Sébastien Boucksom

Jean-Pierre Demailly

Mihai Paun

Thomas Peternell

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

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 pseudoeffective 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 covering family, has non negative Kodaira dimension.

§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 [START_REF] Mori | Flip theorem and existence of minimal models of 3-folds[END_REF] and for non-algebraic Kähler threefolds by [START_REF] Peternell | Towards a Mori theory on compact Kähler 3-folds, III[END_REF], 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 two more tractable parts :

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.

In this paper we give a positive answer to (A) for projective manifolds of any dimension, and a partial answer to (B) for 4-folds. 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 [START_REF] Miyaoka | A numerical criteria for uniruledness[END_REF] (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).

Concerning (B) we prove the following weaker statement. 0.4 Theorem. Let X be a smooth projective 4-fold. Assume that K X is pseudoeffective and there is a 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 ). In order to obtain the full answer to Problem (B) in dimension 4, we would still need to prove that K X is effective if K X is positive on all 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 1,1 R (X) of classes {ω} of Kähler forms (this is an open convex cone).

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

(iii) The Neron-Severi space is defined by NS R (X) := H 1,1 R (X) ∩ H 2 (X, Z)/tors ⊗ Z R.

(iv) We set

K NS = K ∩ N S R (X), E NS = E ∩ N S 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 ∂, one can show the following expected relations between the algebraic and transcendental cones (see [START_REF] Demailly | Singular hermitian metrics on positive line bundles[END_REF], [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF]).

1.2 Proposition. In a projective manifold X, E NS is the closure of the convex cone generated by effective divisors, and K NS is the closure of the cone generated by nef R-divisors.

By extension, we will say that 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 n-1,n-1 R (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 n-1,n-1 R (X) of movable classes to be the closure of the convex cone generated by classes of currents of the form

µ ⋆ ( ω 1 ∧ . . . ∧ ω n-1 )
where µ : X → X is an arbitrary modification (one could just restrict oneself to compositions of blow-ups with smooth centers), and the ω j are Kähler forms on X. 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 n-1,n-1 R (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 NE(X) ⊂ N NS .

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

C = µ ⋆ ( A 1 ∩ . . . ∩ A n-1 )
for suitable very ample divisors A j on X, where µ : X → X is a modification. We let SME(X) to be the convex cone generated by all strongly movable (effective) curves. Clearly 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 1,1 R (X) × H n-1,n-1 R (X) -→ R, (α, β) -→ X α ∧ β.
Then the duality pairing takes nonnegative values (i) for all pairs (α, β) ∈ 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 ⋆ ( ω 1 ∧. . .∧ ω n-1 ) for some modification µ : X → X, where α = {T } is the class of a positive (1, 1)-current on X and ω j are Kähler forms on X.

Then X α ∧ β = X T ∧ µ ⋆ ( ω 1 ∧ . . . ∧ ω n-1 ) = X µ * T ∧ ω 1 ∧ . . . ∧ ω 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∂∂ϕ locally for some plurisubharmonic function in X, we can set µ ⋆ T = i∂∂(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∂∂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∂∂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∂∂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∂∂ψ k -Cω (such a sequence exists by [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF]). Then, writing α = i∂∂v j for some smooth potential v j on U j , we have T = i∂∂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 ∨∨ = 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) NE(X) = N NS . (ii) SME(X) = ME(X) = M NS .
Proof. (i) It is a standard result of algebraic geometry (see e.g. [Ha70]), that the cone of effective cone NE(X) is dual to the cone K NS of nef divisors, hence

N NS ⊃ 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, [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF] gives 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 [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF] conversely shows that the dual of (N ′ ) ∨ is equal to K, so we must have

2.1 Theorem (Demailly-Paun, 2001). If X is Kähler, then the cones K ⊂ H 1,1 R (X) and N ⊂ H n-1,n-1 R (X)
K ∨ = 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 X, under a modification µ : X → 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 [START_REF] Demailly | Holomorphic line bundles with partially vanishing cohomology[END_REF], 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 member of a covering family (C t ), then N C is nef. Now [START_REF] Demailly | Holomorphic line bundles with partially vanishing cohomology[END_REF] 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 SME(X) of strongly movable curves ;

(iii) the closed cone ME(X) of movable curves ;

(iv) the closed cone 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 ) ∨ = 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 [START_REF] Peternell | Kodaira dimension of subvarieties[END_REF]. 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 [START_REF] Miyaoka | A numerical criteria for uniruledness[END_REF]).

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 1 X → 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 [START_REF] Miyaoka | The Chern classes and Kodaira dimension of a minimal variety[END_REF], [START_REF] Shepherd-Barron | Miyaoka's theorems on the generic semi-negativity of T X[END_REF].

Problems.

(1) Does 2.7 generalize to subsheaves F ⊂ T ⊗m X ? (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 (special case of the "abundance conjecture"). If K X is pseudoeffective, then κ(X) 0.

In the last section we will prove this in dimension 4 under the additional assumption that there is a covering family of curves (C t ) such that K X • C t = 0. §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. [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF]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 1 m Z, i.e. there are modifications µ m : X m → X such that

Theorem (Demailly

µ ⋆ m T m = [E m ] + β m
where E m is an effective Q-divisor on X m with coefficients in 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∂∂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) = 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 ).
We should notice that by approximating T -1 m ω instead of T , we can replace β m by β m + 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 decompo- sition 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 : X → X.

By Fujita [START_REF] Fujita | Approximating Zariski decomposition of big line bundles[END_REF] and Demailly-Ein-Lazarsfeld [START_REF] Demailly | A subadditivity property of multiplier ideals[END_REF], if L is a big line bundle, we have

Vol(c 1 (L)) = lim m→+∞ D n m = lim m→+∞ 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 ǫ > 0.

Then there exists a modification µ : X ǫ → 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∂∂ϕ 1 and T 2 = α+i∂∂ϕ 2 are two Kähler currents with logarithmic poles in the class of α, then (3.4)

T = α + i∂∂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∂∂ϕ, ϕ = 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. (Boucksom [Bou02b]). Let X be a compact Kähler manifold. We denote here by H k,k 0 (X) the cone of cohomology classes of type (k, k) which have non-negative intersection with all closed semi-positive smooth forms of bidegree (nk, nk).

Theorem

(i) For each integer k = 1, 2, . . . , n, there exists a canonical "movable intersection product"

E × • • • × E → H k,k 0 (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 ∈ K are nef classes.

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

α 1 • α 2 • • • α n ( α n 1 ) 1/n . . . ( α n n ) 1/n (with α n j = 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 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 [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF], 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 : X → 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 : X 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 X 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 [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF], 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 1,1 R (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 [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF].

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 N S(X) ∩ E ⊂ H 2 (X, Z), otherwise two such divisors D ≡ D ′ would yield a blow-up µ : X → 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 [START_REF] Demailly | Champs magnétiques et inégalités de Morse pour la d ′′ -cohomologie[END_REF] (see also [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF]).

In the remaining cases, the most tractable situation is probably 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).
Especially, if we assume additionally that q(X) = h 0,1 (X) is zero, then mK X is linearly equivalent to an integral divisor for some multiple m, and it follows immediately that κ(X) = 0. The case of a general projective (or compact Kähler) manifold with ν(X) = 0 and positive irregularity q(X) > 0 would be interesting to understand.

The preceeding remarks at least give a proof up to dimension 4 :

3.11 Proposition. Let X be a smooth projective n-fold with n ≤ 4. If ν(X) = 0, then κ(X) = 0.

Proof. The proof is given in (9.1) in a slightly more general situation.

We will come back to abundance on 4-folds in sect. 9. §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 n-1 m • E m ) 2 20 (Cω) n Vol(α) -D n m
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
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 n m + nt D n-1 m • µ ⋆ m (α + Cω) + n k=2 t k n k D n-k m • µ ⋆ m (α + Cω) k -nt D n-1 m • (D m + Cµ ⋆ m ω) -nt 2 n-1 k=1 t k-1 n -1 k D n-1-k m • 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 n m + ntD n-1 m • E m -nt 2 n-1 k=1 2 k+1 t k-1 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 n m + nt D n-1 m • E m -5n 2 t 2 (Cω) n . Now, the choice t = 1 10n (D n-1 m • E m )((Cω) n ) -1
gives by substituting 1 20

(D n-1 m • E m ) 2 (Cω) n Vol(E m + D m ) -D n m Vol(α) -D n m
(and we have indeed t 1 10n by Lemma 4.4), whence Theorem 4.1. Of course, the constant 20 is certainly not optimal.

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 n-1 m ) • (µ m ) ⋆ E m = D n-1 m • (µ m ) ⋆ (µ m ) ⋆ E m D n-1 m • E m C(Vol(α) -D n m ) 1/2 . Since α n-1 = lim(µ m ) ⋆ (D n-1 m ), N (α) = lim(µ m ) ⋆ E m and lim D n m = 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 αεω ∈ (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 n-1 δ

). By the Hovanskii-Teissier concavity inequality

ω • Γ (ω n ) 1/n (D n δ ) (n-1)/n .
On the other hand

α • Γ = α • (µ δ ) ⋆ (D n-1 δ ) = µ ⋆ δ α • D n-1 δ µ ⋆ δ (α + δω) • D n-1 δ = (E δ + D δ ) • D n-1 δ = D n δ + D n-1 δ • E δ .
By the orthogonality estimate, we find

α • Γ ω • Γ D n δ + 20(Cω) n (Vol(α + δω) -D n δ ) 1/2 (ω n ) 1/n (D n δ ) (n-1)/n C ′ (D n δ ) 1/n + C ′′ (Vol(α + δω) -D n δ ) 1/2 (D n δ ) (n-1)/n
. However, since α ∈ ∂E NS , the class α cannot be big so

lim δ→0 D n δ = Vol(α) = 0.
We can also take D δ to approximate Vol(α+δω) in such a way that (Vol(α+δω)-D n δ ) 1/2 tends to 0 much faster than D n δ . Notice that D n δ δ n ω n , so in fact it is enough to take

Vol(α + δω) -D n δ δ 2n .
This is the desired contradiction by (5.1).

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 : X → X and a Kähler metric ω on X such that α • µ ⋆ ( ω) 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 [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF], 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 : X → 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 [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF] and [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF] 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 [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF], 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 C be the strict transform of C in X, so that C = µ ⋆ C. We then find (α + δω) • C = ([E] + β) • C 0 since β 0 and C ⊂ |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 + C + 2 E j = µ ⋆ (F + C) + E j
where C 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 C ∪ E j , although

L • C = (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

In this section we consider pseudo-effective and almost nef vector bundles as introduced in [START_REF] Demailly | Pseudo-effective line bundles on compact Kähler manifolds[END_REF]. 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 [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF]. 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)) -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))

-δ ω and π(Sing(h δ )) ⊂ Z δ X (for some Kähler metric ω on P(E) and arbitrary small δ > 0). We can actually take ω = Θ(A) and ω = 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 1/(1+δε 0 ) δ h δε 0 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) 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 ∂-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. [START_REF] Demailly | Compact complex manifolds with numerically effective tangent bundles[END_REF]) 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 C = H ∩ S j , a section over C = C j . Let V ⊂ E C be the maximal ample subsheaf (see [START_REF] Peternell | Ample vector bundles and branched coverings II[END_REF]). 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 C induces an exact sequence

0 → O C (-mG) → F → F ′ → 0 and therefore O(1)| C 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 [START_REF] Demailly | Pseudo-effective line bundles on compact Kähler manifolds[END_REF] for the general quartic in P 3 and below for any K3-surface.

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 Partial nef reduction

In this section we construct reduction maps for pseudo-effective line bundle which are zero 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 x, y can be joined by a chain of C t . We also say that X is (C t )-connected. Using Campana's reduction theory [START_REF] Campana | Coréduction algébrique d'un espace analytique faiblement kählérien compact[END_REF]94], 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 such that the general (compact) fiber of f is (C t )-connected. f is called the partial nef reduction of L with respect to (C t ).

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 partial nef reduction f : X > Y with respect 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. p(L) = 0 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].
8.5 Proposition. Let X be a projective manifold, L a pseudo-effective line bundle and

(C t ) a connecting family. If L • C t = 0, then κ(L) ≤ 0.
Proof. Supposing the contrary we may assume that h 0 (L) ≥ 2. Let C = C t be a general member of our family. Then we find a non-zero s ∈ H 0 (L) such that s|C = 0. If A ⊂ X is any algebraic set, we let G(A) be the union of all y which can be joined with A by a single C t . Now the general C t through a general point must be irreducible, hence G(C) is generically filled up by irreducible C t , and we conclude that s|G(C) = 0. Define G k (C) = G(G k-1 (C)). Then by induction we obtain s|G k (C) = 0 for all k. Since (C t ) is connecting, we have G ∞ = X, hence s = 0. Therefore h 0 (L) ≤ 1.

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 (8.5) which is prepared by 8.6 Lemma. Let X be a projective manifold, C ⊂ X an irreducible curve with normalization f : C → 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 ) ≤ ct k=0 h 0 (f * (S k (I/I 2 /tor) ⊗ L t )).
Proof. Easy adaptation of the proof of (2.1) in [START_REF] Peternell | Kodaira dimension of subvarieties[END_REF]. 8.7 Corollary. Let X be a projective manifold and C ⊂ X be an irreducible curve with normalisation f : C → C such that f * (I/I 2 ) * is ample. Let L 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 (8.5) 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.

8.8 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 (8.5) resp. (8.7) we have κ(X) ≤ 0. Suppose that κ(X) = -∞. Then X is uniruled by Miyaoka's theorem. Thus K X is not pseudo-effective. Although we will not need it later, we will construct a nef reduction for pseudoeffective 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 [START_REF] Tsuji | Numerically trivial fibrations[END_REF], [START_REF] Eckl | Tsuji's numerical trivial fibrations[END_REF].

8.9 Theorem. Let L be a pseudo-effective line bundle on a projective manifold X. Then there exists an almost holomorphic meromorphic map f : X > 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. The argument is rather standard: start with a covering L-trivial family (C t ) and build the nef partial quotient 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 partial 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 show that covering families which are interior points in the movable cone are connecting: 8.10 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 : X → X be a modification such that the induced map f : X → Z is holomorphic. Let A be very ample on Z and put L = π * ( f * (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 (8.10) 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 is 0 on some covering family of curves, has κ(X) ≥ 0. In the remaining case that K X is positive on all covering 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. [START_REF] Mori | Classification of higherdimensional varieties[END_REF] for further references. This gives κ(X) ≥ κ(F ) + κ(Y ) ≥ 0. 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, dim Z ≥ 1. Then q(Z) ≥ 1, otherwise Z would be rational and hence Y rationally connected by Colliot-Thélène [START_REF] Colliot-Thélène | Arithmétique des variétés rationnelles et problèmes birationnels[END_REF], see also Graber-Harris-Starr [START_REF] Graber | Families of rationally connected varieties[END_REF], and we conclude by (9.1).

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, whose partial 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, however by (8.5) we even have 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 . We will get rid of E by the following construction.

Let Y 0 ⊂ Y be the largest open set such that f is smooth over Y 0 ; let X 0 = f -1 (Y 0 )
. By [START_REF] Kollár | Classification of three-dimensional flips[END_REF] we have a birational model f ′ 0 : X ′ 0 > Y 0 via a sequence of relative contractions and relative flips such that the fibers F ′ of f ′ 0 are minimal, hence nK F ′ = O F ′ for suitable n. Now compactify. Thus we may assume that the general fiber of f is minimal, paying the price that X might have terminal singularities. However these singularities don't play any role since below we will argue on a general surface in X, which automatically does not meet the singular locus of X, this set being of dimension at most 1.

In particular we have E = 0 in ( * ). 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. Now let S ⊂ X be a surface cut out by 2 general hyperplane sections. Let L = mK X |S.

Denoting G i = F i |S and g = f |S, we obtain from ( * ) ( * * ) L = g * (A) + a i G i .
On the other hand, we consider the divisorial Zariski decomposition L ≡ N + Z constructed in [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF]; see (3.5). Here N is an effective R-divisor covering the non-nef locus of L and Z is an R-divisor which is nef in codimension 1. Let l be a general fiber of g. Then L • l = 0 and thus

N • l = Z • l = 0.
So N is contained in fibers of g and Z = f * (O Y (a)), [START_REF] Bauer | A reduction map for nef line bundles; Complex Geometry[END_REF]2.11]; moreover a ≥ 0.

Comparing with ( * * ) we get Z = f * (A) and thus A is nef. Hence ( * ) gives κ(X) ≥ 0.

9.5 Remark. Proposition 9.4 also holds in dimension 5. In fact, in view of (C n,1 ), only two things to be observed. The first is that κ(F ) ≥ 0. But this follows from (9.9) below.

The second is that in case of 4-dimensional fibers we cannot apply [START_REF] Kollár | Classification of three-dimensional flips[END_REF]. However it is not really necessary to use [START_REF] Kollár | Classification of three-dimensional flips[END_REF]. We can also argue as follows. Let E ′ = E|S.

Then necessarily E ′ ⊂ N so that g * (A) + a i G i = N ′ + Z is pseudo-effective, too, and we conclude as in (9.4).

The same remark also applies to the next proposition (9.6).

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. By (9.3) we may assume that we have a holomorphic partial nef reduction f : X → Y = P 2 . Again κ(F ) = 0 for the general fiber F . Then for a suitable large m we have a decomposition

( * ) mK X = f * (A) + a i F i + E 1 -E 2 + D
where the support of a i F i does not contain the support of any divisor of the form f -1 (C), but f (F i ) is 1-dimensional for all i, where E i are effective with dim f (E i ) = 0, and where D is effective, projecting onto Y with D|F = mK F for the general fiber. The divisor E 2 arises from the fact that f * (kK X ) might not be locally free, but only torsion free.

Writing A = O Y (a), we are going to prove that a ≥ 0. Let l ⊂ Y be a general line and

X l = f -1 (l). Let g = f |X l , A ′ = A|l, F ′ i = F i • X l , D ′ = D • X l and L = mK X |X l . Then ( * ) gives L = g * (A ′ ) + a i F ′ i + D ′ .
Passing to a suitable model for X l → l as in the proof of (9.4), we may assume that D ′ = 0. Now consider Boucksom's divisorial Zariski decomposition L = N + Z as in the proof of (9.4). Then we conclude as before that Z = g * (A ′ ) and that N = a i F ′ i . Hence A ′ is nef and thus a ≥ 0 (this can also be verified easily without using the divisorial Zariski decomposition).

Going back to ( * ), the only remaining difficulty is the presence of the negative summand E 2 . This requires the following considerations. We write

f * (mK X ) = I Z ⊗ O(a)
with a finite set Z; being defined by

I Z = f * (O X (E 2 )).
In a first step we claim that I Z ⊗ O(a) is pseudo-effective in the sense that N (I k Z ⊗ O(ka + 1)) has sections for large k and N. Let σ : Ŷ → Y be a birational map with Ŷ smooth and X the normalization of the fiber product X × Y Ŷ with induces maps τ : X-→X and f : X → Ŷ such that f is flat, in particular equidimensional. Adopting the arguments from above, f * τ * (mK X ) is a pseudo-effective line bundle. Now

σ * f * τ * (mK X ) = f * τ * τ * (mK X ) = f * (mK X ), hence I Z ⊗ O(a) is clearly pseudo-effective.
Let x ∈ Y = P 2 be general and let ρ : Ỹ → Y be the blow-up of x, inducing a P 1bundle π : Ỹ → P 1 . Let F be a fiber of π, i. If we let Z m be the subspace defined by I m Z , we obtain in completely the same way (considering

I m Z ⊗ O(ma) and assuming H 0 (I m Z ⊗ O(ma)) = 0) that l(Z m ) = ma(ma + 1) 2 .
Now consider the case that Z is reduced. Then l(Z 2 ) = 3l(Z) and l(Z 3 ) = 10l(Z).

On the other hand, we can compute l(Z i ) by the above formula, and this produces a contradiction. So Z cannot be reduced.

To deal with the general case, choose a deformation (Z t ) of Z = Z 0 such that Z t is reduced for t = 0. Then clearly I Z t ⊗ O(a) is pseudo-effective. Now we apply to Z t the above considerations and obtain a contradiction. 9.7 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 an 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 [START_REF] Kollár | Higher direct images of dualizing sheaves II[END_REF], and we obtain the well-known formula of Q-divisors

( * ) K X = f * (K Y + ∆) + E Here E is an effective divisor such that f * (O X (E)) = O Y . Moreover ∆ = ∆ 1 + ∆ 2 with ∆ 1 = (1 - 1 m i )F i + a k and ∆ 2 ∼ 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 ∈ 1 12 N according to Kodaira's list. Then by a general choice of the divisor ∆ 2 , the pair (Y, ∆ 1 + ∆ 2 ) is klt. Now K Y + ∆ is pseudo-effective. In fact, by Theorem 2.2 it suffices to show that K Y + ∆ • C t ≥ 0 for every covering family of curves. But this is checked very easily as in (9.4/9.6). Hence the log Minimal Model Program [START_REF] Kollaŕ | Flips and abundance for algebraic 3-folds[END_REF] in dimension 3 implies that K Y + ∆ is effective. Hence κ(X) ≥ 0 by (*).

In order to attack the case p(K X ) = 0, i.e. there is a connecting family (C t ) with K X • C t = 0, we prove a more general result. 9.8 Theorem. Let X be a projective manifold of any dimension n, (C t ) a connecting family and L pseudo-effective. If L • C t = 0, then there exists a line bundle L ′ with L ≡ L ′ and κ(L ′ ) = 0. More generally, if L ≡ N +Z is the divisorial Zariski decomposition, then Z = 0. 9.9 Corollary. Let X be a projective manifold with q(X) = 0 such that K X is pseudoeffective. If p(K X ) = 0, i.e. there is a connecting family (C t ) such that K X • C t = 0, then κ(X) = 0.

First we derive 9.9 from 9.8. Proof of 9.9. Applying (9.8), mK X is effective up to some numerically trivial line bundle G. Since we may assume q(X) = 0, the line bundle G is trivial after finite étale cover, hence κ(X) ≥ 0. By 8.5 we get κ(X) = 0.

Proof of 9.8.

First notice that suffices to prove the statement on the Zariski decomposition; then the first statement follows from [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF], see (3.7). In fact, then L ≡ N and N is an effective Q-divisor.

(I) In a first step we assume that dim T = n -1 resp. we can find a connecting (n -1)dimensional subfamily.

Let L = N 1 + Z 1
be the divisorial Zariski decomposition of L with N 1 effective and Z 1 nef in codimension 1 (as real divisors). We consider the graph p 0 : C 0 → X with projection q 0 : C 0 → T ; we may assume T smooth and C 0 normal. Let π : C → C 0 be a desingularisation and put p = p 0 • π, q = q 0 • π. Now p * (Z 1 ) might not be nef in codimension 1, so that we decompose p * (Z 1 ) = Ñ + Z and put N = p * (N 1 ) + Ñ . We end up with p * (L) = N + Z and this is the divisorial Zariski decomposition of p * (L). This equation holds up to the pull-back of a topologically trivial R-line bundle on X. Notice that N does not meet the general fiber of q since L • C t = 0. This decomposition can be rewritten as follows

( * ) p * (L) = a i F i + b j B j + q * (N ′ ) + q * (A) -E ′
where F i are fibered over q(F i ) by parts of reducible 1-dimensional fibers, the B j are irreducible components of the exceptional locus of π with codim q(B j ) ≥ 2, where N ′ is R-effective, where E ′ is effective with codim(q(E ′ )) ≥ 2 and finally Z = q * (A). In particular A is nef in codimension 1. The coefficients a i , b j are positive and real a priori.

First we verify the decomposition

( * * ) N = a i F i + b j B j + q * (N ′ ).
In fact, since p * 0 (N 1 ) • q -1 (t) = 0 for general t, we can decompose the effective R divisor p * 0 N 1 into the components F ′ j with codim q 0 (F ′ j ) = 1 but the F ′ j contain only parts of fibers, and the other components which then consist only of full fibers of q 0 . Therefore we can write

p * 0 (N 1 ) = a ′ i F ′ i + q * 0 (N ′ )
and then

p * N 1 = a ′ i F i + c j B j + q * (N ′ )
with c j ≥ 0. Then ( * * ) follows by adding and decomposing Ñ ; notice here that Ñ cannot contain multisections of q since p * (L) • q -1 (t) = 0.

To get the decomposition ( * ), consider first a line bundle B over C which is q-nef in codimension 1 with B • q -1 (t) = 0 for general t. This means that B is q-nef over a Zariski open subset in T whose complement has codimension at least 2. Then B defines a section in R 1 q * (O C ) over a Zariski open affine set T 0 ⊂ T such that codim(T \T 0 ) ≥ 2. Since R 1 q * (O C ) is a direct sum of a torsion sheaf supported in codimension at least 2 and a reflexive sheaf ( [START_REF] Kollár | Higher direct images of dualizing sheaves II[END_REF]), s extends to a section of R 1 q * (O C ) on all of T (possibly first enlarge T 0 ). By the Leray spectral sequence we obtain a topologically trivial line bundle G on C such that B|q -1 (t) = G|q -1 (t) for general t. Thus, possibly substituting B by B ⊗ G * , we may assume that B|q -1 (t) is trivial for general t. Now consider the canonical map q * q * (B) → B to obtain a decomposition of type ( * ) for B, namely B = q * (B ′ ) + E, where E is a not necessarily effective divisor whose components E k satisfy codim q(E k ) ≥ 2.

Going back to our case, we would like to apply this to B = Z.

However Z ∈ H 1,1 Q (C)⊗R is not a Q-divisor. Therefore we approximate Z by Q-divisors Z j ∈ H 1,1
Q (C) such that Z j •q -1 (t) = 0 for general t. This is possible since the linear subspace {A|A•q -1 (t) = 0} is rationally defined. Now apply the previous considerations to Z j ; but first we have to make sure that the Z j are q-nef in codimension 1. This can be achieved by requiring that the Z j are 0 on the components of general reducible fibers. Now applying our previous considerations, it follows that Z j = q * (Z ′ j ) + E j with divisors E j whose components E k,j satisfy codim q(E k,j ) ≥ 2. Then let Z ′ be the limit of the Z j ; we obtain Z = q * (Z ′ ) + E ′ . Since Z is nef in codimension 1, we conclude that -E ′ is effective. This finally establishes ( * ) (possibly we have to pass from L to mL in order to avoid multiple components in a i F i ).

We also notice that the coefficients a i in ( * ) must be rational. In fact, by ( * ), a i F i is rational and by Boucksom [Bou02b, 2.1.15]; see (3.7), the F i are linearly independent. Since deg p = 1, the map p is birational; let E denote the exceptional locus. Then two curves from the family (C t ) can only meet at points in p(E). Since (C t ) is connecting, E projects onto T , i.e. q(E) = T . To be more precise, we pick some component E i projecting onto T . Thus every t is contained in some subvariety

T (x) := q(p -1 (x) ∩ E i ),
i.e. every C t passes through a point x ∈ p(E i ) and through each such x there exists an at least 1-dimensional subfamily (C t ). Notice that T (x) might not be irreducible, so that for general x ∈ p(E i ) we pick an irreducible component such that (taking closure) we obtain a compact family which we again denote (T (x)). Now consider x ∈ p(E i ) general and let S x ⊂ q -1 (T (x)) be the irreducible component mapping onto T (x). By restricting

p * (L) = a i F i + b j B j + q * (N ′ + A)
to S x ⊂ q -1 (T (x)) we obtain

A|T (x) ≡ N ′ |T (x) ≡ 0
(cut by a general hyperplane section in T (x)). If T (x) = T , then N ′ and A are numerically trivial and ( * * * ) holds.

So suppose dim T (x) < dim T . If the family (T (x)) is connecting, then we find connecting families of curves, say (C s ) such that If (T (x)) is not connecting, then we form the quotient g : T > W . We can choose x, x ′ ∈ p(E) such that T (x)∩T (x ′ ) = ∅. Since C t 1 ∩C t 2 ⊂ p(E) for all choices t j ∈ T , we conclude that x and x ′ cannot be connected by chains of C t 's. This is a contradiction.

N ′ + A • C s = 0, hence N ′ + A is Q-effective
(b) Now let deg p ≥ 2. Take a general C t . Then through the general x ∈ C t there is at least one other C t ′ . Therefore we obtain a 1-dimensional family (C s ) s∈T t through C t . To be more precise, let D ⊂ X be the subspace over which p has positive dimensional fibers. Then let

T t ⊂ q(p -1 (C t \ D))
be the union of all irreducible components of dimension 1 whose q-images still have dimension 1. Putting things together, we obtain a family (T t ), however the general T t might be reducible. We claim that the family (T t ) is connecting. In fact, take t 1 , t 2 ∈ T general. Then we can join the curves C t 1 and C t 2 by an odd number of irreducible curve

C s j , 1 ≤ j ≤ 2n -1. So C t 1 ∩ C s 1 = ∅, C s 1 ∩ C s 2 = ∅ etc. Thus s 1 ∈ T t 1 ∩ T s 2 so that T t 1 ∩ T s 2 = ∅. Moreover s 3 ∈ T s 2 ∩ T s 4 so that T s 2 ∩ T s 4 = ∅.
Continuing, finally T s 2n-2 ∩ T t 2 = ∅, so that t 1 and t 2 can be joined by chains of T s . Thus (T t ) is connecting.

Let S t ⊂ C t be the corresponding surface over T t resp. an irreducible component. Then p * (L) is numerically trivial on the general fiber of q|S t and also on some multi-section. Therefore q * (N ′ + A)|S t = 0 due to the following remark (+), hence (N ′ + A) • T t = 0.

Let L = N ′ + A be a pseudo-effective line bundle over a smooth projective surface or threefold T with N an effective R-divisor and A nef in codimension 1. Assume that there is a map g : T → W to the smooth curve W . Then L is numerically Q-effective.

In fact, N = r i F ′ i with fiber components F ′ i and A is numerically trivial on all fibers since A is nef in codimension 1. Thus A ≡ g * (A ′ ) with A ′ nef on W . This proves (+). Now we apply induction if the general T t is irreducible resp. Lemma 9.10 if the general T t is reducible and obtain that A = 0, hence Z = 0.

(II) If there is no connecting (n -1)-dimensional subfamily, we choose some (n -1)dimensional subfamily C 0 over T 0 ⊂ T . Now consider the p-preimages of the C t corresponding to t ∈ T 0 . Then we obtain a connecting family of multi-sections on which q * (N ′ + A) is numerically trivial and, taking q-images, a connecting family in T 0 on which N ′ + A is numerically trivial. Then argue by induction as in (I.b). 9.10 Lemma. If Theorem 9.8 holds in dimension at most n, then it also holds in dimension ≤ n for arbitrary covering connecting families (C t ) with the general C t being reducible: if L is pseudo-effective and L • C j t = 0 on every movable component C t , then Z = 0 in the divisorial Zariski decomposition.

Proof. Let (C t ) t∈T be a connecting family of reducible curves with graph p : C → X and q : C → T. Let

C = l i C i
be the decomposition into irreducible components and set p i = p|C i , q i = q|C i . Some C i might still have generically reducible q-fibers, so we pass to the normalisation Ci -→C i . Let Ci,j be the decomposition into irreducible components; then every Ci,j defines a family of generically irreducible curves on X. In total we obtain finitely many families Ĉk -→S k of generically irreducible curves, not all of them covering possibly.

Pick one of the covering families, say Ĉ1 . Let f : X > X 1 be the associated quotient (with X 1 smooth). Since we are allowed to blow up, we may assume f holomorphic from the beginning. If Ĉ1 happens to be connecting, i.e. dim X 1 = 0, then (9.8) gives our claim for the given line bundle L. So suppose dim X 1 > 0. Our plan is to proceed by induction on dim X. Thus consider the induced family Ct = f * (C t ) (for t generic; then take closure in the cycle space). Obviously ( Ct ) is connecting. By (9.8) and (8.5) -possibly after tensoring with a topologically trivial line bundle as in the proof of (9.8) -we have κ(L|F ) = 0 for suitable m and the general fiber F of f. Thus f * (mL) has rank 1. Choosing m sufficiently divisible, we obtain

mL = f * (L 1 ) + E 1 -E 2 ,
where E j are effective, the components of E 1 either consists of parts of fibers of f or project onto subvarieties of codimension at least 2 in X 1 and the components of E 2 also project onto subvarieties of codimension at 2 in X 1 . By passing to a suitable model of f (blow up X and Y ), we may assume that f * (mL) is locally free, hence E 2 = 0. Now by cutting down to movable curves, we conclude as in 9.4/9.6 that L 1 is pseudo-effective. In order to apply induction we still need to show that L • Cj t = 0 for the movable components. This is however clear:

0 = mL • C j t = E 1 • C j t + L ′ • Cj t , so that E 1 • C j t = L ′ • Cj t = 0.
Hence we can apply induction: if L 1 = N 1 + Z 1 is the divisorial Zariski decomposition, then Z 1 = 0. Thus the same holds for L.

Combining everything in this section we finally obtain 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 covering family (C t ) of curves such that K X • C t = 0, then κ(X) ≥ 0.

The remaining task is to consider 4-folds X with K X pseudo-effective such that K X •C > 0 for every curve C passing through a very general point of X, i.e. p(K X ) = 0. 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 9.12 below. But it is not at all clear whether K X |S is big, which is of course still not enough to conclude. 9.12 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 Ŝt is a smooth variety of dimension at most 3 and not of general type. Using a minimal model if κ( Ŝt ) = 0, we find in Ŝt a covering family of curves intersection K Ŝt trivially. Thus we find a covering family (C s ) in C, all members being in q-fibers such that K C • C s = 0. Since K C = p * (K X ) + E with E effective, we get K X • p * (C s ) ≤ 0, a contradiction.

Using the Iitaka fibration we obtain 9.13 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 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 nnα n-1 • β > 0. Then {α -β} contains a Kähler current and

Vol(α -β) ≥ α n -nα n-1 • β.
Remarks about the conjecture. If α = c 1 (L) for some holomorphic line bundle L on X, then the inequality ( * * ) was established in [START_REF] Boucksom | On the volume of a big line bundle[END_REF] as a consequence of the results of [START_REF] Demailly | Champs magnétiques et inégalités de Morse pour la d ′′ -cohomologie[END_REF]. 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(αβ, ≤ 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 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 n C ,
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 n C (in particular θ 0 everywhere on X), and put

ψ = log |σ 0 | 2 |σ 0 | 2 + |σ 1 | 2 + . . . + |σ n | 2 .
We also use the standard notation d c = i 4π (∂ -∂) so that dd c = i 2π ∂∂. 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 n ε = 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∂∂ϕ ε ) n = (1δ)ω n + δ X ω n c 1 (A) n α n ε .

By the theorem of S.-T. Yau [START_REF] Yau | On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation[END_REF], there exists a smooth solution ϕ ε , unique up to normalization by an additive constant, such that ω ε := ω + i∂∂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 [START_REF] Boucksom | On the volume of a big line bundle[END_REF] show that its absolutely continuous part satisfies

X T n ac ≥ (1 -δ) X ω n .
We are going to use the same ideas as in [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF], 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 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 X ω n c 1 (A) n U ′ ε 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 n ε ) 1/2 (α ε ∧ θ n-1 ) 1/2 2 U ′ ε ρ 2 . . . ρ n α n ε U ′ ε 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 n ε n X ω n-1 ε ∧ α ε = n X ω n-1 ∧ c 1 (A).
On the other hand, an explicit calculation shows that

α n ε 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 , hence U ′ ε (α n ε ) 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

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 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 n ε ) 1/2 (α ε ∧ θ n-1 ) 1/2 2 √ 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 1 ρ 2 . . . ρ n α ε ∧ θ n-1 4δ (n + 1) 2 (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 4δ (n + 1) 2 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 4δ (n + 1) 2 X ω n X ω n-1 ∧ c 1 (A)

  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.

  e. a line in P 2 . Then the pseudo-effectivity of I Z ⊗ O(a) yields H 1 (F, I Z ⊗ O(a)|F ) = 0 from which we get the vanishing R 1 π * (I Z ⊗ ρ * (O(a)) = 0. Thus we have an exact sequence 0-→π * (I Z ⊗ ρ * (O(a)))-→π * (ρ * (O(a)))-→O R -→0 where R ≃ Z. Now suppose H 0 (I Z ⊗ O(a)) = 0 (otherwise we are done). Since π * (ρ * (O(a)) = S a (O ⊕ O(1)), we deduce by taking H 0 that l(Z) = a(a + 1) 2 (and H 1 (I Z ⊗ O(a)) = 0).

(

  I.a) First suppose that deg p = 1. Then our claim comes down to prove that Z = 0. ( * * * )

  by induction and ( * * * ) holds.

(

  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 n m . (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 ).

  c 1 (A) n . Therefore, as [H] ≡ θ ∈ c 1 (A), we infer c 4δ (n + 1) 2 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(Tc[H])

  case A and B are integral (Cartier) divisors, this is a consequence of the holomorphic Mores inequalities, [De01,8.4]. 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].

4.3 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 X such that each difference β ′ jβ j is pseudo-effective. Then the n-th intersection products satisfy

The specific choice

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 [START_REF] Demailly | Numerical characterization of the Kähler cone of a compact Kähler manifold[END_REF]). 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

Proof. We follow closely the approach of D. Huybrechts in [Huy02], page 44. 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

where the first inequality is a consequence of the semi-continuity of the volume obtained in [START_REF] Boucksom | Cônes positifs des variétés complexes compactes[END_REF], 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.