1 Introduction
1.1 Strominger–Yau–Zaslow
Mirror symmetry stemmed from the study of superstring compactification in the late 1980s. Its first precise formulation was given by Candelas, dela Ossa, Green, and Parkes. They conjectured a formula for the number of rational curves of given degree on a quintic Calabi–Yau in terms of the periods of the holomorphic three form on another “mirror” Calabi–Yau manifold, which is related to the theory of closed strings in physics (see [Reference Candelas, Ossa, Green and ParkesCOPL91]). In the mid-1990s, two developments emerged, inspired by the open string theory: Kontsevich’s proposal of homological mirror symmetry (see [Reference KontsevichKon95]) and the proposal of Strominger–Yau–Zaslow (SYZ; see [Reference Strominger, Yau and ZaslowSYZ96]).
Let us focus on the SYZ’s proposal. Consider a pair of compact Calabi–Yau threefolds M and $\check M$ related by mirror symmetry in the sense: the set of BPS A-branes on M is isomorphic to the set of BPS B-branes on $\check M$ , while the set of BPS B-branes on M is isomorphic to the set of BPS A-branes on $\check M$ . The simplest BPS B-branes on M are points, and their moduli space is M itself. For every point in M, the corresponding BPS A-brane on $\check M$ is a pair $(T,L)$ , where T is a special Lagrangian submanifold of $\check M$ and L is a flat $U(1)$ -bundle on T. Then, there is a family of special Lagrangian submanifolds on $\check M$ parametrized by points of M. According to McLean’s theorem [Reference McLeanMcl98], the deformation space of a special Lagrangian submanifold T is unobstructed and has real dimension $b_1(T)$ (the first Betti number of T). On the other hand, the moduli space of flat $U(1)$ -bundles on T is $H^1(T,{\mathbb R}/{\mathbb Z})$ (a torus of real dimension $b_1(T)$ ). Thus, the total dimension of the moduli space of $(T,L)$ is $2b_1(T)$ . Since the moduli space is M, we must have $b_1(T)=3$ . Then, M is fibered by tori of dim $3$ . Exchanging the roles of $\check M$ and M, we conclude that $\check M$ is also fibered by three-dimensional tori. Motivated by those, SYZ made a conjecture called SYZ conjecture: every n-dimensional Calabi–Yau manifold M admits a mirror $\check M$ (which is also a Calabi–Yau manifold of dim n). And, there exists a real manifold N of dimension n together with two smooth fibrations h, $\check h$
where the generic fiber is a special Lagrangian n-torus. Moreover, h and $\check h$ are dual in the sense that for a common regular point $b\in N$ of h and $\check h$ , we have
Hitchin [Reference HitchinHit01] extended the formulation of SYZ conjecture to Calabi–Yau manifolds with B-fields, where B-fields are flat unitary gerbes in mathematics. Suppose that $\boldsymbol {B}$ is a flat unitary gerbe on a Calabi–Yau X such that the restriction of $\boldsymbol {B}$ to every special Lagrangian torus fiber T is trivial. Since the set of isomorphism classes of flat unitary gerbes on T is $H^2(T,{\mathbb R}/{\mathbb Z})$ , a trivialization of $\boldsymbol {B}$ on T is a $1$ -cochain whose coboundary is $\boldsymbol {B}$ and two trivializations are equivalent if they differ by an exact cocycle. Then, the set $\mathrm {Triv}^{U(1)}(T,\boldsymbol {B})$ of equivalence classes of trivializations of $\boldsymbol {B}$ on T is an $H^1(T,{\mathbb R}/{\mathbb Z})$ -torsor. The SYZ mirror of Calabi–Yau X with a B-field $\boldsymbol {B}$ is defined to be the moduli space of pairs $(T,t)$ where T is a special Lagrangian torus and t is a flat trivialization of $\boldsymbol {B}$ on T. Note that if $\boldsymbol {B}$ is a trivial flat unitary gerbe, we obtain the original SYZ mirror. More precisely, two n-dimensional Calabi–Yau orbifolds X and $\check X$ , equipped with B-fields $\boldsymbol {B}$ and $\check {\boldsymbol {B}}$ , respectively, are said to be mirror partners, if there is an n-dimensional real orbifold Y and two smooth surjections $\mu $ , $\check \mu $
such that for every regular value $x\in Y$ of $\mu $ and $\check \mu $ , the fibers $\mu ^{-1}(x)$ and $\check \mu ^{-1}(x)$ are special Lagrangian tori and dual to each other in the sense that there are smooth identifications
In [Reference Hausel and ThaddeusHT03], Hausel and Thaddeus showed that the moduli spaces of flat connections on a curve with structure groups $\mathop {\mathbf {SL}}\nolimits _r$ and $\mathop {\mathbf {PGL}}\nolimits _r$ are mirror partners in the above sense. Their work has been extended to the $G_2$ case by Hitchin [Reference HitchinHit07] and to all semisimple algebraic groups by Donagi and Pantev [Reference Donagi and PantevDP12]. For the case of parabolic Higgs bundles, Biswas and Dey [Reference Biswas and DeyBD12] proved the SYZ conjecture for full flags parabolic Higgs bundles with structure groups $\mathop {\mathbf {SL}}\nolimits _r$ and $\mathop {\mathbf {PGL}}\nolimits _r$ .
1.2 Moduli spaces of Higgs bundles, Hitchin morphisms, and Norm maps
In [Reference NironiNir08], Nironi constructed the moduli stacks (spaces) of coherent sheaves on projective Deligne–Mumford stacks. We use his construction to study the moduli stacks (spaces) of Higgs bundles on Deligne–Mumford curves. In fact, Simpson used coverings by smooth projective varieties to give description of the moduli stacks of Higgs bundles with vanishing Chern classes on Deligne–Mumford curves (see [Reference SimpsonSim11]). For the stacky curves (or orbifold curves), Biswas–Majumder–Wong [Reference Biswas, Majumder and WongBMW13], Borne [Reference BorneBor07], Nasatyr–Steer [Reference Nasatyr and SteerNS95], and others had considered the problem.
Let ${\mathcal {X}}$ be a complex hyperbolic Deligne–Mumford curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . We show that the moduli stack ${\mathcal M}_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {GL}}\nolimits _r)$ of rank r Higgs bundles on ${\mathcal {X}}$ is locally of finite type over ${\mathbb C}$ . Fix a polarization $({\mathcal E},{\mathcal {O}}_X(1))$ on ${\mathcal {X}}$ , where ${\mathcal E}$ is a generating sheaf (see Section 2.2) and ${\mathcal {O}}_X(1)$ is an ample line bundle on X. We introduce the notion of modified slope for Higgs bundles on ${\mathcal {X}}$ . Using the modified slope, we define semistable(stable) Higgs bundles. As usual, we can represent the moduli stack ${\mathcal M}_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ of semistable Higgs bundles with modified Hilbert polynomial P as a quotient stack. Moreover, we show that ${\mathcal M}_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ admits a good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ .
Fix a line bundle L on ${\mathcal {X}}$ . The $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is a Higgs bundle $(E,\phi )$ with $\mathop {\mathrm {det}}\nolimits (E)=L$ and $\mathop {\mathrm {tr}}\nolimits (\phi )=0$ . We also prove that the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {SL}}\nolimits _r)$ of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is locally of finite type over ${\mathbb C}$ . And, we show that the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ of semistable Higgs bundles with modified Hilbert polynomial P is a quotient stack and admits a good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ which is a closed subscheme of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ .
Recall that for a principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundle ${\mathcal P}$ , there is an associated cohomology class $\alpha \in H^2({\mathcal {X}},\mu _r)$ , which is the obstruction of lifting ${\mathcal P}$ to a principal $\mathop {\mathbf {SL}}\nolimits _r$ -bundle. We call ${\mathcal P}$ with topological type $\alpha $ . A $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundle is said to be with topological type $\alpha $ if the principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundle is with topological type $\alpha $ . In order to show the algebraicity of moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop { \mathbf {PGL}}\nolimits _r)$ of $\mathop { \mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ , we divide two cases: Case I. Assume that the image of $\alpha $ in $H^2({\mathcal {X}},\mathbb G_m)$ is zero. Therefore, there is a line bundle L on ${\mathcal {X}}$ such that $\delta (L)=-\alpha $ in the Kummer exact sequence (23). We prove that $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {SL}}\nolimits _r)$ is a $\mathcal J_r$ -torsor over $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {PGL}}\nolimits _r)$ , where $\mathcal J_r$ is the stack of $\mu _r$ -torsors on ${\mathcal {X}}$ . Hence, $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {PGL}}\nolimits _r)$ is locally of finite type over ${\mathbb C}$ (see [Reference LieblichLie09, Lemma 3.4]). Case II. Suppose that the image of $\alpha $ in $H^2({\mathcal {X}},\mathbb G_m)$ is nonzero. We consider the $\mu _r$ -gerbe $p_{\alpha } : \mathcal G_{\alpha }\rightarrow {\mathcal {X}}$ corresponding to $\alpha $ . Then, we introduce the notion of twisted Higgs bundles and the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop { \mathbf {SL}}\nolimits _r)$ of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with trivial determinant. Then, we prove that $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {SL}}\nolimits _r)$ is locally of finite type over ${\mathbb C}$ . On the other hand, we show that $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {SL}}\nolimits _r)$ is a $\mathcal J_r$ -torsor over $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {PGL}}\nolimits _r)$ . Thus, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {PGL}}\nolimits _r)$ is also locally of finite type over ${\mathbb C}$ (see [Reference LieblichLie09, Lemma 3.4]). In Section 3.4, we consider the case of stacky curves and give a definition of the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ . For further applications, we also consider the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ (resp. $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ ) of stable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles (resp. stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles) with fixed K-class $\xi \in K_0({\mathcal {X}})_{\mathbb Q}$ .
Hitchin morphism was introduced by Hitchin in his study of two-dimensional reduction of Yang–Mills equations (see [Reference HitchinHit87]). We also introduce the Hitchin morphisms in our setup. If ${\mathcal {X}}$ is a hyperbolic stacky curve, then the Hitchin morphism is proper (see [Reference YokogawaYok93]), where we use the correspondence between the Higgs bundles on a stacky curve and the parabolic Higgs bundles on its coarse moduli space (this correspondence is called orbifold-parabolic correspondence in this paper). In Appendix B, we will give a direct proof of the properness of the Hitchin morphisms, following the argument of [Reference NitsureNit91]. As an immediate corollary, the Hitchin morphism $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r, K_{\mathcal {X}})$ is also proper, where ${\mathbb H}^o(r,K_{\mathcal {X}})$ is the affine space associated with the vector space $\bigoplus _{i=2}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ .
For hyperbolic Riemann surfaces, if the rank of Higgs bundles is at least $2$ , then a general spectral curve is integral (see [Reference Beauville, Narasimhan and RamananBNR89, Remark 3.1]). But, for hyperbolic Deligne–Mumford curves, it is not so. Indeed, there is a hyperbolic Deligne–Mumford curve $\mathbb E_5$ such that for any $\boldsymbol a\in H^0(\mathbb E_5,K_{\mathbb E_5})\oplus H^0(\mathbb E_5,K_{\mathbb E_5}^2)$ , the associated spectral curve is reducible (see Example 4.21). With regard to this, we find an optimal criterion for the integrality of spectral curves (see Proposition 4.3 and Remark 4.2).
A partial classification of spectral curves is obtained (Theorem 4.18). We also construct an example satisfying the last conclusion of the above theorem, i.e., for hyperbolic stacky curve $\mathbb P^1_{4,2,2,2}$ , we show that for a general element $\boldsymbol a$ of $\bigoplus _{i=1}^6H^0(\mathbb P^1_{4,2,2,2},K^i_{\mathbb P^1_{4,2,2,2}})$ , the corresponding spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is singular (see Example 4.22). On the other hand, we also show that the coarse moduli space of the spectral curve on a hyperbolic stacky curve ${\mathcal {X}}$ is the spectral curve of the corresponding parabolic Higgs bundle on X under some condition (see Theorem 4.13 and Remark 4.14).
In Section 5, we systematically study the norm theory on Deligne–Mumford stacks. Applying the general theory to the stacky curves, we obtain the Norm maps for stacky curves (see Proposition 5.21). And, there is a connection between the Norm map of a finite morphism of stacky curves and the Norm map of the induced finite morphism of coarse moduli spaces (see Lemma 5.23). With the help of it, the proof of the SYZ duality can be reduced to the usual case.
1.3 Main results
Let ${\mathcal {X}}$ be a hyperbolic stacky curve of genus g with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . The stacky points of ${\mathcal {X}}$ are $p_1,\ldots ,p_m$ , and the stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ , respectively. Assume that the assumptions of Corollary 4.19 (which ensure that a general spectral curve is irreducible and smooth) are satisfied. Suppose that the K-class $\xi $ satisfies (82) and that $\xi =(r,d_{\xi },(m_{1,i})_{i=1}^{r_1-1},\ldots ,(m_{m,i})_{i=1}^{r_m-1})\in K_0({\mathcal {X}})_{\mathbb Q}$ . Fix a line bundle $L\in \mathop {\mathrm {Pic}}\nolimits ^{d^{\prime },(j_1,\ldots ,j_m)}({\mathcal {X}})$ , where $d^{\prime },j_1,\ldots ,j_m$ satisfy (83). Consider the moduli space of $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ of semistable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles with K-class $\xi $ and determinant L. The Hitchin morphism $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ is surjective. Note that the stable locus $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ of $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ is nonempty. Therefore, the properness of $h_{\mathop {\mathbf {SL}}\nolimits _r}$ implies that there is a nonempty open subset ${\mathcal U}\subseteq {\mathbb H}^o(r,K_{\mathcal {X}})$ such that the inverse image $h_{\mathop {\mathbf {SL}}\nolimits _r}^{-1}({\mathcal U})$ is contained in $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ . Then, $M_{\mathop {\mathbf {SL}}\nolimits _r}:=h_{\mathop { \mathbf {SL}}\nolimits _r}^{-1}({\mathcal U})$ is a hyperkähler manifold and $M_{\mathop { \mathbf {PGL}}\nolimits _r}:=h_{\mathop {\mathbf {PGL}}\nolimits _r}^{-1}({\mathcal U})=[M_{\mathop { \mathbf {SL}}\nolimits _r}/\Gamma _0]$ is a hyperkähler orbifold. Furthermore, we obtained two proper morphisms
where $h_{\mathop {\mathbf {SL}}\nolimits _r,{\mathcal U}}$ and $h_{\mathop { \mathbf {PGL}}\nolimits _r,{\mathcal U}}$ are complete algebraically integrable systems. If we perform a hyperkähler rotation, i.e., change to a different complex structure, the generic fiber of $h_{\mathop { \mathbf {SL}}\nolimits _r,{\mathcal U}}$ (resp. $h_{\mathop {\mathbf {PGL}}\nolimits _r,{\mathcal U}}$ ) is a special Lagrangian torus (see Proposition 6.6). Moreover, for a general point $\boldsymbol a\in {\mathbb H}^o(r,K_{\mathcal {X}})$ , $h_{\mathop {\mathbf {SL}}\nolimits _r,{\mathcal U}}^{-1}(\boldsymbol a)$ and $h_{\mathop {\mathbf {PGL}}\nolimits _r,{\mathcal U}}^{-1}(\boldsymbol a)$ are dual (see Corollary 6.8). On the other hand, there are two flat unitary gerbes $\boldsymbol {\mathcal B}$ and $\check {\boldsymbol {\mathcal B}}$ on $M_{\mathop {\mathbf {SL}}\nolimits _r}$ and $M_{\mathop {\mathbf {PGL}}\nolimits _r}$ , respectively (see Section 6.2). We can therefore state our main results (Theorem 6.13).
Theorem 1.1
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(1) Assume that $\lceil \frac {r}{r_k}\rceil \in \{\frac {r}{r_k},\frac {r+1}{r_k}\}$ for all $1\leq k\leq m$ . Then $(M_{\mathop {\mathbf {SL}}\nolimits _r},\boldsymbol {\mathcal B})$ and $(M_{\mathop { \mathbf {PGL}}\nolimits _r},\check {\boldsymbol {\mathcal B}})$ are SYZ mirror partners if one of the following conditions is satisfied:
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(i) $g\geq 2$ ;
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(ii) $g=1$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2$ ;
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(iii) $g=0$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r+1$ ;
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(iv) $g=0$ , $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r$ and $\mathrm {dim}_{{{\mathbb C}}}H^0({\mathcal {X}},K^k_{\mathcal {X}})\geq 2$ for some $2\leq k\leq r$ .
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(2) Suppose that the assumption about $\lceil \frac {r}{r_k}\rceil $ in $(1)$ does not hold. We make the following assumption: if $\lceil \frac {r}{r_k}\rceil \geq \frac {r+2}{r_k}$ for some $1\leq k\leq m$ , then $\lceil \frac {r-1}{r_k}\rceil =\frac {r-1}{r_k}$ . Then $(M_{\mathop { \mathbf {SL}}\nolimits _r},\boldsymbol {\mathcal B})$ and $(M_{\mathop {\mathbf {PGL}}\nolimits _r},\check {\boldsymbol {\mathcal B}})$ are SYZ mirror partners if one of the following conditions is satisfied:
Corollary 1.2 If the K-class $\xi $ satisfies the condition of Proposition A.4, then for a generic rational parabolic weight (Definition A.3), the moduli spaces $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^s(\mathop {\mathbf {SL}}\nolimits _r)$ and $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,\xi }(\mathop {\mathbf {PGL}}\nolimits _r)$ with natural flat unitary gerbes $\boldsymbol {\mathcal B}$ and $\check {\boldsymbol {\mathcal B}}$ , respectively, are SYZ mirror partners.
Remark 1.3 At the end of Section 6.2, we construct a pair of moduli spaces of Higgs bundles with structure group $\mathop {\mathbf {SL}}\nolimits _3$ and $\mathop {\mathbf {PGL}}\nolimits _3$ , respectively, on the stacky curve $\mathbb P^1_{3,2,2,2,2}$ (see Example 6.15 in Section 6.2). Moreover, in this example, under the orbifold-parabolic correspondence, the quasi-parabolic flags of the corresponding parabolic Higgs bundles are not all full flags. Our theorem provides more examples for the SYZ duality.
1.4 Hausel–Thaddeus conjecture
For any two natural numbers d, e coprime to r, Hausel and Thaddeus [Reference Hausel and ThaddeusHT03] conjectured that the mixed Hodge numbers of the moduli space $ M_{\mathop {\mathrm {Dol}}\nolimits }^d(\mathop {\mathbf {SL}}\nolimits _r)$ of stable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles of degree d on a compact Riemann surface are equal to the stringy mixed Hodge numbers of the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits }^e(\mathop {\mathbf {PGL}}\nolimits _r)$ of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles of degree e on the same compact Riemann surface. And, Hausel and Thaddeus proved the conjecture for $r=2, 3$ by direct calculations in the same paper. Only recently, the conjecture was proved by Groechenig, Wyss, and Ziegler in [Reference Groechenig, Wyss and ZieglerGWZ20] via p-adic integration. Maulik and Shen [Reference Maulik and ShenMS21] gave a new proof of the conjecture using perverse filtration on the moduli space and Ngô’s support theorem in [Reference NgôNgo06, Reference NgôNgo10]. The method in [Reference Maulik and ShenMS21] has more applications in the area of Gopakumar–Vafa invariants. For the moduli spaces of parabolic Higgs bundles, Gothen and Oliveira [Reference Gothen and OliveiraGO19] proved the Hausel–Thaddeus conjecture for ranks $2$ and $3$ but gave evidence that the same holds for any rank.
Notations and conventions
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• All schemes and Deligne–Mumford stacks are defined over the complex field ${\mathbb C}$ throughout of the paper. A Deligne–Mumford stack is always assumed to be a global quotient stack with projective coarse moduli space unless otherwise specified.
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• $K_0({\mathcal {X}})_{\mathbb Q}$ is the rational K-group of coherent sheaves on the Deligne–Mumford stack ${\mathcal {X}}$ .
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• For a Deligne–Mumford stack ${\mathcal {X}}$ , let ${\mathcal {X}}_{\acute{\rm e}{\rm t}}$ denote the small étale site of ${\mathcal {X}}$ .
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• Let $U\rightarrow {\mathcal {X}}$ be a morphism from a scheme U to a Deligne–Mumford stack ${\mathcal {X}}$ . We use $U[n]$ to represent the Cartesian product $U\times _{{\mathcal {X}}}U\times _{{\mathcal {X}}}\cdots \times _{{\mathcal {X}}}U$ of $n+1$ copies of U. Let $\mathrm {pr}_i : U[1]\rightarrow U$ be the projection to the ith factor, for $i=1,2$ , and let $\mathrm {pr}_{12} : U[2]\rightarrow U[1]$ , $\mathrm {pr}_{23} : U[2]\rightarrow U[1]$ , and $\mathrm {pr}_{13} : U[2]\rightarrow U[1]$ be the three natural projections. For an étale covering $U\rightarrow {\mathcal {X}}$ , let $Des(U/{\mathcal {X}})$ denote the category of pairs $(E,\sigma )$ , where E is a sheaf of ${\mathcal {O}}_{U}$ -modules on $U_{\acute{\rm e}{\rm t}}$ and $\sigma : \mathrm {pr}_1^{*}E\rightarrow \mathrm {pr}_2^{*}E$ is an isomorphism on $U[1]_{\acute{\rm e}{\rm t}}$ , which satisfies the cocycle condition $\mathrm {pr}_{23}^{*}\sigma \circ \mathrm { pr}_{12}^{*}\sigma =\mathrm {pr}_{13}^{*}\sigma $ .
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• $(\mathrm {Sch}/{{\mathbb C}})_{\acute{\rm e}{\rm t}}$ is the category of schemes over the complex field ${\mathbb C}$ with big étale topology.
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• For a Deligne–Mumford stack ${\mathcal {X}}$ and a ${\mathbb C}$ -scheme T, we denote the fiber product ${\mathcal {X}}\times T$ by ${\mathcal {X}}_T$ . Also, $\mathrm {pr_{{\mathcal {X}}}} : {\mathcal {X}}_T\rightarrow {\mathcal {X}}$ is the projection to ${\mathcal {X}}$ and $\mathrm {pr}_T : {\mathcal {X}}_T\rightarrow T$ is the projection to T.
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• $\mathop {\mathrm {Tot}}\nolimits (E)$ denotes the relative ${\boldsymbol {{\mathrm {Spec}}}}_{{\mathcal {X}}}(\mathrm {Sym}^{\bullet } E^{\vee })$ , where $\mathrm {Sym}^{\bullet } E^{\vee }$ is the symmetric algebra of the dual $E^{\vee } $ of a locally free sheaf E on a Deligne–Mumford stack ${\mathcal {X}}$ .
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• For a locally free sheaf E on a Deligne–Mumford stack ${\mathcal {X}}$ , the associated projective bundle $\mathbb P(E)$ is defined to be the relative ${\boldsymbol {\mathrm {Proj}}}(\mathrm {Sym}^{\bullet } E^{\vee })$ , where $\mathrm {Sym}^{\bullet } E^{\vee }$ is the symmetric algebra of the dual $E^{\vee } $ of E.
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• For any real number $c\in {\mathbb R}$ , we use $\lceil c\rceil $ to denote the ceiling of c.
2 Preliminaries
2.1 Deligne–Mumford curves
We recall some basic definitions of Deligne–Mumford curves. For a detailed discussion of these topics, please refer to [Reference Behrend and NoohiBN06].
Definition 2.1 A Deligne–Mumford curve ${\mathcal {X}}$ is a one-dimensional Deligne–Mumford stack of finite type over ${\mathbb C}$ . A stacky curve (or an orbifold curve) is a Deligne–Mumford curve with trivial generic stabilizers.
Remark 2.2 For a smooth Deligne–Mumford curve ${\mathcal {X}}$ , there is a smooth stacky curve $\widehat {{\mathcal {X}}}$ and a morphism $\mathcal R : {\mathcal {X}}\rightarrow \widehat {{\mathcal {X}}}$ , where $\mathcal R$ is an H-gerbe for some finite group H (see [Reference Behrend and NoohiBN06, Proposition 4.6]).
Definition 2.3 A smooth irreducible Deligne–Mumford curve ${\mathcal {X}}$ is said to be hyperbolic if the degree $\mathrm { deg}(K_{{\mathcal {X}}})$ of the canonical line bundle $K_{{\mathcal {X}}}$ is positive.
Remark 2.4 A Deligne–Mumford curve ${\mathcal {X}}$ is hyperbolic if and only if the associated stacky curve is hyperbolic (see [Reference Behrend and NoohiBN06, proposition 7.4]).
2.2 Semistable sheaves on Deligne–Mumford stacks
On a Deligne–Mumford stack, there is no very ample line bundle on it unless it is an algebraic space. However, Olsson and Starr [Reference Olsson and StarrOS03] have discovered that under certain hypothesis, there are locally free sheaves, called generating sheaves, which behave like very ample line bundles. In the following, ${\mathcal {X}}$ is a Deligne–Mumford stack with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ .
Definition 2.5 Let F be a coherent sheaf on ${\mathcal {X}}$ . F is said to be a pure sheaf of dimension d if the support of every nonzero coherent subsheaf G of F is of dimension d.
Definition 2.6 A locally free sheaf ${\mathcal E}$ on ${\mathcal {X}}$ is said to be a generating sheaf if for any quasicoherent sheaf F on ${\mathcal {X}}$ , the left adjoint of the identity morphism $\mathrm {id} : \pi _{*}(F\otimes {\mathcal E}^{\vee })\rightarrow \pi _{*}(F\otimes {\mathcal E}^{\vee })$ , $\pi ^{\ast }(\pi _{\ast }({{\mathcal E}}^{\vee }\otimes F))\otimes {\mathcal E}\longrightarrow F$ is surjective.
Remark 2.7 A smooth Deligne–Mumford curve ${\mathcal {X}}$ possesses a generating sheaf (see [Reference Kresch, Abramovich, Bertram, Katzarkov, Pandharipande and ThaddeusKre05, Theorem 5.3]).
Definition 2.8 A polarization on ${\mathcal {X}}$ is a pair $(\mathcal {E},\mathcal {O}_{X}(1))$ , where ${\mathcal E}$ is a generating sheaf on ${\mathcal {X}}$ and $\mathcal {O}_X(1)$ is a very ample line bundle on X.
Example 2.9 Suppose that ${\mathcal {X}}$ is a smooth irreducible stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . Let $\{p_1,\ldots ,p_m\}$ be the set of stacky points of ${\mathcal {X}}$ with the orders of stabilizer groups $\{r_1,\ldots ,r_m\}$ . Then, the locally free sheaf
is a generating sheaf, since it is $\pi $ -very ample (see [Reference NironiNir08, Definition 2.2 and Proposition 2.7]). Let ${\mathcal {O}}_X(1)$ be a very ample line bundle on X. Then, $({\mathcal E}_u,{\mathcal {O}}_X(1))$ is a polarization on ${\mathcal {X}}$ .
Definition 2.10 Let $({\mathcal E},\mathcal {O}_{X}(1))$ be a polarization on the ${\mathcal {X}}$ , and let F be a coherent sheaf on it. The modified Hilbert polynomial $P_F$ of F is defined by $P_{F}(m)=\chi (\pi _{\ast }(F\otimes {{\mathcal E}}^{\vee })\otimes \mathcal {O}_{X}(m))$ , where $\chi (\pi _{\ast }(F\otimes {{\mathcal E}}^{\vee })\otimes \mathcal {O}_{X}(m))$ is the Euler characteristic of $\pi _{\ast }(F\otimes {{\mathcal E}}^{\vee })\otimes \mathcal {O}_{X}(m)$ on X.
Remark 2.11 In general, the modified Hilbert polynomial is
where d is the dimension of the support $\mathrm {supp}(F)$ and $a_i(F)$ are rationals.
Definition 2.12 If the modified Hilbert polynomial $P_F$ of F is (3), then its reduced Hilbert polynomial $p_F$ is defined to be $p_F(m)=\frac {P_F(m)}{a_d(F)}$ .
Definition 2.13 The modified slope $\mu _{\mathcal E}(F)$ of F is defined by $\mu _{\mathcal E}(F)=\frac {a_{d-1}(F)}{a_{d}(F)}$ .
Definition 2.14 Suppose that F is a pure sheaf on ${\mathcal {X}}$ . F is said to be semistable (stable) if for every proper coherent subsheaf $F^{\prime }$ of F, we have
2.3 Higgs bundles and stability
Let ${\mathcal {X}}$ be a smooth Deligne–Mumford curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ .
Definition 2.15 A rank n Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ consists of a rank n locally free sheaf E on ${\mathcal {X}}$ and a morphism $\phi : E\rightarrow E\otimes K_{{\mathcal {X}}}$ of ${\mathcal {O}}_{{\mathcal {X}}}$ -modules, where $\phi $ is called the Higgs field.
Definition 2.16 For a scheme T, a T-family $(E_T,\phi _T)$ of Higgs bundles on ${\mathcal {X}}$ consists of a rank n locally free sheaf $E_T$ on ${\mathcal {X}}_T$ and a morphism of ${\mathcal {O}}_{{\mathcal {X}}_T}$ -modules $\phi _T : E_T\longrightarrow E_T\otimes \mathrm {pr_{{\mathcal {X}}}^{*}K_{{\mathcal {X}}}}$ .
Example 2.17 Let ${\mathcal {X}}$ be a smooth irreducible Deligne–Mumford curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . The canonical line bundle is $K_{\mathcal {X}}=\pi ^{*}K_X\otimes L_{\mathcal {X}}$ , for some line bundle $L_{\mathcal {X}}$ on ${\mathcal {X}}$ . In fact, if ${\mathcal {X}}$ is a stacky curve, it is so (see [Reference Voight and Zureick-BrownVB22, Proposition 5.5.6]). Then, we can get the formula for a general Deligne–Mumford curve by Remark (2.2). Let $E=E_1\oplus E_2$ , where $E_1=\pi ^{*}K^{\frac {1}{2}}_{X}$ and $E_2=\pi ^{*}K_{X}^{-\frac {1}{2}}\otimes L_{\mathcal {X}}^{-1}$ . With respect to the decomposition of E, there is a morphism of ${\mathcal {O}}_{{\mathcal {X}}}$ -modules
where $\mathbf {1}$ is the identity morphism in $\mathrm {Hom}_{{\mathcal {O}}_{{\mathcal {X}}}}(E_1,E_1)$ . The pair $(E,\phi )$ is a Higgs bundle on ${\mathcal {X}}$ .
Using the modified slope, we introduce the notions of semistable (stable) Higgs bundles.
Definition 2.18 Fix a polarization $({\mathcal E},{\mathcal {O}}_{X}(1))$ on ${\mathcal {X}}$ . A Higgs bundle $(E,\phi )$ is said to be semistable (resp. stable) if for all proper nonzero $\phi $ -invariant locally free subsheaf $F\subset E$ (i.e., $\phi (F)\subseteq F\otimes K_{{\mathcal {X}}}$ ), we have
If $(E,\phi )$ is not semistable, we say that $(E,\phi )$ is unstable.
3 Moduli spaces of Higgs bundles
In the following, ${\mathcal {X}}$ is supposed to be a hyperbolic Deligne–Mumford curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ .
3.1 Moduli stacks of Higgs bundles
The moduli functor of rank r Higgs bundles is
where ${\mathcal M}_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {GL}}\nolimits _r)(T)$ is the groupoid of T-families of rank r Higgs bundles on ${\mathcal {X}}$ for a test scheme T. Similarly, we can also define the moduli functor $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ of rank r Higgs bundles with modified Hilbert polynomial P on ${\mathcal {X}}$ . Suppose that $\mathcal M_{\mathop {\mathrm {Vec}}\nolimits ,r}$ is the moduli functor of rank r locally free sheaves on ${\mathcal {X}}$ . There is a forgetful functor
defined by forgetting the Higgs fields.
Proposition 3.1 $\mathcal M_{\mathop {\mathrm {Vec}}\nolimits ,r}$ is an algebraic stack locally of finite type over ${\mathbb C}$ .
Proof Since the stack $\mathcal Coh({\mathcal {X}})$ of coherent sheaves on ${\mathcal {X}}$ is an algebraic stack locally of finite type over ${\mathbb C}$ (see [Reference NironiNir08, Corollary 2.27]) and the inclusion of $\mathcal M_{\mathop {\mathrm {Vec}}\nolimits }$ into $\mathcal Coh({\mathcal {X}})$ is represented by open immersion (see [Reference Huybrechts and LehnHL10, Lemma 2.1.8]), $\mathcal M_{\mathop {\mathrm {Vec}}\nolimits ,r}$ is an algebraic stack locally of finite type over ${\mathbb C}$ (see [Reference OlssonOls16, Proposition 10.2.2]).
Proposition 3.2 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {GL}}\nolimits _r)$ is an algebraic stack locally of finite type over ${\mathbb C}$ .
Proof The morphism (4) is representable, which is an abelian cone over $\mathcal M_{\mathop {\mathrm {Vec}}\nolimits ,r}$ . Hence, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop { \mathbf {GL}}\nolimits _r)$ is an algebraic stack locally of finite type over ${\mathbb C}$ (see [Reference OlssonOls16, Proposition 10.2.2]).
Corollary 3.3 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ is an algebraic stack locally of finite type over ${\mathbb C}$ .
Let $\mathcal Y=\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})$ be the projective bundle associated with $K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ , and let ${\mathcal {O}}_{\mathcal Y}(1)$ be the relative hyperplane bundle on $\mathcal Y$ . Due to the universal property of coarse moduli spaces, we have the commutative diagram
where $\pi ^{\prime } : \mathcal Y\rightarrow Y$ is the coarse moduli space of $\mathcal Y$ and the first square is Cartesian. For a polarization $({\mathcal E}_{{\mathcal {X}}},{\mathcal {O}}_X(1))$ on ${\mathcal {X}}$ , there is a polarization $({\mathcal E}_{\mathcal Y},{\mathcal {O}}_{Y}(1))$ on $\mathcal Y$ , where ${\mathcal E}_{\mathcal Y}=\Psi ^{*}{\mathcal E}_{{\mathcal {X}}}$ ( $\Psi ^{*}{\mathcal E}_{{\mathcal {X}}}$ is a generating sheaf on $\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})$ (see [Reference Olsson and StarrOS03, Proposition 5.3])) and ${\pi ^{\prime }}^{*}{\mathcal {O}}_{Y}(1)={(\pi \circ \Psi )}^{*}{\mathcal {O}}_{X}(1){\otimes } {\mathcal {O}}_{\mathcal Y}(m)$ for some $m\gg 0$ . A Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ is equivalent to a compactly supported one-dimensional pure sheaf $E_{\phi }$ on $\mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})$ (see Appendix C).
Proposition 3.4 A Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ is semistable (resp. stable) with respect to $({\mathcal E}_{{\mathcal {X}}},{\mathcal {O}}_X(1))$ if and only if $E_{\phi }$ is Gieseker semistable (resp. stable) with respect to $({\mathcal E}_{\mathcal Y},{\mathcal {O}}_{Y}(1))$ .
Proof $E_{\phi }$ is a pure sheaf on $\mathcal Y$ with modified Hilbert polynomial $P_{E_{\phi }}(n)=\chi ({\mathcal Y},{{\mathcal E}_{\mathcal Y}^{\vee }}\otimes E_{\phi }\otimes {\pi ^{\prime }}^{*}{\mathcal {O}}_{Y}(n))$ . Since the support of $E_{\phi }$ is contained in ${\mathop {\mathrm {Tot}}\nolimits }(K_{\mathcal {X}})$ , we have
where $\psi : \mathop {\mathrm {Tot}}\nolimits ({\mathcal {X}})\rightarrow {\mathcal {X}}$ is the natural projection. Note that $\psi $ is an affine morphism. Hence, the pushforward functor $\psi _{*}$ is exact on the category of quasicoherent sheaves. Then, we have the identity
where we use the fact $\psi _{*}E_{\phi }=E$ . So, $P_{E_{\phi }}=P_{E}$ . On the other hand, the $\phi $ -invariant locally free subsheaves of E are equivalent to the coherent subsheaves of $E_{\phi }$ . We complete the proof.
In order to construct the moduli space of semistable Higgs bundles on ${\mathcal {X}}$ , we recall a lemma (see [Reference Fantechi, Gottsche, Illusie, Kleiman and NitsureFGIKN05, Section 5.6]).
Lemma 3.5 [Reference Fantechi, Gottsche, Illusie, Kleiman and NitsureFGIKN05]
Let $f : \widehat X\rightarrow S$ be a proper morphism of Noetherian schemes. Suppose that $\widehat Y$ is a closed subscheme of $\widehat X$ and F is a coherent sheaf on $\widehat X$ . Then, there exists an open subscheme $S^{\prime }$ of S with the universal property that a morphism $T\rightarrow S$ factors through $S^{\prime }$ if and only if the support of the pullback $F_T$ on $\widehat X\times _ST$ is disjoint from $\widehat Y\times _ST$ .
We need a stacky version of the above lemma. First, we state a technical lemma.
Lemma 3.6 Suppose that $\widehat {{\mathcal {X}}}$ is a proper Deligne–Mumford stack over a Noetherian scheme S and E is a coherent sheaf on $\widehat {{\mathcal {X}}}$ . If ${\mathcal E}$ is a generating sheaf on $\widehat {{\mathcal {X}}}$ , then we have
where $\pi : \widehat {{\mathcal {X}}}\rightarrow \widehat X$ is the coarse moduli space of $\widehat {{\mathcal {X}}}$ and $F_{\mathcal E}(E)=\pi _{*}({\mathcal E}^{\vee }\otimes E)$ . Moreover, $F_{{\mathcal E}}(E)=0$ if and only if $E=0$ .
Proof The proof of this lemma is the same as Lemma 3.4 in [Reference NironiNir08].
Lemma 3.7 Let $\widehat {{\mathcal {X}}}$ be a proper Deligne–Mumford stack over a Noetherian scheme S, and let ${\mathcal W}$ be a closed substack of $\widehat {{\mathcal {X}}}$ . For a coherent sheaf E on $\widehat {{\mathcal {X}}}$ , there exists an open subscheme $S^{\prime }$ of S with the universal property: a morphism $T\rightarrow S$ factors through $S^{\prime }$ if and only if the support of the pullback $E_T$ of E to $\widehat {{\mathcal {X}}}_T=\widehat {{\mathcal {X}}}\times _ST$ is disjoint from ${\mathcal W}_T={\mathcal W}\times _ST$ .
Proof Let $\pi _{{\mathcal W}}:{\mathcal W}\rightarrow W$ and $\pi _{\widehat {{\mathcal {X}}}}:\widehat {{\mathcal {X}}}\rightarrow \widehat X$ be the coarse moduli spaces of ${\mathcal W}$ and $\widehat {{\mathcal {X}}}$ , respectively. Then, by the universal property of coarse moduli spaces, there is a commutative diagram
where i is the closed immersion and $i^{\prime }$ is the induced morphism.
Claim The morphism $i^{\prime } : W\rightarrow \widehat X$ in (6) is a closed immersion. In fact, there is a short exact sequence of coherent sheaves
where $\mathcal I_{{\mathcal W}}$ is the ideal sheaf of ${\mathcal W}$ in $\widehat X$ . By the tameness of $\widehat {{\mathcal {X}}}$ (see Definition 1.1 and Theorem 1.2 in [Reference NironiNir08]), the pushforward of (7) to $\widehat X$ is
By (6), the following two compositions
are the same. By (8) and the isomorphism ${\mathcal {O}}_{\widehat X}\rightarrow \pi _{\widehat {{\mathcal {X}}}*}{\mathcal {O}}_{\widehat {{\mathcal {X}}}}$ , we have the commutative diagram of short exact sequences
where $\mathcal I_W$ is the kernel of the composition in (9). Note that the naturel morphism ${\mathcal {O}}_{W}\rightarrow \pi _{{\mathcal W}*}{\mathcal {O}}_{{\mathcal W}}$ is an isomorphism. By (6) and (10), we have the short exact sequence
On the other hand, the morphism of topological spaces induced by $i^{\prime }$ is a closed embedding. Thus, $W\rightarrow X$ is a closed immersion. By Lemma 3.5, for the coherent sheaf $F_{\mathcal E}(E)$ , there is an open subscheme $S^{\prime }$ of S with the universal property: a morphism $T\rightarrow S$ factors through $S^{\prime }$ if and only if the support of the pullback ${F_{{\mathcal E}}(E)}_T$ of $F_{{\mathcal E}}(E)$ on $\widehat X_T=\widehat X\times _ST$ is disjoint from $W\times _ST$ . On the other hand, for a morphism $T\rightarrow S$ , we consider the Cartesian diagram
By Proposition 1.5 in [Reference NironiNir08], we have $F_{{\mathcal E}}(E)_T\simeq F_{{\mathcal E}_{T}}(E_T)$ , where the pullback ${\mathcal E}_T$ of ${\mathcal E}$ to $T\times _{S}\widehat {{\mathcal {X}}}$ is a generating sheaf (see [Reference Olsson and StarrOS03, Theorem 5.5]). Then, the support of the pullback $E_{T}$ of E on $\widehat {{\mathcal {X}}}_T$ is disjoint from ${\mathcal W}_T$ if and only if the morphism $T\rightarrow S$ factors through the $S^{\prime }$ , by Lemma 3.6.
Recall Theorem 5.1 in [Reference NironiNir08]:
Theorem 3.8 There is an open subscheme $R^{ss}_1$ of $\mathrm {Quot}_{{\mathcal {X}}/{\mathbb C}}(V{\otimes }_{{\mathbb C}}\pi ^{\prime *}{\mathcal {O}}_{Y}(-n){\otimes }{\mathcal E}_{\mathcal Y},P)$ such that the moduli stack of semistable purely one-dimensional sheaves with modified Hilbert polynomial P on $\mathcal Y $ is the quotient stack $[R^{ss}_1/{\mathop {\mathbf {GL}}\nolimits _N}]$ , where $\mathop {\mathbf {GL}}\nolimits _N$ is the general linear group over ${\mathbb C}$ with $N=P(n)$ .
Let $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ be the moduli stack of rank r semistable Higgs bundles with modified Hilbert polynomial P on ${\mathcal {X}}$ . By Proposition 3.4 and Lemma 3.7, we have the following corollary.
Corollary 3.9 There is an open subscheme $R^{ss}$ of $R^{ss}_1$ such that $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ is quotient stack $\big [{R}^{ss}/{\mathop {\mathbf {GL}}\nolimits _N}\big ]$ , where $N=P(n)$ .
We need the following proposition to define the S-equivalence of semistable Higgs bundles.
Proposition 3.10 Suppose that $(E,\phi )$ is a semistable Higgs bundle on ${\mathcal {X}}$ . Then, there is a sequence of $\phi $ -invariant locally free subsheaves $0\subset E_1\subset E_2\subset \cdots \subset E_s=E$ such that $\mu \big (E_i/E_{i-1}\big )=\mu (E)$ and $(E_i/E_{i-1},\phi _i)$ is stable for each $i=1,\ldots ,s$ , where $\phi _i : E_i/E_{i-1}\rightarrow E_i/E_{i-1}\otimes K_{{\mathcal {X}}}$ is induced by $\phi $ . Moreover, the associated graded Higgs bundle $gr(E,\phi )={\bigoplus }_{i=1}^l\big (E_i/E_{i-1},\phi _i\big )$ is uniquely determined up to an isomorphism by $(E,\phi )$ .
Proof This proposition can be proved following the steps in the proof of [Reference NitsureNit91, Proposition 4.1].
Remark 3.11 Under the equivalence (C.1) (see Appendix C), the coherent sheaf corresponding to $gr(E,\phi )={\bigoplus }_{i=1}^l\big (E_i/E_{i-1},\phi _i\big )$ is isomorphic to a Jordan–Hölder filtration of $E_{\phi }$ .
Definition 3.12 Suppose that $(E,\phi )$ and $(E^{\prime },\phi ^{\prime })$ are two semistable Higgs bundles on ${\mathcal {X}}$ . They are said to be S-equivalent if the associated graded Higgs bundles are isomorphic.
In general, the algebraic stacks without finite inertia rarely admit coarse moduli spaces. Alper introduced the notion of good moduli spaces in [Reference AlperAlp13].
Definition 3.13 Let $\varpi : \mathcal S\rightarrow S$ be a morphism from an algebraic stack to an algebraic space. We say that $\varpi : \mathcal S\rightarrow S$ is a good moduli space if the following properties are satisfied:
-
(i) The pushforward functor $\varpi _{*}$ on the categories of quasicoherent sheaves is exact.
-
(ii) The morphism of sheaves ${\mathcal {O}}_S\rightarrow \varpi _{*}{\mathcal {O}}_{\mathcal S}$ is an isomorphism.
Theorem 3.14 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ has a good moduli space $\mathcal Q : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ . More precisely, the following hold:
-
(i) Universal property: for a scheme Z and a morphism $g : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow Z$ , there is a unique morphism $ \theta : M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow Z$ such that the following diagram
-
(ii) ${M}_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ is a quasiprojective scheme over ${\mathbb C}$ .
Proof According to Theorem 6.22 in [Reference NironiNir08], the moduli stack $[R^{ss}_1/{\mathop { \mathbf {GL}}\nolimits _{N}}]$ of semistable one-dimensional pure sheaves with modified Hilbert polynomial P on $\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})$ has a good moduli space $\mathcal Q_1 : [R^{ss}_1/{\mathop {\mathbf {GL}}\nolimits _N}]\longrightarrow M^{ss}_1$ , where $M_1^{ss}$ is the GIT quotient of $R^{ss}_1$ with respect to the $\mathop {\mathbf {SL}}\nolimits _N$ -action. It also satisfies the properties:
-
• Universal property: for every scheme Z and every morphism $g_1 : [R^{ss}_1/{\mathop { \mathbf {GL}}\nolimits _N}]\rightarrow Z$ , there is a unique morphism $\theta _1 : M^{ss}_1\rightarrow Z$ such that $g_1=\theta _1\circ \mathcal Q_1$ .
-
• $M^{ss}_1$ is a projective scheme over ${\mathbb C}$ .
Recall $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {GL}}\nolimits _r)=[R^{ss}/\mathop {\mathbf {GL}}\nolimits _N]$ (see Corollary 3.9). Since $R^{ss}$ is a $\mathop {\mathbf {GL}}\nolimits _N$ -invariant open subscheme of $R_1^{ss}$ , $R^{ss}_1\setminus {R}^{ss}$ is a ${\mathop {\mathbf {GL}}\nolimits _N}$ -invariant closed subset. Let $Q : R^{ss}_1\rightarrow M^{ss}_1$ be the GIT quotient, which is a good quotient. Hence, the image $Q(R^{ss}_1\setminus {R}^{ss})$ is a closed subset of $M^{ss}_1$ . And, $Q(R^{ss})\cap Q({R^{ss}_1\setminus R}^{ss})=\emptyset $ . In fact, two semistable sheaves on $\mathcal Y$ represent the same point in $M^{ss}_1$ if and only if they are S-equivalent (see [Reference NironiNir08, Theorem 6.20]). By Remark 3.11, for a semistable $(E,\phi )$ on ${\mathcal {X}}$ , the support of the graded sheaf associated with some Jordan–Hölder filtration of $E_{\phi }$ is contained in $\mathop {\mathrm {Tot}}\nolimits (K_{{\mathcal {X}}})$ . Denote $Q({R}^{ss})$ by ${M}^{ss}$ . The following diagram is Cartesian:
The universal property of $\mathcal Q : [{R}^{ss}/{\mathop {\mathbf {GL}}\nolimits _N}]\rightarrow {M}^{ss}$ is an immediate conclusion, since Q is a universal categorical quotient. ${M}^{ss}$ is the good moduli space of $[{R}^{ss}/{\mathop {\mathbf {GL}}\nolimits _N}]$ (see [Reference AlperAlp13, Remark 6.2]).
3.2 Moduli stack of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles
Definition 3.15 Fix a line bundle L on ${\mathcal {X}}$ . An $\mathop {\mathbf {SL}}\nolimits _{\mathbf {r}}$ -Higgs bundle $(E,\phi )$ is a rank r Higgs bundle with ${\mathop {\mathrm {det}}\nolimits }(E)\simeq L$ and ${\mathop {\mathrm {tr}}\nolimits }(\phi )=0$ . The stability of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is the same as Definition 2.18.
The moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {SL}}\nolimits _r)$ of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is the stack whose fiber over a test scheme T is the groupoid of T-families of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles on ${\mathcal {X}}$ . Similarly, we have the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {SL}}\nolimits _r)$ (resp. $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ ) of (resp. semistable) $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with fixed modified Hilbert polynomial P.
Proposition 3.16 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }({\mathop {\mathbf {SL}}\nolimits _r})$ is an algebraic stack locally of finite type over ${\mathbb C}$ .
Proof The Picard stack $\mathcal Pic({{\mathcal {X}}})$ of ${\mathcal {X}}$ is an algebraic stack locally of finite type over ${{\mathbb C}}$ (see [Reference AokiAok06]). There is a morphism of algebraic stacks $\mathcal Det : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow \mathcal {P}ic({{\mathcal {X}}})$ , which is defined by taking determinants. By taking the traces of Higgs fields, we can define a morphism $\mathcal Tr : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow {\mathbb H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ , where ${\mathbb H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ is the affine space associated with $H^0({\mathcal {X}},K_{{\mathcal {X}}})$ . On the other hand, L defines a geometric point $[L] : {\mathop {\mathrm {Spec}}\nolimits }({\mathbb C})\rightarrow \mathcal Pic({{\mathcal {X}}})$ and the origin of $H^0({\mathcal {X}},K_{\mathcal {X}})$ defines a closed point $o : \mathrm { Spec}({\mathbb C})\rightarrow \mathbb {H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ . We therefore have the Cartesian diagram
Hence, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }({\mathop {\mathbf {SL}}\nolimits _r})$ is an algebraic stack locally of finite type over ${\mathbb C}$ by Proposition 3.2.
Corollary 3.17 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {SL}}\nolimits _r)$ is an algebraic stack locally of finite type over ${\mathbb C}$ .
Proof Since the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {SL}}\nolimits _r)$ is an open and closed substack of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop { \mathbf {SL}}\nolimits _r)$ , $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {SL}}\nolimits _r)$ is an algebraic stack of locally finite type over ${\mathbb C}$ by Proposition 3.16.
In the following, we will show that the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is a quotient stack. Let $(E_{R^{ss}},\phi _{R^{ss}})$ be the $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant Higgs bundle on ${\mathcal {X}}\times R^{ss}$ , which is the pushforward of the universal quotient sheaf on $\mathrm {Tot}(K_{{\mathcal {X}}})\times R^{ss}$ . Let $\mathrm {det}: R^{ss}\rightarrow \mathrm {Pic}({\mathcal {X}})$ be the classifying morphism defined by $\mathop {\mathrm {det}}\nolimits (E_{R^{ss}})$ , where ${\mathop {\mathrm {Pic}}\nolimits }({\mathcal {X}})$ is the Picard scheme of ${\mathcal {X}}$ (see [Reference BrochardBro12, Corollary 2.3.7(i)]). On the other hand, the trace of the morphism ${\phi _{R^{ss}}}: E_{R^{ss}}\rightarrow E_{R^{ss}}\otimes \mathrm {pr}^{*}_{{\mathcal {X}}}K_{{\mathcal {X}}}$ defines a section $\mathrm {tr}(\phi _{R^{ss}})$ of $\mathrm {pr}^{*}_{{\mathcal {X}}}K_{{\mathcal {X}}}$ . It defines a morphism $\mathop {\mathrm {tr}}\nolimits : R^{ss}\rightarrow {\mathbb H}^0({\mathcal {X}},K_{{\mathcal {X}}})$ . Consider the Cartesian diagram
Theorem 3.18 There exists a $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant line bundle W on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ such that the moduli stack $\mathcal M^{ss}_{\mathop {\mathrm {Dol}}\nolimits ,P}({\mathop {\mathbf {SL}}\nolimits _r})$ can be represented by $[W^{*}/{\mathop { \mathbf {GL}}\nolimits _N}]$ , where $W^{*}$ is the frame bundle associated with W.
Proof First, we consider the Cartesian diagram
Hence, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ can be presented by $[R^{ss,o}/{\mathop {\mathbf {GL}}\nolimits _N}]$ , where $R^{ss,o}=R^{ss}\times _{{\mathbb H}^0({\mathcal {X}},K_{\mathcal {X}})}\mathop {\mathrm {Spec}}\nolimits ({\mathbb C})$ . Moreover, the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is the fiber product
On the other hand, we have the following Cartesian diagram:
where $\mathop {\mathrm {det}}\nolimits $ is the classifying morphism of the determinant line bundle of the universal Higgs bundle on ${\mathcal {X}}\times \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ . The closed substack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {SL}}\nolimits _r)$ of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ can be presented as $[R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}/\mathop {\mathbf {GL}}\nolimits _N]$ , where $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ is the fiber product in the Cartesian diagram (13). Since $\mathcal M^{ss}_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop { \mathbf {GL}}\nolimits _r)$ in the diagram (15) factors through the closed immersion $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {SL}}\nolimits _r)\rightarrow \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)$ in the diagram (16), it is easy to check that the following commutative diagram is Cartesian
where $\mathcal Det$ is the restriction of $\mathcal Det : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow \mathcal Pic({\mathcal {X}})$ to $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss,o}(\mathop {\mathbf {SL}}\nolimits _r)$ . On the other hand, $\det (E_{R^{ss}})|_{{\mathcal {X}}\times R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}}\simeq \mathrm { pr}^{*}_{{\mathcal {X}}}L\otimes \mathrm {pr}^{*}_{R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}}W$ for some line bundle W on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , where $\mathrm {pr}_{{\mathcal {X}}}$ and $\mathrm { pr}_{R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}}$ are the projections to ${\mathcal {X}}$ and $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , respectively. Moreover, W is a $\mathop { \mathbf {GL}}\nolimits _N$ -equivariant line bundle on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , since $\mathrm { pr}^{*}_{{\mathcal {X}}}L$ is a $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant line bundle with the trivial equivariant structure. By the Cartesian diagram (17), $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ can be represented by $[W^{*}/\mathop { \mathbf {GL}}\nolimits _N]$ , where $W^{*}$ is the frame bundle associated with W.
Corollary 3.19 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ has a good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ , which is a closed subscheme of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ .
Proof As the center ${\mathbb C}^{*}$ of $\mathop {\mathbf {GL}}\nolimits _N$ acts trivially on $R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ , the $\mathop {\mathbf {GL}}\nolimits _N$ -equivariant morphism $W^{*}\rightarrow R^{ss}_{\mathop {\mathbf {SL}}\nolimits _r}$ induces a morphism of quotient stacks
On the other hand, there is a Cartesian diagram:
Note that the top morphism in (19) is a $\mu _r$ -gerbe. It follows that the bottom morphism (18) is also a $\mu _r$ -gerbe. So, the good moduli space of $[R^{ss}_{\mathrm {SL_r}}/{\mathop {\mathbf {PGL}}\nolimits _N}]$ coincides with the good moduli space of $[W^{*}/{\mathop {\mathbf {GL}}\nolimits _N}]$ . Note the good moduli space of $[R^{ss}_{\mathrm { SL_r}}/{\mathop {\mathbf {PGL}}\nolimits _N}]$ is a closed subscheme of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ . This completes the proof.
3.3 Moduli stack of ${\mathop {\mathbf {PGL}}\nolimits _r}$ -Higgs bundles
We first recall some basic facts about principal bundles (or torsors) on ${\mathcal {X}}$ . Our main reference is [Reference GiraudGir71]. For an algebraic group G, the set of isomorphism classes of principal G-bundles is denoted by $H^1({\mathcal {X}}, G)$ (if G is abelian, $H^1({\mathcal {X}},G)$ is equivalent to the étale cohomology group with values in G). For a morphism of algebraic groups $G\rightarrow H$ , we have a morphism of pointed sets
where ${\mathcal P}_G\wedge ^GH$ is also denoted by ${\mathcal P}_G\times ^GH$ in some literatures. In general, the morphism (20) is not surjective. We say a principal H-bundle ${\mathcal P}_H$ can be lifted to a principal G-bundle if ${\mathcal P}_H\simeq {\mathcal P}_G\wedge ^GH$ for some principal G-bundle ${\mathcal P}_G$ . For simplicity, we only consider the case when G is a central extension of H by C, i.e., there is an exact sequence of algebraic groups . The obstruction of lifting ${\mathcal P}_H$ to a principal G-bundle is the so-called lifting gerbe $\mathcal G_{{\mathcal P}_H}$ (or G -lifting gerbe). Recall that there is a natural morphism of classifying stacks $BG\rightarrow BH$ defined by
for a test scheme T. The lifting gerbe $\mathcal G_{{\mathcal P}_H}$ is the fiber product ${\mathcal {X}}\times _{BH}BG$ for the Cartesian diagram
where ${\mathcal {X}}\rightarrow BH$ is the classifying morphism of ${\mathcal P}_H$ .
Remark 3.20 The lifting gerbe $\mathcal G_{{\mathcal P}_H}\rightarrow {\mathcal {X}}$ is a C-gerbe on ${\mathcal {X}}$ (see [Reference OlssonOls16, Definition12.2.2]), since the morphism $BG\rightarrow BH$ is a C-gerbe.
The set of isomorphism classes of C-gerbes is equal to $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},C)$ . Then, we have a morphism of pointed sets
which maps the trivial principal H-bundle to the trivial C-gerbe. Indeed, according to the general theory of [Reference GiraudGir71], we have:
Proposition 3.21 A principal H-bundle ${\mathcal P}_H$ can be lifted to a principal G-bundle if and only if the lifting gerbe $\mathcal G_{{\mathcal P}_H}$ is trivial. Moreover, there is an associated exact sequence of pointed sets
Recall the Kummer sequence
For a line bundle L on ${\mathcal {X}}$ , we use $\mathcal G_{L}$ to denote the $\mu _r$ -gerbe defined by the cohomology class $\delta ([L])$ . Consider the central extension
By Proposition 3.21, we have the following exact sequences of pointed sets:
The $\mathop {\mathbf {SL}}\nolimits _r$ -lifting gerbe of ${\mathcal P}_{\mathop { \mathbf {PGL}}\nolimits _r}$ is denoted by $\mathcal G_{{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}}$ .
Proposition 3.22 Let E be a rank r locally free sheaf on ${\mathcal {X}}$ , and let ${\mathcal P}_E$ be the associated frame bundle of E. Then, the two gerbes are equivalent
where ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}={\mathcal P}_{E}\wedge ^{\mathop { \mathbf {GL}}\nolimits _r}{\mathop {\mathbf {PGL}}\nolimits _r}$ .
Proof By repeating the proof of [Reference Huybrechts and SchroeerHS03, Lemma 2.5], but with replacing the analytic topology with the étale topology, the conclusion of the proposition is immediate.
Definition 3.23 A $\boldsymbol {\mathop {\mathbf {PGL}}\nolimits _r}$ -Higgs bundle $({\mathcal P}_{\mathop { \mathbf {PGL}}\nolimits _r},\phi )$ consists of a principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundle ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ and a section $\phi $ of $\mathrm {ad}({\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r})\otimes K_{{\mathcal {X}}}$ , where $\mathrm {ad}({\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r})$ is the adjoint bundle of ${\mathcal P}_{\mathop { \mathbf {PGL}}\nolimits _r}$ . For a scheme T, a T -family of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles $({\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r,T},\phi _T)$ is a T-family of principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundles ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r,T}$ with a section of $\mathrm {ad}({\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r,T})\otimes \mathrm {pr_{{\mathcal {X}}}}^{*}K_{{\mathcal {X}}}$ .
In order to construct the moduli stack of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles, we introduce the following notion:
Definition 3.24 Suppose that $\alpha $ is a cohomology class in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mu _r)$ . Let k be an algebraically closed field containing ${\mathbb C}$ . A principal $\mathop { \mathbf {PGL}}\nolimits _{r}$ -bundle ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r,k}$ on ${\mathcal {X}}_k$ is said to have topological type $\alpha \in H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mu _r)$ if the $\mathop {\mathbf {SL}}\nolimits _r$ -lifting gerbe of ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r,k}$ in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}}_k,\mu _{r})$ is $\mathrm {pr_{{\mathcal {X}}}}^{*}\alpha $ , where $\mathrm {pr_{\mathcal {X}}} : {\mathcal {X}}_k={\mathcal {X}}\times _{\mathop {\mathrm {Spec}}\nolimits {{\mathbb C}}}\mathop {\mathrm {Spec}}\nolimits k\rightarrow {\mathcal {X}}$ is the natural projection. For a scheme T, a T-family of principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundles with topological type $\alpha $ is a principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundle ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r,T}$ on ${\mathcal {X}}_T$ , which restricts to every geometric fiber of ${\mathcal {X}}_T\rightarrow T$ is with topological type $\alpha $ .
The moduli stack $\mathcal M^{\alpha }_{\mathrm {Dol}}(\mathop {\mathbf {PGL}}\nolimits _r)$ of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ is defined by: for a test scheme T, $\mathcal M^{\alpha }_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {PGL}}\nolimits _r)(T)$ is the groupoid of T-families of $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles, in which the principal bundles with topological type $\alpha $ . Suppose that there is a principal $\mathop { \mathbf {PGL}}\nolimits _r$ -bundle ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ satisfying $\partial ([{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}])=\alpha $ . Consider the exact sequence:
The cohomology class in $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mathbb G_m)$ corresponding the $\mathop {\mathbf {GL}}\nolimits _r$ -lifting gerbe of ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ is the image of $\alpha $ in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mathbb G_m)$ (see the proof of [Reference MilneMil80, Proposition 2.7 in Chapter IV]).
Case I: Assume that the image of $\alpha $ in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mathbb G_m)$ is zero. Then, there is a locally free sheaf E on ${\mathcal {X}}$ such that the associated frame bundle is a $\mathop {\mathbf {GL}}\nolimits _r$ -lifting of ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ and $\delta ([\mathop {\mathrm {det}}\nolimits (E)^{\vee }])=\alpha $ . Moreover, by the Kummer sequence (23), we see that $\mathop {\mathrm {det}}\nolimits (E)$ is uniquely determined up to an rth power of some line bundle. We therefore have the proposition:
Proposition 3.25 Suppose that $\alpha $ is zero in $H^2({\mathcal {X}},\mathbb G_m)$ and that L is a line bundle with $\delta ([L])=-\alpha $ in the Kummer sequence (23). For every principal $\mathop {\mathbf {PGL}}\nolimits _r$ -bundle ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ with $\partial ([{{\mathcal P}_{\mathop { \mathbf {PGL}}\nolimits _r}}])=\alpha $ , there is a locally free sheaf E with $\mathop {\mathrm {det}}\nolimits (E)\simeq L$ , whose associated frame bundle ${\mathcal P}_E$ is a $\mathop {\mathbf {GL}}\nolimits _r$ -lifting of ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ .
Let $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }(\mathop {\mathbf {SL}}\nolimits _r)$ be the moduli stack of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with fixed determinant L. By Proposition 3.25, there is a surjective morphism of stacks
defined by
where T is any test scheme and ${\mathcal P}_{E_T}$ is the frame bundle associated with $E_T$ .
Proposition 3.26 The morphism (26) is a $\mathcal J_r$ -torsor, where $\mathcal J_r$ is the stack of $\mu _r$ -torsors on ${\mathcal {X}}$ .
Proof The proof is the same as [Reference LaszloLas97, Lemma 5.1].
Case II: Assume that $\alpha \in H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mu _r)$ is not zero in $H_{\mathop {\acute{\rm e}{\rm t}}\nolimits }^2({\mathcal {X}},\mathbb G_m)$ . The corresponding $\mu _r$ -gerbe is denoted by $p_{\alpha } : \mathcal G_{\alpha }\rightarrow {\mathcal {X}}$ , which is a Deligne–Mumford curve. By the universal property of $\mathcal G_{\alpha }$ , for any ${\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ with topological type $\alpha $ , $p_{\alpha }^{*}{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ has an $\mathop {\mathbf {SL}}\nolimits _r$ -lifting on $\mathcal G_{\alpha }$ , i.e., there exists a locally free sheaf E of rank r with $\mathop {\mathrm {det}}\nolimits (E)\simeq {\mathcal {O}}_{\mathcal G_{\alpha }}$ on $\mathcal G_{\alpha }$ such that the associated principal bundle is an $\mathop {\mathbf {SL}}\nolimits _r$ -lifting of $p_{\alpha }^{*}{\mathcal P}_{\mathop {\mathbf {PGL}}\nolimits _r}$ . The E is a twisted vector bundle. In what follows, we will give the definition of twisted vector bundles. For a quasicoherent sheaf F on $\mathcal G_{\alpha }$ , it admits an eigendecomposition
where $F_{\lambda }$ is the eigensheaf on $\mathcal G_{\alpha }$ with respect to the character $\lambda $ of $\mu _r$ (see [Reference LieblichLie08, Proposition 3.1.1.4]).
Definition 3.27 A quasicoherent sheaf F on $\mathcal G_{\alpha }$ is called a twisted quasicoherent sheaf if $F=F_{\bar 1}$ in the eigendecomposition (27). In particular, if the aforementioned F is a locally free sheaf, we say that F is a twisted vector bundle. A twisted Higgs bundle is a pair $(E,\phi )$ , where E is a twisted vector bundle and $\phi : E\rightarrow E\otimes K_{\mathcal G_{\alpha }}$ is a $\mu _r$ -equivariant morphism of ${\mathcal {O}}_{\mathcal G_{\alpha }}$ -modules.
Consider the moduli stack $\mathcal M_{\mathrm {Dol}}^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ of twisted Higgs bundles on $\mathcal G_{\alpha }$ , whose fiber over a test scheme T is the groupoid of T-families of rank r twisted Higgs bundles on $\mathcal G_{\alpha }$ . $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ is an open and closed substack of the moduli stack of rank r Higgs bundles on $\mathcal G_{\alpha }$ for the decomposition (27). For a modified Hilbert polynomial P, we can also consider the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ of rank r twisted Higgs bundles with modified Hilbert polynomial P, which is an open and closed substack of $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ on $\mathcal G_{\alpha }$ . If there is a polarization on $\mathcal G_{\alpha }$ , we can also introduce the notion of stability for twisted Higgs bundles as usual. The moduli stack of semistable twisted Higgs bundle with modified Hilbert polynomial P is denoted by $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{\alpha ,ss}(\mathop {\mathbf {GL}}\nolimits _r)$ .
Proposition 3.28 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ and ${\mathcal M}_{\mathop {\mathrm {Dol}}\nolimits ,P}^{\alpha }(\mathop {\mathbf {GL}}\nolimits _r)$ are algebraic stacks locally of finite type over ${\mathbb C}$ . Moreover, $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{\alpha ,ss}(\mathop {\mathbf {GL}}\nolimits _r)$ of semistable twisted Higgs bundles with modified Hilbert polynomial P is a quotient stack, whose good moduli space $M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{\alpha ,ss}(\mathop {\mathbf {GL}}\nolimits _r)$ is a quasiprojective scheme.
Proof Since “twisted,” “with fixed modified Hilbert polynomial,” and “semistable” are open conditions, the conclusion of the proposition is immediate by the counterparts in Section 3.1.
Definition 3.29 A twisted $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundle is a twisted Higgs bundle $(E,\phi )$ with $\mathop {\mathrm {det}}\nolimits (E)\simeq {\mathcal {O}}_{\mathcal G_{\alpha }}$ and $\mathop {\mathrm { tr}}\nolimits (\phi )=0$ .
The moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {SL}}\nolimits _r)$ of twisted $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is an open and closed substack of the moduli stack of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundle on $\mathcal G_{\alpha }$ . As Proposition 3.28, we have:
Proposition 3.30 $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {SL}}\nolimits _r)$ is an algebraic stack locally of finite type over ${\mathbb C}$ . Furthermore, $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{\alpha ,ss}(\mathop {\mathbf {SL}}\nolimits _r)$ of semistable twisted $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with modified Hilbert polynomial P is a quotient stack of finite type over ${\mathbb C}$ . Its good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{\alpha ,ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is a quasiprojective scheme.
For a twisted $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundle on $\mathcal G_{\alpha }$ , the associated $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundle is the pullback of a $\mathop { \mathbf {PGL}}\nolimits _r$ -Higgs bundle with topological data $\alpha $ on ${\mathcal {X}}$ . Then, we have a surjective morphism of algebraic stacks
Similar to Proposition 3.26, we also have:
Proposition 3.31 The morphism (28) is a $\mathcal J_r$ -torsor, where $\mathcal J_r$ is the stack of $\mu _r$ -torsors on ${\mathcal {X}}$ .
By [Reference LieblichLie09, Lemma 3.4] and Propositions 3.26 and 3.31, we have the following theorem.
Theorem 3.32 For any $\alpha \in H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mu _r)$ , the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha }(\mathop {\mathbf {PGL}}\nolimits _r)$ of $\mathop { \mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ is a locally finite type algebraic stack over ${\mathbb C}$ .
3.4 Application to the case of stacky curves
In this subsection, ${\mathcal {X}}$ is assumed to be a genus g hyperbolic stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . Fix a polarization $({\mathcal E},{\mathcal {O}}_X(1))$ on ${\mathcal {X}}$ . Since $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mathbb G_m)$ is trivial (see [Reference PomaPom13, Proposition 5.3]), every cohomology class of $H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mu _r)$ satisfies the assumption of Case I (see Section 3.3). Suppose that $\alpha \in H^2_{\mathop {\acute{\rm e}{\rm t}}\nolimits }({\mathcal {X}},\mu _r)$ is in the image of the $\partial $ in (25) and L is a line bundle on ${\mathcal {X}}$ such that $\delta ([L])=-\alpha $ in (23). Note that there are finitely many modified Hilbert polynomials if the rank and the determinant are fixed. Then, the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^s(\mathop { \mathbf {SL}}\nolimits _r)$ of stable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles with determinant L contains finitely many open-closed substacks indexed by the modified Hilbert polynomials. Therefore, $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits }^s(\mathop { \mathbf {SL}}\nolimits _r)$ admits good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits }^s(\mathop { \mathbf {SL}}\nolimits _r)$ , which is finite type over ${\mathbb C}$ . After rigidification, we get an action of the group $\Gamma $ of r-torsion points of $\mathop {\mathrm {Pic}}\nolimits ({\mathcal {X}})$ on the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits }^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ of stable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles (see Proposition 3.26). Specifically, the group $\Gamma $ acts on $M_{\mathop {\mathrm {Dol}}\nolimits }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ via the tensor product
We give the definition of moduli space of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ .
Definition 3.33 The moduli space of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ is defined to be the quotient stack
Remark 3.34 For a modified Hilbert polynomial P, the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ (resp. $M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ ) may have many open-closed substacks (resp. subschemes) indexed by K-classes in $K_0({\mathcal {X}})_{\mathbb Q}$ .
Suppose that the set of stacky points of ${\mathcal {X}}$ is $\{p_1,\ldots ,p_m\}$ and the corresponding stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ . For each $p_i$ , the residue gerbe $\iota _i : B\mu _{r_i}\rightarrow {\mathcal {X}}$ is a closed immersion. On the other hand, $K_0(B\mu _{r_i})$ is isomorphic to the representation ring $\mathbf {R}\mu _{r_i}={{\mathbb Z}[x]}/{(x^{r_i}-1)}$ where x represents the representation defined by the inclusion $\mu _{r_i}\hookrightarrow {\mathbb C}^{*}$ . The following proposition is well known (see [Reference Adem and RuanAR03, Example 5.9] or [Reference Marcolli and MathaiMM99, p. 563]).
Proposition 3.35 We have an isomorphism
Suppose that E is a locally free sheaf on ${\mathcal {X}}$ . If the K-class $[\iota _i^{*}E]=\sum _{k=0}^{r_i-1}m_{i,k}\cdot x^k$ for every i, then the image of $[E]$ under (29) is
According to the rational K-classes of line bundles on ${\mathcal {X}}$ , the Picard group $\mathop {\mathrm { Pic}}\nolimits ({\mathcal {X}})$ is the disjoint union:
Then, the line bundle L belongs to a unique $\mathop {\mathrm {Pic}}\nolimits ^{d,(i_1,\ldots ,i_m)}({\mathcal {X}})$ . Suppose $\xi =(r,d^{\prime },(m_{1,i})_{i=1}^{r_1-1},\ldots ,(m_{m,i})_{i=1}^{r_m-1})\in K_0({\mathcal {X}})_{\mathbb Q}$ and r, $d^{\prime }$ , $m_{1,i},\ldots ,m_{m,i}$ are all integers. We further assume that $\xi $ satisfies:
-
• $i_k$ is the remainder, when $\sum _{i=1}^{r_1-1}m_{k,i}$ divided by $r_k$ for every $1\leq k\leq m$ ;
-
• $d^{\prime } = d+\sum _{k=1}^m\sum _{i=1}^{r_k-1}m_{k,i}\frac {i}{r_k}-\sum _{k=1}^m i_k$ .
Consider the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ (with good moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ ) of stable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with K-class $\xi $ . We give the following definition.
Definition 3.36 The moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ of stable $\mathop {\mathbf {PGL}}\nolimits _r$ -Higgs bundles with topological type $\alpha $ and K-class $\xi $ is defined to be the quotient stack $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)=[M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{s}/\Gamma _0]$ , where $\Gamma _0$ is the group of r-torsion points of $\mathop {\mathrm { Pic}}\nolimits ^0(X)$ and the action $\Gamma _0$ on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ is given by
Remark 3.37 By the decomposition of $K_0({\mathcal {X}})_{\mathbb Q}$ (see, for example, [Reference Adem and RuanAR03, Example 5.9]), the subgroup of $\Gamma $ which preserves the K-class $\xi $ is the image of $\Gamma _0$ under the morphism $\pi ^{*}$ in (73) (see Section 5.3). Then, $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ is an open-closed substack of $M_{\mathop {\mathrm {Dol}}\nolimits }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ .
Remark 3.38 By the orbifold-parabolic correspondence, for a rational parabolic weight, the corresponding parabolic slope can also define a stability condition on the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }(\mathop {\mathbf {GL}}\nolimits _r)$ of Higgs bundles with K-class $\xi $ . In fact, this way supplies more abundant stability conditions than using modified slopes (see Proposition A.1 and Remark A.2). For stacky curve, we will use parabolic slopes to define stability hereafter.
By the standard infinitesimal deformation theory of Higgs bundles on stacky curves (see [Reference Kydonakis, Sun and ZhaoKSZ20, Proposition 3.2 and Corollaries 3.3 and 3.4]), we have following proposition.
Proposition 3.39 If $(E,\phi )$ is a stable ${\mathop {\mathbf {SL}}\nolimits _r}$ -Higgs bundle with K-class $\xi $ , the dimension of the tangent space of $M_{\mathrm {Dol},\xi }^{s}(\mathrm {SL_r})$ at $(E,\phi )$ is
Moreover, $M_{\mathrm {Dol,\xi }}^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ is smooth at $(E,\phi )$ .
4 Spectral curves and Hitchin morphisms
4.1 Spectral curves
Let $(E,\phi )$ be a Higgs bundle on a hyperbolic Deligne Mumford curve ${\mathcal {X}}$ . The characteristic polynomial of $\phi $ is $\mathrm {{det}}(\lambda -\phi )=\lambda ^r+a_1\lambda ^{r-1}+\cdots +a_r$ , where $\lambda $ is an indeterminate variable and $a_i=(-1)^i\mathrm {tr}(\wedge ^i\phi )$ for $1\leq i\leq r$ . It defines the so-called spectral curve associated with the Higgs bundle $(E,\phi )$ . More precisely, the spectral curve is the zero locus of the section
where $\psi : \mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})\rightarrow {\mathcal {X}}$ is the total space of $K_{\mathcal {X}}$ and $\tau $ is the tautological section of $\psi ^{*}K_{{\mathcal {X}}}$ . Since the spectral curve is only dependent on the coefficients of the characteristic polynomial, we can define a spectral curve ${\mathcal {X}}_{\boldsymbol a}$ for any element $\boldsymbol a=(a_1,\ldots ,a_r)\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}}, K_{{\mathcal {X}}}^i)$ . In general, a spectral curve is neither smooth nor integral. Nevertheless, under some mild conditions, for a general element $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the associated spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral (see Proposition 4.3). It is easy to check the following proposition.
Proposition 4.1 Suppose that $f : {\mathcal {X}}_{\boldsymbol a}\rightarrow {\mathcal {X}}$ is the projection. Then, $f_{*}({\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})\simeq \bigoplus _{i=0}^{r-1}K^{-i}_{{\mathcal {X}}}$ and the arithmetic genus of ${\mathcal {X}}_{\boldsymbol a}$ is $\sum _{i=1}^{r}\mathrm {dim}_{{{\mathbb C}}}H^0({\mathcal {X}},K^i_{{\mathcal {X}}})$ .
There is another method to construct spectral curves, which is used in [Reference Beauville, Narasimhan and RamananBNR89]. Recall that $\Psi : \mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})\rightarrow {\mathcal {X}}$ is the projective bundle associated with $K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ . Since $\Psi _{*}{\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1)= K_{{\mathcal {X}}}^{-1}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ , the section $(0,1)$ of $K_{{\mathcal {X}}}^{-1}\oplus {\mathcal {O}}_{{\mathcal {X}}}$ gives a section y of ${\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1)$ . Meanwhile, since $\Psi _{*}(\Psi ^{*}K_{{\mathcal {X}}}\otimes {\mathcal {O}}_{\mathbb {P}(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1))={\mathcal {O}}_{{\mathcal {X}}}\oplus K_{{\mathcal {X}}}$ , $\Psi _{*}(\Psi ^{*}K_{{\mathcal {X}}}\otimes {\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1))$ has a section $(1,0)$ . It gives a section x of $\Psi ^{*}K_{{\mathcal {X}}}\otimes {\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(1)$ . For $\boldsymbol a=(a_1,\ldots ,a_r)\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}}, K_{{\mathcal {X}}}^i)$ , there is a section
of $\Psi ^{*}K_{{\mathcal {X}}}^r\otimes {\mathcal {O}}_{\mathbb P(K_{{\mathcal {X}}}\oplus {\mathcal {O}}_{{\mathcal {X}}})}(r)$ . Note that the zero locus of x and y are $\mathbb P({\mathcal {O}}_{\mathcal {X}})$ and $\mathbb P(K_{\mathcal {X}})$ , respectively. Hence, the zero locus of section (32) is the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ associated with $\boldsymbol a$ .
Remark 4.2 There exist a stacky curve $\widehat {{\mathcal {X}}}$ and a morphism $\mathcal R : {\mathcal {X}}\rightarrow \widehat {{\mathcal {X}}}$ which is an H-gerbe on $\widehat {{\mathcal {X}}}$ for some finite group H (see Remark 2.2). Since $\mathcal R^{*}K_{\widehat {{\mathcal {X}}}}=K_{\mathcal {X}}$ , the spectral curves on ${\mathcal {X}}$ are H-gerbes on the corresponding spectral curves on $\widehat {{\mathcal {X}}}$ .
Proposition 4.3 Let ${\mathcal {X}}$ be a hyperbolic Deligne–Mumford curve, and let $r\geq 2$ be an integer. Suppose that $K_{\mathcal {X}}$ satisfies
Then, for a general element of $\bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the associated spectral curve is integral.
Proof Recall a basic fact: for a gerbe ${\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ , ${\mathcal {X}}_1$ is integral if and only if ${\mathcal {X}}_2$ is so. By Remark 4.2, we can assume that ${\mathcal {X}}$ is a stacky curve in the following discussion. Since ${\mathcal {X}}$ is a hyperbolic stacky curve, there is a smooth projective algebraic curve $\Sigma $ with an action of a finite group G such that ${\mathcal {X}}=[\Sigma /G]$ (see [Reference Behrend and NoohiBN06, Corollary 7.7]). Suppose that $g : \Sigma \rightarrow {\mathcal {X}}$ is the morphism defined by the trivial G-torsor on $\Sigma $ and the G-action. Then, g is a G-torsor over ${\mathcal {X}}$ . As before, let $\Psi : \mathbb P(K_{\mathcal {X}}\oplus {\mathcal {O}}_{{\mathcal {X}}})\rightarrow {\mathcal {X}}$ be the projective bundle of $K_{\mathcal {X}}\oplus {\mathcal {O}}_{\mathcal {X}}$ . Then, there is a Cartesian diagram:
Similar to the second method of the construction of a spectral, we use $x^{\prime }$ to denote the section of ${\Psi ^{\prime }}^{*}K_{\Sigma }\otimes {{\mathcal {O}}}_{\mathbb {P}(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })}(1)$ corresponding to the section $(1,0)$ of ${\mathcal {O}}_{\Sigma }\oplus K_{\Sigma }$ . And, let $y^{\prime }$ be the section of ${\mathcal {O}}_{\mathbb {P}(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })}(1)$ corresponding to the section $(0,1)$ of $K_{\Sigma }^{-1}\oplus {\mathcal {O}}_{\Sigma }$ . For any $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the associated spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is the zero locus of the section s defined by (32). Since $({g^{\prime }})^{*}x=x^{\prime }$ and $({g^{\prime }})^{*}y=y^{\prime }$ , the pullback section of s is
where $a_i^{\prime }=g^{*}a_i$ for all $1\leq i \leq r$ . The zero locus $\Sigma _{\boldsymbol a}$ of $s^{\prime }$ fits into the Cartesian diagram
where the vertical morphisms are closed immersions. Then, $\widehat g : \Sigma _{\boldsymbol a}\rightarrow {\mathcal {X}}_{\boldsymbol a}$ is a G-torsor and ${\mathcal {X}}_a=[\Sigma _{\boldsymbol a}/G]$ . Under the hypothesis of Proposition 4.3, we will show that the $\Sigma _{\boldsymbol a}$ is integral, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ . Consider the injective linear map of complex vector spaces
Let V be the vector subspace of $H^0(\mathbb P(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma }),\Psi ^{\prime *} K^r_{\Sigma }\otimes {\mathcal {O}}_{\mathbb P(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })}(r))$ generated by the section $x^{\prime \otimes r}$ and the image of (37). Note that the zero loci of $x^{\prime }$ and $y^{\prime }$ are disjoint. Since $H^0({\mathcal {X}},K^r_{\mathcal {X}})\neq 0$ , the base locus $\mathcal B$ of the linear system corresponding to V is codimension $2$ . Then, there is a morphism
where $\mathbb P(V^{\vee })$ is the projective space associated with the dual $V^{\vee }$ of V.
Claim The dimension of the image of $\Phi _V$ is 2. We only need to show that the dimension of the image of the restriction
is 2, where $\Pi =\mathbb P(K_{\Sigma }\oplus {\mathcal {O}}_{\Sigma })\setminus (\mathbb P(K_{\Sigma })\cup \mathbb P({\mathcal {O}}_{\Sigma }))$ . For any closed point $x\in \Sigma $ , the fiber of $\Psi ^{\prime }|_{\Pi } : \Pi \rightarrow \Sigma $ over x is
And, the restriction of the morphism $\Phi _V|_{\Pi }$ to $(\Psi ^{\prime }|_{\Pi })^{-1}(x)$ is
where all the $c_{\bullet \bullet }\in {\mathbb C}$ and $n_i=\mathrm {dim}_{\mathbb C}H^0({\mathcal {X}},K^i_{\mathcal {X}})$ for all $1\leq i\leq r$ . If the image $g(x)$ of x in ${\mathcal {X}}$ is not in the base locus $\widetilde {\mathcal B}$ of the complete linear system $|K^r_{\mathcal {X}}|$ , the coefficients of $z^r$ in (40) are not all zero. In this case, the image of the fiber $(\Psi ^{\prime }|_{\Pi })^{-1}(x)$ under the morphism $\Phi _V|_{\Pi }$ is dimension 1. On the other hand, if $K_{\mathcal {X}}$ satisfies the condition (33), there exist two closed points $y_1,y_2\in {\mathcal {X}}^o\setminus (\widetilde {\mathcal B}\cup \widehat {\mathcal B})$ and a section $a\in H^0({\mathcal {X}},K^k_{\mathcal {X}})$ such that
where ${\mathcal {X}}^o$ is the non-stacky locus of ${\mathcal {X}}$ and $\widehat {\mathcal B}$ is the base locus of the complete linear system $|K^k_{\mathcal {X}}|$ (if $r=k$ , then $\widehat {\mathcal B}=\widetilde {\mathcal B}$ ). Therefore, for any $x_1\in g^{-1}(y_1)$ and $x_2\in g^{-1}(y_2)$ , we have
It means that the images of the two fibers $(\Psi ^{\prime }|_{\Pi })^{-1}(x_1)$ and $(\Psi ^{\prime }|_{\Pi })^{-1}(x_2)$ do not coincide. Hence, the image of $\Phi _V$ has dimension 2.
By Theorem 3.3.1 in [Reference LazarsfeldLaz04], for a general element $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^{r}H^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the zero locus $\Sigma _{\boldsymbol a}$ of
is integral. Therefore, ${\mathcal {X}}_{\boldsymbol a}=[\Sigma _{\boldsymbol a}/G]$ is integral for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ .
Remark 4.4 In general, the conclusion of Proposition 4.3 does not hold if the condition (33) is not satisfied (see Example 4.21 in Section 4.3).
By the proof of Proposition 4.3, we get an immediate corollary.
Corollary 4.5 If a hyperbolic Deligne–Mumford curve ${\mathcal {X}}$ satisfies the conditions
then for a general element of $\bigoplus _{i=2}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the corresponding spectral curve is integral.
4.2 The Hitchin morphism
Let $(E_T,\phi _T)$ be a T-family of rank r Higgs bundles on ${\mathcal {X}}$ for a scheme T. Its characteristic polynomial is
where $a_i(T)=(-1)^i\wedge ^i\phi _T\in H^0({\mathcal {X}}_T,\mathrm {pr_{{\mathcal {X}}}}^{*}K^i_{\mathcal {X}})$ . The zero locus of (44) in the total space of $\mathrm {pr_{\mathcal {X}}}^{*}K_{{\mathcal {X}}}$ is a flat family of spectral curves over T. The affine space ${\mathbb H}(r,K_{\mathcal {X}})$ associated with vector space ${\bigoplus }_{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ parametrizes the universal family of spectral curves. Indeed, it represents the functor
since there is a canonical isomorphism
(see [Reference BrochardBro12, Corollary A.2.2]). We therefore have a morphism of stacks
which is called the Hitchin morphism. The following proposition describes the fibers of the Hitchin morphism.
Proposition 4.6 For a nonzero element $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{{\mathcal {X}}}^i)$ , let $\underline {\boldsymbol a}:\mathrm {{Spec}}({\mathbb C})\rightarrow \mathbb {H}(r,K_{{\mathcal {X}}})$ be the closed point defined by $\boldsymbol a$ . Consider the Cartesian diagram
If the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ associated with $\boldsymbol a$ is integral, then $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop { \mathbf {GL}}\nolimits _r)\times _{\mathbb {H}(r,K_{{\mathcal {X}}})}\mathop {\mathrm {Spec}}\nolimits ({\mathbb C})$ is the moduli stack of rank one torsion-free sheaves on ${\mathcal {X}}_{\boldsymbol a}$ with modified Hilbert polynomial P.
Since the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ of semistable Higgs bundles is an open substack of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ , we can restrict the Hitchin morphism $\mathcal H : \mathcal M_{\mathrm {Dol}}(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow \mathbb {H}(r,K_{{\mathcal {X}}})$ to $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ , which is also denoted by $\mathcal H$ . Let $\mathcal Q : \mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow M^{ss}_{\mathop {\mathrm { Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ be the good moduli space of $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ . By the universal property of ${M}^{ss}_{\mathop {\mathrm {Dol}}\nolimits ,P}(\mathop {\mathbf {GL}}\nolimits _r)$ , there exists a unique morphism $h : M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {GL}}\nolimits _r)\rightarrow \mathbb {H}(r,K_{{\mathcal {X}}})$ satisfying $\mathcal H=h\circ \mathcal Q$ . h is also called Hitchin morphism.
Theorem 4.7 If ${\mathcal {X}}$ is a hyperbolic stacky curve, then the Hitchin morphism h is proper.
Proof The proof of the theorem is given in Appendix B.
Restricting the Hitchin morphism h to $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ , we get the Hitchin morphism for the moduli space of semistable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ , where ${\mathbb H}^o(r,K_{\mathcal {X}})$ is the affine space associated with $\bigoplus _{i=2}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ . Let $\xi \in K_0({\mathcal {X}})_{\mathbb Q}$ be a K-class such that the modified Hilbert polynomial is P. The restriction of $h_{\mathop { \mathbf {SL}}\nolimits _r}$ to $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ is also denoted by $h_{\mathop {\mathbf {SL}}\nolimits _r}$ . Since $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ is an open and closed subscheme of $M_{\mathop {\mathrm {Dol}}\nolimits ,P}^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ , the restriction $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ is also proper. Since $h_{\mathop { \mathbf {SL}}\nolimits _r}$ is invariant under the action of the group $\Gamma _0$ of r-torsion points of $\mathop {\mathrm { Pic}}\nolimits ^0(X)$ , we have the morphism $h_{\mathop {\mathbf {PGL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ , which is called the Hitchin morphism of $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r)$ .
Corollary 4.8 If ${\mathcal {X}}$ is a hyperbolic stacky curve, $h_{\mathop {\mathbf {SL}}\nolimits _r}$ is proper. Furthermore, if $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ has no strictly semistable objects, then $h_{\mathop {\mathbf {PGL}}\nolimits _r}$ is also proper.
Proof Due to the properness of h, $h_{\mathop {\mathbf {SL}}\nolimits _r}$ is also proper. If $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{ss}$ has no strictly semistable objects, then $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)=M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{s}(\mathop {\mathbf {SL}}\nolimits _r)$ . By Proposition 10.1.6(v) in [Reference OlssonOls16], $h_{\mathop {\mathbf {PGL}}\nolimits _r}$ is proper.
4.3 Classification of spectral curves
In the following, ${\mathcal {X}}$ is a stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . The set of stacky points is $\{p_1,\ldots ,p_m\}$ and the stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ . Every smooth stacky curve can be obtained by applying root constructions (see [Reference CadmanCad07]) to its coarse moduli space. Recall Theorem 3.63 in [Reference BehrendBeh14].
Theorem 4.9 [Reference BehrendBeh14]
${\mathcal {X}}= \sqrt [r_1]{p_1}{\times }_{X}\sqrt [r_2]{p_2}{\times }_{X}\cdots {\times }_{X}\sqrt [r_m]{p_m}$ , where $\sqrt [r_k]p_k$ is the $r_k$ -th root stack associated with the divisor $p_k$ for every $1\leq k\leq m$ .
For each $1\leq k\leq m$ , let $(L_k,s_k)$ be the pair, which consists of the universal line bundle $L_k$ and section $s_k$ of $L_k$ on $\sqrt [r_k]{p_k}$ . And, let s be the section $\bigotimes _{k=1}^m\mathrm {pr}_k^{*}s_k$ of $\bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k$ , where $\mathrm {pr}_k : {\mathcal {X}}\rightarrow \sqrt [r_k]{p_k}$ is the projection to $\sqrt [r_k]p_k$ for every $1\leq k\leq m$ .
Corollary 4.10 [Reference Voight and Zureick-BrownVB22]
Under the hypothesis of Theorem 4.9, we have
where $D=\sum _{k=1}^mp_k$ .
Proof By Proposition 5.5.6 in [Reference Voight and Zureick-BrownVB22], we get the first formula. The second is obvious.
Lemma 4.11 If the natural number r satisfies $r\leq r_k$ for every $1\leq k \leq m$ , then for any element $\boldsymbol a=(a_1,\ldots ,a_r)\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}},K_{{\mathcal {X}}}^i)$ , there is an element
satisfying
Proof By Corollary 4.10, we have $K_{{\mathcal {X}}}^i=\pi ^{*}K_X^i\otimes \bigotimes _{k=1}^m\mathrm { pr}_k^{*}L_k^{(i-1)r_k+(r_k-i)}$ for any integer i. If i satisfies $i\leq r_k$ for every $1\leq k \leq m$ , then we have
is an isomorphism, where $\overline {a}\in H^0(X,K^i_X\otimes {\mathcal {O}}_X((i-1)D)$ .
Let t be the global section of the line bundle ${\mathcal {O}}_X(D)$ such that $\pi ^{*}t=\textstyle {\bigotimes _{k=1}^m\mathrm {pr}_k^{*}s_k^{\otimes r_k}}$ . Then, we have the following corollary.
Corollary 4.12 Assume that the natural number r satisfies $r\leq r_k$ for every $1\leq k \leq m$ . Then there is an injection of vector spaces
where $\widetilde a_i=\overline {a}_i\otimes t$ and $\overline {a}_i\in H^0(X,K^i_X\otimes {\mathcal {O}}_X((i-1)D))$ is the section associated with $a_i$ in Lemma 4.11, for every $1\leq i\leq r$ .
Proof The section t defines an injection $K_X^i\otimes {\mathcal {O}}_X((i-1)D)\hookrightarrow (K_X(D))^i\quad \text {for every } 1\leq i\leq r$ . Then, we get the injective linear map
Under the morphism (50), the image of $\overline {\boldsymbol a}=(\overline {a}_1,\ldots ,\overline {a}_r)\in \bigoplus _{i=1}^rH^0(X,K^i_X\otimes {\mathcal {O}}_X((i-1)D))$ is
where $\widetilde {a}_i=\overline {a}_i\otimes t$ for every $1\leq i\leq m$ .
Theorem 4.13 Suppose that the natural number r satisfies $2\leq r\leq r_i$ for all $1\leq i\leq m$ and $\boldsymbol a=(a_1,\ldots ,a_r)$ is an element of $\bigoplus _{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ . Then the coarse moduli space of ${\mathcal {X}}_{\boldsymbol a}$ is the curve $X_{\widetilde {\boldsymbol a}}$ , which is the zero locus of the section $\overline {\tau }^{\otimes r}+\varphi ^{*}\widetilde a_1\otimes \overline {\tau }^{\otimes (r-1)}+\cdots +\varphi ^{*}\widetilde a_r$ on the total space $\varphi : \mathop {\mathrm {Tot}}\nolimits (K_X(D))\rightarrow X$ , where $\widetilde {\boldsymbol a}=(\widetilde a_1,\ldots ,\widetilde a_r)$ is the image of $\boldsymbol a$ under the morphism (49) and $\overline {\tau }$ is the tautological section of $\varphi ^{*}K_X(D)$ .
Proof The section $s=\bigotimes _{k=1}^m\mathrm {pr}_k^{*}s_k$ of $\bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k$ defines an injection $K_{{\mathcal {X}}}\hookrightarrow \pi ^{*}K_{X}(D)$ . Let $\pi ^{\prime \prime \prime } : \mathop {\mathrm { Tot}}\nolimits (K_{\mathcal {X}})\rightarrow \mathop {\mathrm {Tot}}\nolimits (\pi ^{*}K_X(D))$ be the corresponding morphism between total spaces. In general, $\pi ^{\prime \prime \prime }$ is not injective. It satisfies the commutative diagram
On the other hand, there is a Cartesian diagram
Composing the diagrams (51) and (52), we get a new commutative diagram
where $\pi ^{\prime }=\pi ^{\prime \prime }\circ \pi ^{\prime \prime \prime }$ . The curve $X_{\widetilde {\boldsymbol a}}$ is the zero locus of the section $\overline {\tau }^{\otimes r}+\varphi ^{*}\widetilde {a}_1\otimes \overline {\tau }^{\otimes (r-1)}+\cdots +\varphi ^{*}\widetilde {a}_r$ on the total space $\mathop {\mathrm {Tot}}\nolimits (K_X(D))$ . And, the spectral curve ${\mathcal {X}}_{\alpha }$ is the zero locus of section $\tau ^{\otimes r}+\psi ^{*}a_1\otimes \tau ^{\otimes {(r-1)}}+\cdots +\psi ^{*}a_r$ , where $\tau $ is the tautological section of $\psi ^{*}K_{{\mathcal {X}}}$ . Since $(\pi ^{\prime })^{*}\overline {\tau }=\tau \otimes \psi ^{*}s$ , we have
We get the commutative diagram
In order to show that $X_{\widetilde \alpha }$ is the coarse moduli space of ${\mathcal {X}}_{\alpha }$ , we only need to check this locally. For each stacky point $p_i$ , there is an affine open subset $U_i=\text {Spec}(A_i)$ of X, such that $U_i$ contains only $p_i$ and $p_i =(f_i)$ as a divisor on $U_i$ for some $f_i\in A_i$ . In the following, we consider the commutative diagram
where
Since
(see [Reference OlssonOls16, Theorem 10.3.10(ii)]), we have
where the action of $\mu _{r_i}=\text {Spec}(\mathbb {C}[z]/(z^{r_i}-1))$ is defined by
Hence, the coarse moduli space of ${\mathcal {X}}_{\alpha }{\times }_X U_i$ is ${\mathop {\mathrm { Spec}}\nolimits }\left (\frac {A_i[ty]}{((ty)^r+ \overline {a}_1f_i(ty)^{r-1}+\overline {a}_2f_i(ty)^{r-2}+\cdots +\overline {a}_rf_i)}\right )$ .
Remark 4.14 Biswas–Majumder–Wong [Reference Biswas, Majumder and WongBMW13], Borne [Reference Beauville, Narasimhan and RamananBNR89], and Nasatyr–Steer [Reference Nasatyr and SteerNS95] established the orbifold-parabolic correspondence, i.e., there is a one-to-one correspondence between the Higgs bundles on stacky curves ${\mathcal {X}}$ and the strongly parabolic Higgs bundles (see [Reference Biswas, Majumder and WongBMW13, Section 3.1]) on its coarse moduli space X with marked points $\{p_1,\ldots ,p_m\}$ . This theorem explain the relationship between the corresponding spectral curves.
Suppose that j is a natural number. It can be written uniquely in the form $j=h_{jk}\cdot r_{k}-q_{jk}$ , where $h_{jk},q_{jk}\in {\mathbb Z}$ satisfy $0\leq q_{jk} <r_k$ for every $0\leq k\leq m$ . More precisely, we have
for every $0\leq k\leq m$ . Let $\widetilde h_{jk}=j-h_{jk}$ for every $1\leq k\leq m $ . Then, we have
We therefore have $K_{{\mathcal {X}}}^j=\textstyle {\pi ^{*}K^j_X\otimes \bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k^{j(r_{k}-1)}= \pi ^{*}K_X^j\otimes \bigotimes _{k=1}^m\mathrm {pr}_k^{*}L_k^{\widetilde h_{jk}\cdot r_k+q_{jk}}}$ . Then, the pushforward of $K^j_{\mathcal {X}}$ is
Hence, there is an isomorphism
Lemma 4.15 Suppose that the aforementioned ${\mathcal {X}}$ is hyperbolic with genus g and the natural number $r\geq 2$ . Furthermore, we assume that
If $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^r))\geq 2g$ and the condition (33) hold, then the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth for a general element $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{{\mathcal {X}}})$ .
Proof By the uniformization of Deligne–Mumford curves, there is a smooth complex projective algebraic curve $\Sigma $ with a finite group G action such that ${\mathcal {X}}=[\Sigma /G]$ . As before, $g : \Sigma \rightarrow {\mathcal {X}}$ is the étale covering defined by the trivial G-torsor and the G-action on $\Sigma $ . For a general element $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ , the spectral curve $\Sigma _{\boldsymbol a}$ defined by $(g^{*}a_1,\ldots ,g^{*}a_r)\in \bigoplus _{i=1}^rH^0(\Sigma ,K^i_{\Sigma })$ is integral (see the proof of Proposition 4.3). On the other hand, since $\pi _{*}(K_{{\mathcal {X}}}^r)\geq 2g$ , the linear system $\vert \pi _{*}(K_{\mathcal {X}}^r)\vert $ has no base points (see [Reference HartshorneHar77, Corollary 3.2]). There are two cases: $\pi _{*}(K_{\mathcal {X}}^r)={\mathcal {O}}_X$ and $\mathrm {dim}\vert \pi _{*}(K_{\mathcal {X}}^r)\vert \geq 1$ .
Case 1 $\pi _{*}(K_{\mathcal {X}}^r)={\mathcal {O}}_X$ . By (57), the $H^0({\mathcal {X}},K_{\mathcal {X}}^r)$ is a one-dimensional complex vector space generated by the section $\bigotimes _{k=1}^m\mathrm { pr_k}^{*}s_k^{\otimes q_{rk}}$ . By assumption, $g^{*}(\bigotimes _{k=1}^m\mathrm { pr_k}^{*}s_k^{\otimes q_{rk}})$ only has simple zeros. Therefore, for a general $\boldsymbol a\in \bigoplus _{i=1}^mH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve $\Sigma _{\boldsymbol a}$ is integral and smooth, by the Jacobian criterion (see [Reference MatsumuraMat86, Theorem 30.3(5)]). Thus, a general spectral curve ${\mathcal {X}}_{\boldsymbol a}=[\Sigma _{\boldsymbol a}/G]$ is integral and smooth.
Case 2 $\mathrm {dim}\vert \pi _{*}(K_{\mathcal {X}}^r)\vert \geq 1$ . In this case, a general element of $\vert \pi _{*}(K_{\mathcal {X}}^r)\vert $ is a reduced divisor, whose support is disjoint with the stacky locus $\{p_1,\ldots ,p_m\}$ . By (57) and the assumptions about $q_{rk}$ for all $1\leq k\leq m$ , for a general section $a_r\in H^0({\mathcal {X}},K_{\mathcal {X}}^r)$ , $g^{*}a_r$ only has simple zeros. As in Case 1, by Jacobian criterion, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , $\Sigma _{\boldsymbol a}$ is integral and smooth. Therefore, a general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is also integral and smooth.
Lemma 4.16 We assume that the aforementioned hyperbolic stacky curve ${\mathcal {X}}$ and the natural number r do not satisfy the condition (58). For that, we make the assumption:
If $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^{r-1}))\geq 2g$ and the condition (33) holds, then a general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth.
Proof As before, ${\mathcal {X}}=[\Sigma /G]$ and $g : \Sigma \rightarrow {\mathcal {X}}$ is the natural étale covering, where $\Sigma $ is a smooth complex projective algebraic curve with an finite group G action. And, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve $\Sigma _{\boldsymbol a}$ defined by $(g^{*}a_1,\ldots ,g^{*}a_r)\in \bigoplus _{i=1}^rH^0(\Sigma ,K^i_{\Sigma })$ is integral (see the proof of Proposition 4.3). Note that $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^r))\geq 2g$ . Hence, the linear system $\vert \pi _{*}(K_{\mathcal {X}}^r)\vert $ is base-point-free (see [Reference HartshorneHar77, Corollary 3.2]). By the proof of Lemma 4.15, for a general $a_r\in H^0({\mathcal {X}},K^r_{\mathcal {X}})$ , the multiple zeros of $g^{*}a_r$ are contained in the preimages of those stacky points $p_k$ for which $q_{rk}\geq 2$ . In order to show that the spectral curve $\Sigma _{\boldsymbol a}$ is smooth for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , we only need to prove that a general section $a_{r-1}\in H^0({\mathcal {X}},K^{r-1}_{\mathcal {X}})$ does not vanish at those stacky points $p_k$ for which $q_{rk}\geq 2$ by Jacobian criterion (see [Reference MatsumuraMat86, Theorem 30.3(5)] or [Reference Beauville, Narasimhan and RamananBNR89, Remark 3.5]). Since the linear system $\vert \pi _{*}(K_{\mathcal {X}}^{r-1})\vert $ is base-point-free, we have $\pi _{*}(K^{r-1}_{\mathcal {X}})={\mathcal {O}}_X$ or $\mathrm {dim}\vert \pi _{*}(K^{r-1}_{\mathcal {X}})\vert \geq 1$ .
Case 1 $\pi _{*}(K_{\mathcal {X}}^{r-1})={\mathcal {O}}_X$ . By (57), $H^0({\mathcal {X}},K_{\mathcal {X}}^{r-1})$ is a one-dimensional complex vector space generated by the section $\bigotimes _{k=1}^m\mathrm {pr_k}^{*}s_k^{\otimes q_{(r-1)k}}$ . The assumptions about $q_{rk}$ for all $1\leq k\leq m$ imply that the zero locus of $g^{*}(\bigotimes _{k=1}^m\mathrm {pr_k}^{*}s_k^{\otimes q_{(r-1)k}})$ does not intersect with the preimage of those stacky points $p_k$ for which $q_{rk}\geq 2$ . Therefore, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve $\Sigma _{\boldsymbol a}$ is integral and smooth. Then, a general spectral curve ${\mathcal {X}}_{\boldsymbol a}=[\Sigma _{\boldsymbol a}/G]$ is also integral and smooth.
Case 2 $\mathrm {dim}\vert \pi _{*}(K_{\mathcal {X}}^{r-1})\vert \geq 1$ . In this case, a general element of $\vert \pi _{*}(K_{\mathcal {X}}^{r-1})\vert $ is a reduced divisor, whose support is disjoint with the stacky locus $\{p_1,\ldots ,p_m\}$ . By (57) and the assumptions about $q_{rk}$ for all $1\leq k\leq m$ , for a general section $a_{r-1}\in H^0({\mathcal {X}},K_{\mathcal {X}}^{r-1})$ , $g^{*}a_{r-1}$ does not vanish at those points whose images are the stacky points $p_k$ for which $q_{rk}\geq 2$ . As Case 1, $\Sigma _{\boldsymbol a}$ is integral and smooth, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ . Then, a general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth.
Lemma 4.17 Suppose that the aforementioned hyperbolic stacky curve ${\mathcal {X}}$ and the natural number r do not satisfy the conditions (58) and (59). If the condition (33) holds, then a general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is singular.
Proof Recall that ${\mathcal {X}}=[\Sigma /G]$ and $g : \Sigma \rightarrow {\mathcal {X}}$ is the natural étale covering, where $\Sigma $ is a smooth complex projective algebraic curve with a finite group G action. For a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve $\Sigma _{\boldsymbol a}$ defined by $(g^{*}a_1,\ldots ,g^{*}a_r)\in \bigoplus _{i=1}^rH^0(\Sigma ,K^i_{\Sigma })$ is integral (see the proof of Proposition 4.3). We will show that for a general $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , $a_{r-1}$ vanishes at the multiple zeros of $a_r$ . Then, for a general $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , $\Sigma _{\boldsymbol a}$ is singular by Jacobian criterion (see [Reference MatsumuraMat86, Theorem 30.3(5)] or [Reference Beauville, Narasimhan and RamananBNR89, Remark 3.5]). If the conditions (58) and (59) do not hold, then we have
By (57), for a general $\boldsymbol a=(a_1,\ldots ,a_r)\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ , the closed points in the preimage of $p_k$ are multiple zeros of $g^{*}a_r$ and zeros of $g^{*}a_{r-1}$ . We complete the proof of the lemma.
Theorem 4.18 Suppose that ${\mathcal {X}}$ is a hyperbolic stacky curve of genus g. Let r be a natural number with $r\geq 2$ , and let ${\mathcal {X}}_{\boldsymbol a}$ be the spectral curve associated with $\boldsymbol a\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}},K_{{\mathcal {X}}}^i)$ .
-
(1) Assume that $\lceil \frac {r}{r_k}\rceil =\frac {r}{r_k}$ or $\lceil \frac {r}{r_k}\rceil =\frac {r+1}{r_k}$ for all $1\leq k\leq m$ . A general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth if one of the following conditions is satisfied:
-
(i) $g\geq 2$ ;
-
(ii) $g=1$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2$ ;
-
(iii) $g=0$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r+1$ ;
-
(iv) $g=0$ , $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r$ and $\mathrm {dim}_{{{\mathbb C}}}H^0({\mathcal {X}},K^i_{\mathcal {X}})\geq 2$ for some $1\leq i\leq r$ .
-
-
(2) Suppose that the assumption in $(1)$ does not hold. We make the following assumption: if $\lceil \frac {r}{r_k}\rceil \geq \frac {r+2}{r_k}$ for some $1\leq k\leq m$ , then $\lceil \frac {r-1}{r_k}\rceil =\frac {r-1}{r_k}$ . A general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth if any of the following conditions is satisfied:
-
(i) $g\geq 2$ ;
-
(ii) $g=1$ and $\sum _{k=1}^m(r-1-\lceil \frac {r-1}{r_k}\rceil )\geq 2$ ;
-
(iii) $g=0$ , $\sum _{k=1}^m(r-1-\lceil \frac {r-1}{r_k}\rceil )\geq 2r-2$ and $K_{\mathcal {X}}$ satisfies (33).
-
-
(3) If $\lceil \frac {r}{r_k}\rceil \geq \frac {r+2}{r_k}$ and $\lceil \frac {r-1}{r_k}\rceil \geq \frac {r}{r_k}$ for some $1\leq k\leq m$ , then the general spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and singular if one of the following conditions occurs:
Proof (1). By (54), the assumption: $\lceil \frac {r}{r_k}\rceil =\frac {r}{r_k}\quad \text {or}\quad \lceil \frac {r}{r_k}\rceil =\frac {r+1}{r_k}\quad \text {for all } 1\leq k\leq m$ , is equivalent to the condition (58). And, by (56), we have $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^r))=(2g-2)r+\sum _{k=1}^m\widetilde h_{rk}$ . On the other hand, by the orbifold Riemann–Roch formula (see [Reference Abramovich, Graber and VistoliAGA08, Theorem 7.21]) and Serre duality, we get
By some elementary computations, we can show that if one of the conditions (i)–(iv) is satisfied, then $\mathop {\mathrm {deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^r))\geq 2g$ and $\mathrm {dim}_{{\mathbb C}}H^0({\mathcal {X}},K_{\mathcal {X}}^r)\geq 2$ . Hence, a general spectral curve is integral and smooth by Lemma 4.15.
(2). By (54), the assumption: if $\lceil \frac {r}{r_k}\rceil \geq \frac {r+2}{r_k}$ for some $1\leq k\leq m$ , then $\lceil \frac {r-1}{r_k}\rceil =\frac {r-1}{r_k}$ is equivalent to the condition (59). Moreover, the assumption of (2) implies $r\geq 3$ . Then, by $\mathop {\mathrm { deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^{r-1}))=(2g-2)(r-1)+\sum _{k=1}^m\widetilde h_{(r-1)k}$ (see (56)) and $\sum _{k=1}^m\widetilde h_{rk}\geq \sum _{k=1}^m\widetilde h_{(r-1)k}$ , we have that: if either one of the conditions (i)–(iii) holds, then $\mathop {\mathrm { deg}}\nolimits (\pi _{*}(K_{\mathcal {X}}^{r-1}))\geq 2g$ and the condition (33) holds. By Lemma 4.16, a general spectral curve is integral and smooth.
(3). As the above discussions, it is easy to check that the assumption of (3) satisfies the hypothesis of Lemma 4.17. The conclusion is immediately obtained.
Corollary 4.19 With the same hypothesis as Theorem 4.18, we have:
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(1) Under the assumption of $(1)$ in Theorem 4.18, for a general $\boldsymbol a\in \bigoplus _{i=2}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth if one of the following conditions is satisfied:
-
(i) $g\geq 2$ ;
-
(ii) $g=1$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2$ ;
-
(iii) $g=0$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r+1$ ;
-
(iv) $g=0$ , $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r$ and $\mathrm {dim}_{{{\mathbb C}}}H^0({\mathcal {X}},K^k_{\mathcal {X}})\geq 2$ for some $2\leq k\leq r$ .
-
-
(2) Under the assumption of $(2)$ in Theorem 4.18, for a general $\boldsymbol a\in \bigoplus _{i=2}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and smooth if any of the following conditions is satisfied:
-
(i) $g\geq 2$ ;
-
(ii) $g=1$ and $\sum _{k=1}^m(r-1-\lceil \frac {r-1}{r_k}\rceil )\geq 2$ ;
-
(iii) $g=0$ , $\sum _{k=1}^m(r-1-\lceil \frac {r-1}{r_k}\rceil )\geq 2r-2$ and $K_{\mathcal {X}}$ satisfies (43).
-
-
(3) Under the assumption of $(3)$ in Theorem 4.18, for a general $\boldsymbol a\in \bigoplus _{i=2}^rH^0({\mathcal {X}},K^i_{\mathcal {X}})$ , the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is integral and singular if one of the following conditions occurs:
Lemma 4.20 Suppose that $f : {\mathcal {X}}_{\boldsymbol a}\rightarrow \mathcal {X}$ is the projection from the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ to ${\mathcal {X}}$ . Under the assumptions of Theorem 4.18 (resp. Corollary 4.19), which ensure a general spectral curve is smooth, for a general ${\mathcal {X}}_{\boldsymbol a}$ , the stacky points of ${\mathcal {X}}_{\boldsymbol a}$ are contained in $f^{-1}(\{p_1,\ldots ,p_m\}\setminus \Omega )$ , where $\Omega $ consists of these stacky points $p_k\in \{p_1,\ldots ,p_m\}$ satisfying
Moreover, for any $p_k\in \{p_1,\ldots ,p_m\}\setminus \Omega $ , there is a unique stacky point $\widetilde p_k$ in $f^{-1}(p_k)$ with stabilizer group $\mu _{r_k}$ .
Proof Under these assumptions (which ensure a general spectral curve is smooth), $\mathop {\mathrm { deg}}\nolimits (\pi _{*}(K^r_{{\mathcal {X}}}))\geq 2$ . Therefore, the linear system $|\pi _{*}(K_{\mathcal {X}}^r)|$ is base-point-free (see [Reference HartshorneHar77, Corollary 3.2]). Then, $\pi _{*}(K_{{\mathcal {X}}}^r)={\mathcal {O}}_{{\mathcal {X}}}$ or $\mathop {\mathrm {dim}}\nolimits |\pi _{*}(K_{\mathcal {X}}^r)|\geq 1$ . By (57), the general section $a_r\in H^0({\mathcal {X}},K^r_{{\mathcal {X}}})$ does not vanish at any stacky point in $\Omega $ . If ${\mathcal {X}}$ is be viewed as the zero locus of $\mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})$ , then the set of the stacky points of $\mathop {\mathrm {Tot}}\nolimits ({\mathcal {X}})$ is $\{p_1,\ldots ,p_m\}$ with stabilizer groups $\mu _{r_1},\ldots ,\mu _{r_m}$ . We complete the proof.
Example 4.21 The condition (33) is an indispensable hypothesis for Theorem 4.18. For example, let $\mathbb E$ be an elliptic curve, and let p be a closed point of $\mathbb E$ . Consider the stacky curve $\mathbb E_5=\sqrt [5]{p}$ . The projection from $\mathbb E_5$ to $\mathbb E$ is denoted by $\pi : \mathbb E_5\rightarrow \mathbb E$ . The canonical line bundle of $\mathbb E_5$ is ${\mathcal {O}}_{\mathbb E_5}(\frac {4}{5}p)$ . Its degree $\mathop {\mathrm {deg}}\nolimits (K_{\mathbb E_5})$ is $\frac {4}{5}$ . So, it is a hyperbolic stacky curve. It is easy to check that
Then, $\mathrm {dim}_{{\mathbb C}}H^0(\mathbb E_5,K_{\mathbb E_5})=1$ and $\mathrm {dim}_{{\mathbb C}}H^0(\mathbb E_5,K^2_{\mathbb E_5})=1$ . Hence, we have
where $\tau _1$ is the universal section of ${\mathcal {O}}_{\mathbb E_5}(\frac {1}{5}p)$ . For a general $\boldsymbol a=(a\tau _1^{\otimes 4},b\tau _1^{\otimes 8})\in H^0(\mathbb E_5,K_{\mathbb E_5})\bigoplus H^0(\mathbb E_5,K^2_{\mathbb E_5})$ , the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is the zero locus of the section
where $a,b\in {\mathbb C}$ . The section (61) can be represented as a product of two sections
Hence, a general spectral curve is not irreducible.
Example 4.22 We will construct an example satisfying the last conclusion of Theorem 4.18. Taking four distinct points $\{p_1,p_2,p_3,p_4\}$ on the projective line $\mathbb P^1$ , we construct a stacky curve $\mathbb P^1_{4,2,2,2}$ as follows:
The canonical line bundle $K_{\mathbb P^1_{4,2,2,2}}=\pi ^{*}K_{\mathbb P^1}\otimes {\mathcal {O}}_{\mathbb P^1_{4,2,2,2}}(\frac {3}{4}p_1+\frac {1}{2}p_2+\frac {1}{2}p_3+\frac {1}{2}p_4)$ , where $\pi : \mathbb P^1_{4,2,2,2}\rightarrow \mathbb P^1$ is the coarse moduli space. And, the degree of $K_{\mathbb P^1_{4,2,2,2}}$ is $\frac {1}{4}$ . Hence, it is a hyperbolic stacky curve. Since $\mathrm {dim}_{{\mathbb C}}H^0(\mathbb P^1_{4,2,2,2},K_{\mathbb P^1_{4,2,2,2}}^6)\geq 2$ , the condition (33) holds. Suppose that $\tau _1$ , $\tau _2$ , $\tau _3$ , and $\tau _4$ are the sections of ${\mathcal {O}}_{\mathbb P^1_{4,2,2,2}}(\frac {1}{4}p_1)$ , ${\mathcal {O}}_{\mathbb P^1_{4,2,2,2}}(\frac {1}{2}p_2)$ , ${\mathcal {O}}_{\mathbb P^1_{4,2,2,2}}(\frac {1}{2}p_3)$ , and ${\mathcal {O}}_{\mathbb P^1_{4,2,2,2}}(\frac {1}{2}p_4)$ , respectively, such that they are the pullback sections of the universal sections on the corresponding root stacks. By Lemma 4.11, any section of $K^6_{\mathbb P^1_{4,2,2,2}}$ can be represented by
Let $\psi : \mathop {\mathrm {Tot}}\nolimits (K_{\mathbb P^1_{4,2,2,2}})\rightarrow \mathbb P^1_{4,2,2,2}$ be the projection from the total space of $K_{\mathbb P^1_{4,2,2,2}}$ to $\mathbb P^1_{4,2,2,2}$ . For a general element $\boldsymbol a$ of $\bigoplus _{i=1}^6H^0(\mathbb P^1_{4,2,2,2},K^i_{\mathbb P^1_{4,2,2,2}})$ , the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ is the zero locus of the section
where $\tau $ is the tautological section of $\psi ^{*}K_{\mathbb P^1_{4,2,2,2}}$ . By the GAGA for Deligne–Mumford curves (see [Reference Behrend and NoohiBN06]), we can assume that $\mathbb P^1_{4,2,2,2}$ is equipped with complex analytic topology. Then, there is a unit disk $\mathbb D\subset \mathbb P^1$ around $p_1$ such that $\pi : \mathbb P^1_{4,2,2,2}\rightarrow \mathbb P^1$ restricting to $\mathbb D$ is isomorphic to $\pi _{\mathbb D} : [\mathbb D/\mu _4]\longrightarrow \mathbb D$ , where the action of $\mu _4$ on $\mathbb D$ is multiplication and the morphism $\pi _{\mathbb D}$ is induced by the morphism $q : \mathbb D\longrightarrow \mathbb D,\ z\longmapsto z^4$ .
Consider the commutative diagram
where $g_{\mathbb D}$ is the natural projection. Pulling back the spectral curve defined by (62) along $g_{\mathbb D} : \mathbb D\rightarrow [\mathbb D/\mu _4]$ , we get
where $\widehat a_2(z)$ , $\widehat a_4(z)$ , and $\widehat a_6(z)$ are holomorphic functions on $\mathbb D$ . It is easy to check that $(0,0)$ is a singular point of (64).
5 Norm maps
In this section, we systematically study the norm theory on Deligne–Mumford stacks. As an application, we apply the general theory to the case of stacky curves which plays a central role in studying the Hitchin fiber of the moduli space of $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles.
5.1 Norms of invertible sheaves on Deligne–Mumford stacks
Let ${\mathcal {X}}$ be a Deligne–Mumford stack, and let $\mathcal A$ be a commutative ${\mathcal {O}}_{{\mathcal {X}}}$ -algebra with unit. Then, $\mathcal A$ is canonically identified with an ${\mathcal {O}}_{{\mathcal {X}}}$ -subalgebra of $\mathscr {H}om_{{\mathcal {O}}_{\mathcal {X}}}(\mathcal A,\mathcal A)$ . In fact, for an object $(T\rightarrow {\mathcal {X}})$ in ${{\mathcal {X}}}_{\acute{\rm e}{\rm t}}$ , a section $s\in \mathcal A(T\rightarrow {\mathcal {X}})$ defines a morphism of ${\mathcal {O}}_{T}$ -modules $\mathcal A|_T\rightarrow \mathcal A|_T$ by multiplication. If $\mathcal A$ is a locally free ${\mathcal {O}}_{{\mathcal {X}}}$ -module of finite rank, then there is a morphism $\mathrm { det}:\mathscr {H}om_{{\mathcal {O}}_{{\mathcal {X}}}}(\mathcal A,\mathcal A)\rightarrow {\mathcal {O}}_{{\mathcal {X}}}$ defined by
The composition $\mathcal A\hookrightarrow \mathscr {H}om_{{\mathcal {O}}_{{\mathcal {X}}}}(\mathcal A,\mathcal A)\overset {\mathrm {det}}{\longrightarrow } {\mathcal {O}}_{{\mathcal {X}}}$ is denoted by $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}$ . Obviously, $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}$ is a morphism of sheaves of multiplicative monoids. Following [Reference GrothendieckEGA2, Section 6.5], it is easy to verify the following proposition.
Proposition 5.1 For an étale morphism $T\rightarrow {\mathcal {X}}$ , we have:
-
(i) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(s_1\cdot s_2)=\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(s_1)\cdot \mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(s_2)$ , for $s_1, s_2\in \mathcal A(T\rightarrow {\mathcal {X}})$ ;
-
(ii) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(1_{\mathcal A})=1$ ;
-
(iii) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(t\cdot 1_{\mathcal A})=t^n$ if $t\in {\mathcal {O}}_{{\mathcal {X}}}(T\rightarrow {\mathcal {X}})$ and the rank of $\mathcal A$ is n.
Therefore, $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}$ induces a morphism of sheaves of abelian groups
where $\mathcal A^{*}$ is the sheaf of invertible elements of $\mathcal A$ .
Definition 5.2 An $\boldsymbol {{\mathcal A}}$ -invertible sheaf L on ${\mathcal {X}}$ is an $\mathcal A$ -module on ${\mathcal {X}}_{\mathop {\acute{\rm e}{\rm t}}\nolimits }$ whose restriction $L|_U$ to some étale covering $U\rightarrow {\mathcal {X}}$ is isomorphic to $\mathcal A|_U$ as an $\mathcal A|_U$ -module.
We will introduce the notion of norm of an $\mathcal A$ -invertible sheaf L. For the notations used in the following, we refer the reader to the section on Notations and conventions. Since L is a coherent sheaf on ${\mathcal {X}}$ , there is an object $(\mathcal A|_U,\sigma )$ in $Des(U/{\mathcal {X}})$ representing L for an étale covering $U\rightarrow {\mathcal {X}}$ . Then the morphism $\widetilde \sigma =\phi _2\circ \sigma \circ \phi _1^{-1} : \mathcal A|_{U[1]}\rightarrow \mathcal A|_{U[1]}$ is an isomorphism of $\mathcal A|_{U[1]}$ -modules, where $\phi _i : \mathrm {pr}_i^{*}(\mathcal A|_U)\rightarrow \mathcal A|_{U[1]}$ are the natural isomorphisms of ${\mathcal {O}}_{U[1]}$ -algebras, for $i=1,2$ . Let a be the image of the unit $1\in \mathcal A^{*}(U[1]\rightarrow {\mathcal {X}})$ under the morphism $\widetilde \sigma $ . On the other hand, there are three isomorphisms of $\mathcal A|_{U[2]}$ -modules
where $\phi _{12} : \mathrm {pr}^{*}_{12}(\mathcal A|_{U[1]})\rightarrow \mathcal A|_{U[2]}$ , $\phi _{23} : \mathrm { pr}^{*}_{23}(\mathcal A|_{U[1]})\rightarrow \mathcal A|_{U[2]}$ , and $\phi _{13} : \mathrm {pr}^{*}_{13}(\mathcal A|_{U[1]})\rightarrow \mathcal A|_{U[2]}$ are three natural isomorphisms of ${\mathcal {O}}_{U[2]}$ -algebras. It is easy to check that the cocycle condition: $\widetilde \sigma _{23}\circ \widetilde \sigma _{12}=\widetilde \sigma _{13}$ is satisfied. Then, we have $\phi _{23}(\mathrm {pr_{23}}^{*}a)\cdot \phi _{12}(\mathrm {pr_{12}}^{*}a)=\phi _{13}(\mathrm {pr_{13}}^{*}a)$ in $\mathcal A^{*}(U[2]\rightarrow {\mathcal {X}})$ . Since $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}$ is a morphism of sheaves of abelian groups, we have
in ${\mathcal {O}}^{*}_{{\mathcal {X}}}(U[2]\rightarrow {\mathcal {X}})$ by Proposition 5.1. Therefore, $({\mathcal {O}}_U,\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(a))$ is an object of $\mathcal Des(U/{\mathcal {X}})$ which defines a line bundle $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L)$ on ${\mathcal {X}}$ .
Definition 5.3 For an $\mathcal A$ -invertible sheaf L, the line bundle $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L)$ is called the norm of L.
We summarize some basic properties of the norms of $\mathcal A$ -invertible sheaves.
Proposition 5.4 The norms of $\mathcal A$ -invertible sheaves satisfy the following properties (up to a canonical isomorphism):
-
(i) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L_1\otimes _{\mathcal A}L_2)=\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L_1)\otimes _{{\mathcal {O}}_{{\mathcal {X}}}}\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L_2)$ , for any two $\mathcal A$ -invertible sheaves $L_1$ and $L_2$ on ${\mathcal {X}}$ ;
-
(ii) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(\mathcal A)={\mathcal {O}}_{{\mathcal {X}}}$ ;
-
(iii) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L^{-1})=\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L)^{-1}$ , for an $\mathcal A$ -invertible sheaf L on ${\mathcal {X}}$ ;
-
(iv) $\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L\otimes _{{\mathcal {O}}_{{\mathcal {X}}}}\mathcal A)=\mathrm {N}_{\mathcal A/{\mathcal {O}}_{{\mathcal {X}}}}(L)^n$ , for an ${\mathcal {O}}_{{\mathcal {X}}}$ -invertible sheaf L on ${\mathcal {X}}$ .
Proof By Proposition 5.1 and the definition of norm, the proposition is immediate.
5.2 Norm maps of finite morphisms of Deligne–Mumford stacks
Suppose that $f : {\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ is a finite morphism of Deligne–Mumford stacks and that $f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}$ is a locally free sheaf of rank n. Then, for any invertible sheaf L on ${\mathcal {X}}_1$ , the pushforward $f_{*}L$ is a $f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}$ -invertible sheaf. In fact, for an étale covering $U\rightarrow {\mathcal {X}}_2$ , there is a Cartesian diagram
${f_U}_{*}(L|_{U\times _{{\mathcal {X}}_2}{\mathcal {X}}_1})$ is an ${f_U}_{*}({\mathcal {O}}_{U\times _{{\mathcal {X}}_2}{\mathcal {X}}_1})$ -invertible sheaf on U (see [Reference GrothendieckEGA2, Proposition 6.I.I2.I]). Then, we can introduce the notion of the norm map of f.
Definition 5.5 The norm map $\mathrm {Nm}_f$ of f is $\mathrm {Nm}_f : \mathrm {Pic}({\mathcal {X}}_1)\rightarrow \mathrm { Pic}({\mathcal {X}}_2),\quad L\mapsto \mathrm {N}_{f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}/{\mathcal {O}}_{{\mathcal {X}}_2}}(f_{*}L)$ .
Proposition 5.6 The norm map $\mathrm {Nm}_f$ satisfies the following properties:
-
(i) $\mathrm {Nm}_{f}(L_1\otimes L_2)=\mathrm {Nm}_{f}(L_1)\otimes \mathrm {Nm}_{f}(L_2)$ , for any two line bundles $L_1$ and $L_2$ on ${\mathcal {X}}_1$ .
-
(ii) $\mathrm {Nm}_{f}({\mathcal {O}}_{{\mathcal {X}}_1})={\mathcal {O}}_{{\mathcal {X}}_2}$ .
-
(iii) $\mathrm {Nm}_{f}(L^{-1})=\mathrm {Nm}_{f}(L)^{-1}$ , for a line bundle L on ${\mathcal {X}}_1$ .
-
(iv) $\mathrm {Nm}_{f}(f^{*}L)=\mathrm {Nm}_{f}(L)^n$ , for a line bundle L on ${\mathcal {X}}_2$ .
-
(v) For a morphism of line bundles $\alpha : L_1\rightarrow L_2$ on ${\mathcal {X}}_1$ , there is a morphism of line bundles $\mathrm {Nm}_f(\alpha ) : \mathrm {Nm}_f(L_1)\rightarrow \mathrm {Nm}_f(L_2)$ . And, it satisfies:
-
• If there is another morphism of line bundles $\beta : L_2\rightarrow L_3$ , then we have $\mathrm {Nm}_f(\beta )\circ \mathrm {Nm}_f(\alpha )=\mathrm {Nm}_f(\beta \circ \alpha )$ .
-
• For two morphism of line bundles $\alpha _1 : L_1\rightarrow L_2$ and $\alpha _2 : L_3\rightarrow L_4$ , we have $\mathrm {Nm}_f(\alpha _1)\otimes \mathrm {Nm}_f(\alpha _2)=\mathrm { Nm}_f(\alpha _1\otimes \alpha _2)$ .
-
Proof By Proposition 5.4, the conclusions of this proposition are immediate.
Remark 5.7 In Proposition 5.6 $(v)$ , if $L_1={\mathcal {O}}_{{\mathcal {X}}_1}$ , we obtain a canonical map
for any line bundle L on ${\mathcal {X}}_1$ .
Proposition 5.8 Suppose that $f : {\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ is a finite morphism of Deligne–Mumford stacks such that $f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}$ is a rank n locally free sheaf. For a morphism of Deligne–Mumford stacks $g : \mathcal Y_2\rightarrow {\mathcal {X}}_2$ and the Cartesian diagram
we have $\mathrm {Nm}_{f^{\prime }}({g^{\prime }}^{*}L)=g^{*}\mathrm {Nm}_f(L)$ for any line bundle L on ${\mathcal {X}}_1$ .
Proof Using descent theory, we can prove this proposition following the proof of the counterpart in [Reference GrothendieckEGA2].
Proposition 5.9 Let $f : {\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ be a finite morphism of Deligne–Mumford stacks, and let L be a line bundle on ${\mathcal {X}}_1$ . Assume that $f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}$ is a locally free sheaf of rank n. Then, we have
Proof There exists an étale covering $U_2\rightarrow {\mathcal {X}}_2$ such that $L|_{U_1}={\mathcal {O}}_{U_1}$ where $U_1=U_2\times _{{\mathcal {X}}_2}{\mathcal {X}}_1$ . Hence, there exists $a\in {\mathcal {O}}^{*}_{U_1}(U_1)$ such that L is defined by the object $({\mathcal {O}}_{U_1}, a)$ of $Des(U_1/{\mathcal {X}}_1)$ . Consider the commutative diagram
in which every square is Cartesian. The pushforward $f_{*}L$ is represented by the object $\big (f_{1*}{\mathcal {O}}_{U_1}, f_{2*}(a\cdot \mathrm {id})\big )$ of $Des(U_2/{\mathcal {X}}_2)$ where $f_{2*}(a\cdot \mathrm {id})$ is identified with the composition
Therefore, $\mathrm {det}(f_{*}L)$ is defined by the object $\big (\mathrm {det}(f_{1*}{\mathcal {O}}_{U_1}), \mathrm { det}(f_{2*}(a\cdot \mathrm {id}))\big )$ of $Des(U_2/{\mathcal {X}}_2)$ , where $\mathrm {det}(f_{2*}(a\cdot \mathrm {id}))$ is the composition
In addition, the dual $\mathrm {det}(f_{*}{\mathcal {O}}_{{\mathcal {X}}_1})^{-1}$ of $f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}$ is represented by the object $\big (\mathrm {det}(f_{1*}{\mathcal {O}}_{U_1})^{-1}, \mathrm {id})$ of $Des(U_2/{\mathcal {X}}_2\big )$ , where $\mathrm {id}$ denotes the composition
Therefore, the line bundle $\mathrm {det}(f_{*}L)\otimes \mathrm {det}(f_{*}{\mathcal {O}}_{{\mathcal {X}}_1})^{-1}$ is represented by the object $\big ({\mathcal {O}}_{U_2}, \mathrm {N}_{f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}/{\mathcal {O}}_{{\mathcal {X}}_2}}(a))$ of $Des(U_2/{\mathcal {X}}_2\big )$ . By the definition of $\mathrm {Nm}_f(L)$ , we have $\mathrm {Nm}_f(L)=\mathrm {det}(f_{*}L)\otimes \mathrm {det}(f_{*}{\mathcal {O}}_{{\mathcal {X}}_1})^{-1}$ .
In the following, for simplicity, we always assume that ${\mathcal {X}}$ is a smooth irreducible Deligne–Mumford stack of finite type over ${\mathbb C}$ .
Definition 5.10
-
(i) A prime divisor on ${\mathcal {X}}$ is a codimension one closed integral substack of ${\mathcal {X}}$ .
-
(ii) A Weil divisor is an element of the free abelian group $\mathrm {Div}({\mathcal {X}})$ generated by the prime divisors on ${\mathcal {X}}$
-
(iii) Let $\mathcal D={\sum }_in_i\mathcal Y_i$ be a Weil divisor, where the $\mathcal Y_i$ are prime divisors and the $n_i$ are integers. If all the coefficients $n_i\geq 0$ , then $\mathcal D$ is said to be effective.
-
(iv) A rational function on ${\mathcal {X}}$ is a morphism ${\mathcal U}\rightarrow \mathbb {A}^1_{{\mathbb C}}$ from a nonempty open substack to the affine line. The rational functions of ${\mathcal {X}}$ form a field $k({\mathcal {X}})$ , which is called the quotient field of ${\mathcal {X}}$ (see [Reference VistoliVis89, Definition 3.4]). By [Reference VistoliVis89, Lemma 3.3], there is a morphism of abelian groups $\boldsymbol {\partial _{{\mathcal {X}}}} : k^{*}({\mathcal {X}})\longrightarrow \mathrm {Div}({\mathcal {X}})$ , where $k^{*}({\mathcal {X}})$ is the group of nonzero elements of $k({\mathcal {X}})$ . By convention, we use the notation div to denote $\boldsymbol {\partial _{{\mathcal {X}}}}$ . A Weil divisor is said to be a principal divisor if it is in the image of div.
-
(v) Two Weil divisors $\mathcal D, \mathcal D^{\prime }\in \mathrm {Div}({\mathcal {X}})$ are linearly equivalent if $\mathcal D-\mathcal D^{\prime }$ is in the image of div.
-
(vi) The cokernel of div is called the divisor class group ${\mathrm {Cl}}({\mathcal {X}})$ of ${\mathcal {X}}$ .
Remark 5.11 In the intersection theory of Deligne–Mumford stacks (see [Reference GilletGil84, Reference VistoliVis89]), the group $\mathop {\mathrm {Div}}\nolimits ({\mathcal {X}})$ of Weil divisors is the same as the group ${Z}_{n-1}({\mathcal {X}})$ of $(n-1)$ -dimensional cycles, where n is the dimension of ${\mathcal {X}}$ . And, the divisor class group $\mathrm { Cl}({\mathcal {X}})$ is the Chow group $A_{n-1}({\mathcal {X}})$ .
The following definition is a modified version of [Reference VistoliVis89, Definition 3.6].
Definition 5.12 Let $f : {\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ be a morphism of n-dimensional Deligne–Mumford stacks, and let $\mathcal Y$ be any closed integral substack of ${\mathcal {X}}_1$ .
-
(i) If f is proper and representable, the proper pushforward is $f_{*} : \mathop {\mathrm {Div}}\nolimits ({\mathcal {X}}_1)\rightarrow \mathop {\mathrm {Div}}\nolimits ({\mathcal {X}}_2)\quad \mathcal Y\mapsto \mathop {\mathrm {deg}}\nolimits (\mathcal Y/\mathcal Y^{\prime })\mathcal Y^{\prime }$ , where $\mathcal Y^{\prime }$ is image of $\mathcal Y$ in ${\mathcal {X}}_2$ and $\mathop {\mathrm { deg}}\nolimits (\mathcal Y/\mathcal Y^{\prime })$ is the degree of the restriction of f to $\mathcal Y$ and $\mathcal Y^{\prime }$ [Reference VistoliVis89, Definition 1.15].
-
(ii) If f is flat, the flat pullback is $f^{*} : \mathop {\mathrm {Div}}\nolimits ({\mathcal {X}}_2)\rightarrow \mathop {\mathrm {Div}}\nolimits ({\mathcal {X}}_1)\quad f^{*}(\mathcal Y)\mapsto \mathcal D_{\mathcal Y}$ , where $\mathcal D_{\mathcal Y}$ is the cycle associated with the closed substack $\mathcal Y\times _{{\mathcal {X}}_2}{\mathcal {X}}_1$ [Reference VistoliVis89, Definition 3.5].
The following proposition is immediately.
Proposition 5.13 There is a morphism of abelian groups
If $\mathcal D$ is a principal divisor, then ${\mathcal {O}}_{{\mathcal {X}}}(\mathcal D)\simeq {\mathcal {O}}_{{\mathcal {X}}}$ . Then, we have a morphism from the divisor class group $\mathrm {Cl}({\mathcal {X}})$ to $\mathrm {Pic}({\mathcal {X}})$ .
Remark 5.14 In Proposition 5.13, the homomorphism $\mathrm {Cl}({\mathcal {X}})\rightarrow \mathop {\mathrm { Pic}}\nolimits ({\mathcal {X}})$ is injective. In general, it is not surjective if the generic stabilizer of ${\mathcal {X}}$ is not trivial.
Proposition 5.15 Assume that $f : {\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ is a finite morphism of smooth irreducible Deligne–Mumford stacks such that $f_{*}{\mathcal {O}}_{{\mathcal {X}}}$ is a locally free sheaf. If a line bundle $L\simeq {\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D)$ for some Weil divisor $\mathcal D$ , then $\mathop {\mathrm { Nm}}\nolimits _f(L)\simeq {\mathcal {O}}_{{\mathcal {X}}_2}(f_{*}(\mathcal D))$ , i.e., the diagram
is commutative.
Proof The proof is divided into two steps. First, we show the conclusion for an effective Weil divisor $\mathcal D$ . Finally, we check the general case.
Case 1 Let $\mathcal D$ be an effective Weil divisor. Then, $\mathcal D=(s)$ for some section $s\in H^0({\mathcal {X}}_1,L)$ . There is an étale morphism $p_2 : U_2\rightarrow {\mathcal {X}}_2$ such that L is represented by an object $({\mathcal {O}}_{U_1},a)$ of $Des(U_1/{\mathcal {X}}_1)$ where $U_1=U_2\times _{{\mathcal {X}}_2}{\mathcal {X}}_1$ and $a\in {\mathcal {O}}^{*}_{U_1[1]}(U_1[1])$ . Thus, s is represented by an element $h\in {\mathcal {O}}_{U_1}(U_1)$ which satisfies $\mathrm {pr_1^{*}}h\cdot a=\mathrm {pr_2^{*}}h$ on $U_1[1]$ . By (67), the restriction of the norm $\mathop {\mathrm {Nm}}\nolimits _f(s)$ to $U_2$ is $\mathrm {N}_{f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}/{\mathcal {O}}_{{\mathcal {X}}_2}}(h)$ . Consider the Cartesian diagram
By the proof of [Reference VistoliVis89, Lemma 3.9], we have
Claim $f_{1*}(\mathop {\mathrm {div}}\nolimits (h))= \mathop {\mathrm {div}}\nolimits (\mathrm {N}_{f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}/{\mathcal {O}}_{{\mathcal {X}}_2}}(h))$ . Without loss of generality, we can assume that $U_2$ is irreducible. And, $U_1$ is the disjoint union of its irreducible components. Due to the irreducibility of ${\mathcal {X}}_1$ and ${\mathcal {X}}_2$ , the restriction of the morphism $f_1$ to each irreducible component of $U_1$ is a surjective finite morphism to $U_2$ . Therefore, $f_{1*}(\mathop {\mathrm {div}}\nolimits (h))= \mathop {\mathrm {div}}\nolimits (\mathrm {N}_{f_{*}{\mathcal {O}}_{U_1}/{\mathcal {O}}_{U_2}}(h))$ (see [Reference FultonFul98, Proposition 1.4]). The flat pullback $p_1^{*}(\mathcal D)=\mathrm {div}(h)$ and (71) implies $p^{*}_{2}(f_{*}(\mathcal D))=\mathop {\mathrm {div}}\nolimits (\mathrm {N}_{f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}/{\mathcal {O}}_{{\mathcal {X}}_2}}(h))$ . In addition, the flat pullback $p_2^{*}((\mathop {\mathrm {Nm}}\nolimits _f(s)))= \mathop {\mathrm {div}}\nolimits (\mathrm { N}_{f_{*}{\mathcal {O}}_{{\mathcal {X}}_1}/{\mathcal {O}}_{{\mathcal {X}}_2}}(h))$ . Thus, $f_{*}(\mathcal D)=(\mathop {\mathrm { Nm}}\nolimits _f(s))$ (see [Reference GilletGil84, Lemma 4.2]). As a result, $\mathop {\mathrm { Nm}}\nolimits _f(L)\simeq {\mathcal {O}}_{{\mathcal {X}}_2}(f_{*}(\mathcal D))$ .
Case 2 If $\mathcal D$ is a Weil divisor on ${\mathcal {X}}_1$ , then there are two effective Weil divisors ${\mathcal D_1,\mathcal D_2\in \mathop {\mathrm {Div}}\nolimits ({\mathcal {X}}_1)}$ such that $\mathcal D=\mathcal D_1-\mathcal D_2$ . So, ${\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D)={\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_1)\otimes {\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_2)^{-1}$ . Thus, $\mathop {\mathrm {Nm}}\nolimits _f(L)\simeq \mathop {\mathrm { Nm}}\nolimits _f({\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_1)\otimes {\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_2)^{-1})$ . By Proposition 5.6, we have $\mathop {\mathrm {Nm}}\nolimits _f({\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_1)\kern0.5pt{\otimes}\kern0.5pt {\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_2)^{-1})\kern0.7pt{=}\kern0.7pt\mathop {\mathrm {Nm}}\nolimits _f({\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_1))\kern0.5pt{\otimes}\kern0.5pt \mathop {\mathrm {Nm}}\nolimits _f({\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D_2))^{-1}$ . Therefore, we have $\mathop {\mathrm {Nm}}\nolimits _f(L)\simeq {\mathcal {O}}_{{\mathcal {X}}_2}(f_{*}(\mathcal D_1))\otimes {\mathcal {O}}_{{\mathcal {X}}_2}(-f_{*}(\mathcal D_2))={\mathcal {O}}_{{\mathcal {X}}_2}(f_{*}(\mathcal D))$ .
5.3 The case of stacky curves
In this subsection, all stacky curves are assumed to be irreducible and smooth. For a stacky curve ${\mathcal {X}}$ with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ , the group of Weil divisors of ${\mathcal {X}}$ is
where $X({\mathbb C})$ is the set of closed points of X and $r_x$ is the order of the stabilizer group of x. Suppose that the stacky points of ${\mathcal {X}}$ are $p_1,\ldots ,p_m$ and that the stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ , respectively. We have the following lemma.
Lemma 5.16 For every line bundle L on ${\mathcal {X}}$ , it can be uniquely (up to isomorphism) expressed as $L=\pi ^{*}W\otimes {\mathcal {O}}_{\mathcal {X}}(\textstyle {\sum _{k=1}^m\frac {i_k}{r_k}p_k})$ , where W is a line bundle on X and $0\leq i_k\leq r_k-1$ for all $1\leq k\leq m$ .
Proof For any line bundle L on ${\mathcal {X}}$ , there is a Weil divisor $\mathcal D\in \mathop {\mathrm { Div}}\nolimits ({\mathcal {X}})$ such that $L={\mathcal {O}}_{\mathcal {X}}(\mathcal D)$ (see [Reference Nasatyr and SteerNS95, Proposition 1.3]). Note that $\mathcal D$ can be written as $\textstyle {\sum _{x\in X({\mathbb C})}n_x\cdot x+\sum _{k=1}^m\frac {i_k}{r_k}\cdot p_k}$ , where $0\leq i_k\leq r_k-1$ for all $1\leq k\leq m$ and $n_x\in {\mathbb Z}$ . We therefore have $L=\pi ^{*}W\otimes {\mathcal {O}}_{\mathcal {X}}(\textstyle {\sum _{k=1}^m\frac {i_k}{r_k}\cdot p_k})$ , where $W={\mathcal {O}}_X(\sum _{x\in X({\mathbb C})}n_x\cdot x)$ .
If two Weil divisors $\mathcal D_1,\mathcal D_2\in \mathop {\mathrm {Div}}\nolimits ({\mathcal {X}})$ are linearly equivalent, then there is a rational function c on X such that $\mathcal D_1=\mathcal D_2+\mathop {\mathrm {div}}\nolimits (\pi ^{*}c)$ . Hence, the line bundle W is unique up to an isomorphism.
Moreover, we also have the following lemma (see [Reference BrochardBro09, Section 5.4]).
Remark 5.18 For every m-tuple $(i_1,\ldots ,i_m)$ of integers, we have the translation
defined by the line bundle ${\mathcal {O}}_{\mathcal {X}}(\sum _{k=1}^{m}\frac {i_k}{r_k}\cdot p_k)$ . By Lemmas 5.16 and 5.17, $\mathop {\mathrm {Pic}}\nolimits ({\mathcal {X}})$ is the disjoint union of open and closed subschemes
where $\mathop {\mathrm {Pic}}\nolimits ^{(i_1,\ldots ,i_m)}({\mathcal {X}})=T_{(i_1,\ldots ,i_m)}(\pi ^{*}(\mathop {\mathrm { Pic}}\nolimits (X)))$ for all $(i_1,\ldots ,i_m)$ . For any integer d, let $\mathop {\mathrm {Pic}}\nolimits ^d(X)$ be the moduli space of line bundles with degree d on X. It is a connected component of $\mathop {\mathrm { Pic}}\nolimits (X)$ . Then, the connected components of $\mathop {\mathrm {Pic}}\nolimits ({\mathcal {X}})$ are
where $d\in {\mathbb Z}$ and $(i_1,\ldots ,i_m)$ satisfy $0\leq i_k\leq r_k-1$ for all $1\leq k\leq m$ . We therefore have the decomposition of $\mathop {\mathrm {Pic}}\nolimits ({\mathcal {X}})$ into connected components
which coincide with the decomposition (30).
In the following, we will consider norm maps for stacky curves. Let ${\mathcal {X}}_1$ and ${\mathcal {X}}_2$ be two stacky curves with coarse moduli spaces $\pi _i : {\mathcal {X}}_i\rightarrow X_i$ for $i=1,2$ . The set of stacky points of ${\mathcal {X}}_1$ is $\{p_1,\ldots ,p_{m_1}\}$ and ${\mathcal {X}}_2$ ’s is $\{\widetilde p_1,\ldots ,\widetilde p_{m_2}\}$ . The stabilizer groups of ${\mathcal {X}}_1$ and ${\mathcal {X}}_2$ are $\{\mu _{r_1},\ldots ,\mu _{r_{m_1}}\}$ and $\{\mu _{\widetilde r_1},\ldots ,\mu _{\widetilde r_{m_2}}\}$ , respectively. Suppose that $f:{\mathcal {X}}_1\rightarrow {\mathcal {X}}_2$ is a finite morphism and $f^{\prime } : X_1\rightarrow X_2$ is the induced morphism between coarse moduli spaces.
Lemma 5.19 The proper pushforward of f is
Proof By Definition 5.12, the conclusion is immediate.
Remark 5.20 Since the finite morphism f is representable, the stabilizer group of x is isomorphic to a subgroup of stabilizer group of $f^{\prime }(x)$ . Hence, $r_{f^{\prime }(x)}/r_x$ is an integer.
Proposition 5.21 The norm map of f is
Proof For smooth stacky curves, the homomorphism (70) is surjective (see [Reference Nasatyr and SteerNS95, Proposition 1.3]). By Proposition 5.15 and Lemma 5.19, we complete the proof.
Corollary 5.22 Assume that $f^{\prime }(p_i)=\widetilde p_i$ for all $1\leq i\leq m_1$ . For any $d\in \mathbb {Z}$ and any $(i_1,\ldots ,i_{m_1})\in {\mathbb Z}^{m_1}\cap [0,r_1]\times \cdots \times [0,r_{m_1}]$ , the restriction of $\mathop {\mathrm {Nm}}\nolimits _f$ to $\mathop {\mathrm {Pic}}\nolimits ^{d,(i_1,\ldots ,i_{m_1})}({\mathcal {X}}_1)$ is
where
Proof For any $L\in \mathop {\mathrm {Pic}}\nolimits ^{d,(i_1,\ldots ,i_{m_1})}({\mathcal {X}}_1)$ , there is a Weil divisor $\mathcal D=\sum _{x\in X_1({\mathbb C})}n_x\cdot x+\sum _{k=1}^{m_1}\frac {i_k}{r_k}\cdot p_k$ with $n_x\in {\mathbb Z}$ such that $L={\mathcal {O}}_{{\mathcal {X}}_1}(\mathcal D)$ . Then, $\mathop {\mathrm { Nm}}\nolimits _f(L)=\textstyle {{\mathcal {O}}_{{\mathcal {X}}_2}(\sum _{x\in X({\mathbb C})}n_xf^{\prime } (x)+\sum _{k=1}^{m_1}i_k\frac {\widetilde r_k}{r_k}\frac {1}{\widetilde r_k}f^{\prime }(p_k))}$ (see Proposition 5.21).
Lemma 5.23 There is a commutative diagram
in which the pushforward morphisms $\pi _{1*}$ and $\pi _{2*}$ are isomorphisms.
Proof Without loss of generality, we only show that $\pi _{1*}$ is an isomorphism. For any $L\in \mathop {\mathrm { Pic}}\nolimits ^{d,(i_1,\ldots ,i_{m_1})}({\mathcal {X}}_1)$ , there is a unique $W\in \mathop {\mathrm {Pic}}\nolimits ^d(X_1)$ such that
Then, $\pi _{1*}L=W\otimes \pi _{1*}{\mathcal {O}}_{{\mathcal {X}}_1}(\sum _{k=1}^{m_1}\frac {i_k}{r_k}\cdot p_k)$ . On the other hand, $\pi _{1*}{\mathcal {O}}_{{\mathcal {X}}_1}(\sum _{k=1}^{m_1}\frac {i_k}{r_k}\cdot p_k)={\mathcal {O}}_{X_1}$ (see [Reference BehrendBeh14, Theorem 3.64]). Hence, $\pi _{1*}$ is an isomorphism. As the proof of Corollary 5.22, we can directly verify $\pi _{2*}\mathop {\mathrm {Nm}}\nolimits _f(L)=\mathop {\mathrm { Nm}}\nolimits _{f^{\prime }}(\pi _{1*}L)$ .
6 SYZ duality
In this section, ${\mathcal {X}}$ is a hyperbolic stacky curve with coarse moduli space $\pi : {\mathcal {X}}\rightarrow X$ . The stacky points are $p_1,\ldots ,p_m$ , and the stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ , respectively. For each stacky point $p_k$ , its residue gerbe $\iota _k : B\mu _{r_k}\hookrightarrow {\mathcal {X}}$ is a closed immersion.
6.1 BNR correspondence
For $\boldsymbol a\in \bigoplus _{i=1}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ , let $\pi ^{\prime } : {\mathcal {X}}_{\boldsymbol a}\rightarrow X_{\boldsymbol a}$ be the coarse moduli space of ${\mathcal {X}}_{\boldsymbol a}$ . There is a commutative diagram
where $f : {\mathcal {X}}_{\boldsymbol a}\rightarrow {\mathcal {X}}$ is the natural projection and $f^{\prime } : X_{\boldsymbol a}\rightarrow X $ is the induced morphism between coarse moduli spaces. Assume that the assumptions of Theorem 4.18 (which ensure that a general spectral curve is irreducible and smooth) are satisfied. Hence, we can assume that ${\mathcal {X}}_{\boldsymbol a}$ is an irreducible smooth stacky curve and satisfies the conclusion of Lemma 4.20 in the following discussion. Without loss of generality, suppose that the set of stacky points of ${\mathcal {X}}_{\boldsymbol a}$ is $\{\widetilde p_1,\ldots ,\widetilde p_{m_1}\}$ such that $f(\widetilde p_k)=p_k$ for all $1\leq k\leq m_1$ . Note that $K_0(B\mu _{r_k})$ is isomorphic to the representation ring $\mathbf {R}\mu _{r_k}$ for every stacky point $p_k$ and $\mathbf {R}\mu _{r_k}={\mathbb Z}[x_k]/(x_k^{r_k}-1)$ , where $x_k$ represents the representation defined by the inclusion $\mu _{r_k}\hookrightarrow {\mathbb C}^{*}$ .
Lemma 6.1 For each $1\leq k\leq m_1$ , the decomposition of the K-class $[\iota ^{*}_k(f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})]$ in $\mathbf {R}\mu _{r_k}$ only consists of the following two cases:
-
(i) If $\lceil \frac {r}{r_k}\rceil =\frac {r+1}{r_k}$ , then $[\iota ^{*}_k(f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})]=m_kx_k^0+(m_k-1)x_k^1+\cdots +m_kx_k^{r_k-1}$ , where $m_k=\frac {r+1}{r_k}$ .
-
(ii) If $\lceil \frac {r-1}{r_k}\rceil =\frac {r-1}{r_k}$ , then $[\iota ^{*}_k(f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})]=(m_k+1)x_k^0+m_kx_k^1+\cdots +m_kx_k^{r_k-1}$ , where $m_k=\frac {r-1}{r_k}$ .
Proof By Proposition 4.1, we have $f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}}=\bigoplus _{i=0}^{r-1}K_{\mathcal {X}}^{-i}$ . Since ${\mathcal {X}}_{\boldsymbol a}$ satisfies the conclusion of Lemma 4.20, by some elementary computation, we get the decomposition of $[\iota ^{*}_k(f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})]$ in $\mathbf {R}\mu _{r_k}$ for every $1\leq k\leq m_1$ .
If $(E,\phi )$ is a rank r Higgs bundle with spectral curve ${\mathcal {X}}_{\boldsymbol a}$ , then there is a line bundle W in some $\mathop {\mathrm {Pic}}\nolimits ^{d_1,(i_1,\ldots ,i_{m_1})}({\mathcal {X}}_{\boldsymbol a})$ such that $f_{*}(W)=E$ (see Proposition C.2).
Lemma 6.2 The K-class $[W]\in K_0({\mathcal {X}}_{\boldsymbol a})_{\mathbb Q}$ is uniquely determined by the K-class $[E]\in K_0({\mathcal {X}})_{\mathbb Q}$ .
Proof By Proposition 3.35, we only need to show that $\mathop {\mathrm {deg}}\nolimits (W)$ and $\{i_1,\ldots ,i_{m_1}\}$ are uniquely determined by $[E]$ . First, note that there is a line bundle $W^{\prime }\in \mathop {\mathrm {Pic}}\nolimits (X_{\boldsymbol a})$ such that
(see Lemma 5.16). Hence, $E=f_{*}(\pi ^{\prime *}W^{\prime })\otimes {\mathcal {O}}_{{\mathcal {X}}}(\sum _{k=1}^{m_1}\frac {i_k}{r_k}\cdot p_k)$ . We therefore have
for each $1\leq k\leq m_1$ . Note that $[\iota _k^{*}(f_{*}(\pi ^{\prime *}W^{\prime }))]=[\iota _k^{*}(f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})]$ in $\mathbf {R}\mu _{r_k}$ for all $1\leq k\leq m$ . By Lemma 6.1, $i_k$ are uniquely determined. On the other hand, by Propositions 5.9 and 5.21, $\mathop {\mathrm {deg}}\nolimits (W)=\mathop {\mathrm {deg}}\nolimits (E)-\mathop {\mathrm { deg}}\nolimits (f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})$ , where $\mathop {\mathrm {deg}}\nolimits (f_{*}{\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}})=\frac {r(1-r)}{r}(2g-2+\sum _{k=1}^{m}\frac {r_i-1}{r_i})$ .
Corollary 6.3 If the spectral curve of $(E,\phi )$ is irreducible and smooth, then there exists $(i_1,\ldots ,i_{m_1})\in {\mathbb Z}^{m_1}\cap [0,r_1-1]\times \cdots \times [0,r_{m_1}-1]$ such that $[E]\in K_0({\mathcal {X}})_{\mathbb Q}$ satisfies
for all $1\leq k\leq m$ .
Denote the K-class $[E]\in K_0({\mathcal {X}})_{\mathbb Q}$ by $\xi $ . Consider the moduli space $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {GL}}\nolimits _r)$ of moduli space of semistable Higgs bundles with K-class $\xi $ . By Proposition C.2, the following lemma is immediate.
Lemma 6.4 The fiber $h^{-1}(\boldsymbol a)$ of the Hitchin morphism $h : M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {GL}}\nolimits _r)\rightarrow {\mathbb H}(r,K_{\mathcal {X}})$ at $\boldsymbol a$ is isomorphic to $\mathop {\mathrm {Pic}}\nolimits ^{d,(i_1,\ldots ,i_{m_1})}({\mathcal {X}}_{\boldsymbol a})$ .
Suppose that $\xi =(r,d_{\xi },(m_{1,i})_{i=1}^{r_1-1},\ldots ,(m_{m,i})_{i=1}^{r_m-1})\in K_0({\mathcal {X}})_{\mathbb Q}$ . Fix a line bundle $L\in \mathop {\mathrm {Pic}}\nolimits ^{d^{\prime },(j_1,\ldots ,j_m)}({\mathcal {X}})$ , where $d^{\prime }$ , $j_1,\ldots ,j_m$ satisfy
We consider the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ of semistable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with K-class $\xi $ and determinant L. Assume that the assumptions of Corollary 4.19 (which ensure that a general spectral curve is irreducible and smooth) are satisfied, then for a general $\boldsymbol a\in \bigoplus _{i=2}^rH^0({\mathcal {X}},K_{\mathcal {X}}^i)$ , the spectral curve ${\mathcal {X}}_{\boldsymbol a}$ satisfies the conclusion in Lemma 4.20. We also assume that ${\mathcal {X}}_{\boldsymbol a}$ satisfies the conclusion in Lemma 4.20.
Lemma 6.5 The fiber of the Hitchin morphism $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ at $\boldsymbol a$ is
Proof Since ${\mathcal {X}}_{\boldsymbol a}$ satisfies the conclusion in Lemma 4.20, we have
Therefore,
(see Proposition 5.9). Note that $\mathop {\mathrm {det}}\nolimits (f_{*}({\mathcal {O}}_{{\mathcal {X}}_{\boldsymbol a}}))^{-1}=K_{\mathcal {X}}^{{r(r-1)}/{r}}$ . This completes the proof.
For $f^{\prime } : X_{\boldsymbol a}\rightarrow X$ in the diagram (81), the Prym varieties $\mathop {\mathrm { Prym}}\nolimits _{f^{\prime }}(X_{\boldsymbol a})$ is defined by
Then, $h_{\mathop {\mathbf {SL}}\nolimits _r}^{-1}(\boldsymbol a)$ is a $\mathop {\mathrm { Prym}}\nolimits _{f^{\prime }}(X_{\boldsymbol a})$ -torsor (see Lemma 5.23). The fiber $h_{\mathop {\mathbf {PGL}}\nolimits _r}^{-1}(\boldsymbol a)$ of the Hitchin morphism $h_{\mathop { \mathbf {PGL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop { \mathbf {PGL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ at $\boldsymbol a$ is a $\mathop {\mathrm { Prym}}\nolimits _{f^{\prime }}(X_{\boldsymbol a})/\Gamma _0$ -torsor, where
Thus, we have the following proposition.
Proposition 6.6 $h^{-1}_{\mathop {\mathbf {SL}}\nolimits _r}(\boldsymbol a)$ is a $\mathop {\mathrm {Prym}}\nolimits _{f^{\prime }}(X_{\boldsymbol a})$ -torsor and $h^{-1}_{\mathop {\mathbf {PGL}}\nolimits _r}(\boldsymbol a)$ is a $\mathop {\mathrm { Prym}}\nolimits _{f^{\prime }}(X_{\boldsymbol a})/\Gamma _0$ -torsor.
For brevity, we introduce the following notations:
-
• ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}=\{W\in \mathop {\mathrm { Pic}}\nolimits ^{d,(i_1,\ldots ,i_{m_1})}({\mathcal {X}}_{\boldsymbol a})\vert \mathop {\mathrm {Nm}}\nolimits _{f}(W)\simeq L\otimes K_{{\mathcal {X}}}^{{r(r-1)}/{2}}\}$ .
-
• $P^d=\{W\in \mathop {\mathrm {Pic}}\nolimits ^d(X_{\boldsymbol a})\vert \mathop {\mathrm { Nm}}\nolimits _{f^{\prime }}(W)\simeq \pi ^{\prime }_{*}(L\otimes K_{{\mathcal {X}}}^{r(r-1)/2})\}$ .
-
• ${\mathcal P}^{0,(0,\ldots ,0)}=\{W\in \mathop {\mathrm {Pic}}\nolimits ^{0,(0,\ldots ,0)}({\mathcal {X}}_{\boldsymbol a})\vert \mathop {\mathrm {Nm}}\nolimits _f(W)\simeq {\mathcal {O}}_{\mathcal {X}}\}$ .
-
• $P^0=\{W\in \mathop {\mathrm {Pic}}\nolimits ^0(X_{\boldsymbol a})\vert \mathop {\mathrm { Nm}}\nolimits _{f^{\prime }}(W)\simeq {\mathcal {O}}_X\}$ .
-
• $\widehat {{\mathcal P}}^{d,(i_1,\ldots ,i_{m_1})}={\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}/\Gamma _0$ .
-
• $\widehat {P}^d=P^d/\Gamma _0$ .
-
• $\widehat {{\mathcal P}^0}={\mathcal P}^{0,(0,\ldots ,0)}/\Gamma _0$ .
-
• $\widehat {P}^0=P^0/\Gamma _0$ .
Obviously, ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ ( $\widehat {{\mathcal P}}^{d,(i_1,\ldots ,i_{m_1})}$ ) is a ${\mathcal P}^0$ ( $\widehat {{\mathcal P}}^0$ )-torsor. By Lemma 5.23, we have the following lemma.
Lemma 6.7 The pushforward
is an isomorphism of abelian varieties. And,
is an isomorphism of torsors with respect to the isomorphism (84). Moreover, (84) induces an isomorphism of abelian varieties
The morphism (85) gives an isomorphism
of torsors with respect to (86).
Corollary 6.8 The dual of ${\mathcal P}^0$ is $\widehat {{\mathcal P}}^0$ .
Proof The dual of $P^0$ is $\widehat P^0$ (see [Reference Hausel and ThaddeusHT03, Lemma 2.3]). Then, the dual of ${\mathcal P}^0$ is $\widehat {{\mathcal P}}^0$ , by the isomorphisms (84) and (86) in Lemma 6.7.
6.2 The proof of SYZ duality
For convenience, the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{s}(\mathop { \mathbf {SL}}\nolimits _r)$ of stable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles is denoted by $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ . In general, the universal Higgs bundle $\boldsymbol {(E,\Phi )}$ does not exist. But we can construct a universal projective bundle $\mathbb P(\boldsymbol {E})$ and a universal endomorphism bundle ${\mathcal End}(\boldsymbol {E})$ , even though $\boldsymbol {E}$ does not exist. There is a universal Higgs field $\boldsymbol {\Phi }\in H^0({\mathcal End}(\boldsymbol {E})\otimes K_{\mathcal {X}})$ . Fix a closed point $c\in {\mathcal {X}}$ . Restricting $\mathbb P(\boldsymbol {E})$ to $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }\times \{c\}$ , we get a projective bundle $\mathbb P$ on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ . The obstruction to lift the $\mathop { \mathbf {PGL}}\nolimits _r$ -bundle $\mathbb P$ to an $\mathop {\mathbf {SL}}\nolimits _r$ -bundle defines a ${\mathbb Z}_r$ -gerbe $\boldsymbol {B}$ on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ .
Lemma 6.9 The restriction of $\boldsymbol {B}$ to each regular fiber of the Hitchin morphism $h_{\mathop { \mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }\rightarrow {\mathbb H}^0(r,K_{\mathcal {X}})$ is trivial as a ${\mathbb Z}_r$ -gerbe.
Proof Suppose that $a\in {\mathbb H}^0(r,K_{{\mathcal {X}}})$ is a closed point such that the associated spectral curve ${\mathcal {X}}_a$ is integral and smooth. Recall that the fiber of the Hitchin morphism $h_{\mathop { \mathbf {SL}}\nolimits _r}$ at a is ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ , where $0\leq i_k\leq r_k-1$ for all k. Let $\boldsymbol L$ be a universal line bundle on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\times {\mathcal {X}}_a$ . The projection of ${\mathcal {X}}_{\boldsymbol a}$ to ${\mathcal {X}}$ is $f : {\mathcal {X}}_a\longrightarrow {\mathcal {X}}$ . The pushforward $((\mathrm {id}\times f)_{*}(\boldsymbol L),(\mathrm {id}\times f)_{*}(\boldsymbol {\widetilde \phi }))$ is a ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ -family of Higgs bundles on ${\mathcal {X}}$ , where $\widetilde \phi : \boldsymbol L\rightarrow \boldsymbol L\otimes _{{\mathcal {O}}_{{\mathcal {X}}_a}} f^{*}K_{\mathcal {X}}$ is defined by the tautological section of $f^{*}K_{\mathcal {X}}$ . It induces an inclusion ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\subseteq M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ . Hence, we have
Since $f : {\mathcal {X}}_a\rightarrow {\mathcal {X}}$ is a finite morphism, we can choose a closed point $c\in {\mathcal {X}}$ such that $f^{-1}(c)$ does not contain any branched points. So, we have
Thus, $\mathop {\mathrm {det}}\nolimits ((\mathrm {id}\times p)_{*}(\boldsymbol L)|_{{\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\times \{c\}})=\otimes _{y\in p^{-1}(c)}\boldsymbol L|_{{\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\times \{y\}}=V$ . On the other hand, the Néron–Severi class of V is divisible by r. Therefore, there exists a line bundle W on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ such that $W^{\otimes r}\simeq V$ . We have
i.e., $(\mathrm {id}\times p)_{*}(\mathrm {pr}_{{\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}}^{*}W^{-1}\otimes \boldsymbol L)|_{{\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\times \{c\}}$ is an $\mathop {\mathbf {SL}}\nolimits _r$ -lifting of $\mathbb P|_{{\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}}$ . So, the restriction of $\boldsymbol {B}$ to ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ is a trivial ${\mathbb Z}_r$ -gerbe.
Remark 6.10 Recall the commutative diagram (81). Therefore, we have the commutative diagram
where $\pi _{*}^{\prime } : {\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\rightarrow P^d$ is the isomorphism in Lemma 6.7. For the universal line bundle $\boldsymbol L$ in the proof of Lemma 6.9, there exists a universal line bundle $\boldsymbol W$ on $P^d\times X_a$ such that
where $W_{i_1,\ldots ,i_{m_1}}={\mathcal {O}}_{{\mathcal {X}}_a}(\sum _{k=1}^{m_1}\frac {i_k}{r_k}\cdot \widetilde p_k)$ . Let $c^{\prime }$ be the image of c in the coarse moduli space X. Since c is a closed point in ${\mathcal {X}}$ , it is easy to check that
Then, the two sets of trivializations are isomorphic
where $\boldsymbol {B}^{\prime }$ is the $\mathop {\mathbf {SL}}\nolimits _r$ -lifting gerbe of $\mathbb P((\mathop {\mathrm { id}}\nolimits \times f^{\prime })_{*}\boldsymbol W)|_{P^d\times \{{c^{\prime }}\}}$ .
By the proof of Lemma 6.9, we see that a trivialization of $\boldsymbol {B}$ on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ is equivalent to give a universal line bundle $\boldsymbol L$ on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\times {\mathcal {X}}_a$ such that
Then the set $\mathrm {Triv}^{{\mathbb Z}_r}({\mathcal P}^{d,(i_1,\ldots ,i_{m_1})},\boldsymbol {B})$ of trivialization of $\boldsymbol {B}$ on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ is identified with the set ${\mathfrak T}$ of universal line bundles $\boldsymbol L$ on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\times {\mathcal {X}}_a$ satisfying (93). Let $\widehat {{\mathcal P}}^0[r]$ be the group of torsion points of order r in $\widehat {{\mathcal P}}^0$ . The set ${\mathfrak T}$ is naturally a $\widehat {{\mathcal P}}^0[r]$ -torsor. On the other hand, we have
Then, the $H^1({\mathcal P}^{d,(i_1,\ldots ,i_{m_1})},{\mathbb Z}_r)$ -torsor $\mathrm {Triv}^{{\mathbb Z}_r}({\mathcal P}^{d,(i_1,\ldots ,i_{m_1})},\boldsymbol {B})$ is isomorphic to the $\widehat {{\mathcal P}}^0[r]$ -torsor ${\mathfrak T}$ .
Since ${\mathbb Z}_r$ is a subgroup of $U(1)$ , any ${\mathbb Z}_r$ -gerbe extends to a $U(1)$ -gerbe. Let $\boldsymbol {\mathcal B}$ be the $U(1)$ -gerbe defined by the ${\mathbb Z}_r$ -gerbe $\boldsymbol {B}$ . The triviality of the ${\mathbb Z}_r$ -gerbe $\boldsymbol {B}$ implies that the $U(1)$ -gerbe $\boldsymbol {\mathcal B}$ is also trivial. The set of all trivialization of $\boldsymbol {\mathcal B}$ on ${\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}$ is denoted by $\mathrm { Triv}^{U(1)}({\mathcal P}^{d,(i_1,\ldots ,i_{m_1})},\boldsymbol {\mathcal B})$ , which is an $H^1({\mathcal P}^{d,(i_1,\ldots ,i_{m_1})},U(1))$ -torsor. Similarly, we have
We have a natural identification
Proposition 6.11 For any $d,e\in {\mathbb Z}$ , there is a smooth isomorphism of $\widehat {{\mathcal P}}^0$ -torsors
Proof By the isomorphism (92), [Reference Hausel and ThaddeusHT03, Proposition 3.2], and [Reference Biswas and DeyBD12, Theorem 4.2], we complete the proof.
Now consider the reverse direction. We need a gerbe $\widehat {\boldsymbol {B}}$ on the global quotient stack $[M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }/\Gamma _0]$ , i.e., a $\Gamma _0$ -equivariant gerbe on $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }$ . In fact, this is just $\boldsymbol {B}$ equipped with a $\Gamma _0$ -equivariant structure. For $\gamma \in \Gamma _0$ , we use $L_{\gamma }$ to indicate the line bundle on X corresponding to $\gamma \in \Gamma $ . Then the action of $\Gamma _0$ on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ is given by
for $\gamma \in \Gamma _0$ . Let $(\boldsymbol E,\phi )$ be the universal Higgs bundle on $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }\times {\mathcal {X}}$ (if the moduli space $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ is not fine, $\boldsymbol E$ is a twisted vector bundle on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ ). We have a canonical isomorphism
on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }\times {\mathcal {X}}$ , for every $\gamma \in \Gamma _0$ . And, for $\gamma _1,\gamma _2\in \Gamma _0$ ,
Hence, $\mathbb P(\boldsymbol E)$ is a $\Gamma _0$ -equivariant projective bundle on $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }\times {\mathcal {X}}$ . The restriction $\mathbb P$ of $\mathbb P(\boldsymbol E)$ to $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }\times \{c\}$ is also a $\Gamma _0$ -equivariant projective bundle on $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }$ . It determines a $\Gamma _0$ -equivariant structure on the ${\mathbb Z}_r$ -gerbe $\boldsymbol {B}$ on $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ . Then, it defines a ${\mathbb Z}_r$ -gerbe $\widehat {\boldsymbol {B}}$ on $[M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }/\Gamma _0]$ . Specifically, the $\Gamma _0$ -equivariant structure of $\mathbb P|_{{\mathcal P}^{d,(i_1,\ldots ,i_m)}}$ is
for every $\gamma \in \Gamma _0$ . By Remark 6.10, there exists a locally free sheaf $(\mathop {\mathrm {id}}\nolimits \times f^{\prime })_{*}\boldsymbol W|_{P^d\times \{c^{\prime }\}}$ on $P^d\times \{c^{\prime }\}$ such that
where $\boldsymbol W$ is a universal line bundle on $P^d\times X_a$ . On the other hand, the projective bundle $\mathbb P((\mathop {\mathrm {id}}\nolimits \times f^{\prime })_{*}\boldsymbol W|_{P^d\times \{c^{\prime }\}})$ admits a $\Gamma _0$ -equivariant structure
for every $\gamma \in \Gamma _0$ , which is induced by the natural $\Gamma _0$ -equivariant structure of $\mathbb P((\mathop {\mathrm {id}}\nolimits \times f^{\prime })_{*}\boldsymbol W)$ on $P^d\times X$ . Obviously, the $\Gamma _0$ -equivariant projective bundle $\mathbb P|_{{\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}}$ is isomorphic to the pullback of the $\Gamma _0$ -equivariant projective bundle $\mathbb P((\mathop {\mathrm { id}}\nolimits \times f^{\prime })_{*}\boldsymbol W|_{P^d\times \{c^{\prime }\}})$ , along the $\Gamma _0$ -equivariant morphism $\pi _{*}^{\prime } : {\mathcal P}^{d,(i_1,\ldots ,i_{m_1})}\rightarrow P^d$ . We therefore have the following proposition (see [Reference Hausel and ThaddeusHT03, Lemma 3.5 and Proposition 3.6]).
Proposition 6.12 The restriction of $\widehat {\boldsymbol {B}}$ to $\widehat {{\mathcal P}}^{d,(i_1,\ldots ,i_{m_1})}$ is trivial as a ${\mathbb Z}_r$ -gerbe. Moreover, there is a smooth isomorphism of ${\mathcal P}^0$ -torsors
where $\widehat {\boldsymbol {\mathcal B}}$ is the $U(1)$ -gerbe obtained by the extension of $\widehat {\boldsymbol {B}}$ and $d,e\in {\mathbb Z}$ .
Assume that the assumptions of Corollary 4.19 (which ensure that a general spectral curve is irreducible and smooth) are satisfied. Suppose that the K-class $\xi $ satisfies (82) and $\xi =(r,d_{\xi },(m_{1,i})_{i=1}^{r_1-1},\ldots ,(m_{m,i})_{i=1}^{r_m-1})\in K_0({\mathcal {X}})_{\mathbb Q}$ . Fix a line bundle $L\in \mathop {\mathrm {Pic}}\nolimits ^{d^{\prime },(j_1,\ldots ,j_m)}({\mathcal {X}})$ , where $d^{\prime },j_1,\ldots ,j_m$ satisfy (83). Consider the moduli space $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }^{ss}(\mathop {\mathbf {SL}}\nolimits _r)$ of semistable $\mathop { \mathbf {SL}}\nolimits _r$ -Higgs bundles with K-class $\xi $ and determinant L. The Hitchin morphism $h_{\mathop {\mathbf {SL}}\nolimits _r} : M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)\rightarrow {\mathbb H}^o(r,K_{\mathcal {X}})$ is surjective. Note that the stable locus $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ of $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{ss}(\mathop { \mathbf {SL}}\nolimits _r)$ is nonempty. By Proposition 3.39, we have
On the other hand, by some elementary computation, we have
i.e., $\mathop {\mathrm {dim}}\nolimits {\mathbb H}^o(r,K_{{\mathcal {X}}})=\frac {1}{2}\mathop {\mathrm {dim}}\nolimits M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }$ . Hence, $h_{\mathop {\mathbf {SL}}\nolimits _r}$ is surjective, since the restriction of $h_{\mathop {\mathbf {SL}}\nolimits _r}$ to a nonempty open subsect of $M_{\mathop {\mathrm { Dol}}\nolimits ,s}$ is an algebraically integrable systems. Therefore, the properness of $h_{\mathop {\mathbf {SL}}\nolimits _r}$ implies that there is a nonempty open subset ${{\mathcal U}\subseteq {\mathbb H}^o(r,K_{\mathcal {X}})}$ such that the inverse image $h_{\mathop {\mathbf {SL}}\nolimits _r}^{-1}({\mathcal U})$ is contained in $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi }$ . Note that $h_{\mathop {\mathbf {SL}}\nolimits _r}^{-1}({\mathcal U})$ is $\Gamma _0$ -invariant, due to the $\Gamma _0$ -equivariantness of $h_{\mathop {\mathbf {SL}}\nolimits _r}$ and the trivial action of $\Gamma _0$ on ${\mathbb H}^o(r,K_{\mathcal {X}})$ . Then, we obtain two proper morphisms
where $h_{\mathop {\mathbf {SL}}\nolimits _r,{\mathcal U}}$ and $h_{\mathop { \mathbf {PGL}}\nolimits _r,{\mathcal U}}$ are complete algebraically integrable systems (see [Reference Logares and MartensLM10, Reference MarkmanMar94]). Moreover, $M_{\mathop {\mathbf {SL}}\nolimits _r}:=h_{\mathop {\mathbf {SL}}\nolimits _r}^{-1}({\mathcal U})$ is a hyperkähler manifold and $M_{\mathop {\mathbf {PGL}}\nolimits _r}:=h_{\mathop {\mathbf {PGL}}\nolimits _r}^{-1}({\mathcal U})=[M_{\mathop {\mathbf {SL}}\nolimits _r}/\Gamma _0]$ is a hyperkähler orbifold (see [Reference KonnoKon93]). Summarizing the above discussion, we get our main result.
Theorem 6.13
-
(1) Assume that $\lceil \frac {r}{r_k}\rceil =\frac {r}{r_k}$ or $\lceil \frac {r}{r_k}\rceil =\frac {r+1}{r_k}$ for all $1\leq k\leq m$ . $(M_{\mathop { \mathbf {SL}}\nolimits _r},\boldsymbol {\mathcal B})$ and $(M_{\mathop {\mathbf {PGL}}\nolimits _r},\widehat {\boldsymbol {\mathcal B}})$ are SYZ mirror partners if one of the following conditions is satisfied:
-
(i) $g\geq 2$ ;
-
(ii) $g=1$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2$ ;
-
(iii) $g=0$ and $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r+1$ ;
-
(iv) $g=0$ , $\sum _{k=1}^m(r-\lceil \frac {r}{r_k}\rceil )\geq 2r$ and $\mathrm {dim}_{{{\mathbb C}}}H^0({\mathcal {X}},K^k_{\mathcal {X}})\geq 2$ for some $2\leq k\leq r$ .
-
-
(2) Suppose that the assumption in $(1)$ does not hold. We make the following assumption: if $\lceil \frac {r}{r_k}\rceil \geq \frac {r+2}{r_k}$ for some $1\leq k\leq m$ , then $\lceil \frac {r-1}{r_k}\rceil =\frac {r-1}{r_k}$ . Hence, $(M_{\mathop { \mathbf {SL}}\nolimits _r},\boldsymbol {\mathcal B})$ and $(M_{\mathop {\mathbf {PGL}}\nolimits _r},\widehat {\boldsymbol {\mathcal B}})$ are SYZ mirror partners if any of the following conditions is satisfied:
Proof From Proposition 6.6, Lemma 6.7, Corollary 6.8, Proposition 6.11, and Proposition 6.12, we conclude the conclusions of the theorem.
Corollary 6.14 If there are no strictly semistable $\mathop {\mathbf {SL}}\nolimits _r$ -Higgs bundles with K-class $\xi $ , then the $(M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^s(\mathop {\mathbf {SL}}\nolimits _r),\boldsymbol {\mathcal B})$ and $(M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }^{\alpha ,s}(\mathop {\mathbf {PGL}}\nolimits _r),\widehat {\boldsymbol {\mathcal B}})$ are mirror partners.
Example 6.15 For five distinct points $\{p_1,p_2,p_3,p_4,p_5\}$ on the projective line $\mathbb P^1$ , we can construct the stacky curve
Its coarse moduli space is $\pi : {\mathcal {X}}\rightarrow \mathbb P^1$ . The canonical line bundle of ${\mathcal {X}}$ is $K_{\mathcal {X}}=\pi ^{*}K_{\mathbb P^1}\otimes {\mathcal {O}}_{{\mathcal {X}}}(\frac {2}{3}p_1+\frac {1}{2}\sum _{k=2}^5p_k)$ . Note that the degree of $K_{\mathcal {X}}$ is $\frac {2}{3}$ . Hence, it is a hyperbolic stacky curve. We can show that
Since $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}H^0({\mathcal {X}},K^2_{\mathcal {X}})=2$ and $\mathop {\mathrm { dim}}\nolimits _{{\mathbb C}}H^0({\mathcal {X}},K^3_{\mathcal {X}})=1$ , the condition (33) is satisfied. Note that $K^3_{\mathcal {X}}={\mathcal {O}}_{\mathcal {X}} (\frac {1}{2}\sum _{k=2}^5p_k)$ . So, $H^0({\mathcal {X}},K^3_{\mathcal {X}})$ is generated by the section $s=\tau _2\otimes \tau _3\otimes \tau _4\otimes \tau _5$ , where $\tau _i$ is the pullback section of the universal section on root stack $\sqrt [2]{p_i}$ , for each i. Consider the spectral curve ${\mathcal {X}}_s$ define by s, i.e., it is the zero locus of section $\tau ^{\otimes 3}+\psi ^{*}s$ , where $\psi : \mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})\rightarrow {\mathcal {X}}$ is the projection and $\tau $ is the tautological section. According to the uniformization of Deligne–Mumford curves (see [Reference Behrend and NoohiBN06]), there exists a smooth projective curve $\Sigma $ with an action of a finite group G such that ${\mathcal {X}}$ is $[\Sigma /G]$ . More precisely, we have the commutative diagram
where $g : \Sigma \rightarrow {\mathcal {X}}$ is the natural étale covering of ${\mathcal {X}}$ and f is a ramified finite covering. By the discussion in Section 4.1, ${\mathcal {X}}_s=[\Sigma _{s^{\prime }}/G]$ , where $\Sigma _{s^{\prime }}$ is the spectral curve on $\Sigma $ defined by the section $s^{\prime }=g^{*}s$ . The divisor defined by s is $(s)=\sum _{k=2}^5\frac {1}{2}p_k$ . Thus, the divisor associated with $s^{\prime }$ is a reduced divisor on $\Sigma $ . Hence, the spectral curve $\Sigma _{s^{\prime }}$ is a smooth irreducible curve. Then, ${\mathcal {X}}_s$ is a smooth irreducible stacky curve. By Proposition 4.1, the genus of ${\mathcal {X}}_s$ is $g({\mathcal {X}}_s)=3$ . The coarse moduli space of ${\mathcal {X}}_s$ is denoted by $X_s$ . Obviously, $X_s$ has four stacky points $\{\widehat p_2,\widehat p_3,\widehat p_4,\widehat p_5\}$ , whose images in $\mathbb P^1$ are $\{p_2,\ldots ,p_5\}$ . Their stabilizer groups are $\mu _2$ . Let W be the line bundle ${\mathcal {O}}_{{\mathcal {X}}_s}(d\cdot \widehat p+\frac {1}{2}\widehat p_2)$ , where $\widehat p\in X_s$ is not a stacky point and $d\in {\mathbb Z}$ . Let $f : {\mathcal {X}}_s\rightarrow {\mathcal {X}}$ be the projection. Then, we obtain a rank $3$ Higgs bundle $(E,\phi _1)$ on ${\mathcal {X}}$ , where $E=f_{*}W$ and $\phi _1$ is the pushforward of the tautological section $\tau $ . In the following, we will determine the representations defined by the action of stabilizer groups on the fibers of E. Consider the Cartesian diagram
where $\iota _1 : B\mu _3\rightarrow {\mathcal {X}}$ is the residue gerbe of $p_1$ . We use $\mathcal Y_1$ to denote $B\mu _3\times _{{\mathcal {X}}}{\mathcal {X}}_s$ . Then, $\mathcal Y_1$ is isomorphic to the quotient stack
where the action of $\mu _3$ is defined by multiplication. It is a free action. Hence, any locally free sheaf of rank r on $\mathcal Y_1$ is isomorphic to ${\mathcal {O}}_{\mathcal Y_1}^{\oplus r}$ . Then, the $\mu _3$ -representation corresponding to $\iota ^{*}_1E$ is $\rho _1^{\otimes 0}\oplus \rho _1\oplus \rho _1^{\otimes 2}$ , where $\rho _1$ is the representation defined by the inclusion $\mu _3\hookrightarrow {\mathbb C}^{*}$ . Similarly, the $\mu _2$ -representation corresponding to $\iota ^{*}_2E$ ( $\iota _2 : B\mu _2\rightarrow {\mathcal {X}}$ is the residue gerbe of $p_2$ ) is $\rho _2^{\otimes 0}\oplus \rho _2\oplus \rho _2$ , where $\rho _2$ is the representation defined by the inclusion $\mu _2\hookrightarrow {\mathbb C}^{*}$ . For another three stacky points $\{p_3,p_4,p_5\}\subset X$ , the corresponding representations are isomorphic to $\rho _2^{\otimes 0}\oplus \rho _2^{\otimes 0}\oplus \rho _2$ . Denote $\pi _{*}E$ by F. There is a strongly parabolic Higgs bundle $(F,\phi _2)$ on $\mathbb P^1$ with marked points $\{p_1,p_2,p_3,p_4,p_5\}$ , corresponding to $(E,\phi _1)$ . The quasi-parabolic structure on F is given by:
-
• $F_{p_1}=F_{p_1,0}\supset F_{p_1,1}\supset F_{p_1,2}\supset F_{p_1,3}=\{0\}$ at $p_1$ ;
-
• $F_{p_2}=F_{p_2,0}\supset F_{p_2,1}\supset F_{p_2,2}=\{0\}$ at $p_2$ ;
-
• $F_{p_3}=F_{p_3,0}\supset F_{p_3,1}\supset F_{p_3,2}=\{0\}$ at $p_3$ ;
-
• $F_{p_4}=F_{p_4,0}\supset F_{p_4,1}\supset F_{p_5,2}=\{0\}$ at $p_4$ ;
-
• $F_{p_5}=F_{p_5,0}\supset F_{p_5,1}\supset F_{p_5,2}=\{0\}$ at $p_5$ .
And, the multiplicities are:
-
• $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_1,0}/F_{p_1,1})=1$ , $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_1,1}/F_{p_1,2})=1$ and $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_1,2}/F_{p_1,3})=1$ ;
-
• $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_2,0}/F_{p_2,1})=1$ and $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_2,1}/F_{p_2,2})=2$ ;
-
• $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_3,0}/F_{p_3,1})=2$ and $\mathop {\mathrm { dim}}\nolimits _{{\mathbb C}}(F_{p_3,1}/F_{p_3,2})=1$ ;
-
• $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_4,0}/F_{p_4,1})=2$ and $\mathop {\mathrm { dim}}\nolimits _{{\mathbb C}}(F_{p_4,1}/F_{p_4,2})=1$ ;
-
• $\mathop {\mathrm {dim}}\nolimits _{{\mathbb C}}(F_{p_5,0}/F_{p_5,1})=2$ and $\mathop {\mathrm { dim}}\nolimits _{{\mathbb C}}(F_{p_5,1}/F_{p_5,2})=1$ .
Denote the K-class of E in $K_0({\mathcal {X}})_{\mathbb Q}$ by $\xi _E$ , and denote the determinant line bundle of E by $L_E$ . By Proposition A.4, for a generic rational parabolic weight (see Definition A.3), the moduli stack $\mathcal M_{\mathop {\mathrm { Dol}}\nolimits ,\xi _E}(\mathop {\mathbf {SL}}\nolimits _3)$ of $\mathop {\mathbf {SL}}\nolimits _3$ -Higgs bundles with determinant $L_E$ has no strictly semistable object. With a generic parabolic weight, the moduli spaces $M_{\mathop {\mathrm {Dol}}\nolimits ,\xi _E}^{s}(\mathop {\mathbf {SL}}\nolimits _3)$ and $M_{\mathop {\mathrm { Dol}}\nolimits ,\xi _E}^{\alpha _E,s}(\mathop {\mathbf {PGL}}\nolimits _3)$ with natural flat unitary gerbes are SYZ mirror partners, where $\alpha _E\in H^2({\mathcal {X}},\mu _3)$ is the image of $L_E^{-1}$ under the morphism $\delta $ in the Kummer sequence (23).
A Comparison of the modified slope and the parabolic slope
Suppose that ${\mathcal {X}}$ is a smooth irreducible stacky curve and that $\pi :{\mathcal {X}}\rightarrow X$ is its coarse moduli space. The stacky points of X are $p_1,\ldots ,p_m$ , and the corresponding stabilizer groups are $\mu _{r_1},\ldots ,\mu _{r_m}$ . Fix a stacky point $p_i\in X$ . The residue gerbe of $p_i$ is a closed immersion $\iota _i : B\mu _{r_i}\rightarrow {\mathcal {X}}$ . Let $({\mathcal E},{\mathcal {O}}_X(1))$ be a polarization on ${\mathcal {X}}$ , and let E be a locally free sheaf on ${\mathcal {X}}$ . The decompositions of $\iota _i^{*}E$ and $\iota _i^{*}{\mathcal E}$ in the representation ring $\mathbf {R}\mu _{r_i}$ are
where $x_i$ represents the representation corresponding to the natural inclusion ${\mu _{r_i}\hookrightarrow {\mathbb C}^{*}}$ . By orbifold-parabolic correspondence, E corresponds to $\pi _{*}(E)$ with quasi-parabolic structure defined by the stacky structure of E at marked points $p_1,\ldots ,p_m$ . The multiplicities of the quasi-parabolic structure at $p_i$ are $(m_{i,0},\ldots ,m_{i,r_i-1})$ for every $1\leq i\leq m$ . The aforementioned quasi-parabolic structure with the parabolic weights
for each $1\leq i\leq m$ is a parabolic structure on $\pi _{*}(E)$ . At this time, $\pi _{*}(E)$ is called a parabolic bundle. The parabolic degree is
Its parabolic slope is
With the parabolic slope, we can introduce the stability condition for parabolic bundle $\pi _{*}(E)$ . By some elementary computations, we have the following proposition.
Proposition A.1 The modified slope $\mu _{{\mathcal E}}(E)$ is equivalent to the parabolic slope of par- $\mu (\pi _{*}E)$ with weights $\{\alpha _{i,j}\}$ . Furthermore, the modified slope $\mu _{\mathcal E}$ and the parabolic slope par- $\mu $ define the same stability condition on E and $\pi _{*}(E)$ , respectively.
Remark A.2 For abundant stability conditions, we can directly use rational parabolic weights to define the stability condition of Higgs bundles on stacky curve ${\mathcal {X}}$ . And, for any rational parabolic weight, all the results in Section 3 about moduli stacks (spaces) of Higgs bundles on hyperbolic stacky curve hold, by orbifold-parabolic correspondence.
Definition A.3 A rational parabolic weight is said to be generic if the induced stability condition on the moduli stack $\mathcal M_{\mathop {\mathrm {Dol}}\nolimits ,\xi }(\mathop {\mathbf {GL}}\nolimits _r)$ of Higgs bundles with K-class $\xi $ has no strictly semistable objects.
Recall Proposition 3.2 in [Reference Boden and YokogawaBY99].
Proposition A.4 [Reference Boden and YokogawaBY99]
For a K-class $\xi \in K_0({\mathcal {X}})$ , there is a generic rational parabolic weight if and only if d and the set of multiplicities $\{m_{i,j}|1\leq i\leq m,\ \ 0\leq j\leq r_i-1\}$ have greatest common divisor equal to one, where d is the degree of the K-theoretical pushforward of $\xi $ under the morphism $\pi $ .
B Proof of the properness of the Hitchin morphism
Let ${\mathcal {X}}$ be a smooth irreducible stacky curve. For a DVR R over ${\mathbb C}$ with maximal ideal $m=(\pi )$ and residue field $k={\mathbb C}\subset R$ , there is a Cartesian diagram
where ${\mathcal {X}}_R={\mathcal {X}}\times \mathop {\mathrm {Spec}}\nolimits (R)$ ; ${\mathcal {X}}_K={\mathcal {X}}\times \mathop {\mathrm {Spec}}\nolimits (K)$ ; ${\mathcal {X}}_{k}={\mathcal {X}}\times \mathop {\mathrm {Spec}}\nolimits (k)$ ; $i:{\mathcal {X}}_K\hookrightarrow {\mathcal {X}}_R$ is the open immersion; $j:{\mathcal {X}}_{k}\hookrightarrow {\mathcal {X}}_R$ is the closed immersion.
Theorem B.1 Suppose $(E_K,\phi _K)$ is a semistable Higgs bundle on ${\mathcal {X}}_K$ with characteristic polynomial $f_K\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}}_K,K_{{\mathcal {X}}_K}^i)$ . If $f_K$ is the restriction of some $f_R\in {\bigoplus }_{i=1}^rH^0({\mathcal {X}}_R,K_{{\mathcal {X}}_R}^i)$ to ${\mathcal {X}}_K$ , then there exists a family $(E_R,\phi _R)$ of Higgs bundles parametrized by $\mathop {\mathrm {Spec}}\nolimits (R)$ such that:
-
• $(E_K,\phi _K)$ is the restriction of $(E_R,\phi _R)$ to ${\mathcal {X}}_K$ .
-
• The characteristic polynomial of $(E_R,\phi _R)$ is $f_R$ .
-
• The restriction of $(E_R,\phi _R)$ to ${\mathcal {X}}_k$ is a semistable Higgs bundle.
If Theorem B.1 is true, then the proof of [Reference NitsureNit91, Theorem 6.1] also works in our case, which shows that Theorem 4.2 is true. The rest of this section is devoted to proving Theorem B.1. ${\mathcal {X}}$ has an open substack ${\mathcal {X}}^o$ such that it is a smooth irreducible curve over k. For convenience, we introduce the following notations: ${\mathcal {X}}_K^o={\mathcal {X}}^o\times \mathop {\mathrm { Spec}}\nolimits (K)$ and ${\mathcal {X}}_k^o={\mathcal {X}}^o\times \mathop {\mathrm {Spec}}\nolimits (k)$ ; $\beta _2 : \Xi \rightarrow {\mathcal {X}}_K^o$ is the generic point of ${\mathcal {X}}_K^o$ and $\xi $ is the generic point of ${\mathcal {X}}_k^o$ ; ${\mathcal {O}}_{\xi }$ is the stalk of ${\mathcal {O}}_{{\mathcal {X}}_R^o}$ at $\xi $ and $\beta _1 : \mathop {\mathrm {Spec}}\nolimits ({\mathcal {O}}_{\xi })\rightarrow {\mathcal {X}}_R^o$ is the natural morphism; ${\alpha : \Xi \rightarrow \mathop {\mathrm {Spec}}\nolimits ({\mathcal {O}}_{\xi })}$ is the open immersion and $\gamma : {\mathcal {X}}^o\hookrightarrow {\mathcal {X}}$ is the open immersion. Then, there is a Cartesian diagram
Lemma B.2 Let $(E_K,\phi _K)$ be a Higgs bundle of rank r on ${\mathcal {X}}_K$ . Suppose that M is a $(\phi _K)_{\Xi }$ -invariant free rank r ${\mathcal {O}}_{\xi }$ -submodule of $(\gamma _K^{*}E_K)_{\Xi }$ with $M{\otimes }_{{\mathcal {O}}_{\xi }}{\mathcal {O}}_{\Xi }=(\gamma _K^{*}E_K)_{\Xi }$ . Then, there exists a unique family $(E_R,\phi _R)$ of Higgs bundles parametrized by $\mathrm {Spec}(R)$ such that $E_R\subseteq i_{*}E_K$ and $\phi _R$ is the restriction to $E_R$ of $i_{*}\phi _K$ .
Proof Using Lemma 3.3 in [Reference HuangHua22], this lemma can be proved following the same steps as in the proof of [Reference NitsureNit91, Proposition 6.5].
Fixing a semistable Higgs bundle $(E_K,\phi _K)$ of rank r on ${\mathcal {X}}_K$ , we can introduce the so-called Bruhat–Tits complex for it. Let $\boldsymbol {\mathfrak M}$ be the set of all rank n free $(\phi _K)$ -invariant ${\mathcal {O}}_{\xi }$ -submodules of $(E_K)_{\Xi }$ . $\boldsymbol {\mathfrak M}$ is not empty (see [Reference NitsureNit91, Lemma 6.6]). An equivalence relation $\sim $ on $\boldsymbol {\mathfrak M}$ is given by: for $M\in \boldsymbol {\mathfrak M}$ , $M\sim \pi ^pM$ for $p\in {\mathbb Z}$ . By Lemma B.2, equivalent modules in $\boldsymbol {\mathfrak M}$ induce isomorphic extensions of $(E_K,\phi _K)$ to ${\mathcal {X}}_R$ . Let $\boldsymbol {\mathfrak Q}$ be the quotient set $\boldsymbol {\mathfrak M}/\sim $ . We can define a structure of an r-dimensional simplicial complex on $\boldsymbol {\mathfrak Q}$ . $\boldsymbol {\mathfrak Q}$ with the simplicial complex structure is called the Bruhat–Tits complex. Two equivalent classes $[M]$ and $[M^{\prime }]$ are said to be adjacent if M has a direct decomposition $M=N\oplus P$ such that $M^{\prime }=N+\pi M$ . In other words, $[M]$ and $[M^{\prime }]$ are adjacent if and only if M has a basis $\{e_1,e_2,\ldots ,e_r\}$ over ${\mathcal {O}}_{\xi }$ such that $\{e_1,\ldots ,e_s,\pi e_{s+1},\ldots ,\pi e_r\}$ is a basis of $M^{\prime }$ over ${\mathcal {O}}_{\xi }$ . If $0\subset N_1\subset N_2\subset \cdots \subset N_t\subset M$ is a sequence of submodules of M such that each $N_i$ is a direct factor of M and $M_i=N_i+\pi M$ is $(\phi _K)_{\Xi }$ -invariant, then the $t+1$ mutually adjacent vertices $[M],[M_1],\ldots ,[M_t]$ form a t-simplex in $\boldsymbol {\mathfrak Q}$ . To prove Theorem B.1, we only need to find a vertex $[E_{\xi }]$ of $\boldsymbol {\mathfrak Q}$ such that the reduction $(E_k,\phi _k)$ of the corresponding extension $(E_R,\phi _R)$ is semistable.
Proposition B.3 Suppose that $[E_{\xi }]$ is a vertex in $\boldsymbol {\mathfrak Q}$ and $(E_k,\phi _k)$ is the restriction of the corresponding extension $(E_R,\phi _R)$ to ${\mathcal {X}}_k$ . Then there is a one-to-one correspondence between edges in $\boldsymbol {\mathfrak Q}$ at $[E_{\xi }]$ and proper $\phi _k$ -invariant subbundles of $E_k$ . Furthermore, if $F\subseteq E_k$ is a $\phi _k$ -invariant subbundle corresponds to the edge $[E_{\xi }]-[E_{\xi }^{\prime }]$ at $[E_{\xi }]$ and $Q^{\prime }\subseteq E_k^{\prime }$ is the $\phi _k^{\prime }$ -invariant subbundle corresponds to the edge $[E_{\xi }^{\prime }]-[E_{\xi }]$ at $[E_{\xi }^{\prime }]$ , then there is a homomorphism $(E_k,\phi _k)\rightarrow (E_k^{\prime },\phi _k^{\prime })$ of Higgs bundles with kernel F and image $Q^{\prime }$ , and a homomorphism $(E^{\prime },\phi _k^{\prime })\rightarrow (E_k,\phi _k)$ of Higgs bundles with kernel $Q^{\prime }$ and image F.
Proof Part 1. Suppose that $E_{\xi }=(e_1,\ldots ,e_r)$ represents the vertex $[E_{\xi }]$ and ${E_{\xi }^{\prime }=(e_1,\ldots ,e_s,\pi e_{s+1},\ldots ,\pi e_r)}$ represents an adjacent vertex. Since $E_{\xi }^{\prime }\subseteq E_{\xi }$ , there is an injection of the corresponding extensions
(see [Reference HuangHua22, Lemma 3.3]). Consider the exact sequence of Higgs bundles
where $(Q,\overline \phi _k)$ is a Higgs bundle on ${\mathcal {X}}_k$ (see the proof of [Reference HuangHua22, Proposition 3.6]). Restricting (B.2) to ${\mathcal {X}}_k$ , we get an exact sequence
where F is the image of the restriction of (B.1) to ${\mathcal {X}}_k$ . We therefore get a Higgs subbundle $(F,\phi _k|_F)$ of $(E_k,\phi _k)$ . Conversely, if F is a $\phi _k$ -invariant subbundle of $E_k$ and $Q=E_k/F$ is a bundle on ${\mathcal {X}}_k$ , then we have an exact sequence of Higgs bundles
Composing the restriction $(E_R,\phi _R)\rightarrow (E_k,\phi _k)$ with the surjective morphism $(E_k,\phi _k)\rightarrow (Q,\overline \phi _k)$ in (B.4), we get a new surjective morphism $(E_R,\phi _R)\rightarrow (Q,\overline \phi _k)$ , i.e., there is an exact sequence
where $\phi ^{\prime }_R$ is the restriction of $\phi _R$ to $E^{\prime }_R$ . Consider the exact sequence of ${\mathcal {O}}_{\xi }$ -modules
Suppose that $(E_k)_{\xi }$ is generated by $\{\overline e_1,\ldots ,\overline e_r\}$ and $\{\overline e_1,\ldots ,\overline e_s\}$ is a basis of $F_{\xi }$ over ${\mathcal {O}}_{{\mathcal {X}}^o_k,\xi }$ . Moreover, $\{\overline e_1,\ldots ,\overline e_r\}$ lifts to a basis $\{e_1,\ldots ,e_r\}$ of $(E_R)_{\xi }$ over ${\mathcal {O}}_{\xi }$ . Then, $({E^{\prime }_R})_{\xi }$ is generated by $\{e_1,\ldots ,e_s,\pi e_{s+1},\ldots ,\pi e_r\}$ and $({E^{\prime }_R})_{\xi }$ is $\phi _K$ -invariant. So, it represents a vertex $[E_{\xi }^{\prime }]$ of $\boldsymbol {\mathfrak Q}$ adjacent to $[E_{\xi }]$ .
Part 2. Since $\pi E_{\xi }\subseteq E_{\xi }^{\prime }$ , there is another injection $(\pi E_R,\phi _R|_{\pi E_R})\rightarrow (E_R,\phi _R)$ . Composing it with the isomorphism $(E_R,\phi _R)\overset {\pi }{\rightarrow }(\pi E_R,\phi _R|_{\pi E_R})$ , we get the injection
The restriction of (B.7) to the special fiber ${\mathcal {X}}_k$ is
The composition of the two morphisms in (B.9) is zero. In fact, the composition of
is zero and $E_k^{\prime }$ is torsion-free. Obviously, the sequence (B.10) is exact at the middle term. By [Reference HuangHua22, Proposition 2.23], the sequence (B.9) is exact at the middle term. Similarly, restricting (B.8) to ${\mathcal {X}}_k$ , we get
We can also show that (B.11) is exact at the middle term. Therefore, we have the following exact sequence:
Definition B.4 Let E be a locally free sheaf with modified Hilbert polynomial $P_E(m)=a_1\cdot m+a_0$ on ${\mathcal {X}}$ . For every locally free sheaf $E_1$ on ${\mathcal {X}}$ , we define the $\boldsymbol \beta $ -invariant $\beta (E_1)$ of $E_1$ with respect to E as follows: $\beta (E_1)=a_1\cdot a_0(E_1)-a_0\cdot a_1(E_1)$ , where $P_{E_1}(m)=a_1(E_1)\cdot m+a_0(E_1)$ is the modified Hilbert polynomial of $E_1$ .
Remark B.5 $(E,\phi )$ is semistable if and only if $\beta (F)\leq 0$ for all $\phi $ -invariant subsheaf $F\subseteq E$ .
Recall some properties of $\beta $ -invariants (see [Reference HuangHua22, Proposition 2.27]).
Proposition B.6
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(i) If $E_1$ and $E_2$ are two $\phi $ -invariant subsheaves of locally free sheaf E on ${\mathcal {X}}$ , then
$$ \begin{align*} \beta(E_1)+\beta(E_2)\leq \beta(E_1\vee E_2)+ \beta(E_1\cap E_2), \end{align*} $$with equality if and only if $E_1\vee E_2=E_1+E_2$ . -
(ii) If is an exact sequence of locally free sheaves on ${\mathcal {X}}$ , then $\beta (F)+\beta (K)=\beta (G)$ .
Proposition B.7 For a Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ , there exists a unique $\phi $ -invariant proper subsheaf $B\subset E$ such that:
-
(i) For every $\phi $ -invariant subsheaf G of B with $\mathrm {rk}(G)<\mathrm {rk}(B)$ , we have $\beta (G)<\beta (B)$ .
-
(ii) For every $\phi $ -invariant subsheaf H of E, we have $\beta (H)\leq \beta (B)$ .
Proof The claim can be proved following the same steps as in the proof of Proposition 2.31 in [Reference HuangHua22]
If the Higgs bundle $(E,\phi )$ is unstable, then the $\phi $ -invariant subsheaf B in the above proposition, will be called the $\boldsymbol {\beta }$ -subbundle of $(E,\phi )$ . Now, assume that we are given a vertex $[E_{\xi }]$ of $\boldsymbol {\mathfrak Q}$ such that the corresponding Higgs bundle $(E_k,\phi _k)$ on ${\mathcal {X}}_k$ is unstable. Let $B\subset E_k$ be the $\beta $ -subbundle of $(E_k,\phi _k)$ . Thus, $\beta (B)>0$ (See Proposition B.7). By Proposition B.3, there is an edge in $\boldsymbol {\mathfrak Q}$ at $[E_{\xi }]$ corresponding to B. Let $[E_{\xi }^{(1)}]$ be the vertex in $\boldsymbol {\mathfrak Q}$ determined by the edge corresponding to B, and let $(E_{k}^{(1)},\phi _k^{(1)})$ be the corresponding Higgs bundle on ${\mathcal {X}}_k$ . Let $F_1\subseteq E_k^{(1)}$ be the image of the canonical homomorphism $E_k\rightarrow E_k^{(1)}$ (= the kernel of the homomorphism $E_k^{(1)}\rightarrow E_k$ ).
Following similar steps as in the proof of [Reference LangtonLan75, Lemma 1], we can show the following lemma:
Lemma B.8 If $G\subset E_k^{(1)}$ is a $\phi ^{(1)}_k$ -invariant subbundle of $E_k^{(1)}$ , then $\beta (G)\leq \beta (B)$ , with equality possible only if $G+F_1=E_k^{(1)}$ .
Now, we are going to define a path ${\mathcal P}$ in $\boldsymbol {\mathfrak Q}$ , starting with a vertex $[E_{\xi }]$ , whose corresponding Higgs bundle $(E_k,\phi _k)$ is unstable. The succeeding vertex is the vertex determined by the edge corresponding to the $\beta $ -subbundle B of $(E_k,\phi _k)$ . If ${\mathcal P}$ reaches a vertex $[E_{\xi }^{(m)}]$ such that the corresponding Higgs bundle $(E_k^{(m)},\phi _k^{(m)})$ is semistable, then the process stops automatically and Theorem B.1 is proved. If the path ${\mathcal P}$ never reaches a vertex corresponding to a semistable reduction, then the process continuous indefinitely. We have to show that the second alternative is impossible.
Denote the $\beta $ -subbundle of $(E_k^{(m)},\phi _k^{(m)})$ by $B^{(m)}$ , and let $\beta _m=\beta (B^{(m)})$ . By Lemma B.8, $\beta _{m+1}\leq \beta _{m}$ and we must have $\beta _m>0$ unless $E_k^{(m)}$ is semistable. Thus, if the path ${\mathcal P}$ is continuous indefinitely, we have $\beta _m=\beta _{m+1}=\cdots $ , for sufficiently large m. Also, by Lemma B.8, for sufficiently large m, $B^{(m)}+F^{(m)}=E_k^{(m)}$ , where $F^{(m)}=\text {Im}(E_k^{(m-1)}\rightarrow E_k^{(m)})$ ( $\text {Ker}(E_k^{(m)}\rightarrow E_k^{(m-1)})$ ). So, $\text {rank}(B^{(m)})+\text {rank}(F^{(m)})\geq r$ . On the other hand, $\text {rank}(B^{(m-1)})+\text {rank}(F^{(m)})=r$ . Therefore, $\text {rank}(B^{(m)})\geq \text {rank}(B^{(m-1)})$ , for sufficiently large m. Since $\text {rank}(B^{(m)})\leq r$ , we must have $\text {rank}(B^{(m)})=\text {rank}(B^{(m+1)})=\cdots $ , for sufficiently large m. Thus, $\text {rank}(B^{(m)})+\text {rank}(F^{(m)})= r$ . So, $B^{(m)}\cap F^{(m)}=0$ and $B^{(m)}\oplus F^{(m)}=E_k^{(m)}$ . Consequently, the canonical homomorphism $E_k^{(m)}\rightarrow E_k^{(m-1)}$ induces isomorphism $B^{(m)}\rightarrow B^{(m-1)}$ . Also, the canonical homomorphism $E_k^{(m-1)}\rightarrow E_k^{(m)}$ induces isomorphism $F^{(m-1)}\rightarrow F^{(m)}$ . If R is a complete discrete valuation ring, the following lemma leads us to a contradiction.
Lemma B.9 Assume that the discrete valuation ring R is complete and ${\mathcal P}$ is an infinite path in $\boldsymbol {\mathfrak Q}$ with vertices $[E_{\xi }]$ , $[E_{\xi }^{(1)}]$ , $[E_{\xi }^{(2)}]$ , $\ldots $ . Let $F^{(m)}=\text {Im}(E_k^{(m+1)}\rightarrow E_k^{(m)})$ . If the canonical homomorphism $E^{(m+1)}\rightarrow E^{(m)}$ induces isomorphism $F^{(m+1)}\rightarrow F^{(m)}$ for every m, then $\beta (F)\leq 0$ .
Proof The lemma can be checked step by step as Lemma 6.11 in [Reference NitsureNit91].
Hence, Theorem B.1 is proved under the assumption that R is complete. The general case can be proved as [Reference NitsureNit91].
C Spectral construction
In this subsection, we recall the spectral construction. Suppose that ${\mathcal {X}}$ is a hyperbolic Deligne–Mumford curve and $\psi : \mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})\rightarrow {\mathcal {X}}$ is the natural projection. For a Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ , the Higgs field $\phi $ defines a morphism of ${\mathcal {O}}_{{\mathcal {X}}}$ -algebras $\mathrm {Sym}^{\bullet }(K_{{\mathcal {X}}}^{\vee })\rightarrow {\mathcal End}_{{\mathcal {O}}_{{\mathcal {X}}}}(E)$ . Then, E is endowed with an $\mathrm {Sym}^{\bullet }(K_{{\mathcal {X}}}^{\vee })$ -module structure. It defines a compactly supported ${\mathcal {O}}_{\mathop {\mathrm {Tot}}\nolimits ({K_{{\mathcal {X}}}})}$ -module $E_{\phi }$ over $\mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})$ . Moreover, $E_{\phi }$ is a pure sheaf of dimension one (see Proposition C.1). Conversely, if F is a compactly supported pure sheaf of dimension one on $\mathop {\mathrm {Tot}}\nolimits (K_{{\mathcal {X}}})$ , then there is a Higgs bundle $(E,\phi )$ on ${\mathcal {X}}$ such that $E_{\phi }=F$ , where $E=\psi _{*}(F)$ and $\phi $ is defined by the tautological section of $\psi ^{*}K_{\mathcal {X}}$ . There is an equivalence of two categories
where $\mathbf {Higgs}({\mathcal {X}})$ is the category of Higgs sheaves on ${\mathcal {X}}$ and $\mathbf {Coh}_c({\mathop {\mathrm {Tot}}\nolimits }(K_{{\mathcal {X}} }))$ is the category of compactly supported coherent sheaves on $\text {Tot}(K_{{\mathcal {X}}})$ (see [Reference Jiang and KunduJK21, Proposition 2.18]).
Proposition C.1 The equivalence (C.1) gives a one-to-one correspondence between Higgs bundles on ${\mathcal {X}}$ and compactly supported pure sheaves of dimension one on $\mathrm {Tot}(K_{{\mathcal {X}}})$ .
Proof The conclusion of this proposition can be proved locally in étale topology as Proposition 2.18 in [Reference Jiang and KunduJK21].
As Remark 3.7 in [Reference Beauville, Narasimhan and RamananBNR89], we have the following proposition.
Proposition C.2 Suppose that $f : {\mathcal {X}}_{\boldsymbol a}\rightarrow {\mathcal {X}}$ is an integral spectral curve and $(E,\phi )$ is a rank r Higgs bundle on ${\mathcal {X}}$ with spectral curve ${\mathcal {X}}_{\boldsymbol a}$ . Then, the rank one torsion-free sheaf $E_{\phi }$ on ${\mathcal {X}}_{\boldsymbol a}$ corresponding to $(E,\phi )$ satisfies
where $\tau $ is the restriction of the tautological section of $\psi ^{*}K_{\mathcal {X}}$ to ${\mathcal {X}}_{\boldsymbol a}$ .
Proof Consider the total space of the canonical line bundle $\psi : \mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})\rightarrow {\mathcal {X}}$ . Similar to [Reference Tanaka and ThomasTT20, Proposition 2.11], it is easy to show that there is an exact sequence
on $\mathop {\mathrm {Tot}}\nolimits (K_{\mathcal {X}})$ , where $\tau $ is the tautological section of $\psi ^{*}K_{\mathcal {X}}$ . On the other hand, there is an exact sequence
Then, we have the commutative diagram
By diagram chasing, we have the exact sequence
Acknowledgements
The author is most grateful to Professor Yunfeng Jiang for suggesting the research program “Hitchin system on DM curves” and helpful discussions in writing this paper. He would like to thank Sheng Chen, Yuhang Chen, Jianxun Hu, Changzheng Li, Zongzhu Lin, Hao Sun and Shanzhong Sun for helpful conversations. He also thanks André Oliveira for comments about this preprint. He thanks the referee for giving him valuable advices which help him to improve the presentations.