STABLE BUNDLES ON NON-KÄHLER ELLIPTIC SURFACES

VASILE BRÎNZĂNESCU AND RUXANDRA MORARU

Abstract. In this paper, we study the moduli spaces M_{δ,c_{2}} of stable rank-2 vector bundles on non-Kähler elliptic surfaces, thus giving a classification these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system.

1. Introduction

Vector bundles on elliptic fibrations have been extensively studied over the past fifteen years; in fact, there is by now a well-understood theory for projective elliptic surfaces (see, for example, [D, F1, FMW]). However, not very much is known about the non-Kähler case. In this article, we partly remedy this problem by examining the stability properties of holomorphic rank-2 vector bundles on non-Kähler elliptic surfaces; their existence and classification are investigated in [BrMo1, BrMo2]. One of the motivations for the study of bundles on non-Kähler elliptic fibrations comes from recent developments in superstring theory, where six-dimensional non-Kähler manifolds occur in the context of type IIA string compactifications with non-vanishing background H-field; in particular, all the non-Kähler examples appearing in the physics litterature so far are non-Kähler principal elliptic fibrations (see [BBDG, CCFLMZ, GP] and the references therein). The techniques developed here and in [BrMo1, BrMo2] can also be used to study holomorphic vector bundles of arbitrary rank on higher dimensional non-Kähler elliptic and torus fibrations.

A minimal non-Kähler elliptic surface X is a Hopf-like surface that admits a holomorphic fibration π : X → B, over a smooth connected compact curve B, whose smooth fibres are isomorphic to a fixed smooth elliptic curve T; the fibration π can have at most a finite number of singular fibres, in which case, they are isogeneous to multiples of T. More precisely, if the surface X does not have multiple fibres, then it is the quotient of a complex surface by an infinite cyclic group (see example 3.1); multiple fibres can then be introduced by performing a finite number of logarithmic transformations on its relative Jacobian J(X). To study bundles on X, a natural operation is restriction to the smooth fibres of π; this gives rise to an important invariant, called the spectral curve or cover, which is an effective divisor on J(X) that encodes the holomorphic type of the bundle over each smooth fibre of π.

Consider the moduli space M_{δ,c_{2}} of stable holomorphic rank-2 vector bundles on X with fixed determinant δ and second Chern class c_{2}. Note that the line bundle δ on X induces an involution i_{δ} of the relative Jacobian J(X); furthermore, the spectral cover of any bundle in M_{δ,c_{2}} is invariant with respect to this involution, thus descending to an effective divisor on the ruled surface 𝔽_{δ} := J(X)/i_{δ} called the graph of the bundle. We can then define a map

The first author was partially supported by Swiss NSF contract SCOPES 2000-2003, No.7 IP 62615 and by contract CERES 39/2002-2004.

2000 Mathematics Subject Classification. Primary: 14J60; Secondary: 14D21, 14D22, 14F05, 14J27, 32G13, 32J15, 14H70.

G : M_{δ,c_{2}} → Div(𝔽_{δ})

that associates to each stable vector bundle its graph in Div(𝔽_{δ}), called the graph map. In [BH, Mo], the stability properties of vector bundles on Hopf surfaces were studied by analysing the image and the fibres of this map; in particular, it was shown [Mo] that the moduli spaces admit a natural Poisson structure with respect to which the graph map is a Lagrangian fibration whose generic fibre is an abelian variety: the map G admits an algebraically completely integrable system structure. In this paper, we adopt this approach to study stable vector bundles on arbitrary non-Kähler surfaces.

The article is organised as follows. We begin with a brief review of some existence and classification results for holomorphic vector bundles on non-Kähler elliptic surfaces that were proven in [BrMo1, BrMo2]. In the third section, we obtain explicit conditions for the stability of rank-2 vector bundles: we show that unfiltrable bundles are always stable and then classify the destabilising bundles of filtrable bundles. The moduli spaces M_{δ,c_{2}} are studied in the last section. We first prove that these spaces are smooth on an open dense subset consisting of vector bundles that are regular on every fibre of π (on a smooth elliptic curve, a bundle of degree zero is said to be regular if its group of automorphisms is of the smallest possible dimension). However, for Hopf and Kodaira surfaces, the moduli spaces are also smooth at points that are not regular; in this case, the moduli are smooth complex manifolds of dimension 4c_{2} − c_{1}^{2}(δ). Then, we determine the image of the graph map; for simplicity, we focus our presentation on non-Kähler elliptic surfaces without multiple fibres, but similar results hold in the multiple fibre case. Furthermore, we give an explicit description of the fibres of the graph map, which follows immediately from the classification results of [BrMo2, Mo] and the stability conditions of the third section; in particular, the generic fibre at a graph G ∈ Div(𝔽_{δ}) is isomorphic to a finite number of copies of a Prym variety associated to G. We conclude by noting that for Kodaira surfaces the graph map is also an algebraically completely integrable Hamiltonian system, with respect to a given symplectic structure on M_{δ,c_{2}}.

Acknowledgements. The first author would like to express his gratitude to the Max Planck Institute of Mathematics for its hospitality and stimulating atmosphere; part of this paper was prepared during his stay at the Institute. The second author would like to thank Jacques Hurtubise for his generous encouragement and support during the completion of this paper; she would also like to thank Ron Donagi and Tony Pantev for valuable discussions, and the Department of Mathematics at the University of Pennsylvania for their hospitality, during the preparation of part of this article.

2. Holomorphic vector bundles

Let X →^{π} B be a minimal non-Kähler elliptic surface, with B a smooth compact connected curve; it is well-known that X →^{π} B is a quasi-bundle over B, that is, all the smooth fibres are pairwise isomorphic and the singular fibres are multiples of elliptic curves [K, Br]. Let T be the general fibre of π, which is an elliptic curve, and denote its dual T^{∗} (we fix a non-canonical identification T^{∗} := Pic^{0}(T)); in this case, the relative Jacobian of X →^{π} B is simply

J(X) = B × T^{∗} →^{p_{1}} B

(see, for example, [K, BPV, Br]) and X is obtained from J(X) by a finite number of logarithmic transformations [K, BPV, BrU]. In addition, if the fibration π has multiple fibres, then one can associate to X a principal T -bundle π′ : X′ → B′ over an m-cyclic covering ε : B′ → B, where the integer m depends on the multiplicities of the singular fibres; note that the map ε induces natural m-cyclic coverings J(X′) → J(X) and ψ : X′ → X.

To study bundles on X, one of our main tools is restriction to the smooth fibres of the fibration π : X → B. It is important to point out that since X is non-Kähler, the restriction of any vector bundle on X to a smooth fibre of π always has trivial first Chern class [BrMo1]. Therefore, a vector bundle E on X is semistable on the generic fibre of π; in fact, its restriction to a fibre π^{−1}(b) is unstable on at most an isolated set of points b ∈ B; these isolated points are called the jumps of the bundle. Furthermore, there exists a divisor S_{E} in the relative Jacobian of X, called the spectral curve or cover of the bundle, that encodes the isomorphism class of the bundle E over each smooth fibre of π; for a detailed description of this divisor, we refer the reader to [BrMo1]. We should note that, if the fibration π has multiple fibres, then the spectral cover S_{E} of E is actually defined as the projection in J(X) of the spectral cover S_{ψ^{∗}E} ⊂ J(X′) of ψ^{∗}E, where ψ : X′ → X is the m-cyclic covering defined above.

2.1. Line bundles. The spectral cover of a line bundle L on X is a section Σ of J(X) such that the restriction of L to any smooth fibre π^{−1}(b) of π is isomorphic to the line bundle Σ_{b} of degree zero on T ≅ π^{−1}(b). Conversely, given any section Σ of J(X), there exists at least one line bundle on X with spectral cover Σ [BrMo1]. Before giving a classification of line bundles on X, we fix some notation. Suppose that π has a multiple fibre mF over the point b in B; the line bundle associated to the divisor F of X is then such that (O_{X}(F))^{m} = O_{X}(mF) = π^{∗}O_{B}(b). Let P_{2} be the subgroup of Pic(X) generated by π^{∗}Pic(B) and the O_{X}(T_{i}), where m_{1}T_{1}, …, m_{r}T_{r} are the multiple fibres (if any) of X; we then have [BrMo1]:

Proposition 2.1. Let Σ be a section of J(X). Then, the set of all line bundles on X with spectral cover Σ is a principal homogeneous space over P_{2}.

2.2. Rank-2 vector bundles. Consider a rank-2 vector bundle E on X; its discriminant is then defined as

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

In this case, the spectral curve of E is a divisor S_{E} in J(X) of the form

S_{E} := (∑_{i=1}^{k}\{x_{i}\} × T^{∗}) + \overline{C},

where \overline{C} is a bisection of J(X) and x_{1}, ⋯, x_{k} are points in B that correspond to the jumps of E. Let δ be the determinant line bundle of E. It then defines the following involution on the relative Jacobian J(X) = B × T^{∗} of X:

\begin{matrix}i_{δ} : J(X) & → & J(X) \\ (b, λ) & ↦ & (b, δ_{b} ⊗ λ^{−1}), \end{matrix}

where δ_{b} denotes the restriction of δ to the fibre T_{b} = π^{−1}(b). By construction, the spectral curve S_{E} of E is invariant with respect to this involution; in particular, the pair of points lying on the bisection \overline{C} over b is of the form \{λ_{b}, δ_{b} ⊗ λ_{b}^{−1}\}, where λ_{b} and δ_{b} ⊗ λ_{b}^{−1} are the subline bundles of E|_{π^{−1}(b)}. Finally, note that the quotient of J(X) by the involution i_{δ} is a ruled surface 𝔽_{δ} := J(X)/i_{δ} over B; let η : J(X) → 𝔽_{δ} be the canonical map. The spectral cover S_{E} of E then descends to divisor on 𝔽_{δ} of the form

G_{E} := ∑_{i=1}^{k} f_{i} + A,

where f_{i} is the fibre of the ruled surface 𝔽_{δ} over the point x_{i} and A is a section of the ruling such that η^{∗}A = \overline{C}.

2.2.1. Bundles without jumps. We begin with some properties of filtrable bundles without jumps. Let E be a rank-2 vector bundle on X with determinant δ, and spectral cover (Σ_{1} + Σ_{2}), where Σ_{1} and Σ_{2} are sections of J(X); there exists a line bundle D on X associated to Σ_{1} such that E is given by an extension

(2.2)  0 → D → E → D^{−1} ⊗ δ → 0.

Consequently,

(2.3)  Δ(E) = −\frac{1}{8} (c_{1}(δ) − 2c_{1}(D))^{2} .

Given the above considerations, we have the following results.

Lemma 2.4. If Σ_{1} = Σ_{2}, then Δ(E) = 0. Furthermore, the extension (2.2) either splits on every fibre of π or else it splits on at most a finite number of fibres.

Proof. Note that c_{1}(D) = c_{1}(D^{−1} ⊗ δ) because Σ_{1} = Σ_{2}; referring to (2.3), we then have Δ(E) = 0. Suppose that there exists at least one fibre T_{b_{0}} of π over which the extension is non-trivial; therefore, h^{1}(T_{b_{0}}, D^{−1} ⊗ E) = 1. But if the extension splits over the fibre T_{b}, then h^{1}(T_{b}, D^{−1} ⊗ E) = 2. The upper semi-continuity of the map b ↦ h^{1}(T_{b}, D^{−1} ⊗ E) thus implies that h^{1}(T_{b}, D^{−1} ⊗ E) = 1 for generic b. □

Lemma 2.5. If Σ_{1} ≠ Σ_{2}, then |Σ_{1} ∩Σ_{2}| = 4Δ(E). In addition, the extension (2.2) splits globally whenever Δ(E) = 0.

Proof. Since Σ_{1} ≠ Σ_{2}, the sheaf π_{∗}(D^{−2} ⊗ δ) vanishes and the first direct image sheaf R^{1}π_{∗}(D^{−2} ⊗ δ) is a skyscraper sheaf supported on the points of Σ_{1} ∩ Σ_{2}. Therefore, c_{1}(R^{1}π_{∗}(D^{−2} ⊗ δ)) = |Σ_{1} ∩ Σ_{2}| and by Grothendieck-Riemann-Roch,

|Σ_{1} ∩ Σ_{2}| = −\frac{1}{2} (c_{1}(δ) − 2c_{1}(D))^{2} ,

which is equal to 4Δ(E) by (2.3). Consequently, if Δ(E) = 0, then Σ_{1} ∩ Σ_{2} = ∅; in this case, the extension (2.2) splits on every fibre of π and R^{1}π_{∗}(D^{2} ⊗ δ^{−1}) = 0. Hence, the Leray spectral sequence gives H^{1}(X, D^{2} ⊗ δ^{−1}) = 0 and the extension splits globally. □

We have seen that we can associate to any rank-2 vector bundle on X a bisection in J(X). Conversely, given any bisection of J(X), there exists at least one rank-2 vector bundle on X associated to it; if the bisection is smooth, the bundles that correspond to it are classified as follows (see [BrMo2] for precise statements).

Theorem 2.6. Fix a line bundle δ on X and its associated involution i_{δ} of J(X). Let C be a smooth bisection of J(X) that is invariant with respect to this involution; it is then a double cover of B of genus 4Δ(2, c_{1}, c_{2}) + 2g − 1. The set of all rank-2 vector bundles on X with spectral cover C and determinant δ is then parametrised by a finite number of copies of the Prym variety P rym(C/B) associated to the double cover C → B. In particular, if X does not have multiple fibres, then this set of bundles consists of a single copy of P rym(C/B).

2.2.2. Bundles with jumps. Consider a rank-2 vector bundle E on X with determinant δ that has a jump of multiplicity μ over the smooth fibre T = π^{−1}(x_{0}). The restriction of E to the fibre T is then of the form λ ⊕ (λ^{∗} ⊗ δ_{x_{0}}), for some λ ∈ Pic^{−h}(T), h > 0; the integer h is called the height of the jump at T. Moreover, up to a multiple of the identity, there is a unique surjection E|_{T} → λ, which defines a canonical elementary modification of E that we denote Ē; this elementary modification is called allowable [F2]. Therefore, we can associate to E a finite sequence {Ē_{1}, Ē_{2}, …, Ē_{l}} of allowable elementary modifications such that Ē_{l} is the only element of the sequence that does not have a jump at T.

Let us now assume that π has a multiple fibre m_{0}T_{0}. One can then associate to X an elliptic quasi-bundle π′ : X′ → B′, over an m_{0}-cyclic covering ε : B′ → B, such that T′_{0} := ψ^{−1}(T_{0}) ⊂ X′ is a smooth fibre of π′, where ψ : X′ → X is the m_{0}-cyclic covering induced by ε. Given this, we say that E has a jump over T_{0} if and only if the restriction of ψ^{∗}E to the fibre T′_{0} is unstable. Naturally, the height and multiplicity of the jump of E over T_{0} are defined as the height and multiplicity of the jump of ψ^{∗}E over T′_{0}. We can now define the following important invariants.

Definition 2.7. Let T be a smooth fibre of π. Suppose that the vector bundle E has a jump over T and consider the corresponding sequence of allowable elementary modifications {Ē_{1}, Ē_{2}, …, Ē_{l}}. The integer l is called the length of the jump at T. The jumping sequence of T is defined as the set of integers {h_{0}, h_{1}, …, h_{l−1}}, where h_{0} = h is the height of E and h_{i} is the height of Ē_{i}, for 0 < i ≤ l − 1.

If the vector bundle E has a jump over a multiple fibre m_{0}T_{0} of π, we define the length and jumping sequence of T_{0} to be the length and jumping sequence of the jump of ψ^{∗}E over the smooth fibre T′_{0} = ψ^{−1}(T_{0}) of ψ, where ψ : X′ → X is the m_{0}-cyclic covering defined above.

Note that if a vector bundle E jumps over a smooth fibre T of π, with multiplicity μ and jumping sequence {h_{0}, …, h_{l−1}}, then μ = ∑_{i=1}^{l−1} h_{i}. For a detailed description of jumps, we refer the reader to [Mo, BrMo2]; moreover, the basic properties of elementary modifications can be found, for example, in [F2]. We finish this section by stating the following existence result [BrMo2]: if X does not have multiple fibres and S is a spectral cover in J(X) (that may have vertical components), then one can associate to S at least one rank-2 vector bundle on X.

3. Stable rank two bundles

3.1. Degree and stability. The degree of a vector bundle can be defined on any compact complex manifold M. Let d = dim_{ℂ} M. A theorem of Gauduchon’s [G] states that any hermitian metric on M is conformally equivalent to a metric, called a Gauduchon metric, whose associated (1,1) form ω satisfies ∂ ¯∂ω^{d−1} = 0. Suppose that M is endowed with such a metric and let L be a holomorphic line bundle on M. The degree of L with respect to ω is defined [Bh], up to a constant factor, by

deg L := ∫_{M} F ∧ ω^{d−1},

where F is the curvature of a hermitian connection on L, compatible with ¯∂_{L}. Any two such forms F differ by a ∂ ¯∂-exact form. Since ∂ ¯∂ω^{d−1} = 0, the degree is independent of the choice of connection and is therefore well defined. This notion of degree is an extension of the Kähler case. If M is Kähler, we get the usual topological degree defined on Kähler manifolds; but in general, this degree is not a topological invariant, for it can take values in a continuum (see below).

Having defined the degree of holomorphic line bundles, we define the degree of a torsion-free coherent sheaf E on M by

deg(E) := deg(det E),

where det E is the determinant line bundle of E, and the slope of E by

μ(E) := deg(E)/rk(E).

The notion of stability then exists for any compact complex manifold:

A torsion-free coherent sheaf E on M is stable if and only if for every coherent subsheaf S ⊂ E with 0 < rk(S) < rk(E), we have μ(S) < μ(E).

Remark. With this definition of stability, many of the properties from the Kähler case hold. In particular, all line bundles are stable; for rank two vector bundles on a surface, it is sufficient to verify stability with respect to line bundles. Finally, if a vector bundle E is stable, then H^{0}(M, End(E)) = ℂ.

Example 3.1. Let X →^{π} B be a non-Kähler principal elliptic bundle over a curve B of genus g and with fibre T. The surface X is then isomorphic to a quotient of the form

X = Θ^{∗}/⟨τ⟩,

where Θ is a line bundle on B with positive 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. In this case, the degree of torsion line bundles can be computed explicitly (for details, see [T]). Every line bundle L ∈ Pic^{τ} (X) decomposes uniquely as L = H ⊗L_{α}, for H ∈ ∪_{i=0}^{d−1}Pic(B) and α ∈ ℂ^{∗}. Taking into account this decomposition, the degree of L is given by

deg L = c_{1}(H) − \frac{d}{ln |τ|} ln |α|.

In particular, deg(π^{∗}H) = deg H, for all H ∈ Pic(B).

We end this example by observing that if X has a multiple fibre m_{0}T_{0}, then we have deg(O_{X}(T_{0})) = 1/m_{0}.

3.2. Stable vector bundles. Let X →^{π} B be a non-Kähler elliptic surface with multiple fibres m_{1}T_{1}, …, m_{r}T_{r} (if any); the canonical bundle of X is then K_{X} = π^{∗}K_{B} ⊗ O_{X} (∑_{i=1}^{r}(m_{i} − 1)T_{i}), giving ω_{X/B} = O_{X} (∑_{i=1}^{r}(m_{i} − 1)T_{i}) as the dualising sheaf of π. Note that deg ω_{X/B} = r − ∑_{i=1}^{r} 1/m_{i} ≥ 0 (see example 3.1). Fix a rank-2 vector bundle E on X and let δ be its determinant line bundle; there exists a sufficient condition on the spectral cover of E that ensures its stability:

Proposition 3.2. Suppose that the spectral cover of E includes an irreducible bisection \overline{C} of J(X). Then E is stable.

Proof. Suppose that there exists a line bundle D on X that maps into E. After possibly tensoring D by the pullback of a suitable line bundle on B, the rank-2 bundle E is then given as an extension

(3.3)  0 → D → E → D^{−1} ⊗ δ ⊗ I_{Z} → 0,

where Z ⊂ X is a locally complete intersection of codimension 2. In fact, Z is the set of points {x_{1}, …, x_{k}} corresponding to the fibres π^{−1}(x_{i}) over which E is unstable. Let Σ_{1} and Σ_{2} be the sections of J(X) determined by the line bundles D and D^{−1} ⊗ δ, respectively. The extension (3.3) then implies \overline{C} = Σ_{1} + Σ_{2}. □

Consequently, the spectral covers of unstable bundles include bisections of the form \overline{C} = (Σ_{1} + Σ_{2}), where Σ_{1} and Σ_{2} are sections of the Jacobian surface.

Proposition 3.4. Suppose that the spectral cover of E is given by

(∑_{i=1}^{k}{x_{i}} × T^{∗}) + (Σ_{1} + Σ_{2}) .

Then, there exist line bundles K_{1} and K_{2} on X (corresponding to the sections Σ_{1} and Σ_{2}, respectively) such that the set of all line bundles that map non-trivially to E is given by

{K_{j} ⊗ π^{∗}H ⊗ O_{X} (∑_{i=1}^{r} a_{i}T_{i}) : H ∈ Pic^{≤0}(B) and a_{i} ≤ 0} .

Also, E is stable if and only if deg K_{1} and deg K_{2} are both smaller than deg δ/2. Note that if Σ_{1} = Σ_{2}, then K_{1} = K_{2}.

The line bundles K_{1} and K_{2} are called the destabilising line bundles of E.

Proof. Let D be a line bundle that corresponds to the section Σ_{1} and suppose that there exists a non-trivial map D → E. We begin by assuming that E is regular on the generic fibre of π. In this case, the direct image sheaf π_{∗}(D^{−1} ⊗ E) is a line bundle on B, say L, of positive degree. Set K = D ⊗ (π^{∗}L)^{−1}; then, K restricts to Σ_{1}_{b} over the smooth fibres π^{−1}(b) of B and π_{∗}(K^{−1} ⊗ E) ≅ O_{B}. However, any line bundle D′ on X corresponding to Σ_{1} can be written as K ⊗ π^{∗}H ⊗ O_{X} (∑_{i=1}^{r} a_{i}T_{i}), for some H ∈ Pic(B) and integers 0 ≤ a_{i} ≤ m_{i} − 1. Moreover, one can easily show that π_{∗} (F ⊗ O_{X} (∑_{i=1}^{r} a_{i}T_{i})) = π_{∗}(F), for any locally free sheaf F on X. Consequently, if D′ also maps into E, then the line bundle π_{∗}(D′^{−1} ⊗E) ≅ H^{−1} has a non-trivial section, implying that H ∈ Pic^{≤0}(B). Note that π_{∗}(K′^{−1} ⊗ E) ≅ O_{B} for any line bundle K′ of the form K ⊗O_{X} (∑_{i=1}^{r} b_{i}T_{i}), where 0 ≤ b_{i} ≤ m_{i} −1. But, the destabilising bundle K_{1} is the line bundle associated to Σ_{1} of maximal degree that maps into E; we therefore set K_{1} = K ⊗ O_{X} (∑_{i=1}^{r}(m_{i} − 1)T_{i}) = K ⊗ ω_{X/B}. Clearly, any line bundle corresponding to Σ_{1} that maps into E is can be written as K_{1} ⊗ π^{∗}H ⊗ O_{X} (∑_{i=1}^{r} a_{i}T_{i}), for H ∈ Pic^{≤0}(B) and integers a_{i} ≤ 0.

We now assume that E is not regular on the generic fibre of π. The direct image sheaf π_{∗}(D^{−1}⊗E) is then a rank-2 vector bundle on B, say F; it must have a subline bundle L such that F/L is torsion free. If we set K_{1} = D ⊗ (π^{∗}L^{−1})^{−1} ⊗ ω_{X/B}, then π_{∗}(K_{1}^{−1} ⊗ E) has a nowhere vanishing section and, as above, any line bundle induced by Σ_{1} that maps into E is of the required form. □

In fact, the destabilising line bundles of filtrable bundles without jumps can be described explicitely as follows:

Proposition 3.5. Let E be a holomorphic rank-2 vector bundle on X with invariants det(E) = δ, c_{2}(E) = c_{2}, and spectral cover (Σ_{1} + Σ_{2}), where Σ_{1} and Σ_{2} are sections of J(X). Let K_{1} be the destabilising line bundle of E induced by Σ_{1}; there is an extension

(3.6)  0 → K_{1} → E → K_{1}^{−1} ⊗ δ → 0.

(i) If the extension is trivial on every fibre of π, then the second destabilising bundle of E is K_{2} = K_{1}^{−1} ⊗ δ ⊗ ω_{X/B}.

(ii) Suppose that Σ_{1} = Σ_{2} and that the extension splits on a finite number n ≥ 0 of fibres of π. In this case, K_{1}^{2} = δ ⊗ π^{∗}(H_{+}) ⊗ ω_{X/B}, where H_{+} is a line bundle of degree n on B that is trivial whenever n = 0.

(iii) If Σ_{1} ≠ Σ_{2} and the extension is non-trivial on a finite number n ≤ 4Δ(E) of fibres, then the second destabilising bundle of E is K_{2} = K_{1}^{−1} ⊗δ⊗π^{∗}(H_{−})⊗ω_{X/B}, for some line bundle H_{−} on B of negative degree −n.

Proof. Let us first assume that the extension (3.6) splits on every fibre of π; the line bundle π_{∗}(K_{1} ⊗ δ^{−1} ⊗ E) has a nowhere vanishing global section and is thus trivial. The second destabilising bundle of E is then K_{2} = K_{1}^{−1} ⊗ δ ⊗ ω_{X/B}, proving (i).

Next, we suppose that Σ_{1} = Σ_{2}. The restriction of K_{1}^{2} ⊗ δ^{−1} is therefore trivial every fibre of π, implying that K_{1}^{2} ⊗ δ^{−1} = π^{∗}(H_{+}) ⊗ O_{X} (∑_{i=1}^{r} a_{i}T_{i}) for a line bundle H_{+} on B and integers 0 ≤ a_{i} ≤ m_{i} − 1; tensoring the exact sequence (3.6) by K_{1}^{−1} and pushing down to B, we obtain a new exact sequence:

(3.7)  0 → H_{+}^{−1} → O_{B} → R^{1}π_{∗}(K_{1}^{−1} ⊗ E) → H_{+}^{−1} → 0.

Suppose that the extension (3.6) splits over n fibres of π (counting multiplicity). Referring to (3.7), the first direct image sheaf R^{1}π_{∗}(K_{1}^{−1} ⊗ E) is then given by the extension

0 → S → R^{1}π_{∗}(K_{1}^{−1} ⊗ E) → H_{+}^{−1} → 0,

where S is a skyscraper sheaf supported on the n points (counting multiplicity) corresponding to these fibres. By Grothendieck- Riemann-Roch,

deg(R^{1}π_{∗}(K_{1}^{−1} ⊗ E)) = −\frac{1}{2}c_{1}^{2}(K_{1}^{2} ⊗ δ^{−1}) = −\frac{1}{2}c_{1}^{2}(π^{∗}(H_{+})) = 0.

The degree of the line bundle H_{+} is thus n; clearly, H_{+} = O_{B} if n = 0. Note that by construction, K_{1} ⊗ δ^{−1} ⊗ π^{∗}(H_{+}) ⊗ O_{X} (∑_{i=1}^{r} a_{i}T_{i}) is the destabilising line bundle of E; however, π_{∗}(K_{1} ⊗ δ^{−1} ⊗ π^{∗}(H_{+}) ⊗ O_{X} (∑_{i=1}^{r} b_{i}T_{i})) = O_{B}, for all integers 0 ≤ b_{i} ≤ m_{i} − 1. Therefore, a_{i} = m_{i} − 1 for all i = 1, …, r, proving (ii).

Finally, let us assume that Σ_{1} ≠ Σ_{2}. If the extension also splits over m ≤ 4Δ(E) fibres of π (counting multiplicity) corresponding to points in Σ_{1} ∩Σ_{2}, then the rank of R^{1}π_{∗}(K_{1}^{−1} ⊗ E) jumps at these m points; in fact, the first direct image sheaf is given by the extension

0 → O_{B} → R^{1}π_{∗}(K_{1}^{−1} ⊗ E) → R^{1}π_{∗}(K_{1}^{−2} ⊗ δ) → 0.

Dualising, we get R^{1}π_{∗}(K_{1}^{−1} ⊗ E)^{∗} = H_{−}, for H_{−} ∈ Pic(B). Let n := 4Δ(E) − m; since the skyscraper sheaf R^{1}π_{∗}(K_{1}^{−2} ⊗ δ) is supported on 4Δ(E) points (see the proof of Lemma 2.5), the line bundle H_{−} has degree −n. Furthermore, by relative Serre duality, π_{∗}(K_{1} ⊗ δ^{−1} ⊗ E) = R^{1}π_{∗}(K_{1}^{−1} ⊗ E)^{∗} = H_{−}; therefore, the second destabilising line bundle of E is K_{2} = K_{1}^{−1} ⊗ δ ⊗ H_{−} ⊗ ω_{X/B} and we are done. □

Corollary 3.8. Every filtrable rank-2 bundle on X that has no jumps and trivial discriminant is unstable.

Proof. Let E be such a bundle; its spectral cover of is then of the form (Σ_{1} + Σ_{2}), for some sections Σ_{1}, Σ_{2} ⊂ J(X). Let K_{1} be the destabilising bundle induced by Σ_{1}. Referring to Proposition 3.5, if Σ_{1} = Σ_{2}, then deg K_{1} ≥ deg δ/2 and E is unstable. If Σ_{1} ≠ Σ_{2}, then the extension (3.6) splits on every fibre of π (because Δ(E) = 0) and the second destabilising bundle of E is K_{2} = K_{1}^{−1} ⊗ δ ⊗ ω_{X/B}; therefore, deg K_{1} + deg K_{2} = deg δ + deg ω_{X/B} ≥ deg δ and at least one of the destabilising bundle has degree greater or equal to deg δ/2. □

Recall that, for surfaces X with multiple fibres, the spectral cover of a vector bundle E on X was defined in section 2 in terms of the vector bundle ψ^{∗}E on an m-cyclic covering ψ : X′ → X, where X′ is an elliptic fibre bundle over a m-cyclic covering B′ → B. Keeping this in mind, we now state the main result of the section.

Theorem 3.9. Consider a filtrable rank-2 vector bundle E on X with determinant δ that has k jumps of lengths l_{1}, …, l_{k}, respectively; furthermore, suppose that j of them occur over multiple fibres m_{i_{1}}T_{i_{1}}, …, m_{i_{j}}T_{i_{j}}, respectively, for some integer 0 ≤ j ≤ k. We set

ν := ∑_{s=1}^{j} l_{s}/m_{i_{s}} + ∑_{t=j+1}^{k} l_{t}.

Let K be one of the destabilising bundles of E. There is an extension

0 → ψ^{∗}K → \overline{ψ^{∗}E} → ψ^{∗}(K ⊗ δ^{−1}) → 0,

where \overline{ψ^{∗}E} denotes the vector bundle on X′ obtained by performing successive elementary modifications to eliminate the jumps of ψ^{∗}E.

(i) If Σ_{1} = Σ_{2} and the extension is trivial on every fibre of π′, then E is stable if and only if ν > deg ω_{X/B}.

(ii) Suppose that Σ_{1} = Σ_{2} and that the extension splits on only a finite number mn of fibres, then E is stable if and only if ν > n + deg ω_{X/B}.

(iii) If Σ_{1} ≠ Σ_{2} and the extension is non-trivial on a finite number mn of fibres of π′, then E is stable if and only if deg K ∈ (deg δ/2 − ν − n + deg ω_{X/B}, deg δ/2).

Proof. Note that any elementary modification of ψ^{∗}E has the same destabilising bundles as ψ^{∗}E (which are the pullbacks to X′ of the destabilising bundles of E). Furthermore, the elementary modification \overline{ψ^{∗}E} has determinant ψ^{∗}δ ⊗ O_{X′}(−D), where D := ∑_{s=1}^{j} l_{s}T_{i_{s}} + (∑_{t=j+1}^{k} l_{t})T. Applying Proposition 3.5 to the bundle \overline{ψ^{∗}E}, we obtain the theorem. □

4. Moduli spaces

Let X be a non-Kähler elliptic surface and consider a pair (c_{1}, c_{2}) in NS(X)×ℤ. For a fixed line bundle δ on X with c_{1}(δ) = c_{1}, let M_{δ,c_{2}} be the moduli space of stable holomorphic rank-2 vector bundles with invariants det(E) = δ and c_{2}(E) = c_{2}. We define the following positive rational number:

m(2, c_{1}) := −\frac{1}{4} max \{∑_{1}^{n} (\frac{c_{1}}{2} − μ_{i})^{2} , μ_{1}, …, μ_{r} ∈ NS(X), ∑_{1}^{n} μ_{i} = c_{1}\} .

Note that, for any c_{1} ∈ NS(X), one can choose a line bundle δ on X such that

(4.1)  c_{1}(δ) ∈ c_{1} + 2NS(X) and m(2, c_{1}) = −\frac{1}{2} (\frac{c_{1}(δ)}{2})^{2};

moreover, if there exist line bundles a and δ′ on X such that δ = a^{2}δ′, then there is a natural isomorphism between the moduli spaces M_{δ,c_{2}} and M_{δ′,c_{2}}, defined by E ↦ a ⊗ E. Therefore, if δ′ is any other line bundle with Chern class in c_{1} + 2NS(X), it induces a moduli space that is isomorphic to M_{δ,c_{2}}. However, the advantage of using such a δ is that its Chern class has maximal self-intersection −8m(2, c_{1}). Hence, we restrict our study to moduli spaces M_{δ,c_{2}} of stable bundles whose determinant δ satisfies (4.1).

4.1. Existence and dimension. A necessary condition for the existence of holomorphic rank-2 vector bundles is Δ(2, c_{1}, c_{2}) := 1/2 (c_{2} − c_{1}^{2}/4) ≥ 0 [BaL, Br]. Also, a theorem of Bănică - Le Potier’s [BaL] states that there exists a filtrable holomorphic rank-2 vector bundle with Chern classes c_{1} and c_{2} if and only if Δ(2, c_{1}, c_{2}) ≥ m(2, c_{1}). Given our choice of line bundle δ, any element E of M_{δ,c_{2}} has discriminant

Δ(E) = m(2, c_{1}) + \frac{1}{2}c_{2} ≥ 0.

Consequently, c_{2} ≥ −2m(2, c_{1}); moreover, if c_{2} < 0, then E is unfiltrable. However, if the vector bundle E is an unfiltrable, then its spectral cover contains an irreducible bisection; it is then stable by Proposition 3.2. Therefore, if a rank-2 vector bundle has second Chern class −2m(2, c_{1}) ≤ c_{2} < 0, then it is stable.

Assume that the moduli space M_{δ,c_{2}} is non-empty. Consider one of its elements E; there is a natural splitting of the endomorphism bundle End(E) = O_{X} ⊕ ad(E), where ad(E) is the kernel of the trace map. By deformation theory, the moduli space has expected dimension h^{1}(X; ad(E))−h^{2}(X; ad(E)) at E. Since the vector bundle E is assumed to be stable, we have h^{0}(X; ad(E)) = 0 and the expected dimension of the moduli space is equal to −χ(E) = 8Δ(2, c_{1}, c_{2}) − 3χ(O_{X}) = 8Δ(2, c_{1}, c_{2}).

4.2. Smoothness. Let us first assume that X is an elliptic fibre bundle over a base curve of genus ≤ 1. Recall that a vector bundle E on a complex manifold X is said to be good if and only if h^{2}(X; ad(E)) = 0, or equivalently, if h^{0}(X; ad(E)⊗K_{X}) = 0 (by Serre duality); furthermore, the moduli space M_{δ,c_{2}} is smooth at E if and only if the vector bundle E is good. Given this, one easily proves the following:

Proposition 4.2. Let X be a non-Kähler elliptic fibre bundle over a curve B of genus ≤ 1, that is, X is a Hopf surface or a primary Kodaira surface. The moduli spaces M_{δ,c_{2}} are then smooth of dimension 8Δ(2, c_{1}, c_{2}).

Proof. It is sufficient to prove that every stable bundle on X with Chern classes c_{1} and c_{2} is good. In this case, the canonical bundle of the surface is K_{X} = π^{∗}(K_{B}). Since the genus of B is ≤ 1, the canonical bundle is given by O_{X}(−D), where D is an effective divisor. There is an inclusion K_{X} = O_{X}(−D) ⊂ O_{X}, which in turn induces an inclusion on the space of global sections H^{0}(X; ad E ⊗ K_{X}) ⊂ H^{0}(X; ad(E)). However, the stability of E implies that h^{0}(X; ad(E)) = 0 and we are done. □

For an arbitrary non-Kähler elliptic surface X →^{π} B, we consider the elements of the moduli space that are regular, that is, vector bundles that are regular on every fibre of π. Note that for such a bundle E, the direct image sheaves π_{∗}(End(E)) and R^{1}π_{∗}(End(E)) are dual locally free sheaves of rank two; therefore, by Grothendieck-Riemann-Roch, we have c_{1}(π_{∗}(End(E))) = 2ch_{2}(E). Given the natural splitting π_{∗}(End(E)) = O_{B} ⊕ π_{∗}(ad(E)), we conclude that

deg(π_{∗}(ad(E))) = 2ch_{2}(E).

The Leray spectral sequence gives us h^{0}(X; ad(E)⊗K_{X}) = h^{0}(B; π_{∗}(ad(E))⊗K_{B}); hence, if the degree of π_{∗}(ad(E))⊗K_{B} is negative, we have h^{0}(X; ad(E)⊗K_{X}) = 0, leading us to the following result:

Proposition 4.3. Let X be a non-Kähler elliptic surface over a base curve B of genus g. Then, if c_{2} − c_{1}^{2}/2 > g − 1, the moduli space M_{δ,c_{2}} is smooth on the open dense subset of regular bundles. □

Remark. We can also give a sufficient condition for smoothness of the moduli space at points that do not correspond to regular bundles. Consider a stable vector bundle E that is not regular over the fibres of π lying over the points x_{1}, …, x_{s} in B. In this case, π_{∗}(End(E)) is again a rank-2 vector bundle, but R^{1}π_{∗}(End(E)) is the sum of a rank 2 vector bundle with a skyscraper sheaf supported on the points x_{1}, …, x_{s}, with multiplicities γ_{1}, …, γ_{s}, respectively. Let γ = ∑_{i} γ_{i}. Then, one easily verifies that a sufficient condition for smoothness of the moduli space M_{δ,c_{2}} at E is given by

c_{2} − \frac{c_{1}^{2}}{2} > g − 1 + \frac{γ}{4} .

Note that γ depends not only on the spectral cover of E, but also on the geometry of its jumps.

4.3. The image of the graph map. Fix any pair (c_{1}, c_{2}) ∈ NS(X)×ℤ such that Δ(2, c_{1}, c_{2}) ≥ 0 and let δ be a line bundle on X such that m(2, c_{1}) = −\frac{1}{2} (c_{1}(δ)/2)^{2}. Referring to section 2.2, this line bundle determines an involution i_{δ} of the Jacobian surface and there is an associated ruled surface 𝔽_{δ} := J(X)/i_{δ}; the quotient map is denoted η : J(X) → 𝔽_{δ}. Furthermore, to any rank-2 vector bundle E on X with determinant δ and second Chern class c_{2}(E) = c_{2}, there corresponds a graph in 𝔽_{δ}. It was shown in [BrMo1] that these graphs are elements of linear systems in 𝔽_{δ} of the form |η_{∗}(B_{0}) + bf|, where B_{0} is the zero section of J(X), b is the pullback to X of a line bundle on B of degree c_{2}, and f is a fibre of the ruled surface.

Let ℙ_{δ,c_{2}} be the set of divisors in 𝔽_{δ} of the form ∑_{i=1}^{k} f_{i} + A, where A is a section and the f_{i}’s are fibres of the ruled surface, that are numerically equivalent to η_{∗}(B_{0}) + c_{2}f. We have a well-defined map

G : M_{δ,c_{2}} ⟶ ℙ_{δ,c_{2}}

that associates to each vector bundle its graph, called the graph map. Let us then describe the image of this map; we begin by noting that it is surjective on the open dense subset of graphs in ℙ_{δ,c_{2}} that correspond to irreducible bisections in J(X). When considering the remaining graphs, we restrict ourselves, for simplicity, to the case where X has no multiple fibres; however, similar results holds if X does have multiple fibres.

Proposition 4.4. Let X →^{π} B be a non-Kähler elliptic fibre bundle. Choose an element c_{1} ∈ NS(X) such that m(2, c_{1}) = 0; in this case, 𝔽_{δ} = B × ℙ^{1} and the elements of ℙ_{δ,c_{2}} are of the form

∑_{i=1}^{k}({b_{i}} × ℙ^{1}) + Gr(F),

where b_{1}, …, b_{k} are points in B and Gr(F) is the graph of a rational map F : B → ℙ^{1} of degree c_{2} − k. We then have the following.

(i) For c_{2} = 0, the moduli space M_{δ,0} is empty.

(ii) Let S be the set of points λ_{0} in T^{∗} such that the degree of any line bundle on X corresponding to the section B × {λ_{0}} in J(X) is congruent to deg δ/2 modulo ℤ. If I is the projection of S onto ℙ^{1} = T^{∗}/i_{δ}, then we denote B × I the set of graphs

{({b} × ℙ^{1}) + (B × {λ̄}) | b ∈ B and λ̄ ∈ I } .

For c_{2} = 1, the image of the graph map G is ℙ_{δ,1}\ (B × I).

(iii) For c_{2} ≥ 2, the graph map is surjective.

Proof. Consider a graph G = ∑_{i=1}^{k}({b_{i}} × ℙ^{1}) + Gr(F), where b_{1}, …, b_{k} are points in B and Gr(F) is the graph of a rational map F : B → ℙ^{1} of degree c_{2} − k; we denote \overline{C} the bisection of J(X) determined by Gr(F). Referring to section 2, we can construct rank-2 vector bundles on X with graph G; therefore, we only have to determine whether or not at least one of them is stable. Let us fix E a bundle with graph G and discuss its stability.

Suppose that c_{2} = 0; consequently, Δ(E) = 0 and the map F is constant. Hence, the bisection \overline{C} is reducible and E is a filtrable bundle that has no jumps and trivial discriminant; by Corollary 3.8, it is unstable, proving (i).

Now, assume that c_{2} ≥ 1. Recall that the vector bundle E may be unstable only if the bisection \overline{C} is reducible; therefore, suppose that \overline{C} = Σ_{1} + Σ_{2} for some sections Σ_{1}, Σ_{2} ⊂ J(X). If Σ_{1} = Σ_{2}, then the bundle has at least one jump (otherwise, Δ(E) = 0 (see Lemma 2.4), contradicting the fact that Δ(E) = c_{2} ≥ 1); in this case, E is always stable by Theorem 3.9. If Σ_{1} ≠ Σ_{2}, then a stable bundle E can be constructed as follows. Choose a line bundle K corresponding to Σ_{1}; after possibly tensoring K by an element of P_{2}, one can assume that deg K ∈ (deg δ/2 − k − (c_{2} − k), deg δ/2), unless c_{1} = 1 and the degree of K is congruent to deg δ/2 modulo ℤ. Then, consider a regular extension of K^{−1} ⊗ δ(kT) by K and perform k elementary modifications (using a line bundle of degree 1 on T) to introduce the jumps. Note that K is one of the destabilising bundles of E; referring to Theorem 3.9, E is then stable. Finally, if c_{1} = 1 and Σ_{1} ≠ Σ_{2}, then a bundle with graph G is stable if and only if the degrees of its destabilising bundles are in the interval (deg δ/2 − 1, deg δ/2); if all bundles corresponding to Σ_{1} have degree congruent to deg δ/2 modulo ℤ, then this is never possible. □

Proposition 4.5. Let X →^{π} B be a non-Kähler elliptic fibre bundle. Choose an element c_{1} ∈ NS(X) such that m(2, c_{1}) > 0, so that we may have c_{2} < 0. Then, the graph map is surjective whenever the moduli spaces are non-empty, except in the following case. Suppose that c_{2} = 0 and m(2, c_{1}) = 1/4. Furthermore, let J be the set of sections A in ℙ_{δ,0} such that η^{∗}A = Σ_{1} + Σ_{2} is a reducible bisection of J(X) and the degree of any line bundle on X associated to Σ_{1} is congruent to deg δ/2 modulo ℤ. In this case, the image of the graph map is ℙ_{δ,0}\J. □

Proof. Consider a graph G = ∑_{i=1}^{k} f_{i} + A and set \overline{C} = η^{∗}A. We know that there exist bundles corresponding to this graph; let us then discuss the stability of a bundle E that has graph G. If c_{2} < 0, then Δ(E) < m(2, c_{1}): the bisection \overline{C} is irreducible and the bundle is stable. If c_{2} = 0, then Δ(E) = m(2, c_{1}) > 0. There are now two possibilities. The first is k ≠ 0, implying that A^{2} < 4m(2, c_{1}); therefore, the bisection \overline{C} = η^{∗}A is irreducible and the bundle is stable. The second is k = 0 and the bisection is reducible; suppose that \overline{C} = Σ_{1} + Σ_{2}, for some sections Σ_{1}, Σ_{2} ⊂ J(X). Note that Σ_{1} ≠ Σ_{2}; otherwise, k = 0 would imply that Δ(E) = 0, which is a contradiction. The vector bundle E is then an extension of K^{−1} ⊗ δ by K, where K is the destabilising bundle of E corresponding to Σ_{1}, that can be assumed to be regular on every fibre of π. Hence, E is stable if and only if deg K ∈ (deg δ/2 − 4m(2, c_{1}), deg δ/2), as stated in Theorem 3.9. Clearly, if m(2, c_{1}) = 1/4 and the degree of every line bundle corresponding to Σ_{1} is congruent to deg δ/2 modulo ℤ, then E is unstable. Finally, by arguments similar to those used to prove Proposition 4.4, the graph map is surjective whenever c_{2} ≥ 1. □

4.4. Fibre of the graph map. If we consider graphs without vertical components, the description of most fibres of the graph map is then straightforward.

Proposition 4.6. Let X be a non-Kähler elliptic surface over a curve B of genus g. Fix a pair (c_{1}, c_{2}) in NS(X)×ℤ and let δ be a line bundle on X, with c_{1}(δ) = c_{1}, such that M_{δ,c_{2}} is non-empty. Consider an element A of ℙ_{δ,c_{2}} that does not contain vertical components and let \overline{C} = η^{∗}A be the corresponding bisection in J(X).

(i) Suppose that \overline{C} is a smooth bisection of J(X). The fibre of the graph map G at A is then isomorphic to a finite number of copies of the Prym variety P rym(\overline{C}/B). Note that if the surface X does not have multiple fibres, then G^{−1}(A) consists of a single copy of P rym(\overline{C}/B) (see Theorem 2.6).

(ii) If the bisection \overline{C} = Σ_{1} + Σ_{2} is reducible, then the components of G^{−1}(A) are parametrised by the set of line bundles on X associated to Σ_{1} that satisfy the conditions of Theorem 3.9. In particular, the component given by the line bundle K consists of extensions of K^{−1} ⊗ δ by K that are regular on at least one fibre π^{−1}(b) where Σ_{1,b} = Σ_{2,b}, if deg K is not congruent to deg δ/2 modulo ℤ, or that are regular on at least two such fibres, otherwise. □

For graphs with vertical components, the fibre of the graph map can be described by examining how jumps can be added to vector bundles, that is, by classifying elementary modifications. This is done in detail in [Mo] for vector bundles on Hopf surfaces. For the sake of completion, we briefly state how this translates to bundles on an arbitrary non-Kähler elliptic surface X.

Let E be a stable rank-2 vector bundle on X with det E = δ, c_{2}(E) = c_{2}, and a jump of length l over the smooth fibre T = π^{−1}(x_{0}). This jump can be removed by performing l successive allowable elementary modifications, thus obtaining a bundle with determinant δ(−lT); note that this procedure is canonical. But, adding a jump to E implies several choices: a jumping sequence {h_{0}, …, h_{l−1}}, a line bundle N on T for each distinct integer of the jumping sequence, and surjections to N that preserve stability. These choices are parametrised by a fibration that we now describe. Let G be a graph that contains a vertical component over x_{0} of multiplicity μ and {h_{0}, …, h_{l−1}} be a jumping sequence such that ∑_{i=0}^{l−1} h_{i} = μ. We set

E^{j}J_{G,\{h_{0},...,h_{l−1}\}}^{c_{2},l} = \{E ∈ M_{δ(jT),c_{2}} | \begin{matrix}G(E) = G and E has a jump \\ of length l at x_{0} with jumping \\ sequence \{h_{0}, …, h_{l−1}\} \end{matrix} \} .

Associating to a bundle E its allowable elementary modification Ē therefore defines a natural map

\begin{matrix}Ψ : E^{j+1}J_{G,\{h_{0},h_{1},...,h_{l−1}\}}^{c_{2},l+1} & ⟶ & E^{j}J_{G(Ē),\{h_{1},...,h_{l−1}\}}^{c_{2}−h_{0},l} \\ E & ⟼ & Ē. \end{matrix}

Proposition 4.7. The fibre of the natural projection Ψ at W is given by: (i) Aut_{SL(2,ℂ)}(W |_{T}), if c_{2} > h_{0} and h_{0} = h_{1}, (ii) Pic^{−h_{0}}(T) × Aut_{SL(2,ℂ)}(W |_{T}), if c_{2} > h_{0} > h_{1} and l > 0, (iii) Pic^{−c_{2}}(T), if c_{2} = h_{0} or l = 0. □

4.5. Integrable systems. A Poisson structure on a surface X is given by a global section of its anticanonical bundle K_{X}^{−1} [Bo]. Suppose that X →^{π} B is a non-Kähler elliptic surface that may have multiple fibres T_{1}, …, T_{r} of multiplicities m_{1}, …, m_{r}, respectively. The anticanonical bundle of X is then π^{∗}K_{B}^{−1} ⊗ (⨂_{i=1}^{r} O_{X}(1 − m_{i})), implying that X admits a Poisson structure if and only if the genus of the base curve is ≤ 1 and if the surface does not have multiple fibres. From now on, we suppose that X is a non-Kähler elliptic surface without multiple fibres over a curve B of genus g = 0 or 1, that is, a Hopf surface or a primary Kodaira surface. Let us fix a Poisson structure s ∈ H^{0}(X, K_{X}^{−1}) on X. A Poisson structure θ = θ_{s} ∈ H^{0}(M, ⊗^{2}T M) on the moduli space M := M_{c_{2},δ} is then defined as follows: for any bundle E ∈ M, θ(E) : T_{E}^{∗}M × T_{E}^{∗}M ⟶ ℂ is the composition

θ(E) : H^{1}(X, ad(E) ⊗ K_{X}) × H^{1}(X, ad(E) ⊗ K_{X}) ⟶^{∘}

H^{2}(X, End(E) ⊗ K_{X}^{2}) ⟶^{s} H^{2}(X, End(E) ⊗ K_{X}) ⟶^{Tr} ℂ,

where the first map is the cup-product of two cohomology classes, the second is multiplication by s, and the third is the trace map.

If the base curve B is elliptic, the canonical bundle of X is trivial and the Poisson structure s is non-degenerate; in this case, θ has maximal rank everywhere, that is, θ is symplectic. If the base curve is instead rational, the Poisson structure s is now degenerate; we denote its divisor D := (s). Then, at any point E ∈ M,

rk θ(E) = 4 dim_{ℂ} M − dim H^{0}(D, ad(E|_{D})).

Suppose that the locally free sheaf O_{B}(2) on B ≅ ℙ^{1} is given by the divisor x_{1} +x_{2}, for some points x_{1}, x_{2} ∈ B; then, D = T_{1} + T_{2}, where T_{i} = π^{−1}(x_{i}) for i = 1, 2. We now see that the rank of the Poisson structure is generically 4 dim_{ℂ} M − 2 and “drops” at the points of M corresponding to bundles that are not regular over the fibres T_{1} and T_{2} (for details, see [Mo]).

Referring to sections 4.2 and 4.4, the moduli space M has dimension 8Δ(2, c_{1}, c_{2}) and the generic fibre of the graph map G : M ⟶ ℙ_{δ,c_{2}} is a Prym variety of dimension 4Δ(2, c_{1}, c_{2}) + g − 1 (see Proposition 4.6). Also, one can show as in [Mo] that the component functions H_{1}, …, H_{N} of the graph map are in involution with respect to the Poisson structure, that is, {H_{i}, H_{j}} = 0 for all i, j. Consequently, the graph map G is an algebraically completely integrable Hamiltonian system.

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Institute of Mathematics ”Simion Stoilow”, Romanian Academy, P.O.Box 1-764, RO-70700, Bucharest, Romania

   E-mail address: Vasile.Brinzanescu@imar.ro

Department of Mathematics and Statistics, Burnside Hall, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6

   E-mail address: moraru@math.mcgill.ca