{1} of ℰ such that \overline{𝒦} ⊂ ℰ_{1} ⊂ ℰ. Moreover, there is a “divisorial Zariski decomposition”

α = {N(α)} + ⟨α⟩

where N(α) is a uniquely defined effective divisor which is called the “negative divisorial part” of α. The map α ↦ N(α) is homogeneous and subadditive, and N(α) = 0 if and only if α ∈ ℰ_{1}.

(v) The components of N(α) always consist of divisors whose cohomology classes are linearly independent, especially N(α) has at most ρ = rank_{ℤ} NS(X) components.

Proof. We essentially repeat the arguments developped in [Bou02b], with some simplifications arising from the fact that X is supposed to be Kähler from the start.

(i) First assume that all classes α_{j} are big, i.e. α_{j} ∈ ℰ^{∘}. Fix a smooth closed (n−k, n−k) semi-positive form u on X. We select Kähler currents T_{j} ∈ α_{j} with logarithmic poles, and a simultaneous log-resolution μ : X̃ → X such that

μ^{⋆}T_{j} = [E_{j}] + β_{j}.

We consider the direct image current μ_{⋆}(β_{1}∧…∧β_{k}) (which is a closed positive current of bidegree (k, k) on X) and the corresponding integrals

∫_{X} β_{1} ∧ … ∧ β_{k} ∧ μ^{⋆}u ⩾ 0.

If we change the representative T_{j} with another current T′_{j}, we may always take a simultaneous log-resolution such that μ^{⋆}T′_{j} = [E′_{j}] + β′_{j}, and by using (3.4′) we can always assume that E′_{j} ⩽ E_{j}. Then D_{j} = E_{j} − E′_{j} is an effective divisor and we find [E_{j}] + β_{j} ≡ [E′_{j}] + β′_{j}, hence β′_{j} ≡ β_{j} + [D_{j}]. A substitution in the integral implies

∫_{X} β′_{1} ∧ β_{2} ∧ … ∧ β_{k} ∧ μ^{⋆}u

= ∫_{X} β_{1} ∧ β_{2} ∧ … ∧ β_{k} ∧ μ^{⋆}u + ∫_{X}[D_{1}] ∧ β_{2} ∧ … ∧ β_{k} ∧ μ^{⋆}u

⩾ ∫_{X} β_{1} ∧ β_{2} ∧ … ∧ β_{k} ∧ μ^{⋆}u.

Similarly, we can replace successively all forms β_{j} by the β′_{j}, and by doing so, we find

∫_{X} β′_{1} ∧ β′_{2} ∧ … ∧ β′_{k} ∧ μ^{⋆}u ⩾ ∫_{X} β_{1} ∧ β_{2} ∧ … ∧ β_{k} ∧ μ^{⋆}u.

We claim that the closed positive currents μ_{⋆}(β_{1} ∧ … ∧ β_{k}) are uniformly bounded in mass. In fact, if ω is a Kähler metric in X, there exists a constant C_{j} ⩾ 0 such that C_{j}{ω} − α_{j} is a Kähler class. Hence C_{j}ω − T_{j} ≡ γ_{j} for some Kähler form γ_{j} on X. By pulling back with μ, we find C_{j}μ^{⋆}ω − ([E_{j}] + β_{j}) ≡ μ^{⋆}γ_{j}, hence

β_{j} ≡ C_{j}μ^{⋆}ω − ([E_{j}] + μ^{⋆}γ_{j}).

By performing again a substitution in the integrals, we find

∫_{X} β_{1} ∧ … ∧ β_{k} ∧ μ^{⋆}u ⩽ C_{1} … C_{k} ∫_{X} μ^{⋆}ω^{k} ∧ μ^{⋆}u = C_{1} … C_{k} ∫_{X} ω^{k} ∧ u

and this is true especially for u = ω^{n−k}. We can now arrange that for each of the integrals associated with a countable dense family of forms u, the supremum is achieved by a sequence of currents (μ_{m})_{⋆}(β_{1,m}∧…∧β_{k,m}) obtained as direct images by a suitable sequence of modifications μ_{m} : X̃_{m} → X. By extracting a subsequence, we can achieve that this sequence is weakly convergent and we set

⟨α_{1} · α_{2} ⋯ α_{k}⟩ = lim ↑_{m→+∞}{(μ_{m})_{⋆}(β_{1,m} ∧ β_{2,m} ∧ … ∧ β_{k,m})}

(the monotonicity is not in terms of the currents themselves, but in terms of the integrals obtained when we evaluate against a smooth closed semi-positive form u). By evaluating against a basis of positive classes {u} ∈ H^{n−k,n−k}(X), we infer by Poincaré duality that the class of ⟨α_{1} · α_{2} ⋯ α_{k}⟩ is uniquely defined (although, in general, the representing current is not unique).

(ii) It is indeed clear from the definition that the movable intersection product is homogeneous, increasing and superadditive in each argument, at least when the α_{j}’s are in ℰ^{∘}. However, we can extend the product to the closed cone ℰ by monotonicity, by setting

⟨α_{1} · α_{2} ⋯ α_{k}⟩ = lim ↓_{δ↓0}⟨(α_{1} + δω) · (α_{2} + δω) ⋯ (α_{k} + δω)⟩

for arbitrary classes α_{j} ∈ ℰ (again, monotonicity occurs only where we evaluate against closed semi-positive forms u). By weak compactness, the movable intersection product can always be represented by a closed positive current of bidegree (k, k).

(iii) The Teissier-Hovanskii inequalities are a direct consequence of the fact that they hold true for nef classes, so we just have to apply them to the classes β_{j,m} on X̃_{m} and pass to the limit.

(iv) When k = 1 and α ∈ ℰ^{0}, we have

α = lim_{m→+∞}{(μ_{m})_{⋆}T_{m}} = lim_{m→+∞}(μ_{m})_{⋆}[E_{m}] + {(μ_{m})_{⋆}β_{m}}

and ⟨α⟩ = lim_{m→+∞}{(μ_{m})_{⋆}β_{m}} by definition. However, the images F_{m} = (μ_{m})_{⋆}E_{m} are effective ℚ-divisors in X, and the filtering property implies that F_{m} is a decreasing sequence. It must therefore converge to a (uniquely defined) limit F = lim F_{m} := N(α) which is an effective ℝ-divisor, and we get the asserted decomposition in the limit.

Since N(α) = α − ⟨α⟩ we easily see that N(α) is subadditive and that N(α) = 0 if α is the class of a smooth semi-positive form. When α is no longer a big class, we define

⟨α⟩ = lim_{δ↓0} ↓ ⟨α + δω⟩, N(α) = lim_{δ↓0} ↑ N(α + δω)

(the subadditivity of N implies N(α + (δ + ε)ω) ⩽ N(α + δω)). The divisorial Zariski decomposition follows except maybe for the fact that N(α) might be a convergent countable sum of divisors. However, this will be ruled out when (v) is proved. As N(•) is subadditive and homogeneous, the set ℰ_{1} = {α ∈ ℰ ; N(α) = 0} is a closed convex cone, and we find that α ↦ ⟨α⟩ is a projection of ℰ onto ℰ_{1} (according to [Bou02b], ℰ_{1} consists of those pseudo-effective classes which are “nef in codimension 1”).

(v) Let α ∈ ℰ^{∘}, and assume that N(α) contains linearly dependent components F_{j}. Then already all currents T ∈ α should be such that μ^{⋆}T = [E] + β where F = μ_{⋆}E contains those linearly dependent components. Write F = ∑ λ_{j}F_{j}, λ_{j} > 0 and assume that

∑_{j∈J} c_{j}F_{j} ≡ 0

for a certain non trivial linear combination. Then some of the coefficients c_{j} must be negative (and some other positive). Then E is numerically equivalent to

E′ ≡ E + tμ^{⋆}( ∑ λ_{j}F_{j}),

and by choosing t > 0 appropriate, we obtain an effective divisor E′ which has a zero coefficient on one of the components μ^{⋆}F_{j_{0}}. By replacing E with min(E, E′) via (3.4′), we eliminate the component μ^{⋆}F_{j_{0}}. This is a contradiction since N(α) was supposed to contain F_{j_{0}}.

3.6 Definition. For a class α ∈ H_{ℝ}^{1,1}(X), we define the numerical dimension ν(α) to be ν(α) = −∞ if α is not pseudo-effective, and

ν(α) = max{p ∈ ℕ ; ⟨α^{p}⟩ ≠ 0}, ν(α) ∈ {0, 1, …, n}

if α is pseudo-effective.

By the results of [DP03], a class is big (α ∈ ℰ^{∘}) if and only if ν(α) = n. Classes of numerical dimension 0 can be described much more precisely, again following Boucksom [Bou02b].

3.7 Theorem. Let X be a compact Kähler manifold. Then the subset 𝒟_{0} of irreducible divisors D in X such that ν(D) = 0 is countable, and these divisors are rigid as well as their multiples. If α ∈ ℰ is a pseudo-effective class of numerical dimension 0, then α is numerically equivalent to an effective ℝ-divisor D = ∑_{j∈J} λ_{j}D_{j}, for some finite subset (D_{j})_{j∈J} ⊂ 𝒟_{0} such that the cohomology classes {D_{j}} are linearly independent and some λ_{j} > 0. If such a linear combination is of numerical dimension 0, then so is any other linear combination of the same divisors.

Proof. It is immediate from the definition that a pseudo-effective class is of numerical dimension 0 if and only if ⟨α⟩ = 0, in other words if α = N(α). Thus α ≡ ∑ λ_{j}D_{j} as described in 3.7, and since λ_{j}⟨D_{j}⟩ ⩽ ⟨α⟩, the divisors D_{j} must themselves have numerical dimension 0. There is at most one such divisor D in any given cohomology class in NS(X) ∩ ℰ ⊂ H^{2}(X, ℤ), otherwise two such divisors D ≡ D′ would yield a blow-up μ : X̃ → X resolving the intersection, and by taking min(μ^{⋆}D, μ^{⋆}D′) via (3.4′), we would find μ^{⋆}D ≡ E + β, β ≠ 0, so that {D} would not be of numerical dimension 0. This implies that there are at most countably many divisors of numerical dimension 0, and that these divisors are rigid as well as their multiples.

The above general concept of numerical dimension leads to a very natural formulation of the abundance conjecture for non-minimal (Kähler) varieties.

3.8 Generalized abundance conjecture. For an arbitrary compact Kähler manifold X, the Kodaira dimension should be equal to the numerical dimension :

κ(X) = ν(X) := ν(c_{1}(K_{X})).

This appears to be a fairly strong statement. In fact, it is not difficult to show that the generalized abundance conjecture would contain the C_{n,m} conjectures.

3.9 Remark. Using the Iitaka fibration, it is immediate to see that κ(X) ≤ ν(X).

3.10 Remark. It is known that abundance holds in case ν(X) = −∞ (if K_{X} is not pseudo-effective, no multiple of K_{X} can have sections), or in case ν(X) = n. The latter follows from the solution of the Grauert-Riemenschneider conjecture in the form proven in [Dem85] (see also [DPa04]).

In the remaining cases, the most tractable situation is the case when ν(X) = 0. In fact Theorem 3.7 then gives K_{X} ≡ ∑ λ_{j}D_{j} for some effective divisor with numerically independent components, ν(D_{j}) = 0. It follows that the λ_{j} are rational and therefore

(∗)  K_{X} ≡ ∑ λ_{j}D_{j} + F  where λ_{j} ∈ ℚ^{+}, ν(D_{j}) = 0 and F ∈ Pic^{0}(X).

By [CP09] it now follows that κ(X) ≥ 0. Thus we obtain:

3.11 Proposition. Let X be a smooth projective manifold with K_{X} pseudo-effective. If ν(X) = 0, then κ(X) = 0.

We will come back to abundance on 4-folds in sect. 9.

The arguments given in (3.10) are actually not restricted to the canonical bundle and show

3.12 Proposition. Let X be a projective manifold and L a pseudo-effective ℝ−divisor on X.

(i) If ν(L) = 0, then L ≡ ∑ λ_{j}D_{j} with λ_{j} positive real numbers and D_{j} irreducible divisors. If L is Cartier, the λ_{j} are rational.

(ii) If L is moreover nef in codimension 1 and if ν(L) = 0, then L ≡ 0.

§4 The orthogonality estimate

The goal of this section is to show that, in an appropriate sense, approximate Zariski decompositions are almost orthogonal.

4.1 Theorem. Let X be a projective manifold, and let α = {T } ∈ ℰ_{NS}^{∘} be a big class represented by a Kähler current T. Consider an approximate Zariski decomposition

μ_{m}^{⋆}T_{m} = [E_{m}] + [D_{m}]

Then

(D_{m}^{n−1} · E_{m})^{2} ⩽ 20 (Cω)^{n}( Vol(α) − D_{m}^{n})

where ω = c_{1}(H) is a Kähler form and C ⩾ 0 is a constant such that ±α is dominated by Cω (i.e., Cω ± α is nef ).

Proof. For every t ∈ [0, 1], we have

Vol(α) = Vol(E_{m} + D_{m}) ⩾ Vol(tE_{m} + D_{m}).

Now, by our choice of C, we can write E_{m} as a difference of two nef divisors

E_{m} = μ^{⋆}α − D_{m} = μ_{m}^{⋆}(α + Cω) − (D_{m} + Cμ_{m}^{⋆}ω).

4.2 Lemma. For all nef ℝ-divisors A, B we have

Vol(A − B) ⩾ A^{n} − nA^{n−1} · B

as soon as the right hand side is positive.

Proof. In case A and B are integral (Cartier) divisors, this is a consequence of the holomorphic Morse inequalities, [Dem01, 8.5]. If A and B are ℚ-Cartier, we conclude by the homogeneity of the volume. The general case of ℝ-divisors follows by approximation using the upper semi-continuity of the volume [Bou02b, 3.1.26].

§4 The orthogonality estimate 15

4.3 Remark. We hope that Lemma 4.2 also holds true on an arbitrary Kähler manifold for arbitrary nef (non necessarily integral) classes. This would follow from a generalization of holomorphic Morse inequalities to non integral classes. However the proof of such a result seems technically much more involved than in the case of integral classes.

4.4 Lemma. Let β_{1}, …, β_{n} and β′_{1}, …, β′_{n} be nef classes on a compact Kähler manifold X̃ such that each difference β′_{j} − β_{j} is pseudo-effective. Then the n-th intersection products satisfy

β_{1} ⋯ β_{n} ⩽ β′_{1} ⋯ β′_{n}.

Proof. We can proceed step by step and replace just one β_{j} by β′_{j} ≡ β_{j} + T_{j} where T_{j} is a closed positive (1, 1)-current and the other classes β′_{k} = β_{k}, k ≠ j are limits of Kähler forms. The inequality is then obvious. End of proof of Theorem 4.1. In order to exploit the lower bound of the volume, we write

tE_{m} + D_{m} = A − B, A = D_{m} + tμ_{m}^{⋆}(α + Cω), B = t(D_{m} + Cμ_{m}^{⋆}ω).

By our choice of the constant C, both A and B are nef. Lemma 4.2 and the binomial formula imply

Vol(tE_{m} + D_{m}) ⩾ A^{n} − nA^{n−1} · B

= D_{m}^{n} + nt D_{m}^{n−1} · μ_{m}^{⋆}(α + Cω) + ∑_{k=2}^{n} t^{k}\binom{n}{k}D_{m}^{n−k} · μ_{m}^{⋆}(α + Cω)^{k}

− nt D_{m}^{n−1} · (D_{m} + Cμ_{m}^{⋆}ω)

− nt^{2} ∑_{k=1}^{n−1} t^{k−1}\binom{n − 1}{k}D_{m}^{n−1−k} · μ_{m}^{⋆}(α + Cω)^{k} · (D_{m} + Cμ_{m}^{⋆}ω).

Now, we use the obvious inequalities

D_{m} ⩽ μ_{m}^{⋆}(Cω),  μ_{m}^{⋆}(α + Cω) ⩽ 2μ_{m}^{⋆}(Cω),  D_{m} + Cμ_{m}^{⋆}ω ⩽ 2μ_{m}^{⋆}(Cω)

in which all members are nef (and where the inequality ⩽ means that the difference of classes is pseudo-effective). We use Lemma 4.4 to bound the last summation in the estimate of the volume, and in this way we get

Vol(tE_{m} + D_{m}) ⩾ D_{m}^{n} + ntD_{m}^{n−1} · E_{m} − nt^{2} ∑_{k=1}^{n−1} 2^{k+1}t^{k−1}\binom{n − 1}{k}(Cω)^{n}.

We will always take t smaller than 1/10n so that the last summation is bounded by 4(n − 1)(1 + 1/5n)^{n−2} < 4ne^{1/5} < 5n. This implies

Vol(tE_{m} + D_{m}) ⩾ D_{m}^{n} + nt D_{m}^{n−1} · E_{m} − 5n^{2}t^{2}(Cω)^{n}.

Now, the choice t = \frac{1}{10n}(D_{m}^{n−1} · E_{m})((Cω)^{n})^{−1} gives by substituting

\frac{1}{20} \frac{(D_{m}^{n−1} · E_{m})^{2}}{(Cω)^{n}} ⩽ Vol(E_{m} + D_{m}) − D_{m}^{n} ⩽ Vol(α) − D_{m}^{n}

(and we have indeed t ⩽ \frac{1}{10n} by Lemma 4.4), whence Theorem 4.1. Of course, the constant 20 is certainly not optimal.

4.5 Corollary. If α ∈ ℰ_{NS}, then the divisorial Zariski decomposition α = N(α) + ⟨α⟩ is such that

⟨α^{n−1}⟩ · N(α) = 0.

Proof. By replacing α by α + δc_{1}(H), one sees that it is sufficient to consider the case where α is big. Then the orthogonality estimate implies

(μ_{m})_{⋆}(D_{m}^{n−1}) · (μ_{m})_{⋆}E_{m} = D_{m}^{n−1} · (μ_{m})^{⋆}(μ_{m})_{⋆}E_{m} ⩽ D_{m}^{n−1} · E_{m} ⩽ C(Vol(α) − D_{m}^{n})^{1/2}.

Since ⟨α^{n−1}⟩ = lim(μ_{m})_{⋆}(D_{m}^{n−1}), N(α) = lim(μ_{m})_{⋆}E_{m} and lim D_{m}^{n} = Vol(α), we get the desired conclusion in the limit.

§5 Proof of the duality theorem

We want to prove that ℰ_{NS} and SME(X) are dual (Theorem 2.2). By 1.4 (iii) we have in any case

ℰ_{NS} ⊂ (SME(X))^{∨}.

If the inclusion is strict, there is an element α ∈ ∂ℰ_{NS} on the boundary of ℰ_{NS} which is in the interior of SME(X)^{∨}.

Let ω = c_{1}(H) be an ample class. Since α ∈ ∂ℰ_{NS}, the class α + δω is big for every δ > 0, and since α ∈ ((SME(X))^{∨})^{∘} we still have α − εω ∈ (SME(X))^{∨} for ε > 0 small. Therefore

(5.1)  α · Γ ⩾ εω · Γ

for every movable curve Γ. We are going to contradict (5.1). Since α + δω is big, we have an approximate Zariski decomposition

μ_{δ}^{⋆}(α + δω) = E_{δ} + D_{δ}.

We pick Γ = (μ_{δ})_{⋆}(D_{δ}^{n−1}). By the Hovanskii-Teissier concavity inequality

ω · Γ ⩾ (ω^{n})^{1/n}(D_{δ}^{n})^{(n−1)/n}.

On the other hand

α · Γ = α · (μ_{δ})_{⋆}(D_{δ}^{n−1})
= μ_{δ}^{⋆}α · D_{δ}^{n−1} ⩽ μ_{δ}^{⋆}(α + δω) · D_{δ}^{n−1}
= (E_{δ} + D_{δ}) · D_{δ}^{n−1} = D_{δ}^{n} + D_{δ}^{n−1} · E_{δ}.

§6 Non nef loci 17

By the orthogonality estimate, we find

\frac{α · Γ}{ω · Γ} ⩽ \frac{D_{δ}^{n} + (20(Cω)^{n}(Vol(α + δω) − D_{δ}^{n}))^{1/2}}{(ω^{n})^{1/n}(D_{δ}^{n})^{(n−1)/n}}

⩽ C′(D_{δ}^{n})^{1/n} + C′′ \frac{(Vol(α + δω) − D_{δ}^{n})^{1/2}}{(D_{δ}^{n})^{(n−1)/n}} .

However, since α ∈ ∂ℰ_{NS}, the class α cannot be big so

lim_{δ→0} D_{δ}^{n} = Vol(α) = 0.

We can also take D_{δ} to approximate Vol(α+δω) in such a way that (Vol(α+δω)−D_{δ}^{n})^{1/2} tends to 0 much faster than D_{δ}^{n}. Notice that D_{δ}^{n} ⩾ δ^{n}ω^{n}, so in fact it is enough to take

Vol(α + δω) − D_{δ}^{n} ⩽ δ^{2n}.

This is the desired contradiction by (5.1).

5.2 Remark. If holomorphic Morse inequalities were known also in the Kähler case, we would infer by the same proof that “α not pseudo-effective” implies the existence of a blow-up μ : X̃ → X and a Kähler metric ω̃ on X̃ such that α · μ_{⋆}(ω̃)^{n−1} < 0. In the special case when α = K_{X} is not pseudo-effective, we would expect the Kähler manifold X to be covered by rational curves. The main trouble is that characteristic p techniques are no longer available. On the other hand it is tempting to approach the question via techniques of symplectic geometry :

5.3 Question. Let (M, ω) be a compact real symplectic manifold. Fix an almost complex structure J compatible with ω, and for this structure, assume that c_{1}(M) · ω^{n−1} > 0. Does it follow that M is covered by rational J-pseudoholomorphic curves ?

§6 Non nef loci

Following [Bou02b], we introduce the concept of non-nef locus of an arbitrary pseudo-effective class. The details differ a little bit here (and are substantially simpler) because the scope is limited to compact Kähler manifolds.

6.1 Definition. Let X be a compact Kähler manifold, ω a Kähler metric, and α ∈ ℰ a pseudo-effective class. We define the non-nef locus of α to be

L_{nnef}(α) = ⋃_{δ>0} ⋂_{T} μ(|E|)

for all log resolutions μ^{⋆}T = [E] + β of positive currents T ∈ {α + δω} with logarithmic singularities, μ : X̃ → X, and μ(|E|) is the set-theoretic image of the support of E.

It should be noticed that the union in the above definition can be restricted to any sequence δ_{k} converging to 0, hence L_{nnef}(α) is either an analytic set or a countable union of analytic sets. The results of [Dem92] and [Bou02b] show that

(6.1′)  L_{nnef}(α) = ⋃_{δ>0} ⋂_{T} E_{+}(T)

where T runs over the set α[−δω] of all d-closed real (1, 1)-currents T ∈ α such that T ⩾ −δω, and E_{+}(T) denotes the locus where the Lelong numbers of T are strictly positive. The latter definition (6.1′) works even in the non Kähler case, taking ω an arbitrary positive hermitian form on X. By [Bou02b], there is always a current T_{min} which achieves minimum singularities and minimum Lelong numbers among all members of α[−δω], hence ⋂_{T} E_{+}(T) = E_{+}(T_{min}).

6.2 Theorem. Let α ∈ ℰ be a pseudo-effective class. Then L_{nnef}(α) contains the union of all irreducible algebraic curves C such that α · C < 0.

Proof. If C is an irreducible curve not contained in L_{nnef}(α), the definition implies that for every δ > 0 we can choose a positive current T ∈ {α + δω} and a log-resolution μ^{⋆}T = [E] + β such that C ⊄ μ(|E|). Let C̃ be the strict transform of C in X̃, so that C = μ_{⋆}C̃. We then find

(α + δω) · C = ([E] + β) · C̃ ⩾ 0

since β ⩾ 0 and C̃ ⊄ |E|. This is true for all δ > 0 and the claim follows.

6.3 Remark. One may wonder, at least when X is projective and α ∈ ℰ_{NS}, whether L_{nnef}(α) is actually equal to the union of curves C such that L·C < 0 (or the “countable Zariski closure” of such a union). Unfortunately, this is not true, even on surfaces. The following simple example was shown to us by E. Viehweg. Let Y be a complex algebraic surface possessing a big line bundle F with a curve C such that F · C < 0 as its base locus (e.g. F = π^{⋆}𝒪(1) + E for the blow-up π : Y → ℙ^{2} of ℙ^{2} in one point, and C = E = exceptional divisor). Then take finitely many points p_{j} ∈ C, 1 ⩽ j ⩽ N, and blow-up these points to get a modification μ : X → Y. We select

L = μ^{⋆}F + Ĉ + 2 ∑ E_{j} = μ^{⋆}(F + C) + ∑ E_{j}

where Ĉ is the strict transform of C and E_{j} = μ^{−1}(p_{j}). It is clear that the non nef locus of α = c_{1}(L) must be equal to Ĉ ∪ ⋃ E_{j}, although

L · Ĉ = (F + C) · C + N > 0

for N large. This example shows that the set of α-negative curves is not the appropriate tool to understand the non nef locus.

§7 Pseudo-effective vector bundles 19

§7 Pseudo-effective vector bundles

In this section we consider pseudo-effective and almost nef vector bundles as introduced in [DPS00]. As an application, we obtain interesting informations concerning the tangent bundle of Calabi-Yau manifolds. First we recall the relevant definitions.

7.1 Definition. Let X be a compact Kähler manifold and E a holomorphic vector bundle on X. Then E is said to be pseudo-effective if the line bundle 𝒪_{ℙ(E)}(1) is pseudoeffective on the projectivized bundle ℙ(E) of hyperplanes of E, and if the projection π(L_{nnef}(𝒪_{ℙ(E)}(1))) of the non-nef locus of 𝒪_{ℙ(E)}(1) onto X does not cover all of X.

This definition would even make sense on a general compact complex manifold, using the general definition of the non-nef locus in [Bou02b]. On the other hand, the following proposition gives an algebraic characterization of pseudo-effective vector bundles in the projective case.

7.2 Proposition. Let X be a projective manifold. A holomorphic vector bundle E on X is pseudo-effective if and only if for any given ample line bundle A on X and any positive integers m_{0}, p_{0}, the vector bundle

S^{p}((S^{m}E) ⊗ A)

is generically generated (i.e. generated by its global sections on the complement X∖Z_{m,p} of some algebraic set Z_{m,p} ≠ X) for some [resp. every] m ≥ m_{0} and p ≥ p_{0}.

Proof. If global sections as in the statement of 7.2 exist, they can be used to define a singular hermitian metric h_{m,p} on 𝒪_{ℙ(E)}(1) which has poles contained in π^{−1}(Z_{m,p}) and whose curvature form satisfies Θ_{h_{m,p}}(𝒪_{ℙ(E)}(1)) ⩾ −\frac{1}{m}π^{∗}Θ(A). Hence, by selecting suitable integers m = M(m_{0}, p_{0}) and p = P (m_{0}, p_{0}), we find that 𝒪_{ℙ(E)}(1) is pseudoeffective (its first Chern class is a limit of pseudo-effective classes), and that

π(L_{nnef}(𝒪_{ℙ(E)}(1))) ⊂ ⋃_{m_{0}} ⋂_{p_{0}} Z_{m,p} ⊊ X.

Conversely, assume that 𝒪_{ℙ(E)}(1) is pseudo-effective and admits singular hermitian metrics h_{δ} such that Θ_{h_{δ}}(𝒪_{ℙ(E)}(1)) ⩾ −δω̃ and π(Sing(h_{δ})) ⊂ Z_{δ} ⊊ X (for some Kähler metric ω̃ on ℙ(E) and arbitrary small δ > 0). We can actually take ω = Θ(A) and ω̃ = ε_{0}Θ_{h_{0}}(𝒪_{ℙ(E)}(1)) + π^{∗}ω with a given smooth hermitian metric h_{0} on E and ε_{0} ≪ 1. An easy calculation shows that the linear combination h′_{δ} = h_{δ}^{1/(1+δε_{0})}h_{0}^{δε_{0}} yields a metric on 𝒪_{ℙ(E)}(1) such that

Θ_{h′_{δ}}(𝒪_{ℙ(E)}(1)) ⩾ −δπ^{∗}Θ(A).

By taking δ = 1/2m and multiplying by m, we find

Θ(𝒪_{ℙ(E)}(m) ⊗ π^{∗}A) ⩾ \frac{1}{2}π^{∗}Θ(A)

for some metric on 𝒪_{ℙ(E)}(m) ⊗ π^{∗}A which is smooth over π^{−1}(X ∖ Z_{δ}). The standard theory of L^{2} estimates for bundle-valued \overline{∂}-operators can be used to produce the required sections, after we multiply Θ(A) by a sufficiently large integer p to compensate the curvature of −K_{X}. The sections possibly still have to vanish along the poles of the metric, but they are unrestricted on fibers of ℙ(S^{m}E) → X which do not meet the singularities.

Note that if E is pseudo-effective, then 𝒪_{ℙ(E)}(1) is pseudo-effective and E is almost nef in the following sense which is just the straightforward generalization from the line bundle case.

7.3 Definition. Let X be a projective manifold and E a vector bundle on X. Then E is said to be almost nef, if there is a countable family A_{i} of proper subvarieties of X such that E|C is nef for all C ⊄ ⋃_{i} A_{i}. Alternatively, E is almost nef if there is no covering family of curves such that E is non-nef on the general member of the family.

Observe that E is almost nef if and only if 𝒪_{ℙ(E)}(1) is almost nef and 𝒪_{ℙ(E)}(1) is nef on the general member of any family of curves in ℙ(E) whose images cover X. Hence Theorem 2.2 yields

7.4 Corollary. Let X be a projective manifold and E a holomorphic vector bundle on X. If E is almost nef, then 𝒪_{ℙ(E)}(1) is pseudo-effective. Thus for some [or any] ample line bundle A, there are positive numbers m_{0} and p_{0} such that

H^{0}(X, S^{p}((S^{m}E) ⊗ A)) ≠ 0

for all m ≥ m_{0} and p ≥ p_{0}.

One should notice that it makes a big difference to assert just the existence of a non zero section, and to assert the existence of sufficiently many sections guaranteeing that the fibers are generically generated. It is therefore natural to raise the following question.

7.5 Question. Let X be a projective manifold and E a vector bundle on X. Suppose that E is almost nef. Is E always pseudo-effective in the sense of Definition 7.1 ?

This was stated as a theorem in [DPS01, 6.3], but the proof given there was incomplete. The result now appears quite doubtful to us. However, we give below a positive answer to Question 7.5 in case of a rank 2−bundle E with c_{1}(E) = 0 (conjectured in [DPS01]), and then apply it to the study of tangent bundles of K3-surfaces.

7.6 Theorem. Let E be an almost nef vector bundle of rank at most 3 on a projective manifold X. Suppose that det E ≡ 0. Then E is numerically flat.

Proof. Recall (cf. [DPS94]) that a vector bundle E is said to be numerically flat if it is nef as well as its dual (or, equivalently, if E is nef and det E numerically trivial); also, E is numerically flat if and only if E admits a filtration by subbundles such that the graded pieces are unitary flat vector bundles. By [Ko87, p.115], E is unitary flat as soon as E is stable for some polarization and c_{1}(E) = c_{2}(E) = 0.

Under our assumptions, E is necessarily semi-stable since semi-stability with respect to a polarization H can be tested against a generic complete intersection curve, and we know that E is nef, hence numerically flat, on such a curve. Therefore (see also [DPS01, 6.8]) we can assume without loss of generality that dim X = 2 and that E is stable with respect to all polarizations, and it is enough to show in that case that c_{2}(E) = 0. Since E is almost nef, E is nef, hence numerically flat, on all curves except for at most a countable number of curves, say (Γ_{j})_{j∈ℕ}. First suppose that E has rank 2. Then the line bundle 𝒪(1) on ℙ(E) is immediately seen to be nef on all but a countable number of curves. In fact, the only curves on which 𝒪(1) is negative are the sections over the curves Γ_{j} with negative self-intersection in ℙ(E|Γ_{j}). Now take a general hyperplane section H on ℙ(E). Then H does not contain any of these bad curves and therefore 𝒪(1) is nef on H. Hence

c_{1}(𝒪(1))^{2} · H ≥ 0.

Now - up to a multiple - H is of the form H = 𝒪(1) ⊗ π^{∗}(G) so that

c_{1}(𝒪(1))^{3} + c_{1}(𝒪(1))^{2} · π^{∗}(G) ≥ 0.

Since c_{1}(𝒪(1))^{3} = c_{1}(E)^{2} − c_{2}(E) = −c_{2}(E) and c_{1}(𝒪(1))^{2} · π^{∗}(G) = c_{1}(E) · G = 0, we conclude c_{2}(E) = 0. If E has rank 3, we need to argue more carefully, because now 𝒪(1) is non-nef on the surfaces S_{j} = ℙ(E|C_{j}) so that 𝒪(1) might be non-nef on a general hyperplane section H. We will however show that this can be avoided by choosing carefully the linear system |H|. To be more precise we fix G ample on X and look for

H ∈ |𝒪(1) + π^{∗}(mG)|

with m ≫ 0, so that 𝒪(1) is nef on H ∩ S_{j} for all j. Given that 𝒪(1)|H and we can argue as in the previous case to obtain c_{2}(E) = 0. Of course for a general choice of H, all curves H ∩ S_{j} will be irreducible (but possibly singular since C_{j} might be singular). Now fix j and set C̃ = H ∩ S_{j}, a section over C = C_{j}. Let V ⊂ E_{C} be the maximal ample subsheaf (see [PS02]). Then we obtain a vector bundle sequence

0 → V → E_{C} → F → 0

and we may assume that F has rank 2, because otherwise 𝒪(1) is not nef only on one curve over C. Now C̃ induces an exact sequence

0 → 𝒪_{C}(−mG) → F → F′ → 0

and therefore 𝒪(1)|C̃ is nef iff c_{1}(F′) ≥ 0. This translates into c_{1}(F) + m(G · C) ≥ 0. Now let t_{0} be the nef value of E with respect to G, i.e. E(t_{0}G) is nef but not ample. Then F (t_{0}G) is nef, too, so that c_{1}(F) ≥ −2t_{0}(G · C). In total

c_{1}(F′) ≥ (m − 2t_{0})(G · C),

hence we choose m ≥ 2t_{0} and for this choice 𝒪(1)|H is nef.

As a corollary we obtain

7.7 Theorem. Let X be a projective K3-surface or a Calabi-Yau 3-fold. Then the tangent bundle T_{X} is not almost nef, and there exists a covering family (C_{t}) of (generically irreducible) curves such that T_{X}|C_{t} is not nef for general t.

In other words, if c_{1}(X) = 0 and T_{X} is almost nef, then a finite étale cover of X is abelian. One should compare this with Miyaoka’s theorem that T_{X}|C is nef for a smooth curve C cut out by hyperplane sections of sufficiently large degree. Note also that T_{X}|C being not nef is equivalent to say that T_{X}|C is not semi-stable. We expect that (7.7) holds in general for Calabi-Yau manifolds of any dimension. Proof. Assume that T_{X} is almost nef. Then by 7.6, T_{X} is numerically flat. In particular c_{2}(X) = 0 and hence X is an étale quotient of a torus. We will now improve (7.7) for K3-surfaces; namely if X is a projective K3-surface, then already 𝒪_{ℙ(T_{X})}(1) should be non-pseudo-effective. In other words, let A be a fixed ample divisor on X. Then for all positive integers m there exists a positive integer p such that

H^{0}(X, S^{p}((S^{m}T_{X}) ⊗ A)) = 0.

This has been verified in [DPS00] for the general quartic in ℙ_{3} and below for any K3-surface.

The next theorem is also proved in [Na04].

7.8 Theorem. Let X be a projective K3-surface and L = 𝒪_{ℙ(T_{X})}(1). Then L is not pseudo-effective.

Proof. Suppose that L is pseudo-effective and consider the divisorial Zariski decomposition ([Bou02b], cf. also 3.5 (iv))

L = N + Z

with N an effective ℝ-divisor and Z nef in codimension 1. Write N = aL + π^{∗}(N′) and Z = bL + π^{∗}(Z′). Let H be very ample on S. By restricting to a general curve C in |H_{H}| and observing that T_{X}|C is numerically flat, we see that L|π^{−1}(C) is nef (but not ample), hence

N′ · C = 0.

Thus N′ = 0 since H is arbitrary. If a > 0, then some mL would be effective, i.e. S^{m}T_{X} would have a section, which is known not to be the case. Hence a = 0 and L is nef in codimension 1 so that L can be negative only on finitely many curves. This contradicts 7.7.

§8 Nef reduction relative to a covering family of curves

In this section we construct reduction maps for pseudo-effective line bundle which have vanishing intersection numbers on large families of curves. This will be applied in the next section in connection with the abundance problem.

8.1 Notation. Let (C_{t})_{t∈T} be a covering family of (generically irreducible) curves (in particular T is irreducible and compact). Then (C_{t}) is said to be a connecting family if and only if two general points x, y can be joined by a chain of C_{t}’s. We also say that X is (C_{t})-connected. By T^{∗} we denote the Zariski-open (non-empty) set of those t for which C_{t} is irreducible.

§8 Nef reduction relative to a covering family of curves 23

Using Campana’s reduction theory [Ca81,94,04], we obtain immediately 8.2 Theorem. Let X be a projective manifold and L a pseudo-effective line bundle on X. Let (C_{t}) be a covering family with L · C_{t} = 0. Then there exists an almost holomorphic surjective meromorphic map f : X > Y with dim Y < dim X such that the general (compact) fiber of f is (C_{t})-connected. f is called the nef reduction of L relative to (C_{t}).

Recall that a meromorphic map is almost holomorphic, if there is an open nonempty set on which the map is holomorphic and proper. Of course, there might be other families (C′_{s}) with L · C′_{s} = 0 leading to different quotients.

8.3 Definition. Let L be a pseudo-effective line bundle on X. The minimal number which can be realised as dim Y with a nef reduction f : X > Y relative to L is denoted p(L). If there is no covering family (C_{t}) with L · C_{t} = 0, then we set p(L) = dim X.

8.4 Remark. The equality p(L) = 0 holds if and only if there exists a connecting family (C_{t}) such that L · C_{t} = 0. If moreover L is nef, then p(L) = 0 if and only if L ≡ 0 [Workshop].

For computing Kodaira dimensions sometimes the notion of a connecting family has to be strenghtened:

8.5 Definition. A covering family (C_{t})_{t∈T} is strongly connecting if any two sufficiently general points x and y can be joined by a chain of irreducible C_{t}, i.e. t ∈ T^{∗}, avoiding any given analytic set A of codimension at least 2.

8.6 Theorem. Let X be a projective manifold, L a pseudo-effective ℝ−divisor. Let (C_{t}) be a strongly connected family of curves. If L · C_{t} = 0, then ν(L) = 0. If L is Cartier, then L is numerically equivalent to a Cartier divisor L′ with κ(L′) = 0.

Equivalently:

8.7 Theorem. Let X be a projective manifold of any dimension n, (C_{t}) a strongly connecting family and L a ℝ-divisor which is nef in codimension 1. If L · C_{t} = 0, then L ≡ 0.

Proof (of Theorem 8.6 from Theorem 8.7). Consider the divisorial Zariski decomposition L = N + Z with N an ℝ−effective divisor and Z nef in codimension 1. Then N ·C_{t} ≥ 0 and Z ·C_{t} ≥ 0, so that L·C_{t} = 0 forces Z ·C_{t} = 0. Hence Z ≡ 0 by Theorem 8.7 so that ν(L) = 0. For the second statement we refer to (3.12).

Proof (of Theorem 8.7). (I) Let T be the parameter space of the family (C_{t}); we may assume dim T = n − 1. In a first step we reduce to the case that through every point x ∈ X there are only finitely many C_{t}. In fact, in the general situation take a birational map σ : X̃ → X from a projective manifold X̃, such that the induced family C̃_{t} has the finiteness property (this can be achieved e.g. by flattening the projection map from the graph of the family to X). Then let L̃ be the mobile part of σ^{∗}(L), i.e. the part in the divisorial Zariski decomposition which is nef in codimension 1. Since we know that the general C_{t} avoids codimension 2 sets, we obtain L̃ · C̃_{t} = 0. By our assumption that Theorem 8.7 already holds if the finiteness condition is verified, we conclude that L̃ ≡ 0. Hence L ≡ 0.

(II) From now on we may assume that only finitely many C_{t} pass through a fixed point of X. Therefore the following holds. If 𝒞 → T is the parameter space of the family (C_{t}) and if Y ⊂ X is any analytic set of codimension at least 2, then the set of points t ∈ T such that dim(C_{t} ∩ Y) = 1 has codimension at least 2 in T. Let B ⊂ X be a general curve. We follow the arguments in [workshop] and fix a general point x. We choose a family (B_{s})_{s∈S} joining x with the curve B by chains of irreducible curves C_{t}. Let p : S → X be the graph of the family with parameter space q : 𝒞 → T. Let S_{j} be the irreducible components of 𝒞. The codimension argument above says that we may assume that dim B_{s} ∩ A ≤ 0, where A is the non-nef locus of L. Thus p^{∗}(L)|S_{j} is nef in codimension 1 for all j, hence nef. But now the arguments of [workshop] work, and consequently L · B = 0. Taking e.g. B be complete intersection curves cut out by arbitrary hyperplane sections, we conclude that L ≡ 0.

Of course the question arises whether 8.6 holds for all connecting families. Unfortunately this is not true, as demonstrated by the following example.

8.8 Example. We produce a smooth projective threefold X, a line bundle L on X which is nef in codimension 1, in particular pseudo-effective, and a connecting family (C_{t}) such that h^{0}(L) = 2 and κ(L) = 1, but

L · C_{t} = 0.

We start with the ℙ_{2}−bundle

p : X_{1} := ℙ(𝒪 ⊕ 𝒪 ⊕ 𝒪(−1)) → ℙ_{1}.

Consider the section

B_{1} = ℙ(𝒪(−1)) ⊂ X_{1}

with normal bundle N_{B_{1}/X_{1}} = 𝒪(−1) ⊕ 𝒪(−1). Now we flop the curve B_{1} : we first consider the blow-up τ : X_{2} → X_{1} of X_{1} along B_{1}. The exceptional divisor E ≃ ℙ_{1}×ℙ_{1} has normal bundle 𝒪(−1, −1) and therefore X_{2} can be blown-down along the other projection to obtain σ : X_{2} → X. Let B = σ(E) and

L = (σ_{∗}τ^{∗}p^{∗}(𝒪(1)))^{∗∗}.

Then clearly h^{0}(L) = 2 and κ(L) = 1. Let F_{s} be the strict transform of p^{−1}(s), s ∈ ℙ_{1} in X so that L = 𝒪_{X}(F_{s}). Observe that F_{s} is ℙ_{2} blown up in one point, that B is the base locus of H^{0}(L), and that B ⊂ F_{s} is the (−1)−curve.

In order to establish the connecting family (C_{t}) with L · C_{t} = 0, consider a section D′ = ℙ(𝒪), disjoint from B_{1} and let D its strict transform in X. In every fiber F_{s} we can consider the family of “lines” meeting D′, i.e., containing the point p_{s} = D ∩F_{s}. A “line” is of course a curve whose images in ℙ_{2} is a line in the usual sense. The family of “lines” inside F_{s} has exactly one splitting element containing the (−1)−curve B. The other component is the strict transform of the line in ℙ_{2} joining p_{s} and the point to be blown up. Varying s, so we obtain a family (C_{t}), which is clearly connecting because of the curve B. Moreover it is clear that L · C_{t} = 0.

It is still possibly to say something for general connecting families. First observe 8.9 Lemma. Suppose κ(L) ≥ 0 and let D ∈ |mL| for some positive integer m. If (C_{t}) is a covering family such that L · C_{t} = 0, then

supp(D) ∩ C_{t} = ∅

for general t.

Proof. Just choose t general so that C_{t} ⊄ supp(D). Then D · C_{t} implies the claim.

8.10 Proposition. Let X be a projective manifold of dimension n and L a line bundle on X such that L · C_{t} = 0 for a connecting family (C_{t}). Then κ(L) ≤ n − 2.

Proof. Suppose the contrary and choose m such that H^{0}(mL) defines a map f : X > Y to a variety of dimension at least n − 1. By Lemma 8.9, f is holomorphic near the general C_{t}. This already reduces to dim Y = n − 1. Since we may assume that f has connected fibers, the C_{t} are just the fibers of f (at least for general t), so that f is almost holomorphic, and it is immediate that the family (C_{t}) cannot be connecting.

Although (8.6) fails in general for connecting families and arbitrary line bundles L, one might hope more in case L = K_{X}.

8.11 Proposition. Let X be a smooth projective threefold and (C_{t}) be a connecting family such that K_{X} · C_{t} = 0. Then κ(X) ≤ 0 unless we are in the following special situation.

(i) K_{X} is not nef, κ(X) = 1;

(ii) there is a sequence ϕ : X > X′ of Mori contractions and flips such that

K_{X′} · C′_{t} = 0

for the induced family (C′_{t}) (ϕ is holomorphic near the general C_{t});

(iii) on X′ we have a flip

X′ > X^{+},

the induced family (C_{t}^{+}) is no longer connecting, moreover K_{X^{+}} · C^{+} = 0;

(iv) the Iitaka fibration is a holomorphic map f : X^{+} → Z^{+} ≃ ℙ_{1} and f is a quotient for the family (C_{t}^{+}).

(v) the exceptional locus E^{+} ⊂ X^{+} of the flip X′ > X^{+} dominates Z^{+}.

Proof. Assume κ(X) ≥ 1, hence κ(X) = 1 (8.8). By [Workshop] K_{X} cannot be nef. Let ϕ : X → X_{1} be a Mori contraction, necessarily birational; let E be the exceptional divisor. Then E · C_{t} = 0; otherwise K_{X_{1}} · ϕ(C_{t}) < 0, and X_{1} would be uniruled. Then we get a connecting family (C_{t}^{1}) in X_{1} and proceed inductively. So we arrive at a flip

X′ > X^{+}.

Let E′ ⊂ X′ resp. E^{+} ⊂ X^{+} be the 1-dimensional “exceptional” sets. ¿From (8.9) and the fact that K_{X} is negative on all components of E′, we deduce that

C′_{t} ∩ E′ = ∅

for general t. Hence

K_{X^{+}} · C_{t}^{+} = 0.

If (C_{t}^{+}) is again connecting, then we proceed with X^{+}. After finitely many steps we must arrive at the case where (C_{t}^{+}) is no longer connecting, because by [Workshop] we cannot arrive at some X^{+} with K_{X^{+}} nef and the induced family being connecting. Therefore we consider the quotient

f : X^{+} > Z^{+}.

Then E^{+} must map onto Z^{+}; otherwise (C′_{t}) would not be connecting. Thus Z^{+} ≃ ℙ_{1} and the almost holomorphic map f must be holomorphic. By (8.8), the general fiber F of f has κ(F) = 0, hence f is the Iitaka fibration.

It is not completely clear whether the situation in (8.11) really occurs. It is also interesting to look at covering families (C_{t}) of ample curves. Here “ample” means that the dual of the conormal sheaf modulo torsion is ample (say on the normalization). Then we have the same result as in (8.6) which is prepared by

8.12 Lemma. Let X be a projective manifold, C ⊂ X an irreducible curve with normalization f : C̃ → C and ideal sheaf ℐ. Let L be a line bundle on X. Then there exists a positive number c such that for all t ≥ 0:

h^{0}(X, L^{t}) ≤ ∑_{k=0}^{ct} h^{0}(f^{∗}(S^{k}(ℐ/ℐ^{2}/tor) ⊗ L^{t})).

Proof. Easy adaptation of the proof of (2.1) in [PSS99].

8.13 Corollary. Let X be a projective manifold and C ⊂ X be an irreducible curve with normalisation f : C̃ → C such that f^{∗}(ℐ/ℐ^{2})^{∗} is ample. Let L be a line bundle with L · C_{t} = 0. Then κ(L) ≤ 0. In particular this holds for the general member of an ample covering family.

Proof. By (8.12) it suffices to show that

h^{0}(f^{∗}(S^{k}(ℐ/ℐ^{2}/tor) ⊗ L^{t})) = 0

for all k ≥ 1. This is however clear since by assumption f^{∗}(ℐ/ℐ^{2})^{∗} is an ample bundle.

8.14 Corollary. Let X be a smooth projective threefold with K_{X} pseudo-effective. If there is a ample covering family or a connecting family (C_{t}) such that K_{X} · C_{t} = 0, then κ(X) = 0.

Proof. By (8.13) we have κ(X) ≤ 0. Suppose that κ(X) = −∞. Then X is uniruled by Miyaoka’s theorem. Thus K_{X} is not pseudo-effective.

To complete the picture, we will construct a nef reduction for pseudo-effective line bundles, generalizing to a certain extent the result of [Workshop] for nef line bundle (however the result is weaker). A different type of reduction was constructed in [Ts00],[Ec02].

§9 Towards abundance 27

8.15 Theorem. Let L be a pseudo-effective line bundle on a projective manifold X. Then there exists an almost holomorphic meromorphic map f : X > Y such that

(i) general points on the general fiber of F can be connected by a chain of L-trivial irreducible curves.

(ii) if x ∈ X is general and C is an irreducible curve through x with dim f(C) > 0, then L · C > 0.

Proof. Start with a covering L-trivial family (C_{t}) and build the relative nef reduction h : X > Z (if the family does not exist, put f = id). Now take another covering L-trivial family (B_{s}) (if this does not exist, just stop) with relative nef reduction g. For general z ∈ Z let F_{z} be the set of all x ∈ X which can be joined with the fiber X_{z} by a chain of curves B_{s}. In other words, F_{z} is the closure of g^{−1}(g(X_{z})). Now the F_{z} define a covering family (of higher-dimensional subvarieties) which defines by Campana’s theorem a new reduction map. After finitely many steps we arrive at the map we are looking for.

Finally we give a criterion when a covering family is actually connecting:

8.16 Theorem. Let X be a projective manifold and (C_{t}) a covering family. Suppose that [C_{t}] is an interior point of the movable cone ℳ. Then (C_{t}) is connecting.

Proof. Let f : X > Z be the reduction of the family (C_{t}). If the family is not connecting, then dim Z > 0. Let π : X̃ → X be a modification such that the induced map f̃: X̃ → Z is holomorphic. Let A be very ample on Z and put L = π_{∗}(f̃^{∗}(A))^{∗∗}. Then L is an effective line bundle on X with L · C_{t} = 0 since L is trivial on the general fiber of f, this map being almost holomorphic. Hence [C_{t}] must be on the boundary of ℳ.

The converse of (8.16) is of course false: consider the family of lines l in ℙ_{2} and let X be the blow-up of some point in ℙ_{2}. Let (C_{t}) be the closure of the family of preimages of general lines. This is a connecting family, but if E is the exceptional divior, then E · C_{t} = 0. So (C_{t}) cannot be in the interior of ℳ.

§9 Towards abundance

In this section we prove that a smooth projective 4-fold X with K_{X} pseudo-effective and with the additional property that K_{X} · C_{t} = 0 for some good covering family of curves (C_{t}), has κ(X) ≥ 0. In other words we deal with problem (B2) from the introduction in dimension 4. In the remaining case that K_{X} is positive on all covering and non-connecting or strongly connecting families of curves one expects that K_{X} is big.

9.1 Proposition. Let X be a smooth projective 4-fold with K_{X} pseudo-effective. Suppose that there exists a dominant rational map f : X > Y to a projective manifold Y with κ(Y) ≥ 0 (and 0 < dim Y < 4). Then κ(X) ≥ 0.

Proof. We may assume f holomorphic with general fiber F. If κ(F) = −∞, then F would be uniruled, hence X would be uniruled. Hence κ(F) ≥ 0. Now C_{n,n−3}, C_{n,n−2} and C_{n,n−1} hold true, see e.g. [Mo87] for further references. This gives

κ(X) ≥ κ(F) + κ(Y) ≥ 0

and therefore our claim.

9.2 Corollary. Let X be a smooth projective 4-fold with K_{X} pseudo-effective. Let f : X > Y be a dominant rational map (0 < dim Y < 4) with Y not rationally connected. Then κ(X) ≥ 0.

Proof. If dim Y ≤ 2, this is immediate from (9.1). So let dim Y = 3. Since we may assume κ(Y) = −∞, the threefold Y is uniruled. Let h : Y > Z be the rational quotient; we may assume that h is holomorphic and Z smooth. Since Y is not rationally connected, we have dim Z ≥ 1 and moreover Z is not uniruled by Colliot-Thélène [CT86], see also Graber-Harris-Starr [GHS03]. Hence κ(Z) ≥ 0 and we conclude by (9.1).

9.3 Conclusion. In order to prove κ(X) ≥ 0 in case of a dominant rational map f : X_{4} > Y, we may assume that Y is a rational curve, a rational surface or a rationally connected 3-fold.

9.4 Proposition. Let X be a smooth projective 4-fold with K_{X} pseudo-effective. If p(K_{X}) = 1, then κ(X) ≥ 0.

Proof. By assumption we have a covering family (C_{t}) with K_{X} · C_{t} = 0, so that the relative nef reduction is a holomorphic map f : X → Y to a curve Y. By (9.3) we may assume Y = ℙ_{1}. We already saw that κ(F) ≥ 0. By (8.11)/(8.12) “mostly” we even have κ(F) = 0 unless we are in a very special situation.

We begin treating the case κ(F) = 0. Choose m such that h^{0}(mK_{F}) ≠ 0 for the general fiber F of f. Thus f_{∗}(mK_{X}) is a line bundle on Y, and we can write

mK_{X} = f^{∗}(A) + ∑ a_{i}F_{i} + E

where F_{i} are fiber components and E surjects onto Y with h^{0}(𝒪_{X}(E)) = 1. The divisor E comes from the fact that F is not necessarily minimal; actually E|F = mK_{F}. By enlarging m we may also assume that the support of ∑ a_{i}F_{i} does not contain any fiber and also that mK_{X} is Cartier. We consider the divisorial Zariski decomposition

mK_{X} = Ñ + Z̃

with Ñ being ℝ−effective and Z̃ nef in codimension 1. The clearly E ⊂ Ñ so that

(∗)  f^{∗}(A) + ∑ a_{i}F_{i} = Ñ′ + Z̃

with Ñ′ again ℝ−effective. Now let S ⊂ X be a surface cut out by 2 general hyperplane sections. Let L = mK_{X}|S and E′ = E|S. Denoting G_{i} = F_{i}|S, g = f|S and N = Ñ′|, Z = Z̃, we obtain

(∗∗)  L = g^{∗}(A) + ∑ a_{i}G_{i} + E′

and from from (∗)

(∗∗∗)  g^{∗}(A) + ∑ a_{i}G_{i} = N′ + Z.

Then N′ is ℝ−effective and not nef, and Z is nef. However N′ might a priori have a nef part; so we consider the divisorial Zariski decomposition N′ = N_{0} + Z_{0} and set Z_{1} = Z_{0} + Z. Let l be a general fiber of g. Then we conclude from (∗∗∗) that (N′ + Z) · l = (N_{0} + Z_{0}) · l = 0 and thus

N_{0} · l = Z_{0} · l = 0.

So N_{0} is contained in fibers of g and Z_{0} = f^{∗}(𝒪_{Y} (a)), [Workshop, 2.11]; moreover a ≥ 0. Comparing with (∗∗∗) and using the fact that ∑ a_{i}G_{i} does not contain the support of a full fiber, A must be nef. Hence (∗) gives κ(X) ≥ 0.

It remains to treat the case κ(F) = 1. Then we can use the relative Iitaka fibration of f and obtain a birational model X̂ of X such that the induced map f̂: X̂ → Y factors as f̂ = h ∘ g, where g|X̂_{y} is the Iitaka fibration of X̂_{y}, so that the general fiber F_{g} of g has κ(F_{g}) = 0. Now we conclude by (9.7).

9.5 Proposition. Let X be a smooth projective 4-fold with K_{X} pseudo-effective. If p(K_{X}) = 3, then κ(X) ≥ 0.

Proof. Here any reduction is a (possibly meromorphic) elliptic fibration. We choose a holomorphic birational model f : X⟶Y (with X and Y smooth), such that

(a) f is smooth over Y_{0} and Y \ Y_{0} is a divisor with simple normal crossings only;

(b) the j−function extends to a holomorphic map J : Y ⟶ℙ_{1}.

By the first property, f_{∗}(K_{X}) is locally free [Ko86], and we obtain the well-known formula of ℚ−divisors

(∗)  K_{X} = f^{∗}(K_{Y} + Δ) + E − G.

Here E is an effective divisor such that f_{∗}(𝒪_{X}(E)) = 𝒪_{Y} and G is an effective divisor such that dim f(G) ≤ 1. Moreover

Δ = Δ_{1} + Δ_{2}

with

Δ_{1} = ∑(1 − \frac{1}{m_{i}})F_{i} + ∑ a_{k}

and

Δ_{2} ∼ \frac{1}{12}J^{∗}(𝒪(1)).

Here F_{i} are the components over which we have multiple fibers and D_{k} are the other divisor components over which there singular fibers. The a_{k} ∈ \frac{1}{12}ℕ according to Kodaira’s list. Then by a general choice of the divisor Δ_{2}, the pair (Y, Δ_{1} + Δ_{2}) is klt. Now K_{Y} + Δ is pseudo-effective. In fact, by Theorem 2.2 it suffices to show that (K_{Y} + Δ) · B_{s} ≥ 0 for every covering family (B_{s}) of curves. But this is checked very easily by restricting to f^{−1}(C_{t}). Hence the log Minimal Model Program [Ko92] in dimension 3 implies that K_{Y} + Δ is effective. If G = 0, then we could conclude immediately κ(X) ≥ 0 by (*). In general we argue as follows. Consider the divisorial Zariski decomposition

K_{X} = f^{∗}(K_{Y} + Δ) + E − G = N + Z

with Z nef in codimension 1 and N the “exceptional” part. Using the log minimal model of (Y, Δ) we can write

K_{Y} + Δ = N′ + Z′,

where Z′ is the movable part and N′ the fixed part; this is automatically the divisorial Zariski decomposition of K_{Y} + Δ. Since f^{∗}(Z′) might not be nef in codimension 1 due to the large fibers of f, we consider the decomposition

f^{∗}(Z′) = N_{0} + Z_{0}

into the movable part Z_{0} and the fixed part N_{0} which is again the divisorial Zariski decomposition. Notice that N′, E, N_{0}, Z_{0} and G are ℚ−divisors. We obtain

f^{∗}(N′) + N_{0} + E + Z_{0} = N + G + Z.

Then f^{∗}(N′) + N_{0} + E is the exceptional part of the left hand side while N + G is the exceptional part of the right hand side. Thus Z = Z_{0}, so that Z is an effective ℚ−divisor. Then also N is an effective ℚ−divisor, and we conclude.

9.6 Proposition. Let X be a smooth projective 4-fold with K_{X} pseudo-effective. If p(K_{X}) = 2, then κ(X) ≥ 0.

Proof. Again we may assume that we have a holomorphic relative nef reduction f : X → Y with Y smooth and that f is smooth over Y_{0} and Y \ Y_{0} is a divisor with simple normal crossings only. By (8.9) κ(F) = 0 for the general fiber F of f. As in Proposition 9.5 we can write - possibly after birational transformation -

K_{X} = f^{∗}(K_{Y} + Δ) + E − G

with E effective, f_{∗}(𝒪_{X}(E)) = 𝒪_{Y}, G consisting of threedimensional fiber components and (Y, Δ) klt. This is proved in [Am04], proof of (5.1). Then we can proceed as in (9.5).

The arguments of (9.6) actually show

9.7 Proposition. Let X be a smooth projective 4-fold with K_{X} pseudo-effective. Suppose X has a holomorphic surjective map (with connected fibers) X → Y whose general fiber F has κ(F) = 0. Then κ(X) ≥ 0.

Proposition 9.4 could have been proved in the same way as (9.6); but maybe the ad hoc proof given above is instructive. The case p(K_{X}) is finally settled by Theorem 8.6 together with (3.11):

9.8 Theorem. Let X be a projective manifold such that K_{X} is pseudo-effective. If there is a strongly connecting family (C_{t}) such that K_{X} · C_{t} = 0, then κ(X) = 0.

9.9 Remark. Let X be a projective manifold of dimension n and (C_{t}) a connecting family which is not strongly connecting. Assume L·C_{t} = for some pseudo-effective line bundle L on X. One may wonder whether L is numerically equivalent to an effective divisor. Let us concentrate on the case n = 4 and L = K_{X}. Fix a general curve C_{t}, consider the possibly non-compact subspace of X filled up by the chains of irreducible C_{s} meeting C_{t} and take closure. If this subspace is reducible, pick some irreducible component, say Y. Since we assume the family not to be strongly connecting, Y ≠ X, so Y is either a surface or a divisor. It is not difficult to see (using (8.8)) that

ν(K_{X}|Y) = 0.

Let us assume that dim Y = 3. Varying the curve C_{t}, we obtain a family (Y_{s}) of divisors. We may assume that the parameter space S of the family is 1-dimensional. If it is not connecting, we obtain a holomorphic map f : X → S. Let F be the general fiber; then ν(K_{F}) = 0 and by abundance for threefolds, κ(F) = 0. Hence we conclude by (9.5). If the family is connecting, we have to pass to the graph: p : 𝒞 → X with projection q : 𝒞 → S. In that case it might happen that ν(p^{∗}(K_{X})) = 1 and it seems likely that the part of p^{∗}(K_{X}) which is nef in codimension 1 comes from S, hence is effective. Then we obtain κ(X) ≥ 0. Details are left to a future paper.

9.10 Definition. Let X be a projective manifold and (C_{t}) a covering family of curves. Then (C_{t}) is a good covering family if it is either non-connecting or strongly connecting.

Using this notation, we may summarize our results as follows.

9.11 Theorem. Let X be a smooth projective 4-fold (or a normal projective 4-fold with only canonical singularities). If K_{X} is pseudo-effective and if there is a good covering family (C_{t}) of curves such that K_{X} · C_{t} = 0, then κ(X) ≥ 0.

9.12 Remark. Let X be a projective manifold of dimension n.

(1) Suppose 1 ≤ κ(X) ≤ n − 1. If X has a good minimal model via contractions and flips, then X carries a covering non-connecting family of curves. If we require only that X has a good minimal model, then at least some blow-up of X carries a covering non-connecting family of curves.

(2) Suppose κ(X) = 0. If X has a good minimal model via contractions and flips, then X carries a strongly connecting family of curves. Again, if we require only that X has a good minimal model, then at least some blow-up of X carries a covering nonconnecting family of curves.

(3) This shows that the program of proving Problem (B) via (B1) and (B2) - in the version with good connecting families - is really meaningful, i.e., we do not try to prove too much.

The remaining task is essentially to consider 4-folds X with K_{X} ·C_{t} > 0 for all good covering families (C_{t}) (but a priori it might happen that K_{X} · C_{t} = 0 for a connecting, but not strongly connecting family (C_{t}). In that case one expects that X is of general type. It is easy to see that every proper subvariety S of X passing through a very general point of X is of general type, i.e. its desingularisation is of general type; see 9.13 below. But it is not at all clear whether K_{X}|S is big, which is of course still not enough to conclude.

9.13 Proposition. Let X be a smooth projective 4-fold with p(K_{X}) = 0. Then every proper subvariety S ⊂ X through a very general point of X is of general type.

Proof. Supposing the contrary, we find a covering family (S_{t}) of subvarieties such that the general S_{t}, hence every S_{t}, is not of general type. Consider the desingularised graph p : 𝒞 → X of this family; by passing to a subfamily we may assume p generically finite. Denoting q : 𝒞 → T the parametrising projection, the general fiber Ŝ_{t} is a smooth variety of dimension at most 3 and not of general type. We have an equation

K_{𝒞} = p^{∗}(K_{X}) + E

with an effective divisor E. Notice that E must dominate T, otherwise q induces (up to finite étale cover of X) an almost holomorphic map X > T which would give rise to a covering non-connecting family (C_{t}) with K_{X} · C_{t} = 0. By further blowing-up and using [KM92] we may assume that the general Ŝ_{t} dominates holomorphically a minimal model. We now easily find a covering family (C_{s}) of curves (sitting in q−fibers) such that K_{𝒞} · C_{s} = 0 but E · C_{s} > 0. Hence K_{X} · p_{∗}(C_{s}) < 0, a contradiction.

Using the Iitaka fibration we obtain

9.14 Proposition. Let X be a smooth projective 4-fold with p(K_{X}) = 0. Then κ(X) ≠ 1, 2, 3.

§10 Appendix: towards transcendental Morse inequalities

As already pointed out, for the general case of the conjecture 2.3 a transcendental version of the holomorphic Morse inequalities would be needed. The expected statements are contained in the following conjecture, which is also discussed in [Dem10].

10.1 Conjecture. Let X be a compact complex manifold, and n = dim X.

(i) Let α be a closed, (1, 1)-form on X. We denote by X(α, ≤ 1) the set of points x ∈ X such that α_{x} has at most one negative eigenvalue. If ∫_{X(α,≤1)} α^{n} > 0, the class {α} contains a Kähler current and

Vol(α) ≥ ∫_{X(α,≤1)} α^{n}.

(ii) Let {α} and {β} be nef cohomology classes of type (1, 1) on X satisfying the inequality α^{n} − nα^{n−1} · β > 0. Then {α − β} contains a Kähler current and

Vol(α − β) ≥ α^{n} − nα^{n−1} · β.

§10 Appendix: towards transcendental Morse inequalities 33

Remarks about the conjecture. If α = c_{1}(L) for some holomorphic line bundle L on X, then the inequality (i) was established in [Bou02a] as a consequence of the results of [Dem85]. In general, (ii) is a consequence of (i). In fact, if α and β are smooth positive definite (1, 1)-forms and

λ_{1} ≥ … ≥ λ_{n} > 0

are the eigenvalues of β with respect to α, then X(α − β, ≤ 1) = {x ∈ X; λ_{2}(x) < 1} and

1_{X(α−β,≤1)}(α − β)^{n} = 1_{X(α−β,≤1)}(1 − λ_{1}) … (1 − λ_{n}) ⩾ 1 − (λ_{1} + … + λ_{n})

everywhere on X. This is proved by an easy induction on n. An integration on X yields inequality (ii). In case α and β are just nef but not necessarily positive definite, we argue by considering (α+εω)−(β +εω) with a positive hermitian form ω and ε > 0 small.

The full force of the conjecture is not needed here. First of all, we need only the case when X is compact Kähler. Let us consider a big class {α}, and a sequence of Kähler currents T_{m} ∈ {α} with logarithmic poles, such that there exists a modification μ_{m} : X_{m} ↦ X, with the properties (10.2′) μ_{m}^{∗}T_{m} = β_{m} + [E_{m}] where β_{m} is a semi-positive (1, 1)-form, and E_{m} is an

effective ℚ-divisor on X_{m}.

(10.2′′) Vol({α}) = lim_{m↦∞} ∫_{X} β_{m}^{n}.

(see Definition 3.2).

A first trivial observation is that the following uniform upper bound for c_{1}(E_{m}) holds.

10.3 Lemma. Let ω be a Kähler metric on X, such that {ω − α} contains a smooth, positive representative. Then for each m ∈ ℤ_{+}, the (1, 1)-class μ_{m}^{∗}{ω} − c_{1}(E_{m}) on X_{m} is nef.

Proof. If γ is a smooth positive representative in {ω − α}, then μ_{m}^{⋆}γ + β_{m} is a smooth semi-positive representative of μ_{m}^{∗}{ω} − c_{1}(E_{m}).

A second remark is that in order to prove the duality statement 2.3 for projective manifolds, it is enough to establish the estimate

(∗)  Vol(ω − A) ≥ ∫_{X} ω^{n} − n ∫_{X} ω^{n−1} ∧ c_{1}(A)

where ω is a Kähler metric, and A is an ample line bundle on X. Indeed, if {α} is a big cohomology class, we use the above notations and we can write

β_{m} + tE_{m} = β_{m} + tμ_{m}^{∗}A − t(μ_{m}^{∗}A − E_{m})

where A is an ample line bundle on X such that c_{1}(A)−{α} contains a smooth, positive representative. The arguments of the proof of 4.1 will give the orthogonality estimate, provided that we are able to establish (∗).

In this direction, we can get only a weaker statement with a suboptimal constant c_{n}.

10.4 Theorem (analogue of Lemma 4.2). Let X be a projective manifold of dimension n. Then there exists a constant c_{n} depending only on dimension (actually one can take c_{n} = (n + 1)^{2}/4 ), such that the inequality

Vol(ω − A) ≥ ∫_{X} ω^{n} − c_{n} ∫_{X} ω^{n−1} ∧ c_{1}(A)

holds for every Kähler metric ω and every ample line bundle A on X.

Proof. Without loss of generality, we can assume that A is very ample (otherwise multiply ω and A by a large positive integer). Pick generic sections σ_{0}, σ_{1}, …, σ_{n} ∈ |A| so that one gets a finite map

F : X → ℙ_{ℂ}^{n}, x ↦ [σ_{0}(x) : σ_{1}(x) : …: σ_{n}(x)].

We let θ = F^{∗}ω_{FS} ∈ c_{1}(A) be the pull-back of the Fubini-Study metric on ℙ_{ℂ}^{n} (in particular θ ⩾ 0 everywhere on X), and put

ψ = log \frac{|σ_{0}|^{2}}{|σ_{0}|^{2} + |σ_{1}|^{2} + … + |σ_{n}|^{2}} .

We also use the standard notation d^{c} = \frac{i}{4π} (\overline{∂} − ∂) so that dd^{c} = \frac{i}{2π} ∂\overline{∂}. Then

dd^{c}ψ = [H] − θ

where H is the hyperplane section σ_{0} = 0 and [H] is the current of integration over H (for simplicity, we may further assume that H is smooth and reduced, although this is not required in what follows). The set U_{ε} = {ψ ⩽ 2 log ε} is an ε-tubular neighborhood of H. Take a convex increasing function χ : ℝ → ℝ such that χ(t) = t for t ⩾ 0 and χ(t) = constant on some interval ] − ∞, t_{0}]. We put ψ_{ε} = ψ − 2 log ε and

α_{ε} := dd^{c}χ(ψ_{ε}) + θ = (1 − χ′(ψ_{ε}))θ + χ′′(ψ_{ε})dψ_{ε} ∧ d^{c}ψ_{ε} ⩾ 0.

Thanks to our choice of χ, this is a smooth form with support in U_{ε}. In particular, we find

∫_{U_{ε}} α_{ε}^{n} = ∫_{U_{ε}} α_{ε} ∧ θ^{n−1} = ∫_{X} θ^{n} = c_{1}(A)^{n}.

It follows from these equalities that we have lim_{ε→0} α_{ε} = [H] in the weak topology of currents. Now, for each choice of positive parameters ε, δ, we consider the Monge-Ampère equation

(10.5)  (ω + i∂\overline{∂}φ_{ε})^{n} = (1 − δ)ω^{n} + δ \frac{∫_{X} ω^{n}}{c_{1}(A)^{n}} α_{ε}^{n}.

By the theorem of S.-T. Yau [Yau78], there exists a smooth solution φ_{ε}, unique up to normalization by an additive constant, such that ω_{ε} := ω + i∂\overline{∂}φ_{ε} > 0. Since ∫_{X} ω_{ε} ∧ ω^{n−1} = ∫_{X} ω^{n} remains bounded, we can extract a weak limit T out of the family ω_{ε}; then T is a closed positive current, and the arguments in [Bou02a] show that its absolutely continuous part satisfies

∫_{X} T_{ac}^{n} ≥ (1 − δ) ∫_{X} ω^{n}.

We are going to use the same ideas as in [DPa04], in order to estimate the singularity of the current T on the hypersurface H. For this, we estimate the integral ∫_{U_{ε}} ω_{ε} ∧θ^{n−1} on the tubular neighborhood U_{ε} of H. Let us denote by ρ_{1} ⩽ … ⩽ ρ_{n} the eigenvalues of ω_{ε} with respect to α_{ε}, computed on the open set U′_{ε} ⊂ U_{ε} where α_{ε} is positive definite. The Monge-Ampère equation (10.5) implies

ρ_{1}ρ_{2} … ρ_{n} ⩾ δ \frac{∫_{X} ω^{n}}{c_{1}(A)^{n}} .

On the other hand, we find ω_{ε} ⩾ ρ_{1}α_{ε} on U′_{ε}, hence

(10.6)  ∫_{U_{ε}} ω_{ε} ∧ θ^{n−1} ⩾ ∫_{U′_{ε}} ρ_{1}α_{ε} ∧ θ^{n−1} ⩾ δ \frac{∫_{X} ω^{n}}{c_{1}(A)^{n}} ∫_{U′_{ε}} \frac{1}{ρ_{2} … ρ_{n}} α_{ε} ∧ θ^{n−1}.

In order to estimate the last integral in the right hand side, we apply the Cauchy-Schwarz inequality to get

(10.7)  ( ∫_{U′_{ε}}(α_{ε}^{n})^{1/2}(α_{ε} ∧ θ^{n−1})^{1/2})^{2} ⩽ ∫_{U′_{ε}} ρ_{2} … ρ_{n}α_{ε}^{n} ∫_{U′_{ε}} \frac{1}{ρ_{2} … ρ_{n}} α_{ε} ∧ θ^{n−1}.

By definition of the eigenvalues ρ_{j}, we have

(10.8)  ∫_{U′_{ε}} ρ_{2} … ρ_{n}α_{ε}^{n} ⩽ n ∫_{X} ω_{ε}^{n−1} ∧ α_{ε} = n ∫_{X} ω^{n−1} ∧ c_{1}(A).

On the other hand, an explicit calculation shows that

\begin{matrix}α_{ε}^{n} ⩾ n(1 − χ′(ψ_{ε}))^{n−1}χ′′(ψ_{ε}) dψ_{ε} ∧ d^{c}ψ_{ε} ∧ θ^{n−1}, \\ α_{ε} ∧ θ^{n−1} ⩾ χ′′(ψ_{ε}) dψ_{ε} ∧ d^{c}ψ_{ε} ∧ θ^{n−1}, \end{matrix}

hence

∫_{U′_{ε}}(α_{ε}^{n})^{1/2}(α_{ε} ∧ θ^{n−1})^{1/2} ⩾ n^{1/2} ∫_{X}(1 − χ′(ψ_{ε}))^{(n−1)/2}χ′′(ψ_{ε}) dψ_{ε} ∧ d^{c}ψ_{ε} ∧ θ^{n−1}

(we can integrate on X since the integrand is zero anyway outside U′_{ε}). Now, we have

\frac{n + 1}{2}(1 − χ′(ψ_{ε}))^{(n−1)/2}χ′′(ψ_{ε}) dψ_{ε} ∧ d^{c}ψ_{ε}
= −d((1 − χ′(ψ_{ε}))^{(n+1)/2}d^{c}ψ_{ε}) + (1 − χ′(ψ_{ε}))^{(n+1)/2}dd^{c}ψ_{ε}
= −d((1 − χ′(ψ_{ε}))^{(n+1)/2}d^{c}ψ_{ε}) + [H] − (1 − χ′(ψ_{ε}))^{(n+1)/2}θ

and from this we infer

\frac{n + 1}{2} ∫_{X}(1 − χ′(ψ_{ε}))^{(n−1)/2}χ′′(ψ_{ε}) dψ_{ε} ∧ d^{c}ψ_{ε} ∧ θ^{n−1}
= ∫_{X}[H] ∧ θ^{n−1} − ∫_{X}(1 − χ′(ψ_{ε}))^{(n+1)/2}θ^{n}
→ c_{1}(A)^{n} as ε → 0.

We thus obtain

(10.9)  ∫_{U′_{ε}}(α_{ε}^{n})^{1/2}(α_{ε} ∧ θ^{n−1})^{1/2} ⩾ \frac{2\sqrt{n}}{n + 1}c_{1}(A)^{n} − o(1)  as ε → 0.

The reader will notice, and this looks at first a bit surprising, that the final lower bound does not depend at all on the choice of χ. This seems to indicate that our estimates are essentially optimal and will be hard to improve. Putting together (10.7), (10.8) and (10.9) we find the lower bound

(10.10)  ∫_{U′_{ε}} \frac{1}{ρ_{2} … ρ_{n}} α_{ε} ∧ θ^{n−1} ⩾ \frac{4δ}{(n + 1)^{2}} \frac{(c_{1}(A)^{n})^{2}}{∫_{X} ω^{n−1} ∧ c_{1}(A)} − o(1).

Finally, (10.6) and (10.10) yield

∫_{U_{ε}} ω_{ε} ∧ θ^{n−1} ⩾ \frac{4δ}{(n + 1)^{2}} \frac{∫_{X} ω^{n}}{∫_{X} ω^{n−1} ∧ c_{1}(A)}c_{1}(A)^{n} − o(1).

As ⋂ U_{ε} = H, the standard support theorems for currents imply that the weak limit T = lim ω_{ε} carries a divisorial component c[H] with

∫_{X} c[H] ∧ θ^{n−1} ⩾ \frac{4δ}{(n + 1)^{2}} \frac{∫_{X} ω^{n}}{∫_{X} ω^{n−1} ∧ c_{1}(A)}c_{1}(A)^{n}.

Therefore, as [H] ≡ θ ∈ c_{1}(A), we infer

c ⩾ \frac{4δ}{(n + 1)^{2}} \frac{∫_{X} ω^{n}}{∫_{X} ω^{n−1} ∧ c_{1}(A)}.

The difference T −c[H] is still a positive current and has the same absolutely continuous part as T. Hence

Vol(T − c[H]) ⩾ ∫_{X} T_{ac}^{n} ⩾ (1 − δ) ∫_{X} ω^{n}.

The specific choice

δ = \frac{(n + 1)^{2}}{4} \frac{∫_{X} ω^{n−1} ∧ c_{1}(A)}{∫_{X} ω^{n}}

gives c ⩾ 1, hence

Vol(T − [H]) ⩾ ∫_{X} ω^{n} − \frac{(n + 1)^{2}}{4} ∫_{X} ω^{n−1} ∧ c_{1}(A).

§11 Concluding remarks 37

Theorem 10.4 follows from this estimate.

10.11 Remark. By using similar methods, we could also obtain an estimate for the volume of the difference of two Kähler classes on a general compact Kähler manifold, by using the technique of concentrating the mass on the diagonal of X × X (see [DPa04]). However, the constant c implied by this technique also depends on the curvature of the tangent bundle of X.

We show below that the answer to conjecture 10.1 is positive at least when X is a compact hyperkähler manifold ( = compact irreducible holomorphic symplectic manifold). The same proof would work for a compact Kähler manifold which is a limit by deformation of projective manifolds with Picard number ρ = h^{1,1}.

10.12 Theorem. Let X be a compact hyperkähler manifold, and let α be a closed, (1, 1)-form on X. Then we have

Vol(α) ≥ ∫_{X(α,≤1)} α^{n}.

Proof. We follow closely the approach of D. Huybrechts in [Huy98]. Consider 𝒳 ↦ Def(X) the universal deformation of X, such that 𝒳_{0} = X. If β ∈ H^{2}(X, ℝ) is a real cohomology class, then we denote by S_{β} the set of points t ∈ Def(X) such that the restriction β_{|𝒳_{t}} is of (1, 1)-type.

Next, we take a sequence of rational classes {α_{k}} ∈ H^{2}(X, ℚ), such that α_{k} → α on 𝒳 as k ↦ ∞. As {α_{k}} → {α}, the hypersurface S_{α_{k}} converge to S_{α}; in particular, we can take t_{k} ∈ S_{α_{k}} such that t_{k} → 0. In this way, the rational (1, 1)-forms α_{k|𝒳_{t_{k}}} will converge to our form α on X.

We have

Vol(α) ≥ lim sup_{k↦∞} Vol(α_{k|𝒳_{t_{k}}} ) ≥
≥ lim sup_{k↦∞} ∫_{𝒳_{t_{k}} (α_{t_{k}} ,≤1)} α_{t_{k}}^{n}
= ∫_{X(α,≤1)} α^{n}

where the first inequality is a consequence of the semi-continuity of the volume obtained in [Bou02b], and the second one is a consequence of the convergence statement above.

10.13 Corollary. If X be a compact hyperkähler manifold, or more generally, a limit by deformation of projective manifolds with Picard number ρ = h^{1,1}, then the cones ℰ and ℳ are dual.

§11 Concluding remarks

We would like to conclude with some new developments and applications of the main theorem. For a general overview article we refer to [Dem07] and also to the Bourbaki talk by Debarre [Deb06].

(1) In the paper [BFJ09], Boucksom, Favre and Jonsson consider the volume function on the big cone of a projective manifold X of dimension n. They showed (Theorem A) that the volume function vol is differentible of class 𝒞^{1} and computed the derivative. Namely, given a big class α and γ ∈ N^{1}(X), then, using the notations of section 3,

(\frac{d}{dt})_{t=0}vol(α + tγ) = n⟨α^{n−1}⟩ · γ.

In particular, setting γ = α, one obtains

⟨α^{n}⟩ = ⟨α^{n−1}⟩ · α,

which implies Corollary 4.5.

(2) Hacon-McKernan [HM08] showed, solving a conjecture of Shokurov, that given a divisorial log terminal pair (X, Δ) and a birational map f : Y → X, then all fibers f^{−1}(x) are rationally chain connected. A crucial point in the proof is to check that the images of rational maps from fibers of f are uniruled. This is done using the uniruledness criterion Theorem 2.6.

(3) As indicated in section 2, given a surjective map (Ω_{X}^{1})_{X}^{⊗m} → Q → 0,

then by [CP09], det Q is pseudo-effective unless X is uniruled. This generalises a theorem of Miyaoka and again Theorem 2.6 is indispensable. At this point we add a remark concerning general complete intersection curves. For many purposes, e.g., in connection with stability, when one wants to use the theorem of Mehta-Ramanathan, those curve play a central role. The movable cone was defined as the closed cone of curves generated by complete intersection curves on birational models of the variety X which dominate X holomorphically. One might ask whether the movable cone actually agrees with the complete intersection on X, i.e., with the cone generated by curves cut out by sufficiently high multiples of ample divisors. Very unfortunately however, these curves do not coincide in general. Some example was already exhibited in [DPS96], Example 4.8 (and then forgotten, so that we worked also an example of a ℙ_{1}−bundle over ℙ_{2}).

(4) As already mentioned Theorem 2.6 in the Kähler case is wide open. However in dimension 3, Brunella [Br06] was able to establish the uniruledness of a (non-algebraic) Kähler manifold X with K_{X} not nef. In fact, if X is a non-algebraic Kähler threefold, then X carries a holomorphic 2−form which can be seen as a foliation F = K_{X} ⊂ T_{X}, possible with singularities in codimension 1. Now, K_{X} being not pseudo-effective, then the canonical bundle of the foliation (by curves) F is not pseudo-effective. In this situation Brunella shows that F is a foliation by rational curves (actually in any dimension).

(5) There is also a version of Theorem 2.6 for the cotangent bundle. In fact, fix an ample line bundle A and suppose that

H^{0}(X, ((Ω_{X}^{1})^{⊗m} ⊗ A)^{⊗N}) = 0

for m, N sufficiently large, then X is rationally connected. A conjecture of Mumford says that one can actually omit the ample line bundle A. See [Pe06] for details; the proof uses once more Theorem 2.6 to get uniruledness.

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Authors:

Sébastien Boucksom, Université de Paris VII, boucksom@math.jussieu.fr Jean-Pierre Demailly, Université de Grenoble I, demailly@fourier.ujf-grenoble.fr Mihai Păun, Université de Strasbourg I, paun@math.u-strasbg.fr Thomas Peternell, Universität Bayreuth, Thomas.Peternell@uni-bayreuth.de