j. differential geometry 63 (2003) 231-277

A KAWAMATA-VIEHWEG VANISHING THEOREM ON COMPACT KÄHLER MANIFOLDS

JEAN-PIERRE DEMAILLY & THOMAS PETERNELL

Abstract

We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kähler space X: if L is a nef line bundle with L^{2} ≠ 0, then H^{q}(X, K_{X} +L) = 0 for q ≥ dim X −1. As an application we complete a part of the abundance theorem for minimal Kähler threefolds: if X is a minimal Kähler threefold, then the Kodaira dimension κ(X) is nonnegative.

0. Introduction

In this paper we establish the following Kawamata-Viehweg type vanishing theorem on a compact Kähler manifold or, more generally, a normal compact Kähler space:

Theorem 0.1. Let X be a normal compact Kähler space of dimension n and L a nef line bundle on X. Assume that L^{2} ≠ 0. Then

H^{q}(X, K_{X} + L) = 0

for q ≥ n − 1.

In general, one expects a vanishing

H^{q}(X, K_{X} + L) = 0

for q ≥ n + 1 − ν(L), where ν(L) is the numerical Kodaira dimension of the nef line bundle L, i.e., ν(L) is the largest integer ν such that L^{ν} ≠ 0.

Received 10/24/2002.

Of course, when X is projective, Theorem 0.1 is contained in the usual Kawamata-Viehweg vanishing theorem, but the methods of proof in the algebraic case clearly fail in the general Kähler setting. Instead we proceed in the following way: Clearly we may assume that X is smooth and by Serre duality, only the cohomology group H^{n−1} is of interest. Take a singular metric h on L with positive curvature current T with local weight function h. By [29, 6] there exists a decomposition

(D)  T = ∑ λ_{j}D_{j} + G,

where λ_{j} ≥ 1 are irreducible divisors, and G is a pseudo-effective current such that G|D_{i} is pseudo-effective for all i. Consider the multiplier ideal sheaf I(h). We associate to h another, “upper regularized” multiplier ideal sheaf I_{+}(h) by setting

I_{+}(h) := lim_{ε→0_{+}} I(h^{1+ε}) = lim_{ε→0_{+}} I((1 + ε)φ).

It is unknown whether I(h) and I_{+}(h) actually differ; in all known examples they are equal. Then in Section 2 the following vanishing theorem is proved:

Theorem 0.2. Let (L, h) be a holomorphic line bundle over a compact Kähler n-fold X. Assume that L is nef and has numerical Kodaira dimension ν(L) = ν ⩾ 0, i.e., c_{1}(L)^{ν} ≠ 0 and ν is maximal. Then the morphism

H^{q}(X, O(K_{X} + L) ⊗ I_{+}(h)) ⟶ H^{q}(X, K_{X} + L)

induced by the inclusion I_{+}(h) ⊂ O_{X} vanishes for q > n − ν.

The strategy of the proof of Theorem 0.2 is based on a direct application of the Bochner technique with special hermitian metrics constructed by means of the Calabi-Yau theorem.

Now, coming back to the principles of the proof of Theorem 0.1, we introduce the divisor

D = ∑[λ_{j}]D_{j}.

Then Theorem 0.2 yields the vanishing of the map in cohomology

H^{n−1}(X, −D + L + K_{X}) ⟶ H^{n−1}(X, L + K_{X}).

Thus we are reduced to show that H^{n−1}(D, L + K_{X}|D) = 0, or dually that

H^{0}(D, −L + D|D) = 0.

This is now done by a detailed analysis of a potential nonzero section in −L + D|D; making use of the decomposition (D) and of a Hodge index type inequality.

The vanishing Theorem 0.1 is most powerful when X is a threefold, and in the second part of the paper we apply 0.1 — or rather a technical generalization — to prove the following abundance theorem:

Theorem 0.3. Let X be a ℚ-Gorenstein Kähler threefold with only terminal singularities, such that K_{X} is nef (a minimal Kähler threefold for short ). Then κ(X) ≥ 0.

This theorem was established in the projective case by Miyaoka and in [27] for Kähler threefolds, with the important exception that X is a simple threefold which is not Kummer. Recall that X is said to be simple if there is no proper compact subvariety through a very general point of X, and that X is said to be Kummer if X is bimeromorphic to a quotient of a torus. So our contribution here consists in showing that such a simple threefold X with K_{X} nef has actually κ(X) = 0. Needless to say that among all Kähler threefolds the simple non-Kummer ones (which conjecturally do not exist) are most difficult to deal with, since they do not carry much global information besides the fact that π_{1} is finite and that they have a holomorphic 2-form.

The first main ingredient in our approach is the inequality

K_{X} · c_{2}(X) ≥ 0

for a minimal simply connected Kähler threefold X with algebraic dimension a(X) = 0. Philosophically this inequality comes from Enoki’s theorem that the tangent sheaf of X is K_{X}-semi-stable when K_{X}^{2} ≠ 0 resp. (K_{X}, ω)-semi-stable when K_{X}^{2} = 0; here ω is any Kähler form on X. Now if this semi-stability with respect to a degenerate polarization would yield a Miyaoka-Yau inequality, then K_{X} · c_{2}(X) ≥ 0 would follow. However this type of Miyoka-Yau inequalities with respect to degenerate polarizations is completey unknown. In the projective case, the inequality follows from Miyaoka’s generic nefness theorem and is based on char. p-methods. Instead we approximate K_{X} (in cohomology) by Kähler forms ω_{j}. If T_{X} is still ω_{j}-semi-stable for sufficiently large j, then we can apply the usual Miyaoka-Yau inequality and pass to the limit to obtain K_{X} · c_{2}(X) ≥ 0. Otherwise we examine the maximal destabilizing subsheaf which essentially (because of a(X) = 0) is independent of the polarization.

The second main ingredient is the boundedness h^{2}(X, mK_{X}) ≤ 1. If K_{X}^{2} ≠ 0, this is of course contained in Theorem 0.1. If K_{X}^{2} = 0, we prove this boundedness under the additional assumption that a(X) = 0 and that π_{1}(X) is finite (otherwise by a result of Campana X is already Kummer). The main point is that if h^{2}(X, mK_{X}) ≥ 2, then we obtain “many” non-split extensions

0 ⟶ K_{X} ⟶ E ⟶ mK_{X} ⟶ 0

and we analyze whether E is semi-stable or not. The assumption on π_{1} is used to conclude that if E is projectively flat, then E is trivial after a finite étale cover.

From these two ingredients Theorem 0.3 immediately follows by applying Riemann-Roch on a desingularization of X.

The only remaining problem concerning abundance on Kähler threefolds is to prove that a simple Kähler threefold with K_{X} nef and κ(X) = 0 must be Kummer.

1. Preliminaries

We start with a few preliminary definitions.

Definition 1.1. A normal complex space X is said to be Kähler if there exists a Kähler form ω on the regular part of X such that the following holds: Every singular point x ∈ X admits an open neighborhood U and a closed embedding U ⊂ V into an open set U ⊂ ℂ^{N} such that there is a Kähler form η on V with η|U = ω.

Remark 1.2. Let X be a compact Kähler space and let f : X̂ ⟶ X be a desingularization by a sequence of blow-ups. Then X̂ is a Kähler manifold. More generally consider a holomorphic map f : X̂ ⟶ X of a normal compact complex space to a normal compact Kähler space. If f is a projective morphism or, more generally, a Kähler morphism, then X̂ is Kähler. For references to this and more information on Kähler spaces, we refer to [31].

A Kähler form ω defines naturally a class [ω] ∈ H^{2}(X, ℝ), see [13] where Kähler metrics on singular spaces were first introduced. Therefore we also have a Kähler cone on a normal variety.

Notation 1.3. Let X be a normal compact complex space.

(1) Let A and B be reflexive sheaves of rank 1. Then we define A ˆ⊗B := (A ⊗ B)^{∗∗}. Moreover we let A^{[m]} := A^{ˆ⊗m}.

(2) A reflexive sheaf A is said to be a ℚ-line bundle if there exists a positive integer m such that A^{[m]} is locally free.

(3) X is ℚ-Gorenstein if the canonical reflexive sheaf ω_{X}, also denoted K_{X}, is a ℚ-line bundle. X is ℚ-factorial, if every reflexive sheaf of rank 1 is a ℚ-line bundle.

Definition 1.4. Let X be a normal compact Kähler threefold.

(1) X is simple if there is no proper compact subvariety through the very general point of X.

(2) X is Kummer, if X is bimeromorphic to a quotient T/G where T is a torus and G a finite group acting on T.

It is conjectured that all simple threefolds are Kummer.

Notation 1.5.

(1) The algebraic dimension a(X) of an irreducible reduced compact complex space is the transcendence degree of the field of meromorphic functions over ℂ. If a(X) = 0, i.e., all meromorphic functions on X are constant, then it is well-known that X carries only finitely many irreducible hypersurfaces.

(2) A line bundle L on a compact Kähler manifold is nef, if c_{1}(L) lies in the closure of the Kähler cone. For alternative descriptions see e.g., [8]. If X is a normal compact Kähler, then L is nef if there exists a desingularization π : X̂ ⟶ X such that π^{∗}(L) is nef. By [26], this definition does not depend on the choice of π.

2. Hodge index type inequalities

We give here some generalizations of Hodge index inequalities for nef classes over compact Kähler manifolds. In this direction the main result is the Hovanskii-Teissier concavity inequality, which can be stated in the following way (see e.g., [7], Prop. 5.2 and Remark 5.3):

Proposition 2.1. Let α_{1}, …, α_{k} and γ_{1}, …, γ_{n−k} be nef cohomology classes on a compact Kähler n-dimensional manifold X. Then

α_{1} ⋯ α_{k} · γ_{1} ⋯ γ_{n−k} ⩾ (α_{1}^{k} · γ_{1} ⋯ γ_{n−k})^{1/k} ⋯ (α_{k}^{k} · γ_{1} ⋯ γ_{n−k})^{1/k}.

We want to derive from these a non vanishing property for intersection products of the form α^{i} · β^{j}. Let us fix a Kähler metric ω on X. By Proposition 2.1 applied with k = i + j and the α_{ℓ}’s being i copies of α followed by j copies of β and γ_{ℓ} = ω, we have

α^{i} · β^{j} · ω^{n−i−j} ⩾ (α^{k} · ω^{n−k})^{i/k} ⋯ (β^{k} · ω^{n−k})^{j/k}.

As all products α^{k} and analogues can be represented by closed positive currents, we have α^{k} ≠ 0 ⇒ α^{k} · ω^{n−k} > 0, hence with k = i + j we find

(2.2)  α^{i+j} ≠ 0 and β^{i+j} ≠ 0 ⟹ α^{i} · β^{j} ≠ 0.

This is of course optimal in terms of the exponents if α = β, but as we shall see in a moment, this is possibly not optimal in a dissymetric situation. Actually, we have the following additional inequalities which can be viewed as “differentiated” Hovanskii-Teissier inequalities:

Theorem 2.3. Let α and β be nef cohomology classes of type (1, 1) on a compact Kähler n-dimensional manifold X. Assume that α^{p} ≠ 0 and β^{q} ≠ 0 for some integers p, q > 0. Then we have α^{i} · β^{j} ≠ 0 as soon as there exists an integer k ⩾ i + j such that

i(k − p)_{+} + j(k − q)_{+} < k,

where x_{+} means the positive part of a number x.

Proof. Assume that α^{i} · β^{j} = 0. We apply the Hovanskii-Teissier inequality respectively with α_{ℓ} = α + εω (i terms), or α_{ℓ} = β + εω (j terms) or α_{ℓ} = ω (k − i − j terms), and γ_{ℓ} = ω. This gives

(α + εω)^{i} · (β + εω)^{j} · ω^{n−i−j} (∗)

⩾ ((α + εω)^{k} · ω^{n−k})^{i/k}((β + ε)^{k} · ω^{n−k})^{j/k}(ω^{n})^{1−i/k−j/k}.

By expanding the intersection form and using the assumption α^{i}·β^{j} = 0, we infer

(α + εω)^{i} · (β + εω)^{j} · ω^{n−i−j} ⩽ O(ε)

as ε tends to zero. On the other hand (α + εω)^{k} · ω^{n−k} is bounded away from 0 if k ⩽ p since then α^{k} ≠ 0, and (α + εω)^{k} · ω^{n−k} ⩾ Cε^{k−p} for some constant C > 0 if k ⩾ p. Hence we infer from (∗) that

Cε^{(i/k)(k−p)_{+}+(j/k)(k−q)_{+}} = O(ε),

and this is not possible if i(k − p)_{+} + j(k − q)_{+} < k. The theorem is proved. q.e.d.

The special case p = 2, q = 1, i = j = 1, k = 2 provides the following result which will be needed later on several occasions:

Corollary 2.4. Assume that α, β are nef with α^{2} ≠ 0 and β ≠ 0. Then α · β ≠ 0.

Finally, we state an extension of Proposition 2.1 in the case when one of the cohomology classes involved is not necessarily nef.

Proposition 2.5. Let α be a real (1, 1)-cohomology class, and let β, γ_{1}, … γ_{n−2} be nef cohomology classes. Then

(α · β · γ_{1} ⋯ γ_{n−2})^{2} ⩾ (α^{2} · γ_{1} ⋯ γ_{n−2})(β^{2} · γ_{1} ⋯ γ_{n−2}).

Proof. By Proposition 2.1, the result is true when α is nef. If we replace β by β + εω and let ε > 0 tend to zero, we see that it is enough to consider the case when β is a Kähler class. Then α+λβ is also Kähler for λ ≫ 1 large enough, and the inequality holds true with α + λβ in place of α. However, after making the replacement, the contributions of terms involving λ in the right and left-hand side of the inequality are both equal to

2λ(α · β · γ_{1} ⋯ γ_{n−2})(β^{2} · γ_{1} ⋯ γ_{n−2}) + λ^{2}(β^{2} · γ_{1} ⋯ γ_{n−2})^{2}.

Hence these terms cancel and the claim follows. q.e.d.

3. Partial vanishing for multiplier ideal sheaf cohomology

Let (L, h) be a holomorphic line bundle over a compact Kähler nfold X. Locally in a trivialization L_{|U} ≃ U × ℂ, the metric is given by ∥ξ∥_{x} = |ξ|e^{−φ(x)} and we assume that the curvature Θ_{h}(L) := \frac{i}{π} ∂\overline{∂}φ is a closed positive current (so that, in particular, L is pseudo-effective). We introduce as usual the multiplier ideal sheaf I(h) := I(φ) where

I(φ)_{x} := {f ∈ O_{X,x}; ∃V ∋ x, ∫_{V} |f(z)|^{2}e^{−2φ(z)} < +∞}

and V is an arbitrarily small neighborhood of x. We also consider the upper regularized multiplier ideal sheaf

I_{+}(h) := lim_{ε→0_{+}} I(h^{1+ε}) = lim_{ε→0_{+}} I((1 + ε)φ).

It should be noticed that I((1+ε)φ) increases as ε decreases, hence the limit is locally stationary by the Noether property of coherent sheaves, and one has of course I_{+}(h) ⊂ I(h). It is unknown whether these sheaves may actually differ (in all known examples they are equal). In any case, they coincide at least in codimension 1 (i.e., outside an analytic subset of codimension ⩾ 2).

Proposition 3.2. Let

Θ_{h}(L) = ∑_{j=1}^{+∞} λ_{j}D_{j} + G

be the Siu decomposition of the (1, 1)-current Θ_{h}(L) as a countable sum of effective divisors and of a (1, 1)-current G such that the Lelong sublevel sets E_{c}(G), c > 0, all have codimension 2. Then we have the inclusion of sheaves

I_{+}(h) ⊂ I(h) ⊂ O (− ∑[λ_{j}]D_{j}) , [λ_{j}] := integer part of λ_{j},

and equality holds on X ∖ Z where Z is an analytic subset of X whose components all have codimension at least 2.

Proof. The decomposition exists by [29] (see also [6]). Now, if g_{j} is a local generator of the ideal sheaf O(−D_{j}), the plurisubharmonic weight φ of h can be written as

φ = ∑ λ_{j} log |g_{j}| + ψ

where ψ is plurisubharmonic and the E_{c}(ψ) have codimension 2 at least. Since ψ is locally bounded from above, it is obvious that

I(φ) ⊂ I(λ_{j} log |g_{j}|) ⊂ O (− ∑[λ_{j}]D_{j}) .

Now, let Y be the union of all sets E_{c}(ψ) (with, say, c = 1/k), all pairwise intersections D_{j} ∩ D_{k} and all singular sets D_{j sing}. This set Y is at most a countable union of analytic sets of codimension ⩾ 2. Pick an arbitrary point x ∈ X ∖ Y. Then x meets the support of ⋃ D_{j} in at most one point which is then a smooth point of some D_{k}, and the Lelong number of ψ_{k} = ψ + ∑_{j≠k} λ_{j} log |g_{j}| at x is zero. Then φ = λ_{k} log |g_{k}| + ψ_{k} and the inclusion

I(h)_{x} ⊃ O (− ∑_{j}[λ_{j}]D_{j})_{x} = O(−[λ_{k}]D_{k})_{x}

holds true by Hölder’s inequality. In fact, for every germ f in O(−[λ_{k}]D_{k})_{x} we have

∫_{V ∋x} |f|^{2} exp ( − (1 + ε)λ_{k} log |g_{k}|) < +∞

for ε > 0 so small that [(1 + ε)λ_{k}] = [λ_{k}], while e^{−ψ_{k}} is in L^{p}(V_{p}) for some V_{p} ∋ x, for every p > 1. Similarly, we have

I_{+}(h)_{x} ⊃ O( − [(1 + ε)λ_{k}]D_{k})_{x} = O( − [λ_{k}]D_{k})_{x}

for ε > 0 small enough. The analytic set Z where our sheaves differ [i.e., the union of supports of I(h)/I_{+}(h) and O( − ∑_{j}[λ_{j}]D_{j})/I(h)] must be contained in Y, hence Z is of codimension ⩾ 2. q.e.d.

The main goal of this section is to prove the following partial vanishing theorem:

Theorem 3.3. Let (L, h) be a holomorphic line bundle over a compact Kähler n-fold X, equipped with a singular metric h such that Θ_{h}(L) ⩾ 0. Assume that L is nef and has numerical dimension ν(L) = ν ⩾ 0, i.e., c_{1}(L)^{ν} ≠ 0 and ν is maximal. Then the morphism

H^{q}(X, O(K_{X} + L) ⊗ I_{+}(h)) ⟶ H^{q}(X, K_{X} + L)

induced by the inclusion I_{+}(h) ⊂ O_{X} vanishes for q > n − ν.

Of course, it is expected that the Kawamata-Viehweg vanishing theorem also holds for Kähler manifolds, in which case the whole group H^{q}(X, K_{X} + L) vanishes and Theorem 3.3 would then be an obvious consequence. However, we will see in Section 4 that, conversely, Theorem 3.3 can be used to derive the Kawamata-Viehweg vanishing theorem in the first nontrivial case ν = 2. Using the same method for higher values of ν would probably be very hard, if not impossible.

Proof. Our strategy is based on a direct application of the Bochner technique with special hermitian metrics constructed by means of the Calabi-Yau theorem.

Let us fix a smooth hermitian metric h_{∞} on L, which may have a curvature form Θ_{h_{∞}}(L) of arbitrary sign, and let ε > 0. Then c_{1}(L)+εω is a Kähler class, hence by the Calabi-Yau theorem for complex Monge-Ampère equations there exists a hermitian metric h_{ε} = h_{∞}e^{−2φ_{ε}} such that

(3.4)  (Θ_{h_{ε}}(L) + εω)^{n} = C_{ε}ω^{n}.

Here C_{ε} > 0 is the constant such that

C_{ε} = \frac{∫_{X}(c_{1}(L) + εω)^{n}}{∫_{X} ω^{n}} ⩾ Cε^{n−ν}.

Let h = h_{∞}e^{−2ψ} be a metric with Θ_{h}(L) ⩾ 0 as given in the statement of the theorem, and let ψ_{ε} ↓ ψ be a regularization of ψ possessing only analytic singularities (i.e., only logarithmic poles), such that

h̃_{ε} := h_{∞}e^{−2ψ_{ε}}

satisfies Θh̃_{ε}(L) ⩾ −εω in the sense of currents. Such a metric exists by the general regularization results proved in [5]. We consider the metric

ĥ_{ε} = (h_{ε})^{δ}(h̃_{ε})^{1−δ} = h_{∞} exp ( − 2(δφ_{ε} + (1 − δ)ψ_{ε}))

where δ > 0 is a sufficiently small number which will be fixed later. By construction,

Θĥ_{ε}(L) + 2εω = δ(Θ_{h_{ε}}(L) + εω) + (1 − δ)(Θh̃_{ε}(L) + εω) + εω
⩾ δ(Θ_{h_{ε}}(L) + εω) + εω.

Denote by λ_{1} ⩽ ⋯ ⩽ λ_{n} and λ̂_{1} ⩽ ⋯ ⩽ λ̂_{n}, respectively, the eigenvalues of the curvature forms Θ_{h_{ε}}(L)+εω and Θĥ_{ε}(L)+2εω at every point z ∈ X, with respect to the base Kähler metric ω(z). By the minimax principle we find λ̂_{j} ⩾ δλ_{j} + ε. On the other hand, the Monge-Ampère equation (3.4) tells us that

(3.5)  λ_{1} … λ_{n} = C_{ε} ⩾ Cε^{n−ν}

everywhere on X. We apply the basic Bochner-Kodaira inequality to sections of type (n, q) with values in the hermitian line bundle (L, ĥ_{ε}). As the curvature eigenvalues of Θĥ_{ε}(L) are equal to λ̂_{j} −2ε by definition, we find

(3.6)  ∥\overline{∂}u∥2 ; ĥ_{ε} + ∥\overline{∂}^{⋆}u∥2 ; ĥ_{ε} ⩾ ∫_{X}(λ̂_{1} + ⋯ + λ̂_{q} − 2qε)|u|2 ; ĥ_{ε}dV_{ω}

for every smooth (n, q)-form u with values in L. Actually this is formally true only if the metric ĥ_{ε} is smooth on X. The metric h_{ε} is indeed smooth, but h̃_{ε} may have poles along an analytic set Z_{ε} ⊂ X. In that case, we apply instead the inequality to forms u which are compactly supported in X ∖ Z_{ε}, and replace the Kähler metric ω by a sequence of complete Kähler metrics ω_{k} ↓ ω on X ∖ Z_{ε}, and pass to the limit as k tends to +∞ (see e.g., [5] for details about such techniques). In the limit we recover the same estimates as if we were in the smooth case, and we therefore allow ourselves to ignore this minor technical problem from now on.

Now, let us take a cohomology class {β} ∈ H^{q}(X, K_{X} ⊗ L ⊗ I_{+}(h)). By using Čech cohomology and the De Rham-Weil isomorphism between Čech and Dolbeault cohomology (via a partition of unity and the usual homotopy formulas), we obtain a representative β of the cohomology class which is a smooth (n, q)-form with values in L, such that the coefficients of β lie in the sheaf I_{+}(h) ⊗_{O_{X}} C^{∞}. We want to show that β is a boundary with respect to the cohomology group H^{q}(X, K_{X} ⊗ L). This group is a finite dimensional Hausdorff vector space whose topology is induced by the L^{2} Hilbert space topology on the space of forms (all Sobolev norms induce in fact the same topology on the level of cohomology groups). Therefore, it is enough to show that we can approach β by \overline{∂}-exact forms in L^{2} norm.

As in Hörmander [15], we write every form u in the domain of the L^{2}-extension of \overline{∂}^{∗} as u = u_{1} + u_{2} with

u_{1} ∈ Ker \overline{∂} and u_{2} ∈ (Ker \overline{∂})^{⊥} = \overline{Im \overline{∂}^{∗}} ⊂ Ker \overline{∂}^{∗}.

Therefore, since β ∈ Ker \overline{∂},

|⟨⟨β, u⟩⟩|^{2} = |⟨⟨β, u_{1}⟩⟩|^{2}

⩽ ∫_{X} \frac{1}{λ̂_{1} + ⋯ + λ̂_{q}} |β|2 ; ĥ_{ε}dV_{ω} ∫_{X}(λ̂_{1} + ⋯ + λ̂_{q})|u_{1}|2 ; ĥ_{ε}dV_{ω}.

As \overline{∂}u_{1} = 0, an application of (3.6) to u_{1} (together with an approximation of u_{1} by compactly supported smooth sections on the corresponding complete Kähler manifold X ∖Z_{ε}) shows that the second integral in the right-hand side is bounded above by

∥\overline{∂}^{∗}u_{1}∥2 ; ĥ_{ε} + 2qε∥u_{1}∥2 ; ĥ_{ε} ⩽ ∥\overline{∂}^{∗}u∥2 ; ĥ_{ε} + 2qε∥u∥2 ; ĥ_{ε},

so we finally get

|⟨⟨β, u⟩⟩|^{2} ⩽ ∫_{X} \frac{1}{λ̂_{1} + ⋯ + λ̂_{q}} |β|2 ; ĥ_{ε}dV_{ω} (∥\overline{∂}^{∗}u∥2 ; ĥ_{ε} + 2qε∥u∥2 ; ĥ_{ε}) .

By the Hahn-Banach theorem (or rather a Hilbert duality argument in this situation), we can find elements v_{ε}, w_{ε} such that

⟨⟨β, u⟩⟩ = ⟨⟨v_{ε}, \overline{∂}^{∗}u⟩⟩ + ⟨⟨w_{ε}, u⟩⟩ ∀u, i.e., β = \overline{∂}v_{ε} + w_{ε},

with

∥v_{ε}∥2 ; ĥ_{ε} + \frac{1}{2qε}∥w_{ε}∥2 ; ĥ_{ε} ⩽ ∫_{X} \frac{1}{λ̂_{1} + ⋯ + λ̂_{q}} |β|2 ; ĥ_{ε}dV_{ω}.

As a consequence, the L^{2} distance of β to the space of \overline{∂}-exact forms is bounded by ∥w_{ε}∥ĥ_{ε} where

∥w_{ε}∥2 ; ĥ_{ε} = ∫_{X} |w_{ε}|_{h_{∞}}^{2}e^{−2(δφ_{ε}+(1−δ)ψ_{ε})}dV_{ω}

⩽ 2qε ∫_{X} \frac{1}{λ̂_{1} + ⋯ + λ̂_{q}} |β|2 ; ĥ_{ε}dV_{ω}.

We normalize the choice of the potentials φ_{ε}, ψ and ψ_{ε} so that

sup_{X} φ_{ε} = 0,  sup_{X} ψ = −1,  −1 ⩽ sup_{X} ψ_{ε} < 0;

in this way φ_{ε}, ψ_{ε} ⩽ 0 everywhere on X (all inequalities can be achieved simply by adding suitable constants). From this we infer

∫_{X} |w_{ε}|_{h_{∞}}^{2}dV_{ω} ⩽ 2 ∫_{X} \frac{qε}{λ̂_{1} + ⋯ + λ̂_{q}} |β|2 ; ĥ_{ε}dV_{ω},

and what remains to be shown is that the right-hand side converges to 0 for a suitable choice of δ > 0. By construction λ̂_{j} ⩾ δλ_{j} + ε and (3.5) implies

λ_{q}^{q}λ_{q+1} … λ_{n} ⩾ λ_{1} … λ_{n} ⩾ Cε^{n−ν},

hence

\frac{1}{λ_{1} + ⋯ + λ_{q}} ⩽ \frac{1}{λ_{q}} ⩽ C^{−1/q}ε^{−(n−ν)/q}(λ_{q+1} … λ_{n})^{1/q}.

We infer

γ_{ε} : = \frac{qε}{λ̂_{1} + ⋯ + λ̂_{q}} ⩽ min (1, \frac{qε}{δλ_{q}} )
⩽ min (1, Cδ^{−1}ε^{1−(n−ν)/q}(λ_{q+1} … λ_{n})^{1/q}).

We notice that

∫_{X} λ_{q+1} … λ_{n} dV_{ω} ⩽ ∫_{X}(Θ_{h_{ε}}(L) + εω)^{n−q} ∧ ω^{q}
= (c_{1}(L) + ε{ω})^{n−q}{ω}^{q} ⩽ C′′,

hence the functions (λ_{q+1} … λ_{n})^{1/q} are uniformly bounded in L^{1} norm as ε tends to zero. Since 1 − (n − ν)/q > 0 by hypothesis, we conclude that γ_{ε} converges almost everywhere to 0 as ε tends to zero. On the other hand

|β|2 ; ĥ_{ε} = |β|_{h_{∞}}^{2}e^{−2(δφ_{ε}+(1−δ)ψ_{ε})} ⩽ |β|_{h_{∞}}^{2}e^{−2δφ_{ε}}e^{−2ψ}.

Our assumption that the coefficients of β lie in I_{+}(h) implies that there exists p′ > 1 such that ∫_{X} |β|_{h_{∞}}^{2}e^{−2p′ψ}dV_{ω} < ∞. Let p ∈ ]1, +∞[ be the conjugate exponent such that \frac{1}{p} + \frac{1}{p′} = 1. By Hölder’s inequality, we have

∫_{X} γ_{ε}|β|2 ; ĥ_{ε}dV_{ω}

⩽ (∫_{X} |β|_{h_{∞}}^{2}e^{−2pδφ_{ε}}dV_{ω})^{1/p} (∫_{X} γ_{ε}^{p′}|β|_{h_{∞}}^{2}e^{−2p′ψ}dV_{ω})^{1/p′} .

As γ_{ε} ⩽ 1, the Lebesgue dominated convergence theorem shows that

∫_{X} γ_{ε}^{p′}|β|_{h_{∞}}^{2}e^{−2p′ψ}dV_{ω}

converges to 0 as ε tends to 0. However, the family of quasi plurisubharmonic functions (φ_{ε}) is a bounded family with respect to the L^{1} topology on the space of (quasi)-plurisubharmonic functions — we use here the fact that the currents

Θh̃_{ε}(L) = Θ_{h_{∞}}(L) + \frac{i}{π} ∂\overline{∂}φ_{ε} ⩾ 0

all sit in the same cohomology class; the boundedness of their normalized potentials then results from the continuity properties of the Green operator. By standard results of complex potential theory, we conclude that there exists a small constant η > 0 such that ∫_{X} e^{−2ηφ_{ε}}dV_{ω} is uniformly bounded; in fact this follows from Prop. 7.1 of [30], p. 389, and from the arguments of its proof. By choosing δ ⩽ η/p, the integral ∫_{X} |β|_{h_{∞}}^{2}e^{−2pδφ_{ε}}dV_{ε} remains bounded and we are done. q.e.d.

4. Kawamata-Viehweg vanishing theorem for line bundles of numerical dimension 2

In this section we prove the Kawamata-Viehweg vanishing theorem for the cohomology group of degree n − 1 of nef line bundles L with L^{2} ≠ 0 on compact Kähler spaces of dimension n. Furthermore we will prove an extended version where L can be a reflexive sheaf. This will be needed for proving the abundance theorem for Kähler threefolds.

Theorem 4.1. Let X be a normal compact Kähler space of dimension n and L a nef line bundle on X. Assume that L^{2} ≠ 0. Then

H^{q}(X, K_{X} + L) = 0

for q ≥ n − 1.

Proof. In a first step we reduce the proof to the case of a smooth space X (this is comfortable but not really necessary; all arguments would also work in the singular setting as well). In fact, let π : X̂ ⟶ X be a Kähler desingularization. Then, assuming our claim in the smooth case, we have

H^{q}(X̂, KX̂ + π^{∗}(L)) = 0.

By the projection formula and the Grauert-Riemenschneider vanishing theorem

R^{j}π_{∗}(KX̂) = 0,

it follows

H^{q}(X, π_{∗}(KX̂) ⊗ L) = 0.

Since π_{∗}(KX̂) ⊂ K_{X} with cokernel supported in codimension at least 2, namely on the singular locus of X, the vanishing claim follows.

So from now on, we assume X smooth. In the case q = n, we have H^{n}(X, K_{X} + L) = H^{0}(X, −L)^{∗} by Serre duality, and for L nef, −L has no section unless L is trivial. Therefore the only interesting case is q = n−1. We introduce a singular metric h on L with positive curvature current T. By [29] and [5, 6] we obtain a decomposition

T = ∑ λ_{j}D_{j} + G,

where λ_{j} ≥ 1 are irreducible divisors, and G is a positive current such that G has Lelong numbers in codimension ⩾ 2 only — so that in particular G|D_{i} is pseudo-effective for all i. Consider the multiplier ideal sheaf I(h). By Proposition 3.2 we have

I_{+}(h) ⊂ I(h) ⊂ O_{X} (− ∑[λ_{j}]D_{j})

with equality in codimension 1. We put

D = ∑[λ_{j}]D_{j}.

We consider the canonical map in cohomology

H^{n−1}(X, −D + L + K_{X}) ⟶ H^{n−1}(X, L + K_{X})

which is vanishing by (3.3). In order to prove our claim it is therefore sufficient to prove

H^{n−1}(D, L + K_{X}|D) = 0.

By Serre duality and the adjunction formula, this comes down to showing

H^{0}(D, −L + D|D) = 0.

Supposing the contrary, we fix a nonzero section

σ ∈ H^{0}(D, −L + D).

We choose p_{1}, …, p_{k} maximal so that

σ ∈ H^{0} (D, −L + D − ∑ p_{j}D_{j}) ,

i.e., we choose D̃ = ∑ p_{j}D_{j} ⊂ D maximal such that σ|D̃ = 0. [In this notation, we view D̃ as the subscheme of X defined by the structure sheaf O_{X}/O_{X}(−D̃)].

Then 0 ≤ p_{i} ≤ [λ_{i}] for all i ∈ I, not all p_{i} = [λ_{i}], and we shall always consider σ as a section of −L + D − ∑ p_{j}D_{j}. Denote

c_{i} = \frac{\{λ_{i}\} + p_{i}}{λ_{i}} .

Then we have 0 ≤ c_{i} ≤ 1. We introduce c = min c_{i} and

I_{0} = {i ∈ I | c_{i} = c}.

Clearly c < 1. Notice that by construction σ|D_{i} ≠ 0 unless c_{i} = 1. Let

E = − (∑({λ_{i}} + p_{i})D_{i}) − G.

Since L = ∑ λ_{i}D_{i} + G, we have

−L + D − ∑ p_{i}D_{i} = − (∑({λ_{i}} + p_{i})D_{i}) − G = E,

so E is effective (possibly zero) on every D_{i} with c_{i} < 1. Since L is nef, also the ℝ-divisor cL = ∑ λ_{i}cD_{i} + cG is nef. Adding this to the divisor E in the last equation, we deduce that

− (∑({λ_{i}} + p_{i} − cλ_{i})D_{i}) − (1 − c)G

is pseudo-effective on every D_{i} with c_{i} < 1. Since {λ_{i}} + p_{i} − cλ_{i} = 0 for i ∈ I_{0}, it follows that

− (∑_{i∉I_{0}}({λ_{i}} + p_{i} − cλ_{i})D_{i}) − (1 − c)G

is pseudo-effective on every D_{i} with c_{i} < 1, in particular for every i ∈ I_{0}. Now D_{j}|D_{i} is effective (possibly 0) for all j ≠ i, and G|D_{i} is always pseudo-effective, hence, having in mind c < 1 and {λ_{i}} + p_{i} − cλ_{i} > 0 for i ∉ I_{0}, we conclude that

(1)  D_{j}|D_{i} ≡ 0

for all (j, i) with j ∉ I_{0} and i ∈ I_{0} and that

(2)  G|D_{i} ≡ 0

for all i ∈ I_{0}. Introducing

D′ = ∑_{i∈I_{0}} λ_{i}D_{i}

and

D′′ = ∑_{i∉I_{0}} λ_{i}D_{i},

we have

L = D′ + D′′ + G

and D′′·D_{i} = G·D_{i} = 0 for all i ∈ I_{0} by (1) and (2). Hence L·D_{i} = D′·D_{i} for i ∈ I_{0}, so that D′|D_{i} is nef, hence D′ is nef by [26]. In total

L · D′ = D′ · D′

and

D′ · D′′ = D′ · G = 0.

As L^{2} ≠ 0 and D′ ≠ 0, Corollary 2.4 implies L · D′ ≠ 0. First recall that

L|D_{i} = ∑_{j∈I_{0}} λ_{j}D_{j}|D_{i}

is nef for i ∈ I_{0}. On the other hand

− ∑_{j∈I_{0}}{λ_{j}} + p_{j})D_{j} = −c ∑_{j∈I_{0}} λ_{j}D_{j}

is of course pseudo-effective on every D_{i} for i ∈ I_{0} (E is effective on those D_{i}). Combining these two facts, we deduce that either c = 0 or that L · D_{i} = 0 for all i ∈ I_{0}, hence L · D′ = 0, contradiction. So we have c = 0. This means p_{j} = 0 and λ_{j} ∈ ℕ for all j ∈ I_{0}.

Claim 4.2. The divisor D′′ + G is nef, and in fact must be equal to zero.

Proof of the claim. We consider the closed positive (1, 1)-current Θ = [D′′] + G. By the results of [26], the proof of nefness of {Θ} just amounts to showing that the restriction {Θ}_{|Z} of the (1, 1)-cohomology class {Θ} to any component Z in the Lelong sublevel set ⋃_{c>0} E_{c}(Θ) is nef. However Z is either a component of D′′ or a component of ⋃_{c>0} E_{c}(G). In the first case, Z is contained in the support of D′′, and as D′ · D′′ = 0, Z must be disjoint from D′. Hence

{Θ}_{|Z} = {D′′ + G}_{|Z} = {L − D′}_{|Z} = {L}_{|Z}

is nef. If Z is a component of ⋃_{c>0} E_{c}(G), then Z has codimension at least 2. Then we know by [6] that the intersection product [D′] ∧ G is well-defined as a closed positive current. Since the cohomology class of this current is zero, we must have [D′] ∧ G = 0. However, we infer from [6] that

ν([D′] ∧ G, z) ⩾ ν([D′], z) ν(G, z) > 0

at every point z ∈ D′ ∩Z, hence Z must also be disjoint from D′ in that case. We conclude as before that {Θ}_{|Z} = {L}_{|Z} is nef. Now we have D′ · (D′′ + G) = 0, with D′, D′′ + G nef and D^{′2} = L · D′ ≠ 0. Hence {D′′ + G} = 0 by Corollary 2.4, and we conclude that D′′ = 0, G = 0 (both [D′′] and [G] being positive currents). q.e.d.

From this we infer L ≡ D′ and I(h) = O_{X}(−D′).

Case 1. We assume that L = D′. Now the sequence

0 ⟶ I(h) ⊗ K_{X} + L ⟶ K_{X} + L ⟶ K_{X} + L|D′ = K_{D′} ⟶ 0

gives in cohomology

0 ⟶ H^{n−1}(X, K_{X} + L) ⟶ H^{n−1}(D′, K_{D′})

≃ H^{0}(D′, O_{D′}) ⟶ H^{n}(K_{X}) = ℂ ⟶ 0.

Thus we need to show

h^{0}(D′, O_{D′}) = 1.

In order to verify this, we first observe that D′ is connected. In fact otherwise write D′ = A+B with A and B effective and A·B = 0. But A and B are necessarily nef, hence the Hodge Index Theorem gives a contradiction to L^{2} = (D′)^{2} ≠ 0. So D′ is connected and if h^{0}(D′, O_{D′}) ≥ 2, then O_{D′} contains a nilpotent section t ≠ 0. Let ∑_{j∈I} μ_{j}D_{j} denote its vanishing divisor (notice that D′ is Cohen-Macaulay!). Then 1 ≤ μ_{j} ≤ λ_{j} for all j. Let

J = \{j ∈ J | \frac{λ_{j}}{μ_{j}} maximal\}

and let c = \frac{λ_{j}}{μ_{j}} be the maximal value. Notice that − ∑_{j∈I} μ_{j}D_{j}|D_{i} is

effective (possibly 0) for all i. First we rule out the case that c = \frac{λ_{j}}{μ_{j}} for all j ∈ I. In fact, then L|D_{i} = c ∑ μ_{j}D_{j}|D_{i} is nef and its dual is effective, hence L|D_{i} ≡ 0 for all i, whence L^{2} = 0, contradiction. Thus we find some j such that

c > \frac{λ_{j}}{μ_{j}} .

By connectedness of D = D′ we can choose i_{0} ∈ J in such a way that there exists j_{1} ∈ I \ J with D_{i_{0}} ∩ D_{j_{1}} ≠ ∅. Now

∑_{j∈I}(λ_{j} − cμ_{j})D_{j}|D_{i_{0}}

is pseudo-effective as a sum of a nef and an effective line bundle (this has nothing to do with the choice of i_{0}). Since the sum, taken over I, is the same as the sum taken over I \ {i_{0}}, we conclude that

∑_{j≠i_{0}}(λ_{j} − μ_{j})D_{j}|D_{i_{0}}

is pseudo-effective, too. Now all λ_{j} − cμ_{j} ≤ 0 and λ_{j_{1}} − cμ_{j_{1}} < 0 with D_{j_{1}} ∩ D_{i_{0}} ≠ ∅, hence the dual of

∑_{j≠i_{0}}(λ_{j} − μ_{j})D_{j}|D_{i_{0}}

is effective nonzero, a contradiction. q.e.d.

Case 2. Now we deal with the case that L ≠ D′. Then we can write

L = D′ + L_{0}

where L_{0}^{m} ∈ Pic^{0}(X) (The exponent m is there because there might be torsion in H^{2}(X, ℤ); we take m to kill the denominator of the torsion part). We may in fact assume that m = 1; otherwise we pass to a finite étale cover X̃ of X and argue there (the vanishing on X̃ clearly implies the vanishing on X). Then the sequence S is modified to

(S) 0 ⟶ I(h) ⊗ (K_{X} + L) ⟶ K_{X} + L ⟶ (K_{X} + L)|D′ = (K_{D′} + L_{0})|D′ ⟶ 0.

Taking cohomology as before, things come down to proving

(∗)  H^{0}(D′, −L_{0}|D′) = 0.

If −L_{0}|D′ ≠ 0, then we see as above that −L_{0}|D′ cannot have a nilpotent section. So if (∗) fails, then −L_{0}|D′ has a section s such that s|redD′ has no zeroes, so that −L_{0}|redD′ is trivial. But then −L_{0}|D′ is trivial. Now let α : X ⟶ A be the Albanese map with image Y. Then L_{0} = α^{∗}(L′_{0}) with some line bundle L′_{0} on A which is topologically trivial but not trivial. Since L_{0}|D′ is trivial, we conclude that α(D′) ≠ Y, and α(D′) is contained in a proper subtorus B of A. Now consider the induced map

β : X ⟶ A/B

and denote its image by Z. Then β(D′) is a point; on the other hand D′ is nef, so that dim Z = 1 and D′ consists of multiples of fibers of β. But this contradicts D^{′2} = L^{2} ≠ 0. q.e.d.

For applications to minimal Kähler 3-folds, 4.1 is still not good enough, because we need to know the vanishing property H^{2}(X, mK_{X}) = 0 on a ℚ-Gorenstein 3-fold (with K_{X}^{2} ≠ 0). We would like to set L = (m − 1)K_{X} to apply 4.1 but this is no longer a line bundle. This difficulty is overcome by:

Proposition 4.3. Let X be a normal ℚ-Gorenstein compact Kähler 3-fold with at most terminal singularities. Let A be a ℚ-line bundle. Suppose A is nef and A^{2} ≠ 0. Then

H^{2}(X, A + K_{X}) = 0.

Proof.

(A) In a first step we show that we may assume X to be ℚ-factorial. (Actually, in our application in Section 5, it will be clear that we may always assume X to be ℚ-factorial, so the reader only interested in the applications may skip (A).)

In fact, if X is not ℚ-factorial, there exists a bimeromorphic map f : Y ⟶ X from a normal ℚ-factorial Kähler space with at most terminal singularities ([18, 4.5′]). Moreover f is an isomorphism in codimension 1 and f is projective since X has only isolated singularities. Now consider the reflexive sheaf

H = f^{∗}(O_{X}(A))^{∗∗}.

Choose a number r such that A^{[r]} is locally free. Then

H^{[r]} = f^{∗}(O_{X}(rA)),

since both sheaves are reflexive and coincide in codimension 1. Thus H is nef (as ℚ-line bundle) with H^{2} ≠ 0. Once we know the result in the ℚ-factorial case, we get

H^{2}(Y, H ˆ⊗K_{Y}) = 0.

So by the Leray spectral sequence, we only have to show

R^{1}f_{∗}(H ˆ⊗K_{Y}) = 0.

This however follows from [19, 1-2-7]. Actually this citation deals with the algebraic case. However first notice that our statement is local around the isolated singularities of X. Now isolated singularities are algebraic by Artin’s theorem, i.e., we can realize an open neighborhood of an isolated singularity as an open set in a normal algebraic variety. So locally on X the map f : Y ⟶ X can be realized algebraically. Now we can approximate H by algebraic reflexive sheaves H_{k} up to high order k and then apply [19, 1-2-7] to get the vanishing R^{1}f_{∗}(H_{k}) = 0. This sheaf coincides with R^{1}f_{∗}(H) to high order, so R^{1}f_{∗}(H) vanishes to high order. For k approaching ∞, we obtain the vanishing we are looking for.

(B) From now on we assume X to be ℚ-factorial. We proceed as in the proof of 4.1. First of all choose r such that A^{[r]} is locally free. Then choose a singular metric h with positive curvature current on A^{[r]}. Now \frac{1}{r}h is a metric at least on A|X_{reg} with positive curvature current T extending to all of X. We argue as in the first part of the proof of 4.1 to obtain the divisor D and the current R, however D is only an integral Weil divisor. By the same arguments as in 4.1 we can still reduce the problem to proving

H^{2}(D, O_{D}(A + K_{X})) = 0.

(Notice that I_{D} ⊗ O_{X}(A + K_{X}) = O_{X}(−D + A + K_{X}) outside a finite set and that by definition O_{D}(A + K_{X}) = O_{X}(A + K_{X})|D.) Now D is Cohen-Macaulay; here we need in an essential way that locally X is the quotient of a hypersurface by a finite group. To be more detailed, we can write locally X = V/G with V a hypersurface singularity and G a finite group (see e.g., [28]). Let π : V ⟶ X be the quotient map and let D̂ = π^{∗}(D). If we can prove that D̂ is Cohen-Macaulay, then D will be Cohen-Macaulay, too, since this property is G-invariant. So we may assume that X = V. Now X is (locally) a compound Du Val singularity [28], i.e., a 1-parameter deformation of a 2-dimensional rational double point. Hence we can find a Cartier divisor H ⊂ X through x_{0} which has just a rational double point at x_{0}. Now consider D ∩ H. This is a Weil ℚ-Cartier divisor on H. Since x_{0} is a quotient singularity of H, we can argue as above to see that D ∩ H is Cohen-Macaulay. Hence D has a hyperplane section through x_{0} which is Cohen-Macaulay. Thus D is Cohen-Macaulay at x_{0} itself.

Therefore we have by Serre duality

H^{2}(D, O_{D}(A + K_{X})) ≃ Hom(O_{D}(A + K_{X}), O_{D}(K_{D})).

Suppose that H^{2} does not vanish. Then we obtain a nonzero homomorphism s : O_{D}(A + K_{X}) ⟶ K_{D}. This s must be generically nonzero. In fact, D is generically Gorenstein. Hence O_{D}(K_{D})_{x} is isomorphic to an ideal in O_{X,x} for all x, in particular K_{D} has no torsion sections, D being Cohen-Macaulay; see [10] in the algebraic case. Let X_{0} be the regular part of X, this means that we eliminate a finite set from X, all singularities being terminal. Let denote D_{0} = D∩X_{0} and let s_{0} = s|X_{0}.

Then by adjunction we have

0 ≠ s_{0} ∈ H^{0}(D_{0}, O_{D}(−A + D)).

From now on we argue as in 4.1 just working on X_{0} instead of X. The only exceptions are calculations of intersection numbers and Hodge index arguments. Here we still need to argue on X — we do not have any problems with singularities since all divisors are ℚ-Cartier. q.e.d.

5. The case K_{X}^{2} = 0, K_{X} ≠ 0

The second ingredient for the proof of the abundance theorem for Kähler threefolds is the following weak analogue of 4.3 in case K_{X}^{2} = 0 (however one should have in mind that we are dealing with a case which does not exist a posteriori).

Theorem 5.1 Let X be a normal compact Kähler threefold with at most terminal singularities such that K_{X} is nef. Suppose that K_{X}^{2} = 0, K_{X} ≠ 0, and that X is simple and not Kummer. Then

h^{2}(X, mK_{X}) ≤ 1

for all m ∈ ℕ.

As already mentioned the essential property derived from X being simple non-Kummer is that π_{1}(X) is finite [3].

5.2 Start of the proof

Using Kawamata’s ℚ-factorialisation theorem (compare with the proof of 4.3), we may assume that X is ℚ-factorial. Suppose h^{2}(X, mK_{X}) ≥ 2. Using Serre duality we get — following Miyaoka and Shepherd-Barron — (many) non-split extensions

0 ⟶ K_{X} ⟶ E ⟶ mK_{X} ⟶ 0 (S)

with reflexive sheaves E of rank 2. We note

(5.2.1)  c_{1}(E) = (m + 1)K_{X}

and

(5.2.2)  c_{2}(E) = 0.

5.3 The unstable case

(5.3.a) Here we will assume that every non-split extension E as in (S) is ω- unstable for some fixed Kähler form ω independent of E. Let A_{S} ⊂ E be the ω-maximal destabilizing subsheaf. Then A_{S} is a ℚ-line bundle and we determine a ℚ-line bundle B_{S} such that E/A_{S} = I_{Z}B_{S} with some subspace Z of codimension at least 2 (actually Z is generically (i.e., on the smooth part of X) locally a complete intersection or finite and supported in SingX). Since K_{X} ≠ 0, we obtain injective maps

ϕ_{S} : A_{S} ⟶ mK_{X}

and

ψ_{S} : K_{X} ⟶ IB_{S}.

Now there are (up to ℂ^{∗}) only finitely many maps ϕ_{S} : A_{S} ⟶ mK_{X} with some ℚ-line bundle A_{S} arising as maximal ω-destabilizing subsheaf for some extension (S). In fact, fix ϕ = ϕ_{S} : A ⟶ mK_{X}. Then by (6.13) there are only finitely many maximal reflexive subsheaves A′ ⊂ mK_{X} such that A′ ⊄ A. So we may suppose A′ ⊂ A. Then

A′ · ω^{2} < A · ω^{2}.

Actually, putting

ϵ := minY_{j} · ω^{2},

where the minimum runs over the finitely many irreducible hypersurfaces Y_{j} ⊂ X, we have

A′ · ω^{2} ≤ A · ω^{2} − ϵ.

On the other hand, restricting ourselves to A′ of the form A′ = A_{S′}, we have by the destabilizing property

(∗)  A′ · ω^{2} ≥ \frac{c_{1}(E) · ω^{2}}{2} .

As there is an integral divisor ∑ λ_{j}Y_{j} such that

A′ = A ⊗ O_{X} (− ∑ λ_{j}Y_{j}) ,

the finiteness of irreducible hypersurfaces in X gives the finiteness claim, since (∗) reads

(A − ∑ λ_{j}Y_{j}) · ω^{2} ≥ \frac{c_{1}(E) · ω^{2}}{2} .

So we have only finitely many possible maps ϕ (up to ℂ^{∗}). In the same way (by dualizing) we have only finitely many maps ψ (up to ℂ^{∗}). In (5.3.b) below we prove that (ϕ, ψ) and (λϕ, λψ) with λ ∈ ℂ^{∗} always define isomorphic extensions (S). Therefore in total (λϕ, μψ) with λ, μ ∈ ℂ^{∗} define just a 1-dimensional space of extensions, whence h^{2}(X, mK_{X}) ≤ 1.

(5.3.b) We shall now prove that the extension class defining (S) is already determined by ϕ and ψ (modulo ℂ^{∗}). So take another extension

0 ⟶ K_{X} ⟶ E′ ⟶ mK_{X} ⟶ 0 (S′)

with the same destabilizing sheaves A and B and with the same morphisms ϕ and ψ (the case of (ϕ, ψ), (λϕ, λψ) is exactly the same). Let D be the divisorial part of

{ϕ = 0} ∪ {ψ = 0} ∪ Sing(X);

then we obtain a splitting of the sequence (S) over X \ D via ϕ :

E ≃ mK_{X} ⊕ K_{X} ≃ A ⊕ B,

and an analogous splitting of E′ over X \ D. Observe also that A = mK_{X} − D and B = K_{X} + D. Thus we obtain an isomorphism

f : E ⟶ E′

on X \D making the two extensions (S) and (S′) isomorphic over X \D :

0 ⟶ K_{X} ⟶ E ⟶ mK_{X} ⟶ 0
∥ ↓ ∥
0 ⟶ K_{X} ⟶ E′ ⟶ mK_{X} ⟶ 0.

It remains to extend the map f to X. Let us notice that we may assume Z = ∅. In fact let Z_{1} be the codimension 1 part of Z. Restricting our two exact sequences describing E to D, we see that (modulo torsion at finitely many points)

A|D = K_{X}|D,

hence

(5.3.1)  (m − 1)K_{X} · D = D · D.

In particular we note that D|D is nef, hence D itself is nef. Now (5.3.1) yields

0 = c_{2}(E) = c_{1}(A)·c_{1}(B)+c_{2}(I_{Z}B) = (mK_{X} −D)·(K_{X} +D)+Z_{1} = Z_{1},

hence Z_{1} = ∅. In particular Z ⊂ Sing X has codimension at least 3. This shows that we may ignore Z in all our following considerations; in what follows restriction will always that we also divide by torsion. Take a local section s ∈ E(U) over a small disc U. We need to show that f(s) ∈ E′(U); a priori we only know f(s) ∈ E′(D)(U). Let

κ : E → mK_{X}, κ′ : E′ → mK_{X}

and

λ : E → B, λ′ : E′ → B

be the canonical maps.

Now consider the exact sequence

0 ⟶ E ⟶^{κ⊕λ} mK_{X} ⊕ B ⟶^{ρ} mK_{X}|D ⟶ 0

Here ρ(u ⊕ v) = u_{D} − τ(v_{D}), where τ : B_{D} → mK_{X}|D is the canonical sequence arising by restricting our sequences and the maps ϕ and ψ to D. Analogously for E′. Suppose we know

(+)  h^{0}(D_{c}, O_{D_{c}}) = 1

for any connected component D_{c} of D. Actually it suffices to know this for D_{c} ∩ regX. Then τ^{′−1} ∘ τ = a id, and of course we can normalize (in the extension class) to a = 1 (our arguments are local around every individual connected component of D). Therefore we can construct a diagram

0 ⟶ E ⟶ mK_{X} ⊕ B → mK_{X}|D ⟶ 0
↓ ∥ ↓
0 ⟶ E′ ⟶ mK_{X} ⊕ B ⟶ mK_{X}|D ⟶ 0.

Here the map f : E → E′ is defined only on X \ D and meromorphic on X, the left square is commutative on X \ D and the right square is commutative on X. It is immediately checked that mK_{X}|D → mK_{X}|D is the identity (consider images of elements u ⊕ 0), hence f(s) ∈ E′(U) and thus f extends to a global isomorphism making the two extensions isomorphic.

(5.3.c) It remains to prove (+) and we may assume D = D_{c} for simplicity of notations. So suppose that h^{0}(D, O_{D}) ≥ 2, resp. that h^{0}(D ∩ regX, O_{D∩regX}) ≥ 2. Since H^{1}(X, O_{X}) = 0, we obtain

H^{1}(X, O_{X}(−D)) ≠ 0.

Let X_{0} be the regular part of X and D_{0} = D|X_{0}. Since the singularities of X are at most finite,

H^{1}(X_{0}, O(−D_{0})) = H^{1}(X, O(−D)) ≠ 0.

Hence Ext^{1}(O_{X_{0}}, O(−D_{0})) ≠ 0, and therefore there exists a non-split extension

0 ⟶ O_{X_{0}} ⟶ F_{0} ⟶ O(D_{0}) ⟶ 0, (E_{0})

with a locally free sheaf F_{0} over X_{0}. Now F_{0} has a unique reflexive extension to X : consider a singular point x_{0} ∈ X and let U be a Stein neighborhood of x_{0}. Then

H^{1}(U \ x_{0}, O(−D_{0})) = H^{1}(U, O(−D)) = 0,

hence (E_{0}) splits over U \ x_{0} :

F_{0} ≃ O ⊕ O_{U\x_{0}}(D_{0}).

Hence F_{0} extends to a reflexive sheaf F. Moreover (E_{0}) extends to

0 ⟶ O_{X} ⟶ F ⟶ O_{X}(D) ⟶ 0. (E)

In particular Ext^{1}(O_{X}(D), O_{X}) ≠ 0. This is easily seen to be equivalent to

Ext^{1}(K_{X} + O_{X}(D), O_{X}(D)) ≠ 0,

hence H^{2}(X, K_{X} + O_{X}(D)) ≠ 0 by Serre duality. Thus

D^{2} = 0

by (4.3).

We observe c_{1}(F)^{2} = c_{2}(F) = 0, therefore F cannot be ω-stable. So let A′ be the maximal ω-destabilizing subsheaf, and we obtain as before for E a sequence

0 ⟶ A′ ⟶ F ⟶ I_{Z′}B′ ⟶ 0, (F) where Z′ has generically dimension 1 or is contained in the singular locus of X. As before, A′ ⊂ O_{X}(D) and O_{X} ⊂ B′ so that there is an effective divisor D′ such that B′ = O_{X}(D′) and A′ = O_{X}(D − D′). By (E) and the fact that a(X) = 0 and H^{1}(X, O_{X}) = 0, we deduce that h^{0}(X, F) = 2. Hence (F) yields

h^{0}(X, A′) = 1.

So we can write A′ = O_{X}(E) with an effective divisor E. Thus D = D′+E. By restricting (E) and (F) to D′, we obtain A′|D′ = E|D′ = O_{D′} and that the 1-dimensional part Z′_{1} of Z′ is empty, so that Z′ is at most finite. Alternatively, we calculate

0 = c_{2}(F) = E · D′ + Z′_{1}

and conclude together with the observation that D′|D′ and thus D′ is nef.

So we have a decomposition D = D′ + E with D and D′ nef. By instability, E ≠ 0. Having in mind that D is connected, we are going to prove that D,D′ and E are proportional (even numerical proportionality would be sufficient, which in our situation (X simply connected with a(X) = 0) gives equality). Assuming this proportionality for the moment, we obtain D′ = aE and D = (a + 1)E. Since A′ destabilizes, we have a ≤ 1. By restricting the sequence (E) and (F) to D′ we obtain:

(∗)  O_{D′}(E) = O_{D′}

up to torsion. For the simplicity of notation we suppress the torsion and agree, when taking a restriction, that we also divide by the torsion. We may assume that h^{0}(D′, O_{D′}) = 1. In fact, if h^{0}(D′, O_{D′}) ≥ 2, then we substitute D by D′ and argue as before; of course this procedure has to terminate after finitely many steps. So h^{0}(D′, O_{D′}) = 1 and consequently

h^{0} (D′, N_{D′/D}^{∗μ}) ≠ 0

for some positive integer; hence h^{0} (D′, N_{D′/X}^{∗μ}) ≠ 0. (We may neglect the torsion, because we may compute over X_{0}).

Now this non-vanishing implies that μD′|D_{0} is trivial, where D_{0} is the reduction of D′. In fact, let μD′ = ∑ a_{i}Y_{i} and take a nonzero section s of O_{μD′}(μD′). Let D̃ = ∑ b_{i}Y_{i} ⊂ μD′ be the maximal subdivisor such that s|D̃ = 0. Introducing c_{i} = a_{i} − b_{i} ≥ 0, we obtain a section

s′ ∈ H^{0} (D_{0}, O_{D_{0}} (− ∑ c_{i}Y_{i}))

such that s′|Y_{i} ≠ 0 for all i. Fix a Kähler form ω and let α_{ij} = Y_{i} · Y_{j} · ω for i ≠ j. Then we obtain for all j

− ∑_{i} c_{i}Y_{i} · Y_{j} · ω = − ∑ c_{i}α_{ij} ≥ 0.

Since D^{2} = 0 and D is nef, we have D · Y_{j} = 0 for all j and therefore

Y_{j}^{2} = −\frac{1}{a_{j}} ∑_{i≠j} a_{i}Y_{i} · Y_{j}

so that we arrive at the inequalities (for each j)

∑_{i≠j} a_{i}c_{j}α_{ij} ≥ ∑_{i≠j} a_{j}c_{i}α_{ij}.

By simple algebraic considerations this is only possible if we have always equality. This means that the divisor D^{∗} = ∑ c_{i}Y_{i} fulfills D^{∗} · Y_{j} = 0 for all j. Hence D^{∗} is nef, and the proportionality arguments below shows that D^{∗} = cD′ for some positive number c. Because of the nonvanishing of s′ on Y_{j}, D^{∗}|Y_{j} is trivial, hence D^{∗}|D_{0} is trivial. Suppose for the moment that μ = 1. Then D^{∗} ⊂ D′ so that c < 1 and in total we have D^{∗}|D_{0} trivial, and D^{∗}|D^{∗} torsion by (∗). Hence [25, 4.1] says that D^{∗}|D^{∗} is trivial. Now the exact sequence

0 ⟶ O_{X} ⟶ O_{X}(D^{∗}) ⟶ O_{X}(D^{∗})|D^{∗} ⟶ 0

implies by (∗) — keeping in mind H^{1}(X, O_{X})=0 — that h^{0}(X, O_{X}(D′)) ≥ 2, contradicting a(X) = 0. So we are left with the case μ ≥ 2. We deal with μ = 2 and leave the trivial modifications in the general case to the reader. The difficulty here is that possibly c > 1 so that D′ ⊂ D^{∗}, otherwise we conclude as before. At least we know that D^{∗} ⊂ μD′ and we are going to show that μE|μD′ is trivial; then we are done. This does not follows directly from restricting (E) and (F); instead we take S^{2} and obtain an injection O_{X}(2E) ⟶ S^{2}(F). Restricting to 2D′, we obtain a nonzero map O_{2D′}(2E) → F|2D′. Then either the induced map O_{2D′}(2E) → O_{D}(D)) is nonzero; this implies H^{0}(D′, O_{D′}(−D′)) ≠ 0 so that h^{0}(X, O_{2D′}) > 1 and we may take D = 2D′ whence μ = 1. Or this map vanishes; then we get a nonzero map O_{2D′}(2E) → O_{2D′}. This map is an isomorphism and settles our claim.

(5.3.d) It remains to settle the proportionality. After passing to a desingularization this comes down to proving the following statement: Let X be compact Kähler, A = ∑_{i=1}^{s} a_{i}Y_{i} and B = ∑_{i=1}^{s} b_{i}Y_{i} be effective divisors, a_{i} and b_{i} positive, with connected supports. Suppose that A and B are nef, and that A · Y_{i} = B · Y_{i} = 0 for all i (in particular A^{2} = B^{2} = A · B = 0). Then A = cB (not only for numerical equivalence). Observe that if X is a surface, then this is nothing than Zariski’s lemma, which is usually formulated for fibers of maps to curves, but which works in this context; therefore the claim also follows for projective manifolds by taking hyperplane sections and applying Lefschetz. If X is merely Kähler, then we consider the vector space V ⊂ H^{2}(X, ℝ) generated by the classes of the hypersurfaces Y_{i} ⊂ X. Let W be the direct sum ⨁ ℝ · Y_{i}; and let Q be the bilinear form

Q(Y_{i}, Y_{j}) = −Y_{i} · Y_{j} · ω.

In this situation we apply [1, Lemma I.2.10] to conclude. q.e.d.

5.4 The stable case. By 5.3 we are reduced to the case that some extension E is ω-stable for some Kähler form ω. By (5.2.1) and (5.2.2) we have in particular

c_{1}^{2}(E) · ω = 4c_{2}(E) · ω,

which implies that E is projectively flat, at least on the regular part X_{0}. In fact, this is well-known if X is smooth and E is locally free. But the proof generalizes to our case since the singularities of X and E are in codimension at least 3. Now we follow the arguments in [22, p. 113/114].

Assume first that the degree of finite étale covers of X_{0} is bounded: π_{1}^{alg}(X_{0}) is finite. After performing a finite étale cover, we may assume π_{1}^{alg}(X_{0}) = 0. Since E|X_{0} is projectively flat, E^{∗} ⊗ E|X_{0} is hermitian flat and therefore given by a unitary representation ρ of π_{1}(X_{0}). Since ρ(π_{1}(X_{0})) is residually finite, it follows that ρ is trivial, hence E^{∗} ⊗ E is trivial. This implies, using the exact sequence

0 ⟶ K_{X} ⟶ E ⟶ mK_{X} ⟶ 0

and dualizing that (over X_{0}) h^{0}(E^{∗} ⊗ K_{X}) = 0, that h^{0}(E^{∗} ⊗ mK_{X}) ≥ 4 and that therefore h^{0}((m − 1)K_{X}) ≥ 3. This contradicts a(X) = 0.

If π_{1}^{alg}(X_{0}) is infinite, we just take over the arguments of [22, p. 114]: since the local fundamental groups of X at the singularities are finite, any finite étale cover h of X_{0} of sufficiently large degree extends to a covering h : X̃ → X which can be written in the form h = g ∘ f, where f : X̃ → X′ is étale and g : X′ → X is étale outside the singular locus. Therefore π_{1}(X′) is infinite, contradicting the fact that X′ is simple non-Kummer. q.e.d.

6. The inequality K_{X} · c_{2}(X) ≥ 0

The aim of this section is to prove:

Theorem 6.1. Let X be a minimal Kähler 3-fold with a(X) = 0. Then K_{X} · c_{2}(X) ≥ 0.

This inequality is an important step in the proof of abundance for Kähler threefolds. In the projective case, it follows from Miyaoka’s inequality K_{X}^{2} ≤ 3c_{2}(X) which in turn is a consequence of his generic nefness theorem for the cotangent bundle (relying on char p methods).

The rest of this section consists of the proof of 6.1 together with some auxiliary propositions (6.9, 6.10, 6.12/6.13).

6.2 Reduction to the unstable case

Suppose that there is a sequence (ω_{j}) of Kähler metrics converging in H^{2}(X, ℝ) to K_{X} such that T_{X} is ω_{j}-stable for all j. Then we have

c_{1}^{2}(X) · ω_{j} ≤ 3c_{2}(X) · ω_{j}

for all j by Proposition 6.9. Taking limits, we obtain

K_{X}^{3} ≤ 3K_{X} · c_{2}(X),

hence our claim results from K_{X}^{3} = 0.

So from now on we shall assume that T_{X} is ω-unstable for all ω near K_{X} (in H^{2}(X, ℝ)).

6.3 The setup

Let S_{ω} ⊂ T_{X} be the maximal destabilizing subsheaf with respect to ω. Let r denote its rank. Then by Corollary 6.13 below, there are only finitely many choices for S_{ω}, hence there exists an open set in the Kähler cone of X having K_{X} as boundary point such that S_{ω} does not depend on [ω] for [ω] ∈ U. We shall write

S = det S

and let Q = T_{X}/S, a torsion free sheaf of rank 1 or 2. We notice

(6.3.1)  c_{1}(Q) = −K_{X} − S

and, if r = 1,

(6.3.2)  c_{2}(Q) = c_{2}(X) + S · (K_{X} + S).

c_{3}(Q) will be irrelevant for us. The instability of T_{X} gives

(6.3.3)  S · ω^{2} ≥ \frac{−K_{X} · ω^{2}}{3} ,

for ω ∈ U in case r = 1 and

(6.3.3a)  \frac{S · ω^{2}}{2} ≥ \frac{−K_{X} · ω^{2}}{3}

in case r = 2. We claim

(6.3.4)  K_{X}^{2} · S = 0.

In fact, (6.3.3) resp. (6.3.3a) gives in the limit K_{X}^{2} · S ≥ 0. Since we may assume K_{X}^{2} ≠ 0, the tangent sheaf T_{X} is K_{X}-semi-stable by Enoki [11]. This implies K_{X}^{2} · S ≤ 0, hence (6.3.4) follows.

The next lemma is a general statement on Kähler 3-folds with a(X) = 0, independent from our setup.

Lemma 6.4. Let X be a normal compact Kähler 3-fold. Let A and B be ℚ-line bundles on X and let A be nef with A^{2} ≠ 0. If A^{2} · B = 0, then A · B^{2} ≤ 0.

Proof. By passing to a desingularization we may assume X smooth. Fix a Kähler class ω and apply (2.5) with α = c_{1}(B), β = K_{X} + εω and γ = K_{X}. Then expand in terms of powers of ε to obtain the claim.

q.e.d.

Corollary 6.4.a. In our setup (6.3) we have K_{X}·S^{2} ≤ 0 if K_{X}^{2} = 0.

Proof. This follows from 6.4 via 6.3.4. q.e.d.

Lemma 6.4 is of course not true in case A^{2} = 0. Thus in order to obtain (6.4.a) also in case K_{X}^{2} = 0 we need more specific arguments:

Lemma 6.5. Let X be a simply connected minimal Kähler 3-fold with a(X) = 0 and K_{X}^{2} = 0. Let L be a ℚ-line bundle on X. Then K_{X} · L^{2} ≤ 0.

Proof. Assume that K_{X} · L^{2} > 0. If a positive integer c satisfies the following condition:

(∗)  2c^{2}K_{X} · L^{2} > −K_{X} · c_{2}(X),

then by Riemann-Roch we easily get asymptotically

χ(X, mK_{X} + cL) ∼ m.

Observe also that (∗) is satisfied for large c since K_{X} · L^{2} > 0 by assumption. So let us fix such a number c. Then we conclude

(∗∗)  h^{2}(mK_{X} + cL) ≥ Cm.

In fact, otherwise h^{0}(mK_{X} +cL) ≥ Cm by (∗), contradicting a(X) = 0. Now, as in Section 5, we obtain “many” extensions

0 ⟶ K_{X} + cL ⟶ E ⟶ mK_{X} → 0.

Observe that E cannot be ω-stable for ω near K_{X}. In fact, in that case we had

c_{1}(E)^{2} · ω ≤ 4c_{2}(E) · ω,

hence c_{1}(E)^{2} · K_{X} ≤ 4c_{2}(E) · K_{X} in the limit. This comes down to

c^{2}K_{X} · L^{2} ≤ 0,

contradicting our assumption.

We proceed exactly in the same way as in Section 5, introducing the divisors D_{m}, and now (∗∗) and the arguments in Section 5 yield

h^{0}(O_{D_{m}}) ≥ Cm,

for large m. On the other hand,

(m − 1)K_{X} + cL^{∗}|D_{m} = D_{m}|D_{m},

again referring to Section 5, hence for large m, the normal bundle N_{D_{m}} gets more and more “nef”. However, to have many functions on D_{m} means to have a tendency to negativity for the normal bundle. So we will show that (+) and (++) are contradictory. By passing to a subsequence — having in mind that X carries only finitely many irreducible hypersurfaces — we can suppose the following:

D_{m} = ∑_{i=1}^{s} a_{m,i}Y_{i} + ∑_{s+1}^{t} a_{j}Y_{j},

where a_{m,i} < a_{m+1,i} and the a_{j} are independent of m. Put R = ∑_{s+1}^{t} a_{j}Y_{j} and Y = ∑_{1}^{s} Y_{i}. Then

N_{D_{m}/D_{m+1}}^{∗} = (∑_{i=1}^{s} a_{m,i}Y_{i} + R) |Y.

Since by (+), h^{0}(N_{D_{m}/D_{m+1}}^{∗}) > 0, the sequence of divisors − ∑ a_{m,i}Y_{i}|Y, suitably normalized, converges to an effective nonzero divisor on Y. Thus N_{D_{m}|X}^{∗}|Y, suitably normalized converges to an effective nonzero divisor on Y. On the other hand, its dual is nef by (++). This is a contradiction. q.e.d.

Corollary 6.5.a. In our setup (6.3) we have K_{X} · S^{2} ≤ 0.

6.6 The Case: rk S = 1 and Q stable

By “Q stable” we mean that there is a sequence of Kähler forms (ω_{j}) converging to K_{X} (as classes) such that Q is ω_{j}-stable for all j. Then by Proposition 6.9 we have

c_{1}^{2}(Q) · ω_{j} ≤ 4c_{2}(Q) · ω_{j},

hence

c_{1}^{2}(Q) · K_{X} ≤ 4c_{2}(Q)K_{X}.

Putting in (6.3.1) and (6.3.2) we obtain

K_{X} · (K_{X} + S)^{2} ≤ (4c_{2}(X) + 4S · (K_{X} + S)) · K_{X},

which in turn yields

K_{X} · c_{2}(X) ≥ −\frac{3}{4}K_{X} · S^{2}.

Thus 6.4.a gives K_{X} · c_{2}(X) ≥ 0.

6.7 The Case: rk S = 1 and Q is unstable

After the previous case it is clear what unstable has to mean: Q is ω-unstable for all ω near K_{X}. Then we obtain a destabilizing sequence

0 ⟶ L_{1} ⟶ Q ⟶ I_{B}L_{2} ⟶ 0

where L_{i} are reflexive of rank 1 and dim B ≤ 1. This sequence is — as usual — independent of ω, if ω is sufficiently near to K_{X} and contained in a suitable open set U as in (6.3). We first claim

(6.7.1)  K_{X}^{2} · L_{1} ≤ 0.

To verify this, let R be the cokernel of

T_{X} ⟶ I_{B}L_{2} ⟶ 0.

Then we have an exact sequence

0 ⟶ S ⟶ R ⟶ L_{1} ⟶ 0.

Of course we may assume K_{X}^{2} ≠ 0. Then by Enoki [En87], T_{X} is K_{X}- semi-stable, hence

c_{1}(R) · K_{X}^{2} ≤ 0.

This implies K_{X}^{2} · (L_{1} + S) ≤ 0 by the last exact sequence. Now (6.3.4) gives our claim (6.7.1).

Next we show

(6.7.2)  K_{X} · L_{1} = 0.

In fact, the destabilizing property for L_{1} reads

L_{1} · ω^{2} ≥ \frac{c_{1}(Q) · ω^{2}}{2} ,

hence

L_{1} · K_{X}^{2} ≥ \frac{c_{1}(Q) · K_{X}^{2}}{2} = \frac{1}{2}(−K_{X} − S) · K_{X}^{2} = 0.

We now conclude by (6.7.1).

Since c_{1}(Q) · K_{X}^{2} = 0, we also have

(6.7.3)  K_{X}^{2} · L_{2} = 0.

Thus Lemma 6.4 applies:

(6.7.4)  K_{X} · L_{i}^{2} ≤ 0

for i = 1, 2.

The final preparation is

K_{X} · L_{1} · L_{2} = \frac{1}{2}(K_{X} · S^{2} − K_{X} · L_{1}^{2} − K_{X} · L_{2}^{2}).

This follows from the two equations

K_{X} · c_{1}^{2}(Q) = K_{X} · (L_{1} + L_{2})^{2}

and

K_{X} · c_{1}^{2}(Q) = K_{X} · (K_{X} + S)^{2} = K_{X} · S^{2}.

After all these preparations we conclude using (6.7.4) as follows:

K_{X} · c_{2}(X) = K_{X} · c_{2}(Q) + K_{X} · S · c_{1}(Q)

= K_{X} · c_{2}(I_{B}) + K_{X} · L_{1} · L_{2} − K_{X} · S^{2}

= K_{X} · c_{2}(I_{B}) + \frac{K_{X} · S^{2}}{2} − \frac{K_{X} · L_{1}^{2}}{2} − \frac{K_{X} · L_{2}^{2}}{2} − K_{X} · S^{2}.

Since K_{X} · c_{2}(I_{B}) ≥ 0 by the nefness of K_{X}, we conclude by virtue of (6.7.4) and Lemma 6.4.

6.8 The Case: rk S = 2

In this case we consider the maximal destabilizing subsheaf Q^{∗} ⊂ Ω_{X}^{1} = (T_{X})^{∗}. Here it is convenient to switch completely the notations: we denote the maximal destabilizing subsheaf of Ω_{X}^{1} again by S and let Q denote the quotient. Then

c_{1}(Q) = K_{X} − S

and

c_{2}(Q) = c_{2}(X) − S · (K_{X} − S).

Now (6.3) yields K_{X}^{2} · S = 0. Applying again 6.4 gives K_{X} · S^{2} ≤ 0. Now (6.6) and (6.7) run in completely the same way; notice that some minus signs are irrelevant because K_{X}^{2} · S = 0.

Proposition 6.9. Let X be a normal compact Kähler n-fold with codim Sing(X) ≥ 3. Suppose a(X) = 0. Let ω be a Kähler form on X and E a torsion free coherent sheaf on X of rank r ≥ 2. If E is ω-stable, then

c_{1}^{2}(E) · ω^{n−2} ≤ \frac{2r}{r − 1}c_{2}(E) · ω^{n−2}.

Proof. For simplicity of notations set μ = \frac{2r}{r−1}.

(1) First we reduce the problem to the case “E reflexive”. So suppose we know the assertion for reflexive sheaves and let E be torsion free. Then we consider the quotient sheaf

Q = E^{∗∗}/E,

which is supported on a complex subspace Z ⊂ X of codimension at least 2. Now

c_{2}(Q) = −mc_{2}(I_{Z}),

for some positive m, and c_{2}(I_{Z}) is an effective cycle supported on Z, hence

(∗)  ω^{n−2} · c_{2}(Q) ≤ 0.

Now c_{2}(E^{∗∗}) = c_{2}(E) + c_{2}(Q), hence (∗) implies

(∗∗)  c_{2}(E)ω^{n−2} ≥ c_{2}(E^{∗∗}) · ω^{n−2}.

Notice that E^{∗∗} is stable because E is ([20, V.7.7]), hence by our assumption

c_{1}^{2}(E^{∗∗})ω^{n−2} ≤ μc_{2}(E^{∗∗}) · ω^{n−2}.

Since c_{1}(E^{∗∗}) = c_{1}(E), the inequality (∗∗) implies our claim follows.

(2) From now on we shall assume E reflexive. Choose a desingularization π : X̂ ⟶ X by a sequence of blow-ups whose centers all lie over the singularities of X and E. Moreover we may assume that Ê = π^{∗}(E)^{∗∗} is locally free (see [14]). Let ω̂ = π^{∗}(ω). By definition of Kähler forms on singular spaces ω̂ — which a priori exists only on a Zariski open part of X̂ — extends to a semipositive (1, 1)-form on all of X̂. We claim

(+)  E is ω̂ − stable.

Indeed, assume we have a subsheaf Ŝ ⊂ Ê of rank s with

\frac{c_{1}(Ŝ) · ω̂^{n−1}}{s} ≥ \frac{c_{1}(Ê) · ω̂^{n−1}}{r} .

Then consider

S = π_{∗}(Ŝ) ⊂ π_{∗}(Ê).

Since π_{∗}(Ê) is torsion free and since E is reflexive, we have π_{∗}(Ê) ⊂ E, hence S ⊂ E. Now

c_{1}(Ŝ) · ω̂^{n−1} = c_{1}(π_{∗}(Ŝ)) · ω^{n−1} = c_{1}(S) · ω^{n−1},

and

c_{1}(Ê) · ω̂^{n−1} = c_{1}(E) · ω^{n−1},

hence

\frac{c_{1}(S) · ω^{n−1}}{s} ≥ \frac{c_{1}(E) · ω^{n−1}}{r} ,

contradicting the ω-stability of E. This proves (+).

Now ω̂ has the disadvantage not to be a Kähler form, but it is on the boundary of the Kähler cone. To circumvent this difficulty, let E_{i} denote the exceptional components of the exceptional set of π, then we can chose a_{i} < 0, such that E := ∑ a_{i}E_{i} is π-ample. Thus

ω̂_{ϵ} := ω̂ + ϵE

is a Kähler class for all small positive ϵ. We claim that Ê is ω̂_{ϵ}-stable for ϵ small enough. Indeed, suppose the contrary. Then there exists a sequence ϵ − k converging to 0 such that Ê is not ω̂_{ϵ_{k}}-stable. Let S_{i} ⊂ Ê be the maximal destabilizing subsheaf with respect to ω̂_{ϵ_{i}}. Since a(X̂) = 0, we find i_{0} such that S_{i} = S_{j} for all i, j ≥ i_{0} (Prop. 6.12), possibly after passing to a subsequence (but even this could be avoided). So let S = S_{i}, i ≥ i_{0}. Then we have

\frac{c_{1}(S) · ω̂_{ϵ_{k}}^{n−1}}{s} ≥ \frac{c_{1}(Ê) · ω̂_{ϵ_{k}}^{n−1}}{r} ,

so passing to the limit,

\frac{c_{1}(S) · ω̂^{n−1}}{s} ≥ \frac{c_{1}(Ê) · ω^{n−1}}{r} .

This contradicts (+).

Thus Ê is ω̂-stable for small positive ϵ. Therefore Ê is Hermite-Einstein with respect to ω̂_{ϵ} and hence

c_{1}^{2}(Ê) · ω̂_{ϵ}^{n−2} ≤ μ c_{2}(Ê) · ω̂_{ϵ},

hence

c_{1}^{2}(Ê) · ω̂^{n−2} ≤ μ c_{2}(Ê) · ω̂^{n−2}.

Since codim(Sing(X) ∪ Sing(E)) ≥ 3, we conclude

c_{1}^{2}(E)ω^{n−2} ≤ μ c_{2}(E) · ω^{n−2}.

q.e.d.

Proposition 6.10. Let L be a line bundle or a reflexive sheaf of rank 1 on the normal compact complex space X. Suppose a(X) = 0. Let S_{i} ⊂ L be reflexive subsheaves, i ∈ I. Then for all i there are only finitely many j such that S_{j} ⊄ S_{i}.

Proof. Of course we may assume X smooth. Since a(X) = 0, the complex space X has only finite many irreducible hypersurfaces Y_{1}, … Y_{r}, therefore we can write

S_{i} = L − ∑_{j=1}^{r} a_{j}^{(i)}Y_{j}

with a_{j}^{(i)} ≥ 0. Thus the claim is clear. q.e.d.

Definition 6.11. Let F be a torsion free coherent sheaf on a normal compact complex space and let S ⊂ F be a reflexive subsheaf with 0 < rkS < rkF. We say that S is maximal, if there is no proper reflexive subsheaf S′ ⊂ F of the same rank as S such that S ⊂ S′ and S ≠ S′.

If ω is a Kähler form on X and if S is the ω-maximal destabilizing subsheaf of the ω-unstable sheaf F, then S is maximal. This is the way we will identify maximal subsheaves.

Proposition 6.12. Let X be a normal compact Kähler space with a(X) = 0 and F a reflexive coherent sheaf on X. Then F admits only finitely many maximal reflexive subsheaves of rank 1.

Proof. Of course we may assume X smooth. Consider now the maximal subsheaves S_{i} ⊂ F of rank 1, i ∈ I = ℕ. Choose m ∈ ℕ and i_{1} < ⋯ < i_{m} such that

S′ = S_{i_{1}} + ⋯ + S_{i_{m}} ⊂ F

has the following property: if j is different from the indices i_{1}, …, i_{m}, then rk S′ = rk(S′ + S_{j}). So things come down to show that there are only finitely many j such that

rk(S_{j} ∩ S′) = 1.

In order to prove this, we assume to the contrary that there are infinitely many j such that rk(S_{j} ∩ S′) = 1. Then we have infinitely many subsheaves

T_{j} := S_{j} ∩ S′ ⊂ S′

of rank 1 (use again the finiteness of hypersurfaces in X.) Now fix j_{0}. Then by (6.10) there are only finitely many j such that T_{j} ⊄ T_{j_{0}}. For all others we have T_{j} ⊂ T_{j_{0}} and for those we write

T_{j} = S_{j} − A_{j}

and

T_{j_{0}} = S_{j_{0}} − A_{j_{0}}

with effective divisors A_{j}. Since X has only finitely many irreducible hypersurfaces, we have A_{j_{0}} ⊂ A_{j} for almost all j, hence we obtain S_{j} ⊂ S_{j_{0}} for almost all j, contradiction to maximality. q.e.d.

Corollary 6.13. Let X be an normal compact Kähler space with a(X) = 0 and F a torsion free sheaf of rank at most 3. Then F contains only finitely many maximal reflexive subsheaves.

Proof. By 6.12 we have only to deal with the case of subsheaves of rank 2. This is done by dualizing and applying 6.12 to F^{∗} using the following trivial remark: if S ⊂ F is maximal with quotient Q, then Q^{∗} ⊂ F^{∗} is maximal. q.e.d.

7. An abundance theorem for Kähler threefolds

Here we want to solve (the remaining part of) the abundance problem for Kähler threefolds:

Theorem 7.1. Let X be a ℚ-Gorenstein Kähler threefold with only terminal singularities such that K_{X} is nef (a minimal Kähler threefold for short). Then κ(X) ≥ 0.

Of course, more should be true:

Conjecture 7.2. Let X be a minimal Kähler threefold. Then K_{X} is semi-ample, i.e., some multiple mK_{X} is spanned by global sections.

Remark 7.3.

(1) In case X is projective, everything is proved by Miyaoka [23, 24] and Kawamata [18].

(2) In the non-algebraic case, 7.1/7.2 is proved in [27] with the important possible exception that X is simple and not Kummer (see 1.4). In particular in this remaining case we have algebraic dimension a(X) = 0 and π_{1}(X) finite. In [9] 7.1 is proved if K_{X} carries a sufficiently nice metric, e.g., if K_{X} is hermitian semipositive.

(3) In case X is Gorenstein, we have the Riemann-Roch formula

χ(X, O_{X}) = −\frac{1}{24}K_{X} · c_{2}(X).

Therefore, if we recall that we may assume a(X) = 0, the inequality (6.1)

(∗)  K_{X} · c_{2}(X) ≥ 0

implies χ(X, O_{X}) ≤ 0 and therefore h^{0}(X, K_{X}) = h^{3}(X, O_{X}) ≠ 0, so that at least κ(X) ≥ 0. In case X is not Gorenstein, this Riemann-Roch formula is not true; instead one has some positive correction term [12] which might correct the negativity of the product −K_{X} · c_{2}(X) and therefore destroy the contradiction.

Proof of Theorem 7.1. As noticed in 7.3 we may assume that X is simple non-Kummer, in particular q(X) = 0. First we reduce ourselves to the case that X is ℚ-factorial by applying Kawamata’s factorialisation f : X̂ ⟶ X as in the proof of 4.4. Since f is small, we have KX̂ = f^{∗}(K_{X}), so KX̂ is nef. Hence we can work on X̂ and thus may assume X to be ℚ-factorial from the beginning. We consider a desingularization

π : X̂ ⟶ X

and compute by Riemann-Roch

(∗)  χ(X̂, π^{∗}(mK_{X})) = \frac{m}{12}K_{X} · c_{2}(X) + χ(X, O_{X})

for all m such that mK_{X} is Cartier. Assume κ(X) = −∞, so H^{3}(O_{X}) = 0. Since X is not projective, we have H^{2}(O_{X}) ≠ 0. In total we obtain:

χ(X, O_{X}) ≥ 2.

If now K_{X} ≠ 0, then by 4.3/5.1, we have h^{2}(X, mK_{X}) ≤ 1, hence (6.1) and (∗) imply h^{0}(X, mK_{X}) ≥ 1, a contradiction. If however K_{X} = 0, take a positive integer m such that mK_{X} is Cartier. If now mK_{X} is not a torsion line bundle, we must have q(X) > 0, contradiction. q.e.d.

Remark 7.4. In order to settle the abundance for Kähler threefolds completely, it remains to show that a simple threefold X with K_{X} nef and κ(X) = 0 must be Kummer. In the following we collect what we know about X. We shall assume that q(X) = 0, otherwise we consider the Albanese and are easily done. Thus we have χ(X, O_{X}) ≥ 1.

(1) K_{X} · c_{2}(X) = 0 and 1 ≤ χ(X, O_{X}) ≤ 2.

The first part follows easily from equation (∗) in the proof of (7.3) together with 4.1/5.1. Hence

(∗)  χ(X, mK_{X}) = χ(X, O_{X})

for all integers m such that mK_{X} is Cartier. Then again 4.3/5.1 gives the inequality for χ(X, O_{X}).

(2) X cannot be Gorenstein. In fact, then the Riemann-Roch formula

24χ(X, O_{X}) = −K_{X} · c_{2}(X) = 0

gives a contradiction.

(3) If χ(X, O_{X}) = 2, then K_{X}^{2} = 0. This is a consequence of (1) via the vanishing 4.3.

(4) If χ(X, O_{X}) = 1, then h^{0}(X, K_{X}) = 1 and h^{2}(X, O_{X}) = 1.

8. Almost algebraic Kähler threefolds

In this section we show that simple non-Kummer threefolds are very far from projective threefolds, in a sense which is made precise in the following definition:

Definition 8.1. Let X be a normal Kähler variety with only terminal singularities. X is almost algebraic if there exists an algebraic approximation of X. This is a proper surjective flat holomorphic map π : X → Δ from a normal complex space X where Δ ⊂ ℂ^{m} is the unit disc, where X ≃ X_{0}, where all complex analytic fibers X_{t} = π^{−1}(t) are normal Kähler spaces with at most terminal singularities such that there is a sequence (t_{j}) in Δ converging to 0 so that all X_{j} := X_{t_{j}} are projective.

Of course, in case X is smooth, all X_{t} will be smooth (after possibly shrinking Δ).

The following problem is attributed to Kodaira:

Problem 8.2. Is every compact Kähler manifold almost algebraic?

From a point of view of algebraic geometry almost algebraic Kähler spaces seem to be the most interesting Kähler spaces. Therefore it is worthwile to notice

Theorem 8.3. Let X be a nearly algebraic Kähler threefold with only terminal singularities. If X is simple and additionally K_{X} nef or X smooth, then X is Kummer.

Proof. Assume that X is not Kummer. Then π_{1}(X) is finite by [3] as already mentioned. Let π : X → Δ be an algebraic approximation of X. Let (t_{j}) be a sequence in Δ converging to 0 such that all X_{j} = X_{t_{j}} are projective. Notice first that κ(X_{j}) ≥ 0 for all j. In fact, otherwise X_{j} would be uniruled for some j and by standard arguments X_{t} would be uniruled for all t which is not possible, X = X_{0} being simple.

(1) We show that κ(X) = κ(X_{0}) ≥ 0. Fix a positive integer m. Then by [21, 1.6], every t_{j} admits an open neighborhood U_{j} such that

h^{0}(X_{t}, mK_{X_{t}}) = h^{0}(X_{j}, mK_{X_{j}})

for all t ∈ U_{j}. Now choose m such that h^{0}(X_{j}, mK_{X_{j}}) > 0 for some j. Then it follows that h^{0}(X_{t}, mK_{X_{t}}) = h^{0}(X_{j}, mK_{X_{j}}) =: d > 0 for all t in an open set in Δ. Let

A := {t ∈ Δ|h^{0}(X_{t}, mK_{X_{t}}) ≥ d}.

Then A is an analytic set in Δ (semi-continuity in the analytic Zariski topology), and it contains a nonempty open set, hence A = Δ. Thus κ(X_{0}) ≥ 0. Since X_{0} is simple, we conclude κ(X_{0}) = 0.

(2) Suppose that κ(X_{j}) ≥ 1 for some j. Then fix m such that h^{0}(X_{j}, mK_{X_{j}}) ≥ 2. Repeating the same arguments as in (1), we conclude h^{0}(X, mK_{X}) ≥ 2, contradicting X being simple. So κ(X_{j}) = 0 for all j.

(3) Here we will show that X_{j} is Kummer for all j. Let X′_{j} be a minimal model of X_{j}. Observe that

h^{2}(X_{j}, O_{X_{j}}) = h^{2}(X, O_{X}) > 0;

in fact, H^{1}(X, O_{X}) = 0, hence H^{1}(X_{j}, O_{X_{j}}) = 0 for large j. Moreover h^{0}(X_{t}, K_{X_{t}}) is constant by [KM92], as shown above. Therefore the equality follows by Serre duality and the constancy of χ(X_{t}, O_{X_{t}}). Hence we also have h^{2}(X′_{j}, O_{X′_{j}}) > 0.

Since K_{X′_{j}} ≡ 0, there exists a finite cover, the so-called canonical

cover, h : X̃_{j} → X′_{j}, étale in codimension 2, such that KX̃_{j} = OX̃_{j}.

In particular X̃_{j} is Gorenstein and Riemann-Roch yields

χ(X̃, OX̃_{j}) = 0.

Since h^{2}(OX̃_{j}) > 0, we must have q(X̃_{j}) > 0. Let α_{j} : X̃_{j} ⟶ A = A_{j} be the Albanese map. By [Ka85] there exists a finite étale cover B ⟶ A such that

X̂_{j} := X̃_{j} ×_{A} B ≃ F × B.

In particular X̂_{j} and X̃_{j} are smooth because of the isolatedness of singularities. We conclude that X′_{j} is Kummer unless F is a K3-surface. To exclude that case, consider the image F′ ⊂ X̃′_{j} of a general F × {b}. Then F′ is K3 or Enriques and does not meet the singularities of X′_{j}. Moreover the normal bundle N_{F′} is numerically trivial. Since F′ moves, it is actually trivial. Now consider the strict transform in X_{j}, again called F′. Then F′ has the same normal bundle in X_{j}, so that N_{F′/X} = O_{X}^{2} . Since H^{1}(N) = 0, the deformations of F′ cover every X_{t} contradicting the simplicity of X_{0}. So F cannot be K3 and X_{j} is Kummer.

(4) Suppose K_{X_{0}} nef. Fix a positive number m such that

mK_{X} = O_{X} (D)

with some effective divisor D. We may assume that D does not contain any fiber of π; denote D_{t} = X_{t}∩D. We want to argue that K_{X_{t}} must be nef, therefore K_{X_{j}} = 0 so that D_{j} = 0 and D = 0 in total. So we will obtain K_{X} = 0. To see that X is Kummer, consider the canonical cover of X and argue as in the proof of (7.1). To prove nefness, we apply [KM92] to deduce that the sequence φ_{j} : X_{j} → X′_{j} = A_{j}/G appears in family X_{U_{j}} ⟶ X′_{U_{j}} over a small neighborhood U_{j} of t_{j}. In particular some multiple N_{D_{t}}^{∗μ} has many sections for t ∈ U_{j} on an at least 1-dimensional family of curves. By semicontinuity, also N_{D_{0}}^{∗μ} has many sections on such a family, contradicting the nefness of D_{0}. Alternatively, K_{X} is negative on a family of rational curves over U_{j}, which converges to a family of rational curves in X_{0} and therefore forces K_{X_{0}} to be non-nef.

(5) Now suppose X_{0} smooth, i.e., π is smooth after shrinking Δ. Take a sequence of blow-ups of smooth subvarieties of X such that the preimage of redD has normal crossings. After shrinking Δ we may assume that the only points and compact curves blown up lying over X_{0} so that all fibers over Δ \ 0 are smooth. Then take a covering h : X̃ ⟶ X such that KX̃ = O(D̃) with X̃ smooth. This is possible e.g., by applying [Ka81]. Then X_{t_{j}} is Kummer and admits a 3-form and therefore must be bimeromorphically a torus (if A/G admits a 3-form, then it is a torus covered by A. This is a consequence of the simplicity of A and the fact that G acts without fix points). Hence every X̃_{t}, t ≠ 0, has 3 holomorphic 1-forms which are independent at the general point and therefore every X_{t}, t ≠ 0, is Kummer. In order to show that X_{0} is Kummer, consider the central fiber X̃_{0} which contains the preimage of the strict transform X′_{0} of X_{0}. More precisely, we have

X̃_{0} = X′_{0} + ∑ a_{i}E_{i}

where the E_{i} are smooth threefolds contracted to points or curves. By semi-continuity, h^{2}(OX̃_{0}) ≥ 3. Now we check easily that

H^{2}(X̃_{0}, OX̃_{0}) = H^{2}(X′_{0}, O_{X′_{0}}),

hence X′_{0} carries three 2-forms coming from X̃. But then it is clear that also some of the holomorphic 1-forms on X̃ give nonzero 1forms on X′_{0}, since the 2-forms are wegdge products of the 1-forms. Hence X′_{0} is Kummer and so does X_{0}. q.e.d.

If Problem 8.2 has a positive answer in dimension 3, Theorem 8.3 excludes the existence of simple non-Kummer threefolds.

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