THE ALGEBRAIC DIMENSION OF COMPACT COMPLEX THREEFOLDS WITH VANISHING SECOND BETTI NUMBER

FRÉDÉRIC CAMPANA, JEAN-PIERRE DEMAILLY, AND THOMAS PETERNELL

Abstract. We investigate compact complex manifolds of dimension three and second Betti number b_{2}(X) = 0. We are interested in the algebraic dimension a(X), which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. The topological Euler characteristic χ_{top}(X) equals the third Chern class c_{3}(X) by a theorem of Hopf. Our main result is that, if X is a compact 3-dimensional complex manifold with b_{2}(X) = 0 and a(X) > 0, then c_{3}(X) = χ_{top}(X) = 0, that is, we either have b_{1}(X) = 0, b_{3}(X) = 2 or b_{1}(X) = 1, b_{3}(X) = 0.

Introduction

In this note we shall investigate compact complex manifolds of dimension three and second Betti number b_{2}(X) = 0. Of course such a manifold cannot be algebraic. Therefore we will be interested in the algebraic dimension a(X) which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. Note that a(X) > 0 if and only if X admits a non-constant meromorphic function. The topological Euler characteristic will be denoted χ_{top}(X) which is also the third Chern class c_{3}(X) by a theorem of Hopf. Our main result is

Theorem

Let X be a compact 3-dimensional complex manifold with b_{2}(X) = 0 and a(X) > 0. Then c_{3}(X) = χ_{top}(X) = 0, i.e. we either have b_{1}(X) = 0, b_{3}(X) = 2 or b_{1}(X) = 1, b_{3}(X) = 0.

Notice that if a(X) = 3, i.e. X is Moishezon, then we have b_{2}(X) > 0, but examples of compact threefolds X with a(X) = 1 or 2 and with the above Betti numbers exist (see sect. 4). We also show by an example that the assumption a(X) > 0 cannot be omitted.

The following corollary was actually our motivation for the Theorem:

Parts of this work were done during one of the author’s stay at MSRI. Research at MSRI is supported in part by NSF grant DMS-9022140.

Corollary

Let X be a compact complex manifold homeomorphic to the 6-dimensional sphere S^{6}. Then a(X) = 0.

In other words, S^{6} does not admit a complex structure with a non-constant meromorphic function.

The proof of the Theorem consists in showing that for generic L ∈ Pic^{0}(X), we have

h^{0}(X, B ⊗ L) = h^{2}(X, B ⊗ L) = 0

for B either the tangent bundle T_{X} or the cotangent bundle Ω_{X}^{1}. This forces \frac{c_{3}(X)}{2} = χ(X, T_{X}) to be 0. If a(X) = 2 or if a(X) = 1 and the algebraic reduction is not holomorphic, we can even prove the last vanishing statements for all vector bundles B on X.

In the last section we study more closely the structure of threefolds X with b_{2}(X) = 0 and algebraic dimension 1 whose algebraic reduction is holomorphic. We show e.g. that smooth fibers can only be Inoue surfaces, Hopf surface with algebraic dimension 0 or tori.

1. Preliminaries and Criteria for the vanishing of H^{0}.

1.0 Notations

(1) Let X be a compact complex manifold, always assumed to be connected. The algebraic dimension, denoted a(X), is the transcendence degree of the field of meromorphic functions over ℂ.

(2) b_{i}(X) = dim H^{i}(X, ℝ) denotes the i−th Betti number of X.

(3) If G is a finitely generated abelian group, then rkG will denote its rank (over ℤ).

(4) If X is a compact space, then h^{q}(X, F) denotes the dimension of H^{q}(X, F).

1.1 Proposition

Let Y be a connected compact complex space (not necessarily reduced), every component Y_{i} of Y being of positive dimension. Let D be an effective divisor on Y such that D|Y_{i} ≠ 0 for all i. Let F be a locally free sheaf on Y. Then there exists k_{0} ∈ ℕ such that

H^{0}(Y, F ⊗ O_{Y} (−kD)) = 0

for k ≥ k_{0}.

Proof. We have natural inclusions

H^{0}(Y, F ⊗ O_{Y} (−(k + 1)D)) ⊂ H^{0}(Y; F ⊗ O_{Y} (−kD)).

Take k_{0} such that this sequence is stationary for k ≥ k_{0}. Then s has to vanish at any order along D|Y_{i} for every i (s can be thought of locally as a tuple of holomorphic functions), hence s|Y_{i} = 0, and s = 0.

1.2 Corollary

Let S be a smooth compact complex surface containing an effective divisor C such that c_{1}(O_{S}(C)) = 0. Let B be a vector bundle on S. Then for a generic L ∈ Pic^{0}(X) we have

H^{0}(S, B ⊗ L) = H^{2}(S, B ⊗ L) = 0.

In particular χ(S, B) = −h^{1}(S, B ⊗ L) ≤ 0.

Proof. The vanishing H^{0}(S, B ⊗ O_{S}(−kC)) = 0 for large k follows from (1.1). Since O_{S}(kC) is topologically trivial for large suitable k, the required H^{0}−vanishing follows from semi-continuity. The H^{2}−vanishing follows by applying the previous arguments to B^{∗} ⊗ K_{S} and Serre duality.

1.3 Corollary

Let X be a smooth compact threefold with b_{2}(X) = 0 carrying an effective divisor D. Then H^{0}(X, B ⊗ L) = 0 for generic L in Pic^{0}(X) and every vector bundle B on X.

Proof. Since c_{1}(O_{X}(mD)) = 0 in H^{2}(X, ℤ), we can apply (1.1) to obtain

H^{0}(X, B ⊗ O_{X}(−kmD)) = 0

for k large. Now we conclude again by semi-continuity.

The last lemma is of course well-known; we include it for the convenience of the reader.

1.4 Lemma

Let X be a compact manifold of dimension n with a(X) = n. Then b_{2}(X) > 0.

Proof. Choose a birational morphism π : X̂ ⟶ X such that X̂ is a projective manifold. Take a general very ample divisor D̂ on X̂ and a general curve Ĉ ∈ X̂. Let D = π(D̂) and C = π(Ĉ). Then D meets C in finitely many points, hence D · C > 0, in particular c_{1}(O_{X}(D)) is not torsion in H^{2}(X; ℤ).

1.5 Lemma Let X be a smooth compact threefold and f : X ⟶ C be a surjective holomorphic map to the smooth curve C. Let F be a locally free sheaf on X. Then R^{i}f_{∗}(F) is locally free for all i.

Proof. (a) Note that local freeness is equivalent to torsion freeness, since dimC = 1. Hence the claim is clear for i = 0.

(b) Next we treat the case i = 2. We shall use relative duality (see [RRV71], [We85]); it states in our special situation (f is flat with even Gorenstein fibers) thatif R^{j}f_{∗}(G) is locally free for a given locally free sheaf G and fixed j, then

R^{2−j}f_{∗}(G^{∗} ⊗ ω_{X|C}) ≃ R^{j}f_{∗}(G)^{∗},

in particular R^{2−j}f_{∗}(G^{∗} ⊗ ω_{X|C}) is locally free. Here ω_{X|C} = ω_{X} ⊗ f^{∗}(ω_{C}^{∗}) is the relative dualising sheaf. Applying this to j = 0 and G = F^{∗} ⊗ ω_{X|C}^{∗} our claim for i = 2 follows.

(c) Finally we prove the freeness of R^{1}f_{∗}(F). By a standard theorem of Grauert it is sufficient that h^{1}(X_{y}, F|X_{y}) is constant, X_{y} the analytic fiber over y ∈ C. By flatness, χ(X_{y}, F|X_{y}) is constant, hence it is sufficient that h^{j}(X_{y}, F|X_{y}) is constant for j = 0 and j = 2. By the vanishing R^{3}f_{∗}(F) = 0, we have (see e.g. [BaSt76])

R^{2}f_{∗}(F)|{y} ≃ H^{2}(X_{y}, F|X_{y}).

Therefore h^{2}(X_{y}, F|X_{y}) is constant by (b). Finally

h^{0}(X_{y}, F|X_{y}) = h^{2}(X_{y}, F^{∗}|X_{y} ⊗ ω_{X_{y}}) = h^{2}(X_{y}, F^{∗} ⊗ ω_{X}|X_{y})

is constant by applying the same argument, now to F^{∗} ⊗ ω_{X}.

2. Criteria for the vanishing of H^{2}(X, B).

2.0 Set-up We fix in this section the following situation:

(1) X is a smooth compact threefold with b_{2}(X) = 0.

(2) V is a normal projective variety of dimension d = 1 or 2

(3) f : X ^{≻} V is a meromorphic map

(4) X̂ is a normal compact threefold with a birational map σ : X̂ ⟶ X such that the induced map f̂: X̂ ⟶ V is holomorphic; X̂ is smooth in case d = 1

(5) f̂ is equidimensional.

In our applications f will be an algebraic reduction.

2.1 Lemma Let f : X ⟶ V be as in 2.0,(1)-(3). Then (4) and (5) can always be achieved, possibly after changing V birationally.

Proof. The claim is obvious if d = 1, by elimination of indeterminacies. So let d = 2. Take a sequence of blow-ups ρ : X_{1} ⟶ X such that the induced map f_{1} : X_{1} ⟶ V is holomorphic. Then however f_{1} will in general not be equidimensional. To achieve this we could either apply Hironaka’s flattening theorem or we can argue in a more elementary way as follows. Let V_{0} = {y ∈ V |f_{1}^{−1}(y) is 1 − dimensional}. We obtain a holomorphic map μ : V_{0} ⟶ C_{1}(X_{1}), the space of 1-cycles on X_{1}. It is a standard fact in the theory of cycle spaces that μ extends to a meromorphic map μ : V ^{≻} C_{1}(X_{1}). Now let V′ = \overline{μ(V)} ⊂ C_{1}(X_{1}) and let X′ ⟶ V′ be the induced family. Modify V′ to V′′ so that V′′ is projective; then take X′′ = X′ ×_{V′} V′′ (base change). Finally let X̂ be the normalisation of X′′ and V̂ the normalisation of V′′. Then the induced map f̂: X̂ ⟶ V̂ has the properties we are looking for.

From now on we substitute V̂ by V.

First we will treat the case d = 2. We fix an ample effective divisor A on V in the set-up (2.0) and let L = σ_{∗}(f̂^{∗}(A))^{∗∗}. Then

f̂^{∗}(A) = σ^{∗}(L) ⊗ OX̂(−E)

with an effective divisor E on X̂ supported on the exceptional locus of σ.

2.2 Proposition H^{2}(X, B ⊗ L^{⊗m}) = 0 for every vector bundle B and m ≫ 0 (depending on B.)

Proof. First we claim

H^{2}(X̂, σ^{∗}(B ⊗ L^{⊗m})(−mE)) = 0  (1)

for large m. For the proof notice σ^{∗}(B ⊗ L^{⊗m})(−mE)) ≃ σ^{∗}(B) ⊗ f̂^{∗}(mA). We consider the Leray spectral sequence associated to f̂. Then

E_{2}^{2,0} = H^{2}(V, f̂_{∗}(σ^{∗}(B)) ⊗ mA) = 0, m >> 0,

and

E_{2}^{1,1} = H^{1}(V, R^{1}f̂_{∗}(σ^{∗}(B)) ⊗ mA) = 0, m >> 0

by Serre’s vanishing theorem. Moreover E_{2}^{0,2} = 0 simply because R^{2}f̂_{∗}(σ^{∗}(B)) = 0, all fibers of f̂ being 1-dimensional. Now the claim (1) follows from the spectral sequence. Let A_{m} be the complex subspace of X defined by the ideal sheaf σ_{∗}(−mE). Then we consider the following commutative diagram, the vertical arrows just being pullback maps (as arising in the Leray spectral sequence)

H^{2}(X̂, σ^{∗}(B ⊗ L^{m})(−mE)) ⟶ H^{2}(X̂, σ^{∗}(B ⊗ L^{m})) ⟶ H^{2}(mE, σ^{∗}(B ⊗ L^{m}))
↑  ↑  ↑
H^{2}(X, B ⊗ L^{m} ⊗ σ_{∗}(−mE)) ⟶  H^{2}(X, B ⊗ L^{m})  ⟶ H^{2}(X, B ⊗ L^{m} ⊗ O_{A_{m}})

Note that

H^{2}(X, B ⊗ L^{m} ⊗ O_{A_{m}}) = 0  (2)

since dim A_{m} ≤ 1. We claim that the pull-back map

α : H^{2}(X, B ⊗ L^{m}) ⟶ H^{2}(X̂, σ^{∗}(B ⊗ L^{m}))

is injective. In fact, let τ : X̃ ⟶ X̂ be a desingularisation. Since R^{1}(σ ∘τ)_{∗}(OX̃) = 0, the Leray spectral sequence yields the injectivity of the pull-back map

β : H^{2}(X, B ⊗ L^{m}) ⟶ H^{2}(X̃, (σ ∘ τ)^{∗}(B ⊗ L^{m})).

Since β factors through α, also α must be injective. Now the diagram together with (1) and (2) gives our claim (2.2).

2.3 Corollary Let X be a compact threefold with b_{2}(X) = 0 and with a(X) = 2. Let B be a vector bundle on X. Then

(a) H^{2}(X, B ⊗ L) = 0 for L ∈ Pic^{0}(X) generic.

(b) χ(X, B) ≤ 0

(c) c_{3}(X) = 0.

Proof. Since b_{2}(X) = 0, some multiple L^{⊗k} with L as in (2.2) is in Pic^{0}(X). Apply (2.2) to see that H^{2}(X, B ⊗ L^{mk}) = 0 for large m. Hence by semi-continuity (a) holds. By Serre duality H^{1}(X, B ⊗ L) = 0 for generic L. From (1.3) we therefore deduce χ(X, B ⊗ L) = 0. Hence χ(X, B) = 0. Now (c) follows from Riemann-Roch for B = T_{X}.

(2.4) We now treat the case that d = 1 and that f is not holomorphic. Again we fix an ample divisor A on V and let L = σ_{∗}(f̂^{∗}(O_{V} (A))^{∗∗}. We will prove :

2.5 Proposition Let X be a smooth compact threefold with a(X) = 1, b_{2}(X) = 0 and with nonholomorphic algebraic reduction f : X ^{≻} V. Then

χ(X, B) ≤ 0

for every vector bundle B on X.

Proof. As in (2.3) it is sufficient to prove

H^{q}(X, B ⊗ H) = 0

for q = 0 and q = 2 and some H ∈ Pic^{0}(X) (which of course can be taken individually in each case). For q = 0 the claim follows from (1.3). Turning to q = 2, we fix a positive integer k such that

H^{2}(F̂, σ^{∗}(B ⊗ L^{k})) = 0  (∗),

where F̂ is the general fiber of f̂. This k can be found since

H^{2}(F̂, σ^{∗}(B ⊗ L^{k})) = H^{0}(F̂, σ^{∗}(B^{∗} ⊗ L^{−k}) ⊗ ω−1 ; F̂),

which vanishes by (1.1), having in mind that σ^{∗}(L)|F̂ = f̂^{∗}(A)⊗OX̂(E)F̂ is effective and non-zero.

Now let

H = L^{m+k}

with m large to be specified in a moment. Arguing as in (2.2) we must prove

H^{2}(X̂, σ^{∗}(B ⊗ L^{m+k})(−mE)) = H^{2}(X̂, σ^{∗}(B ⊗ L^{k}) ⊗ f̂^{∗}(mA)) = 0.

By the Leray spectral sequence for f̂ we need for that

H^{1}(V, R^{1}f̂_{∗}(σ^{∗}(B ⊗ L^{k})) ⊗ mA) = 0

and

H^{0}(V, R^{2}f̂_{∗}(σ^{∗}(B ⊗ L^{k})) ⊗ mA) = 0.

The first is just Serre vanishing; for the second we claim

R^{2}f̂_{∗}(σ^{∗}(B ⊗ L^{k})) = 0.

But this is just (*) together with (1.5), saying that R^{2}f̂_{∗}(σ^{∗}(B ⊗L^{k})) is locally free. This finishes the proof.

3. The case of a holomorphic algebraic reduction

3.0 Setup In this section we fix a smooth compact 3-fold X with holomorphic algebraic reduction f : X ⟶ V to the smooth curve V of genus g. We shall always assume b_{2}(X) = 0. Of course f has connected fibers.

3.1 Theorem Let B be a holomorphic vector bundle on X. Then χ(X, B) = 0.

3.2 Corollary c_{3}(X) = 0.

Proof. This is just Riemann-Roch for χ(X, T_{X}).

Proof of 3.1 Let W = H^{1}(X, O_{X}). Then every element in W is represented as a topologically trivial line bundle. Let A be an ample line bundle on V. Then for m ≫ 0 we have

H^{0}(V, R^{j}f_{∗}(B) ⊗ A^{−m}) = 0  (1)

for all j ≥ 0. We claim that (1) implies

H^{0}(V, R^{j}f_{∗}(B ⊗ L)) = 0  (2)

for L ∈ W general. In fact, consider locally the universal bundle L̂ on X × W. Let F = f × id : X × P ⟶ V × P and B̂ = pr_{X}^{∗}(B). The coherent sheaf R^{j}F_{∗}(B̂ ⊗ L̂) satisfies

R^{j}F_{∗}(B̂ ⊗ L̂)|V × {t} ≃ R^{j}f_{∗}(B ⊗ L̂_{t}),

where L̂_{t} is the line bundle corresponding to t ∈ W. Choose m ≫ 0 and t_{0} such that f^{∗}(A^{−m}) = L̂_{t_{0}}. By (1) and the assumption b_{2}(X) = 0 we have

H^{0}(V, R^{j}f_{∗}(B ⊗ L̂_{t_{0}})) = 0.

Hence it is sufficient to show that R^{j}F_{∗}(B̂ ⊗L̂) is flat with respect to the projection q : V × P ⟶ W, over a Zariski open set of W, then the usual semi-continuity theorem gives the claim (2). Now R^{j}f_{∗}(B ⊗ L̂_{t}) is locally free on V = V × t for every t by (1.5), hence it is clear that there is a Zariski open set U ⊂ W such that R^{j}F_{∗}(B̂ ⊗ L̂) has constant rank over U, hence is locally free over U (observe just that the set where the rank of a coherent sheaf is not minimal is analytic). On the other hand we have for m ≫ 0 and all i ≥ 1, j ≥ 0 by Serre’s vanishing theorem

H^{i}(V, R^{j}f_{∗}(B) ⊗ A^{m}) = 0  (3).

In the same way as (2) follows from (1) we conclude from (3) that

H^{i}(V, R^{j}f_{∗}(B ⊗ L)) = 0  (4)

for general L ∈ P. Now (2) and (4) give by the Leray spectral sequence H^{i}(X, B ⊗ L) = 0 for general L ∈ P and all i ≥ 0. In particular χ(X, B ⊗ L) = 0. Since c_{k}(B ⊗ L) = c_{k}(B), Riemann-Roch gives our claim χ(X, B) = 0.

4. The Main Theorem and Examples

Putting (2.6), (2.8) and (3.2) together, we obtain

4.1 Theorem Let X be a compact complex manifold of dimension 3. If b_{2}(X) = 0 and if c_{3}(X) ≠ 0, then every meromorphic function on X is constant.

Proof. We only have to notice that a(X) < 3 by (1.2) and then to apply the above mentioned propositions.

4.2 Corollary Let X be a compact complex threefold homeomorphic to the sphere S^{6}. Then every meromorphic function on X is constant.

Proof. Note that c_{3}(X) = χ_{top}(S^{6}) = 2 and apply (4.1).

The condition c_{3}(X) ≠ 0 in (4.1) can be translated into 2b_{1}(X) + b_{3}(X) ≠ 2.

4.3 Corollary Let X be a compact threefold with b_{2}(X) = 0 and a(X) = 1 or 2. Let X ^{≻} V be an algebraic reduction to the smooth variety V.

(1) If dim V = 1, then V = ℙ_{1}.

(2) If dim V = 2 and κ(X) = −∞, then V is rational.

Proof. By the Main Theorem we have b_{1}(X) ≤ 1. Let X̂ ⟶ X be a sequence of blow-ups with smooth centers such that the induced map f̂: X̂ ⟶ V is holomorphic. Notice b_{1}(X̂) ≤ 1. Now the Leray spectral sequence gives b_{1}(V) ≤ b_{1}(X̂), since f̂ has connected fibers and therefore f̂_{∗}(ℂ) = ℂ. So V = ℙ_{1} in case dim V = 1.

10 FRÉDÉRIC CAMPANA, JEAN-PIERRE DEMAILLY, AND THOMAS PETERNELL

If dim V = 2 we still need to know that κ(V) = −∞. This comes from [Ue87].

In the rest of this section we give examples of threefolds with a(X) > 0, b_{2}(X) = 0 and c_{3}(X) = 0 so that (4.1) is sharp.

4.4 Example The so-called Calabi-Eckmann threefolds are compact threefolds homeomorphic to S^{3} × S^{3}, see [Ue75]. They can be realised as elliptic fiber bundles over ℙ_{1} × ℙ_{1}. Hence a(X) = 2, b_{1}(X) = b_{2}(X) = 0 and b_{3}(X) = 2. We now show that Calabi-Eckmann manifolds can be deformed to achieve a(X) = 1 or a(X) = 0 and b_{1} = b_{2} = 0, b_{3} = 2. We choose positive real numbers a, b, c and let B = ℂ^{2} \ {(0, 0)}. We define the following action of ℂ on B × B :

(t, x, y, u, v) ↦ (exp(t)x, exp(at)y, exp(ibt)u, exp(ict)v).

One checks easily that this action is holomorphic, free and almost proper so that the quotient X exists and is a compact manifold. If a = 1 and b = c, then X is a Calabi-Eckmann manifold. If however log a ≠ ℚ and b = c resp. log a ≠ ℚ and log b, log c are ℚ-linearly independent, then a(X) = 1 resp. a(X) = 0.

4.5 Example Hopf threefolds of the form

ℂ^{3} \ {(0, 0, 0)}/ℤ,

with the action of ℤ ≃ {λ^{k}; k ∈ ℤ} being defined by

λ(x, y, z) = (αx, βy, γz), 0 < |α|, |β|, |γ| < 1

are homeomorphic to S^{1} × S^{5}. They have a(X) = 0, 1 or 2 and b_{1}(X) = 1, while b_{2}(X) = b_{3}(X) = 0. This realises the other possibility for the pair (b_{1}, b_{3}) when b_{2} = 0 and a(X) > 0, see the Main Theorem.

Notice that the algebraic reduction is holomorphic in (4.4) but not in (4.5).

4.6 Example We finally give other examples of compact threefolds X with a(X) = 0 and b_{1} = b_{2} = 0, b_{3} = 2. Let Γ ⊂ Sl(2, ℂ) be a torsion free cocompact lattice in such a way that X := Sl(2, ℂ)/Γ has b_{1}(X) = 0. This last condition is not automatic. Let Y = SU(2)\ Sl(2, ℂ)/Γ; then Y is a compact differentiable manifold admitting a differentiable fibration π : X ⟶ Y with S^{3} is fiber. Since b_{1}(X) = 0, we also have b_{1}(Y) = 0, hence b_{2}(Y) = 0 by Poincaré duality. Now the Leray spectral sequence immediately gives b_{2}(X) = 0, the fibers of π being 3-spheres. b_{3}(X) = 2 is again clear from the Leray spectral sequence. Finally the fact that X does not carry any non-constant meromorphic function results from [HM83] from which we even deduce that X does not carry any hypersurface (anyway X is homogeneous).

5. On the finer structure of threefolds with b_{2}(X) = 0 and a(X) = 1.

In this section we investigate more closely threefolds X with b_{2}(X) = 0 and holomorphic algebraic reduction f : X ⟶ V to a curve V. We know already that V is rational. We fix these notations. The key to our investigations is

5.1 Theorem The restriction r : H^{1}(X, O_{X}) ⟶ H^{1}(F, O_{F}) is surjective where F is the general smooth fiber of f.

We need some preparations for the proof of (5.1). Let Δ ⊂ V be a finite non empty set such that A = f^{−1}(Δ) ⊂ X contains all singular fibers of f. Let W = V \ Δ, U = f^{−1}(W) so that g = f|U has only smooth fibers, hence is a C^{∞}- fibration. Let D_{i}, 1 ≤ i ≤ r be the irreducible components of A and let s = card Δ be the number of connected components of A. Furthermore we set t = b_{1}(F), F the general smooth fiber of f and recall that g is the genus of V. For a non-compact space Z we let b_{i}(Z) = dim H_{i}(Z, ℝ), if this space is finite-dimensional. We prepare the proof of (3.1) by three lemmas.

5.2 Lemma

(1) The natural exact sequence of groups

1 = π_{2}(W) ⟶ π_{1}(F) ⟶ π_{1}(U) ⟶ π_{1}(W) ⟶ 1

is exact and (non-canonically) split.

(2) b_{1}(U) = b_{1}(W) + b_{1}(F) = 2g + s − 1 + t ≤ b_{1}(X) + s − 1 + t.

(3) r = s − 1 + t + 2g − b_{1}(X).

Proof. (1) Since W is a non-compact Riemann surface, π_{1}(W) is a free group of 2g + s − 1 generators and since W is uniformised by either ℂ or by the unit disc, we have π_{2}(W) = 1. Hence the exact homotopy sequence of the fibration g : U ⟶ W gives the exact sequence of groups stated in (1). Since π_{1}(W) is a free group, the sequence splits.

(2) From (1) we deduce that

H_{1}(U, ℤ) ≃ H_{1}(W, ℤ) ⊕ H_{1}(F, ℤ).

Moreover f_{∗} : π_{1}(X) ⟶ π_{1}(V) is surjective since the fibers of f are connected. Hence (2) follows.

(3) The cohomology sequence with rational coefficients of the pair (X, A) gives

0 = H^{4}(X) ⟶ H^{4}(A) ⟶ H^{5}(X, A) ⟶ H^{5}(X) ⟶ H^{5}(A) = 0.

By duality we have H^{5}(X, A) ≃ H_{1}(U) and H^{5}(X) ≃ H_{1}(X). Hence

r = dim H^{4}(A) = b_{1}(U) − b_{1}(X) = 2g + s − 1 + t − b_{1}(X)

by (2), as claimed.

Now choose an integer m > 0 such that c_{1}(O_{X}(mD_{i})) = 0 in H^{2}(X, ℤ) for all 1 ≤ i ≤ r. Let G = ⨁ ℤ[mD_{i}] ≃ ℤ^{r} be the free abelian group of r generators and let ϕ : G ⟶ Pic^{0}(X) be given by sending D = ∑_{i} a_{i}mD_{i} to O_{X}(D).

5.3 Lemma Let K = Kerϕ and H = Imϕ. Then rk K ≤ s − 1 and rk H ≥ r − s + 1.

Proof. Write Δ = {x_{1}, …, x_{s}} and define G_{0} = ⨁_{j} ℤ[mx_{j}] ⊂ Div(V). Let K_{0} = {g ∈ G|deg(g) = 0}. Then K_{0} ≃ ℤ^{s−1}. Consider the pull-back map f^{∗} : Div(V) ⟶ Div(X) and let K_{1} = f^{∗}(K_{0}). We claim that K ⊂ K_{1}, which will prove the lemma, since then rk K ≤ rk K_{1} ≤ rk K_{0} ≤ s − 1. To show K ⊂ K_{1}, consider D = ∑_{i} a_{i}mD_{i} = ∑ α_{j}mD_{j} − ∑ β_{k}mD_{k} = D′ − D′′ with α_{j}, β_{k} > 0 and D_{j} ≠ D_{k} for all j ≠ k. If D ∈ K, then O_{X}(D) ≃ O_{X}, hence O_{X}(D′) ≃ O_{X}(D′′). Letting L = O_{X}(D′), we see that h^{0}(X, L) ≥ 2, therefore L ∈ f^{∗}(Pic(V) since f is the algebraic reduction of X. Thus D_{i} ∈ K_{1}, hence D ∈ K_{1}, proving the claim.

The last ingredient in the proof of (5.1) is provided by

5.4 Lemma The kernel of the restriction map α : H^{1}(X, O_{X}) ⟶ H^{1}(U, O_{U}) equals the image of f^{∗} : H^{1}(V, O_{V}) ⟶ H^{1}(X, O_{X}).

Proof. The Leray spectral sequence gives the following commutative diagram with exact rows :

0 ⟶ H^{1}(V, O_{V}) ⟶^{f^{∗}} H^{1}(X, O_{X}) ⟶ H^{0}(V, R^{1}f_{∗}(O_{X})) ⟶ 0
↓ ↓ ↓
0 = H^{1}(W, O_{W}) ⟶ H^{1}(U, O_{U}) ⟶ H^{0}(W, R^{1}g_{∗}(O_{W})) ⟶ 0

Now R^{1}f_{∗}(O_{X}) is locally free by (1.5). This implies that the restriction map

α̃ : H^{0}(V, R^{1}f_{∗}(O_{X})) ⟶ H^{0}(W, R^{1}g_{∗}(O_{W}))

is surjective, hence also α : H^{1}(X, O_{X}) ⟶ H^{1}(U, O_{U}) by the diagram. This proves (5.4).

We are now able to finish the proof of (5.1). We consider again H ⊂ Pic^{0}(X), the image of ϕ : G ⟶ Pic^{0}(X). H has rank ≥ r − s + 1 = t + (2g − b_{1}(X)) by (5.2). Let H̃ be the inverse image of H under the natural map H^{1}(X, O_{X}) ⟶ Pic^{0}(X). Then

rk H̃ = rk H + b_{1}(X) ≥ 2g + t.

Let H_{1} = α(H̃) ⊂ H^{1}(U, O_{U}). We claim that

rk H_{1} ≥ t.  (∗)

In fact, by (5.4) H̃_{0} := Ker(α|H̃) = H̃ ∩ f^{∗}(H^{1}(V, O_{V})). Now H̃_{0} is a discrete subgroup of f^{∗}(H^{1}(V, O_{V}) ≃ ℂ^{g}, so that rkH̃_{0} ≤ 2g. Hence rk H_{1} ≥ 2g +t−2g = t, proving (*). Let ζ : H^{1}(U, ℤ) ⟶ H^{1}(U, O_{U}) be the canonical map; then H_{1} ⊂ ζ(H^{1}(U, ℤ)) since O_{U}(D) ≃ O_{U} for all D ∈ G. We therefore conclude that the group H_{2} := β(H_{1}) with

β : H^{1}(U, O_{U}) ⟶ H^{0}(W, R^{1}g_{∗}(O_{U}))

the canonical isomorphism, is of finite index in

Γ := Im(H^{0}(W, R^{1}f_{∗}ℤ)) ⟶ H^{0}(W, R^{1}g_{∗}(O_{U})).

By 5.2(2) we conclude that rk H^{0}(W, R^{1}f_{∗}(ℤ)) = t, hence H_{2} has finite index in Γ. Now take a general smooth fiber F and let λ : H^{0}(W, R^{1}g_{∗}(O_{U})) ⟶ H^{1}(F, O_{F}) be the restriction map. Then we conclude that λ(H_{2}) is of finite index in

Im(H^{1}(F, ℤ) ⟶ H^{1}(F, O_{F})).

Since the linear span of this image in H^{1}(F, O_{F}) is H^{1}(F, O_{F}), we deduce from the construction of H_{2} that the restriction map H^{1}(X, O_{X}) ⟶ H^{1}(F, O_{F}) is surjective proving finally (5.1).

(5.5) We now study the structure of the smooth fibers F of f. Since K_{F} ≡ 0 and c_{2}(F) = c_{2}(X)|F = 0, we conclude from the classification of surfaces that F is one of the following: a Hopf surface, an Inoue surface, a Kodaira surface (primary or secondary), a torus or hyperelliptic (see e.g. [BPV84]). By [Ka69] however, F cannot be hyperelliptic. The reason is the existence of a relative Albanese reduction in that case.

5.6 Theorem The general fiber F of f cannot be a Kodaira surface nor a Hopf surface with algebraic dimension 1.

Proof. Let F_{0} be a fixed smooth fiber and assume that F_{0} is a Kodaira surface or a Hopf surface with a(F_{0}) = 1. Let g_{0} : F_{0} ⟶ C_{0} be ”the” algebraic reduction which is an elliptic fiber bundle. Let L_{0} = g_{0}^{∗}(G_{0}) with G_{0} very ample on C_{0}.

(1) There exists a line bundle L̃ on X with L̃|F_{0} = L_{0}. Proof. By passing to some power L_{0}^{m} if necessary, we have c_{1}(L_{0}) = 0 in H^{2}(X, ℤ). Let

λ_{1} : H^{1}(X, O_{X}) ⟶ Pic(X)

and

λ_{2} : H^{1}(F_{0}, O_{F_{0}}) ⟶ Pic(F_{0})

be the canonical maps, r : H^{1}(X, O_{X}) ⟶ H^{1}(F_{0}, O_{F_{0}}) the restriction map. Choose α ∈ H^{1}(F_{0}, O_{F_{0}}) with λ_{2}(α) = L_{0}. Since r is surjective by (4.2), we find β ∈ H^{1}(X, O_{X}) with r(β) = α. Now let L̃ = λ_{1}(β).

(2) Let F be any smooth fiber of f. Then κ(L̃|F) = 1. In fact, it follows from the local freeness of R^{j}f_{∗}(L̃) (1.5) that

f_{∗}(L̃^{μ})|{y} ≃ H^{0}(F, L̃^{μ}),

where F = f^{−1}(y), see [BaSt76, chap.3, 3.10].

(3) From the generically surjective morphism

f^{∗}f_{∗}(L̃^{m}) ⟶ L̃^{m},

(m ≫ 0), we obtain a meromorphic map

g : X ^{≻} ℙ(f_{∗}(L̃^{m})),

which, restricted to F is holomorphic and just gives the algebraic reduction of F. Let Z be the closure of the image of g. Then f factors via the meromorphic map h_{1} : X ^{≻} Z and the holomorphic map h_{2} : Z ⟶ V. Now h_{2} is the restriction of the canonical projection ℙ(f_{∗}(L̃^{m})) ⟶ V, therefore h_{2} is a projective morphism and Z is projective. Hence a(X) ≥ 2, contradiction.

From (5.6) it follows that F can only be an Inoue surface, a Hopf surface without meromorphic functions or a torus. In order to exclude by a similar method as in (5.6) also tori of algebraic dimension 1, we would need the existence of a relative algebraic reduction (the analogue of h_{2} : X ^{≻} Z) in that case, too. We now look more closely to the structure of f.

5.7 Proposition Assume that F is not a torus. Then

(1) R^{1}f_{∗}(O_{X}) = O_{V}

(2) R^{2}f_{∗}(O_{X}) = 0

(3) dimH^{1}(X, O_{X}) = 1

(4) H^{2}(X, O_{X}) = H^{3}(X, O_{X}) = 0.

Proof. (2) Since H^{2}(F, O_{F}) = 0, the sheaf R^{2}f_{∗}(O_{X}) is torsion, hence 0 by (1.5). Then H^{3}(X, O_{X}) = 0 is immediate from the Leray spectral sequence.

(1) Since h^{1}(O_{F}) = 1, R^{1}f_{∗}(O_{X}) is a line bundle on V. Let

d = degR^{1}f_{∗}(O_{X}).

Then Riemann-Roch gives χ(R^{1}f_{∗}(O_{X})) = d + 1. On the other hand the Leray spectral sequence together with H^{3}(X, O_{X}) = 0 and (2) yields

χ(R^{1}f_{∗}(O_{X})) = h^{1}(O_{X}) − h^{2}(O_{X}) = −χ(O_{X}) + 1.

We conclude d = −χ(O_{X}) = 0. This proves (1). Now (3) and the second part of (4) are obvious.

5.8 Remark In case F is a torus, R^{1}f_{∗}(O_{X}) is a rank 2 bundleand R^{2}f_{∗}(O_{X}) is a line bundle. Using (4.2) it is easy to see that

(1) R^{1}f_{∗}(O_{X}) = O(a) ⊕ O(b) with a, b ≥ 0

(2) R^{2}f_{∗}(O_{X}) = O(a + b). Note that (2) gives dually f_{∗}(ω_{X|V}) = O(−a − b). Usually one expects the degree of f_{∗}(ω_{X|V}) to be semi-positive, but here we are in a highly non-Kähler situation where it might happen that the above degree is negative, see [Ue87].

5.9 Proposition Assume that F is not a torus. Then H^{0}(X, Ω_{X}^{i}) = 0, 1 ≤ i ≤ 3.

Proof. For i = 3 the claim follows already from (5.7) and (5.8).

(1) First we treat the case i = 1. Let ω be a holomorphic 1-form. Let j : F ⟶ X be the inclusion. Then j^{∗}(ω) = 0, hence at least locally near F we have ω = f^{∗}(η), hence dω = 0 near F and therefore the holomorphic 2-form dω = ∂ω = 0 everywhere. Now the space of closed holomorphic 1-forms can be identified with H^{0}(X, dO_{X}) and as it is well known (see e.g. [Ue75]), we have the inequality

2h^{0}(X, dO_{X}) ≤ b_{1}(X),

hence b_{1}(X) ≤ 1 gives our claim.

(2) In case i = 2, we again have j^{∗}(ω) = 0. Let U be a small open set in V such that f|f^{−1}(U) is smooth. z be a coordinate on U and h = f^{∗}(z). Then we conclude that

ω|f^{−1}(U) = dh ∧ α

with some holomorphic 1-form α. Now again j^{∗}(α) = 0 and therefore α = f^{∗}(β). In total ω = f^{∗}(dz ∧ β) = 0.

5.10 Corollary Assume that F is not a torus. Then there exists some x such that f_{∗}(Ω_{X}^{1}) = ℂ_{x}, i.e. a sheaf supported on x with a 1-dimensional stalk at x. In particular f has at most one normal singular fiber and such a fiber has exactly one singularity which has embedding dimension 3.

Proof. The exact sequence

0 ⟶ f^{∗}(Ω_{V}) ⟶ Ω_{X}^{1} ⟶ Ω_{X|V}^{1} ⟶ 0

together with (5.7) and the projection formula leads to an exact sequence

0 ⟶ Ω_{V}^{1} ⟶ f_{∗}(Ω_{X}^{1}) ⟶ f_{∗}(Ω_{X|V}^{1} ⟶^{α} Ω_{V}^{1} ⊗ R^{1}f_{∗}(O_{X}) ≃ Ω_{V}^{1} .

Since F has no holomorphic 1-forms, f_{∗}(Ω_{X|V}^{1} ) is a torsion sheaf, therefore α = 0. Now observe the following facts.

(1) H^{0}(X, Ω_{X}^{1}) = 0 by (5.9)

(2) H^{1}(X, f^{∗}(Ω_{X})) = H^{1}(V, Ω_{V}^{1} )

(3) H^{1}(Ω_{V}^{1} ) ⟶ H^{1}(X, Ω_{X}^{1}) is 0 (since b_{2}(X) = 0). Then taking cohomology of the first sequence, it follows that

h^{0}(X, Ω_{X|V}^{1} ) = 0,

hence the second sequence gives our claim.

We can say something more about the structure of the singular fibers of f.

5.11 Proposition Assume that F is not a torus. Let A be a union of fibers containing all singular fibers of f. Let s = card(f(A)) and r the number of irreducible components of A. Then r = s, i.e. all fibers of f are irreducible and b_{1}(X) = 0.

Proof. Since F is an Inoue surface or a Hopf surface, we have b_{1}(F) = 0, moreover V = ℙ_{1}, hence the claim follows from (5.2).

5.12 Remark In case F is a torus, the same proof gives r = s + 3. It seems rather reasonable to expect that tori actually cannot appear as fibers of f. Observe that f must have a singular fiber in this case because of r = s + 3. So a study of the singular fibers is needed to exclude tori as fibers of f. We hope to come back to this question in a later paper.

5.13 Proposition Assume that F is not a torus. Then h^{1,1} = h^{1,2} = h^{2,1} = 1 (so that we know all Hodge numbers of X.)

Proof.(1) h^{1,2} = h^{2,1} is of course Serre duality.

(2) By (5.9) and χ(X, Ω_{X}^{1}) = 0 it suffices to see h^{1,3} = 0 in order to get h^{1,1} = h^{1,2}. But this follows again from Serre duality and (5.9).

(3) From the exact sequence

0 ⟶ f^{∗}(Ω_{V}^{1} ) ⟶ Ω_{X}^{1} ⟶ Ω_{X|V}^{1} ⟶ 0  (S)

we deduce that it suffices to show

(a) h^{2}(X, f^{∗}(Ω_{V}^{1} )) = 1

(b) h^{2}(X, Ω_{X|V}^{1} ) = 0

in order to get h^{1,2} ≤ 1.

(a) By the Leray spectral sequence and (5.7) we have h^{2}(X, f^{∗}(Ω_{V}^{1} )) = h^{2}(V, Ω_{V}^{1} ) = 1.

(b) Again we argue by the Leray spectral sequence. Since R^{1}f_{∗}(Ω_{X|V}^{1} ) is a torsion sheaf, we need only to show that

R^{2}f_{∗}(Ω_{X|V}^{1} ) = 0.

But, taking f_{∗} of (S), R^{2}f_{∗}(Ω_{X|V}^{1} ) is a quotient of R^{2}f_{∗}(Ω_{X}^{1}) which is 0 by H^{2}(O_{F}) = H^{2}(Ω_{F}^{1} ) = 0 and (1.5).

(4) We finally show h^{1,1} ≠ 0 to conclude the proof. Let (E_{r}^{pq}) be the Frölicher spectral sequence on X. Since b_{1}(X) = 0 by (5.11), we get E_{∞}^{0,1} = 0. Hence E_{2}^{0,1} = 0. On the other hand

E_{2}^{0,1} = Ker∂ : E_{1}^{0,1} ⟶ E_{1}^{1,1}.

Since E_{1}^{p,q} = H^{pq}(X), we conclude that H^{1}(X, O_{X}) injects into H^{1,1}(X). So by (5.7) H^{1,1}(X) ≠ 0.

We finally collect all our knowledge in the case the general fiber of f is not a torus.

5.14 Theorem Let X be a smooth compact threefold with b_{2}(X) = 0 and holomorphic algebraic reduction f : X ⟶ V to the smooth curve V. Assume that the general smooth fiber is not a torus. Then:

(1) b_{1}(X) = 0, b_{3}(X) = 2

(2) any smooth fiber of f is a Hopf surface without meromorphic functions or an Inoue surface.

(3) the Hodge numbers of X are as follows: h^{1,0} = 0, h^{0,1} = 0, h^{2,0} = 0, h^{1,1} = 1, h^{0,2} = 0, h^{3,0} = 0, h^{2,1} = 1 (the others are determined by these via Serre duality).

(4) all fibers of f are irreducible. There is at most one normal singular fiber.

References

[BaSt76] Banica, C.; Stanasila, O.: Algebraic methods in the global theory of complex spaces. Wiley 1976

[BPV84] Barth, W.; Peters, C.; van de Ven,: Compact complex surfaces. Erg. d. Math. (3), Band 4, Springer 1984

[HM83] Huckleberry, A.T.; Margulis, G.: Invariant analytic hypersurfaces. Inv. Math. 71, 235-240 (1983)

[Ka69] Kawai, S.: On compact complex analytic manifolds of complex dimension 3, II.

J. Math. Soc. Japan 21, 604-616 (1969)

[RRV71] Ramis, J.; Ruget, G.; Verdier, J.L.: Dualité relative en géométrie analytique

complexe. Inv. Math. 13, 261-283 (1971)

[Ue75] Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Math. 439, Springer 1975

[Ue87] Ueno, K.: On compact analytic threefolds with non-trivial Albanese tori. Math. Ann. 278, 41-70 (1987)

[We85] Wehler, J.: Der relative Dualitätssatz für Cohen-Macaulay-Räume. Schriftenr. des Math. Instituts der Univ. Münster, 2.Serie, Heft 35 (1985)

Frédéric Campana, Département de mathématiques, Université de Nancy, BP 239, 54506 Vandoeuvre les Nancy, France (frederic.campana@iecn.u-nancy.fr)

Jean-Pierre Demailly, Université de Grenoble, Institut Fourier, BP 74, U.R.A. 188 du C.N.R.S., 38402 Saint-Martin d’Heres, France (demailly@ujf-grenoble.fr)

Thomas Peternell, Mathematisches Institut, Universität Bayreuth, 95440

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