The algebraic dimension of compact complex threefolds with vanishing second Betti number

We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function.


Introduction
The paper [CDP98] studied compact complex threefolds X such that the second Betti number b 2 (X) = 0. The main result is based on Lemma 1.5, which happens to be incorrect in general (but might still hold in the context of the paper). In any case, some of the statements and proofs need to be adapted to fill the possible gaps; this is done in the present corrigendum, with special regard to potential complex structures on the 6-sphere.

Statement of the results
We prove [CDP98, Theorem 2.1] in full generality in the case where X has a meromorphic non-holomorphic map X P 1 . In the remaining case, X has algebraic dimension 1 and the algebraic reduction f : X → C is holomorphic. In this case we prove that c 3 (X) 0; for simplicity, we will assume not only that b 2 (X) = 0 but slightly more strongly that H 2 (X, Z) = 0 and, moreover, that H 1 (X, Z) = 0, hence C P 1 . This suffices to treat the main application of complex structures on S 6 .
In summary, we shall prove the following result.
Another proof has been given in [LRS18]. This is equivalent to saying that H 0 (X c , σ * (B * ⊗ M * ) ⊗ OX c (−kE) ⊗ ωX c ) = 0 for c ∈ C * . Fixing any point c 0 ∈ C * , this number k 0 = k 0 (c) clearly exists; in the case where X c is reducible, we apply [CDP98, Proposition 1.1]. Hence the support of the direct image sheaf R 2 f * (σ * (B ⊗ L k 0 )) is contained in a finite set C k 0 in C. Since σ * (L)|X c is effective, it follows that C k ⊂ C k 0 . Thus, enlarging k 0 if necessary, (9) is verified.
Hence we only need to consider the fibersX c with c ∈ C * . Let P be the set of line bundles M of the form with m i positive integers and S i fiber components of f not meeting E (the S i considered as surfaces in X). Since all line bundles M ∈ P are trivial on X \ f −1 (C * ), our previous considerations imply the existence of a number k 0 such that for all k k 0 and all M ∈ P , We will now construct a line bundle M ∈ P such that for a suitable number k. Fix a point c ∈ C * . Let F 0 ⊂ X c be the sub-divisor ofX c consisting of all components meeting E; let further F 1 ⊂X c be the sub-divisor consisting of all components which meet F 1 but not E. Continuing in this way, we obtain a decomposition Algebraic dimension of compact complex threefolds of sub-divisors F j ⊂ X c which pairwise do not have common components and which have the property that all components of F j meet F j−1 but do not meet F k for k < j − 1. Now choose m r 0 such that H 0 (F r , OX (−m r F r−1 ) ⊗ σ * (B * ) ⊗ ωX c |F r ) = 0. This is possible by our construction. Next choose m r−1 m r such that Continuing in this way, we obtain a line bundle Since the component F 0 meets E, it needs special treatment. We observe that with some effective σ-exceptional divisor E . Hence, choosing m 1 0 and setting Finally, enlarging k, we get this settles (7). As to (8), we fix k as in (7) and then apply Serre's vanishing theorem to the ample divisor A to obtain t.

General structure of the fibers
From now on, for the rest of the paper, we let X be a compact complex manifold of dimension 3 with H 1 (X, Z) = H 2 (X, Z) = 0 and holomorphic algebraic reduction f : X → C to the curve C P 1 . We will freely use the theory of compact complex surfaces, in particular of non-Kähler surfaces, and refer to [BHPV04] as general reference. Deviating from [BHPV04], we call bi-elliptic surfaces hyperelliptic.
An application of Riemann-Roch gives the following lemma. Proof. By Riemann-Roch, Lemma 4.2. Let s be the number of singular fibers and r be the numbers of irreducible components of the singular fibers. Then where X c is a smooth fiber. Moreover, H 1 (X c , Z) is torsion-free for all smooth fibers X c .
Proof. The first assertion is [CDP98, Lemma 3.2]. For the second, fix a smooth fiber X c , let A be a subset of the union of all singular fibers of f , and set X = X \ A. As seen in the proof of [CDP98, Lemma 3.2], hence it suffices to show that H 1 (X , Z) is torsion-free. To do this, we consider the cohomology sequence for pairs, Actually, H 5 (X, Z) = 0. Consequently, Lemma 4.3. Let X c be a smooth fiber of f . Then X c is either a primary Hopf surface, an Inoue surface, a torus, a hyperelliptic surface with torsion-free first homology group or a primary Kodaira surface with torsion-free first homology group.

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Algebraic dimension of compact complex threefolds Proof. Note first that K Xc is topologically trivial, since K X is topologically trivial, due to b 2 (X) = 0. Then we use the tangent sequence and observe that c 2 (X) = 0, since b 4 (X) = 0. Thus c 2 (X c ) = 0. Since the (sufficiently) general fiber of an algebraic reduction has non-positive Kodaira dimension [Uen75,Theorem 12.1], so does every smooth fiber (see, for example, [BHPV04,VI.8.1]. Then we conclude by surface classification and using the torsion-freeness of H 1 (X c , Z) (Lemma 4.2). Note here that H 1 (X c , Z) for a secondary Kodaira surface X c has torsion, since c 1 (X c ) is torsion in H 2 (X c , Z) (then apply the universal coefficient theorem) and that a secondary Hopf surface has torsion in H 1 (X c , Z) by definition. 2 Corollary 4.4. All fibers of f are irreducible unless the general fiber of f is a torus, a hyperelliptic surface with torsion-free first homology or a primary Kodaira surface with torsionfree first homology.
We fix some notation for the rest of the paper.
Notation 4.5. Let S ⊂ X be an irreducible reduced surface. In particular, S is Gorenstein. We denote by ω S the dualizing sheaf, which is a line bundle. Let be the normalization of S; denote by N ⊂ S the non-normal locus, equipped with the complex structure given by the conductor ideal. LetÑ ⊂S be the complex-analytic preimage. Let π :Ŝ →S be a minimal desingularization and σ :Ŝ → S 0 be a minimal model. For the class of ω S we write K S , and analogously for ωS, etc.
Lemma 4.6. By Notation 4.5, we have with an effective divisor E supported on the exceptional locus of π, andN the strict transform ofÑ inŜ. Moreover, there are exact sequences As an immediate consequence, we note the following proposition.
Proof. The first equation in (a) follows from Lemma 4.6. As to the second equation in (a), observe by Serre duality that since ω S ≡ 0. For the same reasons Proof. Consider the exact sequence If H 0 (X c , T Xc ) = 0, the assertion is clear. So it remains to treat the case where X c has no vector fields. Then by Lemma 4.3 and [Ino74], X c is an Inoue surface of type S M or S − N , in which cases H 1 (X c , T Xc ) = 0. Thus X c is rigid and κ is surjective, so that H 0 (X c , T X |X c ) = 0 also in these cases. 2 Corollary 4.9. Let X c = λS be a fiber with S an irreducible singular surface and λ 1. Then there exists a finiteétale cover S → S such that H 0 (S , T S ) = 0.
Proof. We consider the tangent sequence If λ = 1, then N S/X O S . However, κ is not surjective, since S is singular. Hence By semicontinuity and Proposition 4.8, H 0 (S, T X |S) = 0 and we conclude. If λ 2, arguing in the same way, we obtain a torsion line bundle L on X such that Then we pass to a finiteétale coverS → S to trivialize L |S . 2 Remark 4.10. A vector field v ∈ H 0 (S, T S ) canonically induces a vector field v 0 ∈ H 0 (S 0 , T S 0 ). For brevity, we say that v 0 comes from S.
Proposition 4.11. Let X c = λS with S singular and irreducible. Then S is non-normal.

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Algebraic dimension of compact complex threefolds Proof. Suppose S normal and let π :Ŝ → S be a minimal desingularization and σ :Ŝ → S 0 a minimal model. (a) Suppose that S has only rational singularities, hence S has only rational double points. Then KŜ ≡ 0, in particularŜ is a minimal surface containing (−2)-curves. By surface classification,Ŝ is either a K3 surface, an Enriques surface, of type VII or non-Kähler of Kodaira dimension κ(Ŝ) = 1. The first two cases are impossible since IfŜ is of type VII, then it must be a Hopf surface or an Inoue surface, since KŜ ≡ 0, but these surfaces do not contain (−2)-curves. If κ(Ŝ) = 1, then, sinceŜ has a vector field, it does not any rational curve; see, for example, [GH90, Satz 1].
(b) Suppose now that S has at least one irrational singularity. Then Suppose first that S 0 is not Kähler. By classification, S 0 has to be a primary Kodaira surface or κ(S 0 ) = 1. By Corollary 4.9, H 0 (S 0 , T S 0 ) = 0 (up to finiteétale cover). Choose a non-zero vector field v 0 coming from S; then v 0 does not have zeros by classification [GH90, Satz 1]; note that if κ(S 0 ) = 1, S 0 is an elliptic bundle over a curve of genus at least 2. Hence we must haveŜ = S 0 . But thenŜ does not contain contractible curves, so that S is smooth, a contradiction. Thus S 0 is Kähler. Since KŜ = π * (K S ) − E with E a non-zero effective divisor, κ(S 0 ) = −∞ and S 0 is a ruled surface over a curve B of genus g = g(B) 2. Since S has an irrational singularity, π must contract an irrational curve whose normalization necessarily has genus at least g. Thus h 0 (S, R 1 π * (OŜ)) g and therefore 1 − g = χ(Ŝ, OŜ) χ(S, O S ) − g = −g, which is absurd. 2

Kodaira surfaces, hyperelliptic surfaces and tori
In this section we consider the case where the general fiber of f is a Kodaira surface, a hyperelliptic surface or a torus. We rule out the case of Kodaira and hyperelliptic fibers and show in the torus case that for general line bundles L on X, the restriction L |Xc to any fiber is never torsion.
Proposition 5.1. Assume that the general fiber of f is a Kodaira surface or a torus. Then Proof. It suffices to show that h 2 (X c , O Xc ) is independent of c. Indeed, since h 0 (X c , O Xc ) = 1 for all c and since χ(X c , O Xc ) is constant, h 1 (X c , O Xc ) does not depend on c as well, and the assertions follow by Grauert's theorem. By Serre duality, Setting L = f * (ω X ), a locally free sheaf of rank one, we obtain where F i are the non-reduced fiber components. In particular, ω X |Xc = O Xc for all reduced fibers X c and therefore h 0 (X c , ω Xc ) = 1 for all those c. So let X c be a non-reduced fiber and set Y = red(X c ). We consider the complex subspace Z = (m i − 1)F i of X c and have an induced exact sequence Applying f * and observing that shows that the restriction map Corollary 5.2. Assume that the general smooth fiber of F is a Kodaira surface, a hyperelliptic surface or a torus. Then the restriction map Proof. Suppose first that F is a Kodaira surface or a torus. Then the assertion is [CDP98, Theorem 3.1]; the proof works since we now know that R 1 f * (O X ) is locally free. If F is hyperelliptic, then H 1 (F, O F ) is one-dimensional, hence it suffices to show that r F = 0. Let µ : H 1 (X, O X ) → Pic(X) be the canonical isomorphism and write µ(α) = ω X . Then r F (α) = 0. In fact, otherwise ω F = ω X |F = O F , noticing also that c 1 (ω F ) = c 1 (ω X |F ) = 0 since H 2 (X, Z) = 0.
But ω F O F , a contradiction. 2 As a consequence, we obtain the following corollary.
Corollary 5.3. The general fiber of f cannot be a Kodaira or hyperelliptic surface.
Proof. This is [CDP98, Proposition 3.6]. In the proof of Proposition 3.6, Theorem 3.1 is used which is now established by Corollary 5.2. Notice that in step 2 of the proof of Proposition 3.6 in [CDP98], the local freeness of R j f * (L) is used only generically. 2 Remark 5.4. The same arguments also rule out Hopf surfaces of algebraic dimension 1.
From now, for the remainder of this section, we assume that the general fiber of f is a torus.
Proposition 5.5. There is an isomorphism

Algebraic dimension of compact complex threefolds
Proof. By Proposition 5.1, the sheaf R 1 f * (O X ) is locally free of rank two. Write We observe that R 1 f * (O X ) is generically spanned by Corollary 5.2, since Hence b j 0. 2 Proposition 5.6. For general L ∈ Pic(X), the restriction L |Xc is non-torsion for all c ∈ C.
Proof. By Proposition 5.5, h 1 (X, O X ) 2 and the restriction is surjective for all c. Consequently, the kernel of the restriction is discrete for all c plus a linear subspace of codimension 2. Since dim C = 1, it follows that for L ∈ Pic(X) general, the restriction L |Xc is never trivial and thus also non-torsion. 2

Hopf and Inoue surfaces
In this section we assume that the general fiber of f is a Hopf or Inoue surface and show that for general line bundles L on X, the restriction L |Xc is never torsion.
Proposition 6.1. Assume that the general fiber of f is a Hopf or Inoue surface. Let L ∈ Pic(X) be general. Then L |Xc is non-torsion for all c ∈ C, and the restriction map Pic(X) → Pic(X c ) is surjective for any smooth fiber X c .
Moreover, H 2 (X, C * ) is torsion. Consider the canonical morphism Then by the Leray spectral sequence, λ is injective and the cokernel is torsion. Hence non-torsion. This section defines an inclusion Campana, J.-P. Demailly and T. Peternell Let C 0 be the smooth locus of f in C. We claim that Suppose first that claim (10) holds. Then we conclude as follows. Certainly, ι is an isomorphism over C 0 . Thus we obtain a sequence where Q is supported on the finite set C \ C 0 . Since H 0 (C, R 1 f * (C * )) C * and since H 1 (C, C * ) = 0, it follows that H 0 (C, Q) = 0, hence Q = 0. Thus ι is an isomorphism everywhere and consequently u never takes value 1, nor does, by our choice of u, any multiple u m . Hence u defines a line bundle L such that L |Xc is non-torsion for all c ∈ C.
It remains to prove claim (10). As before, set ∆ = C \ C 0 , A = f −1 (∆) and X 0 = X \ A. Then, as in the proof of Lemma 4.2, Since H 4 (A, C * ) (C * ) s , it follows that Since R 1 f 0 * (C * ) is locally constant of rank one, the claim follows. 2 As a consequence we obtain the following corollary.
Proof. (a) Since R 1 f * (O X ) has rank one, we may write By Proposition 6.1, a = 0. So it remains to show that R 1 f * (O X ) is torsion-free. If not, there exists a line bundle M, such that M |Xc 0 is non-torsion for some with an effective divisor D supported on S. Since S is irreducible, D = mS and therefore O X (D) |Xc is a torsion line bundle, which is a contradiction. (b) By (a), h 1 (X, O X ) = 1. Since the general fiber of f has negative Kodaira dimension, we have H 3 (X, O X ) = H 0 (X, ω X ) = 0.
In this case the morphism p 0 : S 0 → B induced a morphismp :S → B. In the language of divisors and using the notation of (4.5) and (4.6), we have −KŜ ≡N +Ê, whereN is the strict transform ofÑ inŜ. Set N 0 = σ * (N ).
We are now using the theory of ruled surfaces as in [Har77, § V.2], also adopting the notation from [Har77]. In particular, we have the invariant e and a section C 0 with minimal self-intersection C 2 0 = −e. Moreover, −K S 0 ≡ 2C 0 + (e + 2 − 2g)F, where F is a fiber of p 0 and g = g(B) the genus of B. SinceS has rational singularities, π cannot contract any curve projecting onto B. Hence we must have N ≡ 2C 0 + aF with a e + 2 − 2g. Taking into account the numerical description of irreducible curves in S 0 , as given in [Har77, § V.2], it follows immediately that e > 0 and that N = 2C 0 + R with an effective divisor R ∼ aF (note that the curve C 0 is the unique contractible curve in S 0 ). Consequently,Ñ has a unique component, sayÑ 1 , projecting onto B, and this component has multiplicity 2. The mapp :S → B induces a holomorphic map p : S → B and a commutative diagramS The general fiber S b of p is a reduced Gorenstein curve with ω S b ≡ 0 whose normalization of S b is a disjoint union of smooth rational curves. Thus, if S b is irreducible, then it is a rational curve with one node or cusp, and if S b is reducible, it is a cycle of smooth rational curves. In the case where S b has a node or is a cycle, the normalization map η is generically 2 : 1 alongÑ 1 . In these cases, however,Ñ 1 would be reduced (see [KW88]), a contradiction. In the remaining case, η has degree 1 alongÑ 1 , hence τ also has degree 1.