Moduli Spaces of Stably Irreducible Sheaves on Kodaira Surfaces

Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic symplectic manifolds by deformation type. This paper studies a natural Lagrangian fibration on these moduli spaces to determine that they are not K\"ahler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.


Introduction
The study of holomorphic symplectic manifolds began with Bogomolov's classification of compact Kähler manifolds with trivial canonical bundle.These manifolds decompose up to finite étale cover as a product of a complex torus, irreducible Calabi-Yau manifolds, and irreducible holomorphic symplectic manifolds [Bea11,Bog74].It is generally very difficult to construct compact examples of holomorphic symplectic manifolds; nearly all constructions make use of the fact that the Hilbert scheme (or Douady space) of points over a holomorphic symplectic surface is holomorphic symplectic [Bea83], as is a smooth and compact moduli space of stable sheaves with fixed Chern character on a hyperKähler surface [Muk84].
By the Enriques-Kodaira classification, all compact holomorphic symplectic surfaces are complex tori, K3 surfaces, or primary Kodaira surfaces.Each of these holomorphic symplectic surfaces generates an infinite family of holomorphic symplectic manifolds via its Hilbert schemes (or Douady spaces) of points [Bea83].These give rise to generalized Kummer varieties in the case of complex tori, and Bogomolov-Gaun manifolds in the case of primary Kodaira surfaces [Bog96,Gua95].For K3 surfaces and complex tori it has been shown that the moduli spaces of stable sheaves with fixed Chern character are deformation equivalent to the product of a Hilbert scheme of points with the Picard group of the surface [O'G97, Yos01] whenever they are smooth and compact.It is an open question whether this result also holds for primary Kodaira surfaces.
In the case of primary Kodaira surfaces, Toma showed that the moduli space of stable sheaves with fixed determinant and Chern character is holomorphic symplectic whenever it is smooth and compact [Tom01], and determined that a sufficient condition to guarantee smoothness and compactness of the moduli space is to take a Chern character corresponding to stably irreducible sheaves.
Aprodu, Moraru, and Toma studied the two-dimensional moduli spaces of rank-2 stably irreducible sheaves over primary Kodaira surfaces, and determined that they are also primary Kodaira surfaces [AMT12].In higher dimensions, it is not yet known whether these moduli spaces are always deformation equivalent to Douady spaces of points over primary Kodaira surfaces.
In this paper, we determine that there are compact moduli spaces of stably irreducible sheaves on Kodaira surfaces of dimension 2n for every n.In addition, we show that these moduli spaces are non-Kähler and have no simply connected components.Douady spaces of points on Kodaira surfaces are the only other known examples of compact holomorphic symplectic manifolds with these properties.An interesting question is to determine whether these moduli spaces are deformation equivalent to Douady spaces of points on Kodaira surfaces or form a new class of examples.Towards answering this question, we analyse a natural fibration on these spaces, which is described in detail for dimensions 4 and 6 in section 5.
Consider a general compact complex surface X with Gauduchon metric g, and consider the moduli space M g r,δ,c2 (X) of g-stable coherent sheaves with rank r, determinant δ, and second Chern class c 2 on X.In his paper [Tom01], Toma gives a sufficient condition for this moduli space to be smooth and compact: Every g-semi-stable vector bundle E with rank(E) = r, c 1 (E) = c 1 (δ), c 2 (E) ≤ c 2 ( * ) is g-stable.
When X has odd first Betti number, this criterion is equivalent to requiring that every bundle E with rank r, c 1 (E) = c 1 (δ) and c 2 (E) ≤ c 2 is irreducible.In this case, the compactification of the moduli space of stable bundles with rank r, determinant δ, and second Chern class c 2 is isomorphic to the moduli space of stably irreducible torsion-free sheaves.Using Brînzȃnescu's sufficient conditions for a sheaf to be irreducible [Brî96], we find a range of invariants for which ( * ) is satisfied when X is a primary Kodaira surface.In particular, we show that the moduli spaces of rank-two sheaves which are smooth and compact can be of any even dimension.
In section 3, we review the theory of spectral curves on primary Kodaira surfaces, as discussed in [BM05a,BM05b,BM06], which classifies bundles over a Kodaira surface based on their restrictions to the fibres of the natural bundle map associated to the Kodaira surface.We then construct the space P δ,c2 of spectral curves for sheaves in M 2,δ,c2 .We also decompose P δ,c2 into a filtration based on the number of "jumps" each spectral curve has.The spectral curves give a finer invariant for torsion-free sheaves than the Chern character and determinant, and in the rank-two case, the graph map M 2,δ,c2 → P δ,c2 sending each sheaf to its spectral curve is an algebraic completely integrable system [BM05a].The fibres of this integrable system are the focus of sections 4 and 5.
We show that P δ,c2 is a P n -bundle and give an explicit description of the P n -bundle structure for each choice of δ and c 2 .In particular, we show that for ∆(2, δ, c 2 ) ≥ 1 2 P δ,c2 is never biholomorphic to the base Sym n (B) of the natural Lagrangian fibration on any Douady space X [n] over a Kodaira surface.Finally, we review properties of elementary modifications of a rank-2 vector bundle, and use elementary modifications to prove that every irreducible component of a spectral curve in P δ,c2 is smooth when (2, δ, c 2 ) correspond to stably irreducible sheaves.
In section 4, we review the construction of Brînzȃnescu and Moraru [BM06] of the fibres of the graph map above spectral curves without jumps, and describe the fibres of the graph map above spectral curves with exactly one jump.Since spectral curves with k jumps can only occur when the moduli space has dimension at least 4k, understanding these cases allows us to describe all of the fibres of the graph map when the dimension of the moduli space is less than 8.In order to look at the fibres above spectral curves with one jump, we use elementary modifications to parameterize the locally free sheaves, and the structure of the multiplicity one Quot scheme to parameterize the non-locally free sheaves.We also determine which non-locally free sheaves can occur as limit points of vector bundles in the same fibre.
In section 5, we use results from section 4 to prove the main result of the paper: Theorem 1.1.Let X be a primary Kodaira surface, and let (δ, c 2 ) ∈ Pic(X) × Z be such that M 2,δ,c2 has positive dimension and contains stably irreducible vector bundles.Then M 2,δ,c2 (X) is a non-Kähler manifold with no simply connected components.
In this section we also describe the fibration structure of moduli spaces with dimension at most 6 in more detail using the results from section 4, as for these dimensions there are no spectral curves with more than one jump.
The remainder of Section 5 discusses comparisons between the moduli spaces M 2,δ,c2 (X) and the Douady spaces X [n] , as well as the graph map corresponding to moduli of stable rank-2 sheaves on an elliptically fibred abelian surface.Any moduli space of stable sheaves on an elliptically fibred abelian surface is birational to a Hilbert scheme of points, and the birational map can be constructed via allowable elementary modifications [Fri98,Chapter 8].A similar situation does not occur in the Kodaira surface case as the general bundle does not have allowable elementary modifications.We conclude with a discussion of possible avenues to reconcile this discrepancy, including an analysis of moduli spaces of vector bundles on a product of elliptic curves C 1 × C 2 where different choices of elliptic fibration structure give a description of the moduli space both in terms of a graph map and the birational map to Pic

Stably Irreducible Sheaves
Let X be a compact complex surface X with Gauduchon metric g.Given invariants r ∈ Z + , δ ∈ Pic(X), and c 2 ∈ Z, we write M g r,δ,c2 (X) as the moduli space of g-stable rank-r torsion-free sheaves on X with determinant δ and second Chern class c 2 .In his paper [Tom01], Toma shows that a sufficient condition for the moduli space M g r,δ,c2 (X) to be smooth and compact is Every g-semi-stable vector bundle E with In the case that the Betti number b 1 (X) is odd, the ( * ) condition is equivalent to the condition that every vector bundle in M g r,δ,c2 (X) is stably irreducible [Tom01].Since any sheaf which is not g-stable is automatically reducible, stably irreducible sheaves are stable for any choice of Gauduchon metric.Because of this, the choice of metric is irrelevant in these cases, so we write M r,δ,c2 (X) instead of M g r,δ,c2 (X) when b 1 (X) is odd.
Remark 2.2.For any line bundle λ ∈ Pic(X), there is an isomorphism between M 2,δ,c2 (X) and M 2,δ⊗λ ⊗r ,c2+a (X) given by E → E ⊗ λ, where In the rank-2 case, we can use this fact along with the fact that the intersection product on a Kodaira surface is negative definite to find an isomorphic moduli space Under this assumption, we have ∆(2, c 1 , c 2 ) < t(2, c 1 ) if and only if c 2 < 0. Because of this, every moduli space of rank-2 coherent sheaves in the stably irreducible range is isomorphic to a moduli space with invariants satisfying c 1 (δ) 2 = −8t(2, c 1 (δ)) and c 2 < 0, so we can restrict to these cases when searching for examples.

Line Bundles on Kodaira Surfaces
Given the data of a smooth genus one curve B, a positive integer d, a line bundle Θ ∈ Pic d (B), and a complex number τ with |τ | > 1, the quotient X := Θ * / τ via the standard C * action on Θ * gives a principal bundle with base B and structure group C * / τ , otherwise known as a primary Kodaira surface.(Here Θ * represents the complement of the zero section in the total space of Θ.) Let π : X → B be the induced projection map.We denote the fibre π −1 (b) of any b ∈ B by T := C * / τ .Primary Kodaira surfaces have the following topological and analytic invariants [BHPvdV03]: Here the torsion part of H 2 (X, Z) is generated by the fibre of π [Tel98].Furthermore, every line bundle on X with torsion first Chern class can be written as π * (H)⊗ L α , where H ∈ Pic(B) and L α is the bundle on X with constant factor of automorphy α, subject to the relation The bundles L α occur as the quotient of Θ * × C by the Z-action with generator (x, t) → (τ x, αt).
The torsion-free component of the Néron-Severi group can be identified with the group Hom(Pic 0 (B), Pic 0 (T )) [Brî96], and the intersection product on N S(X) can be computed using this identification; the self-intersection is given by for any ϕ ∈ Hom(Pic 0 (B), Pic 0 (T )) [Tel98, Theorem 1.10, Remark 1.11].
Proposition 2.3.For any non-negative integer n and positive integer r, there is a Kodaira surface X, a line bundle δ ∈ Pic(X) and an integer c 2 so that M r,δ,c2 (X) is a compact holomorphic symplectic manifold of dimension 2n.
Proof.If r = 1, then M 1,δ,n (X) is isomorphic to the Douady space X [n] for any line bundle δ and Kodaira surface X, and X [n] is a compact holomorphic symplectic manifold of dimension 2n [Bea83].This isomorphism can be written explicitly by mapping each length-n set Z ∈ X [n] to (δ ⊗ I Z ), where I Z is the ideal sheaf of holomorphic functions vanishing at Z.
Such a Kodaira surface will have N S(X)/Tors(N S(X)) ∼ = Hom(Z + αZ, Z + (rn and any generator a of the torsion-free component of N S(X) will satisfy a 2 = −2(rn − n + r).If we consider the t-invariant in this case, we get , > ∆(r, a, c 2 ), so for any line bundle δ with c 1 (δ) = a, the moduli space M r,δ,c2 will be a compact holomorphic symplectic manifold of dimension 2r 2 ∆(r, a, c 2 ) = 2n.

The Douady Space of Points for a Kodaira Surface
The Douady space M [n] of a complex manifold M is an analytic space parameterizing the coherent O M -modules with finite support of length n.In the case that M has dimension 2, M [n] is itself a complex manifold, and a holomorphic symplectic structure on M naturally lifts to In the following, we compile some known information about Lagrangian fibrations and holomorphic invariants of Douady spaces over Kodaira surfaces in order to compare them with the Lagrangian fibrations we will construct for moduli spaces of stably irreducible sheaves over Kodaira surfaces.

Lagrangian Fibration Structure
The following is closely adapted from the discussion in [Leh11] of the Hilbert scheme of points on a K3 surface.
If π : X → B is a principal elliptic surface with fibre T , there is an induced abelian variety fibration on the Douady space X [n] given by the composition π [n] := ̺ • Sym n (π), where Sym n (π) : Sym n (X) → Sym n (B) is the induced map of symmetric products and ̺ : of an indecomposable bundle on B of rank n and degree −1.)We will focus on this fibration in the case of n = 2, as in this case ̺ is simply the blow-up of the diagonal ∆ in Sym n (X).If given by the union of two irreducible components: and Sym 2 (T ).The symmetric product Sym 2 (T ) can naturally be thought of as the set of effective divisors of degree 2, and it has a ruled surface structure given by sending a divisor to its linear equivalence class in Pic 2 (T ) ∼ = T .The intersection of these components is given by the diagonal of Sym 2 (T ) and the section Atiyah bundle, T X| T b is trivial and T X/B | T b is the mapping onto the second factor.

Topology
Using a result of de Cataldo and Migliorini [dCM00] (due to Göttsche [Göt90] in the projective case), we have that the Betti numbers of where p(X, t) = j≥0 b j (X)t j is the Poincaré polynomial.(Note that truncating the product on the right at k = n gives the correct coefficient for q i for each i ≤ n, so the Betti numbers of for a particular choice of n can be computed with this formula.)In particular, if X is a Kodaira surface, the Betti numbers of X [2] are (1, 3, 8, 18, 24, 18, 8, 3, 1) and the Betti numbers of [3] are (1, 3, 8, 22, 50, 87, 106, 87, 50, 22, 8, 3, 1).In addition, we have from We will refer to these results when discussing the fundamental group and Betti numbers of M 2,δ,c2 (X) in section 5.

Spectral Curves
If E ∈ Coh(X) is torsion-free, we can associate to E the sheaf and the p i 's are the morphisms corresponding to the fibred product.The resulting L E is a torsion sheaf supported on an effective divisor SE in X × C * , consisting of points (b, a) with multiplicity ) and all bundles on B pull back to bundles which are trivial on all fibres of π, SE descends to a divisor S E on J(X) := B × T * , where T * := Pic 0 (T ).We call S E the spectral curve of E. We can describe the spectral curve more concretely as with the multiplicity of (b, λ) given by h 1 (T, E| π −1 (b) ⊗ λ).Since the morphism This fact together with the classification of vector bundles on genus 1 curves gives that Thus the spectral curve has the form for some integers µ b , where C is an r-section of J(X) → B and U E := {b ∈ B : E| π −1 (b) is unstable}.
Definition 3.1.We say that a vector bundle E has a jump at b if E| π −1 (b) is unstable, and the

The Graph of a Rank-2 Sheaf
In the case of rank-2 sheaves, for each δ ∈ Pic(X) we can define an involution on J(X).(This involution only depends on the class of δ in Pic(X)/π * (Pic(B)).)The rank-2 sheaves E with det E = δ ⊗ π * (λ) for some λ ∈ Pic(B) are precisely the sheaves whose spectral curves are invariant under the action of ι δ .Thus these spectral curves descend to the ruled surface F δ := J(X)/ι δ with induced projection ρ : F δ → B. By [BM05a], this ruled surface can be described as F δ = P(V δ ), where and q 1 : J(X) ∼ = B × T * → B is the projection map.The bundle V δ is a rank-2 semi-stable vector bundle on B of degree Remark 3.2.A rank-2 torsion-free sheaf E on X is irreducible if and only if Definition 3.3.For any rank-2 torsion-free sheaf E with det(E) = δ, the graph of E is the set Proposition 3.4 (Brînzȃnescu-Moraru [BM05a]).Let E be a rank-2 torsion-free sheaf with determinant δ and second Chern class c 2 , and let G be the graph of E. Then G is an effective divisor linearly equivalent to A δ +ρ * b, where A δ is the graph of O X ⊕δ, ρ : F δ → B is the induced projection map, and b ∈ Pic c2 (B).
Definition 3.5.Given a line bundle δ ∈ Pic(X) and an integer c 2 , the space of graphs P δ,c2 is the set of all divisors in F δ linearly equivalent to (Equivalently, G is the map sending each sheaf to its spectral curve.We use these two definitions interchangeably.) In the stably irreducible case, we have the following result.
Since in the stably irreducible case the graph map is a natural fibration of M 2,δ,c2 (X) over P δ,c2 , understanding the base and fibres of the fibration will allow us to determine topological properties of M 2,δ,c2 in section 5.In the remainder of this section, we investigate the structure of the P δ,c2 by analyzing the divisors of the form Proposition 3.7.In the case that ∆(2, δ, c 2 ) > 0, P δ,c2 ∼ = P(E δ,c2 ), where bundles on B with base point b 0 ∈ B, and the projections π ij , p k , q ℓ are as in the commutative diagram below.
) by the projection formula.Since q 1 • π 23 = p 2 • π 12 , we can apply the base change theorem to obtain For any b ∈ B, is locally free and the fibre above b in E δ,c2 is indeed Proposition 3.9.Let δ ∈ Pic(X) and c 2 ∈ Z be such that ∆(2, δ, c 2 ) ≥ 0.
Proof.We begin by constructing a long exact sequence involving the E δ,c2 which will be helpful in later computations.Note first that there is a natural exact sequence relating Poincaré bundles of adjacent degrees [CC93].Pulling back by π 12 and twisting by the line bundle If we pushforward by π 12 , we get from the projection formula.By the base change theorem, so the previous exact sequence is short exact.Finally, we can pushforward by p 1 to obtain the long exact sequence | {b}×B is semistable of positive degree when ∆(2, δ, c 2 ) > 0, and any rank-2 semistable vector bundle of positive degree on a genus one curve has trivial first cohomology, so we have R 1 (p 1 ) * (P c2 ⊗ p * 2 V δ ) = 0. We also have This gives the exact sequence (3.2) We now consider the case of ∆(2, δ, c 2 ) = 0.The exact sequence (3.2) for E δ,c2+1 is then where and by [Boo21] we have E δ,c2+1 ∼ = O B ⊕O B (−b), which has a non-zero section.This contradicts Lemma 3.8, so V δ is decomposable.Thus V δ ∼ = λ⊗(L⊕L −1 ) for some λ ∈ Pic −c2 (B) and L ∈ Pic 0 (B).
The extension class corresponding to E δ,c2 is an element of The tensor product of a stable bundle of rank r + 1 with a stable bundle of rank r is another stable bundle of rank r(r + 1) [Ati57, Lemma 28], so in particular there is some point b and the corresponding long exact sequence in cohomology induces an isomorphism Similarly, the extension class corresponding to Since p 1 maps a surface to a curve, its Leray spectral sequence degenerates at page 2, giving The base change theorem gives that R i (p 1 ) * (p * 2 (O B (−p))) is the trivial bundle on B of rank . We now have an isomorphism mapping the extension class of P c2 (b 0 ) ⊗ p * 2 V δ to the extension class of E δ,c2 via pushforward by p 1 , so E δ,c2 is the unique non-split extension of E r+1 (b ′ 2 ) by E r+1 (b ′ 1 ).We now show that this extension is a stable bundle.Suppose for a contradiction that F is a non-trivial sub-bundle of E δ,c2 with µ(F ) ≥ µ(E δ,c2 ).Without loss of generality, we can assume that F is stable.If F has degree −2, 1 ) is destabilizing, so there must be a non-zero morphism f : given by the inclusion of F into E δ,c2 followed by projection to 2 ).Then F must be of the form E k (q) for some 0 < k ≤ r + 1 and 2 ) are constant multiples of the identity, this would induce a splitting of E δ,c2 , so f cannot be surjective.Since f is not surjective, then µ(F ) ≤ µ(im(f )) < µ(E r+1 (b ′ 2 )) by indecomposability of F and stability of E r+1 (b ′ 2 ).We now have deg(F ) ≥ −1 and µ(F ).This implies that F cannot be a destabilizing bundle for E δ,c2 , so E δ,c2 is stable.

Spectral Curves and Elementary Modifications
In order to understand vector bundles with jumps, the main method is to study their elementary modifications.Given a rank-2 vector bundle E on a complex manifold X, a smooth effective divisor D, a line bundle λ on D, and a surjective sheaf map g : E| D → λ, the elementary modification E ′ of E by (D, λ) is the unique vector bundle satisfying the exact sequence where ι : D → X is the inclusion map.The invariants of an elementary modification are given by In the case that X is a Kodaira surface and D is a prime divisor, the divisor D is of the form π Proposition 3.12 (Brînzȃnescu-Moraru [BM05a]).If E is a rank-2 torsion-free sheaf on X, then E has finitely many jumps, and where ι : π −1 (b) → X is the inclusion map.Since ∆(E) ≥ 0, we must have k ≤ 2∆(E).We can iterate this process across all unstable fibres to see that there can only be finitely many, as all vector bundles have non-negative discriminant.
We now consider the case where E is not locally free.In this case, E ∨∨ /E is a torsion sheaf supported at m points with multiplicity, and E ∨∨ is a vector bundle satisfying Since ∆(E ∨∨ ) ≥ 0, the support of E ∨∨ /E must be finite, allowing us to reduce to the first case.
By contrast, elementary modifications by positive-degree line bundles are highly non-unique; if Two elementary modifications by λ corresponding to maps g 1 : isomorphic if and only if there is a bundle automorphism ϕ of E so that Finally, for the case of an elementary modification by a degree zero line bundle, the behaviour of the elementary modification depends on whether the initial bundle is regular.Lemma 3.15.Let E be a rank-2 vector bundle with spectral curve C, and let E ′ be an elementary modification where ι : π −1 (b) → X is the inclusion map for some b ∈ B and λ ∈ Pic 0 (T ).Then E ′ is regular at b if and only if E is.
Proof.If b is of the form π(c) where c is a smooth point of C, then every vector bundle with spectral curve C is regular [Mor03], so we can restrict to the case where π −1 (b) contains a singular point of C. In this case, a vector bundle V with spectral curve C has that V | π −1 (b) is an extension of λ by itself, and Let L ∈ Pic 0 (X) be a bundle with constant factor of automorphy so that L| π −1 (b) = λ −1 .Then taking the tensor product of the exact sequence (3.3) with L gives 0 and pushing forward by π gives the long exact sequence If V is any vector bundle on X so that H 0 (T, V | π −1 (p) ) = 0 for at most finitely many p ∈ B, then , where C b is the skyscraper sheaf supported at b. From this, the above exact sequence reduces to If we now consider the exact sequence induced from the stalks at b, we have ). Together with (3.4), this implies that E ′ is regular at b if and only if E is.
Pairing this lemma with Proposition 3.14 leads to the following result: Proposition 3.16.If C is an irreducible bisection of J(X), then every vector bundle with spectral curve C is regular.
Proof.Suppose that E is a bundle with spectral curve C. By [BM06, Theorem 4.1], there is a vector bundle E 0 with spectral curve C which is regular.E 0 is an elementary modification of the pushforward a line bundle on W by a degree-zero bundle, so by Lemma 3.15 we can assume that E 0 = γ * (L 0 ) for some L 0 ∈ Pic(W ).We also have that there is some L 1 ∈ Pic(W ) so that E is an elementary modification of γ * (L 1 ) by a degree-zero bundle.Since γ * (L 0 ) and γ * (L 1 ) have the same spectral curve, there is a line bundle ) is regular by Lemma 3.15, and we similarly get that E is regular since there is a chain of elementary modifications taking Since all of the rank-2 bundles with irreducible spectral curve C can be expressed as the pushforward of a line bundle on the normalization of X × B C, the bundles with determinant δ and spectral curve C are parameterized by Prym(C/B) [BM06, Theorem 4.5].
In the case of a rank-2 vector bundle E with smooth and irreducible spectral curve C, we have for any rank-2 stably irreducible sheaf E.
Proposition 3.17.If E is a rank-2 stably irreducible sheaf, and the spectral curve S E contains no jumps, then S E is smooth.
Proof.Let E be a rank-2 stably irreducible sheaf in M 2,δ,c2 (X), and suppose for a contradiction that S E consists of a singular irreducible bisection with no jumps.Let C be the normalization of S E . Then X) whose spectral curve is smooth.Since the arithmetic genus of a spectral curve depends only on ∆(2, c 1 (E), c 2 (E)), we have g( 4∆(E).Since a general stably irreducible sheaf has smooth spectral curve, the fibres of the graph map G : M 2,δ,c2 → P δ,c2 have dimension 4∆(E) outside a proper Zariski-closed subset of P δ,c2 .This is a contradiction since the fibre dimension of a holomorphic map is upper semi-continuous, so S E is smooth when it has no jumps.
In the next section, we compute the fibres of the graph map in order to study the structure of

Fibres of the Graph Map
In this section, we attempt to parameterize the space G −1 (G) of torsion-free sheaves on X with determinant δ and spectral curve G, where G is of the form C + k i=1 {b i } × T * with b i ∈ B for each i, and where C is an irreducible bisection of J(X) with normalization C.
In the case where G = C, every vector bundle E with spectral curve C can be described as an Proof.The fibre of the graph map above a spectral curve of this type can be decomposed into two irreducible components, with one component containing the non-locally free sheaves, and the other component containing the vector bundles.If E is a vector bundle with determinant δ and spectral curve C + {b} × T * , then E| π −1 (b) is a degree zero vector bundle so that h 1 (T, for every λ ∈ T * by Proposition 3.12, so there is a line bundle There is a unique choice of a vector bundle Ẽ so that det Since C is smooth and irreducible, either for some λ ∈ Pic 0 (T ) with λ ⊗2 = δ| π −1 (b) , or where A is the unique extension of O T by O T and λ 0 ∈ Pic 0 (T ) such that λ ⊗2 0 = δ| π −1 (b) .In both cases, Hom( Ẽ, j * L) ∼ = C 2 , and the non-surjective maps correspond to |{b} × T * ∩ C| 1-dimensional subspaces.Thus the vector bundles with determinant δ and spectral curve C + {b} × T * are parameterized by a fibre bundle with base Prym(C/B) × T and fibre C or C * .
If E is a non-locally free sheaf with determinant δ and spectral curve C + {b} × T * , then since the double dual of a torsion-free sheaf on a surface is locally free, E ∨∨ is a vector bundle with determinant δ and spectral curve C. Since E has exactly one singularity, E ∨∨ /E is a torsion sheaf supported at a point x ∈ π −1 (b).This gives a well-defined projection E → (E ∨∨ , supp(E ∨∨ /E)) ∈ Prym(C/B) × T .
Since for any vector bundle E with spectral curve C and determinant δ the sheaves E with E ∨∨ = E which are singular at a point are parameterized by Quot(E, 1) = P(E) with the projection map sending E to supp(E/E) [EL99], the non-locally free sheaves with determinant δ and spectral curve {b} × T * + C are a P 1 -bundle with base Prym(C/B) × T .
As the union of these two components is the fibre of a holomorphic map between two compact spaces, the union of the components must be compact, and therefore the closure of the locally free component intersects with the non-locally free component along |C ∩ {b} × T * | sections.
In the previous proposition, we showed that for a spectral curve with a single jump, the fibre of the graph map decomposes into two irreducible components.The following result allows us to explicitly compute the intersection locus of these two components.corresponding to a non-zero s ∈ Ext 1 (O p , λ), define the maps f s = ϕ • α and g s = th, where t is chosen so t(ψ • h) = β, and set Ẽs = ker(f s + g s ).Note that any other extension corresponding to s will be given by maps zϕ and 1 z ψ for some z ∈ C * , giving f s = zϕ • α and g s = tzh, so (f s + g s ) is unique up to multiplication by a scalar, and Ẽs is well-defined.As s goes to zero, these maps become f 0 = ι 1 • α and g 0 = ι 2 • β, where ι i are the co-product maps for j * λ ⊕ O x .This gives that Ẽ0 is of the desired form.

Applications
In this section, we use Proposition 3.9 and the results of section 4 to prove some results about the fundamental groups of moduli spaces of rank-2 stably irreducible sheaves, as well as compute explicit data about the graph map fibration in the cases where the dimension of the moduli space is at most 6.For this section, the invariants δ, c 2 are assumed to be such that a rank-2 sheaf E is stably irreducible whenever det(E) = δ and c 2 (E) = c 2 .

The Topology of the Moduli Spaces
The case of ∆(2, δ, c 2 ) = 1 4 was previously studied in [AMT12] leading to the following result: X) is a primary Kodaira surface with the same fibre as X, and their Néron-Severi groups satisfy the relation ord(N S(X))| ord(N S(M 2,δ,c2 (X))).
Proof.For the case of ∆(2, δ, c 2 ) = 0, every spectral curve is smooth by Proposition 3.12, and the genus formula (3.5) gives that each spectral curve C in P 2,δ,c2 is an unramified double cover of B.
From this we can conclude that G −1 (C) ∼ = Prym(C/B) is the group with two elements, giving the desired result.
Recall that given any fibre bundle F ֒→ Y → Z, there is an induced exact sequence of the homotopy groups [BT82, Section 17].
Using (5.1) we see that whenever (δ, c 2 ) are such that 1 2 ≤ ∆(2, δ, c 2 ) < t(2, δ),we have since P δ,c2 is a holomorphic fibre bundle with connected and simply connected fibres.In particular this means that for any section σ : B → P δ,c2 and any element [γ] ∈ π 1 (P δ,c2 ), there is a representative of [γ] contained in σ(B).Let E be a regular rank-2 vector bundle in M 2,δ,c2−1 (X) with spectral curve C, and take the section σ E : B → P δ,c2 given by b → C + {b} × T * .We will show that for any loop γ ∈ σ E (B), there is a loop in M 2,δ,c2 (X) which maps to γ, demonstrating that π 1 (G) is a surjection.Consider the subset Quot(E, 1) ⊆ M 2,δ,c2 consisting of non-locally free sheaves whose double dual is E and which have one singularity counting multiplicity.Since Quot(E, 1) ∼ = P(E), the map G| Quot(E,1) is a fibration over σ E (B).Applying (5.1) again gives the exact sequence Remark 5.9.For the case of Proposition 5.8, since C ′ → B is a degree 2 map from a genus 2 curve to a genus 1 curve, the map has ramification at two points.Thus Remark 5.10.Note that in this case all of the singular fibres of the graph map have a similar complexity, as the singular fibres correspond after allowable elementary modifications to moduli spaces of sheaves of dimensions 6 − 4k, for some k ∈ Z >0 , of which 2 is the only non-negative value.This contrasts with the Douady space X [3] , where the fibres above points of the form 3p ∈ Sym 3 (B) have significantly different behaviour to the singular fibres above points of the form 2p + q ∈ Sym 3 (B).This can be seen by comparing punctual Hilbert schemes of 2 and 3 points as in [Bri77].
The vector bundles with spectral curve S can be described by parameterizing the sequences of elementary modifications taking a vector bundle with spectral curve C to one with spectral curve S.This process is described in detail in [Mor03, Section 4] for Hopf surfaces, and the method for Kodaira surfaces is similar.
Because of this, the fibres of the graph map above spectral curves with jumps must be computed inductively using information about moduli spaces of lower dimensions, so an understanding of the fibration structure of M 2,δ,c2 (X) requires a description of the fibration structure of M 2,δ,c2−k (X) for all 0 ≤ k ≤ 2∆(2, δ, c 2 ).

Further Questions
In the case of a Lagrangian fibration f : M → P with both M and P Kähler, a result from [SV21] (due to Matsushita [Mat05] in the projective case) gives an isomorphism between R i π * O M and Ω i P for integers i, from which the cohomology of O M can be computed from the Hodge numbers of P via the Leray spectral sequence.The proof of the above result uses the Kähler condition mainly to show the isomorphism away from the singular fibres of the Lagrangian fibration, so if M 2,δ,c2 (X) has a Kähler metric away from the singular fibres of its Lagrangian fibration the above result may still hold in this case.Under these hypotheses the Leray spectral sequence would degenerate at the second page, giving With respect to the fundamental group, in addition to determining whether the bounds from [Ara11] extend to higher dimensions, these bounds may also be improved by studying when loops in the smooth fibres are homotopy equivalent to loops in a singular fibre.This would naturally generalize the case of elliptic surfaces with singular fibres but no multiple fibres, where the fundamental group is entirely determined by the base as all loops in smooth fibres are homotopy equivalent to loops inside the simply connected singular fibres.Proving such a result for M 2,δ,c2 (X) would improve the bounds on the number of generators of the fundamental group to Z 8∆(2,δ,c2)−2 π 1 (M 2,δ,c2 (X)) π 1 (P 2,δ ) 0 for ∆(2, δ, c 2 ) ≥ 1 2 .In the ∆(2, δ, c 2 ) = 1 2 case, this would imply that the fundamental group of M 2,δ,c2 has at most four generators, which is exactly the number of generators for the fundamental group of X [2] .
, and F p is the unique non-trivial extension of O B (p) by O B with p ∈ B. Thus P c2 (b 0 ) ⊗ p * 2 V δ fits into the exact sequence 0 −1 (b) for some b ∈ B. Since D has torsion first Chern class, in this case the determinant and second Chern class are related by det(E ′ ) = det(E) ⊗ π * (O B (−b)), c 2 (E ′ ) = c 2 (E) + deg(λ).If a vector bundle E has a jump at b, there is a unique elementary modification of E along π −1 (b) by a negative degree bundle, called the allowable elementary modification of E at b [Mor03, Section 4.1.2];particularly, since E| π −1 (b) is unstable, it is of the form λ ⊕ (λ * ⊗ det(E)| π −1 (b) ) for some λ ∈ Pic −h (T * ) with h > 0, with the map g : E| π −1 (b) → λ given by projection onto the first coordinate.The elementary modification E ′ then has µ(E ′ , b) = µ(E, b) + deg(λ).

Definition 3 .
13.A rank-2 vector bundle E on X is regular at b for some b ∈ B if E| π −1 (b) is semi-stable and not isomorphic to λ ⊕ λ for any λ ∈ Pic(T ).Moreover, E is regular if it is regular at b for every b ∈ B. The regular rank-2 vector bundles with fixed irreducible spectral curve C are classified via the following result: Proposition 3.14 (Brînzȃnescu-Moraru [BM06]).Let C be an irreducible bisection of J(X) with normalization C. If we set W to be the normalization of X× B C with induced projections γ : W → X and ψ : W → C, then i.There is a line bundle L on W such that γ * (L) is a regular rank-2 bundle on X with spectral curve C. ii.If L 1 , L 2 ∈ Pic(W ) are such that γ * (L 1 ) and γ * (L 2 ) both have spectral curve C, then L 1 ⊗ L −1 2 = ψ * (λ) for some λ ∈ Pic(C), and γ * (L 1 ) ∼ = γ * (L 2 ) if and only if L 1 ∼ = L 2 .Using the notation of the above Proposition, we also have that if L ∈ Pic(W ) is such that γ * (L) has spectral curve C and det(γ * (L)) = δ, then for any λ ∈ Pic(C) det(γ * (L ⊗ ψ * (λ)))) ∼ = δ ⊗ η n (λ), where η : C → B is the projection induced from J(X), and η n : Pic(C) → Pic(B) is the norm homomorphism of η, which is the group homomorphism defined by η n (O C (p)) = O B (η(p)) for all p ∈ C. Because of this, the regular rank-2 bundles on X with determinant δ and spectral curve C are of the form γ * (L ⊗ ψ * (λ)), where λ ∈ Prym(C/B) := ker(η n ).(See [BM06, Theorem 4.5] for more details.)An elementary modification of a vector bundle E at b by a degree zero bundle λ exists if and only if E| π −1 (b) is an extension of λ by another degree-zero line bundle λ ′ .If E is regular at b there is a unique surjection from E| π −1 (b) to λ up to composing with an automorphism of λ, so there is a unique elementary modification.If E admits an elementary modification by λ at b but E is not regular at b, then E| π −1 (b) ∼ = λ ⊕ λ The surjections from E| π −1 (b) to λ are parameterized by C 2 , and since constant multiples of a surjection induce the same elementary modification, the elementary modifications of E by λ at b are parameterized by P 1 .
where ψ : W → C is the map induced from the fibred product [BM06, Theorem 4.5].Choose effective divisors D 0 , D 1 on C so that V = O C (D 0 − D 1 ).Since for any c ∈ C, the pushforward γ * (L ⊗ ψ * (O C (−c))) is an elementary modification of γ * (L) by a degree-zero bundle, there are sequences of elementary modifications by degree-zero bundles taking E 0 to γ * (L 1 ⊗ ψ * (O C (−D 1 ))) and and γ : W → X the map coming from the fibred product[AT03].Since γ * L| b is a regular bundle by Proposition 3.16, every elementary modification of the form (4.1) is itself the pushforward of a line bundle on W [AT03, Remarque 5].We then have that G −1 (C) ∼ = Prym(C/B) [BM06, Theorem 4.5].Note that each connected component of Prym(C/B) has dimension g(C) − 1, and when C is an unramified cover of degree 2, Prym(C/B) is the twoelement group.For the next simplest case, where G = C + {b} × T * and C ∼ = C, we can compute the fibres of the graph map as in the following propositions: Proposition 4.1.Take δ ∈ Pic(X), b ∈ B, and let C ⊂ J(X) be a smooth and irreducible ι δ -invariant bisection.Then the torsion-free sheaves with determinant δ and spectral curve C + {b} × T * are parameterized by a union of two P 1 -bundles over Prym(C/B) × T that intersect along |C ∩ {b} × T * | sections.Remark 4.2.Since C is a bisection of pr 1 : B × T * → B, C ∩ {b} × T * will always contain one or two points.
δ b ⊗ L −1 ) 0 is an exact sequence, where j : π −1 (b) → X is the inclusion map.In particular, Ẽ has spectral curve C. We then have a well-defined projectionE → ( Ẽ, L) ∈ G −1 (C) × Pic 1 (T ) = Prym(C/B) × T .To determine the fibres of this map, we notice that for any choice of bundle Ẽ with determinant δ ⊗ π * O B (b) and spectral curve C, L ∈ Pic 1 (T ), and surjective map f : Ẽ → j * L, there is a vector bundle E with determinant δ and spectral curve C + {b} × T * such that E ⊗ π * O B (−b) = ker(f ).

Proposition 4. 3 .
In the context of Proposition 4.1, the intersection of the irreducible components of the fibre consists of sheaves of the form ker(f ⊕ g) for E a vector bundle with spectral curve C and determinant δ ⊗ π * O B (b), f : E → j * λ non-zero such that (b, λ) ∈ C, and g : E → O x with x ∈ π −1 (b) such that g • ker(f ) = 0. Furthermore, these sheaves are determined up to isomorphism by (E, λ, x).Proof.Since the non-locally free component in Proposition 4.1 is compact, we can compute the intersection by finding the limits of families of deformations within the locally free component which are not locally free.Every vector bundle in the locally free component is given by an elementary E is a vector bundle with spectral curve C and determinant δ ⊗ π * O B (b), and L ∈ Pic 1 (B).Fix a choice of E and L. For any λ such that (b, λ) ∈ C, there is a surjective sheaf map α : E → j * λ, which is unique up to multiplication by a scalar.Let p be the unique point in T such that L is an extension in Ext 1 (O p , λ), and set x = j(p).If we now choose sheaf maps h : E → j * L and β : E → O x so that h is surjective and β • ker(α) = 0, we now have that for any map η : L → O p that there is a unique t ∈ C satisfying t(η•h) = β.Using these data, we can construct a deformation over Ext 1 (O p , λ) whose non-zero elements are given as follows:

µ
i {b i } × T * , the non-locally free component of G −1 (S) can be found by describing Quot(E, ℓ) for all vector bundlesE in M 2,δ,c2−ℓ (X) with spectral curve S E = S − k i=1 ν i {b i } × T * for {ν i } with 0 ≤ ν i ≤ µ i and k i=1