TOWARDS A MORI THEORY ON COMPACT KÄHLER THREEFOLDS, II

THOMAS PETERNELL

Abstract. This paper, a continuation of “Towards a Mori theory on compact Kähler manifolds, 1” (written with F. Campana), introduces a non-algebraic analogue to Mori theory in dimension 3.

Introduction

This is the second part to my joint paper “Towards a Mori theory on compact Kähler manifolds, 1”, with F. Campana. With the techniques developped in that paper we shall prove here the following result.

Main Theorem Let X be a non-algebraic compact Kähler threefold satisfying one of the following conditions. (I) X can be approximated algebraically. (II) κ(X) = 2. (III) X has a good minimal model. Assume that K_{X} is not nef. Then

( 1) X contains a rational curve C with K_{X} · C < 0;

( 2) There exists a surjective holomorphic map φ : X⟶Y to a normal complex space Y with φ_{∗}(O_{X}) = O_{Y} of one of the following types.

(a) φ is a P_{1}- bundle or a conic bundle over a non-algebraic surface (this can happen only in case (1))

(b) φ is bimeromorphic contracting an irreducible divisor E to a point, and E together with its normal bundle N is one of the following

(P_{2}, O(−1)), (P_{2}, O(−2)), (P_{1} × P_{1}, O(−1, −1)), (Q_{0}, O(−1)),

where Q_{0} is the quadric cone.

(c) Y is smooth and φ is the blow-up of Y along a smooth curve. φ is called an extremal contraction. Y is (a possibly singular) Kähler space in all cases except possibly (2c). Moreover in all cases but possibly (2c), φ is the contraction of an extremal ray in the cone \overline{NE}(X).

Most parts of this paper have been worked out during a stay at the MSRI in Berkeley.

I would like to thank the institute for its hospitality and the excellent working conditions.

Research at MSRI is supported in part by NSF grant DMS-9022140.

This is therefore a non-algebraic analogue to Mori theory in dimension 3. Of course one expects that in case (2c) we can arrange things such that Y is Kähler, too, and that φ is the contraction of an extremal ray. In principle we would of course like to prove the theorem without any of the assumptions (I),(II) or (III).

We will now explain the theorem and the method of the proof. Nefness of a line bundle L on an arbitrary compact manifold X is defined via metrics : L is nef, if for every positive ϵ there is a metric h_{ϵ} on L whose curvature satisfies Θ_{L,h_{ϵ}} ≥ −ω, for a fixed positive (1,1)-form ω. Passing to the special case L = K_{X}, it is not at all clear that there is any curve C with K_{X} · C < 0, for arbitrary L this is even false. In order to circumvent this difficulty, we introduce the additional alternative assumptions (I),(II) and (III). We first focus on case (I), i.e. X admits an algebraic approximation, This means that there is a family of compact Kähler manifolds (X_{t}) over the unit disc in some C^{m} with X_{0} ≃ X and with a sequence t_{ν} converging to 0 such that all X_{t_{ν}} are projective. Assume now that K_{X} is not nef. Then we will prove that K_{X_{tν}} is not nef for large ν. Therefore a fixed X_{t_{ν}} admits a contraction of an extremal ray. This extremal ray is given by a non-splitting family of rational curves. The family can be deformed to the nearby fibers and hence also to X_{0}, however the limit family may split. Therefore one has to extract from the limit family a non-splitting family which in turn by Part 1 of this paper will define the map φ we are looking for. However it is now not clear whether φ is induced by contractions of extremal rays in the nearby (projective ) fibers. This causes the difficulty in case (2c) together with some projective problem in the limit which will be explained in full detail in sect. 4; see also below for more comments.

The proof that the bundles K_{X_{tν}} are not nef is done by arguing by contradiction. If all K_{X_{tν}} would be nef, then a suitable multiple would be generated by global sections by Miyaoka-Kawamata’s abundance theorem. Now we examine the limit linear system. It turns out that in case of the presence of a base locus, we can find a curve C, in the base locus, with K_{X} · C < 0. Here we make heavily use of the fact that we are working in dimension three (also abundance works at the moment only in this dimension) because we have to analyse line bundles which are not nef in the analytic sense but which are nef in the algebraic sense on nonalgebraic (possibly singular) surfaces. This is possible since the structure of surfaces of algebraic dimension ≤ 1 is not complicated.

If κ(X) = 2 and K_{X} is not nef, we can directly show that there is a rational curve C with K_{X} · C < 0, using the meromorphic map (Iitaka reduction) attached to the linear system |mK_{X}| for large m. If κ(X) = 1, we can at least prove that there is a curve C with K_{X} · C < 0. Once we have a rational curve C with K_{X} · C < 0, we obtain a non-splitting family of rational curves with the same property and if κ(X) ≥ 0, we can apply the main result of [CP94] to obtain an extremal contraction.

An important point for further developments is of course the question whether Y is again Kähler in case φ is birational. For the notion of a singular Kähler space we refer to sect.3. As stated already in the Main Theorem, we are able to prove this in case that the exceptional divisor E is contracted to a point, using various techiques of the theory currents. In case dimφ(E) = 1, this is however not the case; at least there is no a priori reason why that should be true. Therefore this case remains open but one would expect that Y is Kähler if we have chosen the “correct” φ. The “correct” φ should be given by an extremal ray in the cone of effective curves or the dual cone to the Kähler cone.

A Motivation

The final aim of a “Mori theory” of compact Kähler threefolds X would of course be

(a) to construct minimal models unless X is uniruled

(b) to prove abundance for minimal models: if X′ is a compact Kähler threefold with at most terminal singularities and K_{X′} nef, then mK_{X′} is generated by global sections for suitable large m.

We will discuss these topics in section 6. Besides its independent interest, the solution of these problems would give some new insight in the structure of Kähler threefolds which is not connected a priori to Mori theory. The statement we want to deduce is the following

If the problems (a) and (b) have a positive solution, then simple Kähler threefolds are Kummer.

The relevant definitions are :

(1) a compact Kähler manifold is simple, if there is no covering family of positive dimensional subvarieties (hence through a very general point of X there is no positive dimensional irreducible subvariety);

(2) a compact Kähler manifold is Kummer, if it is bimeromorphic to a variety T/G, where T is a torus and G a finite group acting on T.

Proof. Since X is simple, it cannot be uniruled. Hence by (a) X has a minimal model X′. By (b), mK_{X′} is generated by global sections for some m. Hence the Kodaira dimension κ(X′) ≥ 0. Since X′ is simple, we conclude κ(X′) = 0, hence

mK_{X′} = O_{X′}.

Now there exists a covering

X̃⟶X′,

unramified over the regular part regX, such that KX̃ = OX̃, in particular X̃ is Gorenstein, see e.g. [KMM87]. By the Riemann-Roch theorem for Gorenstein threefolds (cp. e.g.[Ka86]), we obtain

χ(X̃, OX̃) = 0.  (∗)

Now let X̂⟶X̃ be a desingularisation. Since X̂ is non-algebraic we have by Kodaira’s theorem H^{2}(X̂, OX̂) ≠ 0. Because X̃ has only rational singularities, we obtain

H^{2}(X̃, OX̃) ≠ 0.

Since h^{3}(X̃, OX̃) = 1, we obtain from (*):

H^{1}(X̃, OX̃) ≠ 0.

Again working with X̂ we therefore have a non-trivial Albanese map

α : X̃⟶Alb(X̃) = Alb(X̂).

Since X̃ is simple, α is surjective. Having in mind that KX̃ = OX̃, we immediately see that α is etale outside the singular locus of X̃. If x_{0} is a singularity of X̃, then some 1-form would vanish at x_{0} and hence we obtain by wedging 1-forms a 3-form with an isolated singularity which is absurd. Hence X̃ is smooth and therefore X̃ is a torus (α is an isomorphism). Compare [Ka85,sect. 8]. Now X̃ being a torus, X is Kummer.

Preliminaries

(0.1) Let X be a compact Kähler manifold. We say that X can be approximated algebraically, if the folowing holds. There exists a family X = (X_{t})_{t∈Δ} of compact Kähler manifolds over the unit disc Δ ⊂ C^{m} such that X_{0} ≃ X and there is a sequence t_{ν} converging to 0 such that all X_{t_{ν}} are projective. The projection map X ⟶Δ is usually denoted π. By a conjecture of Kodaira (or Andreotti?) every compact Kähler manifolds can be approximated algebraically, however at the moment this only known in dimension 2 (via Enriques-Kodaira classification).

(0.2) Here we collect some standard notations. If X is a compact manifold, we let ρ(X) denote the Picard number of X. Define N_{1}(X) to be the linear subspace of H_{2}(X, R) generated by the classes of irreducible curves. We let \overline{NE}(X) ⊂ N_{1}(X) denote the closed cone generated by the classes of irreducible curves. For details on this and on Mori theory in the algebraic case in general we refer e.g. to [KMM87].

(0.3) The algebraic dimension of an irreducible reduced compact complex space X is by definition the transcendence degree of its field M(X) of meromorphic functions over C. An algebraic reduction of X is a meromorphic map f : X ⇀ Y inducing an isomorphism f^{∗} : M(Y)⟶M(X).

(0.4) For the convenience of the reader we state here the main result of [CP94] which we shall use several times. We shall use the notion of a non-splitting family (C_{t})_{t∈T} of rational curves. This means that T is compact and irreducible and every C_{t} is an irreducible and reduced rational curve. Now the main result of [CP94] reads as follows

Theorem Let X be a compact Kähler threefold and (C_{t})_{t∈T} a nonsplitting family of rational curves.

(1) If K_{X} · C_{t} = −4 and dimT = 4, then X ≃ P_{3}.

(2) If K_{X} · C_{t} = −3 and dimT = 3, then either X ≃ Q_{3}, the threedimensional quadric, or X is a P_{2}−bundle over a smooth curve.

(3) Assume K_{X} · C_{t} = −2 and dimT = 2.

(3.1) If X is non-algebraic and the (C_{t}) fill up a surface S ⊂ X, then S ≃ P_{2} with normal bundle N_{S|X} = O(−1) (the same holds for X projective if S is normal)

(3.2) If X is coveredby the C_{t}, then one of the following holds. (3.2.1) X is Fano with b_{2}(X) = 1 and index 2. (3.2.2) X is a quadric bundle over a smooth curve with C_{t} contained in fibers (3.2.3) X is a P_{1}−bundle over a surface, the C_{t} being the fibers. (3.2.4) X is the blow-up of a P_{2}−bundle over a curve along a section. Here the

C_{t} are the strict transforms of the lines in the P_{2}’s meeting the section (4) Let K_{X} · C_{t} = −1 and dimT = 1. Then the C_{t} fill up a surface S. Assume X

non-algebraic.

(4.1) If S is normal, then one of the following holds. (4.1.1) S = P_{2} with N_{S} = O(−2). (4.1.2) S = P_{1} × P_{1} with N_{S} = O(−1, −1). (4.1.3) S = Q_{0} (a quadric cone) with N_{S} = O(−1). (4.1.4) S is a ruled surface over a smooth curve and X is the blow-up of a smooth

threefold along C. (4.2) Let S be non-normal. Then κ(X) = −∞. If moreover X can be approx-

imated by algebraic threefolds, then we have a(X) = 1 and X has the structure of a “generic conic bundle” over a surface Y with a(Y) = 1. The general fiber of the algebraic reduction f which can be taken holomorphic in this case is an almost homogeneous P_{1}−bundle over an elliptic curve. The surface S consists of reducible conics and is contracted by f to a point.

The essential content of the theorem can be rephrased as follows. Assume that C is a rational curve with K_{X} · C = k, −1 ≥ k ≥ −4. If no deformation of C splits, then the conclusions of the theorem hold. The point is that automatically the dimension of the deformation is at least k.

We will have a more careful look at the case (4.2) in sect.5.

The paper [CP94] will be refered to as Part I.

3. Limits of extremal rays

(3.1) For most of this section we fix a family π : X ⟶Δ of compact Kähler threefolds X_{t} = π^{−1}(t) over the unit ball Δ ⊂ C^{m}. Let t_{ν} be a sequence converging to 0. Assume that all X_{t_{ν}} are projective, whereas X_{0} is not. In other words, X is an algebraic approximation of X = X_{0}. Furthermore we assume that all K_{X_{tν}} are not nef. Fix some ν_{0} and consider an extremal ray R_{ν_{0}} on X_{t_{ν_{0}}} generated by the extremal rational curve C_{ν_{0}}. Let φ_{ν_{0}} : X_{t_{ν_{0}}} ⟶Y_{t_{ν_{0}}} be the contraction associated with R_{ν_{0}}. We are going to construct from R_{ν_{0}} a sequence of extremal rays R_{ν} living on X_{ν}. In order to do this we first examine the structure of R_{ν_{0}}. Let m denote the length of R_{ν_{0}}, i.e.

m = min{−K_{X_{tν_{0}}} · C|[C] ∈ R_{ν_{0}}}.

Then we have :

3.2 Lemma Either m = 1 or m = 2.

Proof. Assume m ≥ 3. Because of the a priori bound m ≤ 4 we have only to exclude the cases m = 4 and m = 3. In the first case we have X_{t_{ν_{0}}} ≃ P_{3}, in the second X_{t_{ν_{0}}} ≃ Q_{3}, the threedimensional quadric or X_{t_{ν_{0}}} is a P_{2}− bundle over a curve [Wi89]. Hence always H^{2}(X_{t_{ν_{0}}} , O_{X_{tν_{0}}} ) = 0. Therefore H^{2}(X_{0}, O_{X_{0}}) = 0 and X_{0} is projective by Kodaira’s well-known theorem, contradiction.

The same argument shows

3.3 Lemma The contraction φ_{ν_{0}} cannot be a del Pezzo fibration (i.e. dimY_{t_{ν_{0}}} ≠ 1) nor a P_{1}− or a conic bundle over a surface of Kodaira dimension −∞.

In fact, otherwise we would have H^{2}(X_{0}, O_{X_{0}}) = 0 by virtue of

R^{i}φ_{ν_{0}∗}(O_{X_{tν_{0}}} ) = 0, i ≥ 1, and H^{2}(Y_{t_{ν_{0}}} , O_{Y_{tν_{0}}} ) = 0.

(3.4) From [Mo82] we obtain the following structure of φ_{ν_{0}} :

(a) a P_{1}−bundle over a surface with nonnegative Kodaira dimension

(b) a conic bundle over a surface with nonnegative Kodaira dimension

(c) a birational contraction contracting an irreducible divisor E such that (c.1) E = P_{2} with normal bundle N = O(−1), (c.2) E = P_{2}, N = O(−2), (c.3) E = P_{1} × P_{1}, N = O(−1, −1), (c.4) E = Q_{0}, the quadric cone, with N = O(−1).

(d) a birational contraction which contracts an irreducible divisor E such that dimφ_{ν_{0}} = 1 and φ_{ν_{0}} is the blow-up of the smooth curve φ_{ν_{0}}(E) in the manifold Y_{t_{ν_{0}}} .

We will call the different contractions of type (a), (b), (c.i) and (d), respectively.

Now consider the extremal rational curve C_{ν_{0}} ⊂ X_{t_{ν_{0}}} . We assume that −K_{X_{tν_{0}}} ·C_{ν_{0}} is minimal, C_{ν_{0}} is then smooth. Then the normal bundle N of C_{ν_{0}} in X_{t_{ν_{0}}} is of the form

N = O(a) ⊕ O(b)

with (a, b) ∈ {(0, 0), (0, −1), (1, −1), (1, −2)}.

If a, b ≥ −1, the deformations of C_{ν_{0}} in X_{t_{ν_{0}}} as well as in X are unobstructed, since

N_{C_{ν_{0}} |X} ≃ O(a) ⊕ O(b) ⊕ O.

Let C = (C_{s})_{s∈S} be the family of deformations of C_{ν_{0}} in X , in particular we obtain a limit family (C_{s}^{0}) in X_{0}. However it is not clear whether e.g. a P_{1}−bundle structure on X_{t_{ν_{0}}} converges to a P_{1}−bundle structure on X_{0}. More precisely, the subfamily (C_{s})_{s∈S_{ν_{0}}} of deformations of C_{ν_{0}} inside X_{t_{ν_{0}}} forms a non-splitting family of rational curves in the sense of Part 1; however the limit family may split. The simplest example (in the case of projective families) is the specialisation of P_{1} × P_{1} into the Hirzebruch surface P(O ⊕ O(−2)) where one of the two rulings converges to a splitting family of rational curves.

We also want to have a limit family in case (a, b) = (1, −2). This case occurs if either φ_{ν_{0}} is a conic bundle, here C_{ν_{0}} must be the reduction of a non-reduced conic, or E = P_{2} with N = O(−2). In the first case we just take C_{ν_{0}} not to be the reduction of a non-reduced conic, but an irreducible component of a reducible reduced conic; then N = O ⊕ O(−1) and we can conclude. If E = P_{2} with N = O(−2) then by deformation theory the deformations of E in X are unobstructed, so we can deform E in X in a 1-dimensional family (note N_{E|X} = O(−2) ⊕ O). So our family C exists also here.

3.5 Theorem There exists a surjective holomorphic map

ψ_{0} : X_{0}⟶Z_{0}

to a normal complex space Z_{0} in class C of type (a),(b), (c.1) or (c.2) , as described in (3.4), such that ρ(X) = ρ(Y)+1. Moreover there exists a morphism ψ_{t} : X_{t}⟶Z_{t} for general t of the same type fitting into a family ψ : X ⟶Z outside a proper analytic set A ⊂ Δ not containing 0. The Z_{t_{ν}} are projective except possibly in case (c.2).

Recall that ρ(X) is the Picard number of X and that a normal compact complex space is in class C if it is bimeromorphically equivalent to a Kähler manifold.

Proof. We consider the family C constructed in (3.4). In particular we have the family (C_{s}^{0}) in X_{0}. For simplicity of notations we skip the upper index 0.

(1) Assume first that the family (C_{s}) does not split. Then we apply the Main Theorem of Part 1 (0.4) and obtain the following. If −K_{X_{0}} · C_{s} = −K_{X_{tν_{0}}} · C_{ν_{0}} = 2, then, X_{0} being non-algebraic, the C_{s} define either a P_{1}-bundle structure or a birational contraction of type (c.1) with E = P_{2} and N = O(−1). This defines our morphism ψ_{0} : X_{0}⟶Z_{0}. It is clear that the corresponding families in the nearby fibers defines maps ψ_{t} of the same type and that all maps fit together. Moreover ψ_{t_{ν_{0}}} = φ_{t_{ν_{0}}} . Now let −K_{X_{0}} · C_{s} = 1. Let E_{0} = ⋃ C_{s} ⊂ X_{0}. If E_{0} is normal, then we are in case (c.i) (2 ≤ i ≤ 4) or (d) and everything holds in the same way as above. So let E_{0} be non-normal. Then X_{0} has the structure of a so-called generic conic bundle in the sense of Part 1. If the C_{s} define an extremal ray in \overline{NE}(X_{0}), or, equivalently, if the associated full family of “conics” splits only in the standard way in the sense of (5.1), then we conclude by (5.2) resp. (5.3). Otherwise we consider the general smooth conic and the associated family (C̃_{t}). Then we have a non-standard splitting

C̃_{t_{0}} = ∑_{i=1}^{p} C′_{i}

with p ≥ 3 or with p = 2 and, say, −K_{X_{0}} · C′_{1} ≥ 2. In any case we have a new C′_{i} with −K_{X_{0}} · C′_{i} ≥ 1 which deforms in X_{0}. Then we proceed as in (2) below.

(2) Assume now that (C_{s}) splits :

C_{s_{0}} = ∑ C′_{i}.

Say that −K_{X_{0}} · C′_{1} > 0. Then C′_{1} deforms in an at least 1-dimensional family, say (C′_{t}), in X_{0}. If this family does not split, then we argue as in (1); note that the C′_{t} deform also to the nearby fibers since in the normal bundle we add a trivial factor (see (3.4) for details). If the family splits again, we take a splitting part C′′_{1} with −K_{X_{0}} · C′′_{1} > 0 etc. The only thing left is to prove that this procedure terminates.

(a) First assume that no generic conic bundle comes up in our procedure. Then we consider the cone \overline{NE}(X_{0}) ⊂ N_{1}(X_{0}); note that every effective curve represents an integer point in \overline{NE}(X_{0}), denote this set of integer points by \overline{NE}(X_{0})_{Z}. Now introduce a norm ∥ · ∥ in N_{1}(X_{0}) such that ∥x∥^{2} ∈ N for x ∈ \overline{NE}(X_{0})_{Z}. Let x_{1} be the class of a general element of our first family (C_{s}), x_{2} the one of the second etc. Then x_{1} = x_{2} + x′_{2} with 0 ≠ x_{2} ∈ \overline{NE}(X_{0})_{Z}. Therefore

∥x_{2}∥ < ∥x_{1}∥

(note that \overline{NE}(X_{0}) ∩ −\overline{NE}(X_{0}) = \{0\} since X_{0} is Kähler). After finitely many steps we reach ∥x_{n}∥ = 0, so x_{n} = 0, which means that the family in step n − 1 does not split.

(b) If generic conic bundles come into to the game, it seems at first glance that they cause difficulties because we then pass from x to 2x. However this difficulty disappears if X_{0} carries only finitely many generic conic bundle structures. Indeed, X_{0} can carry at most one generic conic bundle structure X_{0}⟶Z_{0}, because a second one would produce another covering family of rational curves on Z_{0} so that Z_{0} would be algebraic, hence X_{0} would be algebraic. The claim on the Picard number is clear, since by construction a curve C is contracted by ψ_{0} iff C ≡ aC_{s}. Finally, ψ_{0} extends to ψ_{t} for t near 0, because our non-splitting family of rational giving rise to ψ_{0} extends to non-splitting families on the X_{t}, therefore the claim follows rather easily from (0.4).

3.6 Addendum In cases (a), (b),(c.1) and (d) the space Z_{0} is smooth. In particular Z_{0} is a Kähler surface in (a) and (b), and the curves contracted by ψ_{0} form an extremal ray in \overline{NE}(X_{0}). In the case of (c.1) it is classical that Z_{0} is Kähler, too, hence we have the same conclusion.

It is clearly important to know whether Z_{0} is Kähler also in the remaining cases.

3.7 Theorem Let X be a (smooth) compact Kähler threefold, φ : X⟶Y be a birational morphism contracting the irreducible divisor E to a point p. Let N_{E} be its normal bundle. Assume that (E, N_{E}) is one of the following : (P_{2}, O(−2)), (P_{1} ×P_{1}, O(−1, −1)), (Q_{0}, O(−1)). In case E = P_{1} ×P_{1}, assume also that s × P_{1} ≡ P_{1} × t in X_{0} for all s, t. Then Y is a normal Kähler space.

3.8 Explanations (1) A reduced complex space X is Kähler (cp. Grauert [Gr62]) if there is a Kähler form ω on the smooth part regX such that the following holds. For every singular point x ∈ X there exists an open neighborhood U ⊂ X, an open set V ⊂ C^{N}, a biholomorphic map f : U⟶A ⊂ V onto a closed subspace A and a Kähler form ω′ on V with

f^{∗}(ω′)|regX ∩ U = ω.

(2) Let us explain (3.7) in case where X is projective. Then Y is projective, too. In fact, let L be an ample line bundle on X. Then we can write L|E = O_{E}(−kE) with some k > 0 (in case E = P_{2} we eventually substitute L by L^{⊗2}). Note that k exists in case E = P_{1} × P_{1} because of our assumption s × P_{1} ≡ P_{1} × t. Now there exists a line bundle L′ on Y with

L ⊗ O_{X}(kE) ≃ φ^{∗}(L′),

and it is immediately clear that L′ is ample .

(3) Assume X projective and E = P_{1}×P_{1} with s×P_{1} ≢ P_{1}×t. We want to describe the situation; this is of course well known. Consider the blow-down φ_{i} : X⟶Y_{i} contracting E along the two projections to curves C_{i} ⊂ Y_{i}. Then C_{i} ≃ P_{1} with normal bundle N_{C_{i}} = O(−1) ⊕ O(−1). So we obtain birational maps ψ_{i} : Y_{i}⟶Y contracting exactly the C_{i}. Then Y is projective if and only if φ is the contraction of an extremal face which just means that the face generated by the fibers of φ_{i}, i = 1, 2 is extremal. On the other hand there are examples where Y is not projective.

Proof of 3.7 (I) In a first step we construct a semi-positive closed (1,1)-form ω′ on X which is positive on X \ E such that ω′|E = 0. (I.1) Fix a Kähler form ω on X. As in the projective case (3.8(2)) we can choose λ > 0 such that

[ω] + λc_{1}(O_{X}(E))|E = 0  (∗)

in H^{2}(E, R). Fix a representative η of c_{1}(O_{X}(E)). We note that if η′ is another representative of c_{1}(O_{X}(E)), then we have η − η′ = \overline{∂}ρ, but we have even more, namely

η − η′ = ∂\overline{∂}h  (+)

with a C^{∞}−function h on a neighborhood U of E. This is completely standard: since \overline{∂}∂ρ = 0, ∂ρ is a holomorphic 2-form, therefore by the holomorphic d− Poincaré lemma (note that H^{1}(U, C) = H^{1}(E, C) = 0 choosing U such that E is a retract of U) we can write ∂ρ = dφ with a holomorphic 1-form φ. Now put ρ̃ = ρ − φ; then \overline{∂}ρ̃ = η − η′ and ∂ρ̃ = 0. Hence ρ̃ = ∂h by Dolbeault (note H^{1}(U, O_{U}) = 0 if U is strongly pseudo-convex) and (+) follows.

(I.2) Let D denote the space of bidimension (1,1)-currents on X. Let W ⊂ D be the subspace of those closed with suppT ⊂ E. Obviously W is closed. Following [HL83] we let P ⊂ D be the closure of the cone of the positive (1,1)-currents T. Let B ⊂ D be the subspace given by the (1, 1)-parts of the currents of the form dS. Letting κ : D⟶D/W = D̃ be the projection, we set P̃ = κ(P), B̃ = κ(B). Then we claim :

P̃ ∩ B̃ = {0}.  (∗∗)

In fact, let 0 ≠ T ∈ D with κ(T) ∈ P̃ ∩ B̃. We may assume that T ∈ P. Then there are currents T′ and S with suppT′ ⊂ E, dT′ = 0 such that

T = T′ + (dS)_{(1,1)}.

It follows that ∂\overline{∂}T = 0. Decompose T = χ_{E}T + χ_{X\E}T. A difficulty arising is that T is in general only ∂\overline{∂}−closed and not d−closed. Let us first prove (**) in the simpler case dT = 0. Then by [Sk82] we have d(χ_{E}T) = 0. Now observe that T′(ω + λη) = 0 by (+) because we can choose η in such a way that ω + λη = 0 in a neighborhood of E (see I.3). Hence we get

T (ω + λη) = 0.

This is still independent on theclosedness assumption. By the same reason as for T′, we also have χ_{E}T (ω + λη) = 0. Here we have used dχ_{E}T = 0, so that we can take η as we want. Hence χ_{X\E}T (ω + λη) = 0. On the other hand, χ_{X\E}T (ω) ≥ 0 and by (3.7.a), also χ_{X\E}T (η) ≥ 0, because this does not depend on the choice of η ! Therefore χ_{X\E}T = 0 which was to be proved. The difficulty in the general case is to prove

∂\overline{∂}χ_{E}T = 0.  (A)

Once we know (A) we can conclude as before; since (A) implies that χ_{E}T (η) (and hence χ_{X\E}T (η)) does not depend on the choice of η. Clearly (A) will follow from

∂\overline{∂}χ_{E}T ≥ 0,  (B)

where ≥ is to be understood in the sense of currents (just apply (B) to a constant non-zero function).

In order to prove (B) we make use of the following theorem on currents. (C) Let Ω ⊂ C^{n} be an open set, E ⊂ Ω be a complex submanifold and T a positive current with ∂\overline{∂}T ≥ 0 (called plurisubharmonic in the literature). Then ∂\overline{∂}χ_{E}T ≥ 0. This follows from Bassanelli’s paper, [Ba94,1.24,3.5], as was pointed to me by L.Alessandrini [Al96]. Now (C) implies our claim (B) in case E is smooth. In case E is singular, i.e. a quadric cone with vertex x_{0}, we need an additional argument. By (B) we certainly know that

∂\overline{∂}χ_{E}T |X \ \{x_{0}\}

is positive, so does its trivial extension (∂\overline{∂}χ_{E}T)^{0}. Since

χ_{E}T = (χ_{E}T |X \ {x_{0}})^{0}

(note χ_{{x_{0}}}T = 0, [Ba94,1.13]), we obtain from [AB93,5.11]

∂\overline{∂}χ_{E}T = (∂\overline{∂}χ_{E}T)^{0} + λT_{x_{0}},

where λ ≤ 0 and T_{x_{0}} is the Dirac distribution of x_{0}. We want to prove λ = 0. Take a smooth closed (2,2)-form u ≤ 0 such that the current T + u is negative:

T + u ≤ 0.

E.g. consider the Kähler form ω and let u = −kω ∧ ω for some large k. Having in mind χ_{E}(T + u) = χ_{E}T and

∂\overline{∂}χ_{E}(T + u) ≥ 0

on X \ {x_{0}}, we can apply [AB93] and obtain

∂\overline{∂}χ_{E}(T + u) = (∂∂χ_{E}(T + u)|X \ \{x_{0}\})^{0} + μT_{x_{0}}

with μ ≥ 0. Therefore we conclude λ = μ = 0 and we get ∂\overline{∂}χ_{E}T = 0 also in the case of the quadric cone. (I.3) To finish the proof of (**), we finally show that we can choose a representative of ω + λη, which is 0 on U. For this one needs to solve the equation

ω + λη = −\overline{∂}ρ

on U which leads to ask for the injectivity of

H^{1}(U, Ω_{U}^{1})⟶H^{1}(E, Ω_{E}^{1}).

This follows immediately from

H^{1}(E, N_{E}^{∗}) = H^{1}(E, N_{E}^{∗μ} ⊗ Ω_{U}^{1}) = 0, μ ≥ 1.

Note that these arguments also work in the case E is the quadric cone, we leave the details to the reader.

(I.4)) Having verified (**) we find by the Hahn-Banach theorem a linear functional Φ̃ : D̃⟶R which is strictly positive on P̃ \ 0 and 0 on B̃. Lift Φ̃ to a functional Φ : D⟶R. Then Φ is given by a C^{∞}(1, 1)−form ω′ with the following properties.

(a) dω′ = 0

(b) ω′|X \ E is positive, so ω′ itself is semipositive

(c) [ω′|E] = 0 in cohomology.

(b) and (c) together give ω′|E = 0.

(II) We next claim that ω′ is induced by a (1, 1)−form ω_{0} on Y and which is almost a Kähler metric on Y. Of course, ω_{0} exists on Y \ p, where p = φ(E). We show that there is an open neighborhood V of p and a plurisubharmonic function g on V such that ω_{0} = ∂\overline{∂}g on V \ p. Then ω_{0} exists as form on Y. However it might not quite be a Kähler form on Y since g is possible not strictly plurisubharmonic at p. But then take a closed (1, 1)−form λ on Y which is positive at p and let ω̃ = ω_{0} + ϵλ. For sufficiently small ϵ, ω̃ will be a Kähler form on Y. So it remains to prove the existence of g. For this we need to construct a C^{∞}−function f on a suitable neighborhood U of E such that f|U \ E is strictly plurisubharmonic and ω′ = ∂\overline{∂}f on U \ E. Then automatically f = φ^{∗}(g). Let H denote the sheaf of plurisubharmonic functions on X. Taking real parts of holomorphic functions, we have an exact sequence

0⟶R⟶O_{X}⟶H⟶0  (S).

Let U be a strongly pseudo-convex neighborhood of E, such that E is a deformation retract of U. By [HL83] we have

H^{1}(U, H) ≃\frac{\{ψ ∈ A_{R}^{1,1}(U)|dψ = 0\}}{∂\overline{∂}A_{R}^{0,0}(U)},

where A_{R}^{p,q} is the sheaf of real (p, q)−forms. Therefore ω′|U defines a class [ω′] = 0 ∈ H^{1}(U, H_{U}) and we must show [ω′] = 0. Since ω′|E = 0, we have [ω′|E] = 0 in H^{1}(E, H_{E}), and so it is sufficient to verify that the restriction map

H^{1}(U, H_{U})⟶H^{1}(E, H_{E})  (R)

is an isomorphism. By the exact sequence (S) on U resp. E and the obvious facts

H^{i}(U, O_{U}) ≃ H^{i}(E, O_{E}) = 0, i = 1, 2

we have

H^{1}(U, H_{U}) ≃ H^{2}(U, R),
H^{1}(E, H_{E}) ≃ H^{2}(E, R).

Hence (R) follows from the fact that

rest : H^{2}(U, R)⟶H^{2}(E, R)

is an isomorphism, E being a deformation retract of U. So [ω′|U] = 0 in H^{1}(U, H_{U}), and ω′|U = ∂\overline{∂}f with f ∈ A^{0,0}(U). Since ω′ is positive on X \ E, the function f is strictly plurisubharmonic on U \ E.

3.7.a Sub-Lemma Let X be a compact Kähler manifold, D an irreducible divisor on X. Let T be a positive closed current on X of bidegree (n − 1, n − 1), where n = dimX. Assume χ_{D}T = 0. Then the intersection number T · D ≥ 0.

Proof. I am indepted to Jean-Pierre Demailly for communicating to me the following short proof using his approximation theorem for positive closed curents [De92]. We can write D (the current of integration over D) as a weak limit of smooth closed forms Θ_{ε} in the same cohomology class as [D] such that

Θ_{ε} ≥ −λ_{ε}u − O(ε)ω,

where ω is a positive (1,1)-form, u a suitable semi-positive (1,1)-form on X depending on the global structure of X and (λ_{ϵ}) a decreasing family of non-negative smooth functions (0 < ε < 1) converging pointwise to 0 on X \ D and to the multiplicity m(D, x) on D. It follows

D · T = Θ_{ε} · T ≥ − ∫_{X} λ_{ε}u ∧ T − O(ε).

Now the monotone convergence theorem gives convergence to 0, because χ_{D}T = 0.

3.7.b Remark T ·D can be computed either by [T]·[D], where [T] and [D] are the classes in H^{2}(X, R) resp. H^{2n−2}(X, R) or by representing D by a positive closed (1, 1)−form (the curvature form of a metric on the line bundle associated to D), say η, and computing T (η).

If merely ∂\overline{∂}T = 0, then we still can define a class [T], as explained detailed in the proof of (3.11), and everything in (3.7.a) remains true (the regularisation is applied to the current D and not to T!)

3.7.c Remark Theorem 3.7 should be true in a more general context : Let X be a compact Kähler manifold, f : X⟶Y a birational morphism to a normal complex space. Assume that the exceptional set of f is an irreducible divisor E and that ρ(X) = ρ(Y) + 1. Then Y is Kähler. The proof should be the same as before except that one needs a singular version of Bassanelli’s theorem [Ba94,3.5] which must be true.

3.9 Corollary Let φ : X⟶Y be a contraction of type (c). If ρ(X) = ρ(Y) + 1, then Y is Kähler and R = R_{+}[l] ⊂ \overline{NE}(X) is an extremal ray, where l is any curve contracted by φ. If conversely R is an extremal ray, then obviously ρ(X) = ρ(Y)+1 and Y is Kähler.

Proof. (a) If ρ(X) = ρ(Y) + 1, then the case E = P_{1} × P_{1}, dimφ(E) = 0, s × P_{1} ≢ P_{1}×t is excluded, hence 3.7 applies and it is immediately checked that R is extremal.

(b) If R is an extremal ray then the same arguments apply.

3.10 Remarks (1) The case E = P_{1} × P_{1}, dimφ(E) = 0, s × P_{1} ≢ P_{1} × t is not relevant for us, since then the associated contraction is not defined by an extremal ray and not given by one non-splitting family of rational curves, cp. (3.8).

(2) Clearly the proof of (3.7) also works at least in the case where X is a compact Kähler manifold of dimension n and where φ : X⟶Y is an extremal contraction contracting a smooth divisor to a point.

It remains to treat birational contractions of type (d), i.e. blow-ups of smooth curves. Already in case X_{0} is projective, Y_{0} will not be projective “in general”. This happens exactly if the defining ray R is extremal. In order to generalise this to the Kähler case we consider the closed dual cone \overline{NA}(X) to the Kähler cone. So \overline{NA}(X) ⊂ H^{4}(X, R) resp. H^{2,2}(X). In general, \overline{NE}(X) is a proper subcone of \overline{NA}(X) and we can view \overline{NA}(X) as the cone generated by the classes of positive closed currents of bidegree (2, 2).

3.11 Proposition Let X be a compact Kähler threefold, φ : X⟶Y a birational map of type (d), i.e. X is the blow-up of a smooth curve C in the manifold Y. Let E = φ^{−1}(C) and l ⊂ E a ruling line. Then Y is Kähler if and only if R = R_{+}[l] is extremal in \overline{NA}(X).

Proof. (a) Assume that Y is Kähler. In order to show that l is extremal, we take positive closed currents T_{1}, T_{2} with [l] = [T_{1}] + [T_{2}]. Then 0 = φ_{∗}(T_{1}) + φ_{∗}(T_{2}), and since Y is Kähler, we have for a Kähler form ω on Y that φ_{∗}(T_{i})(ω) = 0, hence φ_{∗}(T_{i}) = 0. Therefore the T_{i} are supported on fibers of φ, i.e. the T_{i} are linear combinations of currents “integration over a fiber” and therefore [T_{i}] ∈ R.

(b) Conversely, assume that R is extremal in \overline{NA}(X). By [HL83] it is sufficient to show the following: (*) if T is a positive current of bidegree (2,2) with T = ∂\overline{S} + \overline{∂}S, then T = 0. For the proof of (*) write

T = χ_{C}T + χ_{Y \C}T.

If dT = 0, then by [Si74] χ_{C}T = λT_{C} where T_{C} is the current “integration over C.” In the general case this is due to Bassanelli [Ba94]. Putting T̃ = χ_{Y \C}T, we can write λ[T_{C}] + [T̃] = 0. Let us first assume dT = 0. Then T̃ defines canonically a current T′ on X by taking the trivial extension of T |X \E. By [Sk82], T′ is closed. Letting C_{0} ⊂ E = P(N_{C|Y}^{∗}) be a section with minimal self-intersection, we obtain :

λ[T_{C_{0}}] + [T′] = μ[l].

Obviously μ > 0, since X is Kähler. Since l is extremal, we must have T′ = 0, i.e. T̃ = 0, and λ = 0, hence T = 0. Now we treat the general case. We define T̃ and T′ as before. We prove that ∂\overline{∂}T′ = 0, the analogous statement for T̃ being completely parallel. By [Ba94,3.5], we have

∂\overline{∂}T′ = (∂\overline{∂}T |X \ E)^{0} + R,

where ( )^{0} denotes again the trivial extension and R is a negative current supported on E. Choosing a negative closed (2,2)-form u on X as in the proof of (3.7) so that T′ + u ≤ 0 we have furthermore by loc.cit.

∂\overline{∂}(T′ + u) = (∂\overline{∂}(T + u)|X \ E)^{0} + R′ = (∂\overline{∂}T |X \ E)^{0} + R′,

where R′ is a positive current supported on E. In total we get R = R′ = 0, hence ∂\overline{∂}T′ = 0. It is clear that we still have the equation

λ[C] + [T̃] = 0.

Next we define a class [T′] ∈ H_{R}^{2,2}(X) ad hoc in the following way. By duality we define instead a linear form

[T′] : H_{R}^{1,1}(X)⟶R,

by setting [T′](α) = T′(η), where η is any ∂− ( = \overline{∂}− closed representative, since the class is real) of α. Then there is still an equation

λ[T_{C_{0}}] + [T′] = μ[l]

as before. Cupping with [ω] gives μ ≥ 0 and in fact μ > 0, if T ≠ 0. Noting [T′] ∈ \overline{NA}(X), the extremality of l yields λ = 0, T′ = 0, so T = 0.

Remark Of course one would expect that a stronger version of 3.11 holds, namely that Y is Kähler if and only if l is extremal in \overline{NE}(X).

We go back to our special situation and assume that Z_{0} is not Kähler in our contraction ψ : X_{0}⟶Z_{0}, which blows up the curve C_{0} ⊂ Z_{0}. Then the fibers of ψ_{0} deform into X_{t} and we obtain a family ψ_{t} : X_{t}⟶Z_{t} for small t. Every ψ_{t} is a blow-down of a divisor E_{t} which fit together to a divisor E. In principle we would like to show that by chosing φ_{t_{0}} at the beginning carefully, Y_{0} must be Kähler. However we do not know how to do this at the moment. Therefore we examine a simpler situation, namely that X_{t_{ν}} has only one extremal ray, so that there is no choice. In that case the extremal rays should converge to an extremal ray on X_{0}, i.e. Y_{0} should be Kähler. Let us still simplify the situation, namely that all X_{t} are projective and hence that all Y_{t} are projective, t ≠ 0. Then still we cannot conclude that Y_{0} is Kähler because the Kähler cone on X_{0} might be smaller than the Kähler cone of the nearby fibers. At least we can state the following proposition which seems to be of independent interest.

3.12 Proposition Let π : X ⟶Δ be a family of compact Kähler threefolds, E ⊂ X a family of ruled surfaces over Δ and φ = (φ_{t}) : X ⟶Y be the simultaneous blow-down of E to the manifold Y⟶Δ. Let h^{1,1}(X_{0}) = 2. Assume that there is a sequence t_{ν} converging to 0 such that the Y_{t_{ν}} are Kähler. Then Y_{0} is Kähler (hence all Y_{t_{ν}} are Kähler for small t.)

Proof. Choose a family (ω_{t}) of Kähler metrics on (X_{t}). Assuming Y_{0} to be non-Kähler and letting C_{t} = φ_{t}(E_{t}), we find a positive ∂\overline{∂}−closed current T̃ such that

C_{0} + T̃ = ∂\overline{S} + \overline{∂}S,

compare (3.11). We define its class [T̃] ∈ H_{R}^{2,2}(Y_{0}) as in (3.11). Then [C_{0}]+[T̃] = 0. Now consider the current φ_{0∗}(ω_{0}); it defines a class in H_{R}^{1,1}(Y_{0}), and we obtain

([C_{0}] + [T̃]) · [φ_{0∗}(ω_{0})] = 0.

Let T′ be the trivial extension of T̃|X_{0} \ E_{0} to X_{0}. By the same arguments as in (3.11), we have ∂\overline{∂}T′ = 0 and define its class [T′] ∈ H_{R}^{2,2}(X_{0}). Let l ⊂ E be a fiber of φ_{0}. Then

[T′] = (φ_{0})^{∗}[T̃] + λ[l],

with λ ≤ 0, by application of (3.7.a,b). Therefore from

[T′] · [ω_{0}] = [T̃] · [φ_{0∗}(ω_{0})] + λ[l] · [ω_{0}]

we obtain [T̃] · [φ_{0∗}(ω_{0})] ≥ 0, hence

[C_{0}] · [φ_{0∗}(ω_{0})] ≤ 0.  (∗)

Of course the same considerations hold on Y_{t}. But now [φ_{t_{ν}} (ω_{t_{ν}})] is represented by a Kähler form, since h^{1,1}(Y_{t_{ν}}) = 1. Therefore

[C_{t_{ν}}] · [φ_{t_{ν}} (ω_{t_{ν}})] > 0.

By continuity we obtain from (∗) that

[C_{0}] · [φ_{0∗}(ω_{0})] = 0.

Since h^{1,1}(Y_{0}) = 1, this means [C_{0}] = 0 in H^{4}(Y_{0}, R). Since [C_{t}] is an integer point, we conclude that [C_{t}] = 0 for all t, contradiction.

3.13 Remark If in (3.12) we have h^{1,1}(Y_{t}) ≥ 2, then we cannot conclude because we don’t necessarily have

[C_{t}] · [φ_{t∗}(ω_{t})] > 0.

Certainly we can choose some Kähler form ω′ on X_{t} with this property, but this ω′ might not fit into a family (ω_{t}) of Kähler metrics on all of X . So it seems not impossible that (3.12) does not hold in general. Unfortunately (3.12) does not give anything in our special situation, because

h^{1,1}(X_{0}) = 2

forces X_{0} to be projective :

3.14 Proposition Assume that in (3.12) all the X_{t_{ν}} are projective. Then all Y_{t_{ν}} are projective and X_{0} is projective.

Proof. Assume X_{0} non projective, equivalently Y_{0} non projective (3.12). Since h^{1,1}(Y_{t_{ν}}) = 1, we have

Pic(Y_{t_{ν}})/torsion ≃ Z.

Fix a big generator O_{Y_{tν}} (1) of Pic(Y_{t_{ν}})/≡. Then

K_{Y_{tν}} ≡ O_{Y_{tν}} (α)

for some α ∈ Z. If α > 0, then Y_{t_{ν}} is of general type and so does Y_{0}. Hence X_{0} is projective. If α < 0, then κ(−K_{Y_{tν}}) = 3 and so does κ(−K_{Y_{tν_{0}}} ). The conclusion is as before. So we are left with α = 0. Then χ(O_{Y_{tν}}) = 0 by Riemann-Roch. X_{0} being non-algebraic by assumption, we have H^{2}(O_{X_{0}}) ≠ 0. Since h^{3}(O_{Y_{tν}}) = h^{0}(K_{Y_{tν}}) ≤ 1, we conclude that the irregularity q(Y_{t_{ν}}) = h^{1}(O_{Y_{tν}}) ≥ 1. Hence we have a non-trivial Albanese map

α : Y_{t_{ν}} ⟶Alb(Y_{t_{ν}}).

Since κ(Y_{t_{ν}}) = 0, α is surjective [Be83]. This immediately contradicts h^{1,1}(Y_{t_{ν}}) = 1.

But now an examination of the proof of (3.14) shows that it demonstrates at the same time

3.15 Proposition Assume the situation of (3.14) but instead of an assumption on h^{1,1} assume ρ(X_{t_{ν}}) = 2 for some ν and moreover that X_{0} is not projective. Then all Y_{t} are tori, in particular Y_{0} is Kähler.

In fact, with the arguments in the proof of (3.14) we arrive at the Albanese map α : Y_{t_{ν}} ⟶Alb(Y_{t_{ν}}) which is onto and has connected fibers. Since ρ(Y_{t_{ν}}) = 1, dim Alb(Y_{t_{ν}}) = 3 and α does not contract any divisor. But since Alb(Y_{t_{ν}}) is smooth, α cannot contract any curve either. So Y_{t_{ν}} = Alb(Y_{t_{ν}}) (alternatively apply directly the decomposition theorem [Be83]). Hence our claim follows easily.

Some parts of the arguments of this section do not need the condition on algebraic approximability. Indeed we have

3.16 Theorem Let X be a compact Kähler threefold with κ(X) ≥ 0. Assume that there exists a rational curve C ⊂ X such that K_{X} · C < 0. Then there exists a contraction φ : X⟶Y as described in (3.4)

Proof. As already explained, we construct from the curve C a non-splitting family (C_{t}) of rational curves. Then we can apply again the main theorem of Part 1 and are through. It was necessary to assume that κ(X) ≥ 0 in order to exclude the case −K_{X} · C_{t} = 1 and the C_{t} fill up a non-normal surface (in which case we only have informations if X can be approximated algebraically or if κ(X) = −∞).

4. An Openness Theorem

In this section we shall prove

4.1 Theorem Let π : X ⟶Δ be an algebraic approximation of the non-algebraic compact Kähler threefold X_{0}. Let (t_{ν}) → 0 be a sequence such that X_{t_{ν}} is projective for every ν. If K_{X_{0}} is not nef, then all K_{X_{tν}} are not nef for ν ≫ 0.

We will also prove that on threefolds with κ(X) = 2 and K_{X} not nef, there exists a rational curve C with K_{X} · C < 0 (theorem 4.10 ). If κ(X) = 1, then we can at least prove that there is some curve C with K_{X} · C < 0 (theorem 4.15).

4.2 Remark If X = (X_{t}) is a family of smooth projective threefolds such that K_{X_{0}} is not nef, then all K_{X_{t}} are not nef. This follows e.g. by deforming extremal rational curves in X_{0} to X_{t}. For details and the higherdimensional context see [Pe95a].

The proof of (4.1) will be given in several steps according to the possible values of the Kodaira dimension. We fix a sequence (t_{ν}) as above and assume that K_{X_{tν}} is nef for all ν.

4.3 Proposition We have κ(X_{t}) ≥ 0 for all t ∈ Δ.

Proof. Assume that κ(X_{t_{0}}) = −∞ for some t_{0} ∈ Δ. If X_{t_{0}} is projective, then X_{t_{0}} is uniruled [Mo88],[Mi88],[Ka92], hence all X_{t} are uniruled, contradiction. So let X_{t_{0}} be non-projective; again we want to show that X_{t_{0}} is uniruled which ends the proof (this proof works also in the projective case, making the argument independent from the deep papers cited above). Since K_{X_{tν}} is nef, we have

χ(X_{t_{ν}} , O_{X_{tν}}) ≤ 0

by [Mi87]. Therefore χ(O_{X_{t_{0}}} ) ≤ 0. Since h^{3}(O_{t_{0}}) = h^{0}(K_{X_{t_{0}}} ) = 0 by our assumption, we conclude that the irregularity

q(X_{t_{0}}) = h^{1}(O_{X_{t_{0}}} ) ≥ 1.

So we have a non-trivial Albanese map

α : X_{t_{0}}⟶Alb(X_{t_{0}}).

If dim α(X_{t_{0}}) = 1, then by C_{3,1} (see [Ue87]) the general fiber F has κ(F) = −∞, hence X_{t_{0}} is uniruled. If dim α(X_{t_{0}}), then by C_{3,2} [Ue87], the general fiber of α is P_{1} and again X_{t_{0}} is uniruled. If finally dim α(X_{t_{0}}) = 3, then κ(X_{t_{0}}) ≥ 0, a contradiction.

Next we deal with two easy cases. Observe first that κ(X_{t_{ν_{0}}} ) does not depend on ν_{0} since by [Ka92]

κ(X_{t_{ν_{0}}} ) = max {m ∈ N ∪ {0}|K_{X_{tν}}^{m} ≢ 0}.

4.4 Proposition (1) If K_{X_{tν}} ≡ 0 for some ν, then K_{X_{0}} ≡ 0 and X_{0} is up to finite etale cover either a torus or a product of an elliptic curve with a K3 surface.

(2) If K_{X_{tν}}^{3} > 0 (i.e. K_{X_{tν}} is big) for some ν, then κ(X_{0}) = 3 and X_{0} is projective.

Proof. (1) Just notice that c_{1}(X_{t_{ν}}) = 0 in H^{2}(X_{t_{ν}} , Q) implies that c_{1}(X_{0}) = 0 in H^{2}(X_{0}, Q). Hence we conclude by [Be83].

(2) By Grauert-Riemenschneider or Kawamata-Viehweg vanishing, we have for q ≥ 1 that

H^{q}(X_{t_{ν}} , mK_{X_{tν}}) = 0  (∗)

for all ν and all m ≥ 2. By the coherence of R^{q}π_{∗}(mK_{X}), equation (*) holds for all t ∈ Δ \ S, with S ⊂ Δ a proper analytic subset. Passing to a local 1-dimensional submanifold of Δ through 0, we may assume that (*) holds for all t ≠ 0. By Riemann-Roch we conclude

χ(X_{t}, mK_{X_{t}}) = h^{0}(X_{t}, mK_{X_{t}}) ∼ m^{3}

for t ≠ 0, and this function in t is constant. By semi-continuity of cohomology it follows

h^{0}(X_{0}, mK_{X_{0}}) ∼ m^{3},

so that X_{0} is Moishezon. Since X_{0} is Kähler, it must be projective, contradiction.

We are therefore reduced to the intermediate cases κ(t_{ν}) = 1 or 2. These cases are much more complicated and we shall need some preparations. The first lemma is probably well-known but apparently never stated explicitly.

4.5 Lemma Let X be a compact Kähler manifold and L a line bundle on X. Then L is nef if and only if c_{1}(L) is in the closure of the Kähler cone.

Proof. (a) First assume that L is nef. Take an arbitrary Kähler metric ω on X. Then we must show that c_{1}(L) + [ω] is represented by a Kähler metric. For 0 < ε < 1 choose a metric h_{ε} on L with curvature

Θ_{L,h_{ε}} ≥ −ϵω.

Then Θ_{L,h_{ε}} +ω ≥ (1−ε)ω, hence Θ_{L,h_{ε}} +ω is a Kähler form representing c_{1}(L)+[ω].

(b) Now suppose that c_{1}(L) is in the closure of the Kähler cone. Fix a Kähler metric ω on X. Let ε > 0. By [Su88] we find a Kähler metric ω_{ε} such that

[ω_{ε} − εω] = c_{1}(L).

Choose a metric h_{ε} on L with curvature Θ_{L,h_{ε}} = ω_{ε} −εω. Then Θ_{L,h_{ε}} ≥ −εω. Since ε > 0 was arbitrary, L is nef.

4.6 Lemma Let X be a smooth compact Kähler surface or threefold and A ⊂ X a compact submanifold. Let π : X̂⟶X be the blow-up of A. Let L be a line bundle on X. Then L is nef if and only if π^{∗}(L) is nef.

Proof. For simplicity of notations we only treat the threefold case. One direction being obvious, we assume that π^{∗}(L) is nef. Let KC(X) and KC(X̂) denotes the Kähler cones on X resp. X̂. By the easy part of (4.5) we have

c_{1}(π^{∗}(L)) ∈ KC(X̂).

We claim that this implies

c_{1}(L) ∈ KC(X).  (1)

Then the proof is finished by applying the difficult part of (4.5). For the proof of (1) we need to show that

T · c_{1}(L) ≥ 0  (2)

for every positive closed current of bidegree (2, 2). We know that T̂ · c_{1}(π^{∗}(L)) ≥ 0 for all closed positive currents T̂ on X̂ of bidegree (2, 2). Therefore the proof of (2) is reduced to the following.

(3) For every closed positive current T on X of bidegree (2, 2) there exists a closed positive current T̂ on X̂ such that

π_{∗}(T̂) = T.

(Note that the dual cone to the Kähler cone is the cone of positive closed (1,1)- currents). Write

T = χ_{A}T + χ_{X\A}T.

If dimA = 0, then χ_{A}T = 0, otherwise χ_{A}T = λT_{A} with λ ≥ 0. Now let T′ = T |X \ A. Identifying X \ A with X̂ \ Â, we can consider the trivial extension T_{1} of T′ on X̂. By [Sk82] this is a closed positive current and obviously

π_{∗}(T_{1}) = χ_{X\A}T.

We are thus reduced to dealing with χ_{A}T and may assume dimA = 1. But then we take a section

C ⊂ Â = P(N_{A|X}^{∗})

and let T_{2} = λT_{C}. Hence π_{∗}(T_{2}) = χ_{A}T and we are done, putting T̂ = T_{1} + T_{2}.

Remark. Of course (4.6) should be true in any dimension.

Lemma 4.6 enables us to define nefness on any reduced compact complex space of dimension at most three

4.7 Definition Let X be a reduced compact complex space, dimX ≤ 3. Let L be a line bundle on X. Then L is nef, if there exists a desingularisation π : X̂⟶X such that π^{∗}(L) is nef.

By (4.6) the definition does not depend on the choice of the resolution.

4.8 Problem There is another natural way to define nefness for a line bundle L on a reduced compact complex space X : we require that for any ε > 0 there exists a metric h_{ε} on L such that the curvature satisfies Θ_{L,h_{ε}} ≥ −εω on the smooth part of X, where ω is a fixed positive (1,1)-form on X. These both notions of nefness should coincide.

The next lemma will be very important for the main results of this section; it is of course trivial in the projective case.

4.9 Lemma Let X be a compact Kähler manifold, L a line bundle on X. Assume that there is an effective divisor D on X such that L = O_{X}(D). If L|D is nef, then L is nef.

Proof. Note that the Kähler cone is the dual cone of the cone of of classes of positive closed currents of bidegree (n − 1, n − 1) on X. So by (4.5) we only need to show

∫_{X} c_{1}(L) ∧ T ≥ 0  (∗)

for every positive closed (n − 1, n − 1)− current T. Of course we may assume that D is a reduced prime divisor. By Skoda [Sk82], the current χ_{D}T is closed, hence we obtain from (*) :

∫_{X} c_{1}(L) ∧ T = ∫_{X} c_{1}(L) ∧ χ_{D}T + ∫_{X} c_{1}(L) ∧ χ_{X\D}T.  (∗∗)

Appyling (4.6) to an embedded resolution of singularities for D, we may assume D to be smooth from the beginning. Then it is an easy exercise to conclude

∫_{X} c_{1}(L) ∧ χ_{D}T ≥ 0

(extend metrics on L|D to a neighborhood).

Now the second term in (**) is also non-negative by (3.7.a), hence (*) follows and the claim is proved.

4.10 Theorem Let X be a compact Kähler threefold with κ(X) = 2. If K_{X} is not nef, there exists a rational curve C with K_{X} · C < 0.

Proof. Let m ≫ 0 so that V = |mK_{X}| defines a rational map

f : X ⇀ Y

to a projective surface Y. We choose a sequence of blow-ups π : X̂⟶X such that the induced map f̂: X̂⟶Y is holomorphic. We note that f is almost holomorphic, i.e. there is a non-empty Zariski open set U ⊂ X such that f|U is holomorphic and proper. In fact, otherwise the exceptional set of π would contain a component lying surjectively over S and therefore X would be projective by [Ca81].

Let D_{0} ∈ |mK_{X}| be a general element. By (4.9) K_{X}|D_{0} is not nef. Observe that V must have a positive dimensional base locus, otherwise K_{X} would be nef. We claim that V must even have fixed components:

otherwise the general D_{0} is irreducible. It follows from (4.14) below (applied to a desingularisation) that there is a curve C ⊂ D_{0} with K_{X} ·C < 0. This is impossible since then C would deform in X_{0} in a 1-dimensional family [Ko91] to fill up D_{0} which is clearly absurd.

Hence V has fixed components, D_{0} is reducible and we can write

D_{0} = B + ∑_{i} λ_{i}A_{i}

where B is the movable part of D_{0}, hence irreducible and with multiplicity 1, and the A_{i} are the fixed components.

(4.10.1) Our first aim is to show that there must be an irreducible curve C ⊂ X (not necessarily rational) with K_{X} · C < 0. By the arguments proving the existence of the fixed components it follows also that we can find some i_{0} such that K_{X}|A_{i_{0}} is not nef (apply the above argument for B instead of D_{0}). Moreover it suffices in order to get a contradiction to show that A_{i_{0}} has a non-constant meromorphic function and that the case (4.13(2)) does not occur. Let A = A_{i_{0}} for simplicity of notations, and let μ = λ_{i_{0}}. We can choose π : X̂⟶X such that the strict transform Â of A is smooth. We investigate the structure of Â by distinguishing cases according to dimf̂(Â).

(a) dimf̂(Â) = 0.

Let F̂ denote the fiber of f̂containing Â, equipped with the natural structure. Write F̂ = λÂ + R. Then the conormal bundle N_{F̂|X̂}^{∗}|Â is generated by global sections, but is clearly not trivial, hence

κ(N_{F̂|X̂}^{∗}|Â) ≥ 1,

in particular a(A) ≥ 1. Therefore we only need to exclude the case (4.13(2)) to conclude. So let ĥ : Â⟶Ĉ be the algebraic reduction of Â and assume

π^{∗}(K_{X}|A) ≡ ĥ^{∗}(Ĝ)

with Ĝ^{∗} nef and not numerically trivial, hence ample. Clearly ĥ descends to a map h : A⟶C with g : Ĉ⟶C being the normalisation. Then K_{X}|A ≡ h^{∗}(G) with G^{∗} ample on C. Write

mK_{X} = D_{0} = μA + M.

Then by “adjunction” we have

K_{μA}|A = mK_{X}|A − M′,

where M′ is effective on A and supported on M ∩A. On the other hand by ordinary adjunction

K_{μA}|A = K_{X}|A + N_{μA|X}|A,

hence in total N_{μA|X}^{∗}|A = N_{A}^{∗μ} is effective by the ampleness of G and effectivity of M′. Therefore from K_{A} = K_{X}|A + N_{A} we see that κ(K_{A}) = −∞. Since KÂ is a subsheaf of π^{∗}(K_{A}), it follows κ(Â) = −∞ and Â is algebraic, contradiction.

(b) dimf̂(Â) = 1. Now it is obvious that a(Â) ≥ 1. The remaining case (4.13(2)) is excluded as in (a).

(c) dimf̂(Â) = 2.

Then f̂|Â is onto Y. However Y is algebraic and so does A. Then our claim is obvious. (4.10.1) is thus completely proved.

(4.10.2) We next claim that there is a rational curve C ⊂ X such that K_{X} ·C < 0. If such a rational C does not exist then we find an irrational curve C ⊂ X with K_{X} · C < 0, hence C deforms (with irreducible parameter space) in an at least 1-dimensional family (C_{t})_{t∈T} [Ko91] with C = C_{0}. Moreover we may assume that no deformation of C splits. We choose T maximal (but of course irreducible, as always). Then ⋃_{t∈T} C_{t} is a fixed component of V, say A. Write a general element of |mK_{X}| as

D_{0} = m_{1}A + R

so that R does not contain A. Then by adjunction

m_{1}K_{A} = m_{1}K_{X}|A + m_{1}A|A ⊂ m_{1}K_{X}|A + D_{0}|A = (m_{1} + m)K_{X}|A.

Therefore we obtain the following basic inequality

K_{A} · B ≤ (1 + \frac{m}{m_{1}} )K_{X} · B < K_{X} · B  (I)

for all but finitely many curves B ⊂ A with K_{X} · B < 0. Hence

K_{A} · C_{t} ≤ −2.  (∗)

Let ν : Ã⟶A be the normalisation of A and π : Â⟶A the minimal desingularisation. Then we have

KÃ = ν^{∗}(K_{A}) − E

with E an effective Weil divisor supported exactly on the preimage of the nonnormal locus of A, cp. [Mo82]. Moreover

KÂ = π^{∗}(KÃ) − E′

with an effective divisor E′ which is 0 if and only if Ã has only rational double points. Let t ∈ T be general and Ĉ_{t} be the strict transform of C_{t} in Â. Then (*) yields

KÂ · Ĉ_{t} ≤ −2.  (∗∗)

The general fiber F̂ of f̂ (which is an elliptic curve) induces a unique maximal family (F̂_{t})_{t∈T̂} with graph

q : C⟶T̂, p : C⟶X̂

(taking closure in the cycle space). The map p is clearly bimeromorphic, f being almost holomorphic, and q is an elliptic fibration. Of course we may assume T̂ smooth. Let \overline{A} denote the strict transform of Â in C; we may assume \overline{A} smooth (by possibly choosing an embedded resolution of singularities and then flattening q). Let g = q|\overline{A} : \overline{A}⟶D ⊂ T̂. By (**) we have

κ(Â) = −∞.

Note also that D is a rational or elliptic curve.

(a) Let us first assume that the genus g(C_{t}) ≥ 2 (= genus of the normalisation). Our aim is to construct a new family of elliptic or rational curves (C′_{t}) such that K_{X} · C′_{t} < 0. Let Ĝ_{t} = F̂_{t} ∩ \overline{A} for t ∈ D. Since f is almost holomorphic, we have

p^{∗}π^{∗}K_{X} · F̂_{t} = 0.

Now take a component Ĉ ⊂ F̂_{t} with dim C = dim π ∘ p(Ĉ) = 0. If

p^{∗}π^{∗}K_{X} · Ĉ > 0,

then we find another Ĉ′ ⊂ F̂_{t} with dim p ∘ π(Ĉ′) = 1 such that

p^{∗}π^{∗}(K_{X} · Ĉ′) < 0.

Let C′ = π ∘ p(Ĉ′). Then K_{X} · C′ < 0 and we have found a family of elliptic or rational curves (C′_{t}) with K_{X} · C′_{t} < 0. Therefore we may assume K_{X} · C = 0 for anychoice of Ĉ ⊂ F̂_{t}, t ∈ D and C = π ∘ p(Ĉ). We shall assume now that the general G_{t} is elliptic, the case that G_{t} is rational being even easier (and therefore omitted). We consider the possibly not relatively minimal fibration

g : \overline{A}⟶D ≃ P_{1}.

Furthermore we have a “ruling” h : \overline{A}⟶E, so that the general fiber of h is a smooth rational curve and every fiber is a tree of P_{1}. Let

L = p^{∗}π^{∗}(K_{X}).

Then L·C = 0 for every component of every fiber of g. Since f is almost holomorphic, we even have

π^{∗}p^{∗}(K_{X}) · F̂_{t} = OF̂_{t}

for general t. From semi-continuity it follows then easily that L|C = O_{C} for all C, possibly up to a torsion line bundle (in case C appears with multiplicity in the fiber). After eventually passing from L to L^{m} we get

L ≡ g^{∗}(O_{D}(a))

for some integer a. Now let C_{t} be a general element of our family (C_{t}) and let \overline{C}_{t} be its strict transform in \overline{A}. Then

L · \overline{C}_{t} < 0.

Since \overline{C}_{t} is a multi-section of g, we conclude a < 0. Now consider a general fiber l of h. Then we conclude L · l < 0, and therefore

K_{X} · πp(l) < 0.

Therefore the images of l define the family C′_{t} of rational curves.

(b) We are now reduced to the case that (C_{t})_{t∈T} is a family of elliptic curves and moreover we may assume that the family is non-splitting in the sense that no component of any member is a rational curve (but multiples of elliptic curves are allowed). We take T of course maximal and assume T normal. Denote by D the normalisation of the graph of the family with projections p_{1} : D⟶A and q_{1} : D⟶T. First assume dimT ≥ 2. Note for the following that T is algebraic ( = Moishezon) since A is algebraic and A = ⋃ C_{t}. Choose a general point x ∈ A and introduce

T (x) = {t ∈ T |x ∈ C_{t}}.

Then dimT (x) ≥ 1. Choose an irreducible curve Δ ⊂ T (x). Let p_{Δ} : D_{Δ}⟶A be the normalisation of the induced graph with projection q_{Δ} : D_{Δ}⟶Δ. Then q_{Δ} admits a multi-section Z with dim p_{Δ}(Z) = 0, so that Z ⊂ D_{Δ} is exceptional. This contradicts Sublemma (4.10.a). We are therefore left with the case dimT = 1. Since K_{A} is a subsheaf of K_{X}|A, the inequality

K_{X} · C_{t} ≤ −1

implies

K_{A} · C_{t} ≤ −1.

For general t, let Ĉ_{t} be the strict transform of C_{t} in Â, as before. Then

KÂ · Ĉ_{t} ≤ −1.

If KÂ · Ĉ_{t} ≤ −2, then Ĉ_{t} moves in an at least 2-dimensional family; hence we are reduced to

KÂ · Ĉ_{t} = −1.

Therefore we conclude that the general C_{t} will not meet the non-normal locus of A. Since not all C_{t} pass through a fixed point, we see that (4.10.a) that

C_{t} ∩ SingA = ∅

for general t. Therefore C_{t} is a Cartier divisor in A and we can consider L = O_{A}(C_{t}). Since dim T = 1, we have h^{0}(A, L) = 2. Since |L| is base point free, it defines a holomorphic map α : A⟶P_{1} with C_{t} being a fiber. This contradicts C_{t}^{2} = 1. The proof of (4.10) is now complete modulo the following

4.10.a Sublemma Let S be a smooth projective surface and f : S⟶C a holomorphic elliptic submersion onto a smooth curve C. Then there is no irreducible curve A ⊂ S with A^{2} < 0.

Proof. Let g be the genus of C. By adjunction

degK_{A} = K_{S} · A + A^{2}.

The canonical bundle formula for elliptic fibrations (see e.g. [BPV84,chap.5, 12.1 and chap.3, 18.3]) gives K_{S} ≡ f^{∗}(K_{C}). Let d = degf|A. Then

K_{S} · A = d(2g − 2).

Let ν : Ã⟶A be the normalisation of A; then degKÃ ≤ degK_{A}. Now the formula of Riemann-Hurwitz yields

degKÃ = d(2g − 2) + degR,

where R is the ramification divisor of Ã⟶C. Putting things together we obtain

degK_{A} = d(2g − 2) + A^{2} ≥ d(2g − 2) + degR,

hence degR ≤ A^{2} < 0, which is absurd.

4.11 Corollary Let X be a compact Kähler threefold with κ(X) = 2. Assume that K_{X} is not nef. Then there exists a birational extremal contraction f : X⟶Y (i.e. of type (c) or (d) in (3.4)).

Proof. By (4.10) exists a rational curve C ⊂ X_{0} with K_{X_{0}} · C < 0; then apply (3.16).

(4.12) In particular we have proved (4.1) in case K_{X_{0}}^{3} = 0, K_{X_{0}}^{2} ≠ 0. In fact, if K_{X_{0}} would not be nef, then we find a rational curve C ⊂ X_{0} with K_{X_{0}} · C < 0 and which can be deformed to X_{t}. In fact, taking over the notations of the last proof, we conclude that in all cases but the case of P_{2} with N_{S} = O(−2) we conclude that

N_{C_{t}|X} ≃ ⨁ O(a_{i})

with a_{i} ≥ −1, so that the deformations of C_{t} in X are unobstructed. Moreover

h^{0}(X , N_{C_{t}|X}) > h^{0}(X_{0}, N_{C_{t}|X_{0}}),

so that the C_{t} can be deformed to the neighbouring fibers X_{s}, so that K_{X_{s}} cannot be nef. In the remaining case we see by a similar argument that we can deform directly the P_{2} out of X_{0} and are also done.

In the proof (4.10) we made use of the following

4.13 Proposition Let S be a smooth compact Kähler surface with algebraic dimension a(S) = 1. Let f : S⟶A be the algebraic reduction to the smooth curve A. Let L be a line bundle on S. Let ≡ denote topological equivalence. Then

(1) L is nef if and only if L ≡ f^{∗}(G) with G nef

(2) L is not nef if and only if either (a) there is a curve C ⊂ S with L · C < 0 (b) L ≡ f^{∗}(G′), G^{′∗} ample.

Proof. (1) One direction being obvious, we let L be nef. First we show that L · C = 0 for every curve C ⊂ S. In fact, assume L · C > 0 for some C. Then

(kL + C)^{2} = k^{2}L^{2} + 2kL · C + C^{2}.

Since L^{2} ≥ 0, we obtain by [DPS94] (kL + C)^{2} > 0 for k ≫ 0, hence S would be algebraic. Note that if L ≡ f^{∗}(G), it is clear that G must be nef (otherwise we obtain a nef line bundle whose dual has a section with zeroes which is impossible by [DPS94]). So assume now L · C = 0 for all curves C. By (4.6) we may assume S minimal.

(a) First let κ(S) = 1. Since L · K_{S} = 0, Riemann-Roch gives

χ(S, mL) = χ(S, O_{S}) > 0.

Hence h^{0}(mL) > 0 or h^{0}(mL^{∗} + K_{S}) > 0. In the first case we conclude that mL|F = O_{F} for every fiber of the algebraic reduction f, hence mL = f^{∗}(G). In the second case we can write

K_{S} = mL + D

with D effective. Since D · F = 0, we get D|F = O_{F} , hence

λL = f^{∗}(G).

(b) Now let κ(S) = 0. Then either S is a K3 surface or S is a torus. If S is a K3 surface, then χ(S, O_{S}) = 2 and we can conclude as in (a). So let S be a torus.

Then f is an elliptic fiber bundle over an elliptic curve. The line bundle L defines a section

s ∈ H^{0}(C, R^{1}f_{∗}(O_{S}^{∗})).

Fix a point y_{0} ∈ C. Then we find a topologically trivial line bundle H on S such that

L|f^{−1}(y_{0}) ⊗ H|f^{−1}(y_{0}) ≃ O.

Since S is not algebraic, we automatically have

L|f^{−1}(y) ⊗ H|f^{−1}(y) ≃ O

for all y ∈ C; otherwise we would get a multisection of f. This implies our claim.

(2) Again one direction is clear. So let L be not nef and assume L ≢ f^{∗}(G′). Then by the proof of (1), there is a curve C_{0} with L · C_{0} ≠ 0. If our claim would be false, then L · C_{0} > 0 and L · C ≥ 0 for every curve C ⊂ S. Let F bethe general fiber of f. Then

L · F = 0,

otherwise (L + kF)^{2} > 0 for large k. Since dimf(C_{0}) = 0, we therefore find C_{1} ⊂ f^{−1}f(C_{0}) with L · C_{1} < 0, contradiction.

4.14 Corollary Let X be a non-algebraic compact Kähler surface. Let L be a line bundle on X such that L · C ≥ 0 for all curves C ⊂ X. Assume moreover that some power L^{m} has a section. Then L is nef.

Proof. (a) First we assume that a(X) = 1 and let f : X⟶C be the algebraic reduction which is an elliptic fibration. Assume that L is not nef. By (4.13) we find (posssibly after passing to a multiple of L) a topologically trivial line bundle G on X such that

L = f^{∗}(H) ⊗ G

with some negative line bundle H on C. We may assume that already L has a section. We conclude that

H^{0}(C, H ⊗ f_{∗}(G)) ≠ 0.  (∗)

Note that f_{∗}(G) is a torsion free sheaf, hence locally free. Since G is topologically trivial, we have f_{∗}(G) ≠ 0 if and only if G|F = O_{F} for the general fiber F of f. On the other hand f_{∗}(G) must be non-zero by (*). Moreover (*) proves that f_{∗}(G) and hence G has a section (we may a priori assume that H^{∗} is effective). Now this section cannot have zeroes and therefore G = O_{X}. But then (*) gets absurd.

(b) Assume now that a(X) = 0. Let π : X⟶X′ be the map to the minimal model. We can write, L being effective,

L = ∑ λ_{i}E_{i} + π^{∗}(L′)

where the E_{i} are the exceptional components of π and λ_{i} ≥ 0. Since L is algebraically nef, it is clear that all λ_{i} = 0. So we may assume X minimal and X is either a torus or K3. On the torus we conclude that L = O and in the K3 case we write L = ∑ a_{i}C_{i}. But the C_{i} are all exceptional as well as their union so that again we have a_{i} = 0. (The only curves in a K3 surface without meromorphic functions are (−2)−curves).

4.15 Theorem Let X be a smooth compact Kähler threefold with κ(X) = 1. Assume K_{X} is not nef. Then there exists an irreducible curve C ⊂ X such that K_{X} ·C < 0.

Proof. We follow the lines of (4.10.1). We assume that K_{X} is algebraically nef and show that K_{X} is nef. Let

f : X ⇀ C

be the meromorphic map defined by |mK_{X}|. Let π : X̂⟶X be a sequence of blow-ups such that the induced map f̂: X̂⟶C is a morphism. Write again

mK_{X} = B + ∑ λ_{i}A_{i},

where B is the movable part. Then mK_{X}|B is effective. Hence by (4.14) we conclude that K_{X}|B is nef. It only remains to show that K_{X}|A_{i} is nef for all i. Fix some i and let A = A_{i}. If a(A) ≥ 1, the arguments of (4.10.1) work. So assume a(A) = 0. Let Â be the strict transform of A in X̂; we may assume Â smooth. Let F̂ denote the fiber of f̂ containing Â and write

F̂ = λÂ + R.

It follows from the exact sequence

0⟶O ⊕ O = N_{F̂|X̂}^{∗}|Â⟶N_{λÂ|X̂}^{∗}|Â⟶N_{λÂ|F̂}^{∗}|Â⟶0

that

H^{0}(Â, detN ∗λ ; Â) ≠ 0.

Since κ(Â) = 0 (Â being bimeromorphic to a torus or a K3-surface), it follows from the adjunction formula that

H^{0}(Â, λKX̂|Â) ≠ 0.

Let g : Ã⟶A be the normalisation of A, then π|Â = g ∘ h with h : Â⟶Ã the induced map. Hence

H^{0}(Â, g^{∗}K_{X} − λ ∑ μ_{i}C_{i}) ≠ 0,

where

KX̂ = π^{∗}(K_{X}) + ∑ μ_{i}E_{i}

and C_{i} = Â ∩ C_{i}. Then we obtain by applying h_{∗} that

H^{0}(Ã, g^{∗}(λK_{X})) ≠ 0.

Now apply (4.14)!

4.16 Theorem Let X be a compact Kähler threefold with κ(X) = 1. Assume that K_{X} is not nef and X is algebraically approximable. Then there exists a rational curve C ⊂ X with K_{X} · C < 0.

Proof. If K_{X_{tν}} is not nef for some ν, then the assertion is clear (sect. 3). So assume K_{X_{tν}} nef. We have already seen that this implies K_{X}^{2} = 0. This is the only conclusion we draw from the algebraic approximability. Write as in (4.15)

mK_{X} = B + ∑_{i=1}^{k} λ_{i}A_{i}.

Let L = O_{X}(B) and let f : X ⇀ S be the meromorphic map defined by H^{0}(X, L) to the curve S.

(1) First let us assume that f is holomorphic, i.e. H^{0}(X, L) is base point free. Then

dimf(A_{i}) = 0

for all i. In fact, consider the general fiber F of f. Then K_{F} = K_{X}|F and from K_{X}^{2} = 0 we get K_{F}^{2} = 0. Since κ(F) = 0, the fiber F must be minimal, hence K_{F} ≡ 0. Now if dimf(A_{i}) = 1 for some i, then mK_{X}|F would be non-zero effective, contradiction. By K_{X}^{2} = 0 we obtain

(∑ λ_{i}A_{i})^{2} = 0.

All A_{i} being contained in fibers of f, this implies that ∑ λ_{i}A_{i} ≡ ρF which is impossible (i.e. kK_{X} is generated by global sections).

(2) Now assume that f is not holomorphic. Let π : X̂⟶X be a sequence of blowups such that f̂ : X̂⟶S is holomorphic. Therefore we obtain some exceptional divisor E for f̂ such that dimf̂(E) = 1. It follows that a(F̂) ≤ 1 for the general fiber F̂ of f̂, otherwise X̂ would be algebraically connected, hence projective [Ca 81]. Assume first that a(F̂) = 0 and observe B = π(F̂). Taking another section in H^{0}(X, mK_{X}), we obtain D ∈ |mK_{X}|B| with D^{2} = 0. Hence π^{∗}(D)^{2} = 0. But on a (blown up) torus or K3 surface without meromorphic functions there is no such effective non-zero divisor. If however a(F̂) = 1, then, arguing in the same way, we deduce that (π|F̂)^{∗}(D) consists of multiples of fibers of the algebraic reduction

h : F̂⟶C.

Since h has no multi-sections, there is a map h′ : B⟶C′ to a possibly non-normal curve C′ (with normalisation C) such that D consists of multiples of fibers of h′. Hence kK_{X}|B is generated by H^{0}(X, kK_{X}) for k ≫ 0. This gives a contradiction to dimf(B) = 0.

(4.17) Finally we prove Theorem 4.1 in the only remaining case K_{X_{0}}^{2} ≡ 0 but K_{X_{0}} ≢ 0 by the same reasoning as in (4.12).

Putting together (3.16) and (4.1) we now obtain the Main Theorem for algebraically approximable X as stated in the Introduction. Combining (3.16) with (4.10) and (4.16) we obtain as another part of the Main Theorem:

4.18 Theorem Let X be a smooth compact Kähler threefold with K_{X} not nef. Assume that κ(X) = 1 or 2 and in case κ(X) = 1 assume furthermore that X can be approximated algebraically. Then X carries an extremal contraction.

5. Generic conic bundles

(5.1) In Part 1, (2.11) and (2.12) we had proved the following. Let X be a nonprojective compact Kähler threefold which can be approximated algebraically or assume that X is uniruled. Assume that X admits a non-splitting 1-dimensional family (C_{t}) of rational curves with −K_{X_{t}} · C_{t} = 1, filling up a non-normal surface S. Then the algebraic dimension a(X) = 1, and X has a “generic conic bundle structure” in the following sense. The algebraic reduction is a holomorphic map f : X⟶C to a smooth curve C whose general fiber is an almost homogeneous P_{1}−bundle over an elliptic curve. The surface S is a fiber of f. Moreover there exists an almost holomorphic meromorphic map g : X ⇀ Y to a normal surface Y such that f factorises over g and such that the following holds:

(a) if U = f^{−1}({c ∈ C|f^{−1}(c) is smooth or S = f^{−1}(c)}, then g|U is holomorphic and proper

(b) g|U is a conic bundle

(c) g is a rational quotient of X. The general smooth fiber of g is a P_{1} with normal bundle O ⊕ O and therefore we have a 2-dimensional family (C̃_{t})_{t∈T̃} such that for every t_{1} ∈ T there is a t_{2} ∈ T and a t ∈ T with C_{t_{1}} + C_{t_{2}} = C̃_{t}. In general of course, the map g will not be holomorphic: just perform some birational transformation on a conic bundle. However we shall prove a criterion when g is actually holomorphic. We say that a 2-dimensional family (C̃_{t}) of rational curves splits only in the standard way if the following holds. If C̃_{t_{0}} is a reducible member of the 2-dimensional family, then C̃_{t_{0}} = C′_{t_{1}} + C′_{t_{2}} with smooth rational curves C′_{t_{i}} meeting transversally at one point and with K_{X} · C′_{t_{i}} = −1. So all C̃_{t} are conics. In this terminology we have:

5.2 Theorem Let things be as in the setting (5.1) and assume that the induced family (C̃_{t}) splits only in the standard way. Then Y can be taken smooth so that g is holomorphic and is a conic bundle over Y whose fibers are just the C̃_{t}.

Proof. Let p : C⟶X be the graph of the family (C̃_{t})_{t∈T̃} with projection q : C⟶T̃. Here we consider T̃ reduced. By (5.1) there is a Zariski open set U ⊂ T̃ such that p|q^{−1}(U) is an isomorphism. Hence p is bimeromorphic, and, by considering the normalisation of T̃, it follows that p has connected fibers. Hence C is normal and so does T̃. It now suffices to prove that p is finite, then p is biholomorphic and our claim follows. Assume there exists x ∈ X such that there is a curve B ⊂ T̃ with x ∈ C̃_{t} for all t ∈ B. Then we form the surface

A = ⋃_{t∈B} C̃_{t} = p(q^{−1}(B)).

(a) First assume that no C̃_{t}, t ∈ B splits. Then, A being normal by Part1, (2.3), similar arguments (even easier) as in Part1, (2.1), lead to a contradiction (by the fact that the C̃_{t} can be moved out of A).

(b) So there exists t_{0} ∈ B such that C̃_{t_{0}} splits. By our assumptions the components of the splitting deform in a non-splitting family. Hence it follows from what we have said in (5.1) (i.e. (2.12)) that there is no common point of the C̃_{t}, t ∈ B, since the map g is holomorphic near the reducible conics ( the maps g attached to the various 1-dimensional non-splitting families attached to the full family of conics are of course the same). Hence p is finite, hence biholomorphic. The smoothness of Y comes from deformation theory and the smoothness of X.

5.3 Corollary Assume in (5.1) that the C_{t} define a geometrically extremal ray in \overline{NE}(X). Then g is a conic bundle over a smooth surface Y with a(Y) = 1.

Proof. We have to prove that the family (C̃_{t}) splits only in the standard way. Assume to the contrary that C̃_{t_{0}} = ∑_{i}^{p} C′_{i} with p ≥ 2. Since [C_{t}] is geometrically extremal and since [C̃_{t_{0}}] = 2[C_{t}] ∈ \overline{NE}(X), we have [C′_{i}] ∈ \overline{NE}(X), hence K_{X} ·C′_{i} < 0. Hence the claim is clear.

It is interesting to have a general look at conic bundles over non-algebraic surfaces. The next proposition follows of course from (5.1) and (5.3) but it is instructive to see a direct argument.

5.4 Proposition Let φ : X⟶S be a conic bundle over the Kähler surface S with a(S) = 0 such that ρ(X) = ρ(S) + 1. Then the discriminant locus Δ = ∅, so that φ is an analytic P_{1}−bundle.

Proof. Assume Δ ≠ ∅. By the Kodaira classification S is birationally equivalent to a torus or a K3 surface. In particular all curves in S are smooth rational curves. Now it is a basic fact on conic bundles with ρ(X) = ρ(S) + 1 that every smooth rational component C ⊂ Δ has to meet \overline{Δ \ C} in at least two points (see e.g. [Mi83], the arguments remaining true in the non-algebraic case). This already rules out the torus case. In the K3 case first notice that the above condition implies Δ^{2} = 0. The union of all rational curves, and in particular Δ, is however contractible, therefore Δ^{2} < 0, contradiction.

The same type of argument together with Kodaira’s classification of singular fibers of elliptic surfaces proves

5.5 Proposition Let φ : X⟶S be a conic bundle over a Kähler surface S with a(S) = 1, satisfying ρ(X) = ρ(S) + 1. Let Δ be its discrminant locus. Then Δ = ⋃ F_{i}, where the F_{i} are smooth fibers of the algebraic reduction f : S⟶C (which is an elliptic fibration) or reduction of multiple smooth fibers.

5.6 Proposition Let us assume the situation of (5.5). Then a(X) = 1 and f ∘ φ is an algebraic reduction of X.

Proof. If a(X) ≠ 1, then a(X) = 2. This case is already ruled out in Part 1. The argument is as follows. Let g : X ⇀ Z be an algebraic reduction and F a general fiber of φ. Since g is an “elliptic fibration”, we have dimg(F) = 1. It follows easily that any two points in X can be joined by a chain of curves, i.e. X is algebraically connected. Hence X is Moishezon by [Ca81], contradiction.

6. Minimal Models

(6.1) The weak minimal model conjecture (WMMC) in the Kähler case predicts that every compact Kähler manifold X is either uniruled or birationally equivalent to a Kähler n−fold X′ with at most terminal singularities such that K_{X′} is nef; such an X′ is called minimal model. The abundance conjecture (AC) says that every minimal model is semi-ample, i.e. some multiple mK_{X′} is generated by global sections. A minimal model with semiample K_{X′} is also called good minimal model. The strong minimal model conjecture (SMMC) states that starting from a compact Kähler manifold X one get derive either a Q−Fano fibration or a minimal model by a sequence of birational divisorial contractions or flips.

See [KMM87] for the background in the algebraic case. SMMC holds for algebraic threefolds by [Mo88] and AC by [Mi88] and [Ka92].

(6.2) Both WMMC and AC have been proved by Nakayama [Na88] in case the compact Kähler threefold carries an elliptic fibration, in particular if a(X) = 2.

(6.3) We next describe the structure theorem of Fujiki [Fu83] for non-algebraic non-uniruled compact Kähler threefolds with a(X) ≤ 1.

(a) If a(X) = 1, then either we have a holomorphic algebraic reduction f : X⟶C with the general smooth fiber being a torus or X is birationally (C × S)/G with C a compact Riemann surface, S a torus or a K3-surface with a(S) = 0 and G a finite group acting on both C and S and on C × S by g(x, y) = (gx, gy).

(b) If a(X) = 0, then either X is a Kummer manifold, i.e. X is birationally T/G, where T is a torus and G a finite group or X is simple, i.e. X does not carry a covering family of compact subvarieties and does not admit a meromorphic map to a Kummer manifold.

6.4 Theorem Kummer threefolds with a(X) = 0 have good minimal models.

Proof. Let X be a Kummer threefold with a(X) = 0. So X is birationally Y = T/G with a torus T of algebraic dimension 0. Since T does not carry positive-dimensional subvarieties, Y can have only isolated singularities and these are quotient singularities, since g : T ⟶T/G is unramified outside a finite set. In particular Y has only canonical singularities (see e.g. [KMM87]). By Reid [Re83] there is a partial resolution

f : X′⟶Y

such that X′ has terminal singularities and f is crepant, i.e. K_{X′} = f^{∗}(K_{Y}). Since obviously K_{Y} ≡ 0, X′ is a minimal model. Since K_{T} = g^{∗}(K_{Y}), we even have mK_{Y} = O_{Y} , hence X′ is good.

(6.5) Remark (1) It might also be possible to construct directly a minimal model in case X ∼ (C × S)/G, but the case of the holomorphic torus fibration is certainly harder. Possibly one has to prove SMMC in that case to proceed to a minimal model. See (6.7) for more comments on SMMC.

(2) A consequence of the existence of good minimal models is the non-existence of simple threefolds as in (6.3) defined; see the Motiviation following the Introduction.

We finally show that once we know the existence of good minimal models, then the notions of algebraic nefness and nefness coincide (for threefolds)

6.6 Theorem Let X be a smooth compact Kähler threefold. Assume that X has a good minimal model. Then K_{X} is nef if and only if K_{X} · C ≥ 0 for all curves C ⊂ X.

Proof. Due to (4.10) and (4.15) we could restrict ourselves to the case κ(X) = 0. However we will give a simultaneous proof in all cases making the arguments independent of section 4. One direction being obvious we assume that K_{X} · C ≥ 0 for all curves C ⊂ X. Let

h : X ⇀ X′

be a bimeromorphic map to a good minimal model. Choose a sequence of blow-ups f : X̂⟶X such that the induced map g : X̂⟶X′ is a morphism. Let

D′ ∈ |mK_{X′}|

be a general smooth member and D ⊂ X be its strict transform (in case κ(X) = 0, we have D′ = 0). Then we can write

mK_{X} = D + ∑ μ_{i}A_{i},  (∗)

with mu_{i} > 0. To determine the structure of the A_{i}, we write

KX̂ = g^{∗}(K_{X′}) + ∑ a_{j}F_{j}, a_{j} > 0.

Let D̂ be the strict transform of D in X̂. Then

g^{∗}(D′) + ∑ ma_{j}F_{j} = D̂ + ∑ c_{j}F_{j} ∈ |mKX̂|

with c_{j} ≥ 0. Therefore

D + ∑ c_{j}f(F_{j})_{0} ∈ |mK_{X}|,

where the index 0 indicates that f(F_{j}) is omitted if dimf(E_{j}) ≤ 1. So A_{i} = f(E_{j})_{0} and in particular all A_{i} are algebraic. By (4.9) it is sufficient to show that K_{X}|D is nef and that K_{X}|A_{i} is nef for all i. Since the A_{i} are algebriac, the second statement is clear.

(1) If κ(X) = 0, then D′ = 0, hence D = 0 and there is nothing to prove.

(2) If κ(X) ≥ 1, we note that mK_{X}|D is effective, hence we conclude by (4.14).

6.7 Corollary Assume that WMMC and AC hold in dimension 3. Let X be a compact Kähler threefold with at most terminal singularities. Then K_{X} is nef if and only K_{X} · C ≥ 0 for all curve C ⊂ X.

6.8 Remark The most difficult step in the construction of minimal models for algebraic threefolds is of course the existence of flips. However this is reduced in [Ka88] to a local analytic problem around the rational curves which have to be flipped. [Ka88] shows also that in the analytic this local reduction works. Since Mori proves in [Mo88] the local analytic existence of flips, we have already the existence of flips in the analytic category. The proof of termination is just the same as in the algebraic case.

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TOWARDS A MORI THEORY ON COMPACT KÄHLER THREEFOLDS, II 41

Thomas Peternell, Mathematisches Institut, Universität Bayreuth, 95440

Bayreuth, Germany (peternel@btm8x1.mat.uni-bayreuth.de)