Holomorphic rank-2 vector bundles over non-Kähler elliptic surfaces I

Dedicated to Professor F. Hirzebruch for his 75th birthday.

Vasile Brînzănescu and Ruxandra Moraru

Introduction

In this paper, our goal is to study holomorphic rank-2 vector bundles over non-Kähler elliptic surfaces from the point of view of two classical problems: the existence and classification problems. Let X be a smooth compact complex surface. The existence problem consists in determining which topological complex vector bundles admit holomorphic structures, or equivalently, in finding all triples (r, c_{1}, c_{2}) ∈ ℕ × NS(X) × ℤ, r ≥ 2, for which there exists a rank-r holomorphic vector bundle E on X with Chern classes c_{1}(E) = c_{1} and c_{2}(E) = c_{2}. For projective surfaces, Schwarzenberger [S] proved that any triple (r, c_{1}, c_{2}) ∈ ℕ × NS(X) × ℤ, r ≥ 2, comes from a rank-r holomorphic (algebraic) vector bundle. In contrast, for non-projective surfaces, there is a natural necessary condition for the existence problem:

Δ(r, c_{1}, c_{2}) := \frac{1}{r} (c_{2} − \frac{r − 1}{2r}c_{1}^{2}) ≥ 0

(cf. [B-L] for the general case; for rank-2 case, see [B-F, E-F]). Using extensions of coherent sheaves, one proves the following sufficient condition (cf. [B-L], see also [B-F, LP]):

Δ(r, c_{1}, c_{2}) ≥ m(r, c_{1}),

where

m(r, c_{1}) := −\frac{1}{2r}max \{∑_{1}^{n} (\frac{c_{1}}{r} − μ_{i})^{2} , μ_{1}, …, μ_{r} ∈ NS(X), ∑_{1}^{n} μ_{i} = c_{1}\};

this is true for any surface other than a K3 surface with algebraic dimension zero, for c_{1} divisible by r in NS(X) and Δ(r, c_{1}, c_{2}) = \frac{1}{r}. This result also has a converse [B-L, B-F, LP]: any filtrable rank-r holomorphic vector bundle E on a non-algebraic surface X, with Chern classes c_{1}(E) = c_{1} and c_{2}(E) = c_{2}, satisfies the inequality

Δ(E) := Δ(r, c_{1}, c_{2}) ≥ m(r, c_{1}).

Thus, the only unknown situations are in the range Δ(r, c_{1}, c_{2}) ∈ [0, m(r, c_{1})) . If m(r, c_{1}) ≠ 0, this interval is non-empty and to solve the existence problem, one has to construct holomorphic vector bundles with discriminant Δ(r, c_{1}, c_{2}) smaller than m(r, c_{1}). All these vector bundles will, of course, be non-filtrable and the difficulty of the problem resides in the lack of a general construction method for this case. One is therefore compelled to focus on particular classes of surfaces, to find some specific construction methods.

The existence problem is completely solved in only two cases: the first is the case where the range for non-filtrable bundles is the same as the one for filtrable ones (i.e., m(r, c_{1}) = 0 for all c_{1} ∈ NS(X)), and the second is the case of primary Kodaira surfaces (cf. [A-B-T]). In the rank-2 case, the existence problem was solved in [T3] for complex 2-tori, in [A-B-T] for primary Kodaira surfaces and in [T-T] for surfaces of class VII and K3 surfaces. Since the method used in [T-T] (Donaldson polynomials) seems to work also in the case of (non-algebraic) Kähler elliptic surfaces, only the case of non-Kähler elliptic surfaces remains. This article is divided into two parts. In the first, we close the existence problem in the rank-2 case, by giving necessary and sufficient conditions for the existence of holomorphic rank-2 vector bundles over non-Kähler elliptic surfaces. In the second part, we tackle the classification problem by studying the moduli spaces of stable rank-2 vector bundles over non-Kähler elliptic surfaces.

For any non-Kähler elliptic surface X, there exists a holomorphic map X →^{π} B, over a curve B, whose general fibre is an elliptic curve T. By restricting a vector bundle E on X to the smooth fibres of π, one can consider E as a family of bundles over T; such families can be studied using Atiyah’s classification [A] of vector bundles over elliptic curves. Furthermore, one can construct two objects, which have proven very useful in the study of vector bundles over elliptic surfaces (cf. [B-H, F1, F-M, F-M-W, M, T]). The first is an effective divisor on the Jacobian surface J(X) = B × T^{∗} of X, called the spectral curve of E, that encodes the holomorphic type of E over each smooth fibre of π. In the case of rank-2 bundles, the second is a section of a ruled surface, defined as the quotient of the Jacobian surface by an involution (which depends on E).

The two parts of this article are organised as follows. Part I. In the first section, we establish a correspondence between line bundles over a non-Kähler elliptic surface X and sections of its associated Jacobian surface J(X) = B × T^{∗} →^{p_{1}} B (cf. [F-M] for the algebraic case); this follows from results of [B1, B2, B3, B-U] regarding the Neron-Severi and Picard groups of non-Kähler elliptic surfaces. The construction of the spectral curve of a holomorphic rank-2 vector bundle over X is presented in the second section. Over an elliptic curve, a bundle is said to be regular if its group of automorphisms is of the smallest possible dimension. We show that if a vector bundle E is regular on all fibres of π, then it is completely determined by its spectral curve S and a certain line bundle on S. Finally, the third section is devoted to the proof of the existence theorem. Part II. We now consider the moduli spaces M_{c_{1},c_{2}} of stable holomorphic rank-2 vector bundles over X, with fixed Chern classes c_{1} and c_{2}. In the first section, we define a map from each non-empty moduli space M_{c_{1},c_{2}} to a projective space (the latter is the linear system of associated spectral curves on the Jacobian surface J(X)); we also describe the general fibre of this map. Section 2 is devoted to the study of rank-2 vector bundles, over a non-Kähler elliptic surface X, that are irregular over at least one fibre of π. In section 3, we define a Poisson structure on the moduli spaces of stable bundles over Kodaira surfaces and we prove that the map defined in section 1 is an algebraically completely integrable Hamiltonian system, with respect to this Poisson structure. In the last section, we introduce Fourier-Mukai transforms in the context of non-Kähler elliptic surfaces and study their properties.

Acknowledgements. The first author would like to express his gratitude to the Max Planck Institute of Mathematics for its hospitality and stimulating atmosphere; this paper was prepared during his stay at the Institute. It is a pleasure for both authors to thank Jacques Hurtubise for suggesting a link between the papers [A-B-T] and [M].

1 Line bundles over non-Kähler elliptic surfaces

Let X →^{π} B be a minimal non-Kähler elliptic surface (B a smooth compact connected curve). It is well-known that X → B is a quasi-bundle over B, i.e. all the smooth fibres are pairwise isomorphic and the singular fibres are multiples of elliptic curves (see [K], [B3]).

Let T denote the general fibre of X →^{π} B, which is an elliptic curve and let T^{∗} denote the dual of T (i.e. T^{∗} := Pic^{0}(T) ≅ T non-canonically). It is known that the Jacobian surface associated to X →^{π} B, in this case, is simply

J(X) = B × T^{∗} →^{p_{1}} B

(see, for example [K], [B-P-V], [B1]) and the surface X →^{π} B is obtained from its Jacobian surface B ×T^{∗} by a finite number of logarithmic transformations (see [K], [B-P-V], [B-U] and the considerations below).

In this section we establish a correspondence between line bundles over X and sections of its associated Jacobian surface J(X).

1.1 Principal elliptic bundles

Let us suppose for a moment that X →^{π} B is a principal elliptic bundle, i.e. it has no multiple fibres and so, no singular fibres. By [B1, B2] we have

Tors H^{2}(X, ℤ) = Tors NS(X) = π^{∗}H^{2}(B, ℤ).

Also, by [B3, T], the restriction morphism H^{2}(X, ℤ) → H^{2}(T, ℤ) vanishes. Let b ∈ B and let T_{b} = π^{−1}(b); then, if L ∈ Pic(X) is a line bundle on X, it follows that c_{1}(L|_{T_{b}}) = 0, i.e., any line bundle on X can be seen as a family of line bundles of Pic^{0}(T).

More generally, let T be a n-torus and let X →^{π} B be a principal torus bundle over the curve B. Every T -principal bundle is described by a cohomology class ξ ∈ H^{1}(B, O_{B}(T)), where O_{B}(T) denotes the sheaf of germs of locally holomorphic maps from B to T (see, for example [B-U]). For a Cech 1-cocycle (ξ_{ij}) the function

ξ_{ij} : U_{i} ∩ U_{j} → T

identifies (z, t) ∈ U_{i}×T with (z, t′) = (z, ξ_{ij}(z)+t) ∈ U_{j}×T for all z ∈ U_{i}∩U_{j}.

Now, for any line bundle L ∈ Pic(T), we have the homomorphism φ_{L} : T → Pic^{0}(T) = T^{∗},

φ_{L}(x) = isom.class of(T_{x}^{∗}L ⊗ L^{−1}),

where T_{x} : T → T is the translation with x ∈ T. The T -principal bundle X → B being fixed, we can associate to any line bundle L ∈ Pic(T) an element in H^{1}(B, O_{B}(T^{∗})) in the following way: For the Cech 1-cocycle (ξ_{ij}) defining our T -principal bundle, ξ_{ij} : U_{i} ∩ U_{j} → T, we put

η_{ij}^{L} := φ_{L} ∘ ξ_{ij} : U_{i} ∩ U_{j} → T^{∗}.

Therefore, (η_{ij}^{L}) is a Cech 1-cocycle, which defines a cohomology class in H^{1}(B, O_{B}(T^{∗})) denoted φ̃_{L}(ξ). The T^{∗}-principal bundle over B described by φ̃_{L}(ξ) is called the associate T^{∗}-bundle to L. We have (see [B-U]):

Lemma 1.1 Let L ∈ Pic(T) be a line bundle. Then, the obstruction to extend L to a line bundle on the total space of the fixed T -principal bundle X →^{π} B is the associated T^{∗}-bundle to L, φ̃_{L}(ξ).

Remark 1.2 If L ∈ Pic^{0}(T) then its associated T^{∗}-bundle is trivial (since L is invariant by translations) and, by the above lemma, L can be extended to a line bundle on the total space of the fixed T -principal bundle.

Let X →^{π} B be a T -principal bundle with T a n-torus. Choose a point b ∈ B and fix it. We shall identify the torus T with the fibre π^{−1}(b). Restricting a line bundle L on X to the fibre π^{−1}(b) we have a natural group homomorphism:

Pic(X) →^{r} Pic(π^{−1}(b)) = Pic(T).

Then Ker(r) consists of isomorphism classes of line bundles whose restriction to the fibre π^{−1}(b) is trivial, hence the restriction to each fibre of π is a line bundle of degree zero.

Let (U_{j}) be an open covering of B with trivializations

π^{−1}(U_{j}) ≅ U_{j} × T.

For each line bundle L belonging to ker(r) there exists a holomorphic mapping

φ_{j} : U_{j} → Pic^{0}(T) = T^{∗}

with

L|_{π^{−1}(U_{j})} ≅ (φ_{j} × id_{T})^{∗}(P),

where P is the Poincaré bundle on Pic^{0}(T) × T, by the universal property of the Poincaré bundle and the fact that L|_{π^{−1}(U_{j})} ∈ Pic(U_{j} × T) has the property that L|_{π^{−1}(x)} has degree zero for any x ∈ U_{j}.

Since any line bundle of degree zero on the torus is invariant by the translations, on U_{j} ∩ U_{k} ≠ ∅ we have

φ_{j} = φ_{k}.

Hence, the line bundle L defines a holomorphic mapping

(1.3)  φ : B → T^{∗} ,

which satisfies

(1.4)  φ(b) = [0] ,

since the restriction L|_{π^{−1}(b)} is trivial.

Remark 1.5 If X is a surface, i.e. T is an elliptic curve then, since the restriction L|_{π^{−1}(b)} ∈ Pic^{0}(T), we can obtain a holomorphic mapping φ : B → T^{∗} as above without the normalization condition (1.4).

Note that by the arguments of the proof of Lemma 1.1 (see [B-U]) we can patch together (φ_{j} × id_{T})^{∗}(P) to get a line bundle on X, denoted by φ^{∗}(P), since the line bundles of degree zero on a torus are invariant by translations. The line bundle L and the holomorphic mapping φ are related by

(1.6)  L ≅ π^{∗}(M) ⊗ φ^{∗}(P) ,

where M is a line bundle on the curve B. Also note that there is a one to one correspondence between the set of holomorphic maps (1.3) with property (1.4) and Hom(J_{B}, T^{∗}), where J_{B} is the Jacobian of the curve B (by the universal property of the Jacobian of a curve).

Let us consider a group homomorphism

R : Pic(X) →^{r} Pic(T) →^{c_{1}} H^{2}(T, ℤ).

A line bundle L belonging to Ker(R) is the one whose restriction to each fibre of π is of degree zero (if X →^{π} B is a surface, then Ker(R) = Pic(X)). Each line bundle L ∈ Pic^{0}(T) can be extended to a line bundle L on X in such a way that its restriction to each fibre is isomorphic to L (see Remark 1.2). Hence, there is an isomorphism

Ker(R)/Ker(r) ≅ Pic^{0}(T).

Define subgroups P_{j} of Pic(X) by

P_{0} = Pic(X), P_{1} = Ker(r), P_{2} = π^{∗}Pic(B).

Then, (P_{j}) defines a decreasing filtration of Pic(X) and we have the isomorphisms:

P_{1}/P_{2} ≅ Hom(J_{B}, Pic^{0}(T))
P_{0}/P_{1} ≅ {L ∈ Pic(T)| φ̃_{L}(ξ) = 0} ,

where ξ ∈ H^{1}(B, O_{B}(T)) is the cohomology class corresponding to the T - principal bundle X →^{π} B and φ̃_{L}(ξ) is defined above (see [B-U], Theorem 6).

Remark 1.7 In the case of surfaces we have P_{0}/P_{1} ≅ Pic^{0}(T) since P_{0} = Pic(X) = Ker(R) and holomorphic mappings (1.3) are in one to one correspondence with the sections of the Jacobian surface J(X) = B × T^{∗} →^{p_{1}} B.

1.2 Multiple fibres

Let us now consider quasi T -bundles X →^{π} B over a curve B, where T = ℂ^{n}/Λ is a n-torus. By [B-U], every quasi T -bundle over B is obtained from the product B × T by means of logarithmic transformations as follows:

Choose points b_{1}, b_{2}, ..., b_{k} on B and put B′ = B \ {b_{1}, b_{2}, ..., b_{k}}. For each point b_{i} fix a positive integer m_{i}. Let a_{i} be an element of \frac{1}{m_{i}} Λ such that the order of the point [a_{i}] of the torus T corresponding to a_{i} is precisely m_{i}. Let

D_{i} = {t_{i} ∈ ℂ| |t_{i}| < ε}

be a coordinate neighbourhood of the point b_{i} and put

D̂_{i} = {s_{i} ∈ ℂ| |s_{i}| < ε^{1/m_{i}}}.

By the mapping

λ_{i} : D̂_{i} → D_{i}, s_{i} ↦ s_{i}^{m_{i}},

D̂_{i} is a m_{i}-sheeted ramified covering of D_{i}. A holomorphic mapping g_{i} : D̂_{i} × T → D̂_{i} × T defined by

g_{i} : (s_{i}, [ζ]) ↦ (e_{m_{i}}s_{i}, [ζ + a_{i}])

is an analytic automorphism of order m_{i} and generates the cyclic group G_{i} = ⟨g_{i}⟩ of order m_{i} where

e_{m_{i}} = exp(2π\sqrt{−1}/m_{i}).

Since the automorphism g_{i} has no fixed points, the quotient D̂_{i} × T/G_{i} is a complex manifold. Let

μ_{i} : D̂_{i} × T → D̂_{i} × T/G_{i}

be the canonical mapping. We have a holomorphic mapping

π_{i} : D̂_{i} × T/G_{i} → D_{i} , [s_{i}, [ζ]] ↦ s_{i}^{m_{i}}.

Over the punctured disk D_{i}^{∗} the holomorphic mapping π_{i} gives a T -principal bundle and over the origin 0 the equation π_{i} = 0 defines a divisor of the form m_{i}T_{i} where T_{i} = T/⟨[a_{i}]⟩ is a torus obtained as the quotient by a finite subgroup generated by the point [a_{i}].

The mapping

ℓ_{a_{i}} : D̂_{i}^{∗} × T/G_{i} → D_{i}^{∗} × T , [s_{i}, [ζ]] ↦ (s_{i}^{m_{i}}, [ζ − \frac{m_{i}a_{i}}{2π\sqrt{−1}} log s_{i}])

is a well-defined holomorphic mapping and an isomorphism. Therefore, we can patch together D̂_{i} ×T/G_{i}, i = 1, 2, ..., k and B′ ×T by the isomorphisms ℓ_{a_{i}} to obtain a compact complex manifold X, which is called the manifold obtained from B × T by means of logarithmic transformations. There is a natural holomorphic mapping π : X → B given by π_{i} on D̂_{i} × T/G_{i} and the projection to the first factor on B′ × T. The fiber space π : X → B is a T -principal bundle over B′ and has multiple fibres with multiplicity m_{i}, if m_{i} ≥ 2. Let us assume that our quasi T -bundle X →^{π} B has

m_{i} ≥ 2, i = 1, 2, ..., ℓ, m_{ℓ+1} = ... = m_{k} = 1.

Now, we consider line bundles on X. Choose as above a general point b ∈ B and consider a natural restriction homomorphism

r : Pic(X) → Pic(π^{−1}(b)) = Pic(T).

Note that for the multiple fibre m_{i}T_{i} the line bundle O_{X}(T_{i}) associated with the divisor T_{i} of X is an element of Ker(r) and O_{X}(T_{i})^{m_{i}} = O_{X}(m_{i}T_{i}) is the pull-back of the line bundle O_{B}(b_{i}) on the curve B.

Let P_{2} be a subgroup of Pic(X) generated by π^{∗}Pic(B) and O_{X}(T_{i}), i = 1, 2, ..., ℓ. A line bundle L belonging to P_{2} is characterized by the fact that the restriction of L to each fibre π^{−1}(c), c ∈ B′ is the trivial line bundle.

To a line bundle L ∈ Ker(r), by the same argument as above, we can associate a holomorphic mapping

φ′ : B′ → Pic^{0}(T) = T^{∗}.

The pull-back μ_{i}^{∗}(L|_{π^{−1}(D_{i})}) defines also a holomorphic mapping

φ̂_{i} : D̂_{i} → Pic^{0}(T),

where μ_{i} is the above natural quotient mapping. Then, on D̂_{i}^{∗} we have φ̂_{i} = φ′ ∘ λ_{i}, where λ_{i} is defined above. It follows that the holomorphic mapping φ′ can be extended to a holomorphic mapping

φ : B → Pic^{0}(T) = T^{∗}.

As L|_{π^{−1}(b)} is a trivial line bundle, we have

φ(b) = [0].

Let H be a subgroup of the torus T generated by [a_{1}], [a_{2}], ..., [a_{ℓ}]. The group H is isomorphic to Λ_{0}/Λ where Λ_{0} is the lattice generated by Λ and a_{i}’s. To any H-invariant line bundle L on the torus T, we associate a cohomology class (η_{ij}^{L}) in H^{1}(B, O_{B}(T^{∗})) as follows (cf. [B-U]):

Let (U_{j}) be an open covering of the curve B such that U_{i} = D_{i} for i = 1, 2, ..., ℓ and that b_{i} ∉ U_{i} ∩ U_{j} for i ≠ j. Since the line bundle L is invariant by the translations T_{[a_{i}]}, though

[\frac{a_{i}}{2π\sqrt{−1}} log t_{i}]

is multivalued

T_{[ \frac{a_{i}}{2π\sqrt{−1}} log t_{i}]}^{∗}L ⊗ L^{−1}

is a well-defined line bundle on π^{−1}(U_{i} ∩U_{j}) for i = 1, 2, ..., ℓ and j ≠ i. Then there exists a holomorphic mapping φ_{ij} from U_{ij} = U_{i} ∩ U_{j} to T^{∗} such that the line bundle above is the pull-back (φ_{ij} ×id_{T})^{∗}(P) of the Poincaré bundle. Put

η_{ij}^{L} := \{ \begin{matrix}φ_{ij} : & 1 ≤ i ≤ ℓ, ℓ < j \\ 0 : & ℓ < i, j \end{matrix}

Then, it follows that (η_{ij}^{L}) is a 1-cocycle and defines a cohomology class in H^{1}(B, O_{B}(T^{∗})). We have (see [B-U]) the extension of Lemma 1.1:

Lemma 1.8 If a line bundle L of T can be extended to a line bundle L on X, then L is invariant by H. An H-invariant line bundle on the torus T = π^{−1}(b) can be extended to the one of X if and only if the cohomology class [(η_{ij}^{L})] is zero.

Now, as above, we define a decreasing filtration (P_{j}) of Pic(X) by

P_{0} = Pic(X), P_{1} = Ker(r),
P_{2} = the subgroup generated by π^{∗}Pic(B) and O_{X}(T_{i})′s,

where m_{i}T_{i}, i = 1, 2, ..., ℓ are all the multiple fibres of the quasi T -bundle π : X → B. We have (see [B-U], Theorem 10) the isomorphisms:

P_{1}/P_{2} ≅ Hom(J_{B}, Pic^{0}(T))
P_{0}/P_{1} ≅ {L ∈ Pic(T)^{H}| [(η_{ij}^{L})] = 0}.

Now we consider again the case of surfaces. We have:

Lemma 1.9 Let π : X → B be a non-Kähler elliptic surface (i.e. a quasi T -bundle, where T is an elliptic curve). Let L be a line bundle on X. Then, for any smooth fibre T_{b} = π^{−1}(b), the restriction L|_{T_{b}} has degree zero.

Proof. Let m_{1}T_{1}, m_{2}T_{2}, ..., m_{ℓ}T_{ℓ} be all multiple fibres of π and let b_{i} = π(T_{i}). Let m denote the least common multiple of m_{1}, m_{2}, ..., m_{ℓ} and choose an integer e ≥ 0 such that m divides ℓ+e. Take distinct points b_{ℓ+1}, b_{ℓ+e}, which are different from b_{i} i = 1, ..., ℓ and a fixed point b with T_{b} smooth. Then there exists at least one line bundle M on B with

M^{⊗m} ≅ O_{B}(b_{1} + ... + b_{ℓ+e}).

Such an M defines a cyclic covering ε : C → B of degree m, totally ramified at b_{1}, ..., b_{ℓ+e} (see B-P-V], Chapter I, Lemma 17.1). By the Lemma 3.18 in [B3] we know that there exist a principal T -bundle π′ : Y → C and a cyclic covering with group ℤ_{m}, ψ : Y → X over the covering ε : C → B. Let T̃ be a connected component of ψ^{−1}(T_{b}). Then T̃ is a fibre of π′ and the restriction T̃ → T_{b} of ψ is an isomorphism. Now we have

c_{1}(L|_{T_{b}}) = c_{1}(ψ^{∗}(L)|T̃) = 0

because π′ : Y → C is a principal elliptic bundle. □

Remark 1.10 If X is a non-Kähler elliptic surface, by using the Lemma 1.9 and the construction above, we can obtain for any line bundle L ∈ Pic(X) a holomorphic mapping φ : B → T^{∗} with

L|_{T_{b}} = φ(b)

without the normalization condition (1.4).

Proposition 1.11 Let π : X → B be a non-Kähler elliptic surface, T the general fibre and let J(X) = B × T^{∗} be the associated Jacobian surface of X. Then:

(i) For any section Σ ⊂ J(X) there exists a line bundle L on X whose restriction to every smooth fibre T_{b} is the same as the line bundle Σ_{b} of degree zero on T = T_{b}.

(ii) The set of all line bundles on X which induce the same line bundle of degree zero on every smooth fibre as that corresponding to the section Σ is a principal homogeneous space over Pic^{τ}(X), where Pic^{τ}(X) is the subgroup of line bundles on X whose first Chern class is torsion.

Proof. Let (P_{j}) be the filtration of Pic(X) defined above. Put

N(X) := {L ∈ Pic(T)^{H}| [(η_{ij}^{L})] = 0}

and

Ñ(X) := {c_{1}(L)| L ∈ N(X)}.

By Theorem 17 [B-U] there exists an exact sequence

0 → Hom(J_{B}, T^{∗}) → NS(X)/F̃_{2} → Ñ(X) → 0 ,

where NS(X) is the Neron-Severi group of X and F̃_{2} is a subgroup of H^{2}(X, ℤ) generated by c_{1}(O_{X}(T_{i})), i = 1, 2, ..., ℓ. By [B1] and [B-U] we know that Ñ(X) = 0 and

NS(X)/TorsNS(X) ≅ Hom(J_{B}, T^{∗}).

It follows that N(X) ⊂ Pic^{0}(T) and P_{2} = Pic^{τ}(X). Since any line bundle in Pic^{0}(T) is invariant by translations by Lemmas 1.8, 1.9 we get

N(X) = Pic^{0}(T).

Choose a point b with T_{b} smooth and fix it. Let λ = Σ_{b} ∈ T^{∗} and let Σ^{λ} be the constant section B × λ ⊂ J(X). By the construction above to the line bundle L^{λ}, the extension on X of λ ∈ T^{∗}, corresponds the constant section Σ^{λ}. By the isomorphism

P_{1}/P_{2} ≅ Hom(J_{B}, T^{∗}),

it follows that there exists a line bundle L_{1} ∈ P_{1} = Ker(r) whose corresponding element in Hom(J_{B}, T^{∗}) is Σ − Σ^{λ}. Then, the line bundle L = L_{1} ⊗ L^{λ} has the property that its restriction to every smooth fibre T_{b} is the same as the line bundle Σ_{b} ∈ T^{∗}, hence (i). If the line bundles L′ and L on X have the above property then, by the same isomorphism, it follows that L′ ⊗ L^{−1} ∈ Pic^{τ}(X), hence (ii). □

2 Holomorphic vector bundles over elliptic fibre bundles.

Let X →^{π} B be a principal elliptic bundle over a curve B of genus g and with fibre T. Therefore, X is isomorphic to a quotient of the form

X = Θ^{∗}/⟨τ⟩,

where Θ is a line bundle on B with non-negative Chern class d, Θ^{∗} is the complement of the zero section in the total space of Θ, and ⟨τ⟩ is the multiplicative cyclic group generated by a fixed complex number τ ∈ ℂ, with |τ| > 1. The standard fibre of this bundle is

T ≅ ℂ^{∗}/⟨τ⟩ ≅ ℂ/(2πiℤ + ln(τ)ℤ).

Throughout this section, we assume that d(X) = d > 0. Therefore, the surface X is non-Kählerian and the restriction H^{2}(X, ℤ) → H^{2}(T, ℤ) vanishes (see Corollary 1.5 in [T]). Moreover, we have the following invariants

\begin{matrix}td(B) = 1 + (1 − g)h, \\ td(X) = 1 + (1 − g)π^{∗}h, \end{matrix}

where h is the positive generator of H^{2}(B, ℤ).

2.1 Line bundles.

The set of all holomorphic line bundles on X with trivial Chern class is given by the zero component of the Picard group Pic^{0}(X). Referring to Proposition 1.6 in [T], one has

Pic^{0}(X) ≅ Pic^{0}(B) × ℂ^{∗}.

Any line bundle in Pic^{0}(X) is therefore of the form H ⊗ L_{α}, where H is the pullback to X of an element of Pic^{0}(B) and L_{α} the line bundle corresponding to the constant automorphy factors α ∈ ℂ^{∗}. We illustrate this by constructing the restriction of the universal (Poincaré) line bundle V over X × Pic^{0}(X) to X × ℂ^{∗} := X × {0} × ℂ^{∗}. One starts with a trivial line bundle ℂ̄ on Θ^{∗} × ℂ^{∗}, and applies to it the following ℤ-action

[figure: Θ^{∗} × ℂ^{∗} × ℤ; Θ^{∗} × ℂ^{∗}; (z, α, n); (τ^{n}z, α).]

Since this action is trivial on ℂ^{∗}, the Poincaré line bundle V is obtained identifying s ∈ ℂ̄_{(z,α)} with αs ∈ ℂ̄_{(τz,α)}.

Notation: In the remainder, we shall denote by L_{α} the line bundle corresponding to the automorphy factor α ∈ ℂ^{∗}.

Remarks. 1) For any vector bundle on X, restriction to a fibre of π is a natural operation. Since line bundles are given by constant automorphy factors, their restriction to any fibre is an element of Pic^{0}(T). Referring to the above, we have the exact sequence

0 → Pic^{0}(B) × ⟨τ⟩ → Pic^{0}(X) → Pic^{0}(T) → 0,
0 → Pic^{0}(B) × ℤ → Pic^{0}(B) × ℂ^{∗} → T → 0.

2) Although L_{τ^{m}} is trivial over the fibres of π, one cannot define an action of ℤ on ℂ^{∗} that leaves restriction of the Poincaré line bundle V to X×ℂ^{∗}invariant. If we let ℤ act on ℂ^{∗}, then the multiplication by τ is defined on the fibres of ℂ̄ as

(2.1)  [figure: Θ^{∗} × ℂ^{∗} × ℂ; τ; Θ^{∗} × ℂ^{∗} × ℂ; (z, α, t); (τz, τα, αt).]

On the surface X, z and τz define the same point x. However, (2.1) indicates that τ sends V_{(x,α)} to V_{(x,τα)} ⊗ L_{τ^{−1}}. Hence, the Poincaré line bundle is not invariant under such an action.

Since the principal elliptic fibre bundle X →^{π} B that is topologically nontrivial, we have the following identification (cf. [B1, B2]):

(2.2)  NS(X)/T ors(NS(X)) ≅ Hom(J_{B}, P ic^{0}(T)),

where J_{B} denotes the Jacobian variety of B. The torsion of H^{2}(X, ℤ) is generated by the class of the fibre. In the remainder, the class modulo T ors(H^{2}(X, ℤ)) of an element c ∈ H^{2}(X, ℤ) will be denoted ĉ. Given these considerations, we have (see also [A-B-T]):

Lemma 2.3 For any element c ∈ NS(X), π_{∗}(c) = 0 and c^{2} = −2 deg(ĉ).

Proof: The lemma is certainly true for torsion classes. Let us then assume that c ∉ T ors(NS(X) and choose a line bundle L ∈ P ic(X) such that c_{1}(L) = c. Then ĉ ≠ 0 and, by fixing a base-point in B, the cohomology class ĉ can be considered as a covering map ĉ: B → P ic^{0}(T) such that

ĉ ^{−1}(λ_{0}) = {b ∈ B | L|_{F_{b}} ≃ λ_{0}}.

Since ĉ ≠ 0, we have ĉ ^{−1}(O_{T}) ≠ B. Therefore, the stalk of π_{∗}L is zero at the generic point in B and π_{∗}L = 0. Furthermore, R^{1}π_{∗}L is a torsion sheaf supported on ĉ ^{−1}(O_{T}) and, by Grothendieck-Riemann-Roch, we have:

c_{1}(R^{1}π_{∗}L) = −π_{∗}(c) − \frac{1}{2}c^{2} · h,

where π_{∗}(c) ∈ H^{0}(B, ℤ) and h is the positive generator of H^{2}(B, ℤ). Hence, π_{∗}(c) = 0 and deg(ĉ) = #(ĉ ^{−1}(O_{T})) = −\frac{1}{2}c^{2}. □

2.2 The spectral curve of a rank 2 bundle.

Consider a pair (c_{1}, c_{2}) ∈ NS(X) × ℤ and its corresponding discriminant Δ(2, c_{1}, c_{2}) := \frac{1}{2} (c_{2} − \frac{c_{1}^{2}}{4} ) ≥ 0. In the following, we also associate to such a pair the integer

(2.4)  n(2, c_{1}, c_{2}) := 2Δ(2, c_{1}, c_{2}) − \frac{c_{1}^{2}}{4} .

Referring to lemma 2.3, we immediately obtain Remark 2.5 If Δ(2, c_{1}, c_{2}) ≥ 0, then n(2, c_{1}, c_{2}) ≥ 0.

Let E be a rank 2 vector bundle over X, with c_{1}(E) = c_{1} and c_{2}(E) = c_{2}, where (c_{1}, c_{2}) ∈ NS(X) × ℤ. We fix the following notation:

Δ(E) := Δ(2, c_{1}, c_{2}) and n_{E} := n(2, c_{1}, c_{2}).

Lemma 2.6 Given the above notations, the vector bundle E satisfies

π_{∗}(ch(E) · td(X)) = −n_{E}h,

where h is the positive generator of H^{2}(B, ℤ).

Proof: Given (2.4), we note that n_{E} = −\frac{1}{2}(c_{1}^{2} −2c_{2}) and ch(E) = 2+c_{1} −n_{E}. Therefore,

ch(E) · td(X) = 2 + (c_{1} + 2(1 − g)π^{∗}h) + (−n_{E} + (1 − g)π^{∗}h · c_{1})

and, by the Projection formula,

π_{∗}(ch(X) · td(X)) = π_{∗}(c_{1}) + (−n_{E} + (1 − g)π_{∗}(c_{1})) · h.

The result then follows from the fact that π_{∗}(c_{1}) = 0 (see lemma 2.3). □ Note: The above is also true for rank r vector bundles.

To study bundles on X, one of our main tools will be restriction of the bundle to the fibres π^{−1}(x) ≅ T of the fibration π : X → B. Since the restriction of any bundle on X to a fibre T has first Chern class zero, we consider E as family of degree zero bundles over the elliptic curve T, parametrised by B. Holomorphic vector bundles over elliptic curves have been completely classified by Atiyah [A]. For rank two bundles E with c_{1}(E) = 0 we have:

Proposition 2.7 (Atiyah) Bundles over an elliptic curve T with trivial first Chern class are of the following types :

(i) λ_{0} ⊕ λ′_{0}, with λ_{0}, λ′_{0} ∈ Pic^{0}(T).

(ii) Non trivial extensions 0 → λ_{0} → E → λ_{0} → 0, where λ_{0} ∈ Pic^{0}(T).

(iii) λ ⊕ (λ′)^{∗}, with λ, λ′ ∈ Pic^{−k}(T), k > 0.

Given a rank two bundle over X, its restriction to a generic fibre of π is of type (i) or (ii). More precisely, we have

Proposition 2.8 Let E be a rank 2 holomorphic vector bundle over X; then E|_{π^{−1}(x)} is of type (iii) on at most an isolated set of points x ∈ B.

Proof: Similar to that of Remark 2.4 in [T]; see also [B-H] and Proposition 3.1 of the next section.

Note: We shall call these isolated points the jumps of the bundle E.

Let L = L_{α} be a line bundle in P ic^{0}(X) such that h^{0}(π^{−1}(x), L^{∗}E) = 0, for generic x. Referring to Proposition 2.7, the direct image sheaf R^{1}π_{∗}(L^{∗}E) is a torsion sheaf supported on isolated points x such that E|_{π^{−1}(x)} has L|_{π^{−1}(x)} as a subline bundle and is of type (i) or (ii), or E|_{π^{−1}(x)} is of type (iii). If h is the positive generator of H^{2}(B, ℤ), then

c_{1}(R^{1}π_{∗}(L^{∗}E)) = −π_{∗}(ch(E) · td(X)) · td(B)^{−1} = n_{E}h.

However, if Δ(E) ≥ 0, then n_{E} ≥ 0 (see remark 2.2): the sheaf R^{1}π_{∗}(L^{∗}E) is supported on n_{E} points, counting multiplicity.

To obtain a complete description of the restriction of E to the fibres of π, this construction must be repeated for every line bundle on X; this is done by taking the direct image R^{1}π_{∗} for all line bundles simultaneously. Let π also denote the projection

\begin{matrix}π : X × Pic^{0}(B) × ℂ^{∗} & ⟶ & B × Pic^{0}(B) × ℂ^{∗} \\ (x, a, α) & ⟼ & (π(x), a, α), \end{matrix}

and let s : X × Pic^{0}(B) × ℂ^{∗} → X be the projection onto the first factor. If V is the universal (Poincaré) line bundle over X × Pic^{0}(B) × ℂ^{∗}, one defines

L := R^{1}π_{∗}(s^{∗}E ⊗ V).

This sheaf is supported on a divisor C̃ that is defined with multiplicity. We have the following remarks:

− Let H be the pullback to X of a line bundle of degree zero on B. The restriction of H to any fibre T is then trivial, implying that the support of R^{1}π_{∗}(s^{∗}E ⊗ V ⊗ H) is also C̃. We can therefore restrict ourselves to X × ℂ^{∗} := X × {0} × ℂ^{∗} and the divisor C̃ descends to B × ℂ^{∗}. In the remainder, we will use the same notation for the restriction to X × ℂ^{∗} of the above construction.

− Consider the ℤ-action on B × ℂ^{∗} induced from the one on X × ℂ^{∗}, described in section 2.1. For any (x, α) ∈ B × ℂ^{∗}, multiplication by τ sends the fibre L_{(z,α)} to L_{(z,τα)} ⊗L_{τ^{−1}}, implying that it does not change the support of L: the divisor C̃ is invariant under multiplication by τ. (However, L is not invariant under this action.)

The quotient of ℂ^{∗} by the ℤ-action is Pic^{0}(T) := ℂ^{∗}/⟨τ⟩. By the above remarks, the support C̃ of L descends to a divisor C on B × Pic^{0}(T), as summarised in proposition:

Proposition 2.9 Let (c_{1}, c_{2}) ∈ NS(X) × ℤ be such that Δ(2, c_{1}, c_{2}) ≥ 0. To each holomorphic vector bundle E on X with c_{1}(E) = c_{1} and c_{2}(E) = c_{2}, there is associated a divisor C in B × Pic^{0}(T) called the spectral curve of E. This divisor is an n-to-one cover of Pic^{0}(T), where n = n(2, c_{1}, c_{2}), and a double cover of B, denoted f : C → B, of genus g(C) = b/2 + n(g − 1) + 1, where b is the branching number of f. In other words, the divisor C is an element of the linear system |2B + nPic^{0}(T)| over B × Pic^{0}(T) and we have the following picture:

[figure: Pic^{0}(T); n : 1; C; ⊂ B × Pic^{0}(T); B.; 2 : 1]

2.3 Regular rank two bundles.

Over a smooth elliptic curve, a rank two vector bundle of degree zero is said to be regular if its group of automorphisms is of the smallest dimension. If E is a holomorphic vector bundle over X, then E is regular if and only if its restriction to a fibre T is never isomorphic to λ_{0} ⊕ λ_{0}, with λ_{0} ∈ Pic^{0}(T), or of type (iii) (see section 2.2). Furthermore, if a regular bundle E satisfies the conditions of Proposition 2.9, then the restriction of the torsion sheaf L = R^{1}π_{1}_{∗}(s^{∗}E ⊗ V) to its support C̃ is a line bundle, and the associated spectral curve C is smooth and irreducible. In this case, one also has that the bundle E can be recovered from the sheaf L, as we shall prove in this section.

Let X_{1} and X_{2} be two copies of the elliptic bundle X, with projections onto B denoted by π_{1} and π_{2}, respectively. Hence, X_{1} ×_{B} X_{2} is the fibred product induced by π_{i} : X_{i} ^{}B, for i = 1, 2. Let p_{i} : X_{1} ×_{B} X_{2} ^{}X_{i}, i = 1, 2, be the natural projections associated to this product. For i = 1, 2, we will also denote by p_{i} the following projections

[figure: p_{i}; :; X_{1} ×_{B} X_{2} × ℂ^{∗}; X_{i} × ℂ^{∗}; (x_{1}, x_{2}, α); (p_{i}(x_{1}, x_{2}), α),]

and by π_{i} the projections

[figure: π_{i}; :; X_{i} × ℂ^{∗}; B × ℂ^{∗}; (x_{i}, α); (π_{i}(x_{i}), α).]

We have the following commutative diagram

X_{1} ×_{B} X_{2} × ℂ^{∗} \overline{p_{2}_{}}X_{2} × ℂ^{∗}

[figure: s^{∗}E; X_{1} × ℂ^{∗}; p_{1}; π_{1}; B × ℂ^{∗}; π_{2}]

[figure: E; X_{1}; s; π_{1}; B.]

Moreover, if s̃ is the canonical identification of X_{2} × ℂ^{∗} with X_{1} × ℂ^{∗}

\begin{aligned}s̃ & : & X_{2} × ℂ^{∗} & ⟶ & X_{1} × ℂ^{∗} \\ & & (x_{2}, α) & ⟼ & (x_{2}, α) \end{aligned} ,

we also have the following commutative diagram

X_{1} × ℂ^{∗} ^{_{}s̃} X_{2} × ℂ^{∗}
[figure: σ]
X_{1}. s_{}

Let E be a regular holomorphic rank 2 vector bundle on X such that Δ(E) ≥ 0. We consider the torsion sheaf L = R^{1}π_{1}_{∗}(s^{∗}E ⊗ V). Referring to section 2.2, its support C̃ = supp(L) ⊂ B × ℂ^{∗} induces the spectral curve C associated to E and the double cover C →^{f} B. Let Y = X_{1} ×_{B} C → C be the induced principal elliptic bundle. Then Y → C is also topologically non-trivial and its total space is a double cover of X_{1}, by the natural map Y →^{φ} X_{1}. We then get E back from L in four steps:

Step 1. Pullback L to X_{2} ×ℂ^{∗} and then push down to X_{1} ×ℂ^{∗}: we obtain the torsion sheaf s̃_{∗}(π_{2}^{∗}L) whose support is now π_{1}^{−1}(C̃) = X_{1} ×_{B} C̃.

Step 2. Tensor s̃_{∗}(π_{2}^{∗}L) by V^{∗} to counter the influence of V in L.

Step 3. Finally, push down to X_{1}. The fibre of the map s : X_{1} ×ℂ^{∗} → X_{1} is not proper. However, since s̃_{∗}(π_{2}^{∗}L)⊗V^{∗} = s̃_{∗}(π_{2}^{∗}R^{1}π_{1}_{∗}(s^{∗}E ⊗V))⊗V^{∗}, the presence of both V and V^{∗} implies that the ℤ-action, induced from X_{1} × ℂ^{∗}, is fibre preserving. Taking the quotient with respect to this action, we obtain a torsion sheaf on the compact manifold (X_{1} × ℂ^{∗})/ℤ = X_{1} × Pic^{0}(T) that we denote L := (s̃_{∗}(π_{2}^{∗}L) ⊗ V^{∗})/_{∼}. The support of L is X_{1} ×_{B} C := Y and the restriction of L to Y is a line bundle.

Step 4. Push down L to X_{1} to obtain E ≅ s_{∗}(L) = φ_{∗}(L|_{Y}).

The above construction is summarised in

Proposition 2.10 Let E be a regular holomorphic vector bundle on X_{1} such that Δ(E) ≥ 0 and let C be the spectral curve associated to E. If L = R^{1}π_{1}_{∗}(s^{∗}E ⊗ V), we then consider the sheaf L := (s̃_{∗}(π_{2}^{∗}L) ⊗ V^{∗})/_{∼}, where the quotient is taken with respect to the ℤ-action induced from X_{1} × ℂ^{∗}. Therefore, L is a torsion sheaf with support Y = X_{1} ×_{B} C and the restriction of L to Y is a line bundle. Conversely, E ≅ φ_{∗}(L|_{Y}), where φ : Y → X_{1} is the natural map.

Proof: Almost identical to that of proposition 3.1 in [M].

3 Existence of rank-2 holomorphic vector bundles over non-Kähler elliptic surfaces

Let X →^{π} B be a non-Kähler elliptic surface, i.e., the fibration π : X → B is a quasi-bundle over the curve B with general fibre an elliptic curve T and multiple fibres over points b_{1}, …, b_{ℓ} ∈ B; the multiple fibres are m_{1}T_{1}, …, m_{ℓ}T_{ℓ}, where m_{i} is an integer ≥ 2 and T_{i} is an elliptic curve (corresponding to the quotient of T by a cyclic group of order m_{i}).

Let E be a holomorphic 2-vector bundle over X and let δ = detE be its determinant line bundle. Denote by c_{1} = c_{1}(E) = c_{1}(δ) and c_{2} = c_{2}(E) the Chern classes of E. If we let

Δ(E) := \frac{1}{2}(c_{2} − c_{1}^{2}/4)

be the discriminant of E, then a well-known result states that Δ(E) ≥ 0 (cf. [B-L, E-F, B-F, B3, LP]).

3.1 The bisection of a rank 2 bundle

Referring to Lemma 1.9, the restriction of δ to any smooth fibre of π has degree zero; hence, E restricts to a rank-2 vector bundle of degree zero over an elliptic curve, on the smooth fibres π. Such bundles have been classified by Atiyah [A]. For a smooth fibre T_{b} = π^{−1}(b), we denote

δ_{b} = δ|_{T_{b}} ∈ Pic^{0}(T) = T^{∗}.

Recall that the restriction E|_{T_{b}} is then one of the following three types:

(i) λ_{b} ⊕ (δ_{b} ⊗ λ_{b}^{−1}), λ_{b} ∈ Pic^{0}(T).

(ii) Non-split extensions 0 → λ_{b} → E|_{T_{b}} → λ_{b} → 0, λ_{b}^{2} = δ_{b}, λ_{b} ∈ Pic^{0}(T).

(iii) λ_{b} ⊕ (δ_{b} ⊗ λ_{b}^{−1}), λ_{b} ∈ Pic^{−k}(T), k > 0.

Proposition 3.1 Let E be a rank-2 vector bundle over X →^{π} B with discriminant Δ(E) ≥ 0. Then E|_{π^{−1}(b)} is of type (iii) on at most an isolated (finite) set of points b ∈ B.

Proof: Suppose that b ∈ B is a point such that E|_{π^{−1}(b)} is of type (iii). Let us consider the elementary modification

0 → E′ → E → j_{∗}λ_{b} → 0,

where j : T_{b} → X is the natural inclusion. Referring to [F2] (Chapter II, Lemma 16), the discriminant of E′ is given by

Δ(E′) = Δ(E) + \frac{1}{2}j_{∗}c_{1}(λ_{b}) < Δ(E),

because λ_{b} ∈ Pic^{−k}(T), k > 0. Since Δ(E′) ≥ 0, we get the result. □

Remark 3.2 After performing all (finitely many) necessary elementary modifications, the restriction of the holomorphic rank-2 vector bundle E is of type (i) or (ii) over any smooth fibre of π : X → B (cf. also [A-T]). Hence, to prove an existence theorem, we just have to consider bundles whose restrictions are of type (i) or (ii) over the smooth fibres of π.

Let J(X) = B × T^{∗} →^{p_{1}} B be the Jacobian surface associated to the elliptic surface X and denote by B_{0} = B × 0 ⊂ J(X) its zero-section. Consider the bisection (spectral cover) \overline{C} ⊂ J(X) corresponding to the rank-2 vector bundle E: it has the following geometrical description. For each smooth fibre T_{b} of the fibration π, we associate to the point b the pair of line bundles {λ_{b}, δ_{b} ⊗ λ_{b}^{−1}}, if E|_{T_{b}} is of type (i), or the line bundle λ_{b} with multiplicity 2, if E|_{T_{b}} is of type (ii). More intrinsically, one can obtain the bisection \overline{C} by constructing a local relative Poincaré line bundle on the fibred product J(X) ×_{B} X (cf. [F-M, M]).

Remark 3.3 The bisection \overline{C} is defined on the complement of the fibres of J(X) that correspond to multiple fibres of π, i.e., on p_{1}^{−1}(B \ {b_{1}, ⋯, b_{ℓ}}). In fact, it also extends across such fibres. To see this, it suffices to show that the pull-back of \overline{C} to some finite base change of J(X) →^{p_{1}} B extends to J(X). According to the proof of Lemma 1.9, there exists a principal T - bundle π′ : Y → C, over a cyclic covering ε : C → B with group ℤ_{m}, which in turn induces a cyclic covering ψ : Y → X with group ℤ_{m}. In light of this, the Jacobian surface associated to Y is J(Y) = C × T^{∗} and the covering ε : C → B induces a covering J(Y) → J(X). Therefore, by replacing X → B with Y → C, which has no multiple fibres, the above construction (see section 2) applies over every fibre.

Let us now define on J(X) = B × T^{∗} the following involution:

\begin{aligned}i : J(X) & → & J(X) \\ (b, λ) & → & (b, δ_{b} ⊗ λ^{−1}). \end{aligned}

For fixed b ∈ B, the involution λ ↦ δ_{b} ⊗ λ^{−1} induced on the corresponding fibre of J(X) →^{p_{1}} B has four fixed points (the solutions of λ^{2} = δ_{b}). If we take the quotient of J(X) with respect to this involution, then each fibre of p_{1} becomes T^{∗}/i ≅ ℙ^{1}: the quotient J(X)/i is isomorphic to a ruled surface 𝔽 over B. Let η : J(X) → 𝔽 be the canonical map. Since the bisection \overline{C} is invariant under the involution i, it is the pull-back of a curve A on the quotient 𝔽, which is a section of the ruling. (Indeed, the bisection \overline{C} satisfies \overline{C}.T^{∗} = 2 and we have A · f = \frac{1}{2}η^{∗}A · η^{∗}f = \frac{1}{2} \overline{C} · T^{∗} = 1, for a general fibre f ≅ ℙ^{1} of the ruled surface 𝔽.) As a section, A must be smooth; however \overline{C} need not be smooth. It may be reducible or multiple with multiplicity 2.

We can actually say more about the Jacobian surface J(X) and the ruled surface 𝔽, as stated in the following two lemmas.

Lemma 3.4 Let L be a line bundle over a non-Kähler elliptic surface π : X → B. Let Σ be the section of the Jacobian surface J(X) of X defined by the line bundle L as in section 1. Then

c_{1}^{2}(L) = (Σ − B_{0})^{2},

where B_{0} is the zero section of J(X).

Proof: The invariants of the surface J(X) = B × T^{∗} are

p_{g}(J(X)) = g, q(J(X)) = g + 1, K_{J(X)} = p_{1}^{∗}K_{B},

where g is the genus of the base curve B. For any section Σ of J(X), the adjunction formula gives Σ^{2} = 0; hence,

(Σ − B_{0})^{2} = −2Σ · B_{0}.

By [B1, B-U], there is an isomorphism

NS(X)/TorsNS(X) ≅ Hom(J_{B}, T^{∗}),

as in the case of principal non-Kähler elliptic bundles. If we denote by ĉ_{1} the class modulo torsion of c_{1}(L), then we can think of ĉ_{1} : J_{B} → T^{∗} as a covering map ĉ_{1} : B → T^{∗}; furthermore, the proof of Lemma 1.3 in [A-B-T] implies that ĉ_{1}^{2} = −2deg(ĉ_{1}). But deg(ĉ_{1}) = Σ · B_{0}, and the result follows. □

Lemma 3.5 The ruled surface 𝔽 has the invariant e = 0.

Proof: Let A be a section of the ruled surface 𝔽. Then η^{∗}A is a bisection of J(X). If this bisection is reducible, then η^{∗}A = C_{1} + C_{2}, where C_{1} and C_{2} are sections of J(X), and

2A^{2} = (η^{∗}A)^{2} = (C_{1} + C_{2})^{2} = 2C_{1} · C_{2} ≥ 0.

If the bisection is irreducible, then by the construction given in the proof of Theorem 3.6, there is an induced map γ′ : J(W) → J(X) that is two-to-one and is such that γ′^{∗}(η^{∗}(A)) = C_{1} + C_{2}, where C_{1} and C_{2} are sections of the Jacobian surface J(W) = C × T^{∗} (we use the notation of that proof). Then, we obtain again

4A^{2} = 2C_{1} · C_{2} ≥ 0,

and since

−e = min{A^{2}|A section of 𝔽}

(cf. Proposition 12 of Chapter 5 in [F2]), it follows that e ≤ 0.

Let R be the fixed point set of the involution i : J(X) → J(X) defined above. Given that J(X) is smooth surface with no singular fibres, the fixed point set R is a smooth curve of J(X), which is a 4-section. But R is the ramification divisor of the double covering η : J(X) → 𝔽; therefore, the branch divisor η_{∗}(R) = D′ of η is a smooth 4-section. Referring to [B-P-V] (Proposition 9.1, Chapter V), there is an isomorphism J(X) → J(X) such that the zero section B_{0} is mapped onto the fixed section Σ_{δ} corresponding to the fixed line bundle δ ∈ Pic(X) (cf. Remark 1.10). In fact, the isomorphism is simply given by (b, t) ↦ (b, δ_{b}^{−1}t), for any (b, t) ∈ J(X), and the section Σ_{δ} is now the zero section of the Jacobian J(X): it is fixed by the involution i. Hence, A_{0} = η_{∗}Σ_{δ} is a section of the ruled surface 𝔽 and η^{∗}A_{0} = 2Σ_{δ}; the branch divisor of η is the sum of A_{0} with a smooth 3-section D′′. In other words,

2A_{0}^{2} = (η^{∗}A_{0})^{2} = 4(Σ_{δ})^{2} = 0,

and e = 0. □

3.2 Existence theorems

Let us now consider the case where the bisection \overline{C} is reducible or a section of the Jacobian surface counted with multiplicity 2. Choosing a component of \overline{C} determines a section Σ of J(X), i.e., there exists a holomorphic map φ : B → T^{∗} such that Σ = (id_{B} × φ)(B). By Proposition 1.11, there exists a line bundle D′ on X such that D′|_{T_{b}} = Σ_{b} = φ(b), on any smooth fibre T_{b} of π. By construction, the vector bundle E ⊗ (D′)^{−1}|_{T_{b}} has a section. Therefore, the vector bundle π_{∗}(E ⊗ (D′)^{−1}) ≠ 0 and by tensoring with a very ample line bundle A on B, the vector bundle π_{∗}(E ⊗ (D′)^{−1}) ⊗ A will have a non-zero section. Consequently, the vector bundle E ⊗(D′)^{−1} ⊗π^{∗}(A) also has a non-zero section. By standard arguments, the initial holomorphic rank-2 vector bundle E is given as an extension

0 → D → E → δ ⊗ D^{−1} ⊗ I_{Z} → 0,

where Z ⊂ X is a locally complete intersection of codimension 2. Indeed, if we denote D′′ = D′ ⊗ π^{∗}(A^{−1}), we have an injective map D′′ → E. If the quotient E/D′′ is not torsion-free, then by taking the inverse image of the torsion of E/D′′, we obtain another line bundle D and two injective maps

D′′ → D, D → E.

Furthermore, D ≅ D′′ ⊗ O_{X}(D), where D is an effective divisor on X. However, any divisor on X is a combination of fibres; therefore, the line bundles D′, D′′ and D have the same restriction Σ_{b} to any smooth fibre T_{b} of π : X → B (see Proposition 1.11). Changing D′′ by D we get a torsion free quotient and the above exact sequence.

Let us now consider the case where the bisection \overline{C} is irreducible and let C be the normalisation of \overline{C}. The map p_{1} : J(X) → B induces a map γ : C → B of degree 2. Let W be the normalisation of X ×_{B} C. Since the surface π : X → B has no singular fibres besides the multiple fibres of type m_{i}T_{i} (where T_{i} is a smooth elliptic curve), W is smooth and we have a commutative diagram

[figure: W; γ̃; X; C; ρ; γ; B,; π]

where γ̃ is generically two-to-one. The map C → \overline{C} ⊂ J(X) defines a section Σ̃_{1} of J(X) ×_{B} C, which is the Jacobian surface J(W) of W (W is again a non-Kähler elliptic surface). By using the section Σ̃_{1}, we obtain an exact sequence

0 → D̃ → γ̃^{∗}E → γ̃^{∗}δ ⊗ D̃^{−1} ⊗ IZ̃ → 0,

where Z̃ ⊂ W is a locally complete intersection of codimension 2. Since deg(γ̃) = 2, we have

2c_{1}^{2}(E) = c_{1}^{2}(γ̃^{∗}E), 2c_{2}(E) = c_{2}(γ̃^{∗}E),

which implies

4Δ(E) = 2Δ(γ̃^{∗}E) = c_{1}(D̃).c_{1}(γ̃^{∗}δ ⊗ D̃^{−1}) + ℓ(Z̃) − \frac{1}{4}c_{1}^{2}(γ̃^{∗}δ).

Therefore,

4Δ(E) = −\frac{1}{4}(c_{1}(D̃) − c_{1}(γ̃^{∗}δ ⊗ D̃^{−1}))^{2} + ℓ(Z̃).

By Proposition 1.11, both D̃ and γ̃^{∗}δ ⊗ D̃^{−1} naturally correspond to sections of J(W). In fact, if we denote by γ′ : J(W) → J(X) the obvious map, then the bisection \overline{C} satisfies γ′^{∗}(\overline{C}) = Σ̃_{1} + Σ̃_{2}, where Σ̃_{1} and Σ̃_{2} are sections of J(W) corresponding to the line bundles D̃ and γ̃^{∗}δ ⊗ D̃^{−1}, respectively. By an argument similar to the one used to prove Claims 1.17 and 1.18 in Chapter 7 of [F-M] (and that extends the Lemma 3.4) we show that

(c_{1}(D̃) − c_{1}(γ̃^{∗}δ ⊗ D̃^{−1}))^{2} = (Σ̃_{1} − Σ̃_{2})^{2}.

Referring to the proof of Lemma 3.4, for any section Σ of J(W) we have Σ^{2} = 0. Then

(Σ̃_{1} − Σ̃_{2})^{2} = 2Σ̃_{1}^{2} + 2Σ̃_{2}^{2} − (Σ̃_{1} + Σ̃_{2})^{2} = −γ′^{∗}\overline{C}^{2} = −2\overline{C}^{2},

and it follows that

Δ(E) = \frac{1}{8}\overline{C}^{2} + \frac{1}{4}ℓ(Z̃).

We have proven the following result, which is similar to Theorem 1.3, Chapter VII, [F-M]:

Theorem 3.6 Let π : X → B be a non-Kähler elliptic surface and E be a holomorphic rank-2 vector bundle over X whose restriction to any smooth fibre of π is of type (i) or (ii). Let δ be the determinant line bundle of E. Then E satisfies one of the following two cases:

(A) There exists a line bundle D on X and a locally complete intersection Z of codimension 2 in X such that E is given by an extension

0 → D → E → δ ⊗ D^{−1} ⊗ I_{Z} → 0.

(B) There exists: (i) a smooth irreducible curve C and a birational map C → \overline{C} ⊂ J(X), where \overline{C} is a bisection that is invariant under the involution i on J(X) defined by δ = det(E);

(ii) a line bundle D̃ on the normalisation W of X ×_{B} C, whose restriction to a smooth fibre of W → C is the same as that induced by the section of J(W) that corresponds to the map C → J(X);

(iii) a codimension 2 locally complete intersection Z̃ in W and an exact sequence

0 → D̃ → γ̃^{∗}E → γ̃^{∗}δ ⊗ D̃^{−1} ⊗ IZ̃ → 0,

where γ̃ : W → X is the natural, generically two-to-one map.

Furthermore, we have

Δ(E) = \frac{1}{8}\overline{C}^{2} + \frac{1}{4}ℓ(Z̃).

□

The bound of Δ(E), given in Theorem 3.6, can be improved as follows:

Theorem 3.7 Let π : X → B be a non-Kähler elliptic surface and δ be a line bundle in P ic(X). Furthermore, let i : J(X) → J(X) be the involution defined by δ and suppose that \overline{C} is an irreducible bisection of J(X) → B, which is invariant by the involution i. Then, there exists a rank-2 holomorphic vector bundle V on X such that

c_{1}(V) = c_{1}(δ), Δ(V) = \frac{1}{8}\overline{C}^{2} = \frac{1}{4}A^{2},

where A is a section of the ruled surface 𝔽 with η^{∗}A = \overline{C}.

Proof. Let C → \overline{C} be the normalisation of \overline{C} and let γ : C → B be the two-to-one map induced from the map C → \overline{C} ⊂ J(X) = B × T^{∗}. If W is the normalisation of the fibred product C ×_{B} J(X), then W → C is a non-Kähler elliptic surface with associated Jacobian surface J(W) = C ×T^{∗}. Also, there is a two-to-one covering map γ̃ : W → X, which induces the covering map γ′ : J(W) → J(X). The map C → C ×_{B} J(X) gives a section Σ_{1} of J(W) → C and γ′^{∗}\overline{C} = Σ_{1} + Σ_{2}, where Σ_{2} is another section of J(W), i.e., γ′^{∗}\overline{C} is reducible. By Proposition 1.11, there exists a line bundle L such that the restriction of L to any smooth fibre T_{c} of W is Σ_{2c}. Let D be the line bundle defined by the equality

L ≅ γ̃^{∗}δ ⊗ D^{−1}.

In addition, let \overline{i} be the involution on W that interchanges the sheets of γ̃. If G ⊂ X is the (smooth) branch divisor of the double covering γ̃ : W → X, then there exists a line bundle L_{0} ∈ Pic(X) such that L_{0}^{2} = O_{X}(G).

Define the holomorphic 2-vector bundle V on X by

V := γ̃_{∗}(L).

By Lemma 29, Chapter 2 of [F2] or by [B4], there is an exact sequence:

0 → \overline{i}^{∗}L ⊗ γ̃^{∗}L_{0}^{−1} → γ̃^{∗}γ̃_{∗}(L) → L → 0.

Since the involution \overline{i} on W is induced by interchanging the sheets of the double cover C → B, and \overline{i} is induced by the involution i on J(X) defined by δ, it follows that the restriction of \overline{i}^{∗}L to any smooth fibre T_{c} of W (which is not in the ramification locus of γ̃), is the same as the restriction of D, namely as Σ_{1c}. From the preceding exact sequence, we get

0 → D ⊗ O_{W} (F) → γ̃^{∗}V → γ̃^{∗}δ ⊗ D^{−1} → 0,

where F is a divisor on W (hence a combination of fibres of the non-Kähler elliptic surface W → C).

Furthermore, as in the last part of the proof of Theorem 3.6, we have

Δ(V) = \frac{1}{8}\overline{C}^{2} = \frac{1}{4}A^{2},

where A is the section of the ruled surface 𝔽 defined by the bisection \overline{C}. By [A-B-T], we also get

c_{1}(V) ≡ c_{1}(δ) mod Tors (NS(X)).

To get rid of the torsion, we need to add multiples of classes of fibres. Then, as in [A-B-T], we can modify the line bundle L, by tensoring it with line bundles of the form O_{W} (T_{c}) or O_{W} (T_{i}), and obtain the desired result

c_{1}(V) = c_{1}(δ).

Note that the discriminant remains unchanged (see the formula in [A-B-T] for the direct image of a line bundle). □

Remark 3.8 Let π : X → B be a non-Kähler elliptic surface and E a holomorphic rank-2 vector bundle on X with determinant δ, which satisfies case (B) of Theorem 3.6. If the following conditions are satisfied: (1) the bisection \overline{C} is smooth (i.e., \overline{C} = C), (2) the branch points of the map C → B do not lie under the multiple fibres of π (so that W = X ×_{B} C), and (3) Z̃ = ∅, then one can prove that E ≅ V (where V is the holomorphic 2-vector bundle of Theorem 3.7), as in the proof of Theorem 1.12, Chapter VII of [F-M].

Remark 3.9 The necessary and sufficient condition, for the existence of filtrable holomorphic vector bundles on the compact complex surfaces, was given in [B-L] (see also [B-F],[LP]). Therefore, only the case of non-filtrable holomorphic vector bundles remains to be studied. For non-Kähler elliptic surfaces, the above results imply that the existence problem for vector bundles is reduced to the existence problem of irreducible bisections, in the Jacobian surface associated to the given elliptic surface; moreover, these bisections must be invariant under the involution constructed above. Equivalently, the problem is reduced to the existence of sections of a ruled surface, whose inverse images are irreducible.

We obtained the following existence result:

Theorem 3.10 Let π : X → B be a non-Kähler elliptic surface and let c_{1} ∈ NS(X) and c_{2} ∈ ℤ be fixed such that m(2, c_{1}) > 0 and Δ(2, c_{1}, c_{2}) ∈ [0, m(2, c_{1})). Then, there exists a holomorphic rank-2 vector bundle over X with Chern classes c_{1}, c_{2} if and only if there exists a line bundle δ ∈ Pic(X) with c_{1}(δ) = c_{1} and such that, if i : J(X) → J(X) is the involution defined by δ and η : J(X) → 𝔽 is the quotient map, there exists a section A of the ruled surface 𝔽 with η^{∗}A irreducible in J(X) and A^{2} = 4Δ(2, c_{1}, c_{2}).

Remark 3.11 The case Δ(2, c_{1}, c_{2}) = 0 was studied in [A-B]. In contrast to the case of primary Kodaira surfaces, for non-Kähler elliptic surfaces with base curve of genus g(B) > 1 there are ”gaps” for the discriminant of holomorphic rank-2 vector bundles. For example, if g(B) = 3, there are no sections A of the ruled surface B × ℙ^{1} with A^{2} = 2. Also, just one of the two value 4, 6 is possible for A^{2}, depending on whether B is hyperelliptic or not (see [Ha], Chapter V). Thus, the existence of holomorphic vector bundles on X depends on the geometry of the base curve B.

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Vasile Brînzǎnescu Institute of Mathematics ”Simion Stoilow”, Romanian Academy, P.O.Box 1-764, RO-70700 Bucharest, Romania email: Vasile.Brinzanescu@imar.ro

Ruxandra Moraru Department of Mathematics and Statistics, Burnside Hall, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6 email: moraru@math.mcgill.ca