Abstract
Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of Markushevich–Tikhomirov, extended by Arbarello–Saccà–Ferretti, Matteini and Sawon–Shen. In this article we address a question of Sawon, showing that these integrable systems and their associated natural Lagrangians degenerate, respectively, into fix loci of involutions considered by Heller–Schaposnik, García-Prada–Wilkin and Basu–García-Prada. Along the way we find interesting results such as the proof that the Donagi–Ein–Lazarsfeld degeneration is a degeneration of symplectic varieties, a generalization of this degeneration, originally described for K3 surfaces, to the case of an arbitrary smooth projective surface, and a description of the behaviour of certain involutions under this degeneration.
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1 Introduction
1.1 Context and motivation
By means of non-abelian Hodge theory [14, 19, 30, 53, 54], the moduli space of Higgs bundles carries a hyperKähler structure which naturally defines a triple of symplectic structures on it, each holomorphic with respect to one of the three complex structures. For one of these complex structures, the Higgs moduli space is a quasiprojective variety, further equipped with a proper fibration onto a vector space whose generic fibres are abelian Lagrangians with respect to the corresponding holomorphic symplectic form. These data define the Hitchin integrable system [29].
The moduli space of pure dimension 1 sheaves on a symplectic surface (i.e. K3 or abelian) can be equipped with the Mukai [44] holomorphic symplectic form and with the Le Potier (support) fibration [35], whose generic fibres are Jacobians. After Beauville [5], these fibres are also Lagrangians. In the case of K3 surfaces, the base of the Le Potier morphism is a linear system, hence the moduli space of pure dimension 1 sheaves on a K3 becomes a projective integrable system, named the Beauville–Mukai integrable system. Putting aside that it is projective, the Beauville–Mukai integrable system has a similar description to that of the Hitchin system obtained via the spectral correspondence. These similarities become even more explicit with the construction of the Donagi–Ein–Lazarsfeld [18] degeneration of the first integrable system into the latter.
The cohomological structure of the Hitchin system is very rich and many surprising identities occur within this framework, giving rise to a great number of conjectures. One of them is topological Mirror symmetry conjecture [26], predicting the equality between the stringy E-polynomials of Higgs moduli spaces for a pair of Langlands dual groups, proven for \(\mathrm {SL}(n,{\mathbb {C}})\) and \(\mathrm {PGL}(n,{\mathbb {C}})\) [23, 26]. Another conjectural cohomological identity is the P=W conjecture [17], predicting that the morphism in cohomology induced by the non-abelian Hodge correspondence exchanges the preverse filtration associated to the Hitchin fibration with the weight filtration on the associated character variety. This was proven by de Cataldo–Hausel–Migliorini [17] in the rank 2 case, and by de Cataldo–Maulik–Shen [15, 16] in the case of base curves of genus 2. The Donagi–Ein–Lazarsfeld degeneration in the case of abelian surfaces was a key element in the work of de Cataldo–Maulik–Shen. Using this degeneration, they constructed a specialization morphism in cohomology, and, so, they could apply the powerful machinery developed by Markman [37,38,39] for the study of the cohomology of the moduli of sheaves on a K3, to the Higgs moduli space.
The cohomological \(\chi \)-independence is another astonishing property of the moduli spaces of (twisted) Higgs bundles and pure dimension 1 sheaves on del Pezzo surfaces. It states that the intersection cohomology is independent from the Euler characteristic \(\chi \) of the classified objects and was recently proven by Maulik–Shen [42].
Making use of the hyperKähler structure, [33] Kapustin and Witten introduced branes in the Higgs moduli space, setting that a \(\mathrm {(BBB)}\)-brane is a hyperholomorphic subvariety supporting a hyperholomorphic sheaf, while a \(\mathrm {(BAA)}\)-brane is a flat bundle over a complex Lagrangian subvariety for the holomorphic symplectic form associated to the first complex structure. Substituting the first by the second and third complex structures in this definition, we obtain \(\mathrm {(ABA)}\) and \(\mathrm {(AAB)}\)-branes, respectively. As indicated in [33], the Mirror symmetry conjecture predicts a duality between \(\mathrm {(BBB)}\) and \(\mathrm {(BAA)}\)-branes. This conjecture has motivated many authors to construct and study branes on Higgs moduli spaces. Most of the previous constructions are obtained by considering fixed loci of involutions on the moduli space, we highlight the \(\mathrm {(BBB)}\) and \(\mathrm {(BAA)}\)-branes obtained from natural involutions considered by Heller–Schaposnik [28] and García-Prada–Wilkin [22] out of holomorphic involutions on the base curve, the work of García-Prada and Ramanan [21] who classified the involutions obtained out of a combination of outer automorphisms of the group and tensorization, and the recent work [3], where Basu and García-Prada extended this study to include the action of holomorphic involution of the base curve, a set-up which already appeared in [10] for the case of elliptic curves. As indicated by Gukov in [24], Mirror duality between \(\mathrm {(BBB)}\) and \(\mathrm {(BAA)}\)-branes would imply certain cohomological relations between their support, a direction that was taken by Hausel–Mellit–Pei [27] to provide strong evidence for the pair of branes considered by Hitchin in [31] which are constructed out of the pair of Nadler–Langlands groups [45] given by \(\mathrm {Sp}(2m,{\mathbb {C}})\) and \(\mathrm {U}(m,m)\).
Kapustin–Witten’s definition of branes extends naturally to hyperKähler varieties other than the Higgs moduli space. The case of the moduli space of sheaves over a symplectic surface was considered by the author, Jardim and Menet in [20], where they constructed branes of any type arising as fixed loci of natural involutions on the moduli induced by involutions on the surface, and studied the behaviour of these natural branes under some correspondences.
Within this setting, the natural involution associated to an anti-symplectic involution on a K3 surface is again anti-symplectic, and its fixed locus defines a complex Lagrangian subvariety of the moduli, which we call natural Lagrangian subvariety. One obtains a symplectic involution by composing this natural anti-symplectic involution with the dualizing involution on the fibres (perhaps tensoring with a line bundle), which is also anti-symplectic. Hence, considering the fixed locus of the symplectic involution constructed out of anti-symplectic involution on a K3 surface, one obtains a class of integrable systems whose Lagrangian fibres are Prym varieties. These Prymian integrable systems on K3 surfaces were first considered by Markushevich–Tikhomirov [40], and extended by Arbarello–Saccà–Ferretti [1], Matteini [41] and Sawon–Shen [50, 52]. In [49], Sawon conjectured that these Prymian integrable systems degenerate into integrable systems related to the Hitchin system, leaving open the description of these conjectural systems.
Sawon’s conjecture was the motivation for our work as such degeneration of Prymian integrable system and that of the associated natural Lagrangian subvarieties, could open the door for a cohomological study of pairs of \(\mathrm {(BBB)}\) and \(\mathrm {(BAA)}\)-branes in the Hitchin system, leading perhaps to strong evidence of their duality, by means of the specialization morphism in cohomology given in [15, 16].
In their recent paper [50], Sawon and Shen provided a degeneration of a particular choice of Prymian integrable system into the \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Higgs moduli space.
1.2 Main results
In this article we extend the Donagi–Ein–Lazarsfeld construction [18] to the case of a curve fitting in an arbitrary smooth projective surface, obtaining a degeneration of the moduli of pure dimension 1 sheaves on the surface into the moduli space of Higgs bundles on the curve, twisted by the normal bundle of the curve inside the surface (Theorem 4.1).We highlight that, in particular, our degeneration connects (Remark 4.2) the two moduli spaces for which the cohomological \(\chi \)-independence is known to hold [42]. We also study how certain involutions fit into this degeneration, finding that natural involutions and their composition with the dualizing involution on the moduli space of pure dimension 1 sheaves on the surface, degenerate into involutions on the moduli space of twisted Higgs bundles that we describe explicitly (Lemmas 4.6 and 4.8). When the twist is the canonical bundle of the curve, i.e. for Higgs bundles in the usual sense, these involutions coincide with those studied by Heller–Schaposnik [28], García-Prada–Wilkin [22] and Basu–García-Prada [3]. This allow us to construct a degeneration of the subvarieties given by loci fixed by these involutions (Theorems 4.12 and 4.14).
In the case of a K3 surface equipped with an antisymplectic involution the previous results provide degenerations of Prymian integrable system and their associated natural Lagrangian subarieties of the moduli of pure dimension 1 sheaves on a K3. In particular, this gives a degeneration of the Prymian integrable systems constructed in [1] by Arbarello–Saccà–Ferretti (Sect. 4.4.2), and also (Sect. 4.4.3) by Markushevich–Tikhomirov [40], Matteini [41] and Sawon–Shen [50, 52]. Sawon conjectured in [49] that these Prymian integrable systems degenerate into integrable systems related to the Hitchin system. Every Lagrangian fibration degenerates naturally into the cotangent bundle of one of its smooth fibres, and the later can easily be identified with a rank 1 Hitchin system. Hence, one can always find examples of such degenerations. It is still open whether or not, the previous Prymian integrable systems degenerate into more interesting Hitchin systems. Our work provides an answer to Sawon’s question in the case of sheaves supported on non-primitive curves, obtaining a degeneration of the corresponding Prymian integrable systems into Hitchin systems of higher rank.
We also review the degeneration given by Sawon–Shen [50] into the \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Higgs moduli space, studying as well the degeneration of the associated natural Lagrangian (Sect. 4.4.4). It is worth noticing that we find that this natural Lagrangian degenerates into the fixed locus of an involution associated to \(\mathrm {U}(m,m)\)-Higgs bundles, whose Nadler–Langlands group is \(\mathrm {Sp}(2m,{\mathbb {C}})\).
We also show that the Donagi–Ein–Lazarsfeld degenerations in the case of symplectic surfaces is equipped with a deformation of the symplectic structure of the moduli spaces (Theorem 4.4). This allow us to understand the degenerations of natural Lagrangian subvarieties studied in [20] and the Prymian integrable systems as degenerations of \(\mathrm {(BAA)}\) and \(\mathrm {(BBB)}\)-branes (Corollary 5.2). Finally, we prove that these branes are dual under a Fourier–Mukai transform restricted to the locus of pure 1 dimension sheaves with smooth support curves (Proposition 5.3 and Corollary 5.4).
1.3 Outline of the paper
The paper is structured as follows. In Sect. 2 we review the necessary background for our work. Section 2.1 contains generalities on the moduli space of pure dimension 1 sheaves on surfaces. In Sect. 2.2 we study the involutions on the moduli space naturally induced by pull-back of involutions on the surface. Section 2.3 is dedicated to the study of the involution on the moduli space of pure dimension 1 sheaves induced by dualizing the restriction of a sheaf to its fitting support. We also study the composition of this involution with a natural involution, and study their fixed loci, which is generically a fibration by Prym varieties. In Sect. 2.4 we consider these involutions starting from an anti-symplectic involution on a K3 surface, revisiting the Markushevich–Tikhomirov construction of Prymian integrable systems. Since we are interested in moduli spaces of sheaves with a non-primitive first Chern class, we treat this case in detail. Also, we study the Lagrangian subvarieties that arise from the fixed point locus of the natural involutions on the moduli constructed out of our anti-symplectic involution on the K3 surface. In Sect. 2.5 we collect the necessary facts about ruled surfaces which will be used in Sect. 2.6 to study twisted Higgs bundles, whose spectral data provide a particularly relevant example of pure dimension 1 sheaves.
We describe in Sect. 3 certain involutions on moduli spaces of twisted Higgs bundles. In Sect. 3.1 we describe some involutions on a ruled surface and study their relation with the Poisson structure. We construct the corresponding involutions on the moduli spaces of spectral data of twisted Higgs bundles in Sect. 3.2 and study their behaviour under the spectral correspondence. We obtain involutions on the moduli space of L–Higgs bundles, which in the particular case of the twisting by the canonical bundle, are involutions that have been widely studied by Heller–Schaposnik, García-Prada–Wilkin and Basu–García-Prada.
The main results of the paper are contained in Sect. 4. In Sect. 4.1 we provide a generalization to the case of an arbitrary smooth projective surface of the Donagi–Ein–Lazarsfeld degeneration, originally described for K3 surfaces. Hence, we obtain a degeneration of the moduli space of pure dimension 1 sheaves on a surface into the moduli spaces of Higgs bundles twisted by the normal bundle of a curve inside our surface. In Sect. 4.2 we provide a deformation of the symplectic structure of the moduli spaces involved in the Donagi–Ein–Lazarsfeld degeneration in the case of symplectic surfaces, showing that it provides a degeneration of symplectic varieties. In Sect. 4.3, we study the behaviour under the Donagi–Ein–Lazarsfeld degeneration of the involutions considered in Sects. 2.2 and 2.3. We then obtain a degeneration of the subvarieties described by their fixed loci. In Sect. 4.4 we provide an explicit description of such degenerations in the context of a K3 surface equipped with an anti-symplectic involution. We describe the non-linear degenerations of the Prymian integrable systems constructed by Arbarello–Saccà–Ferretti, Markushevich–Tikhomirov, Matteini and Sawon–Shen, showing that they degenerate into integrable systems related to the Hitchin system. We also describe the degeneration of the associated natural Lagrangian subvarieties.
Finally, in Sect. 5, we consider the natural Lagrangian subvarieties and the Prymian integrable systems in the context of branes, and provide some evidence for their duality.
2 Pure dimension 1 sheaves on surfaces and involutions on their moduli
2.1 Generalities on the moduli space of sheaves on surfaces
Let S be a projective surface and take \(\mathrm {H}\) to be a polarization on it. We say that a coherent sheaf \({\mathcal {F}}\) on S is pure dimension d if its schematic support has dimension d and every subsheaf is also supported on dimension d subschemes. In that case, its Hilbert polynomial \(P({\mathcal {F}},\mathrm {H})\) has degree d and we define its \(\mathrm {H}\)-polarized rank \(\mathrm {rk}({\mathcal {F}}, \mathrm {H})\) to be the leading term of \(P({\mathcal {F}},\mathrm {H})\) multiplied by d!. A coherent sheaf \({\mathcal {F}}\) is \(\mathrm {H}\)-stable (resp. \(\mathrm {H}\)-semistable) if it is of pure dimension, and for every proper subsheaf \({\mathcal {F}}' \subset {\mathcal {F}}\) we have that its Hilbert polynomial satisfy \(P({\mathcal {F}}',\mathrm {H})/\mathrm {rk}({\mathcal {F}}',\mathrm {H}) < P({\mathcal {F}},\mathrm {H})/\mathrm {rk}({\mathcal {F}},\mathrm {H})\) (resp. \(P({\mathcal {F}}',\mathrm {H})/\mathrm {rk}({\mathcal {F}}',\mathrm {H}) \le P({\mathcal {F}},\mathrm {H})/\mathrm {rk}({\mathcal {F}},\mathrm {H})\)) when \(n \gg 0\). As we require n to be big enough, \(P({\mathcal {F}},\mathrm {H})(n) = \chi \left( {\mathcal {F}}(n\mathrm {H})\right) \). A semistable sheaf is polystable if it decomposes as a direct sum of stable sheaves. Simpson provided in [53] the existence of the moduli space \(\mathrm {M}_{S}^\mathrm {H}(P)\) of pure dimension \(\mathrm {H}\)-semistable sheaves with Hilbert polynomial P, whose closed points represent polystable sheaves.
The topological invariants of a sheaf over a smooth surface S determined by the Hilbert polynomial are the \(\mathrm {H}\)-polarized rank \(\mathrm {rk}({\mathcal {F}},H)\), the first Chern class \(c_1({\mathcal {F}})\), and the Euler characteristic \(\chi ({\mathcal {F}})\). To simplify the computations, we introduce the Mukai vector, \(\mathrm {v}({\mathcal {F}}) := \mathrm {ch}({\mathcal {F}}) \cdot \sqrt{\mathrm {Todd}(S)}\). Note that \(\mathrm {v}({\mathcal {F}}) \in H^{2*}(S, {\mathbb {Q}})\) and the cup product in cohomology provides the Mukai pairing which endows \(H^{2*}(S, {\mathbb {Q}})\) with a lattice structure. We abbreviate by \(\mathrm {M}^{\mathrm {H}}_S(\mathrm {v})\) the moduli space of \(\mathrm {H}\)-semistable sheaves S with Hilbert polynomial determined by the Mukai vector \(\mathrm {v}\in H^{2*}(S, {\mathbb {Q}})\).
It follows from deformation theory that the smooth locus of \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) is the locus defined by those sheaves that are simple and that the tangent space to \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) at the point determined by the simple coherent sheaf \({\mathcal {F}}\) corresponds with
and its cotangent space is
Hence, \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) is smooth at those points.
If, further, S is a K3 or abelian surface, its canonical bundle \(K_S\) is trivial and a choice of a (non-vanishing) section \(\Omega _S\) provides a symplectic form on S. In both cases we say that S is a symplectic surface. Thanks to Serre duality, \(H^2(S, {\mathcal {O}}_S)\) is dual to \(H^0(S, K_S) \cong {\mathbb {C}}\).
Theorem 2.1
[44] Let S be a symplectic surface. Then, \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) is equipped with a holomorphic 2-form \(\Omega _\mathrm {M}\) defined by taking the trace of the Yoneda product composed with \(\Omega _S\),
Furthermore, \(\Omega _\mathrm {M}\) is non-degenerate on the smooth locus of \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\), defining a symplectic form there.
Bottacin [13] and Markman [36] generalized this construction to the case of Poisson surfaces, i.e. those equipped with Poisson bi-vector, a non-zero section \(\Theta _S \in H^0(S, K_S^{-1})\).
Theorem 2.2
[13, 36] Let S be equipped with a Poisson bi-vector \(\Theta _S\). Then, one can define a (closed although possibly degenerate) Poisson structure \(\Theta _\mathrm {M}\) on the smooth locus of \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) by taking the trace of the Yoneda product composed with \(\Theta _S\),
Note that a symplectic surface is also Poisson, and, in that case, the Bottacin–Markman form coincides with the Mukai form after identifying the tangent and cotangent spaces on the smooth locus of our moduli (via a trivialization of \(K_S\)).
Proposition 2.3
[13] A smooth Poisson surface is either a symplectic surface (i.e. K3 or abelian) or a ruled surface.
On a smooth surface S, every pure dimension 1 sheaf \({\mathcal {F}}\) has a locally free resolution of length 2 and we define the fitting support, \(\mathrm {supp}({\mathcal {F}})\), of \({\mathcal {F}}\) as the determinant associated to this resolution. Note that the first Chern class is the cohomology class of its fitting support
The Mukai vector of a pure dimension 1 sheaf \({\mathcal {F}}\) on a smooth surface with canonical sheaf \(K_S\) takes the form
and the pairing is simply given by the intersection of the supports,
Pick a smooth curve C of genus \(g_C\) in S and consider the associated curve nC which is non-reduced if \(n>1\). For any integer a we define the Mukai vector
After Le Poitier [35], the corresponding moduli space can be equipped with a fibration to the Hilbert scheme classifying dimension 1 subschemes of S with first Chern class equal to the second component of the Mukai vector,
The fibre of \(\mathrm {h}_S\) over the curve \(A \in \mathrm {Hilb}_S([nC])\) is the Simpson compactified Jacobian classifying \(\mathrm {H}|_A\)-semistable pure dimension 1 sheaves on A of rank 1 and degree a
Remark 2.4
The Picard group of a smooth projective K3 surface is discrete and embeds into its second integral cohomology space. Hence, in this case, the second component of the Mukai vector fixes the determinant and \(\mathrm {Hilb}_S([nC])\) is the linear system |nC|. After the work of Beauville [5] the fibres are Lagrangian with respect to the Mukai form \(\Omega _\mathrm {M}\). Hence, this is an algebraic completely integrable system called the Beauville–Mukai system.
In general, the natural morphism \(\mathrm {Pic}(S) \rightarrow H^2(S,{\mathbb {Z}})\) is not an embedding. Furthermore \(\mathrm {Hilb}_{S}([nC])\) can be reducible and even non-connected. Choose a curve A parametrized by \(\mathrm {Hilb}_{S}([nC])\), and let us denote by \(\{ A \}\) the connected component of the Hilbert scheme containing the curve A. Accordingly, we denote by
the associated connected component of the moduli.
Remark 2.5
Our notation differs from that of Matteini in [41], where, given a smooth curve A, he defined \(\{ A \}\) to be the irreducible component of \(\mathrm {Hilb}_{S}([nC])\) containing A. Since we will allow A to be singular, and even non-reduced, A may be contained in several irreducible components, although it determines uniquely its connected component.
Remark 2.6
When the irregularity of the surface S vanishes, i.e. \(h^1(S, {\mathcal {O}}_S) = 0\), we have that \(\mathrm {Hilb}_{S}([nC])\) is a discrete union of connected components. In this case, \(\{ A_i \} = |A_i|\), each being a linear system.
Let us denote by \({\mathcal {A}}\) the restriction to \(\{ A \} \times S\) of the universal subscheme associated to \(\mathrm {Hilb}_S([nC])\). Note that \({\mathcal {A}}\) is a family of locally planar curves parametrized by \(\{ A \}\). The polarization \(\mathrm {H}\) on S provides a relative polarization on \({\mathcal {A}}\) that we still denote by \(\mathrm {H}\). By means of the Le Poitier fibration (2.2), one can identify the component \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a,A)\) of our moduli space of pure dimension 1 sheaves on S with the relative compactified Jacobian over \({\mathcal {A}}\) of relative torsion free sheaves of rank 1 and degree a,
If \(\{ A \}^\mathrm {sm}\) denotes the open set of smooth curves, and \({\mathcal {A}}^\mathrm {sm}\) the restriction of \({\mathcal {A}}\) there, one has the following identification of open subsets
contained in the smooth locus.
Remark 2.7
We have dropped the polarization from the notation on the Jacobian as pure dimension 1 sheaves whose restriction to its support have rank 1, are always stable, regardless of the choice of polarization.
2.2 Natural involutions from involutions on the surface
In this section we study the involutions on the moduli space naturally induced by pull-back of involutions on the surface.
Suppose that our smooth projective surface S is equipped with an involution
Let C be a smooth projective curve in S which is preserved under \(\zeta _S\) and consider a Mukai vector \(\mathrm {v}_a\) of the form specified in (2.1) built out the \(\zeta _S\)-invariant curve C. Then, one trivially has that \(\mathrm {v}_a\) is \(\zeta _S\)-invariant as well, so one can construct a birational involution on the moduli space,
that we call natural involution associated to \(\zeta _S\). We denote the closure of its fixed point locus by
If, further, the polarization \(\mathrm {H}\) is \(\zeta _S\)-invariant, \({\widehat{\zeta }}_S\) is a biregular morphism and its fixed locus is already closed. The support morphism (2.2) restricts to \(\mathrm {N}_{S,\mathrm {H}}(\mathrm {v}_a)\) with image \(\mathrm {Hilb}_S([nC])^{{\widehat{\zeta }}_C}\) of curves preserved by the involution, which may have several components.
Note that every curve \(A'\) in \(\mathrm {Hilb}_S([nC])^{{\widehat{\zeta }}_C}\) inherits an involution
inducing another birational involution on the compactified Jacobian,
As the fibres of (2.2) are identified with the corresponding compactified Jacobian, one has
It has been shown (see [2, Lemma 9] or [20] for instance) that the involution induced on the moduli space has the same behaviour under the symplectic form as the starting one. The generalization to the case of Poisson surfaces is straight-forward.
Proposition 2.8
Consider a Poisson surface S and suppose it is equipped with a Poisson involution \(\zeta _S^+\) (resp. an anti-Poisson involution \(\zeta _S^-\)). Then, the natural (birational) involution \({\widehat{\zeta }}_S^+\) is also Poisson (resp. \({\widehat{\zeta }}_S^-\) is anti-Poisson).
Proof
We have to prove that whenever our involution \(\zeta _S^\pm \) preserves the (powers of the) canonical bundle \(K_S\) and commutes or anti-commutes with \(\Theta _S\), so does \({\widehat{\zeta }}_S^\pm \) with respect to \(\Theta _\mathrm {M}\) over the open subset \(U \subset \mathrm {M}_S^H(\mathrm {v}_a)\) of smooth points where \({\widehat{\zeta }}_S^\pm \) is biregular.
The proof is very similar to that of [20, Theorem 3.4]. Suppose that \({\mathcal {F}}\) represents a point in U. Then, \((\zeta _S^\pm )^*{\mathcal {F}}\) lies in U too. For each \(i \ge 0\), consider the induced morphism in cohomology \(R^i(\zeta _S^\pm )^*\), and note that it commutes with the Yoneda product. Also, one has that \(R^i(\zeta _S^\pm )^*\) commutes with the trace morphism and Serre duality, appearing in the definition of the Bottacin–Markman form. The Poisson involution \(\zeta _S^+\) commutes with the remaining morphism in that definition, \(\langle \cdot , \Theta _S \rangle \), while the anti-Poisson involution \(\zeta _S^-\) anticommutes with it. Then, we can see that \(\zeta _S^+\) commutes with the composition of all these morphisms, which define the Poisson form \(\Theta _\mathrm {M}\) while \(\zeta _S^-\) anticommutes with it. Hence, the proof is complete. \(\square \)
Suppose now that the quotient surface 
is smooth and denote the quotient map by
Suppose as well that
for some smooth curve \(D \subset T\). We have already seen that the Mukai vector \(\mathrm {v}_a\) is \(\zeta _S\)-invariant as so is C.
Remark 2.9
A Mukai vector of the form \(\mathrm {v}_{2b}\) is the pull-back of the Mukai vector in T,
Pick a curve \(B \subset T\) in the class [nD] and consider its lift \(A = B \times _T S\) to S, lying in the class [nC]. Recall that we denote by \(\{ B \}\) and \(\{ A \}\) the connected components of \(\mathrm {Hilb}_T([nD])\) and \(\mathrm {Hilb}_S([nC])\) containing B and A, respectively. Note that \({\widehat{\zeta }}_S\) restricts to \(\{ A \}\) although the fixed locus \(\{ A \}^{{\widehat{\zeta }}_S}\) might be disconnected. We shall denote by \(\mathrm {q}_S^*\{B \}\) the connected subset of \(\{ A \}^{{\widehat{\zeta }}_S}\) given by those curves obtained by lifting curves in \(\{B \}\). In principle, \(\mathrm {q}_S^*\{B \}\) is not a connected component of \(\{ A \}^{{\widehat{\zeta }}_S}\), only a union of some irreducible components.
Remark 2.10
We have seen Remark 2.6 whenever the surface S has vanishing irregularity, the \(\{ A_i \}\) are the linear systems \(|A_i|\). In this case, the locus \(| A_i |^{{\widetilde{\zeta }}_S}\) of curves preserved by \(\zeta _S\) is disconnected and amounts to the union of \(|A_i|^{{\widetilde{\zeta }}_S}_+\) and \(|A_i|^{{\widetilde{\zeta }}_S}_-\), each being the projectivization of the \(+1\) and \(-1\) eigenspaces for the action of \({\widehat{\zeta }}_S\) on \(H^0(S, {\mathcal {O}}_S(A_i))\). In this context,
We consider the subvariety
By restriction of (2.2), \(\mathrm {N}_{S,\mathrm {H}}(\mathrm {v},B)\) is equipped with a fibration,
whose fibres are
Consider over T the moduli space \(\mathrm {M}_{T,\mathrm {I}}(\mathrm {w}_b)\) of sheaves with a Mukai vector \(\mathrm {w}_b\) and a polarization I. As before, we denote by \(\{ B \}\) the connected component of the Hilbert scheme in T containing B, and by \({\mathcal {B}}\) the family of curves parametrized by \(\{ B \}\) that we obtain from restricting there the universal subscheme associated to \(\mathrm {Hilb}([nD])\). Recall that one has the identification
Denote the pull-back of the polarization by
For these choices, the involution \({\widehat{\zeta }}_S\) is biregular and the image of the pull-back morphism under the corresponding quotient map lies in its fixed locus,
Given the curves \(B \subset T\) and \(A \subset S\) as before, denote by
the projection induced by \(\mathrm {q}_S\). Observe that the restriction of (2.6) to the Hitchin fibres gives,
where \({\widehat{\mathrm {q}}}_A := \mathrm {q}_A^*\).
Let us denote by \(\{ B \}^\mathrm {ssl}\) the locus of \(\{ B \}\) parametrizing smooth curves \(B'\) whose lift \(A' = B' \times _T S \subset \{ A \}\) is also smooth. The notation stands for smooth and smooth lift. Recall the family of curves \({\mathcal {B}}\rightarrow \{ B \}\) and denote by \({\mathcal {B}}^\mathrm {ssl}\) its restriction to \(\{ B \}^\mathrm {ssl}\). Set also \({\mathcal {A}}^\mathrm {ssl}\) to be the restriction to \(\mathrm {q}_S^*\{ B \}^{\mathrm {ssl}}\) of the family of curves \(\text{\AA} \rightarrow \{ A \}\), and observe that it comes equipped with a projection induced by \(\mathrm {q}_S\),
One can see that \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is ramified whenever the intersection of the branching locus \(\Delta \) of \(\mathrm {q}_S\) with generic elements of \(\{ B \}\) is non-empty, and, when \(\zeta _S\) is without fixed points, \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is unramified. Let us denote by \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}}\) the restriction of \({\widehat{\mathrm {q}}}_S\) to \(\mathrm {q}_S^*\{ B \}^\mathrm {ssl}\). Recall from Remark 2.7, that the sheaves supported on smooth (hence irreducible) curves are always stable,
This provides a description of an open subset of \(\mathrm {N}_{S,\mathrm {H}}(\mathrm {v}_{2b}, B)\).
Proposition 2.11
Suppose that \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is ramified. Then
and (2.6) is, generically, a 1:1 morphism.
If, on the contrary, \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is an unramified 2 : 1 cover associated to the relative 2-torsion line bundle \({\mathcal {L}}\rightarrow {\mathcal {B}}^\mathrm {ssl}\), one has that \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}}\) factors through the quotient by \({\mathbb {Z}}_2\) acting by tensor product with \({\mathcal {L}}\), giving
and (2.6) is, generically, a 2:1 morphism.
Proof
In the first case, by Kempf descent lemma, \(\mathrm {Im}\left( {\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}} \right) \) coincides with the left-hand side of (2.9). Since, by hypothesis, \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is ramified, \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}}\) is an embedding. As we find ourselves within the smooth locus of \(\mathrm {M}^{\mathrm {H}}_S(\mathrm {v}_{2b},A)\), \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}}\) provides an isomorphism of its source with its image. As \(\{ B \}^\mathrm {ssl}\) is a dense open subset of \(\{ B \}\), this is, generically, 1 : 1 morphism.
The second case follows since the pull-back of \({\mathcal {L}}\) is trivial over \({\mathcal {A}}^\mathrm {ssl}\). \(\square \)
2.3 The dualizing involution and Prymian fibrations
In this section we study the involution on the moduli space of pure dimension 1 sheaves induced by dualizing the restriction of a sheaf to its fitting support. We also study the composition of this involution with a natural involution, and study their fixed loci, which is generically a fibration by Prym varieties.
Given a Mukai vector \(\mathrm {v}_a\) on S as defined in (2.1), whose second component is the class [nC], let us choose a line bundle \(J \in \mathrm {Pic}(S)\) whose intersection with the second component of the Mukai vector satisfies
Note that not every value of a and n allow for such a J. It is discussed in [1, Sections 3.3 and 3.5], that the moduli space \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) is equipped with an involution
Pick \(A = \mathrm {supp}({\mathcal {F}})\), which is Gorenstein with canonical line bundle \(K_S(A)|_{A}\), Grothendieck–Verdier duality implies that,
Since A belongs to the class [nC], equation (2.11) implies \(2a = \deg (J(A)|_{A})\) and this ensures that \(\xi _{J,S}\) preserves the Mukai vector. Observe that \(\xi _{J,S}^2 = {\mathbf {1}}_M\) holds as every pure dimension 1 sheaf in a smooth surface is reflexive. Also, \(\xi _{J,S}\) restricts to the fibres of the Mukai fibration, \(\mathrm {h}^{-1}_S(A) \cong {\overline{{{\,\mathrm{Jac}\,}}}}^{\, a}_{\mathrm {H}}(A)\), giving the dualizing involution composed with an appropriate tensor product,
where we recall that the dual of a rank 1 torsion-free sheaf on a Gorenstein curve is always well defined and \(\mathrm {H}\)-stability is preserved.
Remark 2.12
Condition (2.11) is always satisfied when we choose \(\mathrm {v}_0\) and \(J_0 = {\mathcal {O}}_S(-nC)\), for instance. In that case \(\xi _{J_0, A}\) is just the dualizing involution \({\mathcal {E}}\mapsto {\mathcal {E}}^\vee \).
Recall that we have chosen the Mukai vector \(\mathrm {v}_a\) to be \(\zeta _S\)-invariant. If further the line bundle J on S satisfying (2.11) is \(\zeta _S\)-invariant too, \({\widehat{\zeta }}_S\) and \(\xi _{J,S}\) commute and, following [1], we consider their composition
which defines a birational involution on \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\). We denote the closure of its fixed locus by
We now study the relation of these involutions with the Poisson (and symplectic) structure. It was proved in [1, Proposition 3.11] that \(\xi _{J,S}\) is anti-symplectic under the assumptions of S being a K3 surface, \(\mathrm {v}_0\) a Mukai vector with \(n=1\) and \(J = {\mathcal {O}}_S(-C)\).
Proposition 2.13
Given a Poisson surface S, for any choice of Mukai vector \(\mathrm {v}_a\) and \(J \in \mathrm {Pic}(S)\) satisfying (2.11), the involution \(\xi _{J,S}\) is anti-Poisson with respect to the Bottacin–Markman Poisson form \(\Theta _\mathrm {M}\) on \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\),
Proof
The proof of [1, Proposition 3.11] extends to the Poisson case with minor changes. \(\square \)
As an immediate consequence of Propositions 2.8 and 2.13 one derives the following corollary which will be used in Sect. 4.4.
Corollary 2.14
Let S be a Poisson surface and \(\zeta _S^+\) a Poisson involution on it (resp. \(\zeta _S^-\) an anti-Poisson involution). For any choice of Mukai vector \(\mathrm {v}_a\) and a \(\zeta _S^+\)-invariant (resp. \(\zeta _S^-\)-invariant) \(J \in \mathrm {Pic}(S)\) satisfying (2.11), the involution \(\uplambda _{J,S}^+ = {\widehat{\zeta }}_S^+ \circ \xi _{J,S}\) is a birational anti-Poisson involution (resp. \(\uplambda _{J,S}^- = {\widehat{\zeta }}_S^- \circ \xi _{J,S}\) is a birational Poisson involution) on \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\).
Let us now assume that the quotient surface T is smooth, and recall the notation introduced in Sect. 2.2. Consider the closed subvariety
Whenever \(\mathrm {H}\) is \(\zeta _S\)-invariant, \(\uplambda _{J,S}\) is well defined as a regular involution and its fixed locus is already closed.
By construction, one has the following commuting diagram
Observe that, over \(A' \in \mathrm {q}_S^*\{ B \}\),
where
is the induced birational involution on \(\mathrm {h}^{-1}_S(A')\). If \(A'\) is irreducible, in particular when it is smooth and connected, every pure dimension sheaf is stable so \({\widehat{\zeta }}_{J,A'}\) is a biregular involution and so is \(\uplambda _{J,A'}\). Hence, the associated fixed locus is already closed.
Recall that we denoted by \(\{ B \}^{\mathrm {ssl}}\), the open subset of \(\{ B \}\) parametrizing smooth curves, whose lift to S is smooth too.
Remark 2.15
For \(B' \in \{ B \}^\mathrm {ssl}\) lifting to \(A'\) smooth one has that \(\mathrm {h}^{-1}_S(A') = {{\,\mathrm{Jac}\,}}_{A'}(a)\) is a torsor for the abelian variety \({{\,\mathrm{Jac}\,}}_{A'}(0)\). After (2.14), one has that the fibre of \(\mu \) is
whose connected components are torsors for the Prym variety
Recall from (2.7) the morphism \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\). If \(\mathrm {q}_S\) is ramified and its branching locus \(\Delta \) intersects B non-trivially, then \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is a ramified morphism. Let consider the case described in Remark 2.12.
Remark 2.16
Consider the Mukai vector \(\mathrm {v}_0\) associated to \(a = 0\) and \(J_0 = {\mathcal {O}}_S(-nC)\). Recall that for ramified maps, the kernel \({\mathbf {1}}_{{{\,\mathrm{Jac}\,}}} + {\widehat{\zeta }}_{A'}\) is connected. It follows immediately from Remark 2.15 and the fact that \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is ramified, that
Since the kernel \({\mathbf {1}}_{{{\,\mathrm{Jac}\,}}} + {\widehat{\zeta }}_{A'}\) has two connected components if \(\mathrm {q}_{A'}\) is an unramified 2 : 1 cover. Then, in that case, \(\mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_0, J_0,B) |_{\{ B \}^{\mathrm {ssl}}}\) has 2 connected components and (2.17) holds after restricting ourselves to the connected component of the identity.
Whenever \(\{ B \}^\mathrm {ssl}\) is a dense open subset of \(\{ B \}\), by Remark 2.16 the generic fibres of \(\mu : \mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_a,J,B) \rightarrow \mathrm {q}_S^*\{ B \}\) are (non-canonically) isomorphic to Prym varieties. In that case, we refer to it as the Prymian fibration associated to the involution \(\zeta _S\).
2.4 Natural Lagrangian and Prymian integrable systems
Following Sawon–Shen [50] (see also [52]) and Matteini [41], we provide a straight-forward adaptation of the Markushevich–Tikhomirov construction of Prymian integrable systems to the case of non-primitive first Chern classes.
Prymian integrable systems are integrable systems fibered by Prym varieties. The class that we consider here arises from an anti-symplectic involution on a K3 surface. These systems were first constructed by Markushevich and Tikhomirov [40] and later generalized by Arbarello, Saccà and Ferretti [1], Matteini [41], Sawon and Shen [50, 52] and others. The case considered in [40] corresponds to an anti-symplectic involution on a smooth K3 surface, whose associated quotient is a del Pezzo surface of degree 2. In [1], Arbarello, Saccà and Ferretti constructed a Prymian integrable system from an anti-symplectic involution on a K3 giving an Enriques surface. Matteini [41] generalized the construction of [40] to the case of anti-symplectic involutions on K3 associated to a del Pezzo surface of degree 3. Sawon and Shen [50, 52] studied the case of an antisymplectic involution on a K3 whose quotient is a del Pezzo surface of degree 1.
Let X denote a smooth K3 surface, \(\zeta _X^-\) an antisymplectic involution on it, and Y the quotient \(Y = X/\zeta _X^-\) which is a either a smooth rational surface or an Enriques surface. Denote by \(\Delta _Y\) the branch locus of the cover \(X \rightarrow Y\), which is a subcurve of Y giving a point in the linear system \(|-2K_Y|\). Over Y, pick a smooth connected curve \(D \subset Y\) of genus \(g_D\) not tangent to the branch locus \(\Delta _{Y}\), and consider
which is smooth, and it is equipped with a double cover \(\mathrm {q}_C: C \rightarrow D\) ramified at \(\Delta _Y \cdot D\) and with a Galois involution \(\zeta _C : C \rightarrow C\) induced by \(\zeta _X^-\). Similarly, consider curves nD and \(nC = nD \times _Y X\) which are non-reduced for \(n>1\), and denote by \(g_{nD}\) and \(g_{nC}\) the genus of the smooth curves in the linear systems |nD| and |nC|. We have for all \(n > 0\), that
where,
by the genus formula. Since \(\chi (Y) = 1\) in all possible cases,
so
Combining this with (2.19) one has
Since the irregularity of Y vanishes, \(h^1(Y,{\mathcal {O}}_Y) = 0\), \(\mathrm {Hilb}_{Y}([nD])\) consists of a disjoint union of linear systems, for instance \(\{ nD \} = |nD|\).
Consider the natural (birational) anti-symplectic involution \({\widehat{\zeta }}_X^{\, -}\) and, having picked \(\mathrm {J}\) for which (2.11) holds with respect to \(\mathrm {v}_a\), the (birational) involution \(\uplambda _{\mathrm {J},X}^- = {\widehat{\zeta }}_X^{\, -} \circ \xi _{\mathrm {J}, X}\), which is symplectic. Pick the subvarieties \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD)\) and \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\) defined as the closure of the fixed locus of \({\widehat{\zeta }}_X^{\, -}\) and \(\uplambda _{\mathrm {J},X}^-\) respectively, given by those sheaves supported on curves parametrized by \(\mathrm {q}_X^*|nD|\).
Recall that we denoted by \(|nD|^\mathrm {ssl}\) the locus of smooth curves in |nD| with smooth lift in |nC|.
Lemma 2.17
Let D be a smooth curve in a smooth surface Y and let \(\mathrm {q}_X : X \rightarrow Y\) be a 2 : 1 covering by the smooth surface X. Then, \(|nD|^\mathrm {ssl}\) is an open subset in |nD|.
Proof
By Bertini’s theorem, the locus of smooth curves, \(|nD|^\mathrm {sm}\), is an open subset in |nD|. If the curve \(B'\) is smooth and does not intersect tangentially the branch locus \(\Delta \) of \(\mathrm {q}_X\), the lifted curve \(A' = B' \times _Y X\) is smooth by construction. The latter is an open condition, so \(|nD|^\mathrm {ssl}\) is an open subset of |nD|. \(\square \)
Hence \(\mathrm {q}_X^*|nD|^\mathrm {ssl}\) is open in \(\mathrm {q}_X^*|nD|\), and so are \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD)|_{\mathrm {q}_S^*|nD|^\mathrm {ssl}}\) in \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD)\) and \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)|_{\mathrm {q}_S^*|nD|^\mathrm {ssl}}\) in \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\), respectively. In that case, \(\mathrm {h}_X^{-1}(A)\) classifies line bundles over A, being stable and smooth as points in \(\mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a)\). Then, both \({\widehat{\zeta }}_X^{\, -}\) and \(\uplambda _{\mathrm {J},X}^-\) restrict to biregular involutions over the locus of smooth curves in \(\mathrm {q}_X^*|nD|\). This allow us to prove the following.
Proposition 2.18
The projective subvariety \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD) \subset \mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a)\) is Lagrangian with respect to \(\Omega _\mathrm {M}\), in particular,
Proof
We have seen that \(\mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a)\) is smooth over \(\mathrm {q}_X^*|nD|^\mathrm {ssl}\). Also, \({\widehat{\zeta }}_X^{\, -}\) is a biregular anti-symplectic involution there. It then follows that its fixed point locus \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD)|_{\mathrm {q}_S^*|nD|^\mathrm {ssl}}\) is smooth and Lagrangian. As it is an open subset of \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD)\), it follows that the latter is a Lagrangian subvariety as well. \(\square \)
In view of Proposition 2.18, we refer to \(\mathrm {N}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,nD)\) as the natural Lagrangian subvariety of \(\mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a)\) associated to \(\zeta _X^{\, -}\).
We study now the fixed locus of \(\uplambda _{\mathrm {J},X}^-\).
Proposition 2.19
The projective subvariety \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J},nD) \subset \mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a)\) has dimension
Proof
Following Remark 2.16 one has a description of the open subset \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)|_{\mathrm {q}_S^*|nD|^\mathrm {ssl}}\) of \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\) in terms of a Prymian fibration constructed with a projection of curves with genus \(g_{nC}\), to curves of genus \(g_{nD}\), having fibres of dimension \(g_{nC} - g_{nD}\). After (2.20), the statement follows. \(\square \)
Remark 2.20
Following Proposition 2.19, whenever
the dimension of \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\) is \(2\chi (nD) - 2\) and the generic fibres of \(\mu \) are half-dimensional.
We next see that the Prymian fibration
endows \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\) with the structure of an integrable system, that we call the Prymian integrable system associated to the anti-symplectic involution \(\zeta _X^-\). This construction has been considered by several authors [1, 40] in the case of a primitive first Chern class, and later in [41, 50] including the non-primitive case. For completion, we include a proof of it instead of just citing the previous articles.
Theorem 2.21
Let X be a smooth K3 surface X equipped with an anti-symplectic involution \(\zeta _X^-\) giving the quotient \(Y = X/\zeta _X^-\). Pick a smooth curve \(D \subset Y\) of genus \(g_D\) and \(n \ge 1\) such that (2.21) holds. Take C as in (2.18), a Mukai vector \(\mathrm {v}_a\) as in (2.1) and \(\mathrm {J}\in \mathrm {Pic}(X)\) satisfying (2.11) with respect to \(\mathrm {v}_a\). Then, one has that
-
(1)
\(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,\mathrm {J},nD)\) is a projective variety of dimension \(2\chi (nD) - 2 = n^2D^2 -nD\cdot K_Y\);
-
(2)
the smooth locus of \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a,\mathrm {J},nD)\) carries a holomorphic 2-form \(\Omega _\mathrm {P}\) which is symplectic on a dense open subset where the fibres of \(\mu \) are Lagrangian; and
-
(3)
the generic fibre of \(\mu \) is a a torsor over a smooth abelian variety of dimension \(\chi (nD) - 1 = \frac{1}{2}(n^2D^2 -nD\cdot K_Y)\), being the Prym of a double cover of smooth curves.
Proof
(1) is an immediate consequence of Proposition 2.19 and Remark 2.20.
Over the smooth locus of \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\), its tangent space embeds into the tangent space of \(\mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a)\) and one can define the holomorphic 2-form by restricting the Mukai form \(\Omega _\mathrm {M}\) there. Also, \(\uplambda _{\mathrm {J},X}^-\) restricts to a biregular symplectic involution over the locus of smooth curves in \(\mathrm {q}_X^*|nD|\), where the symplectic form \(\Omega _\mathrm {M}\) is well defined. There, the open subset of \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\) given by the fixed locus of \(\uplambda _{\mathrm {J},X}^-\) inherits a symplectic form which obviously coincides with the restriction of \(\Omega _\mathrm {P}\) whenever we find ourselves in the smooth locus of \(\mathrm {P}_{X,\mathrm {H}}^{\, -}(\mathrm {v}_a, \mathrm {J}, nD)\). By [5], \(\Omega _\mathrm {M}\) vanishes on the fibres of \(\mathrm {h}_X\) as we have seen in Remark 2.4. As \(\Omega _\mathrm {P}\) is obtained from restricting \(\Omega _\mathrm {M}\), it follows that \(\Omega _\mathrm {P}\) vanishes on the fibres of \(\mu \) over smooth curves. Equivalently, for every \(A \in \mathrm {q}_X^*|nD|\) smooth, \(\mu ^{-1}(A)\) is isotropic with respect to \(\Omega _\mathrm {P}\). Recalling from Remark 2.20 that these fibres are half-dimensional, we conclude the proof of (2).
Finally (3) follows from Remark 2.15. \(\square \)
2.5 Ruled surfaces
Let us recall in this section some facts about ruled surfaces that will be used in Sect. 2.6.
Given a smooth projective curve C of genus \(g_C\) and a line bundle L on C, consider the total space of our line bundle \(\mathrm {Tot}(L)\) and the obvious projection \(p: \mathrm {Tot}(L) \rightarrow C\). Set \(\ell := \deg (L)\), assuming \(\ell \ge 0\), and consider the projective compactification of \(\mathrm {Tot}(L)\), namely the ruled surface with topological invariant \(\ell \),
naturally equipped with the projection that we still denote by \(p : {\mathbb {L}}\longrightarrow C\).
One has, of course,
where \(\sigma _\infty \cong C\) is the curve in \({\mathbb {L}}\) associated to the 0 section of \({\mathcal {O}}_C\). Let us briefly recall some properties of \({\mathbb {L}}\) following [25, Sect. 2, Chap. V]. The curve \(\sigma _\infty \) has negative self-intersection, hence
for \(m \ge 0\). Along with \(\mathrm {Pic}(C)\), \(\sigma _\infty \) generates the Picard group of \({\mathbb {L}}\),
so any divisor D on \({\mathbb {L}}\) is numerically equivalent to \(b_1 \sigma _\infty + b_2 F\), where F denotes the fibre and \(\sigma _\infty \cdot \sigma _{\infty } = -\ell \), \(\sigma _\infty \cdot F = 1\) and \(F \cdot F = 0\). A divisor is ample if and only if \(b_1 > 0\) and \(b_2 > \ell b_1\), and only exist irreducible curves in those classes with \(b_1 > 0\) and \(b_2 \ge \ell b_1\), with the exception of the infinity section \(\sigma _\infty \). The zero section of L defines \(\sigma _0 : C \rightarrow {\mathbb {L}}\) whose image is linearly equivalent to \(\sigma _\infty + \ell F\) and its normal bundle returns
Observe that any divisor D with null intersection with \(\sigma _\infty \) is numerically equivalent to a multiple of \(\sigma _0\). Also, for \(n \ge 0\), it is linearly equivalent to \(n\sigma _0\) if and only if \({\mathcal {O}}_{\mathbb {L}}(D)\) has a non-zero section, as by scaling the fibres of \(\mathrm {Tot}(L)\), D deforms linearly to a multiple curve supported on \(\sigma _0\). Hence, the linear system \(|n\sigma _0|\) has an open subset classified by linear deformations of the curve, which coincides with those curves not intersecting \(\sigma _\infty \). Linear deformations of a curve are classified by the sections of the normal bundle restricted to the curve. Hence, one has the identification
Let r denote the obvious projection of the multiple curve onto its reduced support and observe that \(r_*{\mathcal {O}}_{{\mathbb {L}}}(n\sigma _0)|_{n\sigma _0}\) amounts to \({\mathcal {O}}_{{\mathbb {L}}}(n\sigma _0)|_{\sigma _0} \otimes r_*{\mathcal {O}}_{n\sigma _0}\) by the projection formula. The structural sheaf \({\mathcal {O}}_{n\sigma _0} \cong {\mathcal {O}}_A/{\mathcal {I}}_C^n\) decomposes, as an \({\mathcal {O}}_C\)-module, into \({\mathcal {O}}_C \oplus ({\mathcal {I}}_C/{\mathcal {I}}_C^2) \oplus \dots \oplus ({\mathcal {I}}_C^{n-1}/{\mathcal {I}}_C^{n})\). The intersection of \(\sigma _0\) with the canonical divisor of the surface (see (2.27) below) is \(L^{-1}\) times the canonical bundle of the curve (i.e. the cotangent bundle of the curve). It then follows that the normal bundle is \({\mathcal {I}}_C/{\mathcal {I}}_C^2 \cong {\mathcal {O}}_{{\mathbb {L}}}(-\sigma _0)|_{\sigma _0} \cong L^{-1}\). Therefore,
The canonical line bundle of \({\mathbb {L}}\) is
with canonical class
If \(\ell > 2g_C-2\) or \( L = K_C\), the inverse of the canonical \(K_{\mathbb {L}}^{-1} \cong \bigwedge ^2 {\mathcal {T}}\, {\mathbb {L}}\) always has non-zero sections and we can equip \({\mathbb {L}}\) with a Poisson structure by picking a Poisson bi-vector given by a non-zero section \(\Theta _{\mathbb {L}}\in H^0({\mathbb {L}}, K_{\mathbb {L}}^{-1})\). In general, the Poisson structure of \({\mathbb {L}}\) is not uniquely defined.
2.6 Twisted Higgs bundles and their spectral data
In our last preliminary section we introduce (L-twisted) Higgs bundles, whose spectral data are pure dimension 1 sheaves on ruled surfaces.
As before, C denotes a smooth projective curve and L a line bundle on it. An L-twisted Higgs bundle over C is a pair \((E, \varphi )\), where E is a holomorphic vector bundle on C, and \(\varphi \in H^0(C, \mathrm {End}(E) \otimes L)\) is a holomorphic section of the endomorphisms bundle, twisted by L. When L is the canonical bundle \(K_C\) of the curve, we refer to \(K_C\)-Higgs bundles, simply, as Higgs bundles. An L-twisted Higgs bundle \((E,\varphi )\) is (semi)stable if every \(\varphi \)-invariant subbundle \(F\subset E\) satisfies
A semistable L-twisted Higgs bundle \((E,\varphi )\) is polystable if it is a direct sum of stable bundles \((E,\varphi )= \bigoplus (E_i, \varphi _i)\) (all with the same slope \(\deg E_i/\mathrm {rk}E_i\)). It is possible to construct [46] the moduli space of rank n and degree d semistable (resp. stable) L-twisted Higgs bundles on C which we denote by \({\mathcal {M}}^L_C(n,d)\) (resp. \({\mathcal {M}}^L_C(n,d)^{\mathrm {st}}\)). When \(L=K_C\), we will write \({\mathcal {M}}_C(n,d)\) (resp. \({\mathcal {M}}_C(n,d)^{\mathrm {st}}\)) for the moduli space of semistable (resp. stable) Higgs bundles whose construction follows from [30, 54], being a connected, normal and irreducible variety of dimension \(2n^2(g_C-1) + 2\) [54]. When \(\deg (L)>\deg (K)\) or \(\deg (L)=\deg (K_C)\) but \(L^n \ncong K_C^n\), the moduli space \({\mathcal {M}}^L_C(n,d)\) is a quasi-projective variety of dimension \(2n^2\deg (L) + 1\) [46].
Given a Higgs bundle \((E,\varphi )\), we see that \(\varphi : E \rightarrow E \otimes L\) determines uniquely an action of \(\mathrm {Sym}^\bullet (L^*)\) on E, the Higgs bundle \((E,\varphi )\) can be seen as a \(\mathrm {Sym}^\bullet (L^*)\)-module. Since p is an affine morphism and \(\mathrm {Sym}^\bullet (L^*) \cong p_*{\mathcal {O}}_{\mathrm {Tot}(L)}\), the push-forward under p provides an equivalence of categories between \(\mathrm {Sym}^\bullet (L^*)\)-modules (including Higgs bundles) and \({\mathcal {O}}_{\mathrm {Tot}(L)}\)-modules, called the spectral correspondence [8, 29] where the Higgs bundle \((E,\varphi )\) is sent to the pure dimension 1 sheaf \({\mathcal {E}}\) on \(\mathrm {Tot}(L)\) having rank 1 on each irreducible component of its support, called the spectral datum of \((E,\varphi )\), and defined by \(\ker (p^*\varphi - \tau )\), where \(\tau \) denotes the tautological section of the pullback bundle \(p^*L \rightarrow \mathrm {Tot}(L)\). One can observe that the spectral datum \({\mathcal {E}}\) associated to any Higgs bundle \((E,\varphi )\) is supported on a curve in the linear system \(|n\sigma _0|\), that we call the spectral curve of \((E,\varphi )\), and the restriction of \({\mathcal {E}}\) to any irreducible component of the spectral curve has rank 1, and over the whole spectral curve is a sheaf with degree
where we denote
denoting by \(\ell \) the topological invariant \(\ell \) of the ruled surface \({\mathbb {L}}\). It then follows from (2.28) that the spectral datum of a L-Higgs bundle of rank n and degree d is a sheaf over \({\mathbb {L}}\) with Mukai vector \(\mathrm {v}_{d + \delta }\), where \(C = \sigma _0\) on its definition. Starting from such \({\mathcal {E}}\), one obtains a Higgs bundle \((E,\varphi )\), by setting \(E = p_*{\mathcal {E}}\) and \(\varphi = p_* ({\mathbf {1}}_{\mathcal {E}}\otimes \tau )\), where we have considered the morphism given by tensor product with the tautological bundle,
After [51] one has that the stability notions of a Higgs bundle and the corresponding spectral data coincide. Therefore, the Higgs moduli space is the open and dense subset of the moduli space of sheaves on \({\mathbb {L}}\) with Mukai vector \(\mathrm {v}_{d + \delta }\), where we recall (2.28), given by those sheaves whose restriction to its associated spectral curve has rank 1 on each irreducible component of its support,
We have seen that \({\mathbb {L}}\) is equipped with a Poisson structure \(\Theta _{\mathbb {L}}\) when \(\ell > 2g_C-2\) or \( L = K_C\). In that case, one can define the Bottacin–Markman Poisson structure on \(\mathrm {M}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta })\), as we saw in Theorem 2.2, which restricts to a Poisson structure \(\Theta _0\) on \({\mathcal {M}}^L_C(n,d)\). When \(L = K_C\), \(\Theta _0\) defines a symplectic form \(\Omega _0\) on \({\mathcal {M}}_C(n,d)\). In the recent work [11], Biswas–Bottacin–Gomez showed that this restriction coincides (up to scaling) with that obtained by extending the canonical symplectic form defined on the cotangent of the moduli space of stable vector bundles. Over the locus of smooth spectral curves, such identification is already implicit in [29].
Since any divisor with null intersection with \(\sigma _\infty \) is a multiple of \(\sigma _0\), the restriction of (2.2) provides a morphism,
known as the L-Hitchin fibration, which takes the form
after the identification 2.26. When \(L=K_C\), this provides an algebraic completely integrable system [29] called the Hitchin system.
3 Involutions on Higgs moduli spaces from involutions on ruled surfaces
3.1 Some involutions on ruled surfaces
We say that an involution on a variety equipped with a Poisson structure is Poisson if it preserves the Poisson structure under pull-back, and we call it anti-Poisson if the pull-back returns the inverse of the Poisson structure. In this section we study some involutions on \({\mathbb {L}}\) and their relation with the Poisson structure.
The first involution we consider is
defined by multiplying by \(-1\) the fibres of L. Note that its fixed locus is the disjoint union of the zero and infinity sections,
Lemma 3.1
Every line bundle \(J \in \mathrm {Pic}({\mathbb {L}})\) is preserved by the involution \(\imath \), i.e.
Suppose further that \(\ell \ge 2g_C - 2\), then \(\imath \) is an anti-Poisson involution with respect to any Poisson structure \(\Theta _{\mathbb {L}}\) defined on the ruled surface.
Proof
One naturally has that \(\imath ^*{\mathcal {O}}_{\mathbb {L}}(\sigma _\infty ) \cong {\mathcal {O}}_{\mathbb {L}}(\imath (\sigma _\infty ))\), hence \({\mathcal {O}}_{\mathbb {L}}(\sigma _\infty )\) is preserved by \(\imath \) after (3.2). As \(\imath \) preserves the fibres of \(p : {\mathbb {L}}\rightarrow C\), every line bundle in \(p^*\mathrm {Pic}(C)\) is \(\imath \)-invariant and the first statement follows from (2.24). In particular, \(K_{\mathbb {L}}^{-1}\) is invariant.
As \(\imath \) is an automorphism of \({\mathbb {L}}\), we have that \(d \imath \wedge d \imath \) provides an isomorphism between \(\imath ^* K_{\mathbb {L}}^{-1}\) and \(K_{\mathbb {L}}^{-1}\). Since the eigenvalues of \(d \imath \) are constantly 1 and \(-1\), one has
and all the sections of \(K_{\mathbb {L}}^{-1}\) are \(\imath \)-anti-invariant. This proves the second statement. \(\square \)
Consider an involution on the base curve
We say that a line bundle L on C is \(\zeta _C\)-invariant if there exists an isomorphism
In this case, the action of \(\zeta _C\) lifts, via f, to an action on L that can be easily extended to its completion \({\mathbb {L}}\),
Observe that the involution associated to \(-f\) coincides with the composition \(\imath \circ \zeta _f\). Denote the associated quotient map by
Remark 3.2
Suppose that 
gives a smooth curve. By Kempf’s descent lemma [34], if
\(\zeta _f\) acts trivially on the fibres of the points fixed under
\(\zeta _C\), the line bundle L descends to a line bundle W on D. Let us denote the associated ruled surface by
\({\mathbb {W}}:= {\mathbb {P}}\left( {\mathcal {O}}_D \oplus W \right) \). Under these conditions, the quotient is
Lemma 3.3
The involutions \(\imath \) and \(\zeta _f\) preserve cohomology class of every line bundle on \({\mathbb {L}}\). Therefore, every Mukai vector on \({\mathbb {L}}\) is \(\imath \)-invariant and \(\zeta _f\)-invariant.
Proof
After Lemma 3.1 it is enough to prove the lemma for \(\zeta _f\). Pull-back under \(\zeta _f\) preserves the line bundle \({\mathcal {O}}_{\mathbb {L}}(\sigma _\infty )\) and those line bundles in \(p^*\mathrm {Pic}({\mathbb {L}})\) coming from \(\zeta _C\)-invariant line bundles. Even if not every line bundle in \(p^*\mathrm {Pic}({\mathbb {L}})\) is preserved by \(\zeta _f\), the involution preserves the class of a fibre, F. Then, the first statement follows from (2.24). The second follows immediately from the first. \(\square \)
As \(\zeta _C\) is an automorphism of C, \(\partial \zeta _C\) provides a natural isomorphism between \(K_C\) and \(\zeta _C^*K_C\). Its compactification, \({\mathbb {K}}= {\mathbb {P}}({\mathcal {O}}_C \oplus K_C)\) is a ruled surface of topological invariant \(2g_C -2\). Let us denote by
and
After (2.27) the canonical bundle of \({\mathbb {K}}\) is
which trivializes on \(\mathrm {Tot}(K_C)\), the complement of \(\sigma _\infty \). Recalling (2.23), one has a unique (up to scaling) Poisson structure \(\Theta _{\mathbb {K}}\) on \({\mathbb {K}}\) defining a canonical symplectic structure on \(\mathrm {Tot}(K_C)\).
Lemma 3.4
The involution \(\zeta _{\mathbb {K}}^+\) is Poisson while \(\zeta _{\mathbb {K}}^-\) is anti-Poisson. Their restriction to \(\mathrm {Tot}(K_C)\) provide, respectively, a symplectic and an anti-symplectic involution.
Suppose that \(\mathrm {q}_C : C \rightarrow D := C/\zeta _C\) gives a smooth curve. Then, the quotient \({\mathbb {K}}/ \zeta _{{\mathbb {K}}}^-\) is smooth.
Proof
For the first statement, thanks to Lemma 3.1, it is enough to prove that \(\zeta _{\mathbb {K}}^+\), is a Poisson involution as \(\zeta _{\mathbb {K}}^-\) is the composition of \(\zeta _{\mathbb {K}}^+\) with the anti-Poisson involution \(\imath \). Suppose first that \(\mathrm {q}_C\) is unramified. On an open subset associated to a trivializing subset of \(K_C\), one can easily show that \(d \zeta _{\mathbb {K}}^+\) acts like \(\partial \zeta _C\) along the curve and like \(d\zeta _C\) along the fibres. Hence, \(d \zeta _{\mathbb {K}}^+ \wedge d \zeta _{\mathbb {K}}^+ = d \zeta _C \wedge \partial \zeta _C = 1\) and \(\zeta _{\mathbb {K}}^+ = \zeta _{\partial \zeta _C}\) is Poisson.
After Remark 3.2, the second statement follows from the observation that \(\zeta _{{\mathbb {K}}}^-\) acts, over an open subset associated to a trivializing subset of \(K_C\), as \(d\zeta _C\) along the fibres and \(-\partial \zeta _C\) along the curve. On the points of \(\mathrm {Fix}(\zeta _C)\), one has that \(d\zeta _C = -1\) so \(-\partial \zeta _C= 1\) and \(\zeta _{\mathbb {K}}^-\) acts trivially on the fibres of the line bundle \(K_C\) over \(\mathrm {Fix}(\zeta _C)\). The statement then follows from Remark 3.2. \(\square \)
3.2 Natural involutions under the spectral correspondence
Starting from the involutions \(\imath \) and \(\zeta _f\) on \({\mathbb {L}}\) defined in Sect. 3.1, one can consider the natural involutions \({\hat{\imath }}\) and \({\widehat{\zeta }}_f\) on the moduli space of sheaves \(\mathrm {M}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta })\), as defined in Sect. 2.2. Also, we recall the involutions \(\xi _{J,{\mathbb {L}}}\) and \(\uplambda _{J,f} = \xi _{J,{\mathbb {L}}} \circ {\widehat{\zeta }}_f\) defined in full generality in Sect. 3.1. In this section we describe the counterpart of these involutions on the moduli space of L–Higgs bundles. We finish this section studying the particular case of \(L = K_C\), associated to Higgs bundles (with no twisting). We find in this case that the previous involutions give rise to well known involutions that have been widely studied [3, 21, 22, 28].
Lemma 3.5
Under the spectral correspondence, the involution \({\hat{\imath }}\) corresponds to the biregular involution
If, given the involution \(\zeta _C : C \rightarrow C\), f provides an isomorphism between L and \(\zeta _C^*L\), then \({\widehat{\zeta }}_{f}\) corresponds under the spectral correspondence to the involution
which is biregular.
Proof
We begin by observing that \({\check{\imath }}\) and \({\check{\zeta }}_f\) preserve (semi)stability of Higgs bundles, so both are regular morphisms within the moduli of Higgs bundles. Recall that the spectral correspondence is realized by taking the push-forward under \(p: {\mathbb {L}}\rightarrow C\), obtaining the Higgs field by pushing-forward the morphism given by tensoring by the tautological section \(\tau \), \(\varphi = p_* ({\mathbf {1}}_{\mathcal {E}}\otimes \tau )\). In the first case, observe that \(p \circ \imath = p\), so \({\check{\imath }}(E) = E\) and
what concludes the proof of the first statement. For the second statement note that \(p \circ \zeta _f = \zeta _C \circ p\), which implies that \({\check{\zeta }}_f(E) = \zeta _C^*E\). Then,
Note also that the tautological section satisfies that
so,
from which the proof follows. \(\square \)
Let us denote the fixed point locus of \({\check{\zeta }}_f\) by
which is, naturally, a closed subvariety of \({\mathcal {M}}_C^L(n,d)\).
As we saw in Remark 3.2, Kempf descent lemma says that the line bundle L descends to a line bundle W on D if the action of f is trivial on the ramification locus of \(\mathrm {q}_C: C \rightarrow D\). In that case, the quotient \({\mathbb {L}}/ \zeta _f\) is the ruled surface \({\mathbb {W}}\). Observe, as well, that in that case \(L \cong \mathrm {q}_C^*W\).
Lemma 3.6
Under the spectral correspondence, \({\mathcal {N}}_C^L(n,d)\) embeds into \(\mathrm {N}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta }, nD)\). Furthermore, the Hitchin fibration restricts there to
Proof
We recall from (2.31) that the image of the support (Hitchin) morphism is contained in (an open subset of) a linear system \(|n\sigma _0|\) in \({\mathbb {L}}\). The fixed locus under the action of \({\widehat{\zeta }}_f\) on a linear system is disconnected and given by the union of the projectivization of the \(+1\) and \(-1\) eigenspaces of the associated space of sections. Observe that the component associated to \(+1\) coincides with \(\mathrm {q}_{\mathbb {L}}^*|nD|\), containing the open subset \(\bigoplus _{i = 1}^n \mathrm {q}_C^*H^0(D, W^{\otimes i})\). The rest of the proof follows easily from this observation. \(\square \)
One can consider the morphism
which corresponds to (2.6) under the spectral correspondence. If the spectral curves in \(\mathrm {Tot}(L)\) are ramified over the spectral curves in \(\mathrm {Tot}(W)\), then, Proposition 2.11 implies that \({\widehat{\mathrm {q}}}_C\) is generically 1:1, being an isomorphism when we restrict to the locus of smooth curves over smooth curves (corresponding to the locus indexed by \(\mathrm {ssl}\)). If, on the contrary, \(\zeta _C\) is unramified, Proposition 2.11 says that \({\widehat{\mathrm {q}}}_C\) factors through
where \({\mathbb {Z}}_2\) acts under tensorization by the 2-torsion line bundle on D associated to the unramified 2 : 1 cover \(\mathrm {q}_C : C \rightarrow D\).
Suppose one can pick an \(\zeta _f\)-invariant line bundle J such that condition (2.11) holds with respect to \(\mathrm {v}_{d + \delta }\). Observe that, after (2.24) and since \(\sigma _\infty \cdot \sigma _0 = 0\), one has that the restriction of J to any curve A on \(|n \sigma _0|\) is
for some \(J_C \in \mathrm {Pic}(C)\).
We now consider the involution \(\xi _{J,{\mathbb {L}}}\) and the composition \(\uplambda _{J_0,f} = \xi _{J,{\mathbb {L}}} \circ {\widehat{\zeta }}_f\) which is another involution as \(\xi _{J,{\mathbb {L}}}\) and \({\widehat{\zeta }}_f\) commute due to the \(\zeta _f\)-invariance of J. Let us also describe the counterpart of these involutions in \({\mathcal {M}}^L_C(n,d)\).
Lemma 3.7
Under the spectral correspondence, the involution \(\xi _{J,{\mathbb {L}}}\) corresponds to
Also, \(\uplambda _{J,f}\) corresponds to
both being biregular involutions.
Proof
After (2.13),
where, in this identification, we have made use of (2.25) and (3.11). Thanks to (2.27), we observe that the relative canonical sheaf \(\omega _{A/C}\) is \(p^*L^{n-1}\). Hence,
and the first statement follows. The second statement is a direct consequence of the first and Lemma 3.5. \(\square \)
Remark 3.8
Recall from Remark 2.12 that, choosing \(\mathrm {v}_0\) as Mukai vector and \(J_0 = {\mathcal {O}}_{\mathbb {L}}(-n\sigma _0)\), one gets \(\xi _{J_0,L}\) is the dualizing involution and
Remark 3.9
For \(\delta \) as in (2.29), the pair of Mukai vector \(\mathrm {v}_\delta \) and line bundle \(J_\delta = {\mathcal {O}}_{\mathbb {L}}(-\sigma _0)\) satisfies condition (2.11). In that case, one gets
Let us denote the fixed point locus of \({\check{\uplambda }}_{J_C, f}\) by
Observe that \({\mathcal {N}}_C^L(n,d)\) and \({\mathcal {P}}_C^L(n,d, J)\) are given by the restriction of \(\mathrm {N}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta })\) and \(\mathrm {P}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta },J)\) in (2.30). Recalling the subvarieties \(\mathrm {N}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta }, nD)\) and \(\mathrm {P}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta },J, nD)\) from (2.4) and (2.15), respectively, we study their relation with \({\mathcal {N}}_C^L(n,d)\) and \({\mathcal {P}}_C^L(n,d, J)\).
Lemma 3.10
Under the spectral correspondence, \({\mathcal {P}}_C^L(n,d, J_C)\) embeds into \(\mathrm {P}_{{\mathbb {L}}, H_0}(\mathrm {v}_{d + \delta },J, nD)\) . Furthermore, the Hitchin fibration restricts to
where W is described in Remark 3.2.
Proof
Mutatis mutandis, the proof follows as the proof of Lemma 3.6. \(\square \)
Sometimes in the literature, in the particular case of \(L = K_C\) and \(f = \partial \zeta _C\), one abbreviates \(({\mathbf {1}}_{\zeta _C^*E} \otimes \partial \zeta _C) \circ \zeta _C^*\varphi \) simply by \(\zeta _C^*\varphi \). Therefore, the natural involutions associated to \(\zeta _{\mathbb {K}}^{\pm } = \zeta _{\pm \partial \zeta _C}\), as described in (3.5) and (3.6), would be expressed as
under the spectral correspondence. This immediately implies that the counterpart of \(\uplambda _{J, {\mathbb {K}}}^\pm = \xi _{J,{\mathbb {K}}} \circ {\widehat{\zeta }}_K^\pm \) on the moduli space of Higgs bundles is
The involutions \({\check{\zeta }}_K^\pm \), along with their fixed point locus, have been studied by Heller–Schaposnik [28] and García-Prada–Wilkin [22]. The involutions \({\check{\uplambda }}_{J, K}^\pm \) are covered by the general study of Basu and García-Prada [3] of automorphisms of the moduli space of Higgs bundles. We combine their results in the following corollary, that could also be derived from Propositions 2.8 and 2.13.
Corollary 3.11
The involution \({\check{\zeta }}_{K}^+\) is symplectic and its fixed locus \({\mathcal {N}}_C^+(n,d)\) is a holomorphic symplectic subvariety of the Higgs moduli space \({\mathcal {M}}_C(n,d)\), while \({\check{\zeta }}_{K}^-\) is anti-symplectic, and defines a Lagrangian subvariety \({\mathcal {N}}_C^-(n,d)\) of \({\mathcal {M}}_C(n,d)\).
Similarly, \({\check{\uplambda }}_{J_C, K}^+\) is an anti-symplectic involution and its fixed locus \({\mathcal {P}}_C^+(n,d, J_C)\) a Lagrangian subvariety of \({\mathcal {M}}_C(n,d)\), while \({\check{\uplambda }}_{J_C,K}^-\) is symplectic and fixes a holomorphic symplectic subvariety \({\mathcal {P}}_C^-(n,d, J_C)\).
4 Non-linear degeneration of integrable systems
4.1 Extending the non-linear degeneration of Donagi, Ein and Lazarsfeld
In this section we generalize the Donagi, Ein and Lazarsfeld [18] non-linear degeneration of the moduli space of pure dimension 1 sheaves on a K3 surface into the moduli space of Higgs bundles over a curve inside the K3. Such degeneration was recently generalized to the case of curves in abelian surfaces by de Cataldo, Maulik and Shen [15], being a central step on their proof of the P=W conjecture over genus 2 curves. In this section we provide a degeneration of the moduli space of pure dimension 1 sheaves over any smooth surface, into the moduli space of L-Higgs bundles over a smooth subcurve, being L the normal bundle of our curve inside the surface.
Take a smooth surface S and consider a smooth curve \(C \subset S\) inside it of genus \(g_C\). From now on, let us denote its normal bundle by
that coincides with \(K_C \otimes (K_S|_C)^{-1}\) by adjunction. The projective completion of the normal cone is isomorphic to the ruled surface \({\mathbb {L}}\) defined in (2.22). This implies that the canonical divisor of \({\mathbb {L}}\) is
By the genus formula, and recalling that \(\sigma _0\) does not intersect with \(\sigma _\infty \), we get that
and
Hence, the linear systems |nC| and \(|n\sigma _0|\) contain generic curves of equal genus,
and the topological invariants of the ideal sheaves describing these curves coincide.
Take the trivial family \(S \times {\mathbb {P}}^1\) and consider the blow-up at \(C \times \{ 0 \}\),
which is non-singular as we are blowing-up a non-singular subvariety. Composing the structural morphism of the blow-up with the projection \(S \times {\mathbb {P}}^1 \rightarrow {\mathbb {P}}^1\), one gets the structural morphism
This map is flat and its fibres outside \(0 \in {\mathbb {P}}^1\) are not affected by our construction, so for the generic fibre, at \(t \ne 0\), we have
Meanwhile, the central fibre, at \(t = 0\), is the union
where both irreducible components meet transversely at the image of infinity section \(\sigma _\infty \subset {\mathbb {L}}\) identified with the curve \(C \subset S\). Note that \(\overline{{\mathbf {S}}}_0\) is a complete intersection subvariety. One can also consider the open subset
and observe that the restriction there of the structural morphism gives, again, a flat family of surfaces over \({\mathbb {P}}^1\), but now the central fibre \({\mathbf {S}}|_{t = 0} = \mathrm {Tot}(L)\) is not projective.
Denote by \(p_S\) the composition of the structural morphism of the blow-up with the projection of \(S \times {\mathbb {P}}^1\) onto S. Being \(\overline{{\mathbf {S}}}\) projective, one may choose a relative polarization
For all \(t \ne 0\) in \({\mathbb {P}}^1\), one trivially has that \({\mathbf {H}}|_{\overline{{\mathbf {S}}}_t} = \mathrm {H}\) while \(H_0 := {\mathbf {H}}|_{{\mathbb {L}}}\) corresponds to \(p^*(\mathrm {H}|_C) \otimes {\mathcal {O}}_{{\mathbb {L}}}(\sigma _{\infty })\), since \({\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {L}}\cup S)|_{\mathbb {L}}\cong {\mathcal {O}}_{{\mathbb {L}}}\) as \({\mathbb {L}}\cup S\) is a fibre, so \({\mathcal {O}}_{\overline{{\mathbf {S}}}}(-{\mathbb {L}})|_{\mathbb {L}}\cong {\mathcal {O}}_{\overline{{\mathbf {S}}}}(S)|_{\mathbb {L}}\cong {\mathcal {O}}_{\mathbb {L}}(\sigma _\infty )\).
Since \(C \times \{ 0 \}\) is a Cartier divisor of \(C \times {\mathbb {P}}^1\), one has that
embeds naturally into \(\overline{{\mathbf {S}}}\). One can consider the fibration \({\mathbf {C}}\rightarrow {\mathbb {P}}^1\), whose generic fibres, at \(t\ne 0\), are naturally identified with C,
Furthermore, the restriction of \({\mathbf {C}}\) to \(t = 0\) coincides with the image of the zero section,
inside \({\mathbb {L}}\subset \overline{{\mathbf {S}}}_0\). Note that the restriction to every point of \({\mathbb {P}}^1\) gives \({\mathbf {C}}_t \in H^2(\overline{{\mathbf {S}}}_t, {\mathbb {Z}})\). Using \({\mathbf {C}}\), consider the Mukai vector
Let us consider the relative Hilbert scheme parametrizing ideal sheaves of curves in the fibres of \(\overline{{\mathbf {S}}} \rightarrow {\mathbb {P}}^1\) with first Chern class \([n{\mathbf {C}}_t]\) and let us denote by \(\{ n{\mathbf {C}}\}\) the connected component containing the family of curves \(n{\mathbf {C}}\). Let us consider, inside it, the open subset given by those curves \(A \subset \overline{{\mathbf {S}}}_t\) contained in the \({\mathbf {S}}\subset \overline{{\mathbf {S}}}\),
Take a point of \(\{ n{\mathbf {C}}\}_{\mathbf {S}}\) representing the ideal sheaf \({\mathcal {I}}_A \hookrightarrow {\mathcal {O}}_{{\mathbf {S}}_t}\) of a curve \(A \subset {\mathbf {S}}_t\). We see that \({\mathcal {I}}_A\) is of rank 1 and pure dimension 2 over a smooth surface \({\mathbf {S}}_t\) which either is projective (for \(t \ne 0\)) or an affine subset of a smooth projective surface \({\mathbb {L}}\) (for \(t = 0\)) where \({\mathcal {I}}_A\) extends naturally. It follows that the determinant in cohomology is well define and one can consider the determinant morphism
to the relative Jacobian of \(\overline{{\mathbf {S}}}\), classifying line bundles on the fibres \(\overline{{\mathbf {S}}}_t\) of the family of surfaces.
For \(t \ne 0\), the fibre of (4.11) over \(L^{\otimes n} \in \mathrm {Pic}(S)\) corresponds to the linear system \(|L^{\otimes n}|\) in S. We recall once again that every irreducible projective curve contained in \(\mathrm {Tot}(L)\) can be deformed linearly to a multiple curve supported on \(\sigma _0\). Then, the image of \(\{ n{\mathbf {C}}\}_{\mathbf {S}}|_{t = 0}\) under (4.11) is \({\mathcal {O}}_{\overline{{\mathbf {S}}}_0}(n \sigma _0)\) and \(\{ n{\mathbf {C}}\}_{\mathbf {S}}|_{t = 0}\) is the locus of \(|n \sigma _0|\) inside \({\mathbb {L}}\) which does not intersect \(\sigma _\infty \). After (2.26), this is
coinciding with the Hitchin base in (2.32).
Following Simpson [53, Theorem 1.21] (see [32, Theorem 4.3.7] for a specific treatment of the relative case), let us consider the moduli space \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a)\) of relatively pure dimension 1 \({\mathbf {H}}\)-semistable relative sheaves over the \({\mathbb {P}}^1\) scheme \(\overline{{\mathbf {S}}}\) having Mukai vector \({\mathbf {v}}_a\). By construction, \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a)\) comes equipped with the structural flat morphism to \({\mathbb {P}}^1\), which trivializes over \({\mathbb {P}}^1 - \{ 0 \}\) having generic fibres equal to \(\mathrm {M}_{S,\mathrm {H}_t}(\mathrm {v}_a)\), and whose central fibre over \(t = 0\) is \(\mathrm {M}_{{\mathbb {L}}\cup S, {\mathbf {H}}_0}(\mathrm {v}_a)\). By specifying the relative (Fitting) support, one has the following surjective morphism
commuting with both structural morphisms. Consider the restriction \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a) |_{\{ n{\mathbf {C}}\}_{\mathbf {S}}}\) having generic fibres \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a,nC)\) over \(t \ne 0\) and central fibre \(\mathrm {M}_{{\mathbb {L}}, H_0}(\mathrm {v}_a)|_{\mathrm {supp}\cap \sigma _\infty = \emptyset }\) at \(t = 0\). For the description of the central fibre, note that a sheaf in \(\overline{{\mathbf {S}}}_0 = {\mathbb {L}}\cup S\) whose support fits in \(\mathrm {Tot}(L) \subset {\mathbb {L}}\) is \(H_0\)-semistable (resp. \(H_0\)-stable) as a sheaf in \({\mathbb {L}}\) if and only if it is \({\mathbf {H}}_0\)-semistable (resp. \({\mathbf {H}}_0\)-stable) as a sheaf in the whole \(\overline{{\mathbf {S}}}_0\). Recalling (2.30), we define the open subset
given by the complement of those sheaves at \(t = 0\) having rank different from 1 on some irreducible component of their support.
We summarize all of the above in the following theorem. This is a generalization to arbitrary smooth surfaces of the non–linear deformation that appeared first in [18] for K3 surfaces and in [15] for abelian surfaces.
Theorem 4.1
[18] Consider a smooth surface S and a genus \(g \ge 2\) smooth projective curve C inside S. Denote the normal bundle of C in S by L. Choose a positive integer n and a Mukai vector \(\mathrm {v}_a\) as in (2.1) constructed out of n and C. One can construct \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\) and \(\{ n{\mathbf {C}}\}_{\mathbf {S}}\) flat over \({\mathbb {P}}^1\), and a morphism between them commuting with the structural morphisms,
The horizontal arrow is a surjective fibration, trivial over \({\mathbb {P}}^1 - \{ 0 \}\), and the generic fibre over \(t \ne 0\) is the Le Poitier morphism (2.2),
while the central fibre at \(t = 0\) gives the Hitchin morphism (2.32),
where \(d = a - \delta \) and \(\delta = \frac{1}{2}(n^2 - n)\ell \).
Remark 4.2
Since the normal bundle L of a curve C in a del Pezzo surface has \(\deg (L) > \deg (K_C)\), we observe that the degeneration given in Theorem 4.1 connects the two moduli spaces considered in the work of Maulik and Shen [42], and for which cohomological \(\chi \)-independence is proven.
4.2 Deforming the symplectic structure
In this section we restrict ourselves to the case where S is a symplectic surface. In this context we study the symplectic structure on the moduli spaces involved in the construction obtained in Sect. 4.1, showing that it provides a deformation of symplectic varieties. Although the results in this section seem to be known to experts, we have not be able to find a good reference for them in the literatre.
Since the canonical bundle of S is trivial, one gets, by adjunction, that the normal bundle of \(C \subset S\) is its canonical bundle K. Recall then that the family of surfaces \(\overline{{\mathbf {S}}} \rightarrow {\mathbb {P}}^1\) has central fibre \(\overline{{\mathbf {S}}}_0 = {\mathbb {K}}\cup S\) meeting at the identification of \(\sigma _\infty \subset {\mathbb {K}}\) with \(C \subset S\). We start by studying \(K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}\) and \(K_{\overline{{\mathbf {S}}}_0}\).
Proposition 4.3
Let S be a symplectic surface and pick \(C \subset S\) smooth curve of genus \(g \ge 2\). One has the following
-
(1)
\(K_{\overline{{\mathbf {S}}}_0}\) is the line bundle given by \({\mathcal {O}}_{\mathbb {K}}(-\sigma _\infty )\) and \({\mathcal {O}}_S(C)\) identified along \(\sigma _\infty \cong C\) by a natural isomorphism between \(K \cong {\mathcal {O}}_{\mathbb {K}}(-\sigma _\infty )|_{\sigma _\infty }\) and \(K \cong {\mathcal {O}}_S(C)|_C\);
-
(2)
\(H^0(\overline{{\mathbf {S}}}_0, K_{\overline{{\mathbf {S}}}_0}) = {\mathbb {C}}\) and every non-trivial section of \(K_{\overline{{\mathbf {S}}}_0}\) vanishes on \({\mathbb {K}}\) and only on \({\mathbb {K}}\);
-
(3)
\(H^0(\overline{{\mathbf {S}}}_0, \omega ^{-1}_{\overline{{\mathbf {S}}}_0}) = {\mathbb {C}}\) and every non-trivial section of \(\omega ^{-1}_{\overline{{\mathbf {S}}}_0}\) vanishes on S and only on S;
-
(4)
\(\pi _*K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1} \cong {\mathcal {O}}_{{\mathbb {P}}^1}(1)\);
-
(5)
\(\pi _*\omega ^{-1}_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1} \cong {\mathcal {O}}_{{\mathbb {P}}^1}(-1)\).
Proof
We first observe that \(\overline{{\mathbf {S}}}\) is constructed by blowing-up a smooth subvariety of \(S \times {\mathbb {P}}^1\), which is also smooth. Hence \(\overline{{\mathbf {S}}}\) is smooth and has canonical bundle \(K_{\overline{{\mathbf {S}}}} = r^*K_{S \times {\mathbb {P}}^1}({\mathbb {K}})\), where we denote by r the structural morphism of the blow-up and we recall that \({\mathbb {K}}\) is the exceptional divisor. One can deduce that the singular central fibre \(\overline{{\mathbf {S}}}_0 = {\mathbb {K}}\cup S\), where both components meet transversally, is a complete intersection variety, so it is Gorenstein and its dualizing sheaf \(K_{\overline{{\mathbf {S}}}_0}\) is a line bundle. This implies that the relative dualizing sheaf \(K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}\) is a line bundle too, indeed it is \({\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {K}})\), as \(K_S\) is trivial.
Since \(\overline{{\mathbf {S}}}_0\) is a complete intersection divisor in \(\overline{{\mathbf {S}}}\), one has that \(K_{\overline{{\mathbf {S}}}_0} = K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}({\mathbb {K}}+ S) |_{\overline{{\mathbf {S}}}_0}\), where we abuse of notation by denoting the total transform of \(S \times \{ 0 \}\) simply by S. Then,
where we recall that \({\mathcal {O}}_{\overline{{\mathbf {S}}}}(-{\mathbb {K}})|_{\mathbb {K}}\) is the linearization \({\mathcal {O}}_{\mathbb {K}}(1) \cong {\mathcal {O}}_{\mathbb {K}}(\sigma _\infty )\). Similarly, the restriction of \(K_{\overline{{\mathbf {S}}}_0}\) to S gives
where we recall that \(K_S \cong {\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {K}}+ S)|_S \cong {\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {K}})|_S \otimes {\mathcal {O}}_{\overline{{\mathbf {S}}}}(S)|_S\) is trivial, hence \({\mathcal {O}}_{\overline{{\mathbf {S}}}}(S)|_S \cong ({\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {K}})|_S)^{-1}\). Recalling that \(K_{\mathbb {K}}= {\mathcal {O}}_{\mathbb {K}}(-2\sigma _\infty )\) and \(K_S \cong {\mathcal {O}}_S\), applying adjunction, one gets that \({\mathcal {O}}_{\mathbb {K}}(-\sigma _\infty )|_{\sigma _\infty }\) and \({\mathcal {O}}_S(C)|_C\) both being isomorphic to K, the canonical bundle of \(\sigma _\infty \cong C\), so the restriction of the line bundle \(K_{\overline{{\mathbf {S}}}_0}\) provides a natural identification. This finishes the proof of (1).
Since \({\mathcal {O}}_{\mathbb {K}}(-\sigma _\infty )\) has no non-zero sections, every non-zero section of \(K_{\overline{{\mathbf {S}}}_0}\) vanishes completely on \({\mathbb {K}}\), hence it is given by a section of \({\mathcal {O}}_S(C)\) vanishing identically at C. The set of those sections determines a 1-dimensional subspace of \(H^0(S,{\mathcal {O}}_S(C))\) and (2) follows.
We prove (3) by observing that \(h^0({\mathbb {K}}, {\mathcal {O}}_{\mathbb {K}}(\sigma _\infty )) = 1\) and that \(h^0(S, {\mathcal {O}}_S(-C)) = 0\). It follows that every non-zero section of \(K_{\overline{{\mathbf {S}}}_0}^{-1}\) vanishes on S, so it is given by a section of \({\mathcal {O}}_{\mathbb {K}}(\sigma _\infty )\) which vanishes identically at \(\sigma _\infty \).
Recall that \(\overline{{\mathbf {S}}}|_{{\mathbb {P}}^1 - \{ 0 \}}\) is the trivial fibration \(S \times ({\mathbb {P}}^1 - \{ 0 \})\), so the restriction there of \(K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}\) is the trivial line bundle. After (2) and (3) and upper semicontinuity of cohomology, one has that \(\pi _*K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}\) and \(\pi _*\omega ^{-1}_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}\) are line bundles over \({\mathbb {P}}^1\), inverse to each other. Note that \(K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1} \cong {\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {K}})\) comes naturally equipped with a section. Furthermore, \(h^0({\mathcal {O}}_{\overline{{\mathbf {S}}}}({\mathbb {K}})) = 1\) by the properties of blow-up. It follows that \(\pi _*K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}\) has a single non-zero section (up to scaling), so (4) follows.
Finally, (5) follows from (4). \(\square \)
In view of (2) and (3) of Proposition 4.3 we shall first construct a Poisson structure on \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a)\) and, then, derive the symplectic structure on \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\). Recall that \(\ell = 2g_C - 2\) in this case, so we fix \(d = a - (n^2-n)(g_C - 1)\).
Theorem 4.4
Consider a smooth symplectic surface S and a genus \(g \ge 2\) smooth projective curve C inside S. Then, there exists a relative Poisson structure \(\Theta \) on \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a) \rightarrow {\mathbb {A}}^1\) which defines a relative symplectic form \(\varOmega \) on \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}}) \rightarrow {\mathbb {A}}^1\) coinciding (up to scaling) with the Mukai form \(\Omega \) over the generic fibre \(\mathrm {M}_{S, \mathrm {H}}(\mathrm {v}_a)\) over \(t \ne 0\), and, on the central fibre \({\mathcal {M}}_C(n,d)\) at \(t = 0\), with \(\Omega _0\) obtained by extending the canonical symplectic form on the cotangent of the moduli space of stable vector bundles.
Proof
Starting from (5) of Proposition 4.3, we pick \({\mathbb {A}}^1 \subset {\mathbb {P}}^1\) a trivialization of \(\pi _*K_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}^{-1}\) containing \(0 \in {\mathbb {P}}^1\) and choose a section \(\vartheta \in H^0({\mathbb {A}}^1, \pi _*K_{\overline{{\mathbf {S}}}/{\mathbb {A}}^1}^{-1})\). Following Theorem 2.2 one can construct a relative Bottacin–Markman Poisson structure \(\Theta \) on \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a)\),
Thanks to (3) of Proposition 4.3, we can check that the restriction of \(\Theta \) to the subset \({\mathcal {M}}_C(n,d)\) of the central fibre is non-degenerate, hence defines a symplectic form there, as the tangent and the cotangent spaces to the moduli can be identified thanks to a trivialization of \(K_{{\mathbb {K}}_C} = {\mathcal {O}}_{{\mathbb {K}}_C}(-2\sigma _\infty )\) over \(\mathrm {Tot}(K_C) = {\mathbb {K}}_C - \{ \sigma _\infty \}\). Furthermore, the canonical bundle is trivial over the generic fibres, \(K_S \cong {\mathcal {O}}_S\), so in this case the cotangent space of \(\mathrm {M}_{\overline{{\mathbf {S}}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a)|_t = \mathrm {M}_{S}^{\mathrm {H}}(\mathrm {v}_a)\) is identified with the tangent space. Then, the Poisson form \(\Theta _t\) over \(t \ne 0\) is non-degenerate and defines a symplectic form which coincides with the Mukai form up to scaling. Hence, the restriction of \(\Theta \) to \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {A}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\) defines a relative symplectic form \(\varOmega \). \(\square \)
4.3 Degenerating the fixed locus of involutions
In this section, we study the behaviour under the Donagi–Ein–Lazarsfeld degeneration of the involutions considered in Sections 2.2 and 2.3. By doing so, we generalize to the case of an arbitrary smooth surface equipped with an involution, the construction of Sawon and Shen [50] who treated the case described in Sect. 4.4.4.
Consider a smooth surface S equipped with an arbitrary involution \(\zeta _S : S \rightarrow S\). Pick a smooth curve C preserved by \(\zeta _S\) and denote by \(\zeta _C: C \rightarrow C\) the restriction of the involution to it. As in (4.1), denote the normal bundle by \(L= {\mathcal {O}}_S(C)|_C\), and note that \(d \zeta _S : \zeta _S^* {\mathcal {T}}S {\mathop {\longrightarrow }\limits ^{\cong }} {\mathcal {T}}S\) sends \(\zeta _C^* {\mathcal {T}}C\) to \({\mathcal {T}}C\) inducing the isomorphism
Let us also recall from (3.3) the associated involution
One can provide an explicit description of \(\zeta _{f_0}\) in the particular case of symplectic surfaces.
Proposition 4.5
Suppose that S is a symplectic surface and \(\zeta _S^+\) a symplectic (resp. \(\zeta _S^-\) an antisymplectic) involution on it. Pick a smooth curve C preserved by \(\zeta _C\) and denote by L its normal bundle inside S. Then \(L = K_C\) and \(\zeta _{f_0} = \zeta _{\mathbb {K}}^+\), as defined in (3.5) (resp. \(\zeta _{f_0} = \zeta _{\mathbb {K}}^-\), as defined in (3.6)).
Proof
By adjunction formula and the triviality of the canonical bundle \(K_S \cong {\mathcal {O}}_S\) of a symplectic surface, one gets \(L = K_C\) in this case.
Note that the isomorphism \(\partial \zeta _C : \zeta _C^*K_C \rightarrow K_C\) that lifts the action of \(\zeta _C\) to the canonical bundle \(K_C\), is the inverse of \(d \zeta _C\).
Since \(\zeta _S^+\) is symplectic (resp. \(\zeta _S^-\) is antisymplectic), \(d \zeta ^+_S : \zeta _S^*{\mathcal {T}}S \rightarrow {\mathcal {T}}S\) has eigenvalues whose product is 1 (resp. \(-1\)). Then, over C, \(d \zeta _S\) has eigenvalues \(d \zeta _C\) and \(f_0 = \partial \zeta _C\) (resp. \(f_0 = -\partial \zeta _C\)). This completes the proof after recalling (3.5) and (3.6). \(\square \)
We defined \(\overline{{\mathbf {S}}}\) in (4.4) and \({\mathbf {S}}\) in (4.6), having generic fibres \(\overline{{\mathbf {S}}}_t = {\mathbf {S}}_t = S\) and central fibres \(\overline{{\mathbf {S}}}_0 = S \cup {\mathbb {L}}\) and \({\mathbf {S}}_0 = \mathrm {Tot}(L)\). As our involution \(\zeta _S\) preserves C, \(\zeta _S \times {\mathbf {1}}_{{\mathbb {P}}^1}\) lifts to the blow-up \(\overline{{\mathbf {S}}}\) giving another involution \(\zeta _{\overline{{\mathbf {S}}}} : \overline{{\mathbf {S}}} \rightarrow \overline{{\mathbf {S}}}\) which preserves \({\mathbf {C}}\) as defined in (4.8). Furthermore, at \(t = 0\), \(\zeta _{\overline{{\mathbf {S}}}}\) preserves the irreducible component \(S \subset \overline{{\mathbf {S}}}_0\) and we denote by \(\zeta _{\mathbf {S}}: {\mathbf {S}}\rightarrow {\mathbf {S}}\) the restriction of this involution to \({\mathbf {S}}\). One has commutativity with the structural morphisms,
giving \(\zeta _S\) on every generic fibre at \(t \ne 0\). We now examine the behaviour of \(\zeta _{\mathbf {S}}\) at the central fibre \({\mathbf {S}}_0\).
Lemma 4.6
With the notation as above, \(\zeta _{\mathbf {S}}\) coincides with \(\zeta _S\) on the generic fibres at \(t \ne 0\), and, over \(t = 0\), with the restriction of \(\zeta _{f_0}\) to \(\mathrm {Tot}(L)\).
Proof
As \(\overline{{\mathbf {S}}}\) is defined by blowing-up \(C \times \{ 0 \}\) inside \(S \times {\mathbb {P}}^1\), the involution given by \(\zeta _{\overline{{\mathbf {S}}}}\) is \(\zeta _S\) on a generic fibre, and on the exceptional divisor \({\mathbb {L}}\) is the involution induced by \(d\zeta _S\) on L, i.e. \(\zeta _{f_0}\). \(\square \)
Given \(C \subset S\), we construct \({\mathbf {C}}\) as in (4.8) and we define \(\{ n{\mathbf {C}}\}_{\mathbf {S}}\) as in (4.10). Consider a Mukai vector \({\mathbf {v}}_a\) as in (4.9). Pick a polarization \(\mathrm {H}\) on S, inducing \({\mathbf {H}}\) as described in (4.7), and consider \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\) as defined in (4.14), classifying \({\mathbf {H}}\)-semistable relative sheaves on \(\overline{{\mathbf {S}}}\) with Mukai vector \({\mathbf {v}}_a\), and whose support is parametrized by \(\{n{\mathbf {C}}\}_{\mathbf {S}}\). In particular, their support restricts to \({\mathbf {S}}\), so, taking pull-back under \(\zeta _{\mathbf {S}}\), one obtains a birational involution
compatible with the structural morphism to \({\mathbb {P}}^1\).
Consider an \(\zeta _S\)-invariant line bundle \(J \rightarrow S\) which satisfies condition (2.11) with respect to \(\mathrm {v}_a\). Starting from such J, we construct a line bundle on \({\mathbf {S}}\)
where \(p_S\) is obtained from inclusion \({\mathbf {S}}\hookrightarrow \overline{{\mathbf {S}}}\), the structural morphism of the blow-up \(\overline{{\mathbf {S}}} \rightarrow S \times {\mathbb {P}}^1\) and the projection of \(S \times {\mathbb {P}}^1\) onto S. Note as well that, over \(t \ne 0\), it restricts to the original line bundle,
while, at the central fibre \({\mathbf {S}}_0 = \mathrm {Tot}(L)\), one gets
where p is the structural projection \(\mathrm {Tot}(L) \rightarrow C\), and
We then observe that (2.11) holds fibrewise for \({\mathbf {J}}\) and \({\mathbf {v}}_a\).
The following is inspired by [50, Lemma 4].
Proposition-definition 4.7
There exists a biregular involution
given by sending a sheaf \({\mathcal {F}}\) supported on a curve inside \({\mathbf {S}}|_t\) to \({\mathcal {E}}\! xt^2_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1}({\mathcal {F}}, {\mathbf {J}}(-{\mathbb {L}}))|_{\mathbf {S}}\).
The restriction of \(\xi _{{\mathbf {J}},{\mathbf {S}}}\) to a generic fibre over \(t \ne 0\) is \(\xi _{J,S}\), while at the central fibre on \(t = 0\) it restricts to \({\check{\xi }}_{J_C,L}\).
Proof
Recall that \(\overline{{\mathbf {S}}}\) is smooth with canonical bundle \(K_{\overline{{\mathbf {S}}}} = r^*K_{S \times {\mathbb {P}}^1}({\mathbb {L}})\) Consider the embedding \(\jmath _t : \overline{{\mathbf {S}}}_t \hookrightarrow \overline{{\mathbf {S}}}\). The canonical bundle for this embedding is
being trivial over a generic fibre over \(t \ne 0\). Then, given a sheaf \({\mathcal {F}}\) supported on \({\mathbf {S}}_t\) with \(t \ne 0\), Grothendieck-Verdier duality implies that
so \(\xi _{{\mathbf {J}},{\mathbf {S}}}\) restricts to \(\xi _{J,S}\) there.
Also, observe that for a sheaf supported on the open subset of the central fibre \(\mathrm {Tot}(L) = {\mathbf {S}}_0 \subset \overline{{\mathbf {S}}}_0\), one also has that
so \(\xi _{{\mathbf {J}},{\mathbf {S}}}\) restricts to \(\xi _{J_0,{\mathbb {L}}}\) which further corresponds to \({\check{\xi }}_{J_C,L}\) after Lemma 3.5.
Finally, since \(\xi _{J,S}\) and \({\check{\xi }}_{J_C,L}\) preserves stability, so does \(\xi _{{\mathbf {J}},{\mathbf {S}}}\) fibrewise, defining a biregular involution of the moduli space. \(\square \)
Observe that \({\mathbf {J}}\) is \(\zeta _{\mathbf {S}}\)-invariant. Then, \({\widehat{\zeta }}_{\mathbf {S}}\) and \(\xi _{{\mathbf {J}},{\mathbf {S}}}\) commute and one can consider the composition
Lemma 4.8
With the notation as above, \(\uplambda _{{\mathbf {J}},{\mathbf {S}}}\) coincides with \({\check{\uplambda }}_{J_C,f_0}\) over \(t = 0\), and with \(\uplambda _{J,S}\) over a generic fibre at \(t \ne 0\).
Proof
The lemma is a consequence of Proposition–definition 4.7 and the fibrewise description of \({\mathbf {J}}\) and \({\mathbf {v}}_a\). \(\square \)
We now study the closure of their fixed points and their structure morphism.
Proposition 4.9
The restriction of the structural morphism to \({\mathbb {A}}^1\) of the closed subvarieties \(\overline{\mathrm {Fix}({\widehat{\zeta }}_{\mathbf {S}})}\), \(\mathrm {Fix}(\xi _{{\mathbf {J}},{\mathbf {S}}})\) and \(\overline{\mathrm {Fix}(\uplambda _{{\mathbf {J}},{\mathbf {S}}})}\) is flat.
Proof
Since \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\) is flat over \({\mathbb {P}}^1\), which is an irreducible and one dimensional base scheme, and the structural morphism trivializes (hence, it is flat) over \({\mathbb {P}}^1 -\{ 0 \}\), one only needs to check that the fibre at \(t = 0\) arises as the closure of \(\overline{\mathrm {Fix}({\widehat{\zeta }}_{\mathbf {S}})}|_{{\mathbb {P}}^1 -\{ 0 \}}\), \(\mathrm {Fix}(\xi _{{\mathbf {J}},{\mathbf {S}}})|_{{\mathbb {P}}^1 -\{ 0 \}}\) and \(\overline{\mathrm {Fix}(\uplambda _{{\mathbf {J}},{\mathbf {S}}})}|_{{\mathbb {P}}^1 -\{ 0 \}}\) inside \({\mathbf {M}}_{{\mathbf {S}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\). Note that this follows trivially for \(\xi _{{\mathbf {J}},{\mathbf {S}}}\) as it is biregular, but also for \({\widehat{\zeta }}_{\mathbf {S}}\) and \(\uplambda _{{\mathbf {J}},{\mathbf {S}}}\), which are biregular at the central fibre. \(\square \)
Suppose now that S denotes a smooth surface equipped with involution \(\zeta _S : S \rightarrow S\) whose quotient map \(\mathrm {q}_S : S \rightarrow T := S/\zeta _S\) gives a smooth surface. Pick a pair of smooth curves \(D \subset T\) and \(C \subset S\) such that C is as in (2.18). Denote by \(\zeta _C: C \rightarrow C\) the restriction of \(\zeta _S\) to C, and by \(\mathrm {q}_C : C \rightarrow D=C/\zeta _C\) the restriction there of \(\mathrm {q}_S\). As in (4.1), denote the associated normal bundles by \(L= {\mathcal {O}}_S(C)|_C\), as well as \(W={\mathcal {O}}_T(D)|_D\). Consider as well the ruled surfaces \({\mathbb {L}}\) and \({\mathbb {W}}\) associated to L and W, as defined in (2.22). Note that the composition with \(d \zeta _S : \zeta _S^* {\mathcal {T}}S \rightarrow {\mathcal {T}}S\) provides an isomorphism \(f_0 : \zeta _C^*L {\mathop {\rightarrow }\limits ^{\cong }} L\) and recall from (3.3) the associated involution \(\zeta _{f_0}\), as well as the projection \(\mathrm {q}_{f_0}\) from (3.4).
Lemma 4.10
Lift, via \(f_0\) as above, the action of \(\zeta _C\) to the normal line bundle L of C inside S. Then, L is the pull-back under \(q_C\) of the normal bundle W of D inside T,
Hence,
Proof
The first statement follows easily as the tangent bundles \({\mathcal {T}}\, T\) and \({\mathcal {T}}D\) coincide, respectively, with the \(\zeta _S\)-invariant and \(\zeta _C\)-invariant subbundles of \({\mathcal {T}}S\) and \({\mathcal {T}}C\). The second statement follows from the first and Remark 3.2. \(\square \)
Following (4.4) and (4.6), define \(\overline{{\mathbf {T}}}\) as well as \({\mathbf {T}}\), having generic fibres \(\overline{{\mathbf {T}}}_t = {\mathbf {T}}_t = T\), over \(t \ne 0\), and central fibre \(\overline{{\mathbf {T}}} = T \cup {\mathbb {W}}\) and \({\mathbf {T}}_0 = \mathrm {Tot}(W)\). Since the preimage of \(D \times \{ 0 \}\) under the composition of \(\overline{{\mathbf {S}}}\rightarrow S \times {\mathbb {P}}^1 \) and the projection \(\mathrm {q}_S \times {\mathbf {1}}_{{\mathbb {P}}^1} : S \times {\mathbb {P}}^1 \rightarrow T \times {\mathbb {P}}^1\) is the divisor \({\mathbb {L}}\subset \overline{{\mathbf {S}}}\), one obtains the projection \(\mathrm {q}_{\overline{{\mathbf {S}}}} : \overline{{\mathbf {S}}} \rightarrow \overline{{\mathbf {T}}}\) commuting with the structural morphisms to \({\mathbb {P}}^1\). At \(t = 0\), \(\mathrm {q}_{\overline{{\mathbf {S}}}}\) sends the irreducible component S of \(\overline{{\mathbf {S}}}_0\) to the irreducible component T of \(\overline{{\mathbf {T}}}_0\). Then, by restricting \(\mathrm {q}_{\overline{{\mathbf {S}}}}\) to \({\mathbf {S}}\), we can define \(q_{{\mathbf {S}}} : {\mathbf {S}}\rightarrow {\mathbf {T}}\) commuting with the structural morphisms,
Lemma 4.11
With the notation as above, \(\mathrm {q}_{\mathbf {S}}\) coincide with \(\mathrm {q}_S\) on the generic fibres at \(t \ne 0\), and, with the restriction of \(\mathrm {q}_{f_0}\) to \(\mathrm {Tot}(L)\) over \(t = 0\). Hence, one has that \(\mathrm {q}_{\mathbf {S}}\) is the quotient map of the action of \(\zeta _{\mathbf {S}}\).
Proof
After Lemmas 4.6 and 4.10, one has that the restriction of \(\mathrm {q}_{\overline{{\mathbf {S}}}}\) to \({\mathbb {L}}\) coincides with \(\mathrm {q}_{f_0}\), which is the quotient map associated to \(\zeta _{f_0}\). This concludes the proof, as we knew already that \(\mathrm {q}_{\mathbf {S}}\) is the quotient map of \(\zeta _{\mathbf {S}}\) over the complement of the central fibre. \(\square \)
We construct \({\mathbf {D}}\) as in (4.8) starting from the smooth curve \(D \subset T\). As defined in (4.10), denote the connected components of the corresponding relative Hilbert schemes containing the family of curves \(n{\mathbf {D}}\), once we restrict to \({\mathbf {T}}\). Denote also \(\mathrm {q}_{\mathbf {S}}^*\{ n{\mathbf {D}}\}_{\mathbf {T}}\subset \{ n{\mathbf {C}}\}_{\mathbf {S}}\) the locus of curves lifted from T to S.
Let us consider the restriction to \(\mathrm {q}_{\mathbf {S}}^{-1}\{ n{\mathbf {D}}\}_{\mathbf {T}}\) of the closure of the locus fixed by \({\widehat{\zeta }}_{\mathbf {S}}\),
This provides a degeneration of the fixed locus of a natural involution on the moduli space of sheaves over S.
Theorem 4.12
Let S be a smooth projective surface equipped with an involution \(\zeta _S\) whose quotient T is smooth, pick a smooth curve \(D \subset T\) and consider its lift C to S. Pick as well a positive integer n and, associated to n and C, a Mukai vector \(\mathrm {v}_a\) as in (2.1). Then, there exists a closed subvariety \({\mathbf {N}}_{{\mathbf {S}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {D}}) \subset {\mathbf {M}}_{\overline{{\mathbf {S}}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\), flat over \({\mathbb {A}}^1\), and equipped with a surjective support morphism
such that the structural morphism is trivial over \({\mathbb {P}}^1 - \{ 0 \}\), the generic fibre over \(t \ne 0\) is the closure of the fixed locus of the natural involution on \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) associated to \(\zeta _S\) equipped with the fibration (2.5),
while the central fibre at \(t = 0\) is the fixed locus of the involution \({\check{\zeta }}_{f_0}\) given in (3.8), being \(f_0\) induced from \(d\zeta _S\) as in (4.17), endowed with the Hitchin morphism (3.9),
where \(d = a - \delta \).
Proof
After Lemma 4.6, the fibrewise description is straight-forward since the restriction of \({\widehat{\zeta }}_{\mathbf {S}}\) to a generic fibre over \(t \ne 0\) coincides with \(\zeta _S\), while, at the central fibre over \(t = 0\), \({\widehat{\zeta }}_{\mathbf {S}}\) restricts to \({\check{\zeta }}_{f_0}\). Then, the description of the generic fibres is provided by the definitions (2.4) and (2.5), while the description of the central fibre follows from Lemmas 3.6 and 4.10. Finally, recall Proposition 4.9 for the proof of flatness. \(\square \)
For T, consider the moduli space \({\mathbf {M}}_{{\mathbf {T}}/{\mathbb {P}}^1, {\mathbf {I}}}({\mathbf {w}}_b, n{\mathbf {D}})\) of sheaves with a relative Mukai vector \({\mathbf {w}}_b\) and a polarization \(\mathrm {I}\) inducing \({\mathbf {I}}\) and \({\widetilde{{\mathbf {I}}}} = \mathrm {q}_{\mathbf {S}}^* {\mathbf {I}}\). Since \(\widetilde{{\mathbf {I}}}\) is \(\zeta _{\mathbf {S}}\)-invariant, \({\widehat{\zeta }}_{\mathbf {S}}\) is biregular and the image of the pull-back morphism under the corresponding quotient map lies in its fixed locus,
Note that over the generic fibre, \(t \ne 0\), (4.22) restricts to (2.6), while on the central fibre, \(t = 0\), the restriction of (4.22) is (3.10). Both cases are covered by Proposition 2.11 which provides the following corollary.
Corollary 4.13
Hence, (4.22) is generically 1:1 when the ramification locus of \(\mathrm {q}_S\) intersects C non-trivially, and, otherwise (in particular when \(\zeta _S\) is unramified), generically 2 : 1.
As we did in the case of a natural involution, consider
which gives a degeneration of the fixed locus of \(\uplambda _{J, S}\).
Theorem 4.14
Let S be a smooth projective surface equipped with an involution \(\zeta _S\) whose quotient T is smooth, pick a smooth curve \(D \subset T\) and consider its lift C to S. Choose a positive integer n and take a Mukai vector \(\mathrm {v}_a\) as in (2.1) and a line bundle J on S satisfying (2.11).
Then, there exists a closed subvariety \({\mathbf {P}}_{{\mathbf {S}}/{\mathbb {P}}^1, {\mathbf {H}}}({\mathbf {v}}_a, {\mathbf {J}}, n{\mathbf {D}}) \subset {\mathbf {M}}_{{\mathbf {S}}/{\mathbb {P}}^1,{\mathbf {H}}}({\mathbf {v}}_a,n{\mathbf {C}})\), flat over \({\mathbb {P}}^1\), and equipped with a surjective support morphism
such that the structural morphism is trivial over \({\mathbb {P}}^1 - \{ 0 \}\), the generic fibre over \(t \ne 0\) is the closure of the fixed point locus of the involution \(\uplambda _{J,S}\) on \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) equipped with the fibration (2.16),
while the central fibre at \(t = 0\) is the fixed locus of the involution \({\check{\uplambda }}_{J_C, f_0}\) given in (3.13), being \(f_0\) induced from \(d\zeta _S\) as in (4.17) and \(J_C\) the restriction of J to C, endowed with the Hitchin morphism (3.14),
where \(d = a - \delta \).
Proof
The proof is analogous to the proof of Theorem 4.12 and follows naturally from Lemmas 3.10, 4.6 and 4.10 and Propositions 4.7 and 4.9. \(\square \)
4.4 Degenerations of natural Lagrangians and Prymian integrable systems
Sawon conjectured in [49] that the Prymian integrable systems constructed by Markushevich–Tikhomirov [40], Arbarello–Saccà–Ferretti [1] and Matteini [41], degenerate into integrable systems related to the Hitchin system, leaving open the description of these conjectural systems. It is well known that every Lagrangian fibration degenerates naturally into the cotangent bundle of one of its smooth fibres, and the later can easily be identified with a rank 1 Hitchin system. Hence, one can always find examples of such degenerations, although, perhaps not very interesting ones.
When the curves appearing in the integrable system are not primitive, it is possible to provide a degeneration of the corresponding Prymian integrable systems into Hitchin systems of higher rank. We provide in this section such a construction, providing an answer of Sawon’s question in the case of sheaves supported on non-primitive curves. The question of whether or not these Prymian integrable systems for primitive curves degenerate into more interesting Hitchin systems remains open.
In this section we provide an explicit description of the degenerations of Sect. 4.3 in the cases studied in Sect. 2.4, focusing on the cases specified in [1, 40, 41, 50, 52], which are constructed using primitive first Chern classes. We also review the degeneration given by Sawon–Shen [50] into the \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Higgs moduli space, studying as well the degeneration of the associated natural Lagrangian. It is worth noticing that we find that this natural Lagrangian degenerates into the fixed locus of an involution associated to \(\mathrm {U}(m,m)\)-Higgs bundles, whose Nadler–Langlands group is \(\mathrm {Sp}(2m,{\mathbb {C}})\).
Our work provides an answer to the question posed by Sawon in [49] (which, strictly speaking, refers only to the case of primitive first Chern classes) and to the reformulation of Sawon’s question on the non-primitive case.
4.4.1 The general case
Consider a K3 surface X equipped with an antisymplectic involution \(\zeta _X^- : X \rightarrow X\), whose quotient \(\mathrm {q}_X : X \rightarrow Y := X /\zeta _X^-\) has ramification divisor \(\Delta _X\). Let \(D \subset Y\) be a smooth curve and C as in (2.18), again smooth. We denoted by \(\zeta _C: C \rightarrow C\) and \(\mathrm {q}_C : C \rightarrow D=C/\zeta _C\) the restriction of \(\zeta _X^-\) and \(\mathrm {q}_X\) to C. Note that the ramification divisor of \(\zeta _C\) is \(\Delta _C = C \cap \Delta _X\). Since \(K_X\) is trivial, the normal bundle of C is
giving the ruled surface \({\mathbb {K}}_C\), and the normal bundle of D is
associated to the ruled surface \({\mathbb {W}}\). Observe that \(K_Y^{-1}|_D\) is a square-root of the branching divisor of \(\mathrm {q}_C\), and its pull-back is the ramification divisor.
Recall from Proposition 4.5, that \(d \zeta _X^- : (\zeta _X^-)^* {\mathcal {T}}X \rightarrow {\mathcal {T}}X\) gives the isomorphism \(-\partial _C: \zeta _C^*K_C {\mathop {\rightarrow }\limits ^{\cong }} K_C\) giving the associated involution \(\zeta _{\mathbb {K}}\) described in (3.6). The associated natural antisymplectic involution \({\widehat{\zeta }}_{\mathbb {K}}^-\) is sent to \({\check{\zeta }}_K^-\) under the spectral correspondence as described in (3.15). As a consequence of Proposition 4.5, Theorems 4.4 and 4.12 and Corollary 4.13 one has the following.
Corollary 4.15
There exists a non-linear degeneration \({\mathbf {N}}_{{\mathbf {X}},{\mathbf {H}}}({\mathbf {v}}_a, n{\mathbf {D}})\) of the natural Lagrangian subvariety \(\mathrm {N}_{X,\mathrm {H}}(\mathrm {v}_a, nD) \subset \mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a, nC)\) obtained from the fixed locus of \(\zeta _X^-\), into the Lagrangian subvariety of the Higgs moduli space \({\mathcal {N}}_C(n, d) \subset {\mathcal {M}}_C(n, d)\), with \(d = a - n(n-1)(g_C - 1)\), given by the fixed locus of the involution
For \(a = 2b\) and a \(\zeta _X\)-invariant polarization \({\widetilde{\mathrm {I}}} = \mathrm {q}_X^*\mathrm {I}\), the morphism
given in (4.22), is generically 1:1 when the ramification locus of \(\mathrm {q}_X\) intersects C non-trivially, and generically 2 : 1 otherwise, for instance whenever \(\zeta _X^-\) is unramified. The generic fibres of \({\mathbf {M}}_{{\mathbf {Y}}/{\mathbb {A}}^1,{\mathbf {I}}}({\mathbf {w}}_b, n{\mathbf {D}})\) are the moduli spaces \(\mathrm {M}_{Y,\mathrm {I}}(\mathrm {w}_b, nD)\) and the central fibre is \({\mathcal {M}}_D^{W}(n,d')\), for W described in (4.24) and \(d' = b - n(n-1)(g_D - 1)\).
As a consequence of Proposition 4.5 and Theorems 4.4 and 4.14 one has the following.
Corollary 4.16
Suppose that (2.11) holds for the choice of a line bundle J and a Mukai vector \(\mathrm {v}_a\) as in (2.1) for some integer a and let d be \(a - \delta = a - n(n-1)(g_C - 1)\).
There exists a non-linear degeneration \({\mathbf {P}}_{{\mathbf {X}},{\mathbf {H}}}({\mathbf {v}}_a, {\mathbf {J}}, n{\mathbf {D}})\) of the Prymian integrable system \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, nD) \subset \mathrm {M}_{X,\mathrm {H}}(\mathrm {v}_a, nC)\) constructed out of \(\zeta _X^-\), into the Prymian integrable system \({\mathcal {P}}_C(n, d, J_C) \subset {\mathcal {M}}_C(n, d)\) given by the fixed locus of the involution
Remark 4.17
For \(n=1\), \({\mathcal {P}}_C(n, d, J_C)\) is the cotangent bundle of a Prym abelian variety associated to \(\zeta _C\) (when \(\zeta _X^-\) is ramified) or a disjoint union of them (when \(\zeta _X^-\) unramified).
4.4.2 The Arbarello–Saccà–Ferretti systems and Saccà Calabi-Yau’s
In this subsection, we consider a K3 surface endowed with an antisymplectic involution \(\zeta _X^-\) with empty fixed locus and whose quotient \(Y = X/\zeta _X\) is a smooth Enriques surface. The canonical divisor of an Enriques surface Y is not trivial but satisfies \(2 K_Y \sim 0\). Any smooth curve \(D \subset Y\) with positive self-intersection is big and nef, and so is \(D + K_Y\). Then, by Kodaira vanishing theorem,
hence
where \(g_D\) is the genus of D. The curve \(C := D \times _Y X\) is equipped with an unramified 2 : 1 cover \(\mathrm {q}_C \rightarrow D\) and an involution \(\zeta _C : C \rightarrow C\) without fixed points.
Observe that condition (2.21) is satisfied for every smooth curve \(D \subset Y\). Then, whenever (2.11) holds for a pair \(\mathrm {J}\) and \(\mathrm {v}_a\), \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, \mathrm {J}, nD)\) has 2 connected components each of them an integrable system of dimension \(2n^2(g_D - 1)\) known as the Arbarello–Saccà–Ferretti integrable system, as described by these authors in [1] when \(n=1\). The Lagrangian subvariety \(\mathrm {N}_{X,\mathrm {H}}(\mathrm {v}_a, nD)\) has dimension \(n^2(g_C -1) +1\). Furthermore, Saccà [48] studied the moduli space of pure dimension 1 sheaves over an Enriques surface \(\mathrm {M}_{Y,I}(\mathrm {w}_b, nD)\) showing that it is Calabi–Yau. By Proposition 2.11, \(\mathrm {M}_{Y,I}(\mathrm {w}_b, nD)/{\mathbb {Z}}_2\) is birational to \(\mathrm {N}_{X,{\widetilde{I}}}(\mathrm {v}_{2b}, nD)\).
Observe that \(L \cong K_C\) in this case, and \(W \cong K_D L_\gamma \), where \(L_\gamma \in H^1(D, {\mathbb {Z}}_2)\) is the 2-torsion line bundle obtained by restricting \(K_Y\) to D. After Corollary 4.15\(\mathrm {M}_{Y,I}(\mathrm {w}_b, nD)\) degenerates into \({\mathcal {M}}^{L_\gamma K_D}_D(n,b-\delta /2)\) and \(\mathrm {N}_{X,\mathrm {H}}(\mathrm {v}_a, nD)\) into \({\mathcal {N}}_C(n,a - \delta )\), which is the fixed point locus of the involution (4.25) constructed out of \(\zeta _C\) unramified. It follows from Corollary 4.16 that \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, nD)\) degenerates into \({\mathcal {P}}_C(n,a - \delta , J_C)\), which is the fixed point locus of the involution (4.26), constructed out of \(\zeta _C\) unramified.
4.4.3 Markushevich–Tikhomirov, Matteini and Sawon–Shen systems
Consider now an anti-symplectic involution \(\zeta _X^-\) on the K3 surface X whose quotient \(Y = X/\zeta _X^-\) is a del Pezzo surface of degree d. In that case, by their defining property, the anti-canonical bundle \(-K_Y\) is ample. In particular, \(-(m+1)K_Y\) is big and nef for \(m \ge 0\) and by Kodaira vanishing theorem,
so
Note that the genus of a smooth curve in the linear system \(-mK_Y\) is \(1 + (m^2 - m)d/2\).
We recall that \(\Delta \), the branching locus of \(\mathrm {q}_X\), lies in \(|-2K_Y|\). Picking a smooth curve D in the linear system \(|-K_Y|\), hence \(g_D = 1\), pulling-back to another smooth curve C on X, we obtain a 2 : 1 cover, \(\mathrm {q}_C : C \rightarrow D\), ramified at the intersection with the ramification locus of \(\mathrm {q}_X\), that we denote by R. Observe that the length of R coincides with the intersection of D and \(\Delta \), so \((-K_Y) \cdot (-2K_Y) = 2d\), from which one can derive that \(g_C = 1 + d\).
Markushevich and Tikhomirov [40] gave the first construction of Prymian integrable system \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, nC)\) starting from a del Pezzo of degree \(d=2\) and for \(n=1\). In this case we have that \(\zeta _C\) sends a genus 3 curve C, to D, an elliptic curve and \(\zeta _C\) is ramified at 4 points and \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, C)\) is a 4-dimensional symplectic V-manifold.
Matteini extended the construction of [40], focusing on the cases of del Pezzo surfaces of degree \(d=1\) and \(d = 3\). In the case of \(d=1\), \(\zeta _C : C \rightarrow D\) is the cover of a genus 2 curve over an elliptic curve, and Matteini found that \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, C)\) is an elliptic K3 surface. In the \(d=3\) case, \(\zeta _C : C \rightarrow D\) is a 2 : 1 cover of a genus 4 curve onto an elliptic curve and \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, C)\) is a 6-dimensional symplectic V-manifold. Even in both cases only \(n=1\) was considered by the authors, their construction works for general n as indicated by Matteini in [41, Section 3.6].
Another 6-dimensional Prym integrable system is considered by Sawon and Shen in [50, 52]. In this case Y is a del Pezzo surface of degree \(d=1\) and \(D \subset Y\) lies in \(|-2K_Y|\), so \(g_D = 2\). The ramification divisor R of the 2 : 1 cover \(\mathrm {q}_C : C \rightarrow D\) has length \((-2K_Y) \cdot (-K_Y) = 2\), hence \(g_C = 5\).
After Corollary 4.16 that \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_a, J, nD)\) degenerates into \({\mathcal {P}}_C(n,a - \delta , J_C)\), which is the fixed point locus of the involution (4.26), constructed out of \(\zeta _C\) ramified at \(\Delta _C = \Delta _X \cap X\).
One can also consider the natural Lagrangian subvarieties \(\mathrm {N}_{X,\mathrm {H}}(\mathrm {v}_a, nD)\) associated to the mentioned Prymian integrable systems. Note that in this case \(\mathrm {N}_{X,{\widetilde{\mathrm {I}}}}(\mathrm {v}_{2b}, nD)\) are birational to \(\mathrm {M}_{Y,\mathrm {I}}(\mathrm {v}_{b}, nD)\). From Corollary 4.15 we know that \(\mathrm {N}_{X,\mathrm {H}}(\mathrm {v}_a, nD)\) degenerates into \({\mathcal {N}}_C(n,a - \delta )\), which is the fixed point locus of the involution (4.25), and \(\mathrm {M}_{Y,\mathrm {I}}(\mathrm {v}_{b}, nD)\) degenerates into \({\mathcal {M}}_D^{K_DQ}(n,b-(n^2-n)(g_D - 1))\), where Q is a line bundle on D whose square is \({\mathcal {O}}_D(\Delta _C)\).
4.4.4 Sawon–Shen degeneration into the \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Hitchin system
When \(\zeta _X^-\) is an anti-symplectic involution on the K3 surface X giving a del Pezzo surface \(Y = X / \zeta _X^-\) of degree d, the ramification divisor of \(\mathrm {q}_X : X \rightarrow Y\) is a connected smooth projective curve \(\Delta \) of genus \(g_\Delta = d + 1\). Trivially, the restriction of \(\zeta _X\) to \(\Delta \) is the identity \(\zeta _\Delta = {\mathbf {1}}_\Delta \). Setting \(\mathrm {v}_0 = (0, 2m\Delta ,- 4m^2(g_\Delta - 1))\), \(\mathrm {H}= {\mathcal {O}}_X(\ell \Delta )\) and \(J = {\mathcal {O}}_X(-2m\Delta )\), Sawon and Shen [50] have constructed a degeneration of the Prymian integrable system \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_0, J, 2m\Delta )\) into a subvariety of the Higgs moduli space \({\mathcal {M}}_\Delta (n,-\delta )\) isomorphic to the moduli space of \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Higgs bundles, where \(\delta = 2m(2m-1)(g_\Delta - 1)\). Shifting in degree, taking \(\mathrm {v}_\delta = (0, 2m\Delta ,- 2m(g_\Delta - 1))\), Sawon and Shen’s construction becomes a degeneration of the Prymian integrable system \(\mathrm {P}_{X,\mathrm {H}}(\mathrm {v}_\delta , J, 2m\Delta )\) into a compactification of the \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Hitchin system over \(\Delta \), given by the fixed locus of the involution
Note that Corollary 4.15 shows that the natural Lagrangian \(\mathrm {N}_{X,\mathrm {H}}(\mathrm {v}_\delta )\) degenerates into the fixed locus of the involution
The morphism obtained by extension of structure group sends the moduli space of \(\mathrm {U}(m,m)\)-Higgs bundles maps to the fixed locus of \({\check{\zeta }}_{K}^-\). It is worth noticing that \(\mathrm {U}(m,m)\) and \(\mathrm {Sp}(2m,{\mathbb {C}})\) are related under the Nadler-Langlands correspondence [45]. In [31] Hitchin proposed that a \(\mathrm {(BBB)}\)-brane constructed over the moduli space of \(\mathrm {Sp}(2m,{\mathbb {C}})\)-Higgs bundles is Mirror dual to a \(\mathrm {(BAA)}\)-brane supported on the moduli space of \(\mathrm {U}(m,m)\)-Higgs bundles (see Section 5). Evidence for this duality was given by Hitchin himself in [31] and by Hausel, Mellit and Pei in [27].
5 Branes and duality
Non-abelian Hogde theory [14, 19, 30, 53, 54] implies that \({\mathcal {M}}_C(n,d)\) is equipped with a hyperkähler structure \((g, \Gamma _1, \Gamma _2, \Gamma _3)\), where \(\Gamma _1\) the complex structure coming from the moduli space of Higgs bundles and \(\Gamma _2\) from the moduli of flat connections. The associated Kahler forms \(\omega _i(\, \cdot \, , \, \cdot \, ) := g(\, \cdot \, , \Gamma _i (\, \cdot \,) )\) combine into a \(\Gamma _1\)-holomorphic symplectic form \(\omega _2 + \mathrm {i} \omega _3\), which coincides with \(\Omega _0\) up to scaling. Similar constructions can be given for \(i=2\) and \(i=3\).
Following [33], a \(\mathrm {(BAA)}\)-brane in \({\mathcal {M}}_C(n,d)\) is a pair \((N, {\mathscr {G}}, \nabla _{\mathscr {G}})\) where N is a subvariety which is complex Lagrangian with respect to \(\Omega _0\), and \(({\mathscr {G}}, \nabla _{\mathscr {G}})\) is a flat bundle over N. A \(\mathrm {(BBB)}\)-brane in \({\mathcal {M}}_C(n,d)\) is given by a pair \((P, {\mathscr {F}}, \nabla _{\mathscr {F}})\), where P is a hyperholomorphic subvariety(i.e. holomorphic with respect to \(\Gamma _1\), \(\Gamma _2\) and \(\Gamma _3\)) and \(({\mathscr {F}}, \nabla _{\mathscr {F}})\) a hyperholomorphic sheaf on P. This means that the connection \(\nabla _{{\mathscr {F}}}\) on the sheaf \({\mathscr {F}}\) is of type (1, 1) with respect to all three Kähler structures \(\Gamma _1\), \(\Gamma _2\), \(\Gamma _3\). It is conjectured in [33] that mirror symmetry interchanges \(\mathrm {(BBB)}\)-branes with \(\mathrm {(BAA)}\)-branes within the Higgs moduli space and in a certain limit, mirror symmetry is enhanced via a Fourier-Mukai transform relative to the Hitchin fibration.
Remark 5.1
When \(({\mathscr {G}}, \nabla _{\mathscr {G}})\) appearing in the definition of a \(\mathrm {(BAA)}\)-brane (resp. \(({\mathscr {F}}, \nabla _{\mathscr {F}})\) in the definition of a \(\mathrm {(BBB)}\)-brane) is the trivial bundle, we simply denote the brane indicating the Lagrangian subvariety N (resp. the hyperholomorphic subvariety P). Equivalently, when no bundle is specified in the description of a brane, we are implicitly assuming that it is the trivial one.
Note that the definition of \(\mathrm {(BAA)}\) and \(\mathrm {(BBB)}\)-branes extends naturally to any other hyperkähler varieties. The case of the moduli space \(\mathrm {M}_{S,\mathrm {H}}(\mathrm {v}_a)\) of sheaves over a smooth projective symplectic surface S was considered in [20] by the author, Jardim and Menet. It is also described in [20] the construction of the \(\mathrm {(BBB)}\)-brane \(\mathrm {N}_{S,\mathrm {H}}^+(\mathrm {v}_a)\) arising from the fixed locus of a natural involution associated to a sympletic involution on the surface. Also, the authors studied the behaviour of these natural branes under some correspondences. As a consequence of Propositions 4.5 and 4.9, we can describe these degenerations in the context of branes, extending the description to \(\mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_a, J)\).
Corollary 5.2
The \(\mathrm {(BBB)}\)-brane \(\mathrm {N}_{S,\mathrm {H}}^+(\mathrm {v}_a)\) (resp. the \(\mathrm {(BAA)}\)-brane \(\mathrm {N}_{S,\mathrm {H}}^-(\mathrm {v}_a)\)) obtained by the fixed locus of a natural involution associated to a sympletic (resp. anti-symplectic) involution on a symplectic surface degenerate into the \(\mathrm {(BBB)}\)-brane (resp. \(\mathrm {(BAA)}\)-brane) inside the Hitchin system \({\mathcal {N}}_C^+(n,d)\) (resp. \({\mathcal {N}}_C^-(n,d)\)), obtained as the respective fixed locus of the involution \({\check{\zeta }}_K^+\) (resp. \({\check{\zeta }}_K^-\)), described in (3.15).
Similarly, the \(\mathrm {(BBB)}\)-brane obtained from \(\mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_a, J)\) degenerates into the \(\mathrm {(BBB)}\)-brane given by \({\mathcal {P}}_C(n,d, J_C)\).
We now discuss the duality under Mirror symmetry of the branes associated to a Prymian integrable system and the natural Lagrangian.
Consider the Mukai vector \(\mathrm {v}_0\) on S, where we assume \(a = 0\), and recall from Remark 2.9 that it is the pull-back of the Mukai vector \(\mathrm {w}_0\) on T. Following Remark 2.12, choose \(J_0 = {\mathcal {O}}_S(-nC)\). In this section we show that certain open subsets of the subvarieties \(\mathrm {N}_{S,\mathrm {H}}(\mathrm {v}_0,B)\) and \(\mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_0,J_0,B)\) are dual under Fourier–Mukai transform.
Observe that the diagram,
restricts, after Proposition 2.11 and Remark 2.16, to
where we have dropped the systematic reference to degree 0 in our notation for Jacobians.
Let us recall that one can equip \({{\,\mathrm{Jac}\,}}_{{\mathcal {A}}^\mathrm {sm}/\{A \}^\mathrm {sm}} \times _{\{ A \}^\mathrm {sm}} {{\,\mathrm{Jac}\,}}_{{\mathcal {A}}^\mathrm {sm}/\{A \}^\mathrm {sm}}\) with a relative Poincaré bundle \({\mathcal {U}}_{{\mathcal {A}}^\mathrm {sm}}\). Denoting by \(\pi _i\) the projection from \({{\,\mathrm{Jac}\,}}_{{\mathcal {A}}^\mathrm {sm}/\{A \}^\mathrm {sm}} \times _{\{ A \}^\mathrm {sm}} {{\,\mathrm{Jac}\,}}_{{\mathcal {A}}^\mathrm {sm}/\{A \}^\mathrm {sm}}\) to the i-th factor, one can consider the relative Fourier–Mukai transform [43]
and, after the choice of \(J_0\), its inverse
One can provide a similar construction for \({\mathcal {B}}^\mathrm {ssl}\) in T, giving rise to the Fourier–Mukai transform \(\Psi _{{\mathcal {B}}^\mathrm {ssl}}\). Recalling the morphism \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}}\) from (2.8), both transforms satisfy (see [47, (11.3.3)] for instance),
where the norm map \(\mathrm {Nm}(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}})\) is the dual of \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}}\).
Proposition 5.3
Whenever \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) has ramification, the structural sheaves of the subvarieties \(\left. \mathrm {N}_{S,\mathrm {H}}(\mathrm {v}_0, B) \right| _{\{ B \}^\mathrm {ssl}}\) and \(\left. \mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_0, J_0, B) \right| _{\{ B \}^\mathrm {ssl}}\) are dual under a relative Fourier–Mukai transform.
If \(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}}\) is unramified, the structural sheaf on a connected component of \(\left. \mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_0, J_0, B) \right| _{\{ B \}^\mathrm {ssl}}\) is Fourier–Mukai dual to \(\left. \mathrm {N}_{S,\mathrm {H}}(\mathrm {v}_0, B) \right| _{\{ B \}^\mathrm {ssl}}\) equipped with the pair of line bundles obtained from pushing forward the structural sheaf under \({\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}.}\) direct sum of the trivial sheaf and the pull-back of a torsion 2 line bundle \({\mathcal {L}}\rightarrow {\mathcal {B}}^\mathrm {ssl}\).
Proof
We start by the ramified case. Observe that \(R {\widehat{\mathrm {q}}}_{{\mathcal {A}}^\mathrm {ssl}, *} {\mathcal {O}}_{{{\,\mathrm{Jac}\,}}_{{\mathcal {B}}}}\) coincides with the trivial sheaf over \(\mathrm {N}_{S,\mathrm {H}}(\mathrm {v}_0, B)|_{\{ B \}^\mathrm {ssl}}\) after Proposition 2.11. Also, \(\Psi _{{\mathcal {B}}^\mathrm {ssl}} ({\mathcal {O}}_{{{\,\mathrm{Jac}\,}}_{{\mathcal {B}}}})\) is the relative sky-scraper sheaf at the identity, so its pull-back under \(\mathrm {Nm}(\mathrm {q}_{{\mathcal {A}}^\mathrm {ssl}})\) is precisely the trivial sheaf over \(\mathrm {P}_{S,\mathrm {H}}(\mathrm {v}_0, J_0, B)|_{\{ B \}^\mathrm {ssl}}\). Then, the first statement follows easily from (5.1).
The second statement follows from the diagram (2.10) and the fact that the preimage of the Norm map has two connected components. \(\square \)
Recalling Proposition 5.3, one immediately deduces the following.
Corollary 5.4
Whenever \(\mathrm {q}_C\) has ramification, the structural sheaves of \({\mathcal {N}}_C(n,d)|_{\{ B \}^\mathrm {ssl}}\) and \({\mathcal {P}}_C(n,d,J)|_{\{ B \}^\mathrm {ssl}}\) are dual under a relative Fourier–Mukai transform.
If \(\mathrm {q}_C\) is unramified, the latter is Fourier–Mukai dual to the sheaf supported on \({\mathcal {N}}_C(n,d)|_{\{ B \}^\mathrm {ssl}}\) given by the direct sum of the trivial sheaf and the line bundle associated to the unramified 2 : 1 cover of the family of spectral curves.
In this context, Corollaries 4.15 and 4.16 can be seen, respectively, as a degeneration of \(\mathrm {(BAA)}\) and \(\mathrm {(BBB)}\)-branes, from the moduli space of sheaves on symplectic surfaces into the Hitchin system.
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Acknowledgements
The author is thankful to Marcos Jardim for his encouragement to address this project, and specially to Gregoire Menet and Justin Sawon, for their comments, questions and suggestions. The author wants to thank Justin Sawon also for having shared a draft of the article [50], which was useful for detecting some previous errors.
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Franco, E. Degeneration of natural Lagrangians and Prymian integrable systems. Math. Z. 302, 2469–2511 (2022). https://doi.org/10.1007/s00209-022-03135-z
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DOI: https://doi.org/10.1007/s00209-022-03135-z